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The Homotopy Type of a 4-Manifold
with finite ~"Sandamental Group
by Stefan Bauer*
A B S T R A C T : ... is determined by its quadratic 2-type, if the 2-Sylow subgroup has 4-periodic cohomology.
The homotopy type of simply connected 4-manifolds is determined by the intersection form. This is a well-known result of J.H.C. Whitehead and 3. Mitnor. In the non-simply connected case the homotopy groups ~rl and 7~ and the first k-invariant k E H3(71, v2) give other homotopy invariants. The quadrat ic 2-type of an oriented closed 4-manifold is the isometry class of the quadruple [71(M), ~r2(M), k(M),~/(~)], where 7(It:/) denotes
~
the intersection form on 72(M) = H2(M). An isometry of two such quadruples is an isomorphism of 71 and 72 which induces an isometry on 7 and respects the k-invariant.
Recently [ H - K] I. Hambleton and M. Kreck, studying the homeomorphism types of 4-manifolds, showed that for groups with periodic cohomology of period 4 the quadratic 2-type determines the homotopy type.
This result can be improved away from the prime 2.
Theorem: Suppose the 2-Sylow subgroup of G has 4-periodic cohomology. Then the homotopy type of an oriented 4-dimensional Poincarfi complex with fundamental group G is determined by its quadratic 2-type.
I am indebted to Richard Swan for showing me proposition 6. Furthermore I am grateful to the department of mathematics at the University of Chicago for its hospitality during the last year.
* Supported by the DFG
Let X be an oriented 4-dimensional Poincar6 complex with finite fundamental group, f : X --~ B its 2-stage Postnikov approximation, determined by ~rl, 7r2, and k, a.nd let ~,(X) denote the intersection form on /72(2). Then S PD (B, 7(X)) denotes the set of homotopy types of 4-dimensional Poincar~ complexes Y, together with 3-equivalences g : Y ---* B, such that f and g induce an isometry of the quadratic 2-types. The universal cover /) is an Eilenberg-MacLane space and hence, by [MacL], H4(/)) P(~r2(B)), the ZTh(B)-modute F(Tr2(B)) being the module of symmetric 2-tensors, i.e. the kernel of the map (1 - T): ~r 2 (B) ® 7r2 (B) -* 7c~ (B) ® 7r2 (B), (1 - 7-)(a ® b) = a ® b - b ® a. The intersection form on 2 corresponds to L[2] of the fundamental class [21 • Hal2; z) . Le t / : / . denote Tate homology.
P r o p o s i t i o n 1: If X is a Poincarfi space with finite fundamental group G, then there is a. bijection [Io( G; ~ra( X) ) , , sPD ( B, "y( X)).
The proof uses a lemma of [H-K]:
L e m m a 2: Let, ( X , f ) and ( Y , g ) b e elements in sPD(B, 7(X)). Then the only obstruction for the existence of a homotopy equivalence h : X ~ Y over B is the vanishing ofg.[Y] - f . [X] • H4(B).
L e m m a 3: Given a diagram
Z ~ M
Z
such that the torsion in the cokernel of ~ is annihilated by n, then the torsion subgroup in the pushout K is isomorphic to the torsion subgroup of coker(c~).
P r o o f of 3: Since the torsion subgroup of M maps injectively into K as well as into coker(a), we may assume it trivial. Then M is isomorphic to NO < z > with a(1) = mz for an integer m dividing n. The pushout then is isomorphic to ( N @ Z @ Z ) / < (0, m, n) > ~ M • Z/m. &
P r o o f of p r o p o s i t i o n 1: Let (X, f ) and (Y, g) be elements in SPD(B) such that f and g induce an isometry of the quadratic 2-types. Let 7(X) = "y(Y) = "y denote the inter- section form on H2(X ~) and H2(]'z). By [W] one has ~r3(X) - F(~r2(X))/(7) -- H4(/~,)~ ~)
and ~ra(X) ®zv Z ~ Ha(B, X). In the pushout diagramm:
0 0 0
o ~ Ha(2) ® z a Z ~ Ha(/)) ® ~ Z , Ha(/), 2 ) ® z c Z ¢ 1 1 1 ~-
0 ~ H4(X) , Ha(B) .... ~ Ha(B, X) J. .~ +
0 ~ Ha(X, 2) ~-; Ha(B,~)) , 0
0 0
, 0
0
the torsion subgroup of H4(B, X) is isomorphic to the torsion subgroup of H4(B) by lemma 3: The module Ha(/), 2 ) is torsion free. Hence the torsion subgroup of Ha(/), 2 ) ®zc Z is annihilated by the order n of the group G. Note that ¢ is just multiplication by n. In particular one has
Torsion(Ha(B)) ~- Torsion(Ha(B, X)) ~ [/o(G;Tra(X))
Since X and Y have the same quadratic 2-type, • [X] = ~.[1~], hence we have L [ X ] - g.[~'] • To~,io,~(HaB). This gives an injection
sPD( B, 7) ~-~/:/o(G;Trs(X)).
What about surjectivity? Let K C A" denote a subspace, where one single orbit is deleted. Let a e ~ra(K) map via the surjection ~r3(K) --* 7ra(X) --* ~ra(X) ® z c Z to a given element & 6 f/o(G;~rs(X)). Let f¢ be the image of 1 6 ZG ~ tIa(f(, K) ~- 7r4(2, K) '--* 7ra(K). Now let k : S 3 ---* K represent c~+fl and define X~ := ( K U k ( G x Da))/G. One has to show that X~ is an orientable Poincar4 space. Orientability is clear, since Ha(X~) ~- Ha(X~,K) ~- Z. Let f : Xa --* B extend fIK/a. The intersection form on ) ~ is determined by
.~.[Xc,] - trf(f,~.[X~]) e Ha(2). But we have fa.[Xa] = f ,[X] + o~: In the following diagram 1 6 Z ~ 7ra(X, B) is mapped to f ,[X] e Ha(B).
H4(X) ~ H4(X,K/G) ~,-- Ha(X,K) '~ m ( X , K ) - - , ~ra(K) f t l 1 l l=
H~(B) = ~ H~(B, K / a ) , - -~ H~([3, K) = ~ ~ ( B , K) ~- ,~ (K)
If the upper row is replaced by the corresponding row for X~ and the vertical maps by the ones induced by f~, then t E ZG is mapped (counterclockwise) to f~.[X~] on the one hand, on the other hand (clockwise) to f . [X] + o~. Since the torsion element o~ lies in the kernel of the transfer, one immediately gets /o.[x~] = f.[x].
In the sequel all ZG-modules have underlying a free abelian group.
The short exact sequence
0 ~ Z ~-L r ( ~ X ) , ~3(X) --* 0
gives rise to an exact sequence in Tate homology:
H 0 ( a ; Z ) , H 0 ( C ; r ( ~ X ) ) , H0(a ;~3(X) ) ~ H _ I ( C ; Z ) ~, H _ ~ ( a ; r ( ~ x ) )
Here/: /0(G; Z) = 0 and / : /_I(G;Z) ~ Z / I Gl .The sequence above gives the connection to [H-K], theorem(]. l) .
In order to analyze this sequence, I recall some facts from [H-K],§g2 and 3.
Facts: 1)
2)
3)
F ( Z G ) = (~i Z[G/Hi] ® F, ,,,here the summation is over all subgroups Hi of order 2 and F is a free ZG-module. r(zG) r([) • z a r(z*) • ZG.Here I denotes the augmentation ideal, I* its dual. The modules D3Z and SaZ are (stably!) defined by exact sequences
O--~QsZ-~ F2-~ FI..-~ Fo--~ Z--+O and
O--+ Z ~ FI ~ F2 ~ F3 ~ S3Z ~ O with free modules F,. There is an exact sequence
0 ~ f~az , ~r~(X) • r Z G ~ S3Z ~ 0
L e m m a 4: If 0 ---* A --~ B --* C ~ 0 is a short exact sequence of ZG-modules, which are free over Z, then there are short exact sequences
0 ~ P ( A ) , F(B) , , , , D ~ 0 and
0 ~ A ® z C , D - - - * F ( C ) ~ 0 .
P r o o f i Given Z-bases {ai}, {c/} and {ai,~j} of A, C and B, the map h : a, @c/ ---* ai ® ci + cJ ® ai is well-defined and equivariant modulo F(A). &
To prove the theorem, it suffices to show that H0(G; 7r3(X)) = 0. This in turn can be done separately for each p-Sylow subgroup Gp of G.
P r o p o s i t i o n 5: The map 7. : H - l ( G p ; Z ) ~ / - / - l (@; F(,r2(X))) is injective, if either p is odd or r e sg 7r2(X) ~ A @ B splits such that the rank of B over Z is odd. In general the kernel is at most of order 2.
Proof i For the sake of brevity, let 7r denote 7r2(X) and also let F denote the module r(Tr). Now look at the following sequence of maps:
a *~ t race ¢ : Z "Y, P ~ ~r @ ~r ~ gom(Tr', ~r), gom(~r, ~r) Z.
A genera.tor of Z is mapped in Hom(Tr*, ~) to the Poincar6 map a : 7r* -- H2(2) --~ H : ( 2 ) = rr, and then to the element id e Horn(or, 7:). So we have ¢(1) = rankz(Tr).
Fact 3) gives rankz(Tr) =- - 2 mod I G h hence the i~duced selfmap ¢ , of Z/I@I af-/_ ~(@; Z) is multiplication by -2. This proves, that the kernel is at most of order 2. In particular it is trivial, if p is odd.
In case p = 2 and resaa rr -~ A @ B, such that the rank of the underlying group
of B is odd, one can replace the map IIom(rr, 7r) t2-~ ~ Z by the map Hom(rr, Tr) , . . i*
Horn(B, B) tra,¢ Z in the defining sequence for ¢. A similar argument as above for p odd gives the claim. &
R e m a r k : The module resa G ~r~(X) ahvays splits, if H4(G;Z) ~ Ext~a(SaZ,fl3Z) has no 2-torsion, in particular if ~2 has 4-periodic cohomology.
P r o p o s i t i o n 6: Let A denote either ~'~Z or S'~Z and let r be the selfmap of A ® A which permutes the factors. Then (-1)n~ - induces the identity on [ /0(G;A N A).
P roof i Let F. --~ Z be a free resolution of Z and let _F. be the truncated complex with Fi = Fi for i < n - 1, ~',, = f/'~ and/~ = 0 else. There is an obvious projection f : F. --~/~'.,
such that fn = 0n. The tensor product F.®F. = F. ~ again is a free resolution of Z and/~.2 is a truncatedfree resolution of Z with/v~,~ = f~Z @ OZ. The chain map f N f induces an
isomorphism of H. (/,-'.2 @z a Z) and - 2 H.(F. @zaZ) in the dimensions * _< 2n. The selfmap of F~., as usual defined by t (z ® y) = (--1)deg(x)deg(y)x ® y, is a chain automorphism,
inducing the identity on the augmentation, hence on all derived functors, in particular on /~.2. H . (/7. 2 ® z c Z) = H. (G; Z). In the same way an involution t can be defined on and
f ® f commutes with t. Obviously ~;,, = (--1)'~r. Hence (-1)'%" induces the identity on
® z c z ) = ::o(a;z). The proof for S '~ Z is dual. &
P r o o f o f t h e t h e o r e m : By proposition 1, it suffices to show that / t0(G;za(X)) vanishes. By proposition 4 and the remark following it, this group is isomorphic to /;/0(G; F(Tr~ (X))). In order to show that this group vanishes it suffices, by lemma 3, to show that [to(G;A) vanishes for A e {[ ' (aaZ),F(SaZ),f23Z ® SaZ) But / I0 (G;aaZ ® SaZ) - f /0(G;Z) = 0. Given a module B (with underlying free abelian group), there is a short exact sequence
0 .--, F(B) - - , B ® B . . , A2(B) ---+ 0.
The map % which flips the both factors, induces, if applied to B E {f~aZ,S3Z} the following diagram:
--* / ) I (G;A(B)) ~ !flo(G;r(B)) ~ H o ( G ; B ® B ) -* I ( - i d ) I ie ~ ( - i d )
The right vertical map is ( - i d ) by proposition 5. This diagram shows that any element in H0(G;I'(B)) is annihilated by 4.In particular this group vanishes, if G is a p-group for an odd prime p. That //0(G2; F(B)) vanishes, if G~ has 4-periodic cohomology, follows at once from the facts 1 - 3, since in this case f'taZ = I* @ n Z G and SaZ = I @ n Z G
Final R e m a r k : An elementary but lengthy computation shows F(SaZ) - Z/2e Z/2 and F(f~3Z) = 0 for G = Z/2eZ/2. In particular the group fIo(Z/2®Z/2;F(~3Z®S3Z)) is nontrivial. Hence the argument above won't work in general.
REFERENCES
[B 1]
[H-K]
[MacL]
[w]
K.S. Brown: CohomoIogy of groups. GTM 87, Springer-Verlag, N.Y. 1982 R. Brown: Elements of Modern Topology. McGraw- Hill, London, 1968 I. Hambleton and M. Kreck: On the Classification of Topological 4-Manifolds with finite Fundamental Group. Preprint, 1986 S. MacLane: Cohomology theory of abelian groups. Proc. Int. Math. Congress, vol. 2 (1950), pp 8 - 14 J.H.C. Whitehead: On simply connected 4-dimensionM polyhedra. Comment. Math. Helv., 22 (1949), pp .18 - 92.
Sonderforschungsbereich 170 Geometrie und Analysis Mathematisches Institut Bunsenstr. 3 - 5 D-3400 GSttingen, FRG
Rational Cohomology of
Configuration Spaces of Surfaces
C.-F. B6digheimer and F.R. Cohen
I. Introduction. The k-th configuration space ck(M) of a manifold M is
the space of all unordered k-tuples of distinct points in M. In previous
work [BCT] we have determined the rank of H.(ck(M) ;~ ) for various
fields ~ . However, for even dimensional M the method worked for F =F 2
only. The following is a report on cal~ulations of H*(ck(M) ;Q) for M
a deleted, orientable surface. This case is of considerable interest
because of its applications to mapping class groups, see [BCP].
Similar results for (m-1)-connected, deleted 2m-manifolds will appear
in [BCM].
2. Statement of results. The symmetric group I k acts freely on the space
~k(M) of all ordered k-tuples (z I ,. ..,z k) , zi6M, such that z± # z; for
i # j. The orbit space is ck(M). As in [ BCT] we will determine the
rational vector space H~(ck(M) ;Q) as part of the cohomology of a much
larger space. Namely, if X is any space with basepoint x o, we consider
the space
where (z I, .... Zki;Xl, .... x k) ~ (z I ..... zn_1;x I ..... Xk_ I) if x k :x o-
The space C is filtered by subspaces
[Ikl Cj x j (2) FkC(M;X) = \j_--j[ (M) ~j ) /~
and the quotients FkC/Fk_IC are denoted by Dk(M;X) -
Let M denote a closed, orientable surface of genus g, and M is g g g
minus a point. We study C(Mg;S 2n) for nZl. H ~ will always stand for
rational cohomology, and P[
exterior algebras over ~.
8
] resp. E[ ] for polynomial resp.
Theorem A. There is an isomorphism of vector spaces
(3) HwC(Mg;S 2n) ~ P[v,u I ..... U2g]®H~(E[w,z I, .... Z2g
with Ivi=2n, iuiJ=4n+2, lwl=4n+1,
d i_ss @iven b_y d(w) = 2(ZlZ 2 + ... + z
],d)
Izil=2n+1, and the differential
2g-lZ2g )"
Giving the generators weights, wght (v) = wght(zi) = I and wght(u i)
= w g h t ( w ) = 2, m a k e s H C i n t o a f i l t e r e d v e c t o r s p a c e . We d e n o t e t h i s
w e i g h t f i l t r a t i o n by FkH~C. The l e n g t h f i l t r a t i o n FkC o f C d e f i n e s a
second filtration H FkC of H C.
Theorem B. As vector spaces
2n e s2n) (4) H FkC(Mg;S ) =FkH C(Mg; .
It follows that H Dk(Mg;S 2n) is isomorphic to the vector subspace
of H~(g,n) = P[v,u i] ®H (E[w,zi],d) spanned by all monomials of weight
exactly k. To obtain the cohomology of ck(Mg) itself, we consider the
vector bundle
(5) < ~k(Mg)~km k ~ ck : (Mg)+
which has the following properties. First, the Thom space of m times
~k is homomorphic to Dk(M ;sm). Secondly, it has finite even order, g
see [CCKN]. Hence
(6) Dk (Mg;S2nk) i2nk "k = C k (Mg) +
for 2n k =ord(~). Thus we have
9
Theorem C. As a vector space, H~ck(Mg) i__ss isomorphic to the vector
subspace generated b~ all monomials of weight k in H~(g,nk ) ,
desuspended 2nkk times.
Regarding the homology of E =El w,z I .... ,Zig] we have
Theorem D. The homology H (E,d) is as follows:
(7)
(8)
rank Hi(2n+1)= -\i-2/ for i:O,I .... g, and all (non-zero)
elements have weight i;
{2gh {2g rank Hi(2n+1)+4n+£ \ i / - \i+2] for i =g ..... 2g, and all
(non-zero) elements have weight i+2;
(9) rank H. :0 in all other deqrees j. 3
Note the apparent duality rank Hj =rank HN_ j for N = 2g(2n+1)+4n+1.
We will give the proof of Theorem A in the next section. The proof
of Theorem B is the same as for [ BCT, Thm.B]. By what we said above
Theorem C folows from Theorem B. And Theorem D will be derived in the
last section.
3. Mapping spaces and fibrations. Let D denote an embedded disc in Mg.
There is a commutative diagram
(Io) C (D;S 2n) >~2s2n+2
C(M $;$2n) $ ;s2n+2) >map o (Mg
C (Mg, D$; s2n) > (&2n+2) 2g
where maPo stands for based maps. The right column is induced by
restricting to the l-section, and is a fibration. The left column is
a quasifibration. Since S 2n is connected, all three horizontal maps
10
are equivalences, see [M], [B] for details.
The E2-term of the Serre spectral sequence of these (quasi)fibrations
is as follows. From the base we have 2g-fold tensor product of
(11)
where
(12)
H ~QS2n+2 : H~(s2n+IxQs 4n+3) =E[ zi] ~P[ul] (i : I,...2g),
Izil = 2n+I and lui[ : 4n+2. From the fibre we have
He~2S 2n+2 : H~(~s2n+Ix ~2s4n+3) : He(~S 2n+I x S 4n+I)
: P[ v] m E[w] ,
where Ivl = 2n and lwl = 4n+I. The following determines all differentials
in this spectral sequence.
Lemma. The differentials are as follows:
(13) d2n+1(v) : O
(14) d4n+2(w) : 2ZlZ 2 + 2z2z 3 + ... + 2Z2g_iZ2g
Proof: Assertion (13) follows from the stable splitting of C(Mg;S2n) ,
on [ B]. (14) results from symmetries of M and of the fibrations (10) g
which leave d invariant. •
S 2n) The lemma implies E4n+3 :E = HwC(Mg; . Furthermore, E4n+3 is
a tensor product of the polynomial algebra P[v,u I .... U2g] and the
homology module H (E,d) of the exterior algebra E = E[w,z I , ...,Z2g]
with differential d. This proves Theorem A.
4. Homology of E. Let us write x i = z2i_1 and Yi = z2i for i = 1,...g.
The form d(w) = 2ZlZ 2 + 2z2z 3 + ... + 2Z2g,Z2g is equivalent to the
standard symplectie form xlY I +x2Y 2 + ... +Xgyg. The vector space
11
E[g] =L[g] ewL[g] with L[g] :E[XlY I, .... XgiYg]. The differential is
zero on the first summand, and sends the second to the first. Hence
we regard d as an endomorphism of L[g], given by multiplication with
d(w) = xlY I + + ... Xgyg.
Let Lk[ g] denote the vector subspace spanned by all k-fold products
. . . . . . with I ~ i I < i 2 < ... < i k < 2 . (15) ZllZl2 Zln _ g
Since d(w) is homogeneous of weight 2, we have
(16) d=d[g] : 6 g dk[g], dk[g]: Lk[g] > Lk+2[g] k=O
The(co)kernelsof dk[g] is determined by the(co)kernel of dl[g-1]
and dl[g-1]2 for 1 = k,k-l,k-2. The (co)kernelOf dl[g-1 ]2 in turn is
determined by the (co)kernels of din[g-2]2 and din[g-2]3 for m= 1,1-1,1-2.
Therefore we will study all powers dk[g]r and prove the following
(Lefschetz) lemma by simultaneous induction on g, k and r.
Lemma. For 0 N k ~g the differential
dk [ g]r : Lk[g ] > Lk+2r[g] is
(17)
(18)
(19)
monomorphism for O N k <g-r,
an isomorphism for k =g-r
an epimorphism for g-r < k ~ 2g
g = r-1 =Z we have k = k + and k r I r + Proof: For kg i=I xiYi g g-1 Xgyg g g-1 rAg-lXgYg'
in particular k g = g!~ where ~ is the volume g g g =XlYlX2Y 2 ... Xgyg
element. To facilitate the induction, we decompose Lk[g] further by
partitioning the canonical basis elements (15) into four types.
(20) i k _< 2g-2
(21) ik_ I _< 2g-2 and i k = 2g-I ,
12
(22) ik_ I ~ 2g-2 and i k = 2g,
(23) ik_ I = 2g-I and i k = 2g.
Hence Lk[g] = Lk[g-1] e Lk_1[ g-1]Xg e Lk_l[g-1]yg ® Lk_2[g-1]Xgyg.
With respect to this decomposition dk[g] r has the following matrix form
(24)
dk[g]r
dk[g-1 ]r 0 0 rdk[g-1 ]r-i
0 dk_1[ g-l# 0 0
0 0 dk_ I [g-1 ]r 0
O O 0 dk_2[ g-1 ]r
[~ 0 0 A
BOO I ]0 0 B 0
Looo c
To start the induction consider the case g = I. The only non-zero
differential do[ I] : Lo[ I]~L2[ I ] is an isomorphism. For g ~ 2 and k =O,
do[g] r sends the generator of Lo[g] to A r, and thus is monic. Assume g
the lelmma holds for g-1. We distinguish three cases.
Case k <@-r: Then A, A', B as well as C in (24) are all monomorphisms
by hypothesis. Hence, from O =dk[g]r(a,bl,b2,c) = (A(a), B(bl), B(b2),
A' (a) +C(c)) we conclude a =b I =b 2 =0, an~ so c =O as well. Thus
dk[g]r is a monomorphism.
Case k=g-r: Here A is an epimorphism, A' and B are isomorphisms,
and C is a monomorphism. Assume O=dk[g]r(a,bl,b2,c) = (A(a) , B(bl),
A' (a) +C(c)). First, b I =b2 =0. We now have A(a) =dk[g-1]ra =0
and dk_2[g-1 ]rc =-rdk[g-1]r-la; writing this as dk[g-1]r(-ra)=A(-ra)
r+1 =O . Thus, since dk_2[g-1] is an isomorphism, c =O. Therefore,
-rdg_r[g-1]r-la =O, and a =O since dg_r[g-1]r-1 is an isomorphism. We
see that dk[g]r is a monomorphism between vector spaces of equal
dimensions, hence an isomorphism.
13
Case k >g-r: This time A, A', B, C are epimorphisms. Given (a,bl,b2,c)
6Lk+2r[g] we can first find a, b I, b 2 satisfying A(a) =a, B(bl) =61
and B(b2) =62 . Then we choose c such that C(c) =c -A' (a). Hence
dk[ g]r is epimorphic. •
The lemma completely determines H.(E,d) as a vector space over
Q. Theorem D now follows.
References
[B] C.-F. B6digheimer: Stable splittings of mapping spaces.
Proc. Seattle (1985), Springer LNM 1286, p. 174-187.
[BCM] C.-F. B~digheimer, F.R. Cohen, R.J. Milgram: On deleted
symmetric products. In preparation.
[BCP] C.-F. B~digheimer, F.R. Cohen, M. Peim: Mapping spaces and
the hyperelliptic mapping class group. In preparation.
[BCT] C.-F. B~digheimer, F.R. Cohen, L. Taylor: Homology of
configuration spaces. To appear in Topology.
F.R. Cohen, R. Cohen, N. Kuhn, J. Neisendorfer: Bundles
over configuration spaces. Pac. J. Math. 104 (1983), p. 47-54.
[M] D. McDuff: Configuration spaces of positive and negative
particles. Topology 14 (1975), p. 91-107.
[CCKN]
C.-F. B~digheimer
Mathematisches Institut
BunsenstraBe 3-5
D-34OO G~ttingen
West Germany
F.R. Cohen
Department of Mathematics
University of Kentucky
Lexington, KY 40506
USA
An S1-Degree and S1-Maps Between Representation
Spheres
by
Grzegorz Dylawerski
Abstract ,. Let V be an orthogonal representation of G=S I and let
S(V) , S(V~R) be the unit spheres in V , V~R respectively. In this
paper we classify SS-equivariant maps S(V~R) , S(V) . Mors preci-
sely we construct an isomorphism [S(V~),S(V)] G ~ A(V) where A(VI =
= KS(VeR)G,s(w G] ~ (~ Z) , HcS 1 runs over all isotropy subgroups
of V different from S I .
Introduction. Let V be an orthogonal finite-dimensional represen-
tation of G=S I ,~c(V.R) an open bounded invariant subset and f:(~,~)
(V,V~[O}) an equivariant map. For the above f an St-degree ,
denoted Deg (f,~) , was defined in work [31. This is an element of
the group Z 2 • (O Z) , where H C S I runs over all the isotropy sub- s
groups of V different from S I . It is natural to ask whether this
degree classifies the homotopy classes of equivariant maps (B,~B) ....
-~(V,V-{O}) , where B denotes the unit ball in V + R . This is not
true ingeneral. Anyway slightly modifying the first coordinate of
Deg (f~) we get a new invariant which classifies these G-homotopy
classes . Since [(B,~B) , (V,V - ~o~ G~ KS(V ~ R),S(~]G , the new
St-degree classifies the Ghomotopy classes of G-maps between spheres.
The problem of classification of G-maps between representation
spheres was studied by G.B. Segal [5] , T. tom Dieck [2] ,
R.Rublnszteln[4S. We would llke to mention that we came to the
method used here in a result of studying ~2S .
In Section O. we introduce notations , compile some basic facts con-
cerning group actions and the obstruction theory . In Section 1. we
15
recall the properties of Deg(f,~) , sketch the definition of Deg(f,&)
in the special case and define a new S1-degree D(f,B) , (Deg(f,B)
and D(f,B) are distinct on the first coordinate only) .
In Section 2. we relate Deg(f,~) to the obstruction theory . In
Section 3. we describe the group structure on KS(V @ R) ,S(V;~G and
define a homomorphism D • ~S(V + R) ,S(V;~G ..... sA (V) . Section 4.
is devoted to the proof of the main theorem of this paper . This
theorem says that homomorphism D is an isomorphism .
O. Preliminaries. We begin by recalling some terminology and
facts concerning group action and obstruction theory .
Let G be a compact Lie group , and X a left G-space . We
shall denote by G x the isotropy subgroup of x ~ X and by Gx the
orbit of x . For each subgroup H of G let X ~ denote the fixed
point set of H i.e. X H = ~xeX , HCGx~ . The set of points of X
for which G x is precisely H will be denoted by X H .
Let V be an orthogonal representation of G and x sV . We de-
note by N x the normal space to Gx at x and B N (xgr) = ~yeN x ,ly-xl(
~r~ . If XCV is an invariant subset such that G(XH) = X , then
the projection IT : X ~ ~ X/G is an N(H)/H - bundle and the homeo-
morphlsm
~: C x H U~ ~ {G/H)~ U .~ G(U~) ='~-I (U.)
(where U~ = X~ BN(X,r; , ~ = (x,r) , r is sufficiently small and
we identify U~ with S(Ua)) form a family of local triviallzations
of bundle IT : X ~ X/G see t1~ II 5.2,II 5.8 .
Suppose that the action of G on X is free and Y is a left G-
space . Consider the associated G-bundle P " Y ×G X --~ X/G . If
: G x U ~ W'~(U) is a local trivialization of the bundle
then ~: Y~U ~(Y×GG)×U ,YXG(G×U ) ~YxG~-I(U)~p -1 (U)
(y,u) = [y,v(e,u)J
16
is a local trlvi~llzation of p . Moreover , there is a one-to-one
correspondence between G-maps f : X ~ Y and cross-sections sf
of p given by sf(~(x} 1 = ~f(x),x] see [I~ II 2.4 , II 2.6 .
Now we assume that G is a compact connected Lie group , G acts
freely on X and X/G is triangulable ~ X/G =[Ki , K - triangula-
tion of X/G . Let Y be an n-slmple G-space and let L be a sub-
complex of K • Consider a partial cross-sectlon s : Knu L .... > Y x G X°
Since G is connected , the bundle of coefficients associated
with the bundle p is trivial . So the obstruction z(s) to exten-
ding s on Kn+lv L lles in C n+S (K,L, Rn (Y)) . From the above
facts it follows :
O.1 Lemma. Let f : ~-I ~KnuL) ~ Y be a G-map and let
sf : KnuL ~ Y x G X be the partial cross-sectlon of p corresponding
to f . Then z(sf~ (~) = ~f.Ve/~] ~ ~n (Y) where is an n+1 - sim-
plex contained in U , ~ : G~U ~-I ~U) is a local trivialization
of W and ~e(X) -- ~(e,x) .
I Let V be a real orthogonal representation of G S I
O(V) = ~H~S I 18x ~V G x -- H) and ~ cV @ R an open bounded in-
variant subset . We denote by CG(&,~) the space of sl-equivariant
maps f : (~,~) .... ~ (V,V~O~) with the standard metric [fl-f2[=
= x~su If I (X) -f2(x)l . We say that fo,fl ~ CG(~I,~Ol) are G-homotop£c
if there exists a G-homotopy h:[o~ ~ [O,I~ , a~l~[O,1]) , QV,V~O~)
( i.e. h(gx,t) = g.h(x,t) ) such that ht',Ol = fo hQ',1) = fl "
For maps f ~CG(5~,O~) we can define the sl-degree Deg(f,~)
Z2 ~(H~ (v) Z) . We denote by degH(ft~)~Z the H-coordlnate of
Deg(f,ol) and by degs~(f,~) ~ Z 2 its first coordinate .
1.1. Theorem. Let ~I~ ~o' ~1' 812 c V @ R be open bounded In-
variant subsets and f ~ CG(&,~) • Then the following properties
hold:
17
a) If degH(f,~) ~ 0 , then f'1(O)n~H ~ # .
b) If f'1(O) c ~oC~ , then Deg(f,~) = Deg(f,~qo) .
C) If f-1(O) c 21U~2C~I and ~Q1n~2 =~ , then
Deg(f,~) = DegCf,~ I) + Oeg(f,~) .
d) If h : (~[o.I], ~a~[0,1~) , (V.V,~O~) is a C-homo-
topy then Deg(ho,6q) = Deg(hl,~) .
e) Suppose W is another representation of G=S I and let
U be an open bounded invariant subset of W such that
O~U . Define F : Ux~I---~WeV by F(x,y I = ~x,f(y)) •
Then Deg(F,U~) = Deg(f,~l) .
f) Let He0(V}u [$1~ and fH : [~H ~6~H ) ; (VH,VH~{0~)
demot~s the restriction of f . Then degH(f,0%) =
= degH(fH, aH) .
g) If ~H = # ' then degH(f,~Q) = 0 .
The properties a-e have been proved in ([3] Theorem 1.2 ) .
The properties f,g follow immediately from ([3] Definition 3.6 ,
3.7 ) .
Let K = H~O(V}C% H . Assume now that ~K = ~ . We recall the
definition of degK(f,~l) .
Suppose a~V ® R and G a ~ S I . Let v be a tangen~ vector
to M = Ga at a and N a = Ix ~V @ R , <x.v) = O~ the normal space
to Ga at a . Note that a£N a and V @ R = N a @ span~v~ . Let
A : N a .... ~ V be a linear isomorphism . Define A ̂ : V ~ R - ~. V ~ R
by N a @ span~v~ ~ (x, lv) ~ (A (x), A)~V @ R and sgn A =
= sgn(det A n ) • Recall the notation : BN(a,r ) = {x~N a ,Ix-al < r~
B(r) = [x£V ,Ixl < r } .
1.2. Definition. Let aEV ~) R and G a ~ S I . We say that a
continuous map T : B(2) , V ~ R is a Slice map at a if :
a) ~(0) = a
18
b) there exists r)0 such that BN(a,r) is a slice and
W (B (2)} c BN (a,r)
c) there exists a linear isomorphism A : N a ~ V such
= A "1 that ~(x) a + (x) and sgn A = +I .
1.3 Definition.
is #lementary if there exists a finite family
of open invmriant subsets of cQ such that :
al ~o c £1u"'~r
b) 61 i n 61j = # for i ~ J
c;
We say that an open invarlant subset ~ c3~
Oq1' .... ' ~r
for each i , 1.4 i.< r , there is a slice map ~i:B(2)
) oQ such that 5t i c G. ~i (B (I)) .
q.4.Definition. We say that f ¢ CG(~,~) is an elementary map
if there exists an elementary subset ~o c o% such that
f-l(0 ) c ~o "
In ([3] Proposition 2.11) the following lemma has been
proved :
Ix5 Lemma, The set of all elementary maps is an open and
dense subset of CG(~,o~) .
Assume now that f ~CG(~ ,0~) is an elementary map and {5~[} -1
~ satisfy the conditions of Definition 1.3 • Let Ui=~i (~i)
-I and F i = f ° ~i : Ui ~ V . Clearly F~ [0) is a compact subset
of U i , thus the Brouwer degree deg(Fi,U i) is well defined .
Define
degK (f, ~l) _- ~ deg (Fi,UiJ
Let now f ~ C G (~,b~) be any equlvarlant map and ~ ffi minlf~)l.~
By Lemma 1.5 . there exists an elementary map fe c C G (~,~) such
that If.fel< ~ . Define
19
1.7 Definition. degK(f,~) = degK(fe,~)
Remark. We would like to point out that in Definitions 1.2 -
- 1.7 and in Lemma 1.5 the set ~satisfled the condition ~K =~ "
For the general case see ([3] Section 3.) •
Let B = [xeV @ R , Ixt < I~ . Now we define a new invariant
for f ¢ CG(B,BB) . It will be needed in Section 3,4 .
1.8.Definition.
Define :
I [O~ @( ~ Z)
Heo(v)
A(V)= Z @( @ Z) H~O(V)
Z 2 e ( ~ Z) H~O(V)
Let V be a real representation of G S I
if dim V G = I
if dim ~ = 3
if dim V G >, 4
or 2
For given f eCG[B,~B) , by the same letter f we denote the
induced map f/Iff : S(V ~ R~ ----~B(V) , and the same for the re-
striction map fG6 C [BG,~B G) . Let [fG] denete the homotopy class
of the map fG : S(V ~ R) G ~ S(V) G .
1.9 Definition. Let f~CG(B,0B)
Define
ahd
and H E O(V) u [81~ .
I degH(f,B) if H ~ S I
dH(f,B ) = 0 if H = S I and dim vG= 1 or 2
[fGj if H = S I and dim vG~3
D(f,BI -- [dH(f°Bl ~ ~ A~Vl .
20
2. Connection of degHSf,~) with the obstruction theory .
Throughout this section we shall make the following assump-
tions.
Let V be an orthogonal representation of S I , dim V = n+l
~ c V ~ R an open bounded invariant connected subset such 2.1
that ~ Is a smooth St-manifold with boundary , and ~H =~
where (H) is the main orbit type on V .
Under the above assumptions the orbit space ~/S I is a smooth
manifold , hence ~/S I is triangulable , Let K denote a triangu-
lation of ~/S I such that each simplex ~K is contained in a chart
(U,p) of the Sq-bundle ~ ) ~/S I (see Section 0.) . Note
%hat dim K = n+l . Let ~ be any n+1 - simplex contained in a
chart (U.v) , U = ~ (BN(a.r)o ~) and
: B(2) ~ BNCa,r) a slice map . We define ~/e: U ., BN(a,r}
Pc(X) = ~(e,x) and V = ~-I (~e(#)) = ~'I (]T-1(~)) c B(1) .
Since ~ is n+1-simplex there exists an orientation-preserving
homeomorphism h:B(1) ~ ~ . Let $+ be the simplex oriented by
-1 the homeomorphism ~ e ~ ~ ° h .
Suppose f e CG(~,~I) Is an equlvarlant map such that
f-1(O)(~ W-1(IKnl) -- @ (K n denotes n-skeleton) . From defini-
tion 1.6. and Theorem 1.1c we have
~e lo = ~ d e g (foq~oh,B) -- ~" deg(f-veo ~ o h , B)
where 6 runs over all n+1 -simplexes of K . Let [f-~e/~÷] e
~n(V~{O~) denots the homotopy class of the map f°~e/a~ s 6+
; V-[O~ . Identifying ~n(VX[O~) ~ Z by
~[n(VX[O~) = [3B.V,{0}] ~ [(B,SB);(V.V-[0~)] de~ Z
we obtain
21
2-2 Lemma. Let 6~ c V @ R satisfy the assumptions 2.1 and
f eCG(0I,~I) be an equlvarlant map such that f'l(o)nW-1(Kn) =
Then
deg~(f,~) -- ~ fo
We shall denote : Voffi V-~O~ , V oXGc~ = E ) K - the bundle
associated with the bundle IF : ~ ---, K and v = ~K = 3~/S 1 .
Any map f ~ CG(~,~) induces a partial cross-sectlon sf: L ) E .
Since the fibre of the bundle p , V o is n-1 - connected , there
exists an extension ~f : K n u L ) E of sf.
The cross-sectlon s~ induces a cocycle z(s~) cn+1(K,L;~n (Vo))
2.3 Lemma. Let cqcV ~ R satisfy 2.1 and f~CG(Oq,0ol) .
Then degH(f,~) -- 7- z(s~)(6 +) .
Proof. Let f^: 17 -I (K n) ~ V o be the equivariant map corre-
spondlng to S~ and ~" : (~,~01) ~ (V,V-~O~) be an extension of
f~ . Since f aud ~ are equal on ~I , degH(f , 61) = degH(T,~).
From Theorem 1.1c we have degH(r,~) = ~degH(~',~'~+)) . We
shall show that degH(~',W'I(~)) = z(s~)(~ ~) . Let (U,~) be a
chart of the bundle ]T which contain ~ . From Lemma 2.2 and 0.1 it
follows that degH(~,~-1(~)) = [~'°~e/~-~] ffi ~f^~Ve/~ = z(s~ )(~+)"
Consider a homomorphlsm ~ : cn+I(K,L;~n(Vo)) ) En(V o)
(z) = >--z(~ +) . Since I Kl is a compact connected and orlentable
manifold with boundary f Ll , the homomorphlsm 7- induces an isomor-
hlsm Y-*..n+1 n(Vo) .
Let f E CG(~,D0~ ) and sf , s~ denote partial cross-sectlons
as above . We denote by of the cohomology class of z(s~ !
cf ~ Hn+I(K,T;~n(Vo~) . From the obstruction theory it is known
that cf is independent of the choice of extension s~ . The coho-
mology class cf is called the first obstruction . It is well~known
22
that there exists an extension of sf on Kn+1- - K if and only if
cf = O . From the above considerations we deduce :
2.4 Corollary. ~-~cf = degH(f , ~) .
2.5 Corollar~. degH(f,~) = 0 if and only if of = O .
2.6 Theorem. Let ~ c V ~ R satisfy 2.1 and f a CG(~,~ ) .
If degH(f,~) = O then there exists fo ~ CG~'~) such that
fo(X) = f(x) for x ~ and fo(~ c V~O~ .
3. The group structure on [S(WJ, S(V~]Du .
Let W,V be orthogonal representations of a compact Lie
group G ; S(W) , S(V~ the unit spheres in W and V respectively
Xo~ S(W) G , yo ~ S(V) G fixed points . Let ~S(W),S(VSG denote the
set of G-homotopy classes of G-maps f : S(W) , S(VI and let
~S(W}, x o ; S(V), YoBG denote the set of G-homotopy classes (rel.
Xo) of G-maps f : S(WI .... ~ S(V) with f(x o) = Yo •
Suppose dim wG~ I and dim VG~ 2 . Let L=span{Xo~ ¢ W and
let W I = L ± be the orthogonal complement of L in W . We may iden-
tify S(W) with a non-reduced suspension ~S(W 1) = [0,1] x S(Wll/~ .
Under this identification Xo=[O,x~ , -Xo=~S,x~ for x eS(W 1) .
Let ~fI] , ~f2~ E Is(w} , s(v)~ G " We can choose fl 'f2 : S~W) ......
~ S(V) in such a way that fs{-Xo) = Yo and f2(Xo) = Yo "
Define f3 : S(W) ~ S(V)
3.1 f3[t,x ] = ~ fs[2t,x] for O~(t ~(I/2 , x~S(W I)
I f2~2t-l,x] for I/2~ t~(1 , xaS(W I)
Now we define a group structure on KS(W),S(V)~ G by
[fl] + [f2 ] = If31
The following lemma shows that the operation "+" is well defined
23
3.2 Lemma. If f1'f2 : S(W) ; S(V) are G-homotoplc and
f1(Xo) = f2(xo ) ' then they are G-homotopic (tel. x o) .
The proof of this lemma is given in [41 . The standard compu-
tations show that the operation "+" yields a group structure
on Ks(w) , s(v)] G . The G-homotopy class of the constant map
f = Yo is the neutral element .
Now consider the case dim wG>~ 2 , dim V G = I . Let
f : (S(W~,Xo~ ~; (S~V),y o) be a G-map . Observe that f(S[w)G)=
= Yo " Therefore we can define in the same way a group structure
on [s(w),x o ) s(v),Yo] G .
The following lemma will be needed in the next section .
3.3 Lemma. If dim wG>r 2 and dim V G = I, then there exists
a bljectlon
%u : Z2X[S(W), x o : S~V), Yo]G ) [S(W), StV)] G .
Proof, Let L = span[Yo} , V I = L ± and ~:SQV) ~ S(V)
be given by ~(A,Y) = (-A.y) , where (A,Y) ~ LEVI= V . Define
V(o*[f]) =[fl . y(1~[f]) : [~-f]
In the remainder of this section we assume that W = V ~) R
and G = S I . Let B denote the unit ball in VeR . Consider an
St-map f : S(V ~ R) ) StY) . Let f^: B ~ V denote an exten-
sion of f . For f^ the degree D(f^,B) ~A(V) has been defined in
Definition 1.9 . It Is easily seen that D(f^,B) is independend of
the choice of the extension f^ . Therefore we can define .
3.4 Definition. D : Ks (v ® R),SQV)] G
DrfJ = D(f^,B)
d H d H ^ We shall denote : ~f] = (f ,B) .
, A(V)
24
3.5 Theorem. I) If dim vG~ 2 then
D : [S(V@R),S(V)] G ~ A(V) is a group homomorphlsm .
li) If dim V G = 1 then
D : [S(V@R),x 0 ; S(V),Y~G ---~AQV) is a group homomorphlsm .
Proof. i) Let If1] , If2 ] ~ [S(V+R),S~V)] G . We have to
prove that dH([f 1] + If2 ]) = dH[f I] + dH[f2] for l~O(V) u [81~.
It is evident for H = S I (see Definition 1.9, 3.4) • Assume now
that H £O(V) . We identify B with [0,1] x B(W 1) /N where
W 1 = span[Xo~l , B(W I) - the unit ball in W I . Let fl,f2.f 3 i% J%
be as in 3.1 . We denote by fl.f2 : B ~ V the S1-extenslons
of fl,f2 , respectively . Define fB:B ~ V by
fB It,x] = J f; K2t,x~
L fB [2t-I . x ]
Consider the sets :
~I = {[t,x]~B ~ o<t<I/2
~2 = l i t , x ] ~B ; 1 / 2 < t <1
From 1.1c, 1.9 , 3.4 we have
for 0 ¢ t ~ 1/2 , x EB[W1)
for I/2%t~I , xeB(W 1)
, x alnt B(WI))
, x ~int B(W1) ~
dH([f d [f3 =
= degH(fB,21) + degH(f~,~ 2) . Let us define two maps
, B v
% o flit'x] = flf2t'x] for O.<t,< 1/2 x sB(W1 )
Yo for 1/2~t~ I , x~S(W I)
I y 0 for 0 ~t ~1/2 , x~ B(W 1)
f~[t.x] = f2f2t'l,x] for I/2~t~I , x~Bt~ I)
Observe that f~ , f~ : (B,BB) : (V,V~[O~) are S1-homotoplc )
the maps f~ , f~ are equal on ~I and f~ , f~ are equal on
~2 . Therefore from 1.1 b,d it follows that degH(f~,~ I) +
+ degH(fB,~ 2) = degH(f~,~ I) + degH(f~,~ 2) = degH(f~,B) + degH(f~,B) A
= degB(f~,B ) + degH(f2,B ) = dH[fl] + dM[f2~ . This proves i) .
The same proof works in the case ll) .
25
4. A classlflcatlo~ of S!-meps S(V~R) S(V) .
In thls sectlon V denotes an orthogonal representation of the
group G = S I . Let Xo,Y o be fixed points of S(V@R) G ,
S(V) G respectively . We now formulate our main result .
4.1 Theorem. i) If dlm vG~2 then
D : [S(V@R),S(V)] G ---~A(V) is a group isomorphism .
li) If dim V G = 1 then
D : [S(V@R),x o I S(V),Y~G ~ A(V) is a group isomorphism ,
The proof of Theorem 4.1 is based on the following two lemmas .
4.2 Lemm~. Let f : (B,BB) .~ (V,V~0~) be an sl-map . If
D(f,B) = 0 then there exists an S1-map fo : (B,~B) ~ (V,V~{O~)
such that f(x) = fo~X) for x ~ aB and fo(B) c V-~O~ .
Proof.
~SI~ u O(V) such that if
We will define st-maps
such that fi(Bi) c V~[O~
Suppose first that
Choose an ordering H t ,H2 , ... ,H k for the set
H IcHj then j % i . Let Bi= ~ B H~ .
fi:(B,aB) ~ (V,W~0~) , i=It ... ,k
and fi(x) = f(x) for xcBB .
I=I . since ds~[foB) -[fZ~Ba]=O,
there exists an extension ~:B G .... ~ vG~0~ of f/~B G . Define
f1: gBuB I .- V~[0~ by fl (x) = f~x) for x~ ~B and fl ~x) =
= ~'(x) for x ~ BG=BI . The map fl can be extended on B by
Tietze-Gleason Theorem .
Assume the map fl is defined . We will define fi+1 " Let
= Hi+ 1 . Since fi(Bi) c V-QO~ , we can choose an open invarlant
connected subset ~I of (V@R) H such that f;1 (0)H c ~ ¢ ~ c B H
and ~ is a smooth S1.manlfold with boundary . Denote h~j
g • (~,~) ~ (VH,VH~O~) the restrlctlon of fi " Theorem 1.1 b,f
26
implies that degH(g,4Z) = degH(f~,BH) = degH(fi,B) = dH(fi,B) = 0 .
From Theorem 2.6 it follows that there exists an sl-map
go: (~,o~) , (vH,v H~ [O~) such that go(X) = g(x) for x ~ ~
and go(~) c VH~ . We define fi+s : ~Bu Bi+ 1 ~ V~ by fi+l[X~=
= fi{x~ for x E ~B u (Bi+1~) and fi+1 (x) = go(X) for x~ .
The map fi+1 extends on B by Tietze-Gleason Theorem .
Observe that B k = B and fk(B) c V~[O~ . We put fo = fk
and the proof is completed .
4.3.Lemma. i) Let K ~ O(V) . Then there exists a G-map
f : S(V@R) , S(V) such that
1 for H = K dH[f] =
0 for H ~ K
ii) If dim vG>~ 3 , then there exists a G-map
f : S(V@R) ~ S(V) such that
(I0 f°r H--St d H f =
for H ~ S I
Remark. If dim V G = I or 2 then ds~[f] = 0 for any
G-map f (see Definition 1.9) .
Proof. i) Choose x o~ int B K , r> 0 and Yo ~ SIv;G such
that IXol+r (I , Ixoi+(3/2) r>1 and D(Xo,2r) c(V@R) K is a
slice in the space (V@R) K at the point x O ( D(xo,2rl denotes a
disc in the space (Nx)K ) . Let x I = Xo+I3/2)r[xo/Ixo~) ,
U2r= G-D(Xo,2r) . We shall define a G-map ~' : VK@R • V K such
that ~(x) = Yo for x ~vK@R~U2r , r 1(0) = GXoUGX 1 and
I I for H=K dH(~'/B K ,B K) =
0 for H & K
It is easy to construct a map g : D(Xo,2r ) ~ V K such that
g(aD(Xo,2r)) = Yo , g-1(O) =[Xo,X1~ and deg(g,D(xo,r)) = I
27
(Brouwer degree) o Define [[z-x) = Yo for z.x ~U2r and [(z.x) =
= z.g(x) for xe D(xo,2r ) , z e31 . We now extend ~' on VeR by
the formula ~'(x,y) = ~'(x) +y where (x,y) E (V~DR)K@(vK)/- = VeR
and we define a G-map f : 3(VG)R) ~ ~ S(V} by f(x) = ~'(x) /l~(x)l .
and Definition 1.9 , 3.4 , it follows immediately From Theorem 1.1
that I if H=K
dH~fS -- o if H#K
ll) Choose a map fG : S(VeR)G ) S(V) G such that [fG] =
= I g [S(V~R)G,s(V) G] . Let ~ ." B G ~ V G denote an extension of
fG We extend ~ on B by formula ~(x,y) = ~'(x)+y where e
(X,y) g (V~R) G Q) (vG) I . The G-map f ~ S(VeR} ---* S(V) is given
by f(x) = [(x)/~(x) I . This ends the proof .
Proof of Theorem 4.1 . (Mono) . Suppose that D[f] = 0 .
From Lemma 4.2 , it follows that there exists a G-extenslon
~':(B, SB) ----, (V,V~O~) of f such that l'(B) c Vx£O~ . We define
a G-homotopy H : S(V~R)×[0,1] .... ~ S(V) by H(x,t) = ~'(tx)/l~'(tx)l
It is easy to check that the homotopy H Joins the map f and the
constant map H(.,O) = ~'(0)/1}'(O)j . Therefore we have [f] -- 0 .
(Epi) . It follows immediately from Lemma 4.3
Oorollary 4.4
3 V+R ,S V G =
Eye ( • HeO(V)
• z HeO (V)
Ze( • Z) H~ 0 (V)
Z 2 e( @ Z) HeO ~V)
z) ~f dlmV~--1
if dim V G = 2
if dim v G = 3
if dim V G ~ 4
In this paper we have studied Deg[f,~) of an S I- maps .
Nevertheless we are able to define an analogous invariant of T n-
T n the torus ~oreover, the statement of equivariant maps , -
28
Theorem 4.1 extends on this case in following manner
[S(V~R),S(V)B T n = [S(V@R)Tn,S(v)T~ @ (~B Z) H
where the last sum is taken ever all Isotropy groups H
with one dlmensionsl orbits .
Con S(V))
Institute of Mathematics
University of Gda6ek
Wita Stwosza 57
80-952 Gda£sk
References
[I] G.E. Bredon . Introduction to Compact Transformation Group ,
Academic Press , New York and London 1972 .
[2] T. tom Dieck , Transformation Groups and Representation Theory,
Lect. Notes in Math. 766 , Springer , Heldelberg-New York,1979.
[3] G.Dylawerski,K.G~ba, J.Jodel,W.Marzantowicz , An SI-Equlvariant
Degree An The Fuller Index, Preprlnt No 64 , University of
Gda£sk . 1987 .
[4] R.L. Rublnsztein , On the equlvariant homotopy of spheres ,
Dissertationee Mathematlcae , No 134 , Warszawa 1976 .
[5] G.B. Segal , Equlvarlant stable homotopy theory , Actes ,
Congres Inten. Math. Nice 1970 , Tome 2 , p. 59-63 •
[6] N.Steenrod , The topology of fibre bundles , Princeton
University Press , 1951 .
ON C~RTAIN SIEGEL MODULAR VARIETIES OF GENUS TWO AND LEVELS ABOVE TWO
Ronnie Lee* and Steven H. Weintraub**
In our previous work, we have studied spaces M A which are
modull spaces of stable curves (i.e. Riemann surfaces) of genus 2
with level A structure, for two particular subgroups A of
PSP4(Z) •
In general, M A is a three-dimenslonal complex projective
variety. It is usually non-singular, though for some choices of A
it has finite quotient singularities. (In Satake's language, it
is then a V-manifold). It is the Igusa compactification of the
variety M A = S2/A , the quotient of $2, the Siegel space of
• 0 where degree 2, under the action of A. Further, M A m M A m MA,
0 (a Zariski open set in M A) is the moduli space of non- M A
singular curves of genus 2 with level A structure.
We follow our previous notation and write the complement * 0
M A - M~ = ~A u OA, where ~A and @A are unions of components
(each a complex surface) and are themselves moduli spaces for the
two kinds of singular but stable curves of genus 2 (see [LWI] ,
section 8.4).
In our papers [LWI] , [LW2] we considered the case A = F(2),
the principal congruence subgroup of level 2. (In this case a
level A structure is more commonly known as a level 2 structure.)
* Partially supported by the National Science Foundation.
** Partially supported by the National Science Foundation and the Sonderforschungsberelch fur Geometrle und Analysis (SFB 170).
30
In [LW3] we considered the case A = F, where F is a certain
subgroup of F(2). We define F precisely in (1.1) below. Here
we just observe that F(2) m F D F(4), and [F(2): F] = 2 6 ,
[F(2): F(4)] = 29 . The quotient F(2)/F(4) is an elementary
abelian 2-group, and hence so is F(2)/F.
We wish to investigate the topology of these spaces M A for
various A.
In [LWI] , [LW2] we proved the following theorem in case
A = F(2) (see also [G]).
Theorem O.I. a) Hi(M A) = 0 for i odd.
b) The map Hi(~ A u OA) --+ Ni(MA)
for i < 6.
is an epimorphism
c) H4(M A) has a basis consisting of algebraic
c y c l e s ( s o , in p a r t i c u l a r , in the Hodge decom-
*(M~) H p'q position of H , = 0 unless p = q).
d) The i n t e g r a l homology of M h i s t o r s i o n - f r e e .
In [LW3] we proved the same theorem in case A = F.
In addition to this qualitative ("soft") information, in the
above-mentioned papers we have the following quantitative ("hard")
information.
Theorem 0.2. a) In case A = r(2), rank H4(M A) = 16.
b) In case h = F, rank H4(M h) = 79.
(Of course, by Poincare duality, rank H 4 =
rank H2).
Our main result in this paper is the determination of the
homology of M A (at least up to 2-torsion) for all F = I~ = F(2).
The line of argument is given to us by the following theorem:
31
Theorem 0.3. a) - d) The conclusions of theorem 0.I a) - d) are
valid for any F = A = r(2), except that homology must be taken
in Z[$], rather than in Z. with coefficients z
e) For any such A there is an exact sequence
(with (co)homology having coefficients in Z[~]) z
0 --+ HI(M~) --* H4(3 A u OA) --* H4(M:) --+ 0.
Proof: Recall the following general fact: If a finite group G
acts on a space M, and if ~ is any field of characteristic 0
or prime to the order of G, then (see [B, theorem 111.2.4]):
H,(M/G: IF) = H,(M: IF) G
where ( )G denotes the elements fixed under the action of G.
Here we have the 2-group A/F acting on M F with quotient M A.
Let us for the moment take coefficients in 9- Then a) holds im-
mediately, as does b), since the quotient (~r u OF)/A is ~A o 0 A.
Furthermore, c) holds as well, as the required basis of algebraic
cycles for H4(MA) will be images in M A of those elements of the
basis in M F whose fundamental classes are fixed under the action
of A/F. As the reader will see, if in the arguments in this paper
we replace 9 by any field ~ of characteristic not equal to two,
we obtain the same dimensions for all spaces, so in particular
dim H,(MA: IF) = dim H,(MA: 9)
and so the homology of M A has no odd torsion and d) holds,
Of course , i t i s the proof of pa r t e) t h a t r e q u i r e s work.
Consider the exact sequence of the pair (Mr' ~r u Or), again with
coefficients in 9 (or ~, char ~ ~ 2)
H5(~FU @r ) --+ H5(MF' ~F u OF) --+ H4(~ F u OF) --+ H4(M F)
32
The first of these groups is zero, as ~ u 0 F is a union of com-
I * plex surfaces, and the second is isomorphic to H (M F - (~F u @F))
= HI(M~) by Alexander duality. Thus we have
0--+ HI(M~) --+ H4(~ F u @F ) --+ H4(M~)
In fact, as we showed in [LW3] , this last map is an eplmorphism
(and indeed, this is part of 0.I b)). This follows from the compu-
tation of dim H4(MF) = 79, dim HI(M ) = 27, and dim H4(~FU O F)
= 106 there. We shall see below how to make these last two com-
putations (cf. 1.16 and 3.1, and 1.12, 2.1 and 2.2).
Now let A/F act on this short exact sequence.
obtain a short exact sequence
We again
0--+ HI(M~) A/F --+ H4(~ F u 0r)A/r--+ H4(M~) A/F--+ 0
which is nothing other than the sequence
0 --+ HI(M~) --+ H4(3 A u @A ) --+ H4(M ~) --+ 0
as desired.
Corollar~ 0.4. dim H4(M A) = dim H4(~ A u @A ) - dim HI(M ), where
(co)homology is taken in an arbitrary field ~ of characteristic
not equal to two (and these numbers are independent of the choice
of ~ . ) (Note t ha t by Poincare d u a l i t y t h i s i s a l so dim H2(M A)
as M A is an ~-homology manifold.)
This corollary tells us how to compute H4(M A) - compute the
two terms on the right-hand side and subtract. In prlnciple~ this
is the approach we follow.
In practice, as there are very many subgroups A, we choose a
somewhat different line of attack. Namely, what we actually deter-
mine is the action of G = r(2)/r on the space R = H4(MF).
33
Knowing R as a representation space of G of course tells us ~)A/F *
dim H4(M = dim H4(MA) for any F = A = r(2). (It turns out
as well that the final result is much easier to state in terms of
R.) How do we compute R as a representation of G? It turns out ,
that we may do so by computing dim H4(M A) for a relatively small
number of subgroups A (of a kind we call "un-twlsted").
Thus our paper is arranged as follows: In section 1 we es-
tablish our notation, recall some basic results of [LW3], and
establish precisely what we what we need to compute. In section 2
and 3 we compute the two terms on the right-hand side of 0.4 for
certain A. In section 4 we assemble this information to obtain
our main result, theorem 4.2, which gives the action of G on R.
As a specific application we then give dim H4(M A) for the extreme
cases r of index 2 in A, and A of index 2 in r(2).
All (co)homology henceforth is to be understood as having
coefficients in ~. As we have remarked, this gives us information
complete except for 2-torsion. We discuss this question in 4.7.
In many cases we can show that the (co)homology of M A is torsion-
free.
As this manuscript is being photo-offset from typescript, the
reader will be able to appreciate the marvelous typing job done on
it by Nell Castleberry, to whom the author extends his deepest
thanks.
34
I. T'ne situation at level F.
We begin by establishing notation and recalling some of the
results of [LW3].
Definition I.I. Let F(n) = {M ~ PSP4(Z) I M E I mod n}.
Let F = {M e PSP4(Z) I M ~ I + 2 0
a b 0 mod 4}.
c -a
(Observe that F(4) c F c F(2), and [F: F(4)] = 23 , IF(2): F] = 26.)
Definition 1.2. Let $2 = ~M ~ M2(C) I M = tM and Im(M)
is positive definite }.
z 0 Let O 0 = S2 = {(0 w )} I Im(z) > 0, Im(w) > 0}
together with the union of its translates under the action of 0
PSP4(Z). We call 00 the Humbert surface in S 2 and set S2 =
S 2 - 0 0"
Definition 1.3. For any subgroup A of PSP4(Z), set
0 S~/A, 0 = O0/A M h = S2/A , M h = 0 h -
0 is the For any A = F(2), A acts freely on S . Also, M 2
moduli space of non-singular Riemann surfaces R of genus 2 with a
level A structure (i.e. a choice of symplectic basis for HI(R: Z)
modulo the action of A.)
Deflnitlou 1.4. Let M A
Set a A = N A - ~ , and l e t
be the Igusa compactification of M A. ,
0 in M A- @A be the closure of O A
The space M A is the moduli space of stable (in the sense of
Mumford [M]) Riemann surfaces of genus 2 with level A structure.
We call 0 A t h e Humber t s u r f a c e i n MA, a n d , by a b u s e o f l a n -
g u a g e , a A the boundary of M A (even though, as a projective
35
variety, this space has no boundary in the topological sense). Of
course, both the boundary and the Humbert surface are the union of
many irreducible components.
We now recall our results on the structure of M F. We begin 0
with M F .
First we exhibit a 4-fold cover f: ~ + ~, branched over 3
points, each of whose inverse images has cardinality two.
Le~mn 1.5. Let X = {Im z > 0}/FI(4) and X 0 = {Im z > O}/FI(2) ,
where Fl(n) is the principal congruence subgroup of level n of
PSL2(Z). Then X is a 4-fold cover of X 0 with group FI(2)/FI(4)
= (Z/2) + (Z/2). Also, we may identify X with ~ - {±i,O,~,±l}
and X 0 with ~ - {0,I,~}, and the covering projection f: X ÷ X 0
with the function
z = f(w) = ((w 2 + l)/(w2-1)) 2.
Furthermore, f extends to a branched cover of X = ~ to XO =
with f(±i) = 0, f(O) = f(~) = i, and f(~l) = =.
Proof. [LW3] , 2.1 and 2.2.
The group of the cover (Z/2) + (Z/2) is generated by the
loops around 0 and I in the base, each of which has order 2. We
denote them by p and 2 respectively. Of course, the loop
around ~ is then r = pq, the other non-trivial element of this
group.
N o t a t i o n 1.6. Let V denote the Klein 4-group (isomorphic to
(Z/2) + (Z/2)) with elements {l,p,q,r} which appears in 1.5.
Theorem 1.7. Let Z 0 = {(Xl,X2,X 3) E X O I x i not all distinct}
and Z = {(Xl,X2,X 3) ~ X I (f(xl),f(x2),f(x3)) E Z0}.
0 O Then Mr(2) = X 0 x X 0 x X 0 - Z0, and M r = X x X x X - Z,
0 + 0 is covering projection f x f x f. where the map M r MF(2)
36
0 is a 26-fold cover of 0 Thus M F MF(2) with group
V I x V 2 x V3 , with each V i naturally isomorphic to V.
G --
Proof. This is theorem 3.9 of [LW3].
We let V i have the non-trlvial elements pi,qi,ri = piqi ,
where the isomorphism with V is the obvious one suggested by the
notation. We shall frequently identify V i with V via the iso-
morphism. Of course G is generated by {pi,qi }, i = 1,2,3.
In order to do our computations, we need specific matrices
representing the generators (and hence elements) of G. These we
also obtained in the proof of this theorem in [LW3] , and we quote
the result here. (In [LW 3] we denoted PI' say, by opl, as it is
given by the effect of a "Dehn twist" around a lift of a loop repre-
senting Pi' but for simplicity of notation we drop the o here.)
C oI Lemmn 1.8. Pl = I + 2 0 0 0 0 O0 ql = I+ 2
Oil
P2 = I + 2 0 0 q2 = I + 2
1 1
p 3 = i + 2 O 0 I 0 0 0 0 0 0
O 0 I I I I
°111 i 0 1 I I 0 I i i
q3 = I + 2
i01) 1 0 1 1 0 1 0 0 0
We now make a useful observation, which will reduce the number
of cases which we have to consider. Recall that the automorphism
group of V is E3, which acts by permuting the non-trivlal ele-
ments.
Lemma l.g. Let E 3 x E 3 operate on G = V 1 x V 2 x V 3 as follows:
An element of G is a word in pl,...,r3. The first factor oper-
ates on such a word by permuting the symbols p, q, and r (as
above). The second factor acts by permuting the subscripts I,
2, and 3.
37
If A 1 and A 2 are two subgroups of G equivalent under
this ac t i on , then MA1 and MA2 are equ iva len t as complex algebraic varieties.
Proof. The action of E 3 on V may be realized by the unique
automorphlsm of X0 = ~ permuting O, I, and ~ as specified,
giving an action of the first factor on X O x X 0 × X0, and the
second factor acts on this product by permuting the factors. These 0 *
are clearly automorphlsms of Mr, which extend to MF, as their
effect on ~F u ~F is to permute components. This automorphism of
M F then descends to an equivalence between and MA 1 MA 2"
We shall denote this E 3 × E 3 subgroup of the automorphlsm
group of G by A(G).
• 0 Now we consider M A - M~ = ~A u O A. Each of ~A and O A is
a union of irreducible components which are complex surfaces. ~A
is a union of complex surfaces DA(£) , where the indexing set is
{±£ I £ a non-zero primitive vector in ~4}/actlon of A (l.lO)
where A acts on £ by ordinary matrix multiplication, (£)~ = £~.
(In case A = F(n), the principal congruence subgroup of level n,
this set is just the set of non-zero primitive vectors in (Z/n) 4,
taken up to sign.)
Similarly, @A is a union of complex surfaces HA(A) where
the indexing set is
{A = {6,61 } I A = ± ~i^%2 , 6 i = ± £{^~, with 6 and 6 i
mutually orthogonal anlsotroplc subspaces
of Z 4, 6 + 81 = ~4}/action of A .
( l . l l )
We refer to such a A as a anisotroplc pair. Again A acts by
matrix multiplication, and in case A = F(n) this is the set of
pairs of such subspaces in (Z/n) 4.
We call the components D(%) of 3 A boundary components and
the components H(A) Humbert surfaces. The Justification for all
these remarks can be found in [LW2] , for A = F(2), and in [LW3],
38
for A = F, but the set-up holds generally. The set in (1.5)
indexes one type of vertex in the Tits building for A. We do not
need to consider the Tits building here, (or the "Tits building
with scaffolding" of [LW3]) for it contains the further information
on how the various components of a A (or ~A u @A) intersect, and
that is superfluous here.
Le, n~ 1.12. H4(~ A u OA) is the free abelian group on the gener-
ators {[D(%)]} of (1.5) and {[H(A)]} of (1.6), where [ ] de-
notes the fundamental homology class.
Proof. As we are dealing with complex surfaces, which have a
canonical orientation, [ ] is well defined. Since the intersec-
tion of two irreducible components is a complex subvariety (perhaps
singular or empty) it has real codimension at least two so the
lemma is immediate from the Mayer-Vietoris sequence.
1.13. As a representation space of G, H4(a F u @F) is
isomorphic to + Z[%] + + Z[A], where 7 ¢ G acts on the latter A
by ~--+ %7, A--+ AY.
Proof. The action of Y
H(Ay), so it induces a map
takes D(£) to D(%y) and H(A) to
7, on H4(a F u OF) by Y,([D(%)]) =
i[D(%y)], y,([H(A)]) = ±[H(Ay)]. Since the action of y is a
complex automorphism, it preserves the canonical orientations, so
both signs are +, and the lemma follows.
We rephrase theorem 1.7 in a way that will be slightly more
useful.
Definition 1.14. Let ZO = {(Xl'X2'X3) e XO x XO x X01(Xl,X2,X3) ¢ Z 0
or x i = 0, I, or ~ for some i}.
Let Z = {(Xl,X2,X3) E X x X × XI(f(xl),f(x2),f(x3)) ~ Z0 }.
0 ~×~×~_~ Theorem 1 . 1 5 . M r =
0 Proof. This is immediate - compared with the description of M F
in 1.7, we are adding and then subtracting all points (Xl,X2,X3)
with (f(xl),f(x2),f(xq)) not in X O.
39
This latter description has the advantage that X × X × X is
a compact manifold (in fact 7 x 7 x 7) and Z is a union
(not disjoint) of irreducible components, each of which is compact.
Typical components are [(x,y,O)} or {(x,x,y)}, so each com-
ponent of Z is 7 × 7, and different components intersect in a
7 or ~, or not at all.
The group G = F(2)/F has an obvious action (by covering
translations on each factor X) extending its previously defined
0 ~A action on M F. For any F(2) = A m F, we will let denote the
quotient of ZF = ~ under the action of the subgroup A/F of G.
ProposltlOn 1.16. For any subgroup A of r(2), HI(M~)O is a
free abellan group of rank 3 less than the number of irreducible
components of ZA"
HI(M 0) = H5(X x X x X,Z). We then Proof. By Alexander duality,
have the exact sequence of the pair:
H5(X x X x X) --+ H5(X x X × X,Z) --+ H4(Z ) --+ H4(X x X × X)
The first group above is obviously zero; the last map is obviously
onto a free abelian group of rank 3, and, by the same argument as
in 1.12, H4(Z) has rank equal to the number of components of Z,
so the proposition follows for A = F.
Now consider a group A with A/F non-trlvial. Certainly
H I is free abellan, so we need only compute its rank. Thus let us
take homology with coefficients in ~.
Then we have
0--+ H5(X x X x ~,~)A/r --+ H4(~)A/F __+ H4(~ × ~ x ~)A/F__+ 0
Now, by the argument of 0.3, we may identify the first two terms
with HI(M~)~, and H4(Z A) respectively. Furthermore, A/r acts
trivially on the homology of X x X x X (as each generator
pl,...,q 3 of G does) so the last term has rank three and the
proposition follows.
40
We single out a special class of subgroups of G.
Definition 1.17. A subgroup of G is called untwisted if it has a
set of generators which are a subset of {pl,P2,P3,ql,q2,q3,rl,r2,r3}.
We call F c A c F(2) untwisted if A/F c G is.
Such subgroups are naturally distinguished. Also, we will see
that in order to determine the action of G on the homology of
MF, it suffices to consider relatively few subgroups, all of which
are untwisted.
41
2. The action on lines and anlsotropic planes.
By 1.13, the action of A on the fundamental classes of the
components of the boundary (resp. the Humbert surface) of M r is
given by the action of A/r on lines £ (resp. anisotroplc pairs
A), In this section we determine these actions for certain
(enough) subgroups A.
These are long computations, so rather than working them out
in full we indicate how to do them in one or two illustrative
cases.
The boundary components (resp. Humbert components) at level A
are in i-1 correspondence with the A-equlvalence classes of lines
(resp. anistropic pairs), so it is this number we need to compute.
Of course, for A = F or r(2) we already know the answer; it is
for the intermediate levels that work must be done.
Proposition 2.1. For the following untwisted groups A, the
number of equivalence classes of lines at level A is as stated:
[A: F] Generators of A/F Number of equlvalence class@s
1 - 54
2 Pl 40
4 PI' P2 32
4 PI' ql 33
4 PI' q2 29
8 PI' P2' P3 28
8 PI' P2' ql 25
8 PI' P2' q3 23
64 PI' P2' P3' ql' q2' q3 15
Proof. To determine the action of A/F on lines at level F we
of course must know the latter. They are given by [LW3], theorem
3.6 (and there are 54 of them). Recall they arise as follows:
There are 15 lines at level 2, given by (al,a2,bl,b 2) where
a i and b i are defined mod 2 and not all are zero. Each llne
at level 2 is covered by 8 lines at level 4.
42
For example, (I,0,0,0) is covered by
{(i,0,0,0), (1,0,0,2), (1,0,2,0), (1,0,2,2),
(1,2,0,0), (1,2,0,2), (1,2,2,0), (1,2,2,2)}
and (0,1,0,0) is covered by
{(0,I,0,0), (0,1,0,2), (0,1,2,0), (0,1,2,2),
(2,1,0,0), (2,1,0,2), (2,1,2,0), (2,1,2,2)}
and (I,I,0,0) is covered by
{(i,I,0,0), (1,1,0,2), (i,-i,0,0,), (1,-1,0,2),
(1,1,2,0), (1,1,2,2,), (I,-1,2,0), (I,-1,2,2)}
mod 4,
mod 4,
mod 4.
(Note we have resolved the ambiguity in sign by choosing some
entry to be +I mod 4.)
Now on lines over (I,0,0,0), F/F(4) acts trivially, so there
are 8 equivalence classes of such lines at level F. On lines over
(0,I,0,0), F/F(4) acts by interchanging the first and second,
third and fourth, fifth and sixth, and seventh and eighth, so there
are 4 equivalence classes of such lines at level F. On lines over
(I,I,0,0), F/F(4) acts transitively on the first four, and tran-
sitively on the last four, so there are 2 equivalence classes of
such lines at level F.
Now let us consider the action of a typical element Pl of
G. On the equivalence classes of lines over (l,0,O,0) it acts as
follows: Interchanging (I,0,0,0) and (1,2,2,0), (I,0,0,2) and
(1,2,2,2,), (1,0,2,0) and (I,2,0,0), (1,0,2,2,) and (I,2,0,2).
Thus Pl has 4 orbits on these. (On the other hand, as the reader
may check, there are also 8 lines at level F covering the line
(0,0,1,0) at level 2 and Pl acts trivially on these, so there
are 8 orbits of Pl there.)
On the four equivalence classes of lines over (0,1,0,0) Pl
acts trivially , giving 4 orbits. Also, Pl acts by interchanging
the two equivalence classes of lines over (1,1,0,0), giving I
orbit here. Then adding the number of orbits over each of the 15
lines at level 2 gives 40 orbits for A with A/F generated by
PI' giving the second line of the table.
From this point on the computation is routine.
43
proposition 2.2. For the following untwisted groups A, the
number of equivalence classes of anisotropic pairs at level A is
as stated:
[A: F] Generators of A/F ............. Number of equivalence classes
1 - 52
2 Pl 36
4 PI' P2 26
4 PI' ql 28
4 PI' q2 25
8 PI' P2' P3 22
8 Pl' P2' ql 20
8 PI' P2' q3 18
64 PI' P2' P3' q1' q2' q3 I0
Proof. This is entirely analogous to the proof of the preceding
proposition (only slightly more complicated as there are more
choices of representatives). By theorem 3.7 of [LW3] we know all
the anisotropic pairs of level F. (Each of the I0 anistropic
pairs at level 2 is covered by 16 at level 4, and by either 4 or 16
at level F. There are a total of 52 of these at level f.) For
example, the pair A = {6,6±} at level 2, with
= (i,I,0,0)^(0,0,I,0), is covered by the four equivalence
classes of pairs at level F whose representatives we may take
to be ~ and its orthogonal complement ~±, with
= (l,l,O,O)A(O,O,l,O), (l,l,O,O)A(O,O,l,2),
(l,l,0,0)A(0,2,1,0), (1,1,0,0)^(0,2,1,2) mod 4.
Then Pl acting on the first one of these planes sends it to
(1,-1,2,0)^(0,0,1,0). But adding twice the second vector to the
first, we see that this is the same plane as (l,-l,0,0)A(0,O,l,0).
Also, in the proof of 2.1, we observed that the line (1,-1,0,0)
is equivalent to (i,I,0,0), so we conclude that this plane is
equivalent to (i,I,0,0)^(0,0,I,0), i.e. Pl acts trivially on
(1,1,0,0)A(0,0,1,0).
Otherwise the computation is straightforward.
44
3. The action on excised components.
We see from 1.16 that we must count the number of components
of ZA' i.e. the number of orbits of components of Z = ZF under
the action of A/F.
Proposltion 3.!. For each untwisted subgroup A of F(2),
number of irreducible components of A A is as follows:
the
[A: P] Generators of A/P .............. Number of cgmponents
1 - 3 0
2 Pl 24
4 Pl ' P2 20
4 Pl ' ql 21
4 Pl ' q2 19
8 Pl ' P2' P3 18
8 Pl ' P2' ql 17
8 Pl ' P2' q3 16
64 Pl ' P2' P3' q l ' q2' q3 12
Proof. Again there are many cases and we shall merely indicate a
few.
Recall f: X--+ X0 is a branched covering with group
(Z/2) + (Z/2) generated by p and q, with each of f-l(0),
f-l(1), and f-l(=) having cardinality two. The following sche-
matic represents this cover:
I I
2. Z.
3
+ ~.
Q Q • o | ore,
45
Let X (resp. X ) denote the quotient of X by the action of p P q
(resp. of q). Then Xp (resp. Xq) is a 2-fold branched cover of
X0, branched over 1 and ~ (resp. 0 and ~). We continue to let
f denote the covering projection. Then f-l(0), f-l(1), f-l(~)
have cardinallty 2, I, I in Xp (resp. cardinality i, 2, i in X2).
An irreducible component of AA projects onto one of the
following types of components in A0: (*,x,y), (x,*,y), (x,y,*),
(x,x,y), (x,y,x), or (y,x,x), where * = 0, I, or ~ and x
and y are arbitrary. Thus we have six kinds of components, and
we will gather the number of each kind into a 6-tuple, whose sum is
the number of components of AA" For example, when A = F(2),
AA = A0 has the 6-tuple (3,3,3,1,1,1,) and so has 12 components.
a) The case A = F. The 6-tuple is (6,6,6,4,4,4,), as
follows: Here the covering space is X × X x X. F-l(*,x,y) has 6
components as f-l(o) u f-l(1) u f-l(~) has cardinality 6, giving
the first entry, and the second and third are identical. The
fourth entry is the number of components of X , the inverse of
the diagonal in X x X. But this inverse consists of {(Xl,X 2) I x 1
and x 2 differ by a covering translation}, and so has 4 components
(as the group of covering translations has 4 elements), and the
fifth and sixth entries are identical.
b) The case A/F generated by PI" Now the 6-tuple is
(4,6,6,2,2,4). Here the covering space is X × X x X. F-l(*,x,y) P
has 4 components as f-l(0) u f-I u f-l(~) has cardinality 4 in
giving the first entry. The second and third are as in a). P,
The fourth and fifth entries are the number of components of the
inverse image of the diagonal in X x X. But this inverse image P
is the quotient of X (as in a)) by the group generated by
p x id: X x X--+ X x X, and this quotient has two components. The
sixth entry is as in a).
e) The case A/F generated by Pl and P2" The 6-tuple is
× X x X. In par- (4,4,6,2,2,2) and the covering space is Xp P
tlcular note that the fourth entry is 2 by the same argument as
in a) •
46
d~ The case A/r generated by Pl and ql"
is (3,6,6,1,1,4). Here the covering space is
the argument is similar to b).
Now the 6-tuple
X0 x X x X, and
e~ The case A/F generated by Pl and q2" The 6-tuple is
(4,4,6,1,2,2) and the covering space is X x X x X. The only P q
(subtle) difference between this and case c) is the following: The
inverse image of the diagonal of X0 x X0 in X x X has 4 com-
ponents. Under the action of the group generated by Pl and
P2 they are identified to two components in X x X (i.e. this P P
group acts on the 4 components with Pl and P2 each acting non-
trivially but giving the same identification), while here, under
the action of the group generated by Pl and q2 they are identi-
fied to one component in X x X (i.e. this group acts on the 4 P q
components with Pl and P2 each acting non-trivlally but giving
different identifications).
The remaining cases are similar.
47
4 . The r e p r e s e n t a t i o n o f G on the homology o f M r .
In this section we obtain our main result. We use the
c a l c u l a t i o n s of s e c t i o n 2 and 3, which give dim H4(M A) fo r some
A, to decompose H4(M F) into a sum of irreducible representations
of G = r(2)/r.
First we assemble some information.
Prpposlt~gn 4.1. For the following untwisted subgroups
F c A c F(2), dim H4(MA: ~) is as stated:
[A: F] .................... Generators of A/F ......... Dimensio ~
1 - 79
2 Pl 55
4 PI' P2 41
4 PI' ql 43
4 PI' q2 38
8 PI' P2' P3 35
8 PI' P2' ql 31
8 PI' P2' q3 28
64 PI' P2' P3' ql' q2' q3 16
Proof. Immediate from 0.4, 2.1, 2.2, 1.16 and 3.1.
Now G has 64 irreducible representations, all I-dimensional,
which are obtained by letting each of the six generators PI' P2'
P3' ql' q2' q3 act by multiplication by ±i. We will denote an
irreducible representation of G by e = (el,...,e 6) where each
e i is + or - according as the corresponding generator acts by
+I or -I.
Let R = H4(M F) regarded as a representation space of G.
Let R(e) be the subspace on which G acts by the representation
e. Our problem is to determine dim R(e) = the multiplicity of e
in R. The answer is this:
1"neorew 4.2. The multiplicities of the irreducible representations
of G in its action on H4(M F) are given by the fo l lowing t ab le :
48
e l , e 2 , e 3
e 4 , e s , e 6
+++
+÷--
÷--+
--÷÷
÷++ +÷-- ÷--+ --÷+ +---- --+-- ---+ ----
16 3 3 3 3 3 3 1
3 3 0 0 0 0 0 0
3 0 3 0 0 0 0 0
3 0 0 3 0 0 0 0
3 0 0 0 3 1 1 0
3 0 0 0 1 3 1 0
3 0 0 0 1 1 3 0
1 0 0 0 0 0 0 1
(Thus for example, the multiplicity of the representation where
pl,P2,P3 (resp. ql,q2,q3 ) act by (+I,-I,-I) (resp. (-I,+i,-i))
is the intersection of the column labelled +-- and the row
labelled -+- and is I. (Note in this representation (rl,r2,r 3)
act by (-I,-I,+I).)
Remark 4.3. The reader will observe that the multiplicity of each
non-trivial representation is one less than a power of two. Why
this should be so, or what it means, is a complete mystery to us.
Gathering the irreducible representation of G into A(G)-
equivalence classes, we may rephrase the theorem as follows. (Note
that when we compare representations of different Vi's, we are
using their identification to V.)
Theorem 4.4. As a representation space of
R = H4(M F) decomposes as f o l l o w s :
G =V 1 xV 2 xV 3,
Type of irreducible No. of irreducibles Multiplicity Total of this type of each in R Dimension
in R
Trivial 1 16 16
V i acts non-trlvially 9 3 27
for one value of i
49
V i acts non-trivially
for two values of i - both act the same way
V i acts non-trivlally
for two values of i - they act differently
V i acts non-trivially
for all values of i - all act the same way
V i acts non-trivially
for all i - two act same, one different
V i acts non-trivially
for all i - all act differently
9 3 27
18 0 0
3 1 3
18 0 0
6 1 6
Proof. Since all representations of a given type are A(G)-
equivalent, they occur with the same multiplicity, so we must
determine this common value for each type. Let the multiplicities
of these types be m0,...,m 6
trivial representation, m I
representation in which V i
i, etc).
It is easy to check that the number of each type of irreduc-
ible appearing in R is as claimed. Thus by counting dimensions
we obtain the equation
(i.e. m 0 is the multiplicity of the
the multiplicity of each irreducible
acts non-trivially for one value of
m 0 + 9m I + 9m 2 + 18m 3 + 3m 3 + 81m 5 + 6m 6 = dim R = 79.
Now consider the action of Pl on R. By 4.1, the dimension
of the subspace of R on which Pl acts trivially is 55. This
subspace is a sum of copies of 32 of the 64 irreducible represen-
tations of G, and it is easy to see that the number of these of
type 0 is I, of type 1 is 7, of type 2 is 5, etc.
50
Proceeding in this fashion for all the subgroups A given in
4.1 yields the linear system
II 9 9 18 3 18 6 1
1 7 5 I0 I 6 2
i 5 3 4 1 2 0
1 6 3 6 0 0 0
1 5 2 5 0 2 1
1 3 3 0 I 0 0~
/ I 4 1 2 0 0 0
i 3 i 2 0 i 0
I 0 0 0 0 0 0
m 0
m 1
m 2
m 3
m 4
m 5
m 6
This (consistent) system has rank 7, and hence a unique solu-
tion, (mo,ml, .... m 6) = (16,3,3,0,I,0,i), yielding the theorem.
From this theorem we may of course determine dim H4(M A) for
any F c A c F(2). There are very many such A (even up to A(G)-
equivalence) so we content ourselves with listing the extreme cases.
Corollary 4"5" Let F c A c F(2) be any subgroup with [A: f] = 2.
Then A is a A(G)-equlvalent to one of the following, and
dim H4(M A) is as stated:
Generator of A/F Dimension of H4(M A)
Pl 55
plP2 51
plq2 45
plP2P3 59
plP2q3 43
plP2qlq3 41
Corollary 4.6. Let F c A c F(2) be any subgroup with
[F(2): A] = 2, so A is the kernel of a homomorphism
~A: F(2)/F --+ {±I}. Then A is a A(G)-equivalent to one of the
following, and dim H4(MA) is as stated:
51
Generators not in Ker(~A) Dimension of H4(M A)
Pl 19
PI' P2 19
PI' q2 16
PI' P2' P3 17
PI' P2' q3 16
PI' P2' ql' q3 17
(Note that here H4(MA) will be a sum of two types of irre-
ducible representations of G, the trivial one and one other. The
six cases of this corollary correspond, in order, to the six non-
trivial types of irreducibles in theorem 4.4.)
We close by considering the question of torsion in the ho-
mology of M A. As we have seen, the only possible torsion is
2-torsion.
Theorem 4.7. Suppose A is untwisted. Then H,(M A)_ _ is torsion
free.
P r o o f . If A is untwisted, then A/r may be written as a product
W I x W 2 x W 3 with W i c V i. (The different W i need not be iso-
morphic.)
From theorem 1.15, we see that M r is rational, and indeed, 0 *
this theorem shows that MF, a Zariski open set in Mr, is iso-
morphic to a Zariski open set in X x X x X = ~ x ~ x ~.
But then M~ is isomorphic to a Zariski open set in
(X/WI) x (X/W2) x (X/W3) , which is itself isomorphic to
x ~ x ~, so M A is rational.
Then by [AM, proposition I], H,(M A) is torsion-free.
52
R e f e r e n c e s
[AM]
[B]
[G]
[LW I ]
[LW2]
[LW 3 ]
IN]
Artin, M. and Mumford, D. Some elementary examples of uni- rational varieties which are not rational, Proc. Lond. Math. Soc. 25 (1972), 75-95.
Bredon, G. Introduction to compact transformation groups, Academic Press, New York, 1972.
van der Geer, G. On the geometry of a Siegel modular three- fold, Math. Ann. 260 (1982), 317-350.
Lee, R. and Weintraub, S. H. Cohomology of a Siegel modular variety of degree two, in Group Actions on Manifolds, R. Schultz, ed., Amer. Math. Soc., Providence, RI, 1985, 433-488.
Cohomology of Sp4(Z) and related groups and spaces, Topology 24 (1985), 291-310.
Moduli spaces of Riemann surfaces of genus two with level structures, to appear in Trans. Amer. Math. Soc.
Mumford, D. Stability of projective varieties, L'Enseignement Math. 23 (1977), 39-110.
Yale University Louisiana State University and Universit~t GSttingen
THE RO(G)-GRADED EQUIVARIANT ORDINARY COHOMOLOGY OF COMPLEX PROJECTIVE SPACES WITH LINEAR 2[/p ACTIONS
L. Gaunce Lewis, Jr.
INTRODUCTION. If X is a CW complex with cells only in even dimensions and R is a ring, then, by an elementary result in cellular cohomology theory, the ordinary eohomology H*(X;R) of X with R coefficients is a free, 7/-graded R-module. Since this result is quite useful in the study of well-behaved complex manifolds like projective spaces or Grassmannians, it would be nice to be able to generalize it to equivariant ordinary eohomology. The result does generalize in the following sense. Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and R be a ring-valued eontravariant coefficient system JILL]. Then the G-equivariant ordinary Bredon cohomology H*(X; R) of X with R coefficients may be regarded as a coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free module over R in the sense appropriate to coefficient systems. Unfortunately, this theorem does not apply to complex projective spaces or complex Grassmannians with any reasonable nontrivial G-action because these spaces do not have the right kind of G-CW structure. In fact, if G is ~/p, for any prime p, and r / is a nontrivial irreducible complex G-representation, then the theorem does not apply to S ~, the one- point compactification of r 1. Moreover, the 2~-graded Bredon cohomology of S n with coefficients in the Burnside ring coefficient system is quite obviously not free over the coefficient system.
The purpose of this paper is to provide an equivariant generalization of the "freeness" theorem which does apply to an interesting class of G-spaces and to use this result to describe the equivariant ordinary cohomotogy of complex projective spaces with linear :Y/p actions. These results are obtained by regarding equivariant ordinary cohomology as a Mackey functor-vatued theory graded on the real representation ring RO(G) of G [LMM, LMSM]. To obtain such a theory, we take the Burnside ring Mackey funetor as our coefficient ring. Instead of using cells of the form G/H x e n, where H runs over the subgroups of G, we use the unit disks of real G-representations as cells. Our main theorem, Theorem 2.6, then has roughly the following form.
THEOREM. Let G be 2[/p and let X be a G-CW complex constructed from the unit disks of real G-representations. If these disks are all even dimensional and are attached in the proper order, then the equivariant ordinary cohomology H~X of X is a free RO(G)-graded module over the equivariant ordinary eohomology of a point.
To show that this theorem is not without applications, we prove in Theorem 3.1 that if V is a complex G-representation and P(V) is the associated complex projective space with the induced linear G-action, then P(V) has the required type of cell structure. Theorems 4.3 and 4.9, which describe the ring structure of H~P(V), follow from the freeness of H~P(V) . As a sample of these results, assume that p = 2 and V
54
is a complex G-representation consisting of countably many copies of both the (complex) one-dimensional sign representation ,~ and the one dimensional trivial representation 1. Then P(V) is the classifying space for G-equivariant complex line
*p bundles. As all RO(G)-graded ring, i t G (V) is generated by an element c in dimension 1t and an element C ill dimension 1 + A. The second generator is a polynomial generator; the first, satisfies the single relation
c 2 = e2c + ~C,
where e and ~ are elements in the cohomology of a point. If, instead, V contains an equal, but finite, number of copies of A and 1, then the only change in HOP(V ) is that the polynomial generator C is truncated in tile appropriate dimension. If the number of copies of 1 in V is different from the number of copies of A in V, or if p is odd, then the ring structure of H~P(V) is more complex.
Equivariant ordinary Bredon cohomology with Burnside ring coefficients is just the part of RO(G)-graded equivariant ordinary cohomology with Burnside ring coefficients that is indexed on the trivial representations. All of the generators of HOP(V ) occur in dimensions corresponding to nontrivial representations of G. This behavior of the generators offers a partial explanation of the difficulties encountered in trying to compute Bredon cohomology. All that can been seen of HOP(V ) with Z-graded Bredon eohomology is some junk connected to the RO(G)-graded cohomology of a point whose presence in H~P(V) is forced by the unseen generators in the nontrivial dimensions.
Using t t~P(V), tt is possible to give an alternative proof of the homotopy rigidity of linear 2~/p actions on complex projective spaces [LIU]. Moreover, the "freeness" theorem should apply to complex Grassmannians with linear Z /p actions, and it should be possible to compute the ring structure of the equivariant ordinary cohomology of these spaces. Of course, it would be nice to extend the main theorem to groups other than Z/p. Unfortunately, the obvious generalization of this theorem fails for groups other than 7//p. The counterexamples have some interesting connections with the equivariant Hurewicz theorem [LE1]. All of these topics are being investigated.
All of the results in this paper depend on the observation that equivariant cohomology theories are Mackey functor-valued. Therefore, the first section of this paper contains a discussion of Mackey Mnctors for the group 7//p. In the second section, we discuss the RO(G)-graded cohomology of a point, precisely define what we mean by a G-CW complex, and prove our "freeness" theorem. The G-cell structure of complex projective spaces with linear 2~/p actions is discussed in section 3. There the cohomology of these spaces is shown to be free over the cohomology of a point. Section 4 is devoted to the multiplicative structure of the eohomology of a point. The multiplicative structure of the cohomology of complex projective spaces is discussed in section 5. The results stated in this section are proved in section 6. The results on the cohomology of a point stated in sections 2 and 4 are proved in the appendix.
A few comments on notational conventions are necessary. Hereafter, all homology and cohomology is reduced. If X is a G-space and we wish to work with
55
the unreduced cohomolgy of X, then we take the reduced cohomology of X +, the disjoint union of X and a G-trivial basepoint. In particular, instead of speaking of the cohomology of a point, hereafter we speak of the cohomology of S °, which always has trivial G action. If V is a G-representation, then SV and DV are the unit sphere and unit disk of V with respect to some G-invariant norm. The one-point compactification of V is denoted S V and the point at infinity is taken as the basepoint. If X is a based G-space, then NVx denotes tile smash product of X and S V. Unless otherwise noted, all spaces, maps, homotopies, etc., are G-spaces, G-maps, and G-homotopies, etc. We will shift back and forth between real and complex G-representations; in general, real representations will be used for grading our cohomology groups and complex representations will be used in discussions of the structure of projective spaces. If the virtual representation c~ is represented by the difference V - W of representations V and W, then lal = dim V - dim W is the real virtual dimension of a and a G = V G - W G is the fixed virtual representation associated to a. The trivial virtual representation of real dimension n is denoted by n. Recall that the set of irreducible complex representations of G forms a group under tensor product. If 7/is an irreducible complex representation, then r1-1 denotes the inverse of r] in this group. The tensor product of r / and any representation V is denoted 77 V. Many of our formulas contain terms of the form A/p, where A is some integer-valued espression. The claim that A is divisible by p is implicitly included in the use of such a term.
I would like to thank Tammo tom Dieck, Sonderforschungsbereich 170, and the Mathematisches Institut at GSttingen for their hospitality during the initial stages of this work. I would especially like to thank Tammo tom Dieck for suggesting the problem which led to this paper and for invaluable comments, especially on the main theorem, Theorem 2.6.
Equivariant cohomology theories graded on RO(G) are not universally familiar objects, so a few remarks about what this paper assumes of its readers seem appropriate. Equivariant ordinary cohomology with Burnside ring coefficients assigns to each virtual representation c~ in RO(G) a contravariant functor I t~ from the homotopy category of based G-spaces to the category of Mackey functors. It also assigns a suspension natural isomorphism
t t~+v( ,~Vx) =~ tiGX~( )
to each pair (a ,V) consisting of a virtual representation o~ and an actual representation V. The isomorphisms associated to the three pairs (c~, V), (c~, W), and ((~,V + W) are required to satisfy a coherence condition. The functors H~ are required to be exact in the sense that they convert cofibre sequences into long exact sequences. The dimension axiom requires that H~S ° be the Burnside ring Mackey functor and that n 0 tIGS be zero if n E 7/ and n@0. If a is a nontrivial virtual representation, then ~ 0 IIGS need not be zero, but it is uniquely determined by the axioms. Note that because • 0 ttGS is nonzero in dimensions other than zero, the assertion that the cohomology of certain spaces is free over the cohomology of S O is very different from the assertion that the cohomology is free over the coefficient ring. Our cohomology theory is ring calued; that is, any pair of elements drawn from tI~X
56
n~+ZX and t t~X have a cup product wtfich is in a, o . We will also work with
RO(G)-graded, Mackey functor-valued, reduced equivariant ordinary homology with Burnside ring coefficients. This homology theory satisfies the obvious analogs of the cohomology axioms. Also, it has a Hurewicz map, which we use to convert various space level maps into homology classes. Finally, we assume that S O and the free orbit G + satisfy equivariant Spanier-Whitehead duality [WIR, LMSM]; that is, for any e~ in RO(G) there are isomorphisms
I-I~S ° ~H-6~S ° and It~G~ + --~ tI-C~G +.
The proofs of all our results flow from these basic assumptions. In fact, most of the proofs are simple long exact sequence arguments which would be left to the reader in a paper dealing with a g-graded, abelian group-valued, nonequivariant cohomology. One of the points of this paper is that these simple techniques work perfectly well in RO(G)-graded, Mackey functor-valued, equivariant cohomology theories and yield useful results. The one serious demand made of the reader is a willingness to work with Mackey functors. When the group is g /p , these are really very simple objects. Section one is intended as a tutorial on them.
1. MACKEY FUNCTORS FOR Z/p. Since the language of Mackey functors pervades this paper, this section contains a brief introduction to Mackey functors for the groups 7//p. For any finite group G, a G-Mackey functor M is a contravariant additive functor from the Burnside category B(G) of G to the category Ab of abelian groups [DRE, LE2, LIN]. However, since we are only concerned with G = g /p , rather than describing B(G) in detail, we simply note that a g/p-Mackey functor M is determined by two abelian groups, M(G/G) and M(G/e); two maps, a restriction map
and a transfer map
p : M(G/G) -+ M(G/e)
r : M ( G / e ) + M(G/G);
and an action of G on M(G/e). The trace of this action and the composite p r are required to be equal by the definition of the composition of maps in B(G); that is, if x 6 M(G/e) , then
p (x) = gx. geG
The abelian groups M(G/G) and M(G/e) are the values of the Mackey functor M at the trivial orbit and the free orbit; or, if one prefers to think in terms of subgroups instead of orbits, the values of M at the group and at the trivial subgroup. For convenience, we abbreviate G / G to 1 and write M(e) for M(G/e). Frequently the G-action on M(e) is trivial; in these cases the composite pr is just multiplication by p.
A map f : M + N between Mackey functors consists of l lomomorphisn~s
f(1): M(1) ~ N(1) and f(e) : M(e) -+ N(e)
57
which commute with p and r in the obvious sense. The map f(e) must also be G-equivariant. The category ~fft of Mackey functors is a complete and cocomplete abelian category. The limit or colimit of a diagram in ~ is formed by taking the limit or colimit of the corresponding two diagrams consisting of the abelian groups associated to G/G and to G/e. The maps p and r and the group action on the limit or colimit are the obvious induced maps and action.
We wilt describe Mackey functors diagramatically in the form
M(1)
l M(e)
t¢ 0
where M(1) and M(e) will be replaced by the appropriate abelian groups, p and r may be replaced by explicit descriptions of the restriction and transfer maps, and 0 may be replaced by an explicit description of the group action. If p or r is replaced by a number (usually 0, 1, or p), then the map is just multiplication by that number. If 0 is omitted or replaced by 1, then the group action on M(e) is trivial. If p = 2 and 0 is replaced by -1, then the generator of G = Z/2 acts by multiplication by -1.
EXAMPLES 1.1 The following Mackey functors and maps appear repeatedly in our cohomology computations.
(a) The Burnside ring Mackey functor A is given by
Z®Z
(1,P) l l(o,1)
2[
where the notation (1,p) means that the restriction map p is the identity on the first component and multiplication by p on the second. Similarly, (0,1) means that the transfer map is the inclusion into the second factor. For any Mackey functor M, there is a one-to-one correspondence between maps f : A ~ M and elements of M(1). The correspondence relates the map f to the element f(1)((1,0)) of M(1). It follows from this correspondence that A is a projective Mackey functor.
(b) The d-twisted Burnside ring Mackey functor Aid] is given by
ZOZ
(d,P) 1 7 ( 0,1 )
58
where d E g. Note tha t A = A[1]. If d _= + d ' mod p, then there is an i somorph i sm f : Aid] ~ A[d'] of Mackey functors. The m a p f(e) is the ident i ty and if d ' = + d + np, then
f(1)(1,0) = ( + l , n ) E igGT/
f (1)(0 , i ) = (0,1).
I f d = 0 m o d p, then Aid] decomposes as the sum of two other Mackey functors; thus A[d] is only of interest when d ~k 0 rood p. In this ease, it is a project ive Mackey functor . An a l te rna t ive ~-basis for A[d](1) will be used in some of our cohomology calculat ions. To dist inguish the two bases, we denote (1,0) and (0,1) in the present basis by # and r respectively. Select integers a and b such tha t ad + bp = 1. The a l te rna t ive g-basis consists of ~ = a# + b r and ~ = p# - d r . Note t ha t p(~r) = 1, p(~) = 0, and r (1 ) = r . In fact, ~ generates the kernel of p, and r generates the image of the m a p r for which it is named. Of course, c~ depends on the choice of a and b; in our applicat ions, these choices will a lways be specified.
(c) If C is any abel ian group, then we use (C) to denote the Mackey functor described by the d i ag ram
C
°I l° 0
(d) If d 1 and d2 are integers pr ime to p, then there is an i somorph ism
g12: A[dm] ® (77} - , A[d2] G (g}.
Let #i and r i be the s tandard generators for A[di], and let z 1 and z 2 be genera tors of (77)(1) in the domain and range of g~2. Select integers a i and b i such t ha t aid i q- bip = 1, for i = 1 or 2. The m a p g12(e) : 77 -~ 77 is the ident i ty map , and the m a p g12(1) is given by
g l ; ( 1 ) ( t q ) = d 1(a2/*2 + b2r2) + (bl + b2 - blb2P )z2
g12(1)(r l ) = r 2
and
g12(1)(zl) = P#2 - d2r~ - a ld~z>
T h e inverse of g12 is jus t g~l- The existence of this i somorph i sm will explain an appa ren t inconsistency in our descript ion of the equivar iant cohomology of project ive spaces.
(e) Associated to an abel ian group B with a G-act ion, we have the Mackey functors L(B) and R(B) given by
59
L(B) R(B) B /G B G
I B B t2 0 0
Here, ~ : B c -* B is the inclusion of the fixed point subgroup and 7r : B -* B / G is the projection onto the orbit group. The two maps tr are variants of the trace map. The
map t r : B - * B c takes x E B to ~ g x E B e. I f x E B and Ix] is the associated
equivalence class in B/G, then tr : ~ f G --* B is given by
tr([x]) = 2 g x E B. gcG
These two constructions give functors from the category of 7/[G]-modules to the category of Mackey functors. These functors are the left and right adjoints to the obvious forgetful functor from the category of Mackey functors to the category of 2r[G]-modules. We will encounter these functors most often when B is 7/ with the trivial action or, if p = 2, with the sign action. Denote the resulting Mackey functors by L, R, L_, and R_. These functors are described by the diagrams
L R Z
7/ 7/
1 1
L_ P~_ z/2
-1
0
Z U" -1
If C is any abelian group, there is an obvious permutation action of G on C p, the direct sum of p co~es of C. Unless otherwise indicated, this action is assumed
I1 ~ P when we refer to L(C ) or R(C ). These two functors are isomorphic and are described by the diagram
60
C
C p
d 0
where A is the diagonal map. V is the folding map, and 0 is the permutation action.
(f) If M is a Mackey functor, then L(M(e) p) ~ R(M(e) p) is denoted M G.
There are two reasonable choices of a G action on M(e) p, the permutation action or
the composite of the permutation action and the given action of G on each factor M(e). These actions yield isomorphic g[G]-modules, so the choice is not important. The simple permutation action is always assumed here. The assignment of M G to M is a special case of an important construction in induction theory [DRE, LE2] that assigns a Mackey functor M b to each object b of B(G) and each Mackey functor M.
The restriction map p : M ( 1 ) - + M ( e ) ~ M G ( t ) and the diagonal map
A: M(e)-4 M(e )P~MG(e ) form a natural transformation p from M to M G.
Similarly, r : MG(1) ~ M ( e ) + M(1) and the folding map V: MG(e ) ~ M ( e ) p + M(e)
form a natural transformation r : M G -+ M. The Mackey functor A c = L(77 p) is characterized by the fact that, for any Mackey functor M, there is a one-to-one correspondence between maps f : Ao -+ M and elements of M(e). This correspondence relates the map f to the element f(e)((1,0,0 . . . . . 0)) of M(e). It follows that A G is a projective Mackey functor.
G (g) If Y is a G-space, M is a Mackey functor, c, 6 RO(G), and H~(Y;M) and Ha(Y; M) denote the abelian group-valued equivariant ordinary cohomology and homology of Y with coefficients M in dimension a, then the Mackey functor valued cohomology H~(Y; M) and homology H~(Y; M) are described by the diagrams
M) Hg(Y; r,1)
Ha(G × Y; M) H~(G x Y; M)
6
where the maps rr*and rr. are induced by the projection r r : G x Y-+ Y, and the maps rr t and rr! are the transfer maps arising from regarding the projection rr as a covering space. The group H~(G x Y;M) is isomorphic to the nonequivariant cohomology group HIm(Y;M(e)). If r~ is represented by the difference V - W of representations V and W, then, under this isomorphism, the action of an element g of
61
G on H~(G x Y;M) may be described as the composite of multiplication by the degrees of the maps g : S V ~ S V and g : S W - * S W and the actions of g on
HIm(Y; M(e)) induced by the action of g on M(e) and the action of g-1 on Y. Similar
remarks apply in homology. If no coefficient Mackey functor M is indicated in equivariant cohomology or homology, then Burnside ring coefficients are intended.
(h) For any Mackey functor M and any abelian group B, the Mackey functor M ® B has value M(G/H) ® B for the orbit G / H and the obvious restriction, transfer, and action by G. If M* is an RO(G)-graded G-Mackey functor and B* is a Z-graded abelian group, then M* ® B* is the RO(G)-graded G-Mackey functor defined by
(M*® B*) c~ = ~ M z ® B '~. /~+n=a
If a CW complex Y with cells only in even dimensions is regarded as a G-space by assigning it the trivial G-action, then there is an isomorphism of RO(G)-graded Mackey functors
* ~ ® H * ( Y ; 77) H e Y = ~H* S o
which preserves cup products.
For any finite group G, there is a box product operation [] on the category ~Jl of G-Mackey functors which behaves like the tensor product on the category of abelian groups. In particular, ~0l is a symmetric monoidal closed category under the box product. The Burnside ring Mackey functor A is the unit for •. If G = 27/p, then the box product M [] N of Mackey functors M and N is described by the diagram
I-M(1) ® N(1) ® M(e) ® N ( e ) ] / ~
M(e) ® N(e)
0
The equivalence relation ~ is given by
x ® r y ~ p x ® y
r v ® w ,,~ v ® p w
for x C M(1) and y e N(e)
for v C M(e)and w C N(1).
The action 0 of G on M(e) @ N(e) is just the tensor product of the actions of G on M(e) and N(e). The map r is derived from the inclusion of M(e)® N(e) as a summand of the direct sum used to define M rlN(1). The map p is induced by p ® p on the first summand and the trace map of the action 0 on the second.
EXAMPLES 1.2(a) For any integers d I and d2, there is an isomorphism
A[dl]DA[d2] ~-- A[dld2]
62
of Mackey functors.
(b) isomorphism
(c) diagram
If B is a 7][G]-module and M is any Mackey functor, then there is an
L(B)IqM ~ L(B®M(e)).
For any Mackey functor M, the product ROM is described by the
M(1)/(p - rp)
pt ~ 7 r! M(e)
0
where M(1)/(p - rp) is the cokernel of the difference between the multiplication by p map and the composite rp. The maps pt and r t are induced by the restriction and transfer maps for M. In particular, if M = R(B) for some 7[G]-module B, then R[]R(B) ~ l%(B). Also, for any abelian group C, R • < C > ~ < C / p C > .
(d) If p = 2, then for any Mackey functor M, the product R_DM is described by the diagram
M(e)/(image p)
1 - u ~ ) r r
Ct -0
Here ~r: M(e) -~ M(e)/(image p) is the projection onto the cokernel of the restriction map and v: M(e) --* M(e) describes the action of the nontrivial element of G on M(e). The action -0 is the composite of the given action 0 of G on M(e) and the sign action of GonM(e) . In particular, R_[]R ~ L.
(e) For any abelian group C and any Mackey functor M,
< :C:>•M ~ <C®(M(1)/ image r )> .
A Mackey functor ring (or Green functor [DRE, LE2]) is a Mackey functor S together with a multiplication map # : S 13 S --* S and a unit map r/: A ~ S making the appropriate diagrams commute. A module over S is just a Mackey functor M together with an action map ~:SDM--* M making the appropriate diagrams commute. Since the Burnside ring Mackey functor A is the unit for [], it is a Mackey functor ring whose multiplication is the isomorphism A []A ~ A and whose unit is
63
the identity map A --* A. Every Mackey functor is a module over A with action map the isomorphism A [ ] M -~ M. Note that if S is a Mackey functor ring and R is a ring, then the Mackey functor S ® R of Examples 1.1(h) is a Mackey functor ring. Similar remarks apply in the graded case. The cohomology of any G-space Y with coefficients a Mackey functor ring S is an RO(G)-graded Mackey functor ring whose multiplication is given by maps
'~ Y" • t t ~ ( Y ; S ) --* .o.~ t ; ,, I I G ( , S ) • ~+Z/y S~
for ~ and /3 in RO(G).
The following result characterizes maps out of box products and allows us to describe a Mackey functor ring S in terms of S(1) and S(e). This is the approach to Mackey functor rings used in our discussion of the ring structure of the cohomology of complex projective spaces.
P R O P O S I T I O N 1.3 For any Mackey functors M, N and P, there is a one-to-one correspondence between maps h : M [-1N --* P and pairs H = (H1, He) of maps
n 1 : M(1) ® N(1) -* P(1)
H e : M(e) ® N(e) --* P(e)
such that, for x E M(1), y E N(1), z E M(e), and w E N(e),
He(pX ® py) = pIq(x ® y) HI(Tz ® y) = T He(z ® py)
H l ( x ® r w ) : T H e ( p x O w ) .
The second and third of these equations are called the Frobenius relations.
PROOF. The maps H e and h are related by H e = h(e). Given h, H 1 is derived in an obvious way from h(1) using the definition of MV1N. Given H 1 and He, h(1) is constructed from the maps H I and T H e on the two summands used to define M E N ( l ) .
It follows easily from the proposition that, if S is a Mackey functor ring, then S(1) and S(e) are rings, p: S(1) ~ S(e) is a ring homomorphism, and r : S(e) --* S(1) is an S(1)-module map when S(e) is considered an S(1)-module via p. Moreover, if M is a Mackey functor module over S, then M(1) is an S(1)-module and M(e) is an S(e)- module. If we regard M(e) as an S(1)-module via p:S(1)--* S(e), then the maps p: M(1) --* M(e) and T: M(e) --* M(e) are S(1)-module maps.
2. H* ~0 AND SPACES W I T H FREE COHOMOLOGY. Here, we recall Stong's G ° unpublished description of the additive structure of the RO(G)-graded equivariant ordinary cohomology of S °. We use this to show that if X is a generalized G-cell complex constructed from suitable even-dimensional cells, then H~X and H G x are
* 0 free over t t G S . The additive structure of the cohomology HOG + of the free orbit is also described. This is used to show that FI~X and tt.GX are projective over H~S °
64
when X is constructed from a slightly more general class of even-dimensional cells.
* 0 Since 72/2 has only one nontr ivial irreducible representat ion, I t e S is very easy to describe when G = 7//2.
T H E O R E M 2.1. If G = 22/2 and s E RO(G), then
c¢ 0 H e S =
r A, if lal = Is el = o, R, if Isl = 0, [ s e l < 0, and let el is even, R_, if tetl = 0, t a e i _< 1, and t a e i i s o d d , L, if letl = 0, Is el > o, and let Gi is even, L_, if IetI = 0, let et > 1, and let el is odd,
(~), if Isl # 0 arid let eI = 0, (72/2), if letl > 0, I s e l < 0, and ietel is even, (7//2), if le t l < 0, Io, e l > 1, and l e t e l is odd,
~. 0, otherwise.
• o FIGS for various a on The most effective way to visualize tIGS is to display a 0 a coordinate plane in which the horizontal and vertical coordinates specify lete[ and lad respectively. In such a plot, given as Table 2.2 below, the zero values of HeS* 0 are indicated by blanks. The only values in this plot with odd horizontal coordinate are the R_ and L_ on the horizontal axis and the (7//2} in the fourth quadrant .
• . + (7//2) (;~/2} (7/12) (7//2) (~/2) (7//2) {27/2} (7//2) {77/2)
(7/12) (7/lZ) 9ei2) R R_ R R_ R
(z)
R_ A
(z)
(7/)
9')
R_ L L_ L L_ L -.-
(7//2) (~ /2) ...
(7//2) (7//2) . . .
(~12) (~12) . . .
(~/2) (7//2) . . .
T A B L E 2.2. H~S ° for p = 2.
Even though the representat ion ring of G is much more complicated when p :/= 2, I-I~S ° is completely determined by the integers a and ]o~GI except in the special case where Isl = c~ c ---- 0. In this special case, II~S ° is Aid] for some integer
65
d which depends on a. Unfortunately, because of the isomorphism described in Examples 1.1(b), d is only determined up to a multiple of p. The major source of unpleasantness in the description of the multiplicative structure of the equivariant cohomology of a point and of complex projective spaces is this lack of a canonical choice for d. To explain the relation between a and d, we introduce several relatives of the representation ring. Let R(G) be the complex representation ring of G and RSO(G) be the ring of SO-isomorphism classes of SO-representations of G. Since any real representation of G is also an SO-representation, the difference between RO(G) and RSO(G) is that, in RSO(G), equivalences between representations are required to preserve underlying nonequivariant orientations on the representation spaces. The difference between R(G) and RSO(G) is that elements of RSO(G) may contain an odd number of copies of the trivial one-dimensional real representation of O. Let R0(G ), RO0(G ), and RSO0(G ) denote the subrings of R(G), gO(G), and RSO(G) containing those virtual representations a with Ic~l = [aGt = O. Note that R0(G ) = RSO0(G ). Let R0(G) be the free abelian monoid generated by the formal differences C-r] of complex isomorphism classes of nontrivial irreducible complex representations. Note that R0(G ) is the quotient of R0(G) obtained by allowing the obvious cancellations and that RO0(G ) is the quotient of R0(G ) obtained by identifying conjugate representations. Let A be the irreducible complex representation which sends the standard generator of 7//p to e 2'~i/p. The monoid R0(G) is generated by elements of the form Am _ An where 1 < m , n _< p - 1 . Define a homomorphism from R0(G) to 77, regarded as a monoid under multiplication, by sending the generator A m - A n to m(n-1), where n -1 denotes the unique integer such that 1 _< n -1 _< p - 1 and n(n - 1 ) - 1 mod p. Define functions from RSO0(G ) and RO0(G ) into 77 by composing this homomorphism with sections of the projections from R0(G ) to RSOo(G ) or RO0(G ). Let d~ denote the integer assigned to the virtual representation oe by either map. The sections can not be chosen to be homomorphisms, so the assignment of dc~ to a will not be a homomorphism from RSO0(G ) or RO0(G ) to the multiplicative monoid g. However, the assignment of da to o~ does give a homomorphism from R0(G ) to the group of units (77/p)* of g / p and a homomorphism from RO0(G ) to the quotient (77/p)*/{+1} of (g/p)*. For later convenience, we select our sections so that d o is 1.
Stong's description of the additive structure of * 0 HGS can now be translated into the Mackey functor language of section one.
THEOREM 2.3. If p is odd, then
A[d~J R L
0 JaG s = (7/)
<77/p) (X/p)
0
if if if if if if lal < 0, otherwise
lai = laGt = 0 Ic~l = 0 and taG] <: 0 lal = 0 and la G ] > 0 tal :fi 0 and la c] = 0 tal > 0, lae] < 0, and la Gl is an even integer
laGI > 1, and let GI is an odd integer
66
As in the case p = 2, H~S ° is best visualized by plotting it on a coordinate plane whose horizontal and vertical axes specify faGl and lal respectively, In this plot, given as Table 2.4 below, the zero values • 0 of ItGS are indicated by blanks. The vertical and horizontal coordinates of all the nonzero values, except the (2r/p) values in the fourth quadrant, are even, Notice in the plots for both the odd primes and 2 that the vanishing of * 0 2) is ttGS on the vertical line laGl = 1 (for I~l ¢ 0 if p = unlike its behavior on the vertical lines corresponding to the other odd positive values for lc*c[. These unusual zeroes for H~S ° are the key to our freeness and projectivity results. When G = ?7/pn for n > 1, the corresponding values are not zero, so our techniques do not extend to these groups.
Hereafter, we will often describe elements in H~S ° by their position in these plots• For example, we may refer to the torsion in the fourth quadrant or the copies of (7]} on the positive vertical axis.
(Z/p) (Z/p) (Z/p) (~)
<~/p> (~/p> <~/p} (~>
... (~/p} (~/P) (~/P) (~)
R R R A[d~] L L L
(iV/p} (~/p}
9z) (~/P} (~/P}
(Z/p> {~/p)
TABLE 2.4. H~S ° for p odd.
Recall, from Examples 1.1(f), the new Mackey functor M G which can be derived from any Mackey functor M, and the observation that A 6 = L(~ 'p) = R(gP).
. + r ~ ~S0~ and from this, to compute ttGG . It is easy to check that I-I~G + is ~G~ JG,
67
C O R O L L A R Y 2.5. For any prime p,
• + f A o if lal : 0 HGG =
0 otherwise
, + Proposition 4.12 tells us that IIGG is an RO(G)-graded projective module
* 0 over IIGS , and that. any map
f: * + M* HGG -*
of RO(G)-graded modules over H~S ° is completely determined by the image of (1,0,0 . . . . ,0) E gP = H~(G+)(e) in M°(e).
A generalized G-cell complex X is a G-space X together with an increasing sequence of subspaces X,~ of X such that X 0 is a single orbit, X = tO Xn, X has the colimit (or weak) topology from the X,~, and Xn+l is formed from X,~ by attaching G-cells. We will allow two types of G-cells. If V is a G-representation and DV and SV are the unit disk and sphere of V, then the first type of allowed cell is a copy of DV attached to X , by a G-map from SV to X,~. The second type of cell is a copy of G x e "~, where e rr̀ is the unit m-disk with trivial G action, at tached to Xn by a G-map from G x S m-1 to Xn. For each n, we let J , + l denote the set of cells added to X , to form X,+ 1. Regard a cell DV of the first type as even-dimensional if IV] and ]V G] are even. Regard a cell G x e m as even dimensional if m is even.
T H E O R E M 2.6. Let X be a generalized G-cell complex with only even-dimensional cells.
(a) Assume that X 0 = • and all the cells of X are of the first type; that is, disks DV of G-representations V. Assume also that IV c] >_ IwGI whenever DV 6 J,~, DW 6 ak, 1 < k < n, and IVI > Iwl. Then * + t t c X is a free RO(G)-graded module
• 0 over H~S with one generator in dimension 0 and one generator in dimension V for
each DV 6 J, , , n > 1. The homology I-I,~X + of X is also a free RO(G)-graded • 0 ; " module over H o S vlth generators in the same dimensions.
(b) If X contains cells of both types and all the cells of X of the first type satisfy the condition in part (a), then * + H c X is a projective RO(G)-graded module
l- l* X + l-l* X + • 0 over H~S °. Moreover, ~ G is the sum of one copy of ~ G 0, which is I-ItS or u. G + . o ~ G , in dimension 0, one copy of HGS in dimension V for each DV 6 Jn , and one
. + copy of HGG in dimension 2k for each G x e 2k E J , , n > I. The homology I-I,GX + of X is also a projective RO(G)-graded module over H~S -d and decomposes into the s a n l e summands.
PROOF. Abusing notation, we let J,~+l denote both the set of cells to be added to X~ and the space consisting of the disjoint union of those cells. Let OqJn+ 1 denote the space consisting of the disjoint union of the boundaries of the cells in J,~+l- Associated to the cofibre sequence
68
+ X~ + -. X~+ 1 ~ Jn+l/0J,~+i,
we have the long exact sequences
G + • . . - . . - , a X _ , . . .
and
o~ + l~c~X+ ~ c~+1 • "" ~ ItGX-+~ ~ .~O '~ t tG (J~+i/cgJ,~+l) ~ . . . .
The space J,~+i/cgJ~+l is a wedge of one copy of S V for each DV E J~+l and one
• j copy of G+^ S 2k for each G x e 2k E Jn+l. Thus, HG(n+l/O.Jn+l) and
H.Gj( n+i/cgjn+l ) are projective modules over H~S ° with generators in dimensions
corresponding to the cells added to X~ to form X~+ 1. Moreover, if J~+l contains
only type, t ] G ( , J n + l / ( * J n + l ) cells of the first then and HG(j~+l/cgJ,~+i) are free
I:l* X + modules over H~S °. The space X 0 is either a point or tile free orbit G, so ~G 0 and
H.GX0+ are projective, and perhaps free, modules over I t , S ° generated by single
elements in dimension 0.
We will show inductively that the boundary maps 0 in both long exact
sequences are zero. The long exact sequences rnust then break up into short exact * +
sequences which split by the projectivity of HG(Jn+l /0Jn+l) and t IGX, . Thus, by
I I * X + G + , 0 H. X,~ are projective, as appropriate, over HGS , with induction, -~G n and free or
the indicated generators. It follows by the usual colimit argument that HGx + is free,
or projective, with the appropriate generators. Since the map
I-I oe X + l.[ce X + i k G n + l -* .l~/. G n
is always a surjection, the appropriate lim 1 term vanishes, and the cohomology of X,
being the limit of the cohomologies of the X,~, is free (or projective) with the
appropriate generators.
The graded Mackey functors H;(J~+l/cgJ~+l) , I-I.6(J~+i/cgJ~+i) , n* X + S~G 0
G + , + and t I . X 0 are sums of copies of • 0 ttGS and HGG in various dimensions. By
induction, we may assume that . + G + HGX~ and It. X~ are also of this form. To show
that the maps 0 are zero, it therefore suffices to show that they are zero from each
summand of the domain to each summand of the range. For the cohomology
sequence, the four possibilities for the summands and the map between them are:
and
• - 2 k + H G G ~-- H~(G+^S 2k)
• - w 0 ~ t I ~ t t G S = S w
H , - 2 k r _ + ~ H~(G+^ ) G "J : S2k
t t5-Ws ° =~ t i e s w
69
T r * + l z , ~ + c~2rn\ l ~ . + l - 2 r n G + - ~ 116 t, t J A ~ ) ~ " ~ G
T T * + I / , ~ + c~2m\ ~ u * + l - - 2 m ( 2 + - ~ r l G t, kJr A O ) ~ " ~ G "~
* + 1 V ~ I T * + I - V N 0 tt G S = -~G
*+1 V ~ U * + I - V s 0 tIG S = ~-G
lff* X + Here, we use I-I~(G+A S 2k) and H~S w to denote summands of .u. G n isomorphic to
H* G + in dimension 2k or H~S ° in dimension W. The four maps above are all maps G
• 0 of RO(G)-graded modules over I-IGS . Any such map out of * 0 t tGS is determined by . +
the image of 1 E A(1) = H~(S°)(1). By Proposition 4.12, such a map out of IIGG
is determined by the image of (1,0,0 . . . . . 0) E 7/P = H~(G+)(e) . Thus, to show that
T r 2 k + l - - 2 r n / , ~ + x / x the four maps are zero, it suffices to show that the groups zl. G ~o )~,e),
W+l-2m + 1 W+I-V 0 tt G ( G ) ( ) , H ~ + l - V ( s ° ) ( e ) , and tt G (S ) (1 ) are zero. The integers
1 2 k + l - 2 m l and I W + l - 2 m ] are odd and ~ + ttGG vanishes whenever Ictl is odd, so the
first two groups are zero. The integer 12k+l VI is odd and ~ 0 t tG(S )(e) vanishes when
lal is odd, so the third group is zero. For the fourth group, if IVI <IWI, then
tt GW+I-VS0 is zero because ] w G + I - V G] is odd and ] W + I - V ] is positive. Otherwise,
W + l - V o IvGI _> IwGI, and tt G S is zero because Iw + -vq is at most one. An
analogous proof shows that the map (9 in the homology sequence is zero. Note that if
n W + l - V s ° is a result of the Ivl>lwl and IVGI=Iw% then the vanishing of ~G
anomalous zeroes on the I GI = 1 line in the graph of H~S °.
In order to compute the ring structure of the equivariant cohomology of X, we must compare it with more familiar objects, such as the nonequivariant ordinary cohomology of X and X G. If X is a generalized G-cell complex satisfying the conditions of either part of Theorem 2.6, then so is X G. Thus, Examples 1.1(h)
describes H ~ ( X 6 ) + in terms of the nonequivariant cohomology of X G. Since the
tIG(X )(e) is just the nonequivariant ordinary cohomology of X with 7/ group * +
coefficients, the map
p G i* : H~(X+)(1) --* H~(X+)(e) O H ~ ( ( x G ) + ) ( 1 )
70
offers a comparison between H~(X+)(1) and two more easily understood cohomology
rings. This map does not detect the torsion in H~(X+)(1) coming from the fourth
* 0 * G + quadrant torsion in HGS . Moreover, the torsion in tiG((X ) )(1) makes it hard to
compute the image of p @ i*. These difficulties suggest that we also consider the
image of * + ~ + tIG(X )(1)/ torsion in ( H ~ ( X ) ( e ) @ H ~ ( ( X a ) + ) ( 1 ) ) / t o r s i o n . Since * +
I-I6(X )(e) contains no torsion, in the range we are only collapsing out the torsion in
I4~((X6)4-)(1). The most useful comparison map is produced by also collapsing out
* G + the image of the transfer map r from t Ic ( (X ) )(e). The quotient
t I ; ( (xG)+) (1 ) / ( t o r s ion @ im r )
consists of copies of 2 in various dimensions; there is one ~' in the quotient for each
A[d] or (77) which appears in I-I~((XG)+)(1).
For many spaces, including complex projective spaces with linear actions, the cells can be ordered so that Ivl_> Iwl whenever DVEJ,~ , D W ~ J k, and k < n .
* X + t When the cells can be so ordered, there is no torsion in ItG( )( ) in the dimensions of the generators of * + H6X as a module over H~S °. Therefore, the collapsing we have done causes a minimal loss of information. The following result describes the extent
, X + to which tIG( )(1) is detected by p ® i*.
COROLLARY 2.7. Let X be a generalized G-cell complex satisfying the conditions of either part of Theorem 2.6 and let i: X G ~ X be the inclusion of the fixed point set. Then, for any a E RO(G) with Ic~l even, the map
1 4 - _ ~ p • i* : J G( ja&(x4-)(e)
is a monomorphism. Moreover, for any a E RO(G), the map
p@i*: (H~(X+)(1)) / tors ion ~ ~ X4- HG( )(e) @ (H~((XG)+)(1)) / ( torsion @ im T)
is a nmnomorphism.
PROOF. Since the equivariant cohomology of X is the limit of the cohomologies of the Xn, it suffices to show that the result holds for every X,~. It is easy to check the second part for X 0. Assume the second part for X~, and let x be an element of I-IG(X,~+l)(1)/torsmn vanishing under the map into
X + H a ( , ~ + l ) ( e ) ® (H~((xG+l)+)(1)) / ( tors ion @ im r )
X + 1 induced by p G i*. We must show that x is zero. The group ttG( n+l)( ) is the
71
direct sum of the groups I-I~(Jn+z/0J,+l)(1 ) and ~ + t t6(Xn)(1) , and this decomposition
is respected by the map p ® i*. Thus, x is the sum of classes y and z in
o¢ X + • It~(J,~+z/C0Jn+~)(1)/torsion and t i c ( ,~)(1)/torsmn, respectively, which vanish
under the analogous maps. By our inductive hypothesis, z is zero. Since J,~+l/COJ,~+l is a wedge of copies of S V and G+^S ~k for various V and k, y vanishes by our remark about X 0. Thus, x is zero. The proof of the first part is similar. For this part, we must assume that ic~l is even because the map p O i* does not detect the torsion in the fourth quadrant of H~(S°)(1).
3. THE COHOMOLOGY OF COMPLEX PP~OJECIVE SPACES. As an application of the results from section two, we show that the cohomology of a complex projective
" * S O Let V be a finite or countably infinite space with a linear action is free over .u. G . dimensional complex G-representation and let C* be C - {0}. The complex projective space P(V) with linear G-action associated to V is the quotient G-space ( V - {0})/C*. Note that if W C V, then P(W) may be regarded as a subspace of P(V). If V is infinite dimensional, then we topologize V as the eolimit of its finite dimensional subspaces W; the quotient topology on P(V) is then the same as the colimit topology from the associated subspaces P(W). To describe the cohomology of P(V), we must
write V as the sum ~ ¢i of irreducible complex representations (including possibly i = 0
the trivial complex representation). Of course, if V is infinite dimensional, then n = oo. Points in P(V) will be described by homogeneous coordinates of the form
<x0, xl, x2 . . . . . x~), xi e ¢i
with the conventions that not all of the x i are zero, and if V is infinite dimensional, that all but finitely many of the x i are zero. Each element of the group G acts on each homogeneous coordinate of P(V) by multiplication by a complex number. Therefore, if all the irreducibles in V are isomorphic, then the action of G on P(V) is trivial. Moreover, if r] is any irreducible complex representation, then P(V) and P(r 1V) are isomorphic G-spaces. If 7/ and ¢ are irreducible complex representations, then P(rl) is just a point and P(r I O ¢) is G-homeomorphic to the one-point compactification of either r] -1 ¢ or 7/¢-1.
Since we have selected a eolimit topology on P(V) when V is infinite, to show that P(V) is a generalized G-cell complex for any G-representation V, it suffices to
k - 1 show this when V is finite dimensional. Let V k be the representation ~ ¢i and let
i = 0 W be the representation -1 ¢,, V,~_ 1. Describe points in the unit disk DW by complex
-1 coordinates (x0, xl, ... ,xn_l) , with x i C Cn ¢i" Define a map f: DW -~ P(V) by
f((xo, x 1 . . . . . Xn_l) ) = <Xo, Xz, x 2 . . . . . xn_l, 1 - E txil2)" i = 0
The tensor product with ~ n I is inserted in the definition of W to ensure that the map f is equivariant. The image of SW in P(V) lies in the subspace P(Vn_I) of P(V), and f is a homeomorphism from D W - SW to its image in P(V). Thus P(V) is formed
72
from P(Vn_l) by adjoining the G-cell DW along the map f l S W : SW-* P(V~_~). Working backwards through the sequence of representations Vk, we conclude that P(V) is a generalized G-cell complex with cells the unit disks of the representations
¢~lVk for 1 _< k _< n.
In order to show that the equivariant cohomology of P(V) is free over H~S °, we must show that the cells of P(V) can be attached in an order satisfying the condition in Theorem 2.6(a). This proper ordering of cells is derived from a careful ordering of the set q~ of irreducible summands of V. Since the remainder of our discussion focuses on ~, we write P(~) for P(V). An ordering ¢0, ¢~, ¢2 . . . . of the
elements of • is said to be proper if the number of irreducibles in the se t {¢i}O<_i<_k-1 isomorphic to ¢~ is a nondecreasing function of k. For example, if ¢ and r} are distinct complex irreducibles and q5 consists of two copies of ¢ and one of q, then r], ¢, ¢ and ¢, r], ¢ are proper orderings of ~, but ¢, ¢, 7/ is not. The dimension of
lk--1 the fixed subrepresentation of the representation ¢~- ~ ¢ i is the number of
i=0 irreducibles in the s e t {¢i}o<i<k-1 isomorphic to Ck- Thus, if q~ is properly ordered, then the cell structure described above satisfies the conditions of Theorem 2.6.(a).
PROPOSITION 3.1. If ¢0, ¢2, ¢2, -.. is any ordering of the elements of a set • of irreducible representations, then p(dp) is a generalized G-cell complex with cells the
unit disks of the G-representations ¢ - tk-1 k ~ ¢~, for k > I. Moreover, H~P(~5) + and i = 0 * 0
I I~P(~) + are free RO(G)-graded modules over I t , S . If the ordering of ~ is proper, then the homology and cohomology of P(~) are each generated by one element in
k-1 dimension zero and one in each of the dimensions 6~-1 ~ ¢i , for k _> 1.
i=0
The G-fixed subspace of P(q~) is a disjoint union of complex projective spaces, one for each isomorphisnl class of irreducibles in ~5. The (complex) dimension of the complex projective space in P(e)) a associated to the irreducible ¢ is one less than the multiplicity of ¢ in ~. Thus, the effect of properly ordering the irreducibles is that the maximal dimension of the components of the G-fixed subspace of P({¢i}0</_<k) increases as slowly as possible with increasing k.
REMAl~KS 3.2. Our description of the eohomology of P(~) contains one apparent anomaly. Suppose that (, r/, and 4) are distinct complex irreducible representations and • = {(, q, 6}. If we assign the proper ordering (, r/, ¢ to ~, then we find that the generators of * + HGP((P ) are in dimensions 0, r] -1 ¢, and ¢-1 (¢ • r/). However, if we select the proper ordering 6, (~, q, we find that the generators are in dimensions 0, ¢-1 6, and r] -1 (6 O (). In particular, the cohomology in dimension r]-1¢ must be A ® <2[) ® <2[) if we use the first set of generators, and A[d] ® <N> ® <7/) if we use the
second, where d is the integer associated to the element rl - t ( - ( - 1 ¢ of RO0(G).
There is no contradiction in these two claims about the cohomology in dimension
7 3
r / - l ( because these two Mackey functors are isomorphic by Examples 1.1.(d). The apparent difficulties in all the other dimensions are resolved in exactly the same way.
This example illustrates the latitude that one has in selecting the dimensions of the generators of the cohomology of P(~) for almost any set • of irreducibles. This latitude is necessary because, for most ~, there are many proper orderings and a choice of a proper ordering corresponds to a selection of the dimensions of the generators.
It would be nice to have some simple cohomology invariants of P((I,) which could be used for problems like comparing the cohomology of projective spaces with different G-actions. The fact that the dimensions for the cohomology generators don' t provide such an invariant is a disappointment. However, one invariant related to the dimensions of the generators is readily available. Select a proper ordering of ¢5 and plot the dimensions c~ of the resulting set of generators of I-I~P(~) + on a
coordinate plane whose horizontal and vertical axes indicate laGI and I~1,
respectively. The dimensions lie on a stair-step pattern whose foot is at the origin. This plot is an invariant of P(q@ The height of the steps in the plot decreases, or remains constant, as one goes up the steps (that is, moves in the direction of increasing laGI and loci). The height remains constant only if irreducible types appearing in (I) have equal multiplicity. The step-like structure of the plot reflects a filtration on (I) which plays an important role in our discussion of the ring structure
, + ofttGP((I) ) . An increasing filtration
0 = qS(0), ~(1), (P(2) . . . . . ~(r) . . . .
of the set • is said to be proper if, for every r and every complex irreducible ¢, the number of irreducibles in (I)(r) isomorphic to ¢ is the lesser of r mad the number of irreducibles in • isomorphic to ¢. Any two proper filtrations of ~5 differ only by an interchange of isomorphic irreducible complex representations, so there is essentially only one proper filtration of ~. The steps in the plot of the dimensions of the generators are in a one-to-one correspondence with the stages in the filtration of ~. The height of the step corresponding to filtration level r is the number of elements in ~5(r) - ~(r - 1).
4. CUP PRODUCTS IN • 0 H G S . Here we describe the multiplicative structure of • 0 ttGS . We begin with the case p = 2, which is due to Stong.
DEFINITIONS 4.1. Let ( be the real one-dimensional sign representation of
G = ?7/2. The identity element 1 in A(1) = H~(S°)(1) is the identity element of the
RO(G)-graded Macl(ey functor ring tIGS* 0. Let n E It~(S°)(1) be 2 - rp(1) . Observe
that n ' - = 2n. Let ~ E H~(S°)(1) be the Euler class; that is, the image of
1 E H~(S°)(1) under the map induced by the inclusion S ° C S ¢. Select a
74
nonequivariant identication of S C with S 1 and let h - ( • H~-i(S°)(e) -~tt~(S~)(e)
and L(_ 1 • H~-a(S°) (e)~H~(S~)(e) be the images of p ( 1 ) • H~(S° ) (e )~H~(S~) (e )
under the maps induced by this identification. Let ( • tt~<-2(S°)(1) be the unique
element with p ( ( ) : Q_> The elements 1 and n generate the abelian group
H~(S°)(1) and the Mackey functor H~S °. Each of the elements e ~, (~ , and era( ~, a 0 for m, n _> 1, generates the abelian group ttG(S )(1) and the Mackey functor ~Gn~S° in
the appropriate dimension a. For m > 1, the element L~_ i or L~_ 1 generates the
abelian group H~(S°)(e) in the appropriate dimension and L~_< generates the Mackey
functor . o I-IGS in the appropriate dimension. For m > 2, r(L~_~) generates the
• 0 1 abelian group I-IG(S )( ) in the appropriate dimension.
LEMMA 4.2.
, 2 n + 1 , (2n+l)(l-() 0 r t h - < ) 6 t t G ( S ) ( 1 )
are infinitely divisible by e 6 H~(S°)(1); that is, for m elements
and
The class g • H~(S°)(1) and, tbr n _> 1, the classes
_> 1, there are unique
e - ' ~ 6 tt~'~¢(S°)(1)
- m / 2 n + l ~ . 2 n + l - e T i L l _ ~ ) 6 H G (2n+m+l)¢(sO)(1)" "" "
such that
e m ( e - m ~ ) ~ a n d . . . . . . . 2~+1,, , 2~+1, = e (,e r(q_¢ )) = riq_ ().
Moreover, each of the elements e-m,~ o r e--r*~T(/~ 2 n + 1 ) generates the abelian group H~(S°)(1) and the Maekey functor * 0 tlGS in its dimension.
THEOREM 4.3. The elements
e 6 H~(S°)(1)
q_< ~ ~-((S°)(e)
q-1 e l{~-~(S°)(e) 2 ( - 2 o ettG (S)(1)
~-~ ,~ e t t ~ ' ~ ( s ° ) ( 1 ) ,
and
f o r m _> i,
- - m z 2 n + I \ ~ 2 n + 1 g r t , t l _ ( ) E H G - ( 2 n + m + l ) ( ( S ° ) ( 1 ) , f o r m , n _> 1,
75
• 0 RO(G) -g raded Mackey functor a lgebra over the Burnside generate I tGS as an Mackey functor ring A. The only relat ions a m o n g these elements, other than those forced by the Frobenius relat ions or the vanishing of I-I~S ° in various dimensions, are genera ted by the relat ions
p(~) = o
*1-¢ ~¢-I = p(1)
r(q_<) = 0 / 2 r e + l \ r(~¢_ ~ ) = O,
{0 ~-(,7_~) ~(,?_~) = 2~(,7._~),
2 e ~ = O
p ( c - ~ ) = O,
( ~ - ~ ) ( , - ~ ) = 2~-(~+~>~,
/ 2 r~+lx 2e - m r ( t l _ ~ ) = O,
p(C.-m z 2 n + l ~ \ rk t l _ ¢ )) = O,
- m 2 n + l = ( ~ l - r n / 2 r t + l x
(¢-m , 2 ~ + 1 , , ( - % ) 0, r t t l - ¢ )) =
and
-rr~ 2 n + l (~ ~-('1-~ )) 6 - m / 2 n - l x
T ( L I - ( ) ,
for m _> 0,
f o r m _> 1,
if m or n is odd,
if m and n are even,
f o r m > 0,
for m >__ 1,
for m, n > 0,
for m _> 0 and n > 1,
for m _> 0 and n _> 1,
for m, n _> 1,
f o r m , q >_ 0 a n d n > 1,
f o r m >_ 0 a n d n >_ 2.
R E M A R K S 4.4. (a) The last relation indicates that , for m _ 0 and n _> 1, ~ - m i 2 r t + l x r t , l _ ¢ ~ is infinitely divisible by ~. Thus, we can think of all the e lements in
the four th quad ran t of the graph of * 0 t tG(S ) as being derived f rom r(L3 ¢) via division by powers of e and ~. One mnemon ic for the effect of e and ~ on the e lements in the
four th quad ran t is to denote the nonzero e lement in H~-"~ -~" (¢ -1 ) (S° ) (1 ) , for m _ 2
and n _> 1, by e - m ~ - " w, where w is regarded as a ficti t ious e lement in d imension 1. T h e reason for selecting a fictit ious e lement in dimension 1, instead of the actual e lement in d imension 3 - 34, is discussed in Remarks 4.10(b).
(b) For p = 2, the e lements + ( 1 - r p ( 1 ) ) in A(1) are units, and l - r p ( 1 ) appears in the fo rmula describing the a n t i c o m m u t a t i v i t y of cup products . For any
I-I i + J ( X + I-l-m + n i X + then G-space X, if a E z~ G and b E -~'G ,
76
a b = ( - 1 ) i v * ( 1 - r p ( 1 ) ) J n b a .
The generators L1_¢, t¢_1, e, e - n n , and c - m , 2n+l, r~q_¢ ) are in dimensions where the
behavior of this nontrivial unit matters. Of course, since e - ~ ; 2~,+1, rt~l_ ¢ ) has order 2, any unit acts trivially on it. It is easy to check that
(1-rp(1))~l_¢ = - t1-¢ and ( 1 - r p ( 1 ) ) L ¢ _ 1 = -#¢-1"
This action of 1 - rp(1) on LI_ ¢ aI]d re_ 1 never affects cup products in tI~S ° because it is always balanced by the ( -1) 'm term in the commuta t iv i ty formula. However,
n* S o where the effects of this unit on q_¢ and re_ 1 are there are algebras over -~-G visible. The unit 1 rp(1) acts trivially on e and e - ' ~ . This shows up dramatical ly in ~a* S O The elements e and e->~+l~ are odd-dimensional, so our intuition about
a ,a , G •
graded algebras Dom the nonequivariant context suggests that their squares should vanish, or at least be 2-torsion. In fact, the squares are not torsion elements, an apparent anomaly possible only because the action of 1 - rp(1) is trivial. The overall effect of the actions of the units of A on tile generators of * 0 I-IGS is that t t~S ° is commuta t ive in both the graded and the ungraded seuse.
When p is odd, several complications ill the multiplicative structure of I-I~S ° arise from the greater complexity of RO(G). Tile most obvious are a host of sign problems coming fl'om the identification of representations with their complex conjugates. Initially,. we resolve these sign problems by grading HG S , 0 on RSO(G) instead of RO(G). In Remark 4.11, we explain steps which must be taken to pass back to an RO(G)-grading. The most serious complication arises from the misbehavior of the integers da associated to the virtual representations c, in RSO0(G ). One way to deal with this complication is to avoid it. This can be done very nicely if one is only interested in • 0 HGS . Because of the intuition this approach offers, we outline it as an introduction to the odd primes case.
The stable homotopy groups reds °, for ~ E RSO0(G ), have been studied extensively by tom Dieck and Petrie [tDP], and the stable Hurewicz map
h: ~r~_~S °-~tIG_oS ° ~--u~S° - - .~,,,L G
is an isomorphism [LE1] if 3 C RSO0(G). Thus, many of their results can be applied to homology in the appropriate dimensions. They have shown that the multiplication map
° + ze+ G s o
is an isomorphism for any /3 E RSO0(G ) and any 7 E RSO(G). By similar reasoning, the multiplication map
0 01-1 y 0 H~+~S0 I-t~S ttGS -~
is an isomorphism under the same conditions on /3 and 7. Thus, to understand all of I-I~S °, it suffices to understand the part of * 0 tlGS which tom Dieck and Petrie have already described and the part indexed on some subset of RSO(G) complementary to RSO0(G ). Recall that k is the irreducible complex representation that takes the
77
standard generator of 7]/p to e 2'ri/p. Let RSOz(G) be the additive subgroup of RSO(G) generated by 1 and I . As an additive group, RSO(G) is the internal direct sum of RSO0(G ) and RSOx(G). To complete our description of II~S °, it suffices to describe that part of it indexed on RSOx(G). This part is almost identical to H~S °
* 0 for G = ~/2. Consider the description given above of that part of I l l s for p = 2 indexed on the additive subgroup of RO(i7/2) generated by 1 and 2C. Replace 24 by
f . . . . 0 for p ,k and all the other 2's by p's. The result is a description of the part o r t6~ odd indexed on RSO:~(G). This approach describes * o t t c S as the graded box product of two subrings indexed on complementary subsets of RSO(G). The unpleasant behavior of the integers d~ is buried in the computations of the box products.
Unfortunately, because of peculiarities in the dimensions of the algebra generators of H~P(V) +, this description • 0 of HaS as the box product of two subrings can not be used to describe the ring structure of the cohomology of complex projective spaces. Thus, we offer an alternative description of the ring structure of It* S O for p odd. In section 2, we defined a function from RO0(G ) to Z using a G section of the projection from R0(G) to RO0(G ). Since we are now working with RSO0(G ) instead of RO0(G), we define an analogous function from RSO0(G ) to 7] using a section s: RSO0(G ) -* Ro(G) of the projection from t~0(G ) to RO0(G). We insist that s(0) = 0 and that our original section RO0(G ) -* R0(G) factor through s.
DEFINITIONS 4.5. (a) If ct E RSO0(G ) and s(a) = ~ ¢ i - r h , then we wish to
define an equivariant map #~: S ~ i - * S ~¢i with nonequivariant degree d~. If
a = I m - I '~ with 0 < m,n < p and n -1 is the unique integer such that 1 _<n -1 _<p- 1
and nn -1 ~ 1 rood p, then ~t~ is the extension to one-point compactifications of the
complex power map z -~ z "~('~-1), for z E C. In general, we identify S x t i and S "~ni
with Ai Sci and A S ~i, respectively, and take the smash product of the appropriate
complex power maps to obtain the equivariant map #~ from S ~¢i to S ~ ' i with
nonequivariant degree d~. Also denote by #~ the image of this map in It~(S°)(1)
under the Hurewicz map. Clearly, if the 8i and the ~?i were paired off in a different
order, then a different map from S ::el to S ~"i would be obtained. However, the
maps coining from the two pairings would be equivariantly homotopic and so would
give the same element in H~(S°)(1).
(b) Let a be an element of RSO(G) with lal = 0. Then a must be
represented by a s u m E~i-?]i, where the ¢i and r h are irreducible complex i
representations, some of which may be trivial. Since the 8i and r h are complex
representations, they have canonical nonequivariant orientations. Combine these to
produce a nonequivariant identification t~ of S ~¢i with S ~ni which is unique up to
78
homotopy. Let Le also denote the image of this identification in I-I~(S°)(e). The
resulting cohomology classes L~ are then independent of the ordering of the ¢i and
the Vi. The abelian group H~(S°)(e) is generated by L~. If lc~GI > 0, then 7"(~)
generates the abelian group H~(S°)(1) and L~ generates the Mackay functor ltGS~ 0.
(c) If c~ E RSO0(G), then in HGS ~ 0,
p(#~) = d~ ~ and pr(L~,) = p ~ .
We have already asserted that I t , S ° is A[d~]. Under this identification, #~ and
r ( t~) become the elements /~ and r of A[d~](1) and ~ becomes 1 E 7/ = A[d~](e).
There is a unique integer b ~ s u c h t h a t d _ ~ d ~ + b~p = 1. Let ~ = p # ~ - d ~ T ( L ~ )
and G~ = d - ~ # ~ + b~T(L~). Then, cr~ and ~ form an alternative Z-basis for
tt~(S°)(1).
(d) Let /3 be an element of P~SO(G) with 1/31 > 0 and 19GI = 0 There exist
an c~ in RSO0(G ) and a G-representation V such that V G = 0 and /5 = c~ + V. Let
ep E linG(S°)(1) be the image of #c~ E t t ; (S°) (1) under the map from It~(S°)(1) to
ttZG(S°)(1) induced by the inclusion S o C S v. In Lamina A.11, it is shown that this
Euler class eZ is independent of the choice of the decomposition of/5 into the sum of
the representation V and the element c~ of RSO0(G). The class eZ generates the
abelian group HZG(S°)(1) and the Mackay functor tt~GS °.
(e) If I~l = 0 and < 0, let be the unique element of I-I~(S°)(1) with
p({~) = L~; this class generates the abelian group H~(S°)(1) and the Mackay functor
tt s °.
When p is odd, it is harder to pick a multiplicative basis for the torsion in the fourth quadrant of the graph of u* S o In each dimension there is a choice of " ~ ' G "
p - 1 generators, instead of a single nonzero element. Moreover, since these torsion elements are not tied by an Euler class to elements on the positive horizontal axis, there is no way to base the choice of a generator on choices already made for the axis. The following lemma justifies the procedure we employ to select multiplicative generators for the fourth quadrant.
LEMMA 4.6. Let 13 be an element of RSO0(G ) and let a, 7, and 6 be elements of RSO(G) such that
79
and
lal = I~GI = 0,
I~1, la~l < 0,
Ivl > 0,
Is~l > 3,
Io~GI is odd.
If x is any nonzero element in I-IG(S~ 0)(1), then p z x is a generator in ttG+Z(S°)(1). Moreover, x and #Z x are uniquely divisible by both c-~ and ~e"
DEFINITIONS 4.7. Select a generator in I-I~-~a(S°)(1) and denote it by ~'3-~a- If
~ = 1 - m ( ) , - 2 ) - n l , for m, n_> 1, then let us be the unique element in H~(S°)(1)
such that
For any o~ C RSO(G), there are unique integers m, n, and q such that q = 0 or 1
and
c t - [ q - m ( , k - 2 ) - n & ] E RO0(G).
Denote by < a > the element q - m ( , k - 2 ) - n , k associated to a by these conditions. If
ct E RSO(G) with lal < 0, laGI _> 3, laGI odd, and a :~ < a > , then define
uo" E tt~(S°)(1) by
P'o" : t / o ' _ < a > b ' < o ' > .
The element uo" generates the abelian group H~(S°)(1) and the Mackey functor a 0 ttGS .
LEMMA 4.8. If a E RSO0(G), then /go" E It~(S°)(1) is divisible by ee, for any
fl e RSO(G) with ]fl[ > 0 and ]fiG] = 0; that is, there is a unique element
~71/gs e ~ 9(s°)(1)
such that
~5,8 ((T~ 1 /gO,) = /go',
The element e~ 1/go" generates the abelian group I-I~-P(S°)(1) and the Mackey functor
~; -e s0 .
80
T H E O R E M 4.9. The e lements
and
c~ o ~ e ttG(s )(1), 0 ~ ~ t tG(s )(e),
% • t t~ (S°) (1 )
~ - 2 • t t~-~(S°) (1)
~:_~ e e~-~(s°)(e)
~;~,~o • I~;~(s°)(1),
a 0 ~,~ c t tG(s )(1),
for c~ = : t : ( A n - A ) , with 1 < n < p,
for o~ =: t : ( )~n- ,x) , with t < n < p,
f o r m > 1,
dimensions, are generated by the relat ions
p ( ~ ) = d ~ ,
p(~e) = 0,
T(~) = p¢~ ,
~- ¢~ = ¢~+e,
~ ¢ ~ = d~4~+~,
for a c RSOo(G);
for c~, fl E RSO0(G);
for 191 > 0 and 191 = 0;
for lal, Ifl] > 0 and
I~1 =1~[_=0;
for ~ e a S O o ( 6 ) , f~l > o,
and I~ l - - O;
for lal = 0 and laG[ < 0;
for Iod = 0 and I~I < 0;
for la'I = I~I = 0 and
l~el, b q < 0;
for a E RSO0(G) , t/31 = 0,
and triG] < 0;
• 0 RSO(G) -g raded Mackey functor a lgebra over the Burnside generate H a S as an Maekey functor ring A. All of relat ions a m o n g the e lements of * o I t6S , o ther than those forced by the Frobenius relat ions or the vanishing of • 0 I-I6S in var ious
for a = l - - i n ( A - 2 ) - n A , w i t h i n , n_>1,
p ee {~ = 0,
e/~ {~, = da_ ~ e-r {~,
p ( { ~ ; l ~ a ) = O,
/*3' ( e ; 1 /~oe) = ,£;1 K:°c+7 '
eft (£.;1 /~0,) ~- KOe,
e7 (e~ 1 ~ ) = e~l_~ ~c~,
-1 (e~l~a)(c;l '~6) = PQ~+-r ha+o,
pu~ = O,
p ( ~ ) = o,
81
for ~ = t g q : 0, i ~ q < 0,
~nd I,< > 0;
for Iod = [81--1t3GI = 17GI = 0,
I~q, leq < 0, I~1,1:1 > o, and a + / 3 = 7 + 6 ;
for a, ~5 E RSOo(G),
Ieq=i:q=o, I~t, 171 > 0, and
a + 7 = , ~ + 6 ;
for ~ 6 RSO0(G), I s g = 0 ,
~nd t~1 > 0;
for a, 7 E RSOo(G), I~g ;o, and 191 > o;
['or c~ E RSOo(G ), I f lGl=0,
and I~l > 0;
for a E RSOo(G),
I ~ q - - I ~ q = o , and
]/~1 > 171 > 0;
for a, 6 E RSOo(G),
19[ = I'~g ; o ,
and 1/31, [7I > 0;
for t~1 < 0, ic~GI > 3, and
l~q odd;
f o r l a l < O , [aG[>a, and
I~q odd;
for /3 E RSOo(G), lat < O,
IaGI _> 3, and ioeGI odd;
82
(3 ~'c~ --~ /"c,+~,
( E ; 1 //;7) /JOe = O, for
tc~ I.fi ~ tcx+fi ,
for Io~-4- ~1 < 0, IsGI ~ 3,
lonG] odd, I'~1 > 0, and
I~GI--0;
for ,~, < 0, Io~ ~ + ,~GI > 3,
[czGI odd, I~1 = o, and
19Gi < 0;
7 E RSO0(G), Ic~l < 0,
IsGI >_ a, I~GI odd,
[fiG[= 0, and I~1 > 0;
i~1 = l ~ l = 0 . for
REMARKS 4.10. (a) For p odd, the only units in A(1) are +1. The only generators in odd dimensions are the v~. Since v~vZ is zero for any c~ and fl, no sign problems occur in commuting products in H~S °. Thus, H~S ° is commutative in both the graded and ungraded senses.
(b) As an alternative to using the v~ as a basis in the fourth quadrant, one
may define elements e71~1co in ttG-~-QS°)(1), for l a l = [ / 3 G ] = 0 , l aG]<0 , and
191 > 0, by
( 7 1 ~ 1 C 0 = d~,_<~>//1-a'--fi"
Here, aa is regarded as a fictitious element in dimension 1 which is divisible by any product ~<, e/~. We employ a fictitious element because there is no canonical choice for the dimension of an actual element. The relations satisfied by the elements e~z~lo~ are
- 1 - 1 for ,~, = I~ G] : I 'r~ I : O,
191 > 171 > o,
and [a G] < O;
for I~, = rTJ = f~GJ = o,
I~1 < I,r~ I < 0,
and ]fi[ > O;
83
#7 (e71~7'1w) = d<v>-~ e7 l ~ a !~ w '
for 7 E RSO0(G),
= 1/3 1 = o, l s e i < o,
and 1/31 > 0;
for 7 e RSO0(G),
and 1/31 > 0.
The one difficulty with this alternative basis is that if a + / 3 = 7 + 5, then e~l~gl~o and e}-l~71w are in the same dimension, but they need not be equal. In fact,
(c) Observe that in the formulas for the product of #~ with any of e/~, e~ 1 ~ , or v5 there is no de,, but there is such a constant in the formula for the product p~ f~ . On the other hand, er~ f/~ = ~ + ~ , but there is a d -~ in the formula
for the product of ~ , with any of e/~, e~ 1 ~v, or t@. This difference in the behavior
• 0 1 of the elements #~ and er~ of HG(S )( ) reflects the fact that there is a conjugacy
class of subgroups of G associated to any well chosen element of any G-Mackey functor M for any finite group G. This association is based on the splitting of M which occurs when M is localized away from the order of G. This splitting can not be observed directly before localization, but it can be seen indirectly in the association of subgroups to well chosen elements in the Mackey functor. The elements #~ , e~,
e~ 1 ~y, and v~ are all associated to the subgroup G of G, and products of pairs of
them behave nicely. The elements era and ~Z are associated to the trivial subgroup, and their product is nice. However, the product of elements associated to two different subgroups will either be zero or involve some fudge factor like a d a . We have introduced both #a and era so that, when one of these elements is needed in our description of the relations in H~P(V) +, we can always choose the one that will give us the simpler formula.
• 0 REMARKS 4.11. In order to explain the passage from an RSO(G) grading on HGS
to an RO(G) grading, we must first clarify what is meant by the assertion tha t HGS* 0
is RO(G)-graded. The assertion does not mean that , for a C RO(G), ~ 0 ] t6S can be described without reference to a choice of a representative for a. Rather it means that if V I - W 1 and V 2 - W2 are two representatives for a and I-I 1 and H 2 are the
values of ~ 0 FIGS obtained using these representatives, then we can construct an
isomorphism between I-I 1 and H 2 in a natural way from any isomorphism
f: V2 O W1 ~ V1 G \¥~ of representations illustrating the equivalence of V 1 - W1
and V 2 - W 2 in RO(G). This is exactly what we mean when we say that nonequivariant homology is 7/ graded. To define the nonequivariant homology group H~X, we must pick a standard n-simplex. Different choices of the n-simplex lead to
8 4
different groups, as anyone who has been embarrassed by an orientation mistake knows all too well.
Let fl---- V 2 G W 1 -V~ ® W 2 and let f denote the image of f in Ji~(S°)(1). Then the isomorphism from H 1 to H 2 is just multiplication by f. To provide a means of computing the effect of this isomorphism, we write f in terms of the
s tandard generators of tI~(S°)(1). The map f induces a map 1¢ 3 between the fixed
point subspaces of the representations. If nonequivariant orientations are choose for their domains and ranges, then the maps f and fG have well-defined nonequivariant degrees. It follows from Lemma A.12 that
(deg f ) - (deg fG)d~ = (deg fG)/2~ + p r(L/~).
The structure of • + • 0 HGG as an algebra over JIGS follows easily from our • 0 results on ttGS and the description of the additive structure of * + JIGG given in
section 2.
. 0 • + P R O P O S I T I O N 4.12. As an RO(G)-graded module over ttGS , is HGG generated
by the single element ~ = (1, 0, 0 . . . . . 0) of J I ~ ( G + ) ( e ) = Z p. Moreover, for any
RO(G)-graded module M* over H~S °, there is a one-to-one correspondence between
maps f: * + M* HGG --* of RO(G)-graded modules over H~S ° and elements in M°(e).
This correspondence associates the map f with the element f(e)(~) of M°(e). Thus,
• + * 0 JIGG is a projective RO(G)-graded module over ttGS .
PROOF. Unless lal 0, ~ + ~ + = ItG(G ) = 0 . If l a l = 0 , then t ~ generates HGG as a
module over A. Thus, ~ generates H* G + • 0 ~-G as an RO(G)-graded module over JIGS , , +
and any RO(G)-graded module map f: HGG + M* is determined by f(~). On the
other hand, recall the observation from Examples 1.1(f) that a map from A G to any
Mackey funetor N can be specified by giving the image of (1, 0, 0 . . . . . 0) E AG(e ) in
N(e). Let m be an element of M°(e). For each o E R O ( G ) with I c d = 0 , Lore is in
c~ + M~(e) and there is a unique map f ~ : H G G + M ~ of Maekey functors sending
tc~b C I-I~(G+)(e) to t a m C M~(e). These maps fit together to form a map
f: * + M* tt* S o The projectivity of * + JIGG ~ of RO(G)-graded modules over z~ G . JIGG
follows immediately.
85
5. TH E MULTIPLICATIVE STRUCTURE OF t t~P(V) +. We assume that there are at least two distinct isomorphism classes of irreducibles in V; otherwise, the
. + mnltiplicative structure of l tGP(V ) is completely described in Examples 1.1.(h). As in section 3, we take 4) to be the set of irreducible summands of the complex representation V. Let 4)(0), 4)(1), (I)(2), ... be a proper filtration of 4). Then 4)(1) consists of exactly one representative of each of the isomorphism classes of irreducibles that appears in (I). Let ¢0, ¢1, ¢2, -.. , ¢,~ be an enumeration of the elements in 4)(1), and let n i be the number of elements of 4) isomorphic co ¢i (with n i ----(x> allowed). Arrange the enumeration of the elements of 4)(1) so that n o > n 1 _> ... _> nm. Extend the ordering of 4)(1) to (I) by selecting the unique proper ordering of 4) which is consistent with the filtration and in which, for each r > 1, the ordering of the representations in 4)(r+1) (l)(r) is the same as the ordering of the corresponding representations in dp(1). If the irreducibles which appear in 4) appear with equal multiplicity, then, regarded as an ordered set, • is a sequence of blocks, each of which is a copy of 4)(1). If the multiplicities are not equal, then 4) is still a sequence of blocks, but each block after the first will be either a copy of 4)(1) or of an initial segment of 4)(1). The lengths of the initial segments in the sequence can not increase. We will abuse notation by writing ¢i E 4 ) ( r + l ) - 4 ) ( r ) to mean that 4)(r+ 1)-4)( r ) contains an irreducible representation isomorphic to ¢i- We say that two sets of irreducible representations are equivalent if they contain the same number of irreducibles in each isomorphism class. Moreover, we sometimes identify equivalent sets of irreducibles.
Corollary 2.7 will be used to derive the multiplicative structure of H~P(V) + . +
from the multiplicative structures of I-IG(P(V ) )(e) and I-I~((P(v)G)+)(1). The
group I-I~(P(V)+)(e) is isomorphic to the nonequivariant cohomology group
HI~(P(V)+;7/), and we will think of the restriction map p as a map from
H~(P(V)+)(1) to KI~I(p(v)+;7/). Select an algebra generator x E tt2(P(V)+;7/) for
H*(P(V)+;7/). The fixed point space of P(V) is the disjoint union of the spaces
P(ni ¢i) ~ P(ni)- Let qi denote both the inclusion of the subspace P(ni) into P(V)
and the map H~(P(V)+)(1) * + --*I-IG(P(ni) )(1) induced by this inclusion. By
Examples 1.1.(h), * + HGP(ni) is a truncated polynomial algebra over H~S ° generated
2 + by an element x i in Hc(P(n i ) )(1). Let
c i: * + --* HG(P(ni) ) (1) /( torsion G im p)
denote the composition of qi and the projection onto the quotient. If y is in * + .
H~(P(ni)+)(1) , then [y] denotes its image in t tG(P(ni) ) (1) / ( tors ion ® i m p ) .
Throughout this section, we will index H~P(V) + on RSO(G) to simplify the selection of the integers d~. The comments in Remarks 4.11 on the passage from RSO(G)-grading to RO(G)-grading for * 0 I-IGS apply equally well to I-I~P(V) +. Recall that ,~ is the irreducible complex representation that sends the standard generator of
86
7]/p to e 2'~i/p and that ( is the real one-dimensional sign representation of 7//2. If p is 2, then 1, regarded as a real representation, is just 2~.
We begin with the case p = 2. Any complex irreducible representation is isomorphic to either the complex one-dimensional trivial representation or the complex one-dimensional sign representation A. Since P(V) and P(AV) are G-homeomorphic, we may assume that there are at least as many copies of the trivial representation in ~5 as there are copies of the sign representation. Thus, we may take ¢0 to be the trivial representation and ¢1 to be the sign representation.
* + * 0 By Theorem 3.1, t tGP(V ) , regarded as a module over ttGS , has one generator in each of the dimensions
2k + 2 k ~ and 2k + 2(k + 1)~,
for 0 < k < n 1 , and one in each of the dimensions
2k + 2 n 1 ( ,
for n I < k < n o . If one assumes n o = n 1 , or ignores the generators special to the case n o >111, then one might guess that, as an algebra, H~P(V) + had an exterior generator in dimension 2~ and a truncated polynomial generator in dimension 2(1 + ~). Except for the fact that the generator in dimension 2~ is not quite an exterior generator and for some difficulties in the higher dimensions when n o > nl, this guess is a good description of H~P(V) +. However, in order to describe the behavior in the higher dimensions as simply as possible, we adopt a notation that does not immediately suggest this.
r-r* S 0 H~P(V) + is generated by an T H E O R E M 5.1. (a) As an algebra over ~G ,
element c of H~(P(V)+)(1) in dimension 2 I and elements C(k) of H~(P(V)+)(1) in
d imens ions2k + 2min(k, n l ) ( , for l _ < k < n 0.
(b) For any positive integer k, let k denote the min imum of k and n I. Then the generators c and C(k) are uniquely determined by
0(e) = [0]
p(c) = x E H2(P(V)+; 7/)
and
; ( C ( k ) ) = x
Moreover,
87
and
2C + qo(c) = e x o e H G ( P ( n o ) ) ( 1 )
2~ + ql(C) ~-- e 2 -~" ~X 1 E H G ( P ( n l ) ) ( 1 )
qo(C(k)) = xok(e 2 + ~xo) i e t t~k+kt)(P(no)+)(1)
q](C(k)) = x~(e 2 + ~X1) ~ e H~(k+ki)(P(nl)+)(1 ).
n i If n i is finite, then x i = 0 and some of the terms in the last two sums above may vanish.
(c) The generators c and C(k) satisfy the relations
c 2 = e2c + ~C(1),
cO(k) = ~ C ( k + l ) , for k_>n, ,
and
C O + k ) , C@)C(k) = ?+~:_ ,~ . . ._
C ( j + k + i ) ,
fo r j + k _ < n l ,
fo r j + k > n I •
In these relations, we take C(i) to be zero if i >_ n o .
REMARKS 5.2. (a) By iteratively applying the third relation, we obtain
C(k) = (C(1)) k, for k _< n 1,
so that below the dimensions where we run short of copies of the sign representation, • p +
tIG (V) is generated by c and C(1). Moreover, in these dimensions, C(1) acts like a polynomial generator.
(b) If n 0 = n l , then H~P(V) + is generated by c and C(1). The only relations satisfied by these two generators are the relation
c 2 = e2c + ~C(1)
and, if n o < ec, the relation
C(1) n° = 0.
REMARKS 5.3. Notice that the maps q0 and ql behave differently on the generator
c. The element ~ = c + e 2 - n c of t tGP(V ) may be used as a generator in the
place of c and its behavior with respect to q0 and q1 is exactly the reverse of the behavior of c. To understand the geometric relation between these elements, observe that c and ~ can be detected in the cohomology of any subspace P(1 + A) of P(V) arising from an inclusion 1 + A C V. The space P(1 + A) is G-homeomorphic to S ~, but unlike S ~, it lacks a canonical basepoint. Either choice for the basepoint of P(1 + A) determines a splitting of I-I~P(1 + A) + into the direct sum of one copy of
8 8
• 0 • A I-IGS and one copy of PIGS . The canonical generator o f * A PIG S in dimension 2~ is identified with c by one of the two splittings and with ~ by the other.
, + When p is 2, the multiplicative structure of t tGP(V ) does not really exhibit
any complexities beyond those one might experience in a Z-graded ring. However, when p is odd, there are quirks in the multiplicative structure of H~P(V) + which are only possible because of the RSO(G)-grading. For tile odd prime case, recall the stairstep diagram obtained by plotting the dimensions a of the generators of I-I~P(V) + in terms of tctl and I~GI. Looking at this diagram in the special case where the irreducibles appearing in V appear with equal multiplicity, one might guess that I /~P(V) + was generated by two truncated polynomial generators, one in a dimension a with levi=2 and I a G t = 0 and one in a dimension ~ with Ifil=2m + 2 and lfiGl = 2. Unfortunately, such a guess would badly underestimate the complexity of I-I~P(V) +. The set of dimensions for a full set of additive generators must generate a larger additive subgroup of RSO(G) than can be accounted for by a pair of truncated polynomial generators. For example, recall that the first two additive generators of t t~P(V) + are in dimensions ¢~-1¢ 0 and ¢ ~ ( ¢ 0 + ¢1). If the additive generator in dimension ¢{1¢0 were to serve as a truncated polynomial generator, then the additive generator in the next higher dimension would need to be in dimension 2¢~-1¢0 instead of ¢~i(¢ 0 + ¢1). Any replacement of these two generators by an element and its square requires the introduction of further generators in some other dimensions inconsistent with a simple truncated polynomial structure. To provide a better feeling for the multiplicative structure of H~P(V) +, we give two sets of multiplicative generators. The first is a natural set with a great deal of symmetry. It does not exhibit a preference for any one ordering of ~. Unfortunately, this set is much too large. By selecting an ordering on ~, we are able to construct a much smaller, but very asymmetrical, set of algebra generators.
In order to describe the effect of the maps qi on our algebra generators, we must introduce more notation related to the integers d~.
DEFINITIONS 5.4. (a) For any two distinct integers i and j with 0_<i, j _< m, let
~i j denote the irreducible representation ¢}-1¢j, and let di-{ denote the integer d~, i j
for c~ = f i i j - f i r s . Note t h a t dij is 1 for any pair of distinct integers i and j. For any
integer i and any distinct pair of integers r and s such that 0 _ i, r, s _< m, let d/ / be
zero. The integers di.{ satisfy the relations
dirt d~; _= d / j mod p,
d~.{ + d¢~ _= d ~ mod p,
and
d~ j t~, t,, ~j dvw -= d ~ d,,~ rood p.
89
(b) If ¢i E t ( r + l ) - I ( r ) , then let c~i(r) denote the representation
¢ ~ 1 ~ ¢, and let ~l~j b e d o , f o r a = a i ( r ) - c ~ j ( r ) . Note that, i f ¢ i E i ( r + l ) - i ( r ) ,
d~i = 1. If either ¢i or Cj is not in i ( r + l ) - i ( r ) , then let d~j be zero. If ¢ i , ¢5'
and Ck are in I ( r + 1 ) - i ( r ) , then the integers d~j satisfy the relations
d~jd~k =_ dik mod p
and, if i :/= j,
d~'5 : (cti'J~r ik ak - - , - . 7 ~ ~ ( d j k ) m o d p, O<k<rn k ¢ i , j
where a k is the multiplicity of Ck in if(r).
THEOREM 5.5. (a) If i and j are distinct integers with 0 < i, j < m, then there is a unique element el5 in HZGiJ(P(V)+)(1) such that
I kJ jl, f o r 0 < k < m , ~k(cij) = di je~ i
and
P(eih) = x.
If r_>0 and ¢5 E I ( r + l ) - I ( r ) , then there is a unique element Ch(r ) in
I-I;J(~)(P(V)+)(1) such that
~'tk(Cj(r)) = e j%j (~ )_ rx , f o r 0 < k < m ,
and
p ( % ( r ) ) = x
The elements Cij , for 0 < i,j _< m and i ¢ j , and the elements Ck(r), for r_> 1 and * + * 0 Ck E i ( r + 1) - i ( r ) , generate HGP(V ) as an algebra over ItGS .
(b) For 0 < i , j , k < m and i :~ j ,
~j qk(cij) : dij fJ3ij "t- ~ ~ i j - 2 Xk"
(c) For r > 1 and Ck e I ( r + 1 ) - i ( r ) ,
90
1-[ (e&i + 2xk) 1 qk(Ck(r)) = x~ | ¢ i c #(r) ¢#/~i- •
If Cj e ~ ( r + 1 ) - ~ ( r ) and j ¢ k, then
~[,<i )~I I-I (d"e#ji+~#ji_2xk) l %(Cj(r)) = xkka~k co5 ~ + Ce~_ xk ¢i~#(~) k
r [ A k J ~ r ki ak~ - ~'vk) 1-I (d~) ¢i e ~(r) iSkj,k
%j(~)_~x~.
+
If ¢~ ~ ~ ( r + 1)-49(r), then qk(C/(r)) is zero.
(d) For 1 _<j _< m, let 7j be the representation ¢71 ¢i and let Dj be the i=0
j -1 element l-I cji in H;J(P(V)+)(1). Then the elements D j , for 1 <__j _< m, the elements
i=0 C0(r), for r>_l and ¢0 • ¢5(r+l)-qS(r) , and the elements DjCj( r ) , for 1"_>1 and
* p + n , S O ¢bj • ~ ( r + 1)-¢5(r), generate H G (V) as an algebra over . ,G .
REMARKS 5.6. In order to simpli~ our indexing, we define D O and C j(0), for
0_<j_<m, to be 1 C H~(P(V)+)(1). We also define 70 and oej(0) to be 0. Our
second set of generators for H~P(V) + is then just the set of elements DjCj( r ) , for
r _> 0 and Cj • ¢5(r+ 1 ) - q~(r). This set of elements of I-I~(P(V)+)(1) is also a set of • 0 additive generators of H~P(V) + as a module over t i cS . One might hope that a set
of multiplicative generators could be much smaller than a set of additive generators, but if the various irreducibles in • appear with very different multiplicities, then small sets of multiplicative generators do not exist.
We will order the set of generators Dj Cj( r ) by the dictionary order on r and then j. On the stairstep plot of the dimensions of these generators, moving in the direction of increasing order corresponds to moving up and to the right.
REMARKS 5.7. Nothing that has been said in the discussion of the odd prime case actually depends on p being odd; rather, mod 2 arithmetic is so simple that most of the technicalities necessary when p is odd are unnecessary when p = 2. The elements c and ~ in the case p = 2 are c10 and c01. The element C(j) is C0(j).
91
In order to describe the relations among the generators in H~P(V) + in a palatable form, we must introduce one more batch of elements in H~(P(V)+)(1).
DEFINITION 5.8. Observe that, for 1 _<j < m, tcDj is divisible by e.~j. Moreover, and
I -'d l o + qk(e~ ~Dj) = Pi~0 ji e HG(P(nkCk) ) / ( tors ionOim r). j - 1 j -1
Since l~ d~! is zero if k < j and 1 if k = j , the coefficients p I-t dk.! which appear in i = 0 jz i = 0 O~
the O k ( ~ ~Dj) form a matrix which is p times an upper triangular matrix with 1%
on the main diagonal. Applying the obvious analog of the process for diagonalizing
an upper triangular matrix to the elements e ~ D j
H~(P(V)+)(1) characterized by the conditions
and
p(k ) = 0,
( [p],
[ 0,
These elements can be described inductively by the equations
km ~ ¢-r I ~Dm
and, for l < j < m , kj = ¢-1 ~cDj m ~ I d k i
7j k=~j+l(i~O ji) ~k" m
Define k 0 ¢ H~(P(V)+)(1) to be ~ - E k j .
for j ¢ 0 then also characterize k 0 .
produces elements kj of
if k ----j,
otherwise.
The equations above characterizing kj j = l
Moreover,
f p, if k = j , q k ( k j )
0, otherwise.
kj Cj(r). These elements kj(r) are characterized by the equations H;J(~)(p(v)+)(1) For r_>l and Cj e • ( r + l ) - ¢ ( r ) , define &j(r)E to be
= 0,
and
p%5(~)_~x~ ,
O,
if k = j ,
otherwise.
Moreover,
92
qk(kJ (r)) = { p%j(,.)_~x~, i f k = j ,
0, otherwise.
For convenience, we define kj(0) to be k j . Observe that, for r > I, the elements kj(r) can also be constructed from the elements KDj Cj(r) in the same way that the elements kj are constructed from the ~cDj.
We begin our list of relations with the relation between any two of the Cij and the relation between any two of the Cj(r).
PROPOSITION 5.9. (a) Let i, j, r, and s be integers with 0_<i, j, r, s ~ m and i @ j , r ¢ s . Then
kj sj rs ks sj dij - dij - dij drs
Cij : (7flij-flrsCrs -~- dij ~Zij -1- E p ~[3ij kk" kCs
(b) Let r>_l and let i and j be integers such that ¢i and Cj are in • (r+l)-~(r). Then
-dkjdji Ci(r) = ~ i ( , - ) -~ j ( r ) Cj(r) -t- ~ p /~i(~)_~k(r ) kk(r ) .
kCj
An obvious initial response to this result is to assume that H~P(V) + can be generated as an algebra over H~S ° by any one of the cij and, for each r with q~(r+l ) -qS(r ) nonemepty, any one of the Cj(r). The k k and kk(r ) in the formulas spoil this simplification, especially since they are defined in terms of precisely the generators one would hope to omit. Solving this by taking the elements k k and kk(r ) as part of a generating set is hardly satisfactory since, from a Mackey functor point of view, these are torsion elements (because P(kk) and p(kk(r)) are zero).
The remaining results in this section describe the products of pairs of elements from either of the generating sets in terms of the smaller generating set. All of the relations in I-I~P(V) + follow from the relations in Proposition 5.9 and the relations below. If V is finite, then some of the elements appearing on the right hand
• V + side of these relations may not appear in the list of generators of It G (V) . Any such element is to be regarded as zero. We begin with the products which land in dimensions where there is no torsion. These are easily computed using the maps ~ and p.
PROPOSITION 5.10. (a) Let i, j, r, and s be integers with 0 < i , j , r , s _ < m and i @ j , r @ s . If I n > 2 , then
c i jcr ' = d0Jd0;~ lj is A0 j 0 s ) g -1310 C10 + a~D2 + ~j ~ij+/~rs -}- (d i j d r , ~ u i j drs Pij+~r s
kj k5 A0JA0~ [,41 j Is Oj Os kO kO kl - - drs)dl0 d-~ ~ d i j drs "-~ij "~rs -- k'~ij drs d i j - d20 d21
k=2 P e~3i j+flrs kk ,
93
where a = f l i t + ~ - 72"
If m = l , then
Oj O~ eij Crs = di j dr~ f~ij+~r s
Co(l). ~S i j+J3r~-OeO(1)
(b) Let i, j, and r be integers with- 0 _< i, j < m, i @j, and
( , l j 1~ ,Oj ,0s, -Jr- ~ d i j d r s - d i j dr~)e/3i j+~rs_fl loclO q-
rj e i j Dr = d i j ~'~.. D~ + o-~ D~+ t +
~3
(dkJ drj)rI:[1 "k~ r dk ~ m ,--ij - - - - i j , xxdrs - d - a 1-I r+l,s
E s=0 s=0 p k=r+l
l < r < m . Then
^
¢ ~ij+Tr ~k '
where ~ = t3ij + 7,- - 7~+a.
(e) Let i, j be integers with 0 _< i, j _< m and i =)kj. Then
"~J Dm + { Co(1 ). ei j Dm = d i j £~ij flij+'~rn--aO(1 )
(d) Let i , j , r, and s be integers with 0_< i , j , s_<m, i ¢ j , r_>l, and ¢~ E O ( r + l ) - ~ ( r ) . I f¢1 E ~ ( r + l ) - d p ( r ) , t h e n
Oj ~r e i jC, ( r ) = di j do, e&j+~,(~)_~o(~)Co(r) + a ~ D 1Cl(r ) +
k j ~ r _,.tOJ~ir r AkO~Ir d-cr ~] d i j d k , "*ij=o, d k o - ~ l o ~ k l P k>l
Ck e ~(r+l)-~(r) e ~ij+as(r)-ak(r ) kk(r),
where a = f l i j + a s ( r ) - 7 1 - %(r).
If 81 ~ ~(r + 1 ) - ~i'(r), then Oj ~r
e i jC . ( r ) = d i j d o , e&jCo(r ) + 4&j+a0(r)_~0(r+l)C0(r+l ).
(e) Let i , j , r , a n d s be integers with 0_< i , j , s_<m, ig : j , 8~ e ¢ 5 ( r + l ) - ~ ( r ) . If ¢,+1 E ~ ( r + l ) - e ( r ) , then
cij D, C~(r) 'J = di j epi j D, Cs(r) q- ere, Ds+ 1Cs+l(r) q-
dk.s+ 1 d_a I~I d kt Z a L (d:} - d:~)t*=Iff°d:: - - ~ t=O s+l,t
P k>s+l
r_> 1, and
% ~k(r),
94
where c ~ = / 3 i j + 7 ~ + o ~ , ( r ) - 7 , + l - c % + l ( r ) and 6 k = / 3 i j + 7 " + ~ , ( r ) - c ~ k ( r ).
If ¢s+1 ff ~(r -t- 1) - ~(r), then
*J Cs(r) + ~/3ij+.rs+as(r)_c~o(r+l)Co(r+l ). Cij D , C s ( r ) : dij ePijDs
(f) Let r, s_> 1 and assume that i < j _< m. O(r + s) appear with equal multiplicities, then
~ r ~ S . ~ r + s
%(r)%(s) = %(r+s) + E dkjdk - p Ck e q~(r+s+l )--q~(r+s)
If the irreducibles that appear in
Poj(r+,)-ak(r+, ) kk(r +s)"
Moreover, the integers "~kj may be selected to be the products dk jdk j so that the kk(r + s) correction terms are not needed.
Since the elements kk(r ) appear in so many formulas, we include a description of products involving them.
LEMMA 5.11. Let i, j, k, r, and s be integers with 0_<i,j ,k_<m, ¢k E ~ ( s + 1 ) -~ ( s ) .
(a) If i C j, then kj
cijkk(s ) = dij e&jkk(s ).
r, s>O, and
and
(b) If ¢j E ¢(r + 1) - ~(r) and ¢~ E ~(r + s + 1) - qS(r + s), then
C/(r) kk(s ) "" -= dkj ¢oej(r)+ok(r)_ok(r+s) k~(r + s)
f j -1 :l ~t r k t
Dj Cj(r) kk(s ) = kj t~=odj e T j + o j ( r ) + ok(s ) - ok(r+s ) ;¢k(r+s)"
In the formula for Cj(r) kk(s), replace %j(r) + %(s) - ok(~+s ) by
#oj(r) + % ( , ) - %(~+s) if ]aj(r) + C%(S) - ak(r+s)] is zero.
(c) If ¢j C ¢(r + 1 ) - ¢ ( r ) and ¢k 6 ¢(r + s + 1 ) - ¢ ( r +s ) , then Cj(r) kk(s )
and Dj Cj(r) kk(s ) are zero.
To complete our description of the multiplicative structure of HOP(V) + we
need to describe the products of various pairs made from elements of the types Ci(r),
95
DjCj( r ) , and D k, If we use the convention that D O = Cj(O) = 1, then the products
we must describe are all special cases of the general product (Di, Ci( r ) ) (Dj , Cj(s)),
where r , s_>0, ¢i E ~ ( r + l ) - ~ ( r ) , ¢j C ~ ( s + l ) - ~ ( s ) , i' is 0 or i, a n d j ' is 0 or
j. We may assume that i' > j ' . Recall the formula given in Theorem 5.1(c) for the
product C(j)C(k) when p = 2 and j + k > n I. Observe that this formula may be
o b t a i n e d from the binomial expansion of (¢ 2 + ~x) j+k-nl by replacing the powers of
x by various generators C(t). The formula for our general product is related in a
similar way to the expansion of an expression of the form l~I (ai + b/x). The i r a0
summands in this expansion are indexed on the subsets of the set {0, 1, ... ,n}. The
summand corresponding to the subset I is
( l ~ a i ) ( ] ~ b i ) x [I[, i~I i~I
where ]I] denotes the number of elements in I. To describe the analogous part of our
formula for (Di, Ci(r))(D j,Cj(s)), we must specify the indexing set which replaces
{0, 1 . . . . ,n}, the factors which replace y [ a i and l~b i , and the procedure for
replacing the powers of x by the appropriate D~ Ck(t ).
In the p = 2 case, describing how the powers of x are to be replaced by the
generators C(j) is very simple because, if j > n~, then the next generator after C(j) is
always C(j + 1). However, when p is odd, the generator after D k Ck(r ) may be either
Dk+ 1Ck+l(r ) or C0(r+ 1). To handle this complication, we introduce two functions f
and g from the nonnegative integers to the nonnegative integers. These functions are
to be chosen so that, for any i_> 0, Cf(i+l)(g(i+ 1)) is the generator immediately
following Cf(i)(g(i)) in our stairstep ordering. If Cf(,~)(g(n)) is the last generator in • ~ + H~P( ) , then we define f ( i ) = 0 and g ( i ) = g ( n ) + i - n for i > n and use the
convention that Dj Cj(r) is to be regarded as zero if it does not appear in the list of
generators of I I~P(~) +. Each time we use this notation, the initial values, f(0) and
g(0), of the functions will be specified to suit the particular application.
The indexing set which replaces the set {0, 1 . . . . . n} is related to the
difference in dimension between the product (Di, Ci(r))(Dj, Cj(s)) and the lowest
dimensional generator D i , C i ( r + s ) which should appear in its description. If r_> 0
96
and 0 < j _< m, then define the subset ~j(r) of d2(r+ 1) by
(I)j(r) = ~(r) U {¢i: i < j and ¢i E ~ ( r + l ) - ~ ( r ) } .
Let ~i,(r) U ¢sj,(s) denote the disjoint union of the sets ~i,(r) and ~/,(s). Our
replacement for the set {0, 1 . . . . ,n} is the set • obtained by deleting from
• i,(r) U ~j,(s) a subset equivalent to the set (I)i,(r+s). We abuse notation by
writing ~ as ~i,(r) 1_1 ~ j , ( s ) - ( I ) i , ( r+s ). Observe that ~j,(s) is equivalent to the
disjoint union of • and ~ i , ( r + s ) - ¢5i,(r). Let u be l ~ l - 1 and number the elements
of ~ from 0 to u. Let h be a function from the set {0, 1 , . . . , u } to the set
{0, 1 . . . . , m} such that the i th element of • is isomorphic to the irreducible
representation q ) h ( i ) "
One of the coefficients appearing in our formula is determined by a certain
element a of RSO(G) with Ic~l = 0 and tc,~I _< 0. This coefficient will be go if
Ic~GI < 0 or c o if laGI = 0. To simplify our notation, we write )~o for either of these,
relying on Ic~cl to indicate whether ~ or ao is intended. Another coefficient will
depend on a certain element f3 of RSO(G) with 19 I- 0 and I~31 _> o. This coefficient
will be e5 if I/3I > 0 a n d / ~ if It31 = 0. We write 0~ for either of these, relying on 1/31
to indicate which is intended.
PROPOSITION 5.12. Let i, i', j, j ' , r; and s be integers with r,s_>0,
¢ i E ~ ( r + l ) - q S ( r ) , C j E ~ ( s + l ) ~(s), i ' = 0 or i, j ' = 0 or j, and i ' > j ' . Let
= ¢5i,(r ) t_J ~j,(s) - (I)v(r + s ). Initialize the functions f and g by
and
i', i f¢ i , E ¢ ( r + s + l ) , f(0) =
0, otherwise,
r+s , i f¢ i , C ( I ) ( r+s+l ) ,
g(0) = r + s + i, otherwise.
Let u : l q * l - 1 and number the elements of 9 from 0 t o u . Let A C 9 and let s ' a n d
97
s" be the number of elements isomorphic to Cj in A and q~i , ( r+s ) -~ i , ( r ) ,
respectively. If the subset A of • contains the elements numbered J0, J l , ... , j~o,
with J0 < J l < . . . <J~ , then let
and
I df(Jt-~)'h(Jd~ I df(J~-t)'J~ d~ = t~=o j ,h ( j t )~ t~=o jk ~, h ( j t ) ¢ j h ( J t )=J
t = 0 j , h ( j t t = 0 /3j h ( j t ) ~ j h ( J t ) = J
where
Xz~ Xa ,
a = c - i F ~ ' 1
Lh( j t )7~ j C t ~ O i¢( r + s ) - ~ ir( r
¢ i Ct eOi~(r)
e ~f(l AI)(g(IAI))
The tag j :/= 0
present only if j ~ 0.
trivial representation.
on the bracket about the (s' + s " ) ¢ j l ¢ 0 indicates that this term is
The 2s term in a indicates 2s copies of the real one-dimensional
If a E RSO0(G), then let.
d a = d o.
I f A = ~ , t h e n l e t d~x, e ~ , ( t , a , a n d X be 1. A
where
I f i ' < k < m a n d ¢ ~ E ~ ( r + s + l ) - ~ ( r + s ) , l e t
O k = 0/~,
98
= ~'dr) + %.(s) + -y~, + L , - ~ , ( r + s ) ,
and let A k be
1 ~ i ~ i ' -1 kJ(t~=od~tt)( I-I dkt ~ -" ~ [-'~r+s(v-I kt\ 0 rid.) E - t=0 J'*] v=i' t=0 lal=v-i'ac~'
Then
0 0 Cf(taf)(g(1Al)) (Di, Ci ( r ) ) (Dj , Cj(s)) = ~ d~,_~ ee-za X Df(Iz~f) ~ c ~ z~
Ak 0k kk( r+s) . k=i I
+
REMARKS 5.13. (a) Let r_> 1. If d2(r) contains r copies of every irreducible complex G-representation, then (~i(r) is independent of i and it is easy to see that C i ( r ) = C j ( r ) for every i and j such that ¢ i , 4~j E ~ ( r + l ) - q ) ( r ) . Moreover, Cj(r) = C j(1) r. Thus, if ~ contains every irreducible complex G-representation and these representations appear with equal multiplicities in ~, then Ci(r) generates a polynomial, or truncated polynomial, subalgebra of H~P(O) +. In this case, the elements Dj , for 1 _<j _< m, and Ci(1), for any i, generate I-I~P((I)) + as an algebra
* 0 over HaS .
(b) If p = 3, then we may choose the integers d , so that da = +1 for every a in RSO0(G ). When this is done, the assignment of d~ to a is a homomorphism fl'om the additive group of RSO0(G ) to the multiplicative group {5:1}. With this
choice of the integers d~, all the relations among the d~ j and the d~;j given in
Definitions 5.4, except the one involving a sum, hold in 7/as well as in g/3. If r _> 1 and ¢i , Cj E ~ ( r + 1), then
CAr) = %d~) -~¢~) C~(r).
Thus, the only elements of the form Cj(r) needed to generate H~P((I)) + as an algebra over H~S ° are the elements C0(r ) for r > 1. Also, a pair of elements cij and c~, will generate D 1 and D 2 if (tk(cijcr~) is nonzero for only one value of k. In particular, c01 and %o generate D 1 and Du. When all three irreducible complex G-representations of 7//3 appear in • with equal multiplicities, c01 , c~0 , and C0(1 ) generate I-I~P(~) + as an algebra over I t , S °.
6. PROOFS. Tile results stated in section 5 are proved here. As indicated in Remark 5.7, our results for p = 2 are a special case of the results asserted for odd
9 9
primes. They have been presented separately only because they can be stated so simply. The proofs given here are independent of whether p is 2 or odd. We begin by construct the elements c~j and Cj(r). We then show that they generate H~P(V) +
• 0 as an algebra over H~S . Finally, the relations stated at the end of section 5 are verified. Throughout this section, • is a set of irreducible complex representations of 7//p and ~(0), ~(1) . . . . is a proper filtration of ~. We order the elements of q~ in the standard proper ordering introduced in section 5. Recall the maps qi and Cti and the cohomology classes x and x i from the introductory remarks in section 5 and the representations a~(r), /3~j, and 7j from Definitions 5.4 and Theorem 5.5(d). If A C q, then x also denotes the image of x E t t~(P(¢)+)(e) in H~(P(A)+)(e); thus, the powers of x are thought of as the standard additive generators for the nonequivariant cohomology of all the sub-projective spaces of P(q) . For each integer j with 0_<j < m , let pj(4~) be the component of the fixed point space of P(¢) associated to the irreducible representation 6 j .
The classes cij and Cj(r) are constructed by defining them on the smallest possible projective space and then inductively lifting them to larger projective spaces.
CONSTRUCTION 6.1. (a) Let i and j be distinct integers with 0_< i, j _< m. The
space P({¢j}) is just a point and the space P({¢i , C j}) is G-homeomorphic to S #ij. The inclusion of P({¢j}) into P({¢i , ¢j}) induces the cofibre sequence
p({¢j})+ q4 5 s ei . ~ i j fig *
Let ciy ~ ttG (P({¢g, ¢i})+)(1) be the image of 1 e A(1) ~ H iJ(sZiJ)(1) under ~r .
Then q j ( c ~ j ) = 0 by exactness and q~(cij ) = e#ij by the commutativity of the diagram
p({qSi})+ qi , p({q~i,q~j})+
S O efliJ~ 813ij.
These are the correct values for qi(cij) and qj(cij ) because x i and xj are zero. Since
the map 7r*: H~iJ(S#iJ)(e).-, H~iJ(P({6i ,6j})+)(e) is an isomorphism in dimension
/3~j, p(c~j) = x.
Let ~ be a subset of (I, which properly contains the set {¢i, ¢j} and assume that, for every proper subset A of • containing {6 i ,6 j} , cij has been defined in
HGiJ(P(A)+)(1) and has the proper images under the maps qk and p. Pick an
irreducible representation 6t which appears in • at least as often as any other irreducible. If no irreducible appears more than once in tit, then we may also insist
100
that t ¢ i , j. Let Z X = ~ - { ¢ t } , and let V be the representation ~b~-lEqS.
inclusion of A into • induces the cofibre sequences
The
p(A)+ 0 p ( , ) + -~ S v
and
Pt(A) + ~ P,(•)+ -~ S VG.
We will lift the class cij ¢ H~iJ(P(A)+)(1) along the map
e*(1): It G (P(~)+)(1) -, H~iS(P(A)+)(1)
induced by 0. To distinguish the class ci5 and its lifting, we will denote the class in r ~
H~'J(P(&)+)(1) by cij. The maps qk, for k 5Lt, factor through 0"(1), so any lifting
of cij along 0"(1) will have tile right image under q~, for k ~ t. Moreover, since 0*(e) is an isomorphism in dimension /3ij , any lifting of aij will also have tile right image under p.
It remains to show that we can choose a lifting of cij with the correct image under qt . We have chosen t so that the long exact cohomology sequences associated
to our cofibre sequences have zero boundary maps. If IvGt > 2, then HGiS(sV)(1)= 0
and we take cij to be the unique lifting of e O. If IVGt > 2 , then 0 t induces a cohomology isomorphism in dimension /3ij and this lifting of cij along 0"(1) must have the correct image under q,. If IvGI = 2, then the short exact, sequence
splits. The end terms are
tt "s ~ at " (zx) + ~ = R and Pt = •
The image of 1 ¢ ?7 = R(1) in H ~ J P t ( ~ ) + is {&j_2xt. By our induction hypothesis,
tj O;(1)qt(cij) = qt(aij) = dij e&j.
Since P ( c i j ) = x , pq,(ei j) is the generator of H~iJ(P,(O)+)(e). It follows ttlat tj
q t ( c i j ) = dij e~i j + ~i3ij_2 x t "
If IvGI = 0, then no irreducible appears more than once in ~ and we have selected qS, so that t :/: i, j. In the diagram
L ev I q~ ~ q,
o - . + o
-~ 0
I0t
comparing the cohomology sequences of our two cofibre sequences, we have that
I-I~ ijS V and tt~ ijS ° are (g) and the map e V is multiplication by p. Thus, if z is a
lifting of ci j , then by adding elements from the image of I-I~GiJs V to Z, we can adjust
qt(z) by any multiple of p. It now suffices to show that there is a lifting z with
tj mod p. The lifting problems for P(g/) and P({¢i , Cj, Ct}) can be q~(z) ~ dij e~i j compared via the cohomology maps induced by the inclusion of {¢i, Cj, Ct} into ~.
This comparison indicates that it suffices to show that the lifting problem can be
solved when k0 = {¢i, C j , Ct}. In this case, consider the diagram
0 --+ H P G i J s V /3 i j + O* "" H G P ( ~ ) -~ l t ~ ' e ( z x ) + -+ 0
l e [ q ~qj
0 -~ It~iJs ~tj 7 ; i j e -* tt G })+ -+ H ({¢d' Ct})+ qj i jp({¢j -'+ 0
comparing the cohomology exact sequences for the pairs (p(ko), p(A)) and
(P({¢ j ,¢ t} ) , P({¢j})). Let a = (3ij - /3 t j . If z is a lifting of eij along 0"(1), then _~_ LI fl i J [ ~ fl qj(z) q j q ( z ) = 0 . Thus, q ( z ) = 7 ( y ) for some y e .~G ,~" tJ)(1) ' Since pq(z) is
the generator x of H~ (P({¢j,¢¢})+)(e), p(y) must generate H~iJs&J(e), and y
must be a~ + na~ for some integer n. The diagram
G
*e -[qt
commutes and gives that qt(z) = qqt(z) = e(y) _= e(c~) mod p. By the definition of tj o'~,, e(o'~) = dij e&j.
(b) Let r_> 1 and let Cj E q~(r+l). The cofibre sequence associated to the inclusion of P(~(r)) into P(O(r) W {¢~}) is
P(~(r)) + -* P(~(r) U {¢j})+ -~ S ~'j(').
Define C j ( r ) e t t G (P((I)(r)U{¢j})+)(1) to be the image under ~*(1) of
1 E A ( 1 ) = H Since Tr* is an isomorphism in dimension c~j(r),
p(Cj(r)) = x The cohomology diagram in dimension c~j(r) induced by the
diagram
102
Pj(¢(r) u {%})+ 7rj
S~
qJ P(@(r) U {¢j})+
l- e ~j(~)
---* S
indicates that qj(Cj(r))=ec, j(r)_~x~j. If k@j , q k ( C j ( r ) ) = 0 for dimensional
reasons. As we did with the definition of cij in part (a), we extend the definition of Cj(r) to H~P(@) + by working inductively along a sequence of subsets of ¢5 between ~5(r) U {¢j} and ~. The only difference between the argument given for cij and the one which should be used for Cj(r) is that the liftings of Cj(r) should be chosen to behave properly with respect to p and ~1~ instead of p and qk. This change is necessary because %(Cj( r ) ) is more complicated than qk(cij). The behavior of the Cj(r) with respect to the maps qk is established in the lemma below.
LEMMA 6.2. Let r _ > l a n d ¢~ E d 2 ( r + l ) - ~ ( r ) . Then
r q/c(Ck(r)) = X k 1-I (~9~, + )]
¢i e q~(r) ~ fiki - 2 x k •
i¢:k
If Cj e ~ ( r + 1)-4p(r) and j # k, then
r d ~i e = + . %,_
' k ~¢j,k
+
r [dkJ~ r ki 21 ~kj ~ jk)
i=/=j,k
%j(~)_~x~.
If Ck ~ q?(r+ 1) q)(r), then qk(Cj( r ) ) i s zero.
PROOF. If ¢~ ~ 4p(r+ 1) - qS(r), then %(Cj( r ) ) vanishes for dimensional reasons. Therefore, assume that Cj, Ck ¢ ~5(r+l)-dp(r) . Let
= ¢(r) u {¢: ¢ • ¢ ¢(r) and ¢ ~- Ck}"
The {mage of the class Cj(r) in II;P(dp) + under the map
H~P(¢~) + + ItSP(k~ U {¢j})+
may be computed using the maps p and ~!i- It is the class Cj(r) in t t ~ P ( ~ U {¢j})+. The image of this class under the map
HEP('I ' U {Ca})+ -~ J.[~P(~)+
103
is the class O'aj(r)_c~k(r)Ck(r ). Thus,
qk(Cj(r)) = qk((raj(~)_ak(~) Ck(r)) : aaj(r)_~k(r ) qk(Ck(r)),
since Pk(~) = Pk(9) and the map qk for P(~) factors as the composite of the map
I t~P(~) + -* I-I~P(9) + and the map qk for 9. Observe that
(r~J (~)-~k(~) ---- (crPjk-Pkj ¢i t~c~ j ( r ) - ~ k ( r )
for some integer a. With this description of O~c~j(r)_c~k(r), it is easy to derive the
formula for qk(Cj(r)) from the formula, for qk(Ck(r)). The formula for qk(Ck(r) is
derived using an iterative procedure. Let s > r and pick Ct E • with t :~k. The
image of Ck(s ) E I-I~(P(~)+)(1) under the map I t~P(~) + --* I-I~P(~ - {¢~})+ is
czktCk(s ) + {Zkt_2Ck(s+ 1).
Iterating this process to eliminate from q~ all the irreducible representations not isomorphic to Ck, we move from H~P(~) + to It~P(n~ ¢~)+ ~H~P~(gl ) + and from C~(r) to the expansion of
(%,+ t [¢i ~ ~(r) ~ i -
On the other hand, the image of Ce(r) under this sequence of transformations must be q~(Ck(r)).
Now that we have defined the classes cij and Cj(r), we must show that they • 0 generate H~P(¢) + as an algebra over ttGS .
PROPOSITION 6.3. The classes cij , for ¢ i , Cj E (I)(1), and the classes Cj(r), for
r > 1 arid Cj C ~(1" + 1) - ~(r), generate H~P((P) + as an algebra over H~S °.
PROOF. If • is infinite, then, by the proof of Theorem 2.6, H~P(~) + is the limit of the ~ P ( A ) + where A runs over the finite subsets of ~. Thus, it suffices to prove the result for • finite. Recall the functions f and g and the subsets ~j(r) of defined in the remarks preceding Proposition 5.12. For this proof, initialize f and g by f(0) = 0 and g(0) = 0. We will show, by induction on n, that the classes cij and
, + Cj(r) which are defined in HGP(~f(,~)(g(n)) ) generate that Mackey functor as an
• 0 algebra over t t G S . The result is obvious for n = 1, since (I)f(1)(g(1))= {¢0} and
104
P({¢50} ) is a point. Assume the result, for n. Denote c~f(n+l)(g(n+l)) + 7f(n+l) by
ct. The boundary map is zero in the cohomology long exact sequence associated to
the cofibre sequence
p(~f(,0(g(n)))+ O p(~f(n+l)(g(n+l)))+ + S ~.
Thus, we have a split short exact sequence
0 --~ fIGS --. " + t tGP(~f(n+l)(g(n+l)) ) -+ I-IGP(~f(n)(g(n)) ) ~ 0.
. + All of the classes ci# and Cj(r) which are defined in ItGP(¢f(,~)(g(n))) are also
defined in * + HGP(~f(n+l)(g(n+l)) ) . Moreover, 0* takes these classes in . +
H~P(<I)f(n+l)(g(n+l))) + to the corresponding classes in HGP(¢f(,~)(g(n)) ) . Thus, to • + * 0 generate HcP(45f(,~+~)(g(n+l)) ) as an algebra over HGS , it suffices to add to these
classes the image z of the canonical generator of A ( 1 ) = H~(S~)(1). Clearly, p(z) is O' + the generator of HG(P(~f(=+l)(g(n+l)) ) )(e). Moreover, for k : f i f (n+l ) , ~ k ( z ) = 0
since qk factors through H~P(<l)f(,~)(g(n))) +. Finally,
qf(n+1)(Z) z ~(:c~_g(n+1)(Xf(n+1)) g(n+l)}
since the diagram
pf(,~+~)(~f(~+~)(g(n+l))) + qf('~+})p(@f(~+~)(g(n+l))) +
Sg(~+1) e , S ~
commutes. The elements z and Df(n+l)Cf(n+u(g(n+l)) must be equal since they
have the same image under the maps qk and p.
The equations in Propositions 5.9 and 5.10 describe elements in dimensions where there is no torsion. As a result, these equations can be checked easily by applying the maps p and qk to both sides. The equations in Lemma 5.11 are easily checked using the maps # and qk because the images of the classes kj(r) under the maps qk are so simple. However, the formula in Proposition 5.12 is more difficult to verify.
PROOF OF PROPOSITION 5.12. We may assume that l~I _> ]~i,(r) I + qsj,(s) so
105
that all of the Df(lal)Cf(iz~l)(g(lAI)) on the right hand side of the equation are
nonzero. If I¢1 is too small, then form a sufficiently large set ~ ' by adding enough copies of ¢0 to ~. The proof below applies to 4p'; the result for ~ is obtained using the cohomotogy map induced by the inclusion of • into ~' . We show the equality of the images of the two sides of the equation under the maps p and q~. Since the map p preserves products, p(D i, C i ( r ) D , Cj(s)) is the generator of H~(P((IS)+)(e) in the appropriate dimension. The only term on the right hand side of the equation in Proposition 5.12 which is not in the kernel of p is the summand corresponding to
regarded as a subset of itself. This term is X Df(u)Cff~)(g(u)) and its image under p
is the generator of H~(P((I))+)(e) in the same dimension. Thus, the expressions on the two sides of the equation have the same image under p.
Let k be an integer with 0 < k < m . If Ck ~ ~ ( r + s + 1 ) - <I)(r+s), then both sides of the equation vanish under % . If Ck C (I ) ( r+s+ 1) <I)(r+s), then expand the polynomial obtained by applying qk to Di, Ci(r)Dj, Cj(s ). Each term in the expansion consists of the product of an integer, a power of x k , and an element of the
, HaS . We classify these terms according to the factor form ~ , ~ or e ~ from * 0
from . 0 HaS . There is exactly one term with a ~ ; its integer coefficient is one. There
is exactly one term with an c~; its integer coefficient may be zero. This term is
exactly the part of qk which is detected by qk. There may be any number, including zero, of terms containing a product e~ ~ . These terms are all torsion elements of order p.
Expand the polynomial obtained by applying qk to the right hand side of the equation and observe that the same three types of terms appear. The su mma n d
indexed on q~ regarded as a subset of itself is the only source of a ~ . It is easy to
see that this ~ term exactly matches the corresponding term from the left hand side of the equation. If i ' > k, then the expansion of the image of the right hand side under qk will contain no ~ term. In this case, ~k(Di~) is zero and the image of the left hand side under qk also lacks an ~ term. If i ' < k, then numerous summands contribute to the c~ term of the left hand side, but the coefficient of the k k ( r + s ) term is explicitly designed to ensure that the c z terms of the expansions of both sides match. The only problem here is that it is not obvious that the coefficient A k of k k ( r + s ) is an integer. To show that A k is an integer, it suffices to show that, modulo p, the image under qk of the left hand side is equal to the image of the part of the right hand side indexed on the subsets of ~. Since the e ~ terms are all
O~. torsion of order p and the k t ( r + s ) summands on the right hand side contribute nothing to them, proving the equation
q~ }--~ d~,_A ~ _ A xADf(i,al)Cf(l~l)(g(]A]))) m o d p qk(Di, Ci(r) D j, Cj(s)) -- (~,c,,
also shows that the c~ ~ terms of the two sides agree and so completes the proof of the proposition.
We prove this equation modulo p by transforming the right hand side into
106
the left. In Theorem 5.5(c), %(Cj ( r ) ) is described as a sum of two terms when j ¢ k. The second term can be ignored in this t ransformation process because it vanishes
modulo p. Recall that each * a is a ~ , for some virtual representation a'. We
accomplish our transformation by writing a as a sum of differences r / - ¢ of
irreducible complex representations. We then rewrite X a = Xa as the product of the
elements X _ ¢ . To see that such a rewriting is justified, recall that if /3 and 7 in
RSO(G) are chosen so that the elements below are defined, then in H~(S°)(1)
( ~ 7 = {0+y {~ % = 0 e 0 % = pc0+ 7
and
(r 0 o " = ~ 0 + ~ + A~0+~,
where A is some integer depending on /3 and 7. Now observe that every summand in
the expansion of %(Df(l~l)Cf(lai)(g(IAt)) ) contains either an c a or a ~ . Thus, the
as the product of the X ~ + ~ error terms that might arise in the rewriting of Xa ~-¢
are killed by the e~ and ~ from %(Df(l,~t ) Cf(lz~l)(g(lAt)) ).
We perform our transformation of the left hand side in four stages. During the first three stages, we think of the left hand side as a sum indexed on the subsets of q* and work on each summand separately. Therefore, fix a subset A of • and let
a be the virtual representation such that ;(a = )~" Recall that s' and s" are the
number of elements isomorphic to ¢j in A and ~ , ( r + s ) ~ , ( r ) , respectively. Recall
that u = Ig'l- 1, that the elements of q~ are numbered from 0 to u, and that h is a function from the set {0, 1 . . . . . u} to the set {0, 1 . . . . . m} such that the i th element in • is isomorphic to Oh(i)" Assume that the elements of • numbered J0, J l , --- , jw, with J0 < J l < - - - < J * ~ , are in A and that the elements numbered i0, i 1 . . . . , i~, with i 0 < i 1 < . . . < i ~ , are in g~ -A . For any integers q and t, with 0_<q, t_<m, abbreviate e~q t and ~ q t _ 2 by eqt and 4q~. Define the elements a l ,
a2 , and a a of RSO(G) by
z f(I,al)J| ¢~ '~i,(") |
Lt.~f( ~ ),i,k J
[(r -F 6)(qSi -I CfdA] ) - qSf-(i~[ ) Ck)]f([ A] ) ¢ i,k -+"
107
[~71¢k_¢f--(~/q) ¢k]f(,z~l)>i>k q- [ (~7"l¢k-2]i>f(l~'),k
)[ 2 < tef(lal),j ,k _]
[ ( s - s' - s") (¢7 ' Ck - ¢7 t ¢ o ) ~ o , 5, k +
-~ ¢ + [Stt((~71~k-'~f(IAl) J)~j~f(IAI),k
and
where
=
Ct 3 ~ O~ C~ 1 -- O~ 2
1, if i' > f(lA[),
O, otherwise.
In the first stage of our transformation~ Xc~I is used to convert
into the product of
and
o eo (Di, Ci(r))
108
g(Im)-"-<~' l I'-[ (d~t ~f(l~l),* x~ ¢~ ~ @f(l~l)(g(i al))_ @i,(,.)k
f(tAI),~
~#f(t ,-al),~
H- ~f(!al),t x~) I -
I(dS<,f(l~l) t, f(lal)>~ ef(l~i),~
x~)g( la l ) -~-6] [~ f(13l),~ x~]
i' >f( lal)> k or f(Iz~l) > k > i'
Here, 6 is as in the definition of 6% 1 and
~- 1, if i' > f ( l~ l ) , k; 6'
0, otherwise.
In the second stage of the transformation, ; ~ is used to convert this product into
the product of d~_~ ~+_/, ~,o, ~ (>~, C~(~))with the three factors
H )][( Xk Ct e ~i,(#+.~s)_ '~i '(r) \ at ~jt + ~jt djk (jk 4- ~jk , .~j#k
t ¢ j , ~
xkg(l~I) . . . . . ~' tdf(t,a l),f(t) cf(l~l),f(t) 4- ~f(lal),f(t) xk , -j
and
f/~ k,f(Iz~l) ~Gf(lal),~ ef(Izal),k 4-
~f(lal/~ _lf(Izll) ¢ k k i' >f(l~l)> k or f(I~l) > k _> i'
. . . . . d ~ e k factor. Observe that the (1S ~ e S a factor has been transformed into a ~ - ~ ~ - a
This is accomplished by the [ ( s - s ' s " ) (4 ;71% ¢)-1 ~50)]0 # j,k summand in a~. If
k = 0, then obviousiy no such transformation is needed. If j = 0, then there will not
k k will not depend be any elements of 9 isomorphic to 4D j , and the value of d ~ _ a e e _ a
on k. In the description of the factor above indexed on t, for 0 < t < w, and
throughout the third stage of the transformation, the set Ofd~[)(g(IA[)) - (I)i,(r 4- s) is
109
identified with the set {¢f(t) : 0 < t < W). By this identification, constructions that
would naturally be indexed on (I)f(t~l)(g(]AI))-¢i,(r + s) may be indexed on t. The
description of the set {¢f(t) : 0 < t < w} involves our usual abuse of notation in that,
whenever q ¢ t and f(q) = f(t), the representations el(q) and ef(t) are intended to be
distinct, but isomorphic, elements of the set.
The factor
qk(Di ' C~(r)) x~ I¢t ]-I (d kt
¢¢it(ra'+~s)-q~it(r)\ jt (jr tJj,k
"4- ~ jt Xk djk 6jk $ t t
appears in every summand of the transformation of the right hand side of the
equation. We therefore factor it out of the sum and ignore it for the rest of the
transformation. Observe that this factor consists of qk(Di, Ci(r)) and that part of \ /
% ( D / C j ( s ) ) which is associated with the set ~ i , ( r+s ) - ( I ) i , ( r ) when q) i , ( s ) i s
regarded as the disjoint union of • and ~ i , ( r+s ) -ep i , ( r ) . Thus, we must transform
what remains of the sum after this factor is removed into the part of q ~ ( ' ~ , % I s ) )
coining from ~.
In the third stage of the transformation, X 3 is used to transform the
remaining part of the A summand into
{, x,) n + ~'-z.4 q~-A I t~3 \ j,h(jt) j,h(jt) ~- Lh(57)~5 J,,(h) _] Lh (}7)o= j
For the fourth stage of the transformation, consider the subsets A of ~ that
contain the last element eh(u) of ql. The summands indexed on A and A-{¢h (u )}
contain the common factor
h(~t)O j Lh (: =)~: j j,h(it: ~(i~)=j j
110
te--i dk,f(t ) ) i w-i ~,f(t) ) l
Lh(J?i#j J'h(Jt) Lh(Jt)=j
which we have written down using the i, and Jt numbering of the elements in ~ - A
and A. Each of the two summands contains exactly one term not in this common
factor. If h(u) ¢ j , then these terms are
= d k,h(u) dfj(hg'~(*')ej,h(,~ ) + d k ' ~ ) e h~ + x~. , J, ( ) J, ( ) ~j,h(u)Xk j,h(u) ej,h(u) 4- ~j,h(u)
If h(u) = j , then these terms are
df(~),/ dk,f(w) k,j j,k e l k + j,k e lk + ~j,k xk = dj,~ ej,k + ~j,k xk"
In either case, the result is independent of A and may be factored out of the sum.
Moreover, this factor is exactly the contribution that Ch(~) should make to
%(, Dj, C/(s))~ when Ch(~) is regarded as an element of Cj,(s) under the identification %
of Cj,(s) with the disjoint union of • and ~i,(r + s ) - ~pi,(r).
The sum that remains after the factor associated to Ch(~) is removed may be
regarded as one indexed on the subsets A of * - {¢h(~)}" We now pair the summand
indexed on a subset A containing the last element Ch(~-i) of q - { ¢ h ( ~ ) } with the
sumrnand indexed on A - { ¢ h ( ~ _ i ) } to obtain the factor of % ( D j , Cj(s) ) associated %
%
to Oh(u-i)" Repeating this process until the elements of ~ are exhausted, we recover
the part of q JD\ j, Cj(s)]] associated with kl/.
APPENDIX. Computing H~S °. Here, we outline the calculation of * o HGS . The computation of the additive structure and, for G = 7//2 or 7]/3, the computation of the multiplicative structure are unpublished work of Stong.
Three cofibre sequences suffice for the computation of the additive structure of H~(S°). Recall that ( is the real 1-dimensional sign representation of 7//2. Let r/ be a nontrivial irreducible complex representation of G = Z/p, for any prime p. Let G+--* Sr] + be the inclusion of an orbit and let Srj+~ S O and S~+~ S O be the maps collapsing the unit spheres Sr] and S( to the non-basepoint in S o . The cofibre sequences associated to these maps are
G + ~ St/+ -, EG +
S + -, S O e S ~
111
and
G + ~ S¢+_+S O e_, S("
The first step in the computation is obtaining the values of H,GSrj + and * +
H GSq from the first cofibre sequence.
LEMMA A.1. For any nontrivial irreducible complex representation rl of G,
t L, g _ ,
t t 0 s ~ + = < a ,
R_,
0,
iflc~l = 0 a n d ] G]iseven,
if lal = 0 and ]c~ G] is odd,
if Ic~l = 1 and ]e~GI is odd,
if lal = 1 and la G ] is even,
otherwise,
t t ~ S v + =
"R,
R_,
L,
L_,
0,
if Ic~l = 0 and [c~GI is even,
if Ic~l = 0 and I~GI is odd,
if Ic~l = 1 and I~GI is odd,
if Ic~l --= 1 and I~GI is even,
otherwise.
PROOF. The next map E G + + EG + in the first cofibre sequence is l - g , the difference of the identity map and the multiplication by g map, for some element g of G which depends on 7/. The homology and cohomology long exact sequences associated to the first cofibre sequence have the form
G G + O G + . . . - . t t G G + - Jag G + - J t~ s,7 + - , H~_~ - . ~ _ ~ - ~ . . .
and
"~ ~GT][(~--I"~+~ T T ~ - - I ~ + c~ + G + . . . . • .. - * x t G t . - . H ~ S 7 1 + - + I t c G - . H ~ - .
The Mackey functor t t ~ G + may be identified with the Mackey functor (I t~S°)G defined in Examples 1.1(0. The difference 1 - g may be regarded as a map in B(G). Under the identification of H ~ G + with ( I t ' S ° ) 6 , the first map in the part of the homology long exact sequence displayed above becomes the map from (H~S°)G to (H~S°)G induced by the map 1 - g in B(G). It follows that the cokernel of the map
G 0 ( 1 - g ) , : l i ~ G + -+HGG + is the Mackey functor L(I-Io(S )(e)) defined in Examples 1.1(e). Similar observations reduce the homology and cohomology long exact sequences of the first cofibre sequence to the short exact sequences
o - . L ( H ~ ( S ° ) ( e ) ) --, r I~ s ~ + - . R ( H ~ < ( S ° ) ( e ) ) --, 0
a n d
0 --+ L(H~-l(S°)(e)) --+ H~ Sq + --+ R(H~(S°)(e)) --+ 0,
1 1 2
Since I - I G ( s ° ) ( e ) " 0 = HI<(S ;?7), L( I IG(s° ) (e ) ) is zero if lal :/: 0. If lal = 0, then
L(I-I~(S°)(e)) is L(7/) for some act ion of G on 7/. This act ion is the sign act ion of 7//2
on 7/ when p = 2 and a contains an odd number of copies of ~; otherwise, the act ion
is tr ivial . Similar r emarks app ly to L(H~-I (S0) (e ) ) , G 0 R ( t t a _ , ( S )(e)), and
R ( H S ( S ° ) ( e ) ) .
T*Ot S + Notice the frequency with which £t e 7? and I-IaGS~ + vanish. F r o m the
d imension ax iom, we also obta in tha t ~ + t-leG = t t ~ G + = 0 if l a I ¢ 0 . These
vanishing results de te rmine mos t of the homological and cohomological behavior of the m a p s e in our second and the third cofibre sequences.
L E M M A A.2. Let a C RSO(G) .
(a) T h e r n a p e*: ~-~ 0 ~ ~ TJ~(S % t t e S ~ H ~ ( S ) - * * , C ~ J
is mono f o r l a l @ l , 2, epi for Ic~l @0, 1, iso for tc~l @0, t, 2.
a ~ ~ 0 ~ - < S 0 ~ t te(S ) (b) If p = 2, then the m a p e*:,u. G ~ HG(S )
is mono for lal @ 1, epi for Ic~I @ 0, iso for lal @ 0, 1.
The divisibil i ty results involving Euler classes in L e m m a s 4.2, 4.6, and 4.8 of ~ n <0 follow f rom this l emma. Moreover, from this l e m m a and the vanishing ~G~, , for
n E 7/ and n @ O, one can derive all of the zeroes in the first and third quadran t s of o f H * ~0 our s t andard plot ~ e o .
L E M M A A.3. Let a e RSO(G) . Then H~S ° = 0 if lal and la e] are both posit ive or bo th negative.
L e m m a A.2 indicates tha t all of H ~ S ° can be deterrrfined f rom the values of a 0 I tGS for the a in RSO(G) with - 2 _ < ! a t < 2 . If p = 2 , it suffices to know ~ 0 t tGS for
the a in R S O ( G ) with -1 _< Ial _< 1. The next l e m m a describes * 0 H e S on the edges of these two ranges of values for lal.
L E M M A A.4. Let a E R S O ( G ) and let r/ be any nontr ivia l irreducible complex representa t ion of G.
(a) If lal = 2, then
0 ~ ~ , ~ - , , ~ + H~-~S0) . HGS = coker ( r : ~ ,~ -~
113
(b) If laJ = - 2 , then o~ 0 ~ + r l 0 o~+r; +
HGS ~ ker ( p : t t G S - 4 H G G ).
(c) If p = 2 and lal = 1, then
a, 0 t tGS = coker ( r : tI G~-¢G+ -4 I-I~-¢S °).
(d) If p = 2 and lal = - 1 , then
,.~ a + ( 0 a + ( + "~Gu~S° : ker (p : It G S - 4 I t G G ).
Moreover, in all four cases, I-I~(S°)(e) = 0.
P R O O F . Pa r t (d) follows immedia te ly f rom the cohomology long exact sequence associated to the third cofibre sequence. Pa r t (c) follows via dual i ty f rom the homology long exact sequence associated to the third cofibre sequence. For par t (b), consider the d i ag ram
ua+nS0 f TTe+n S + 0 -4 I-I~S ° -4 ~.G .0. G 77
l h
H~+nG+ G
in which the row is f rom the cohomology exact sequence of the second cofibre sequence and the vertical arrow comes f rom the inclusion of an orbit G into St/. Clearly, ~ 0 . . . . S + ttGS = k e r f . By our c o m p u t a t i o n of n G 7/ , the m a p h is mono, so k e r f ~ ker h f . The composi te h f is jus t p. The proof for pa r t (a) is s imilar , but
uses the homology long exact sequence to describe t t - ~ S o as the cokernel of the m a p
I-I~_~ G + -4 ItG_~ S o induced by the collapse m a p G + -4 S °. Dualizing the homology
Mackey functors to cohomology Mackey functors gives the result since the t ransfer is the dual of the collapse map . In all four cases, the group H~(S° ) (e ) is zero either because r (e) is SUljective or because p(e) is injective.
Most of the values of tIGS~ 0 for lal = 0 and la G] @ 0 follow immed ia t e ly f rom the cohomology long exact sequence of the second cofibre sequence and L e m m a s A.1 and A.3.
L E M M A A.5. Let a E RSO(G) with lal = 0 . Then
i ,
u ~ S O R_, ~ G = L,
if laG[ _< 2 and laG] is e v e n ,
if laG[ < - 1 and laG[ is odd,
if laG[_ 2 and laGI is even,
if laGI > 3 and laGI is odd.
114
PROOF. Let r/ be any nontrivial irreducible complex representation. If Ja'GI < 0, then consider the portion
a - r / 0 ,'~ Hc~ S 0 a + ,'~ H G S = c~ r2 i ~ + l S r / i_ia+l--r/¢/0 H G S ~ ~ ' G -~ t l G S r / -~ "~-G =-~'G
of the cohomology long exact sequence of the second cofibre sequence. The left hand term is zero by Lemma A.3 and the right hand term is zero by the same lemma
unless l GI is -1. If -1, then p = 2, a = 4 - 1, I-IGSr ] is R_ by Lemma A.1,
Lia+l--~O and ~G ~, is (:~) by Lemma A.4. The last identification is based on the
observations that ~ must be 2~ and H~S ° is A. By inspection, there are no
nontrivial maps from R_ to {77). Thus, if I GI < 0, the middle arrow must be an
isomorphism.
If JaG1 _> 2, then consider the portion
a + r / - 1 0 , T a + r / - 1 S + r . j ra+r /s r l ~ i ~ a S 0 ~ i a + r / S 0 H G S -+ .LI. G r/ ~ ~-~G =aaG -+ "~G
of the cohomology long exact sequence for the second cofibre sequence. The left and right hand terms in this portion of the sequence nmst be zero by Lemma A.3. Therefore, the middle arrow is an isomorphism.
If p = 2, then the results above reduce the computat ion of I-I~S ° to the
determination of t t~S °, which is A by the dimension axiom, and It~-¢S °, which is
given by the following lemma.
LEMMA A.6. If p = 2, then II~-<S ° ~ R_.
PROOF. Consider the portion
s o G + ° s o
of the cohomology long exact sequence of the third cofibre sequence. By the dimension axiom, the right hand term is zero and the first two terms from the left
are A and AG, respectively. The value of ttG-~S° follows by computat ion.
If p ~ 2 , then we must still determine the value of a 0 HGS when tal = 4-1 or c~ E RSO0(G ). The next three lemmas dispose of the c~ with I~1 = ±1 which are not already covered by Lemma A.3.
LEMMA A.7. Let M be a Mackey functor and f : L - ~ M be a map. If f(e) is a monomorphism, then so is f.
PROOF. The composite f(e) p is a monomorphism and pf(1) = f(e) p.
115
LEMMA A.8. If p 5/= 2, ee e RSO(G), lee, = 1, and leeGI < 0, then I-IGS a 0 = 0.
PROOF. Consider the portion
a--r1 0 ~ a ~ c~ 0 c~ + f 14a+l~ ~.~ ~+i--~7 0 H G S H G S = t t G S r I aa G o H G S ~ H G S -., =
of the cohomology long exact sequence associated to the second cofibre sequence. The left hand term must be zero by Lemma A.3. By Lemma A.1, ~ + I-IGS ~ ~ L. Since
lee+ 1 - ql = 0, I-I~+l- '(S°)(e) is g. The map f: ttGSr/° + + i_iG~+l-nS0 is induced by
the geometric map S ~ + ESr/+ which identifies the points 0 and oo in S ' . From this description, it follows that f(e) is an isomorphism. By the lemma above, f is a monomorphism. Therefore, ~ 0 t tGS must be zero.
LEMMA A.9. Assume that p ¢ 2 , ee e RSO(G), leel = - 1 , and JeeGJ > 0. any nontriviat irreducible complex representation r/,
a 0 ,~, a+rl--1 0 - - a + r l - 1 S + \ I"IGS = coker (tIG S ~ 1 t G r/ ).
Moreover, if jaGI > 1,
H a S ° ~ / p .
Then for
PROOF. Consider the portion
at'tGIla+rl-ls0 h tlG'*a+v-1 ST] + -4 Z'tGLla+rlsrl =HGS,,-, c~ 0 _+ "~GtlC~+~S°
of the cohomology long exact sequence for the second cofibre sequence. The right hand term must be zero by Lemma A.3. The first part of the lemma follows immediately. By Lemma A.1, gl G u = R. The map h is induced by the collapse map Sq + + S °. Since J a + r l - 11 = 0,
= ttG ( S , ) ( ) Z.
The map h(e) is an isomorphism by an obvious computation in nonequivariant c~+rl--1 0 ,',., cohomology. If JaGj > 1, then by Lemma A.5, I t G S = L. The only two maps h
from L to R with h(e) an isomorphism have cokernel {g/p}.
If d ~ 0 mod p, then the only maps h: Aid] -* P~ with h(e) an isomorphism
are surjective. Therefore, once we have shown that I t , S ° is A[dz] when
fl E RSO0(G), it will follow from the lemma above that ~a~S° a, G = 0 w h e n l a I = 1 and
lofiJ = 1.
Lemma 4.6 follows from Lemma A.9.
PROOF OF LEMMA 4.6. Let a and fl be elements of RSO(G) with leel = - 1 , G Gt>0, 1~ t=0 , and ]flGj_<0. Let q be a nontrivial irreducible complex
116
representation. Consider the diagram
a + r / - 1 0 LI~S0 R=I - t c S --, , ,G -4 0
1 1 ilia +3+r / - - l~0 a+fl 0
R - - ~ c ~, ~ It G S ~ 0
in which the vertical arrows are given by multiplication by {~ or #~. The rows of ~+~-~ o 1 this diagram are exact by the proof of Lemma A.9. Let y C H G ( S ) ( ) be a
generator and let x 6 H~(S°)(1) be its image. Since p preserves products, p ( f~y ) must be a generator. Thus, @/~y must be a generator and so must @ox. Similarly, p(#~ y) is d/~ times a generator, so #~ y is d~ times a generator. It follows that #~ x is a generator. This proves Lemma 4.6 in the special case where Ic~1---1 and Iofil > 0. The general case follows from the special case and Lemma A.2.
Let a be an element of RSO0(G ). The main difficulty in identifying HGSa 0 with A[d~] is that we must select a representative for a in R0(G) in order to define ~ and d~. To circumvent this difficulty, we work primarily with elements of R0(G) instead of elements of RSO0(G ) in the remainder of our discussion of the additive structure of * 0 t{GS . If c~ is in R0(G), we write H~S ° for the cohomology Maekey functor associated to the image of c~ in RSO(G). To work with elements of I~0(G), we must introduce variants of Definitions 4.5(a) and 4.5(d).
DEFINITION A.10. Observe that the procedure used to produce the element #4 in
Definitions 4.5(a) actually associates a map p: S 2~i ~ S ~¢i to any element ~ ¢ i - ~ i
of R0(G). If a is a nonzero element of R0(G), denote this map, and its image in a 0 I-IG(S )(1), by ~ , Let n0 denote the identity map of S O and 1 E H~(S°)(1). If ¢ is
a nontrivial irreducible complex representation, then let ea,¢ : S 2~i ~ S ¢+~¢i denote
the smash product of the map e : S ° + S ¢ and the map no . We also use %,¢ to
denote the corresponding element in/ t~+*(S°)(1) .
If a and /3 are elements in R0(G) which represent the same element in RS00(G), then n~ and n0 need not be the same class in H~(S°)(1). However, the
class e~,~ in tt~+~(S°)(1) is uniquely determined by the sum a + ¢ in RSO(G). This
uniqueness can be exploited to resolve the problems caused by dependence of ~ on Ct.
LEMMA A.11. Let a and /3 be in R0(G) and let ¢ and r / be nontrivial irreducible complex representations such that a + ~b and /3 + 71 represent the same element in
~+~ S O 1 RS0(G) . Then the cohomology classes %,e~ and e~,~ in H G ( )( ) are equal.
t17
PROOF. We establish the result for three special cases and then argue that the
general case follows from them. Let r/, r/1 , r/2, ¢, ¢1, and ¢2 be nontrivial
irreducible complex representations and let c: S¢1+'2-+ S ¢2+¢1 be the switch map.
Regard a l = ¢ l - r / , a 2 = ¢ 2 - r / , and a = ¢ l + ¢ 2 - 2 r / as elements of R0(G). Let
e : S ° - + S ~ be the usual Eulerclass. The two maps l ^ e a n d e ^ l f r o m S ~ t o S ~+~ are
obviously equivariantly homotopie. On the level of maps,
%2,¢1 = ~ (e ^ 1) and %1,¢2 = c/*~ (1 ^ e).
Therefore, ea2,¢ I and c c~1,¢ 2 are equivariantly homotopic. Thus, e~.2,¢1 and e~1,¢2,
regarded as cohomology classes, are equal. Here, the map c is, of course, absorbed in
the passage to an RSO(G)-grading for tt~S °.
If r/ and ¢1 are equal and e': S o --, S ¢2 is the inclusion, then the trick used
above can also be used to show that 1 ̂ e' : S n --* S ¢1+e2 is equivariantly homotopic to
¢~ 0 1 %2,¢q" Thus, if ct 3 = ¢1 - q51 e I~0(G ), then e' and e~a,¢ 2 are equal in I-IG"(S ) ( ) .
Regard /31 = (¢1 - rh) + (¢2 - r/u) and /32 = (¢1 - r/e) + (¢2 - 711) as elements
of t~0(G ). By three applications of the result just proved for e~2,¢ 1 and %1,¢2' it is
possible to show that e~1,¢ and e/)2, ¢ are equal in HZGI+¢(S°)(1).
If c~ and /3 are in R0(G) and ¢ and r/ are nontrivial irreducible complex representations such that c, + ¢ and /3 + r~ represent the same element in RSO(G), then we can convert the pair (o~, ¢) into the pair (/3, r/) by some combination of the three basic transformations for which the lemma has already been proved. Thus,
%,¢ and eZ,,~ must be equal in H~+¢(S°)(1).
This lemma establishes that the element e z of Definition 4.5(d) does not depend on the choice of c~ and V used in its definition.
LEMMA A.12. If a E RSO0(G ), then ~ 0 ItGS ~A[dc~]. Moreover, if r l is any nontrivial irreducible complex representation, then #a is the unique element of t t~(S°)(1) such that e~ >~ = e~+, and p ( # ~ ) = d~ ~ .
PROOF. Recall the map s: RSO0(G ) --* R0(G) introduced in section 2. Let n
o~ E RSO0(G) and assume that s ( a ) = ~ ¢ i - r / i . Let c% be 0 C R0(G) and, for i = 1
118
k l < k < n , let ak be the element ~ ¢ i - r / i of R0(G ). Denote by d(c%) the integer
i = 1 associated to ak by our homomorphism from l~0(G ) to 7/. For 0 < k < n, let /3 k be the element a k + ¢~+, of RSO(G). We will show by induction on k that
i) Ta%S° is isomorphic to A[d(ak)], a.L G
ii) ~ ~nd ~ ( ~ k ) generate H~k(S°)(1),
iii) t-I~kS ° is isomorphic to (7/), and
iv) e ~ generates H~GkS °.
By the dimension axiom and Lemma A.4, these statements are true for k = 0. Consider the portion
~k-~ + l t~s~k+~ ~ ~k+~ 0 H~kS 0
of the cohomology long exact sequence of the second cofibre sequence. By Lemma A.1, The left hand term is isomorphic to L and the right hand term is zero. By Lemma A.7, the left hand arrow is a monomorphism. Thus, we have a short exact sequence
0 ~ L f l'lC~k+l~0a.a. G o -+ I-I~GkS 0 "+ 0.
Assume that the assertions above hold for some integer k. The element #k+l in a k + l 0 (S)(1) hits the generator e ~ in H~k(S°)(1) by Lemma A.11. Since f(e) is H~ an
isomorphism, we may assume that f(e) takes the generator 1 E 7/= L(e) to the a k + l 0 generator t%+ 1 of H~ (S)(e). It follows that #%+1 and r ( t%+l ) generate
t t ;k+l(s°)(1). Since
P (#%+l )=d(c~k+l ) t%+l and p r ( t % + l ) = p L % + l ,
C~k+l 0 H G S is isomorphic to A[d(ak+l) ]. By Lemma A.4, It@k+lS° is isomorphic to (7/)
= I-IGS is isomorphic to and is generated by e/~+ I. Since Po,~ / ~ and d ( a n ) = do, ~ 0 Aid@
Replacing c%+ 1 by c~, r/k+1 by % and /3 k by a + 7? in the cohomology long exact, sequence above, we obtain the short exact sequence
0 -~ L - . I4~S ° h nO+,S0 ~G -* 0.
Our characterization of #~ in terms of e, #o = h(/~o) and p(#~) follows directly from this sequence.
Two general observations suffice for the proofs of many of the multiplicative
119
relations. Any product involving at least one element in the image of the transfer map v is easily computed using the Frobenius property
xv(y) = r(p(x) y).
Any relation involving an element, like e-m~, obtained by divided some other element by an Euler class may be checked by eliminating the division by the Euler class and checking the resulting relation. The original relation then follows by Lemma A.2.
PROOF OF THEOREM 4.1. We will describe the individual Maekey functors H~S ° of H~S ° by their positions in our standard plot of H~S °. Since
H~(S°)(e) ~Hlal(S°; 7/), it is easy to check that the elements ~1-~ and re_ 1 generate
H~(S°)(e) and satisfy no relations in H~(S°)(e) other than the obvious relation
tl_ ¢ re_ t = p(1). It follows immediately from the structure of the Mackey functors
R_, L, and L_ that the elements r(t~_~), for n > 1, generate the part of tt~(S°)(1) on
the positive horizontal axis. For any positive integer n, p(~,~) = t~_12n. Therefore, ~'~
must generate H2"(;-1)(S%q~ /~ /. The relation r(t~_l) = 2 ~'~ follows from the additive
structure. No other relations involving only ( and t¢-I are permitted by the additive structure. Lemmas A.2 and A.4 ensure that the powers of e generate the part of II~(S°)(1) on the positive vertical axis. These two lemmas also indicate that the
• 0 1 elements e m 4'~, for m, n _> 1, generate the part of HG(S )( ) in the second quadrant.
The same two lemmas indicate that the elements e -m~ and the elements - r n / 2 n + l ~ e r ( t l_ ~ ) generate the parts of H~(S°)(1) on the negative vertical axis and in the
fourth quadrant, respectively. The relations not already verifed follow easily from the additive structure of * 0 I-IGS or from our general observations. The additive structure of H~S ° eliminates the possibility of any unlisted relations involving a single element. Since we have described every possible nonzero product of a pair of generators in terms of the generators, no further relations involving products are possible.
PROOF OF THEOREM 4.9. Again, we describe the individual Mackey functors a 0 * 0 HGS in terms of their positions in our plot of t t G S . Since H~(S° ) ( e )~ Hill(S°; ~),
it is easy to check that the relation ~= ~# = %+# holds for any c~, /3 E RSO(G) with
]od = ]/3] = 0 and that no other relations in H~(S°)(e) hold among the L~. Therefore, for any /3 G RSO(G) with I j31 = 0, L z can be written as a product of the Lc~ included in the proposed list of generators of * 0 t-IGS . The elements LZ, for /3 E RSO(G) with I/3t = o, generate H~(S°)(e) and the elements r(L#), for /3 E RSO(G) with 1/31 = 0 and
[fiG[ > 0, generate the part of H~(S°)(1) on the positive horizontal axis.
Let a and /3 be in RSO0(G ) and let 7 be an element of RSO(G) such that 17t > 0 and 17GI = 0 . The relation pa¢.~ = %+7 follows from Lemma A.11. The relation
120
#o #8 = #~+~ + [(d~d8 - do+/~)/P] r(ta+/~)
follows from ore" characterization in Lemma A.12 of /%+~ as an element, of }t~+5(S°)(1). From this relation, it follows that all of the elements #~ can be constructed from the Iz~ and ~ in our proposed list of generators. By Lemma A.12, the elements #0 and ~ generate all of the u ~ S ° which are plotted at the origin. The
a.~ G
relation #~ e~ = e~+~ indicates that we can construct all the elements e7 from our proposed list of generators. By Lemmas A.2 and A.4, these elements generate all of the o 0 fIGS on the positive vertical axis.
Let ~ E RSO0(G ) and 13,7 • RSO(G) with I~/I =171 = 0 and I/3(31, 1 (31 < 0 . The element ~ro can be obtained from #4 and t~. The relations
P(/J~ ~8) = do to+ 8 = p(do ~ + 8 ) ,
and
follow- from the fact that p is a ring homomorphism. They imply the relations #o ~4 = d ~ + 8 , ~°~8 = 4o+~, and ~p ~ = ~ + ~ since p is a monomorphism in dimensions c~ + fl and fl + 7- These relations indicate that all of the elements ~ can be produced from our proposed list of generators. These elements generate the part of • 0 H(3S on the negative horizontal axis. By Lemmas A.2 and A.4, the elements
T~* S 0 e 6 ~ generate the part of ~G in the second quadrant.
The relations / ~ (e5 i ~ ) = e~ i ~;~+8 and e~ i ~ca = e~ 1*%, for c~ + 7 = fl + ~, may be checked by our general procedure for relations involving division by an Euler class. Together, these relations indicate that our proposed set of generators suffices to construct all of the elements e~ i Ks and therefore to generate the part o f H ~ S ° on the negative vertical axis.
Let fl E RSO0(G) and let cr • RSO(G) with Icrl < 0 and I GI > 0 Recall the class ~o and the virtual representation <c~> from Definitions 4.7. By definition, < ~ + fl> = <c~>, and by the Frobenius relation, ~'<o> r(~o+Z) = 0. Therefore,
# 8 ~'a = ptfl # o - < o > z /<o>
Ptc~+8-<c~>/ ;<0>
b 'o+ ~ .
This relation indicates that our proposed set of generators suffices to produce all of the elements ~'o and therefore the part of H~S ° in the fourth quadrant.
We have now shown that our proposed set of generators does generate I-I~S °. Seven of the relations we have not already established deserve comments . The relation eoep = %+p follows easily from the definition of the guler classes, the Frobenius relation and the product relation for the classes /l~. The relation eO ~ = de_ ~ e~ ~a, for ct + ~ = 7 + ~, follows from the sequence of equations
121
= e-r #~_~ {~
= da_oe~ ~a"
The relations ~ ~, = P ~ + e and ~7 v~ = 0 can be confirmed from the definitions, the Frobenius property, and the relations which have already been established. Given these equations, the relations
e~ (e51 ~) = ~l_~ ~ ,
(G ~ ~)(~ ~ ) -~ = Pe~+v ~;~+~'
and (~71 ~ ) . ~ = 0
follow from our general procedure for checking relations involving classes divided by Euler classes. For the relations eZus = u~+p and ~Zus =d<z>_Z~,~+Z, observe that ~Z can be written as c~¢_<p>~<Z> and that eZ can be written as #vena , for some 7 E RSO0(G ) and some positive integer n. The relations now follow by straightforward computat ions using the definitions, the Frobenius property, and the previously established relations. All of the remaining relations in the theorem follow
* 0 directly from the definitions or the additive structure of H G S . The additive structure of • 0 I-IGS eliminates the possibility of any unlisted relations involving a single element. Since we have described every possible nonzero product of a pair of generators in terms of the generators, no further relations involving products are possible.
122
REFERENCES
[tDP]
[DRE]
JILL]
[LE1]
[LE2]
[LMM]
[LMSM]
[LIN]
[LIU]
[MAT]
[WIR]
T. tom Dieck and T. Petrie, Geometric modules over the Burnside ring. Inventiones Math. 47 (1978), 273-287.
A. Dress, Contributions to the theory of induced representations. Springer Lecture Notes in Mathematics, vol. 342, 1973, 183-240.
S. Illman, Equivariant singular homology and cohomology I. Memoirs Amer. Math. Soc. vol. 156, 1975.
L. G. Lewis, Jr., The equivariant Hurewicz map. Preprint.
L. G. Lewis, Jr. An introduction to Mackey functors (in preparation).
L. G. Lewis, Jr., J. P. May, and J. E. McClure, Ordinary RO(G)-graded cohomology. Bull. Amer. Math. Soc. 4 (1981), 208-212.
L. G. Lewis, Jr., a. P. May, and M. Steinberger (with contributions by a. E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics, vol. 1213, 1986.
H. Lindner, A remark on Mackey functors. Manuscripta Math. 18 (1976), 273-278.
A. Liutevicius, Characters do not lie. Transformation Groups. London Math. Soc. Lecture Notes Series, vol. 26, 1976, 139-146.
T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead. J. Fac. Sci. Univ. Tokyo 18 (1971/72), 363-374.
K. Wirthmiiller, Equivariant homology and duality. Manuscripta Math. 11 (1974), 373-390.
THE EQUIVARIANT DEGREE
by
Wolfgang L~ck
O. Introduction
Abstract. In this paper we study the possible values deg fH,
H c G for a G-map f : M ~ N if M and N are compact smooth G-
manifolds and G a compact Lie group. We generalize results about
maps between spheres of G-representations. We give applications
to one-fixed point actions and G-surgery. We prove that the un-
stable H-homotopy type of the sphere of the H-normal slice
SQ (MH,M)x for x 6 M H is a G-homotopy invariant of M.
Survey. As an illustration we state a consequence of our main re-
sult in a very special situation where it is easy to formulate.
Let G be finite. Consider a compact smooth G-manifold M such that
M H is non-empty, connected and orientable for all H c G. Assume
either that G is nilpotent or that dim M H ~ dim M K -2 holds for
H c K, H, K ~ Iso(M) = {G x I x £ M} Here and elsewhere G denotes " x
the isotropy group {g 6 G I gx = x} of x 6 M. The set of finite G-
sets S with Iso(S) c Iso(M) is an abelian semi-group under disjoint
union. Let A(G,Iso(M)) be its Grothendieck group. The cartesian
product induces the structure of a commutative ring with unit on it.
Let Con(G) be the set of conjugacy classes of subgroups of G and
C(G) be the ring ~ ~. Then A(G,Iso(M)) is a subring of C(G) Con(G)
by identifying S with (card sH I (H) £ Con(G)). For a G-selfmap
f : M ~ M define DEG(f) 6 C(G) by (deg fH I (H) £ Con(G)).
Theorem A.
a) DEG(f) £ A(G,Iso(M) c C(G).
b) If H c G is a p-group then:
deg f ~ deg fH mod p.
t24
c) If G has odd order and deg fH 6 [+1} for each H c G, then we
have for all H c G:
deg f = deg fH. o
This theorem is well known for M as the one-point compactification
V c of a G-representation V. The proof for V c uses the equivariant
Lefschetz index and Smith theory. These methods do not suffice
for M a G-manifold. Our main tool is quasi-transversality and the
notion of a local degree.
The notion of the degree is used to classify G-hcmotopy classes
of G-maps f : V c ~ W c (see tom Dieck [6'], p. 213, Laitinen [14],
Tornehave [21]) and G-homotopy types of G-homotopy representations
(see tom Dieck-Petrie [8]). It plays also a role in equivariant
surgery theory (see for example Dovermann-Petrie [11], LHck-
Madsen [17]). We give a survey over the various sections.
In section one we define the fibre transport tPM of the tangent
bundle of a G-manifold and the notion of an O(G)-transformation
: f tp N ~ tPM for a G-map f : M ~ N. Roughly speaking,
assigns to each point x in M a G-map (not necessarily a G -homo- x
topy equivalence) TN~x ~ TM~ such that certain compatibility
conditions hold. Using ~ we get a one-to-one-correspondence between
local orientations of M H at x and N H at fx for each H c G and
x 6 M H. This enables us to define the equivariant degree
DEG(f,~) 6 C(N) = ~ ~ ~ in section two. (H) E Con(G) , (N H)/WH
o
In section three the Burnsidering A(G,~ ) of a compact Lie group
with respect to a family ~ is treated. We identify [vC,vC] G and
125
A(G,Iso(V)) for a G-representation V. We introduce in section
four a multiplicative submonoid Endtp N c C(N) and prove
DEG(f,~0) 6 Endtp N for any f and ~0 in section five. We will see
that Endtp N does not involve f and ~ but depends only on the compo-
nent structure of N. The main idea of the proof is best explained
in the special case where G is finite and all N H are non-empty
and connected. Then C(N) = C(G) = ~ ZZ and Endtp N is A(G,Iso(N)). Con (G)
Choose y in N G and make f quasi-transverse to y. Then f-1 (y) is
finite and for each x 6 f-1 (y) f looks like a (not necessarily
linear) norm-preserving Gx-ma p TxM ~ TyN in a Gx-neighbourhood
of x. Consider a G-orbit c of f-1 (y). For each x in c we obtain
Gx-maps TM c ~ TNy by f and TNy ~ TM c by ~. Their composition
TNy ~ TNy defines an element in A(Gx,ISo(TNy)). Its image in
A(G,Iso(N)) under the induction homomorphism for G x c G is inde-
pendent of the choice of x and denoted by d(c). Let d be the sum
Zd(c) running over c 6 f-1 (y)/G. Since the global degree can be
computed by local degrees, DEG(f,~0) 6 C(G) is just d 6 A(G,Iso(N)).
Roughly speaking, we have counted the local degrees orbitwise in
the Burnside ring to get the global degree.
Section six contains some examples to illustrate our results. We
give an elementary proof of the following known statement (see
Atiyah-Bott [I], Browder [4], Ewing-Stong [12]).
Corollary B. There is no closed G-manifold M with dim M ~ I such
that each M H is connected and orientable and M G a single point
if G is the product of a p-group and a torus, m
126
It is of special interest to choose ~ : f tPN ~ tPM as an
O(G)-equivalence i. e. all TN~x ~ TM~ are Gx-homotopy equi-
valences. Then another choice of ~ would change the equivariant
degree only by a unit. Moreover, we have:
Theorem C. A normal G-ma~ f : M ~ N can be change d into a G-
homotopy equivalence by equivariant surgery only if there is an
O(G)-equivalence ~ : tPN ~ f tPM with DEG(f,~) = I. o
The existence of an O(G)-equivalence ~ is related to the notion
of the equivariant first Stiefel Whitney class w M of a G-mani-
fold. In section seven we relate tPM und w M and show that the
existence of an O(G)-equivalence ~ : f tPN ~ tPM is equivalent
to f w N = w M. We prove:
Theorem D. I_ff f : M ~ N is a G-homotopy equivalence w~e have
f w N = w M. o
This implies the unstable version of the stable result in Kawakubo
[13].
Corollary E. I_~f f : M ~ N is a G-homotopy equivalence, w_~e @et
for x 6 M:
TMC ~G TN~x " [] x
Our setting and proofs would be much simpler if we supposed that
all fixed point sets are non-empty, connected and orientable. Un-
fortunately, such conditions are unrealistic in the study of G-
manifolds. Hence we make no assumptions about the existence of
127
G-fixed points or about the connectivity or orientability of the
fixed p o i n t s e t s a n d do n o t d e m a n d ~ ( fH) b e i n g b i j e c t i v e . o
Our notion of the equivariant degree using O(G)-transformations
has some advantages compared with the one using fundamental
classes. It is in this generality much easier to state elemen-
tary properties like bordism invariance or the computation by
local degrees in our language. We have the global choice of
instead of the various choices of fundamental classes [M H] and
[NH]. Notice that the choice of [M H] is independent of the one
of [M K] for (K) • (H) and [NK]. Hence in the case of fundamental
classes the interaction between the various fixed point sets are
not taken into account, what is done in our setting. It seems to
be difficult, or even impossible, to state some of our results
by means of fundamental classes. For example, the statement of
example 6.5 makes no sense if it is formulated with fundamental
classes and in example 6.3 there must appear signs because we
can substitute [M H] by -[M H] and thus change the corresponding
degree by a sign. The advantages of our approach for the notion
of an equivariant normal map is worked out in L~ck-Madsen [17].
(see also theorem C above and example 2.8).
Conventions: We denote by G a compact Lie group unless it ex-
plicitly is stated differently. Subgroups are assumed to be
closed. A G-representation is always real, A G-manifold M is a
compact smooth G-manifold with smooth G-action and possibly
non-empty boundary. We call a component C of M H an isotropy
component if there is a x in C with isotropy group G x = H. We
say that M fullfills condition (~) if it satisfies the conditions
128
i) and ii) or the conditions i) and iii) below.
i) C # {point} for all C 6 ~ (MH), H c G. o
i i ) I f C £ ~ (M H) i s an i s o t r o p y c o m p o n e n t , C >H i s o
H {x £ C 1G x ¢ H} a n d H c G we h a v e d i m C > H + 2 ~ d i m C
iii) G is finite and nilpotent.
A G-map f : M ~ N respects always the boundary and we assume
6 Mo(MH) , D £ Mo(NH), H c G with fH(c) c D. dim C dim D for all C
Acknowledgement. The author wishes to thank the topologists at
Arhus for their hospitality and support during 1985 - 1986 when
the main part of this paper was written. The author is indebted
to Ib Madsen and Erkki Laitinen for their useful comments.
I. The fibr e transport.
We organize the book-keeping of the components of the varbus fixed
point sets and their fundamental groups for a G-space as follows.
We recall that an object of the fundamental groupoid ~ (Y) of a
space Y is a point in y and a morphism Yo ~ Yl is a homotopy
class of paths from Yl to Yo" The orbit category O(G) has the ho-
mogenous spaces G/H as objects and G-maps as morphisms.
Definition 1.1. The fundamental O(G)-groupoid uGx of a G-space X
is the contravariant functor uGx : O(G) ~ {groupoids} sending
G/H to U(X H) = H(map(G/H,x)G).
In general an O(G)-category resp. O(G)-groupoid is a contravariant
129
functor from O(G) into the category of small categories resp.
groupoids. We recall that a groupoid is a category whose mor-
phisms are all isomorphisms. An O(G)-functor F : C ~ D between
O(G)-categories is a natural transformation. Let I be the cate-
gory of two objects O and I and three morphisms ID : O ~ O,
ID : 1 ~ I and u : O ~ I. We define an O(G)-transformation
: F ° ~ F I between O(G)-functors F ° and F I : C ~ D as an
O(G)-functor ~ : C x I ~ D with C I i = F i. Given a second O(G)-
transformation 4: F 1 ~ F2, let the composition ~ ~ ~ : F o ~ F 2
be determined by ~ ~(id,u) = ~(id,u) o ~(id,u) : (x,O) ~ (x,1)
for all x 6 C. One should think of an O(G)-functor F : C ~ D as
a collection of functors F(G/H) : C(G/H) ~ D(G/H) and of an
O(G)-transformation ~ : F o ~ F I as a collection of natural trans-
formations ~(G/H) : Fo(G/H) ~ FI(G/H) fitting nicely together.
An O(G)-transformation ~ : F ° ~ F 1 is called an O(G)-equivalence
if there is an O(G)-transformation ~ : F I ~ F ° with both compo-
sitions the identity.
A G-map f : X ~ Y induces an O(G)-functor HGf : HGx ~ Gy
whereas a G-homotopy h : X x I ~ Y between f and g determines an
O (G) -equivalence HGf ~ HGg.
G n A -S -Hurewicz-fibration q%X is called locally linear if there
exists a G -neiqhbourhood U for each x in X such that U is G - x x x x
× SV for some G -representation fibre homotopy equivalent to U x x x
• ~ n . V x We call a locally linear J-S -Hurewlcz-fibration briefly a
a G-Sn-fibration. An example is the fibrewise one-point comDacti-
fication ~c of a G-~n-bundle ~. Denote by bfG,n(X) the category
130
of G-Sn-fibrations over X with G-fibre homotopy classes of
fibrewise G-maps as morphisms. We obtain an O(G)-category bfG, n
by letting X vary over all homogenous spaces. One should notice
that bfG,n(G/H) is equivalent to the category with spheres of H-
representations and H-homotopy classes of H-maps as morphisms.
We prefer bfG,n(G/H) because of its better transformation be-
haviour in view of O(G).
The fibre transport of a G-Sn-fibration q%X defines an O(G)-
functor tpq : uGx ~ bfG, n analogously to the non-equivariant
case (see [19], p. 343). The functor tp(G/H) : U(X H) ~ bfG,n(G/H)
sends a point in X H given by x : G/H ~ X to x n. Let
h : G/H x I ~ X be a G-homotopy from y to x representing a mor-
phism x ~ y. Choose a solution ~ of the G-homotopy lifting
problem
x n × l ...... % n
/
~ /
x q×I ~ X h e (p x id)
Define x ~ ~ y ~ by the pull-back property and ho"
Definition 1.2. We call tpq : ~GM ~ bfG, n the fibre transport
o_ff q%X. The fibre transport tPM of a G-manifold M i_ss tPTMC.
2. The equivariant degree .
we consider a G-map f : M - N between G-manifolds an an O(G)-
131
transformation ~ : f tPN ~ tPM with f tPN := tPN~ Gf. We want
to define its e q u i v a r i a n t d e g r e e DEG(f ,~) l y i n g i n a c e r t a i n r i n g
C(N).
We consider the case G = I and both M and N connected first. Re-
call that we always assume dim M = dim N. Suppose that ~(x) :
C
TM~ is not nullhomotopic for one (and hence all) x 6 M. TNfx
Otherwise define DEG(f,~) 6 ~ to be zero. Let u be any loop in M
at x. By functoriality of ~ we get ~(x)~ tPN(f~ u) ~ tPM(U) o ~(x).
Since the first Stiefel-Whitney class w1(M) 6 HI(M,~/2) = HOM(~I(M) ~/2)
sends u to deg tPM(U) we have f w1(N) = Wl(M). Let p : ~ ~ M be
the orientation covering if w1(M) is non-trivial and the identity
= M ~ M otherwise and define p : ~ ~ N analogously. Then
and N are orientable connected manifolds and we can choose a lift
: M ~ N. If f(M) c ~N let DEG(f,~) be zero. Otherwise choose a
^ ^ ^c point x 6 ~ ~M with fx 6 ~ ~N. Write x = px. Let c : ~ TM x
^ ^
and c : N ~ TN~x be the collaps maps uniquely determined up to
homotopy by the property that the differentials at x and fx are the
identity. Let d be the degree of the following endomorphism of ~.
ZZ = Hn(M,~M) ...... ~ Hn(N,~N )
ic I c H n (TM c) H n (TN~f~)
i i i~ (Tp~), (Tp~:~),
H n(TMxc) < H n(TNfx) ~(x).
A straightforward calculation shows that d is independent of the
132
choices of and x. Now define DEG(f,~) as 2d if w1(M) = 0 and
w1(N) • O, and as d otherwise. The factor 2 in the case w1(M) = O
and Wl(N) • O is due to the fact that then M is only one of the
two components of the pullback of the orientation covering of N.
The global degree has an easy description by local degrees. Let y
be a point in N~ ~N. Assume that f-1(y) is finite and f looks in
a neighbourhood of x like a proper map k(x) : (TMx,O) ~ (TNy,O)
with k(x)-1(O) = O if we identify the tangent space with neigh-
bourhoods by an exponential map. Then:
Proposition 2. I
DEG (f,~) = Z_I x£f
deg(k(x)Co ~0(x)C : TN c ~ TNy) (y) Y
Proof. Use [9], p. 267. Q
As an illustration consider the example of a n-fold covering
p : M ~ N between connected manifolds. Its differential induces
an O(1)-transformation Up : p tp N ~ tPM. By proposition 2.1
s2m ~p2m DEG(p,~p) is n. This applies in narticular to p : ~ .
Notice that S 2m is orientable but ~p2m not.
Now we treat the general case. Let Con(G) be the set of conjugacy
classes of subgroups of G. The set of isomorphism classes x of
objects x in a category C is denoted by C. Given an O(G)-groupoid
L, ~ ' we write CON(%) for ~ ~ i G/H)/WH and C(~ for the (H) 6 Con(G) ~r
ring of functions CON(k) ~ ~. Let CON(X) and C(X) be CON(uGx)
and c(uGx) for a G-space X.
133
We will define DEG(f,~) in C(N) by specifying integers
DEG(f,~) (D,H) for all H c G and D c ~o(NH). Let C1,...,C r be the
components of M H with fH(c i) c D and fl : Ci • . ~ D be the map in-
duced by fH. Because of (TM r MH) H = T(M H) we obtain from ~ non-
equivariant transformations ~i : fitPD ~ tPc i by restriction
and taking the H-fixed point sets. We have introduced DEG(fi,~ i)
above. Define:
r DEG(f,<0) (D,H) = E DEG(fi,~ i) .
i=I
The sum shall be zero for r = O.
Definition 2.2. We call DEG(f,~) inn C(N) the equivariant degree
of f with respect to ~. o
Finally we state the ~tarv properties. Consider a G-map of
triads (F,f,f+) : (P,M,M+) ~ (0,N,N+) and O(G)-transformations
: f tPN ~ tPM and ~ : F tpQ ~ tpp. Identifying TP x = TMx~
using the inward normal we get tpp ! M = tP(TM ~)c and analogously
tpQ~N= tp(TN ~)c. We assume that e ! N and ~ fit together under
these identifications
Let j : C(Q) ~ C(N) be the ring homomorphism
given by composition with the obvious map CON(N) ~ CON(Q). Then
the equivariant degree turns out to be a bordism invariant.
2.3 DEG(f,~) = j DEG(F,~)
The equivariant degree is a homotopy invariant in the following
sense. Given a G-homotopy h : M x I ~ N between f and g we get
134
an O(G)-equivalence ~h : g tPN
Then:
f tPN by the fibre transport.
2.4 DEG(f,~) = DEG(g,~ ~h )
Consider G-maps f : L ~ M and g : M ~ N and O(G)-equivalences
: f tPM ~ tPL and ~ : g tPN ~ tPM. Provided that
no(gH) (M H) ~ ~ (N H) is bijective for all H c G, we obtain : no 0
the qomposition formula:
* *) -I 2.5 DEG(g~ f,~0z f ~) = DEG(g,~) • (g (DEG(f,~))
The following examples illustrate our definitions.
Example 2.6. Let f : V c ~ W c be a G-map for two G-representations
V and W with dim V G, dim W G _> 1. Any G-map # : W c ~ V c can be
interpreted as an O(G)-transformation ~ : f tPvC ~ tPwC using
the facts that TvC~ (V c x~) = V c x (V~ ~) holds and the suspension
[vC,wC] G ~ [ (V~ ~)c, (W~ ~)c]G is bijective. Then DEG(f,~) lies
in C(W c) = C(G) and DEG(f,~) (H) is just deg(~H¢ fH) for (H) 6 Con(G).
Example 2.7. Let M be a G-manifold such that the components of M H
are orientable for all H c G. If f : M ~ M is a G-map with
no(fH) : ~o(M H) ~ ~O(M H) the identity for all H c G we can define
its degree DEG(f) £ C(M) by the collection (deg(fHIc : C ~ C)
C 6 no(MH) , H c G). The orientability condition ensures that we
get a well-defined O(G)-equiva!ence ~ : f tPM ~ tPM uniquely
determined by the property that
• (G/H) (X)eH : tPM(G/H) (fX)e H ~ tPM(G/H) (X)eH is given by the fibre
135
transport of the H-bundle TMIM H along any path in M H from x to
fx. One easily checks DEG(f) = DEG(f,~). o
The following non-equivariant example indicates the advantage
of our notion of the degree with the one usinq fundamental
classes for surgery.
Example 2.8. Let M and N be closed orientable connected manifolds
with fundamental classes [M] and [N] and let M- be M with -[M].
Consider a normal map f:M , N, f: TM @ IR k , ~ of degree one
taken with respect to the fundamental classes. If M O is M + M- + M
disjoint union gives a normal map of degree one g = f + f + f :
M ..... ) N, g = f + f + f. The reader should figure out by himself that o
it is impossible with these bundle data and orientations to convert
f by surgery into a normal map f+ : M+ ~ N of degree one with
connected M+. We can see this using our degree as follows. Fix an
0(I) - equivalence ~ : tp~ -- tPT N @ IR k . Let ~g : g tPN ~ tPM
be the 0(I) - equivalence uniquely determined by the property that
its suspension is (~ o tp~) -1 : f tPTN @ IR k ~ tPTM @ IR k.
Since "normally bordant" includes the bundle data, DEG (g,~) is
a normal bordism invariant. But DEG (g ~) is ± 1 by Proposition 3.1.
136
3. The Burnside ring of a compact Lie group.
The Burnside ring of a compact Lie group G was introduced and
examined by tom Dieck [5] and [6], p. 103 ff. Since we need
some modifications of this material and want to keep the paper
self-contained we make some remarks about it in this section.
A prefamily ~ is a subset of g(G) = {H!H c G} closed under
conjugation. We call ~ a family if it is also closed under
intersection and finite if {(H) 6 Con(G) I H 6 ~ } is finite.
The set of isotropy groups Iso(X) = {Gx r x 6 X) of a finite
G-CW-complex X is a finite prefamily. If X is a G-manifold
with connected fixed point sets, Iso(X) is a finite family for
finite G, but not in general. A counterexample is the sphere
in the SO(3)-representation ~3 ~3 if SO(3) acts in the ob-
vious way on both summands. If ~ is a prefamily and × denotes
the~uler characteristic let A(G,~ ) be the set of equivalence
classes of finite G-CW-complexes X with Iso(X) c ~ under the
equivalence relation X~Y ~ X(X H) = x(Y H) for all H c G. The
disjoint union defines an abelian group structure. Moreover,
the cartesian product induces the structure of an associative
commutative ring with unit if ~ is a family containing G. We
can identify A(G) := A(G,S(G)) with the Burnside ring in [~ p. 103.
Let C(G,~ ) be the ring of functions {(H) 6 Con(G) i H £ ~ } ~
and C(G) = C(G,S(G)). For each K c G we obtain a ring homomorphism
ch K : A(G,~) ) ZZ IX] ; ~ X (XH) •
Since WH acts freely on G/H K and WH contains a circle for infinite
I37
WH we get ChK(G/H) = O for all K if WH is infinite. For any pre-
family ~ let • f be {H 6 ~ I WH finite}. Using the ideas in [6]
p. 3, 4, 104, 119 one proves that ch isgiven by the product of
the ChK-S:
Proposition 3.1. Let~ be a finite prefamily. Then {[G/H]IH 6 ~ f}
is a ~-base of A(G,~ ). The homomorphism
ch : A(G,~ ) * C(G,~ f)
i_~s injective with a finite cokernel of order U {(H) I H 6~ f}
Moreover, each ch(G/H) is divisible by 'WHI and
{ I ch(G/H) I H 6 ~ f} is a ~-base for C(G,~f) m TW~7
I WH!,
Now we introduce the equivariant Lefschetz index fo!lowinq [14],
chapter I to produce a bijection [vC,vC] G * A(G,Izo(V)) for
an appropriate G-representation V.
Consider a G-self map f : X ~ X of a finite G-CW-complex X. Let
L(fH,f>H) be the Lefschetz index of the self map (fH,f>H) of the
pair of CW-complexes (xH,x>H).
Definition 3.2. The equivariant Lefschetz index LG(f) i__nn A(G,Iso(X))
is defined as
I f>H) LG(f) = Z I~--~-T" L(fH' • [G/HI {(H) IH £ Iso(X)~} ~
Since (xH,x >H) is WH-free, L(fH,f >H) is divisible by IWHI.
Proposition 1.8 in [14] extends to compact Lie groups~
1 3 8
Lemma 3.3. ChK(LG(f)) = L(f K) for K c G.
Proof. Since the Lefschetz index is additive ([9], p. 213) one
can reduce the problem by induction over the orbit bundles and
dimensions to the case X = I±G/H x Dn~±G/H x S n-1 where one has r r
to show with * the obvious base-point:
L(fK,,) =
I IWHI
O
• L(fH,,) • x(G/H K) if WH is finite
otherwise
The second case follows from the fact that WH acts freely relative
* on X and X K and contains a circle. The canonical inclusions and
projections of the wedge X yield a pair of inverse isomorphisms
between H.(X,*) and ~ H.((G/H x sn)/(G/H x .),.) where • denotes r
the various base points. Now an easy homological computation re-
duces the proof of the first case to X = (G/H x sn)/(G/H x .) with
WH finite. Then fH is a self-map of (WH x sn)/(WH x .). The KHnneth
formula and the obvious map G/H x (WH x sn)/(WH x .) ~ X
induce a chain homotopy equivalence such that the following dia-
gram commutes up to homotopy
C(G/HK) ® ~WH C(WH x sn/WH x .,.) } c(xK,,)
I id ®~WH c(fH'*) I C(fK'*)
C(G/HK) ® ~WH C(WH x sn/WH x .,.) ) c(xK,.)
Notice that C(WM x sn/wH x .,.) is concentrated in dimension n and
is ZWH there. Let Za w • w 6 ~WH be the element determined by c(fH,.).
139
Then L ( f H , , ) i s IWHI-a 1 and L ( f K , , ) i s x(G/H K) - a 1 s i n c e C(G/H K)
is ZWH-free. This finishes the proof, m
A G-homotopy representation X of G is a finite-dimensional G-com-
plex of finite orbit type such that for each subgroup H of G the
fixed point set X H is an n(H)-dimensional CW-complex homotopy equi-
valent to S n(H). If dim X G _> I and Iso(X) is a family, we equip
[X,X] G and A(G,Iso(X)) with the monoid structure given by compo-
sition and multiplication. If I denotes [G/G} and xG(X) : = LG(idx )
we have the unit xG(x) -I in A(G,Iso(X)) and maps
I : IX,X} G A(G,Iso(X)) If} ~ (LG(f) - I) ( G × (X) -I )
DEG : [X,x]G ----~ C(G) [f] ....... ~ [deg fH I (H) 6 Con(G)}
The main result of this section is:
Theorem 3.4. Let X be a G-homotoDy representation pith dim X G ~ I
satisfyinq ~ondition (.) defined in the introduction.
a) L G- I : IX,X} G ~ A(G,Iso(X)) i~. bijective.
b) If_f_Iso(X) is a f~amily the monoid ma_~ I : [X,X] G ~ A(G,Iso(X))
is bijective an__d ch ~ I = DEG.
Theorem 3.4 follows from proposition 3.1, lemma 3.3 and the equi-
variant Hopf theorem 3.5 below. For its proof and further expla-
nations we refer to [6] p. 213, [7] II.4. , [14], [18] and [21].
Theorem 3.5. Le___t x an___d Y be G-homotopy representations with
dim X H = dim yH for all H c G sa_ t_isfyinq condition (*). C_h_cose
140
fundamental classes for X H an___d yH such that deg fH for a G-map
f : X ~ Y is defined.
Then [X,Y] G is non-empty. Elements [f] are determined b_16 the set
{deg fH I H 6 Iso(Y)f}. The degree deg fH is modulo IWHI determined
b~ th__@e deg fK, K m H, and fixing these degrees deg fK the__ possible
deg fH fill the whole residue class mod IWHI.
We end with some remarks about induction and restriction for an
inclusion j : H ---9 G of compact Lie groups.
Let ~ be a prefamily for H. Then j,~ = {g-lj(K)g ! g 6 G,K 6~ }
is a prefamily for G. We want to define an abelian group homomor-
phism
indj : A(H,~ ) ~ A(G,j,~ )
by sending [X] to [G xj X]. The following formula and proposition
3.1 show that this is well-defined.
-I 3.6 x((G×jX) K) = E x(X gKg
gH £ G/H K ) for K c G, WK finite.
Notice that G/H K has only finitely many WK-orbits ([2], p. 87)
and is therefore finite if WK is finite.
Given a prefamily ~ for G, we have the prefamily
j*~ = {j-I(K) I K 6 $ } for H. We obtain an abelian group homo-
morphism
141
resj : A(G,~) ~ A(H,j ~- )
by restriction: [X] ~ [resj X]. If ~ is a family containing
H and G then j~ is a family with H 6 j% and res. is a ring 3
homomorphism.
4. The monoid of endomorphisms of the fibre transport.
If we want to examine the dependency of DEG(f,~) on ~ we have
to compute in view of the composition formula 2.5 the O(G)-trans-
formations ~ : tPN ~ tPN and the possible values DEG(ID,~) in
C(N).
More generally we consider the monoid End(tp) of O(G)-transfor-
mations ~ : tp ~ tp of any O(G)-functor tp : ~ ~ bfG, n. The
g r o u p o f i n v e r t i b l e e l e m e n t s E n d ( t p ) c o n s i s t s o f t h e O ( G ) - e q u i -
valences ~ : tp ~ tp.
Consider C(~ ) as monoid by its multiplicative structure. The
monoid map
DEG : End(tp) ~ C(~)
maps ~ to DEG(~) specified by the following function CON(~ ) ~ ~.
For H c G and x in ~ (G/H) we get a G-fibre map ~(G/H) (x). Let
DEG(~) (x,H) be the degree of the induced self map on the H-fixed
H of the fibre over eH. Recall that tp(G/H)(X)eH point set tp(G/H) (X)e H
is H-homotopic to SV for some H-representation V. We want to show
that DEG : End(tp) ~ C(~ ) is an embedding of monoids and describe
its image.
142
We say that an O(G)-transformaticn tv : ~ ~ bfG, n satisfies
condition (~) if for any H c G an~ x 6 ~ (G/H) tp(G/H) (X)eM does
and has an H-fixed point. If furthermore Iso(tp(G/H) (X)eH) is a
family we call tp admissible. Consider a G-manifold N satisfying
condition (~). Then tPN satisfies condition (~) and is even ad-
missible if G is finite. If G is finite nilpotent and N a G-mani-
fold such that no component of N H is a point for H c G then tPN
is admissible.
We recall the noticn of the homoto~y colimit F(~ ) (see [20]
p. 1625). Objects are pairs (x,H) with x 6 ~ (G/H) and H c G. A
morphism (o,u) : (x,H) ~ (y,K) consists of a G-map
: G/H ~ G/K and a morphism u : x ~ ~ y with
= ~ (~) : ~ (G/K) ~ ~(G/H). Composition is defined by the
"semi-direct product formula" (~,v) ~ (o,u) = (To ~,~v~ u). Notice
that ~ is Con(~ ) (see section 2). The fundamental group cate-
gory of a G-space X appearing in [7] p. 57 and [15] is F(~Gx). We
now introduce contravariant functors Atp, C~ and Etp and relate
their inverse limits to End(tp) and C(~). The contravariant func-
tor into the category of monoids
Etp : F(9) ~ MONO
]H maps (x,H) to [tp(G/H)(X)eH, tp(G/H) (X)eH . Given a morDhism
(~,u) : (x,H) ~ (y,K) choose g in G with ~(eH) = gK so that we
obtain a group homomorphism c(g) : H ~ K h ~ g-lhg. If
l(g -1) is multiplication with g-1 we qet a H-homotopy equivalence
a : tp(G/H) (X)eH ~ reSc(g)tp(G/K) (Y)eK by l(g -I) ~ tp(G/H) (U)eH.
143
Define Etp(~,u) : [tp(G/K) (Y)eK , tp(G/K) (Y)eK ]K ~ [tp(G/~(X)eH,
tp(G/H) (X)eH ]Hby restriction with c(q) and conjugation with a.
This is well defined since conjugation within H-self-equivalence
induces the identity on [X,X] H for a G-homotopy representation X
(theorem 3.4).
The contravariant functors
Atp : F (~) ~ MONO
send (x,H) to A(H,Iso(tp(G/H) (X)eH)) and C(H). Given a morphism
(o,u) : (x,H) ~ (y,K) let g 6 G and c(g) : H ~ K be as above.
Define Atp(O,u) and C~(o,u) as the restriction with c(g).
Let the transformation
D : Etp ~ C~
A : Etp ~ Atp
CH : Atp ~ C
be induced by the degree and the maps of section three
i : [tp(G/H) (X)eH,tp(G/H) (X)eH ]H ~ A(H,Iso(tp(G/H) (X)eH))
ch : A(H,Iso(tp(G/H) (X)eH)) ~ C(H)
The inverse limit of a contravariant functor F : C ~ MONO is
the submonoid inv F of N F(x) consisting of those elements x6C
144
(a x J x 6 C) such that F(f) (ax) = ay
f : y ~ x.
holds for any morphism
We define a monoid map
: inv lim C~ ~ C(~ ) = U
as follows. Let pr H : C(H) ~ ZZ be the projection onto the
factor belonging to (H) £ Con(H). An element in the inverse
limit given by {u(x,H) £ C(H) ! (x,H) 6 F(~)}~ is sent to
{PrH(u(x,H)) 6 ~ ! ~-~,H) 6 F(--~-~}.
Let a(x,H) : End(tp) ~ Etp(X,H) be the monoid map sending
to ~(G/H) (X)eH. We obtain a homomorphism of monoids
: End(tp) ~ inv lim Etp
Theorem 4.1.
a) I_ff tp : ~ ~ bfG, n fullfills condition (~), the following
diagram of monoids commutes. All maps are injective and
is bijective.
End(tp) > inv lim Etp
DEG inv lim D
C(~ ( ( inv lim C~ S
145
b) I_~f tp is admissible the following dia@ram of monoids commutes.
All maps ar___ee injective and inv lim A i_~s bi~ective.
inv lim A inv lim inv lim h / Etp ~ ~/i Atp
nv lim D tjj j~ inv lim CH
inv lim C~
Proof. Everything follows directly from theorem 3.4 and the de-
finitions.
Let • : MONO ~ GROUPS be the functor "invertible elements"
Since the inverse limit is compatible with • and End(tp) is
the group Aut(tp) of O(G)-equivalences tp ~ tp we conclude:
Corollary 4.2. For admissible tp the followin~ diagram of abelian
groups commutes. The maps a and inv lim A are bijective the
others injective.
146
Aut(tp) . ~ inv lim Etp
II inv lim i
DEG inv lim Atp v I
I inv lim CH
C(~ ) < < inv lim C
Corollary 4.3. Let N be a connected G-manifold satisfying con-
dition (w
a) If N H is connected and non-empty for all H c G and Iso(N) a
family then:
End(tPN) = A(G,Iso(N)) c C(N) = C(G)
b) Let G be a torus. Assume that any component of N H contains a
G-fixed point for H c G. Then we have for y 6 N the bijection
End(tPN) ~ 2Z ~ ~ deg(~(G/1) (y))
c) I_~f G is finite of odd order we get for y E N a__nn isomorphism.
Aut (tPN) ~ {-+1} <0 ~ deg (~0 (G/I) (y))
Proof:
a) If x is a G-fixed point, we have for any object (y,H) in F(~Gx)
a morphism (y,H) ~ (x,G). Two such morphisms define the same
147
map EtPN(X,G) ~ EtPN(Y,H) and inv lim EtPN(X,G) is End(tPN).
Hence End(tPN) = EtPN(X,G) = A(G,Iso(N)).
b) If X is a G-homotopy representation of the torus G with
dim X G ~ I ther~ [X,X] G ~ ~ [f] ~ deg f is bijective by
proposition 3.1 and theorem 3.4.
c) ch I = A(G) ~ {±1} is a isomorphism by [6] p. 8 if G has odd
order.
If N is a G-manifold and tPN is admissible, End(tPN) ~ inv lim Atp
depends only on the c~mponent structure of N and the sets Iso(TNx) f
for all x 6 N which can be read off from the dimension function.
5. The degree relations.
In this section we state the central result of this paper. In the
following we identify End(tPN) with its image in C(N) under the
embedding DEG.
Theorem 5.1. Let f : M ~ N be a G-map of n-dimensional G-mani-
folds satisfying condition (,) and ~ : f tPN ~ tPM be an O(G)-
transformation. Then:
DEG(f,~) 6 End(tPN) c C(N) o
The rest of this section deals with its proof. Examples to
illustrate its meaning are given in the next section. The most
important ingredients are the concept of quasi-transversality which
we will extend to compact Lie groups (see [10], chapter 3 for
finite G) and local degrees.
148
We call the G-map f : M ~ N of G-manifolds quasi-transverse to
y in N if the fo!lowinq is true.
i) The preimage f-1(y) consists of finitely many orbits Gy/H
with all W G H finite. Y
ii) Equip the G-normal bundle ~ (f-I(G/Gy),M) and ~(G/Gy,N)
with equivariant metrics. There is a norm preserving G-
fibre map
k (f-1 (G/Gy) ,M) .......... ) (G/Gy,N)
f-1 (G/Gy) ) G/Gy
such that f looks like k in a tubular neighbourhood.
Lemma 5.2. We can change f u_~ t_2o G-homotopy such that f i_~s quasi-
transverse to y. (see also [10, ch. 3] ).
Proof. Let KI,K2,...,K r be a complete system of representatives
of conjugacy classes (K) of subgroups of G with K occuring as
isotropy group in M and K c G . We construct inductively an open Y
G-set U i containing MKI,...,MKi such that i) and ii) hold if one
substitutes f-1(y) and f-1(G/Gy) by their intersections with U i -
We can assume (Ki) c (Kj) ~ i a 9.
The induction begin i = O is trivial: U = ~. In the induction step
from i -1 to i write U = Ui_1, K = K i. By possibly shrinking U we
can suppose the existence of a closed G-set V with int(V) ~ clos(U)
149
and f-1(G/Gy) n V~U = @. Let M O be M Kx (U N MK). By induction
hypothesis WK acts freely on M . If f is fKIM consider the non- o o o
equivariant map
(fox id)/WK : Mo/WK ~ (N K x Mo)/WK
We can change it homotopically relative V N Mo/WK into fl such
that fl is transverse to (G/Gy x Mo)/WK. By a cofibration argument
we can assume (fox id)/WK = f1" Now G/G K is a finite disjoint r Y
union of WK-orbits ±± WK-(giGy) (see [2], p. 87). One easily i=I
checks dim(fo/WK) -I (WK.giGy) = dim(f ° x id/WK) -1(WK.giGy x Mo/WK ) =
-dim NK N Ggiy/K. Hence (fo/WK) -I (WK.giGy) consists of finitely
many points if NK N giGy/K is finite and is empty otherwise. In
other words f-1 (y) n GM O consists of finitely many orbits Gy/Kj
such that WG/j = NKj N Gy/Kj is finite. We can treat any such orbit
separately.
Consider any x in GM o with f(x) = y so that f maps G/G x ~ G/Gy
by the projection. We identify ~(G/Gx,M) with a tubular G-neigh-
bourhood of G/G x and analogously for y. we have
dim ~(G/Gx,M)L ~ dim ~(G/Gy,N)$ for all L c G so that we can x G x
extend any non-equivariant map SQ(G/Gx,M)x x ~ S~ (G/Gy,N)~x
to a Gx-ma p 5~ (G/Gx,M)x ~ $~(G/Gy,N)y. Since ~(G/Gx,M) =
G x s ~(G/Gx,M) x, ~(S/Gy,N) = S x G ~(S/Gy,N)y and (K) = (G x) x y
holds we can construct a norm preserving fibre map
k (G/Gx, M) > ~k (G/Gy,N)
G/G x 3 G/Cy
150
such that the restriction of k to the K-fixed point set agrees
with fK By a cofibration argument we can change f in a small
G-neighbourhood of G/G x relative to M K such that k coincides
with f on all ~ (G/Gx,M). Now one easily enlarges U to the de-
sired U i. This finishes the proof of lemma 5.2. D
Proof of theorem 5.1. We have to construct ~ 6 End(tPN) such that
DEG : End(tPN) ~ c(uGN) sends A to DEG(f,~). Let
e(y,H) £ CHGN(Y,H) := C(H) for (y,H) 6 F(NGN) be defined by
Con(H) ~ Z (K) ~ DEG(f,m) (y,K). One checks directly that
{~(y,H) i (y,H) £ F(HGN)} determines an element in inv lim CuG N
mapped by B : inv lim CNG N ~ c(HGN) to DEG(f,~). Suppose that
we can construct for each (y,H) 6 F(NGN) a H-self-map 6(y,H) of
tPN(G/H) (Y)eH = TN~ such that D(y,H) : EtpN(y,H) ~ C GN(Y,H)
sends 6(y,H) to ~(y,H). Then {~(y,H) ! (y,~)6 F~GN)} defines an el~ent 4'
in inv lim Etp N. By theorem 4.1 there is A £ End(tPN) such that
: End(tPN) ...... , inv lim Etp N maps A to 4', and A has the de-
sired property.
Let ch' : A(H,Iso(TNy)) ~ C(H) be the composition of the in-
clusion A(H,Iso(TNy)) ~ A(H), multiplication with the unit
xH(TN~)-I : A(H) ~ A(H) and ch : A(H) ~ C(H). The map
LH-I : [TNC,TNC] H y y ~ A(H,Iso(TNy)) is a bijection (theorem 3.4.)
and ch' o (LH-I) = DEG. Hence theorem 5.1. is true if we can
construct for any (y,H) 6 F(NGN) an element d £ A(H,Iso(TNy))
satisfying:
5.3. ch~(d) = DEG(f,~) (y,K) for all K c H.
151
Now we construct d. We can assume in view of 2.4 and lemma 5.2
that f is quasi-transverse to y. Furthermore we can suppose
H = Gy. We want to assign to each H-orbit c in f-1(y) an ele-
,Iso(TNy) . = ment d(c) in A(H ) Choose x in c. Then TM x
= = ~ ~(G/Gy,N)y. Split (TG/Gx) x • ~(G/Gx,M) x and TNy (TG/Gy)y
TPx : (TG/Gx) x ~ (TG/Gy)y induced from the projection p as
0 ~ qx : (TGy/Gx)x ~ V ~ (TG/Gy)y with qx a Gx-linear isomor-
phism• Checking the dimensions and using elementary obstruction
theory we can extend the Gx-ma p k x : ~ (G/Gx,M) x ~ ~(G/Gy,N)y
coming from k appearing in the definition of quasi-transverse to
a norm preserving Gx-ma p k'x : ~ (G/Gx'M)x ~ (TGy/Gx)x ~ ~(G/Gy,N)
Since k' ~ qx : TM ~ TN is norm-preserving we obtain a G -self x x y x
' . As H/G~ contains map i x : TN~ ~ TN~ by (k x ~ qx )c o ~(G/Gx)eG x
only finitely many WHK-orbits (see [2], p. 87) H/G~ is finite and
(TGy/Gx)Kx = {0} for K c H with finite WHK. Hence
Y
5.4. IK =x (TpK ~ kK)C ° ~(G/Gx)(x)KG if WHK is finite. x
G G Denote the image of i x under L x-1 : [TN~,TN~] x- ~ A(Gx,iSo(TNy))
by d(x) and the image of d(x) under ind~ : A(Gx,ISo(TNy)) x
A(H,Iso(TNy)) by d(c). For u 6 A(H) and v 6 A(G x) one easily checks
ind~ (res~ (u).v) = u.ind~ (v). We obtain from 3.6. and 5.4. and X X X
G H x
res G (xH(TN~) -I) = X (TN)-I x
5 . 5 . ch~ c ind H (d(x)) = x
_ hKh -1 khKh -1 c , ,hKh -1 _ deg((TPx e x ) 0 ~(G/Gx) tX;e G ) =
hG 6 H/G K x x x
K K c K Z K d e g ( ( T P z i~ kz ) o ko(G/Gz) (Z)eGz) i f WHK i s f i n i t e .
z 6 c
152
This shows in particular using proposition 3.1. that d(c) does
not depend on the choice of x. Define d = Z d(c) . c 6 f-1 (y)/H
If fK M K N K K K * : ~ and : f tPNK ~ tPMK are induced by f and
DEG(fK, K) is the non-equivariant degree we have by definition
DEG(f,~) (y,K) = DEG(fK,~K) (y) and by proposition 2.1.:
5.6. DEG(f,~) (y,K) = [
z 6 (fK)-1
K deg((TpK(~ kKz ) o ~0(G/Gz)(Z)e G
(y) z
if WHK is finite.
Combining 5.5. and 5.6. gives
5.7. ch~(d) = DEG(f,~) (y,K) if WHK is finite.
Let K c H be any subgroup. We can find a bigger subgroup K' with
K c K' c H such that K'/K is a torus T, WHK' is finite and
ch K = ch K, and hence ch~ = ch~, holds (see [6] p. 113). If we can
show DEG(f,~) (y,K) = DEG(f,~) (y,K') the assertion 5.3 is a conse-
quence of 5.7. Since T acts on N K with fixed point set N K' this
follows from:
Lemma 5.9. Let g : P ~ Q be a T-map between T-manifolds and
: g tpQ ~ tpp a,O(T)-transformation. Then we @et
DEG(g,~) (y,T) = DEG(g,~)(y,1) for a T-fixed point y.
Proof. We can assume that ~ is quasi-transverse to y. Since WL
for L c T is finite only for L = T the preimage f-1(y) is a
finite set of T-fixed points x11..x r. By proposition 2.1 we obtain
153
for certain T-self maps I i : TNy ~ TNy that
DEG(f,~) (y,T) = £ deg(l~) and DEG(f,~) (y,1) = ~ deg(li). Now
apply proposition 3.1 and theorem 3.4. This finishes the proof
of lemma 5.9 and of theorem 5.1. o
6. Some examples
The theorem 5.1 is very general so that it is necessary to give
some examples to explain its meaning. The general problem is to
calculate inv lim Atp as a subring of C(~ ). This can be done in
special cases where F(~ ) is rather simple or A(H,~ ) c C(H) is
well understood for all subgroups H of G. We recall that DEG(f,~)
lies in C(N) = N ~o(NH)/wH for a G-map f : M ~ N and a
Con(G) O(G)-transformation ~ : f tPN ~ tPM and DEG(f,~) (z,H) is the
integer belonging to the component of N H containing z £ N H.
In the following we always assume that N fulfills condition (,)
and is connected.
Example 6.1. Assume that N H is non-empty and connected for all
H c G. Suppose that Iso(N) is a family. This follows already
from our assumption if G is finite. Then we have C(N) = C(G)
and by corollary 4.3 and theorem 5.1.
DEG(f,~) 6 A(G,Iso(N)) c C(G)
Hence we obtain the same relations as in the special case
M = N = V c with V a G-representation (see theorem 3.4.). The
assumption N G # @ is essential. If N is connected and free we
get End(tPN) = A(1) = C(N) = Z. Indeed, each integer d can be
154
realized as the degree of a self-map of some connected free
orientable G-manifold N. Take any connected free orientable
G-manifold N and a map f : S I ~ S I of degree d then o
i~ x f : N O x S I I N O x S is an example. However~ if N is the
sphere of a free G-representation the degree of f is I modulo IGI
for finite G and 1 for infinite G. One ex-
planation for this phenomenon is that the suspension of a mani-
fold is not a manifold in general but the suspension of a homo-
topy representation is again a homotopy representation. []
Example 6.2. Let G be a torus T n and assume that each component
of N H contains a G-fixed point for H c G.
We get from corollary 4.3 and theorem 5.1
DEG(f,~) (z,H) = DEG(f,~) (y,1)
for all H c G and z 6 N H. o
Example 6.3. If H is a p-group the homomorphism ch I and
ch H : A(H) ~ ~ fulfill ch I ~ ch H mod p. If H is a torus ch I
and ch H agree. Hence we get for each (z,H) by theorem 5.1 (see
5.3).
DEG(f,~0) (z,H) = DEG(f,~0)(z,1) rood p, if H is a p-group
DEG(f,~) (z,H) = DEG(f,~)(z,1) , if H is a torus
If G is itself a p-group we obtain for all (z,H) (see also [3],
[11] p. 10).
155
DEG(f,~) (z,H) m DEG(f,~0) (z,1) rood p []
Remark 5.4. Now we give the proof of corollary B stated in the
introduction. Assume the existence of M. Since M has finite
orbit type (see [6] p. 121) we can find a finite p-group L c G
with M G = M L. Hence we can suppose that G itself is a finite p-
group. We use induction over IGI. The induction begin G = Z/p
is done in [I] or by the following argument reflecting the re-
sults of example 6.3. Let c : M ~ TM~ be the collaps map. If
is the point at infinity c-I(~) D M G is empty. Since c-I(~)
is contained in the free part of M we can use non-equivariant
transversality to change c up to G-homotopy such that c is
transverse to ~ in the non-equivariant sense and still
-I M G c (~) D = ~ holds. We can assume that G acts orientation
preserving, otherwise consider M x M. Hence the local degree of
-I c at x and gx for x 6 c (~) and g 6 G agree. Each orbit in the
-I finite set c (~) consists of p elements. Therefore the degree
of c must be divisible by p. A contradiction, since computing
deg c by its local degrees at O 6 TM$ yields one. In the induction
step choose a central subgroup C in G with C = Z/p. If M C # M G
we ~et a contradiction to the induction hypothesis. Namely, con-
sider the G/C-action on M C. But M C = M G is impossible by the in-
duction begin applied to the C-action on M. This finishes the
proof of corollary B.
If one drops the assumption in corollary B that all M H are
connected the result remains true for G an abelian p-group and
is false for G a non-abelian p-group provided that D is odd
(see [4], [12]). A complete classification of compact Lie groups
156
with one-fixed point actions on (orientable) closed G-manifolds
is given in [12]. D
Example 6.5. Let G be a finite group of odd order. If DEG(f,~)
lies in C(N) we get from corollary 4.3. and theorem 5.1.
DEG(f,~) (z,H) = DEG(f,~0) (y,1)
for all (z,H). []
Remark 6.6. All the relations we get for DEG(f,~) also hold in
the case of an endomorphism f : M ~ M of a G-manifold with
~o(f H) : ~o(M H) ~ ~o(M H) the identity and all components of
M H orientable for H c G. Then we can define DEG(f) without speci-
fying ~ (see theorem A in the introduction and example 2.7.). G
Finally we mention a consequence of theorem 4.1. and theorem 5.1.
and the composition formula 2.5.
Corollary 6.7. If DEG(f,~) lies in C(N) for some O(G)-transfor-
mation ~ then there is an O(G)-equivalence ~ with
DEG(f,#) [ 1 o
7. The fibre transport and the first equivariant Stiefel-Whitney
class.
In this section we analyze the fibre transport from a bundle
theoretic point of view. We relate it and the question when an
O(G)-equivalence ~ : f tPN ~ tPM exists to the equivariant
157
analogue of the first Stiefel-Whitney class.
We have introduced the notion of an O(G)-groupoid, O(G)-functor
and O(G)-transformation in section one. We call two O(G)-func-
tors F ° and F 1 : ~ o ~ ~ I homotopic if there exists an O(G)-
equivalence q9 : F O ~ F 1. Let [~o' ~I ]O(G) be the set of homo-
~ ~ AG-map f : X ~ Y topy classes of O(G)-functors o 1"
induces an O(G)-functor Gf : nGx ~ Gy. A G-homotopy
h : X x I ~ Y defines an O(G)-equivalence ~Gh ° ~Gh I . Hence
we get a w=~l-defined map [X,Y] G ~ [ GX, Gy]O(G ) [f] ~ [ Gf].
Let ~ = q(G,n) ~ BF(G,n) be the classifyinq G-Sn-fibration. It is
characterized by the property that the map [X,BF(G,n)] G ~ bfG, n(X)
sending [f] to the G-fibre homotopy class of f ~ is bijective.
Definition 7.1. Let ~ ~ X be a G-Sn-fibration and f~ : X ~ BF(G,n)
be a classifying map. We call w E = [ Gf] £ [ GX, GBF(G,n)]O(G) th_~e
first equivariant Stiefel-Whitney class of ~. Let w M be WTM c for a
G-manifold M. D
This notion reduces for G = I to the ordinary definition of the
first Stiefel-Whitney class w1(M) E HI(M,Z/2) = Hom(~I(M),~I(BF(n))-
It is related to the fibre transport by:
proposition 7.2. Let tp : ~GB(G,n) ~ bfG, n be the fibre trans-
port of the universal G-Sn-fibration.
a) For each H c G tp (G/H) : ~(B(G,n) H) ~ bfG,n(G/H) is an
equivalence of categories.
158
b) For any G-complex X we ~et a bijectio n
(tPq) , : [~Gx,~GB(G,n)]O(G) ~ [~Gx,bfG,n ]O(G)
c) If ~ ~ X is a G-Sn-fibration (tp n) sends w~ to [tp~].
d) Let $I and ~2 be G-Sn-fibrations over the same one-dimensional
G-complex X. Then ~I and ~2 are G-fibre homotopy equivalent if
and only __if w~1 = w~2 holds.
Proof:
a) We must show that tp (G/H) induces a bijection between the sets
of isomorphism classes of objects and for any object x 6 n(B(G,n) H)
an isomorphism Aut(x) ~ Aut(tp (G/H) (x)) u ~ tp (G/H) (u). The
first assertion follows directly from the universal property. The
second f~>llows from the observation that H-fibre homotopy classes
of H-Sn-fibrations over S I equipped with the trivial H-action are
in one to one correspondence to H-homotopy classes of self H-maps
of the fibre by the fibre transport, b) and d) follow directly
from [16] whereas c) is obvious, o
Given two G-Sn-fibrations ~ and ~ over X we want to analyze when
w~ = wq holds. If w1(~H) and w1(q H) are the (non-equivariant) first
Stiefel-Whitney classes in HI(xH,z/2) we have the following obvious
conditions for w~ = wq:
i) ~x ~G q x for each x 6 X. x
ii) w I ($H) = Wl (H) for H c G.
The following example shows that they are not sufficient in general.
If ~ is the trivial and ~_ the non-trivial ~/2 representation and
159
, S I ~c. ~/2 acts freely on S I consider [ = S I × ~c and ~ = × _
Under certain conditions, however, i) and ii) are sufficient.
Theorem 7.3. Let ~ and n b_ee G-Sn-fibrations over X. Then w~
hold____~s if on___ee o_ff th___ee following conditions is satisfied.
= w q
i) X H is connected and wl (~H) = w1(~ H) for all H c G. There is a
x i n X G w i t h ~x ~G ~x" x
ii) The group G is finite of odd order and ~x ~G n x for all x 6 M. x
We have after forgetting the group action w1(~) = w1(n).
Proof: We have to specify for each H c G and x £ X H a G-fibre ho-
motopy equivalence ~(G/H) (x) : tp (G/H) (x) ~ tp (G/H) (x). We do
this by determining a H-homotopy equivalence ~(G/H) (X)eH:
~x ~ ~x between the fibres over eH. The independence of the
choice of the path u below follows from the assumptions about the
first Stiefel-Whitney classes.
i) Fix y in X G and a G-homotopy equivalence ~ : ~y ~ ny. Define
~(G/H) (X)eH by requiring that the following diagram commutes
up to H-homotopy for a path u from y to x in X H.
• (G/H) (X)eH {x .... ) nx
tp (G/H) (U)eH ~! ~ I tp (G/H) (u)
,, ) qy
eH
ii) Without loss of generality we can suppose that X is connected.
Fix a point y in X and a non-equivariant homotopy equivalence
: ~y ~ ~y. By assumption there is a H-homotopy equivalence
160
Sx ~ n x and there are only two up to homotopy because of
A(H) = A(H,ISO(~ )) = {±I} (see [6], p. 8) and theorem 3.4. x
Let ~(G/H) (X)eH be t h e one m a k i n g t h e f o l l o w i n g n o n - e q u i -
variant diagram commutative up to homotopy for a path u be-
tween y and x.
tp(G/1) (u)
~ (G/H) (x) ~x eH ~x
e I ~ 1 tp(G/I) (u)e
~y ~ ny D
Now we examine whether w M is a homotopy invariant. If f : M ~ N
c is a G-homotopy equivalence f (TN ~ V) and (TM ~ V) c are G-fibre
homotopy equivalent for appropriate V by [13], theorem 2.3. If
this could be destabilized to f TN c ~G TMc we would in particular
obtain f w N = w M. Unfortunately, this is not possible in general,
theorem 3.4. gives counterexamples over a point. Now we prove the
unstable result f w N = w M.
Theorem 7.4. Let f : M ~ N be a G-ma~ between G-manifolds such
~o(f H) : ~o(M H) ~ ~o(N H) is bi~ective for all H c G. Suppose that
for C 6 ~o(M H) and D 6 ,o(N H) with fH(c) c D that (fHIc)~wI(D) is
w1(C). Then the (non-equivariant) degree deq(fHIc : C ~ D) £ ~/{±I}
is defined. Assume deg (fHIC) = ±I and that M and N fulfill condition
(~).
Then there is an O(G)-equivalence ~: f tPN ~ tPM uniquely deter-
mined b_~ the property that DEG(f,~) ~ I.
161
Proof. Denote by MH(x) the component of M H containing x for
H c G, x 6 M H. Let MH(x) be MH(X) if w1(MH(x)) is zero and the
orientation covering otherwise so that MH(x) is an orientable A
connected manifold. Choose a lift X 6 MH(x) of x and a lift
~H ~H ~ N H : (x) ~ NH(fx) of fIMH(x) : MH(x) (fx). Write ^
Y = ~H(~) and y = fH(x). Let ~p(x,H) : (TN;)H ~ (TM~)H be a
(non-equivariant) map for x 6 M H, H c G making the following
diagram commutative where c denotes the collaps map and p the
projection and n is dim MH(x) = dim NH(fx).
~H
H (MH(x)),~MH(x)) ~ ~ H (NH(y)),~NH(y)) n n
c i c
H n (TM H (X)X~), H n (TN H (y)y),
Hn((TMC)H ) *(x,HIM c H <' H n ((TNy))
One easily checks that the homotopy class of ~(x,H) depends only ^
on (x,H) but not on the choice of x and ~H.
Lemma 7.5. There is up to Gx-homotopy exactly one Gx-ma p
T c ~(x) : Nfx ~ TM c for each x i_nn M such that ~(x) H and 9(x,H)
are non-equivariantly homotopic for all H c G x. Each ~(x) is a
Gx-homotopy equivalence.
Let m be the product ~W G H I running over {(H) 6 Con(G x) I H 6 ISO(TNy), x
W G H f i n i t e } . We g e t f r o m [ 7 ] p . 173 + 174 t h e e x i s t e n c e o f G - m a p s x
162
~) : TN ~ TM and ~' : TM ~ TN with deg((~' o ~)H) = I mod m y x x y
for all H c G. Because of the equivariant Hopf theorem 3.5. and
theorem 3.4. lemma 7.5. follows from:
7.6. The element d = {deg(e H o ~(x,H)-1) T (H) 6 Con(Gx)} 6 C(G x) G
lies in the image of DEG : [TNy,TNy] x ~ C(Gx).
We firstly give the proof of 7.6. under the assumption that N H
is connected for all H c G . Consider f : M ~ N as a G -map so X X
that x is a Gx-fixed point in M. As in the proof of theorem 7.3.
ii) we get ~O(Gx)-transformation ~ : f tPN ~ tPM uniquely de-
termined by the property that ~(Gx/Gx) (x) is just (~). Note that
for any z 6 M H, H c G the case w1(MH(z)) = O and w1(NH(z)) % O
never occurs because of deg(fHIMH(z)) = ,I. Now one checks direct-
ly that DEG(f,~) 6 C(G x) is just d. By theorem 5.1. we have G
d 6 image(DEG : [TNC,TN c] x ~ C(Gx)) " Y Y
In the general case one has the problem that ~ does not determine
an O(Gx)-transformation ~ : f tPN ~ tPM if there is a non-connected
N H for some H c G x. But we can restrict everything to the O(Gx)- G
subgroupoid uGXMx of ~GxM with ~XM(Gx/H) := ~(MH(x)) sO that we con-
only the component of M H containing x. Then we get an O(Gx)- sider G G
transformation ~ : f t p N t ~xXM ~ t p M I ~xXM b y c0 a s b e f o r e . As i n G
section two we can at least define DEG(f,~) in C(~xXN) = C(G x) and
get d = DEG(f,~). The same argument as in the proof of theorem 5.1. G
gives d 6 image (DEG : [TN~,TN~];v x ~ C(Gx)). This shows 7.6. and
finishes the proof of Lemma 7.5.
163
Let ~ : f tPN ~ tPM be defined by the property that
~ TM c is the restriction from G to H of ~(G/H) (X)eH : TN x x x
9(x). We leave it to the reader to check that ~ is an well-
defined O(G)-equivalence. By construction DEG(f,~) s I holds.
The uniqueness follows from theorem 4.1. This finishes the
proof of theorem 7.4. o
We obtain as a corollary the homotopy invariance of the first
equivariant Sti~fe~Whitney class and the unstable version of
the result in [13], corollary 2.4.
Corollary 7.7. Let f : M ) N be a G-map ~ the assumption
of Theorem 7.4. Then we have f~ w N = w M and the spheres of the normal
slices of M at x and N at fx are G x - homotopy equivalent fo r al__!l
x 6 M.
Proof We derive f~ w N = w M and TM~ ~ G x
Now apply theorem 3.5. o
TNfx from theorem 7.4.
If G x is connected TM x and TNfx are even isomorphic as Gx-represen-
tations (see [22]). For finite G there are non-isomorphic G-represen-
tations V and W with V c W c (see [6] p. 249). ~-- S
Now we give a necessary condition for converting a G-map
f : M ~ N between G-manifolds into a G-homotopy equivalence by
surgery. Notice that this would imply the existence of a bordism
appearinq below. Theorem 7.8. motivates the approach to equi-
variant surgery given in [17].
164
Theorem 7.8. Let (f,f,f+) : (P,M,M+) ~ (Q,N,N+) be a G-map
between G-triads of G-manifolds such that f+ is a G-homotopy
H pH equivalence. Assume that M+ ~ i_ss l-connected for H c G.
Then we can find an O(G)-equivalence ~ : f tPN ~ tPM with
DEG(f,~) = I.
Proof. We get the existence of an O(G)-equivalence
• + : f+tPN+ ~ tPM + with DEG(f+,~+) ~ 1 by corollary 7.7. Using
the inward normal we get identifications tp I M+ = tP(TM÷~)c
and tpp [ M = tp(TM @~)c and analogously for Q. For two G-repre-
sentations V and N with (V ~)c (W ~)c and dim V G = dim W G > I ~S
the suspension [vC,wC] G ~ [V ~C,w ~c]G is bijective (theorem
3.5.). Hence we obtain O(G)-equivalences ¢ : F tpQ ~ tpp and
: f tp N ~ tPM such that ¢IM corresponds to ~ and %IM+ to ~+
under the identification above. Now apply the bordism invariance
2.3.
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165
[4] Browder, W.: Pulling back fixed points, inv, math,
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[5] tom Dieck, T.: The Burnside ring of a compact Lie group I.
Math. Ann. 215, 235 - 250 (1975).
[6] tom Dieck, T.: Transformation groups and representation
theory. Lect. notes in math. 766, Springer Verlag, Berlin-
Heidelberg-New York (1979).
[7] tom Dieck, T.: Transformation groups, de Gruyter (1987).
[8] tom Dieck, T. and T. Petrie: [{omotopy representations of
finite groups, Publ. Math. IHES 56 (1982), 337 - 377.
[9] Dold, A.: Lectures on algebraic topology, Springer Verlag,
Berlin-Heidelberg-New York (1972).
[10] Dovermann, K. H.: Addition of equivariant surgery obstructions,
algebraic topology, Waterloo (1978), lecture notes in math.
741, Springer Verlag (1979), 244 - 271.
[11] Dovermann, K. H. and T. Petrie: G surgery II. Mem. of the AMS
Vol. 37, no. 260 (1982).
[12] Ewing, J. and R. Stong: Group actions having one fixed point.
Math. Zeitschrift 191, 159 - 164 (1986).
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[16]
[17]
LHck, W.: Equivariant Eilenberg-MacLane spaces K(~ ,~,1)
with possibly non-connected or empty fixed point sets,
manuscr, math. 58, 67 - 75 (1987)
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Wolfgang L~ck Mathematisches Institut der Georg-August-Universit~t BunsenstraSe 3 - 5
3400 G~ttingen
Bundesrepublik Deutschland
SURGERY TRANSFER
by W.Luck and A.Ranicki
Introduction
Given a Hurewicz fibration F ,E P ,B with fibre an t
n-dimensional geometric Poincare complex F we construct
algebraic transfer maps in the Wall surgery obstruction
groups
!
p" : Lm(Z[~I(B) ] ) ~ Lm+n(Z[~l(E) ] ) (m~>0)
and prove that they agree with the geometrically
defined transfer maps. In subsequent work we shall
obtain specific computations of the composites p p! ,
p!p with p! :Lm(Z[~I(E) ]) ~Lm(Z[~I(B) ]) the change of
rings maps, and some vanishing results.
!
The construction of p" is most straightforward in
the case when F is finite, with L. the free L-groups '
L . In ~9 we shall extend the definition of p" to
finitely dominated F and the projective L-groups L~, as S
well as to simple F and the simple L-groups L,, and
also to the intermediate cases.
There are two main sources of applications of the
surgery transfer. The equivariant surgery obstruction
groups of Browder and Quinn [ I ] were defined in terms
of the geometric surgery transfer maps of the normal
sphere bundles of the fixed point sets. An algebraic
version will necessarily involve the algebraic surgery
transfer maps. (In this connection see LHck and Madsen
[8].) The recent work of Hambleton, Milgram, Taylor and
Williams [3] on the evaluation of the surgery
obstructions of normal maps of closed manifolds with
finite fundamental group depends on the factorization
of the assembly map by twisted product formulae which
are closely related to the algebraic surgery transfer.
Our construction of the quadratic L-theory
transfer maps is by a combination of the algebraic
168
surgery theory of Ranicki [ 1 4 ] , [ 1 9 ] and the method used
by L~ck [7} to define the algebraic K-theory transfer
maps p :Km(Z[~I(B) ]) ,Km(Z[~I(E) ]) (m=O,l) for a
fibration with finitely dominated fibre F.
The algebraic surgery transfer maps p for a
fibration are a special case of transfer maps !
(C,~,U) " :Lm(A) ,Lm+n(B) (m~0) defined in abstract
algebra. Here, A and B are rings with involution, C is
an n-dimensional f.g. free B-module chain complex with i
a symmetric Poincare duality chain equivalence
* cn-* ~ma :C ...... ~ , and U:A :R=H0(HOmB(C,C))°P is a
morphism of rings with involution from A to the
opposite of the ring of chain homotopy classes of
B-module chain maps f :C ,, ,JC, with the involution on R
defined by T(f)=~-I f ~. An element of L2i(A) is
represented by a nonsingular (-)l-quadratic form
(M,~:M ~M ) on a f.g. free A-module M=~A. We define k
!
(C,~,U) " (M,@)=(D,0)~Ln+2i(B) to be the cobordism class s
of the (n+2i)-dimensional quadratic Poincare complex
(D,@) given by
U ( ¢ ) ( ~ a -1 ) i f s=O = k :
Os 0 if s~0
D n+2i-r-s = ~C n+i-r-s ~ D = ~)Cr_ i k r k
There is a similar formula in the case m=2i+l, for
which we refer to ~4.
The algebraic transfer maps of fibration F :E P JB J
with fibre an n-dimensional geometric Poincare complex
F are given by
! !
p" = (C(F~),~,U) ' : Lm(Z[KI(B) ]) ' Lm+n(Z[~l(E) ])
with C(F) the cellular Z[~I(E) l-module chain complex of
the cover F of F induced from the universal cover E of
E, ~=([F]O-) -I n-* :C(F) ~C(F) the Poincare duality
169
chain equivalence,
transport.
and U determined by the fibre
H e r e
algebraic
the identification of
transfers in L~ck [7] how
of CW structures from the
fibration. We use the
Ranicki [ 16, ~7.8 ] both to
data in the base spaces as
the algebraic surgery
structures
is the main idea Jn the identification of the
and geometric surgery transfer. We know from
the corresponding K-theory
to handle ~n algebra the lift
base to the total space of a
ultraquadratic L-theory of
encode the algebraic surgery
CW structures, and to decode
data from the lifted CW
in the total spaces.
The paper was written during the second named
author's visit in the academic year 1987/1988 to the
Sonderforschungsbereich SFBI70 in G~ttingen, whose
support is gratefully acknowledged.
The titles of the sections are:
Introduction
~I. The algebraic K-theory transfer
~2. Maps of L-groups
~3. The generalized Morita maps in
~4. The quadratic L-theory transfer
~5. The algebraic surgery transfer
~6. The geometric surgery transfer
~7. Ultraquadratic L-theory
~8. The connection
~9. Change of K-theory
Appendix I. Fibred intersections
Appendix 2. A counterexample in
References
L-theory
symmetric L-theory
170
~1. The al~ebraic K-theory transfer
We recall from LUck [7] the construction of the
algebraic K-theory transfer maps, and the connection
with topology.
Given a ring R let R °p denote the opposite ring,
with the same elements and additive structure but with
the opposite multiplication.
Definition I. 1 A representation (A,U) of a ring R in an
additive category ~ is an object A in ~ together with a
morphism of rings U:R JHomA(A,A) °p.
[3
Given an associative ring R with I let ~(R) be the
additive category of based f.g. free R-modules R n
(n~O). A morphism f :R n ~R m is an R-module morphism,
corresponding to the mXn matrix (aij) l~i~m, l~j~n with
entries aijER , such that
n f = (aij) : m n , m m ; (xj) ~ ( ~ x .a )
j=1 J ij
Example I .2 The universal representation (R,U) of R in
~(R) is defined by the ring isomorphism
U : R ~ HomR(R,R)°P ; r " ( s , sr ) ,
which we shall use to identify R=HOmR(R,R) °p.
[3
A functor of additive categories F:A
required to preserve the additive structures.
Proposition [.3 Given a ring R and an additive category
there is a natural one-one correspondence between
functors F:~(R) ,A and representations (A,U) of R in
A.
Proof : Given a functor F define a representation (A,U)
by
A = F(R) ,
U : R = HomR(R,R)°P
171
Hom~(A,A) °p
Conversely, given
functor F=-~(A,U) :~(R)
(~:R ,R) ....... , (F(~) :A
a representation
F(R n ) = A n ,
F((a~j):R n
~ by
~R m = (U(a ij
,A)
(A,U) define a
) ) : A n l A m
[]
, A ;
HomA(A,A)°P J A
The functor associated to a representation (A,U) of R
in A is the composite
Ut F = -~(A,U) : ~(R) " , B(HomA(A,A)°P)
embedding
-@(A, I) : ~(HomA(A,A)°P)
Example 1.4 A morphism of rings f :R JS determines a
representation (S,U) of R in B(S) with
U = f : R l HOms(S,s)°P = S
such that -~(S,U)=f! :~(R) ~B(S) is the usual change of
rings functor.
O
For any object A in an additive category A there
is defined a representation (A, i) of the ring
HomA(A,A) °p in A. The corresponding functor is the full
172
- O ( A , 1 )
Given chain complexes C,D in A let HomA(C,D) b e
the abelian group chain complex defined by
dHomA(a,D ) : HomA(C,D) r HomA(Cp,D q) q-p=r
Hom~(C,D)r_ 1 ; f - - . dDf + (-)qfd C
There is a natural one-one correspondence between chain
maps f:C ........ ,D and O-cycles f'EHom~(C,D)0 , with
f, = (_)nf : C .......... D (nEZ) n n
Similarly for chain homotopies and l-chains. Thus
H0(Hom~(C,D)) is isomorphic to the additive group of
chain homotopy classes of chain maps C : ~D.
A chain complex C is finite if C =0 for r<O and r
there exists n~O such that C =0 for r>n. r
Definition I. 5 Given an additive category ~ let D(~) be
the homotopy category of ~, the additive category of
finite chain complexes in ~ and chain homotopy classes
of chain maps with
HomD(A)(C,D) = H0(HomA(C,D))
For a ring R we write D(~(R)) as D(R).
[]
We refer to Ranicki [17],[18] for an account of
the algebraic K-groups Km(~ ) (m=0, I) of an additive
category ~ with the split exact structure, and the
application to chain complexes. In particular, the
class of a finite chain complex C in ~ is defined by
[C] = m (-)r[c ] ~ K0(A) r= O r '
and the torsion of a self chain equivalence f :C ~C is
173
defined b y
T(f) = T(d+r:C(f)od d :C(f)even) 6 KI(A)
for any chain contraction r:Oml :C(f)
algebraic mapping cone C(f).
~C(f) of the
Definition 1.6 The generalized Morita maps
~:Km(~(~)) JKm(~) (m=0, I) are defined for any additive
category A by:
for m=O ~ sends the class [C]6Ko(~(A)) of an
object C in D(A) to the class [C]6Ko(A) ,
for m=l ~ sends the torsion T(f)6KI (D(A)) of an
automorphism f:C pC in ~(A) to the torsion T(f)6KI (~)
of any representative self chain equivalence.
[]
A morphism in D(~) is a chain homotopy class and
the definition of ~ involves a choice of representative
chain map. The generalized Morita maps ~ are therefore
not induced by a functor D(~) 'A.
Example 1.7 (L~ck [7]) A Hurewicz fibration F :E ~,B
with the fibre F a CW complex determines a ring
morphism
U : Z[~I(B)] " , H 0 ( H o m z [ N I ( E ) ] ( C ( F ) , C ( F ) ) ) ° P
w i t h C ( F ) t h e c e l l u l a r b a s e d f r e e Z [ N l ( E ) ] - m o d u l e c h a i n
c o m p l e x o f t h e p u l l b a c k F t o F o f t h e u n i v e r s a l c o v e r
of E, and U the chain homotopy action of
H0(OB)=Z[~I(B) ] on C(F) determined by the homotopy
action of the loop space QB on F. For finite F this
defines a representation (C(F),U) of Z[~I(B) ] in
D(Z[~I (E) ]). For the identity map p=l :E ,B=E with
F=< ~} this is the universal representation (R,U) of 1.2
f o r R=Z[~ 1 ( B ) ] = Z [ K 1 (E) 1 •
174
[]
The transfer map in the torsion groups associated
to a representation (C,U) of a ring R in D(~) is the
composite
! (C,U) " : KI(R ) = KI(B(R))
U~ , KI(D(A)) , KI(A)
of the map U! induced by the functor
(C,U)~-:B(R) B~(A) and the generalized Morita map ~.
The torsion ?(f)~Kl (R) of an automorphism f :R k ~R k is !
sent by (C,U) " to the torsion T(U(f))~KI(~) of the se] f
chain equivalence U(f) :~C ,~C. k k
The idempotent completion of an additive category
is the additive category ~ with objects pairs
( A = object of A , p = p2 : A ....... , A )
and morphisms f : (A,p) ,(A' ,p') defined by morphisms
f :A *A' in ~ such that p' fp=f :A ~A' The evident
functor D(~) ~(A) is an equivalence of additive
categories, since every chain homotopy projection in
splits (LHck and Ranicki [9]).
For any ring R the additive category ~(R) of f.g.
projective R-modules is equivalent to the idempotent
completion ~(R) of the additive category ~(R) of based
f.g. free R-modules, with an equivalence
~ ( R ) : P ( R ) ; ( R k , p ) ..... , i m ( p )
For any representation (C,U) of a ring R in D(~)
the functor (C,U)~- :~(R) ,D(~) extends to a functor
~(R) '~(~) (cf. Lemma 9. 3) , and so determines a
transfer map in the class groups
(C,U)' U, ' : K0(R) = KO(~(R)) K0(D(~))
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176
Z[~I(B) l-module chain complex. Let F ,E P JB be a
Hurewicz fibration such that the fibre F is a CW
complex. Let F ,(Y',Y) ....... ~(X',X) be the fibration
obtained from p by pul ]back along the map X' ~B, with
(Y' ,Y) the pullback to (Y' ,Y) of the universal cover
of E. Then (Y' ,Y) is homotopy equivalent to a relative
CW pair ( a ] so denoted by (Y' ,Y)) with cellular based
free Z[~I(E) l-module chain complex
C(~Q' ,~Q) = s r c ( p # ( d ) : ~ C ( F ) , ~ C ( F ) ) J I
the r-fold suspension of the algebraic mapping cone of
a chain map in the chain homotopy class
# # p (d) : p (@Z[~I(B) ]) = @C(F ~)
J J
# , p (~Z[~ I (B) l ) = @C(F)
I I
Proof : See L(fck [ 7 ] .
[]
~2. Maps o__f_f L-group s
We refer to Ranicki [ 14] , [ 19] for the definition
of the quadratic L-groups Ln(~) (n~0) of an additive
category ~ with involution * :~ ~, as the cobordism J
groups of n-dimensional quadratic Poincare complexes
(C,~EQn(C)) in ~, and for the proof that these groups
are 4-periodic, with L2i(~ ) (resp. L2i+l(~)) the Witt
group of nonsingular (-)i-quadratic forms (resp.
formations) in ~.
We now put an involution on the notions of ~I.
Definition 2.1 An involution on an additive category
is a contravariant functor
* : ~ : ~ ; M l M ,
177
(f :M ,N) , (f :N ,M )
together with a natural equivalence
e : id A , ** : ~ ~ ~ ;
M ....... ~ (e(M):M ~M )
such that
e(M ) = (e(M) -I ) : M ~ M
[]
We
e(M):M
shall use the
~M to identify M =M.
natural
Example 2.2 Given a ring R with involution
: R , R ; r - - ~
isomorphisms
let the additive category
~(R) = {based f.g. free R-modules}
have the duality involution
* Rn (R n ) =
such that
(aij) = (a3i) ,
Ln(B(R)) = Ln(R ) (n~O)
d
By definition, a quadratic Poincare complex over R is I
the same as a quadratic Poincare complex in ~(R) .
[3
Notation 2.3 Let ~ be an additive
involution.
category with
178
i) A chain complex C in ~ is n-dimensional if C =0 for r r<0 and r>n.
ii) The n-dual of an n-dimensional chain complex C is
the n-dimensional chain complex C n-* in ~ with
dcn-* = (-)r(dC) :
(C n-* = C n-r )* )r = (Cn-r
, , ( C n - * ) r - 1
iii) For n~>O let Dn(A) be the additive category of
n-dimensional chain complexes in ~ and chain homotopy
classes of chain maps, with the n-duality involution n-*
T=n-*:Dn(A) ,Dn(A) ;C ,C
D
A functor of additive categories with involution
F:~ ,~ is a functor of the underlying additive
categories together with a natural equivalence
G:F* '*F:A '6, such that for any object M in A there
is defined a commutative diagram in
e ~ ( F ( M ) ) ~ F ( M ) . . . . . . . . . . . . . . . . . . . . . . F ( M )
F(eA(M)) I , [ G(M)
~ G(M ) ~ F(M ) , F(M )
Notation 2.4 A functor F : ~ ~ of additive categories
with involution induces morphisms of the quadratic
L-groups which we write as
F! ; Ln(~ ) , Ln(~ ) ;
( C , ¢ ) ...... , ( F ( C ) , F ( ~ ) ) (n~O)
D
Example 2.5 A morphism of r~ngs with involution f:R ~S
179
determines functors of additive categories with
involution f ! :~(R) ,~(S) which induces change of rings
m o r p h i s m s i n t h e q u a d r a t i c L - g r o u p s f t : L n ( R ) . .... , L n ( S )
(n~0).
Definition 2.6 Given a
[]
nonsingular symmetric form
(A,~=~ :A ,A ) in an additive category with involution
let the ring Hom~(A,A) °p have the involution
: Hom~(A,A) °p ) Hom~(A,A) °p ;
(f :A )A) ) (Cl-lf ~CI:A )A ~A ,A)
By analogy with Definition I. 1 :
[]
[]
In particular, (A,U) is a representation of R in
the additive category ~ in the sense of I.I.
By analogy with Example 1.2:
Example 2.8 The universal symmetric representation
(R,~,U) of a ring with involution R in ~(R) is defined
by
: R J R ; r , ( s ~ sr )
with U the isomorphism of rings with involution
Definition 2.7 A symmetric representation (A,~,U) of a
ring with involution R in an additive category with
involution ~ is a nonsingular symmetric form (A,~) in
together with a morphism of rings with involution
U:R ,Hom~(A,A) °p
180
U : R ~ HomR(R R) °p , ; r .... , ( s ~ s r )
We sha] ] use IJ as an ident~ fication of rings with
involution R--HOmR(R,R) ° p .
By analogy with Proposition 1.3:
[]
A = F(R) ,
(I = G(R) : F(R ) = A " F(R) = A ,
U : R = HomR(R,R)°P , HomA(A,A)°P ;
(p:R :R) , (F(~) :A ,A)
Conversely, given a symmetric representation (A,~,U)
define a functor F=-~(A,~,U) :~(R) ,A by
F(R) = A ,
G(R) = (I : F(R ) = A " F(R) = A ,
R n A n A m F((ai3) : ~R m) = (U(aij)) : ,
[]
By definition, a nonsingular symmetric form (C,~)
in Dn(A ) is an n-dimensional symmetric complex C in
together with a self dual chain homotopy class of chain
equivalences ~=T~:C ,C n-*
Proposition 2.9 G~ven a ring with involution R and an
additive category with involution ~ there is a natural
one-one correspondence between functors of pairs of
additive categories with involution F:~(R) ~A and
symmetric representations (A,a,U) of R in ~.
Proof: Given a functor F define a symmetric
representation (A,~,U) by
181
Proposition 2.10 A symmetric representation (C,~,U) of
a ring with ~nvolution R in Dn(A) determines a functor
F=-~(C,~,U) :~(R) ,~n(~) inducing morphisms in the
quadratic L-groups
F! = -@(C,g,U) : Lm(R) ............ , Lm(~n(A)) (m~>0)
Proof: Immediate from 2.4 and 2.9.
[]
Given a ring with involution S let ~n(S)=~n(~(S)) ,
the additive category of n-dimensional chain complexes
of based f.g. free S-modules and chain homotopy classes
of chain maps with the n-duality involution C ~C n-* A
symmetrJ c representation (C,~,U) of a ring with
involution R in Dn(S) determines by 2.10 a functor
F=-~(C,~,U) :~(R) . . . . . . . . ~Dn(S) inducing morphisms in the
quadratic L-groups
F! = -~(C,~,U) : Lm(R ) , Lm(Dn(S) ) (m~O)
~3. The 8eneralized Morita maps i___n_n L-theory
By analogy with the algebraic K-theory generalized
Morita maps ~:Km(D(A)) ,Km(~ ) (m=O,l) of ~i we define
generalized Morita maps in the quadratic L-groups
~:Lm(Dn(A)) ~Lm+n(A) (m,n~0) by passing from
nonsingular quadratic forms and formations in Dn(A) to l
quadratic Poincare complexes in A- The L-theory ~ is
the identity for n=0, since D0(A)=A. For n~l the maps
are not isomorphisms and are not induced by functors of
additive categories with involution: a morphism in
Dn(A) is a chain homotopy class and as in K-theory the
definition of ~ involves a choice of representative
chain map.
Proposition 3.1 i) A nonsingular (-)i-quadratic form in
182
Dn(A)
(M,0~coker(l-(-)iT:HOmDn(A) (M,M ) JHom (M M ))) Dn(A ) '
is represented by an n-dimensional chain complex M in
together with a chain map ~:M JM n-* such that
(l+(-)iT)~=e+(-)n+i~ :M ,M n-
ii) The cobordism class
(n+2i)-dimensional quadratic
(C,~) defined by C=M n+i-* and
H if s=0 =
~S 0 if S~I
is a chain equivalence.
(C,~)~L +2i(A) of the
Poincare complex in
C n+2i-r-s = M ~ C = M n+i-r r-i-s r
depends only on the class
n _~ ~ coker ( I - (-)iT : H0 (Hom~% (M, M )) ~H0 (HomA (M, Mn-* ) ) )
= coker(l-(-)iT:Hom (M,Mn-~)----~HOmDn Dn(A ) (A) (M,Mn-~))
iii) Suppose given (C,~) as in ii), an n-dimensional
chain complex L, a chain map j:L ~M and n+l-r
(xEHOmA(Lr,L ) try0> defining a chain homotopy
X+(_)n+i+l * X : j #j m 0 : L ~ L n-*
such that the chain map (3 (I+T)~0 O) :C(3) ~L n-* is a
c h a i n e q u i v a l e n c e , w i t h C ( j ) t h e a l g e b r a i c m a p p i n g c o n e
of 3. Then (C,~)=0ELn+2i(~) .
Proof: i) Trivial.
ii) The isomorphism of abelian groups
Q( )i(M) = coker(l ( )iT - - : t t o m ( M , M ) - IDn ( A )
~H°m~)n(A) (M,M ))
' Qn+2i (C) ;
183
[0:M
defined by
,M n- ] J (~s6HomA(cn+2i-r-S,Cr) ]r,s~>0 }
sends the
~6Qn+2i(C)-
~0 = @ ' Ks = 0 for s~l
class of O t o the quadratic structure
iii) Define an (n+2i+l)-dimensional quadratic Poincare
pair in A (f:C ~D,(5~,~)) by
* Mn+i_* f = j : C = • D = L n+i-* ,
= D n+2 = L n+i-r ~0 X : i+|-r = Lr_i_l ~ Dr ,
5~s = 0 for s~l
[]
of
We refer to ~2 of Ranicki { 19] for the definition
a nonsingular (-)i-quadratic formation
(F,G)=(F,|r'|G) in an additive category with involution
•, and fo~ £he result that (F,G)=06L2i+I(A) if and only
if there there exist a (-) i+l-quadratic form in A (H,~)
and a morphism j :F----~H such that the morphism defined
in A by
j [+(_)i+l [* : F~)H ........ ~ G ~H
is an isomorphism.
Proposition 3.2 i) A nonsingular (-)i-quadratic
I
II q~ I +
I I
¢% + I
II
! ,.r] ! + I
II
r
I v
o + +
o o
II
I +
I I
0 D
o +
II
+ ~
, ?
~ v
II ~1
1 ~
~ v
~ X
I
II
I +
! I
1 !
ii ! i +
I I
II
1
II "{-
"J
0o
ro
~o
~"
o ~"
r ~
m
~ m
°%
m
o +
~ ~
'.
~ + v o
o "o
+ ~
v ~
+ ~ o
~ D
'-d
I(-
D
+ I 11-
.o 1 D
i
v
o
x "o
~
" o
I, ~
~
+ ~
n
I + •
¢I5
I lt-r
~
~ o r, I
o
,-~ I x-
I
rl.
o :y
=
~=~
-~
D"
0
I""
0
0
? ,
I ~
v
I 1 !
o f't
f~
e~
o c =
I ~'~
0 ~
• 0
N
.m
| o.
~ n
1 °
c ~J o ~h
II
I
+ I +
~.
~
+ X-
@ I
@ I x'-
~ m
i o
f*
~ v
N m
I
0 II
0 "D
~-
' 0
0 0
0
v II
~"
I
Ii o
~
/ o
1 o
~
o
~ e
~ ~
~v
U
I '~.
~ 0"
O~
o I?
r'l
"
(~,
v.,.
I ~
D.1
t~
I ~,
o ~
o ~
o
0~
i'b
~.
m
°o + !
v 1
o
o~
o
II
~"
+ +
,..,
!
I
1::
+
11.
+ I
~.
+
I "
-(
x.. I('
I : o N o
II-
c'l
v
+ ! J " rr
o
-'~
o
x-
'o
~,
:3"
~..'.
I
ii
t~r
r-t
+ ,--x
¢~
+
@
o o rt
~ o
,~ + !
~h
I!
+ A I +
J~
lt-
v
+ I 1 cl
1::
v
f~
e~
,<
x o
(n
~3
v
÷ ~ "1
~
o
I o
,.~
o I
rt r~
Oo
187
i i)With I~ ~J as in i)there exist
: G n-* r~ G n-* ) F , // :
: G n-* ) G
c h a i n maps i n
) F n-* ,
and a chain homotopy
X : Y I1 - + ( x j n+i+l~* G n-* ) G
such that the chain map
[ ~ + ( _ ) n + i ~'* ~ * ] X Y "'* n+l-* "~*
: C(~ ) , C(~ ) 0
is a chain equivalence in A. Let (C,~) be the i
(n+2i+l)-dimensional quadratic Poincare complex derived r~
from (F)Gn-*,y,~)0,X) in the way (C,%#) is derived from
(F)G)Y)~,0,X) . Define an (n+2i+2)-dimensional quadratic
Poincare cobordism ((f f) :C(9C ,D, (5~,~-~)) by
D = S i+IF , 5~ = 0 ,
= G n - r + i ~ F r _ i _ l ) D = F f = ( 0 i ) : C r r r - i - I
f~ ( 0 1 ) : C~r r - i - I r r - i - I = = G i~)Fr_ ) D = F
Thus (C,~)=(C,~)~Ln+2i+I (A) . Since 0 and X can be
chosen independently of 0 and X it follows that the
cobordism is independent of these choices also. Given
(F,G,Y,~,@,X) and chain equivalences h:F )F' , k:G )G'
it is possible to define (F' ,G' ,y' ,~' ,@' ,X') such that
the corresponding quadratic Poincare complex (C' ,~' ) is
homotopy equivalent to (C,~), and so
(c' ~')=(c, ) ~)~Ln+2i+1 (~) •
J
iii) Define an (n+2i+2)-dimensional quadratic Poincare
pair (f :C )D) (~)~)) by
188
n+i+l-* D=H
f = ( 0 j ) :
= Gn-r+i(~F Hn+~+l-r Cr r-i-I ~ Dr = '
D n+2i+2-r = H ~ D = H n+i+l-r r-~-I r
~ = 0 for s)l
I
This is a quadratic Poincare null-cobordism of (C,~),
so that (C,~)=OELn+2i+l(~) .
[]
Definition 3.3 For any additive category
involution A define the generalized Morita maps
with
~/ : Lm(~n(A)) , Lm+n(A) (m,n~O)
for m=2i (resp. 2i+i) by sending a nonsingular
(-)i-quadratic form (M,~) (resp. formation (F,G)) in
~n(~) to the cobordism class of the (m+n)-dimensional S
quadratic Poincare complex (C,~) in ~ defined in
Proposition 3. I ii) (resp. 3.2 ii)). The verification
that the maps ~ are well-defined is contained in
Propositions 3. I iii) (resp. 3.2 iii)).
gl
For a ring with involution R apply 3.3 to ~=~(R)
to obtain generalized Morita maps ~:Lm(~n(R)) ~Lm+n(R)
( m , n ) O ) .
~4. The quadratic L-theory transfer
As before, let ~ be an additive category with
189
involution, and let ~n(A) be the chain homotopy
category of n-dimensional chain complexes in A with the
n-duality involution.
Definition 4.1 The ~uadratic L-theory transfer maps of
a symmetric representation (C,~,U) of a ring with
~nvolution R in Dn(A)
!
(C,~,U)" : L m ( R ) , Lm+n(A) (m>~O)
are the composites
!
(C,~,U) " : Um(R) = nm(~(R)) -~(c,a,u)
, Lm(Dn(A))
' Lm+n (A)
of the maps -~(C,~,U) of 2.10 and the generalized
Morita maps ~ of 3.3.
[]
Example 4.2 Let ~ be the additive category ~(S) of
based f.g. free S-modules with the duality involution,
for a ring with involution S. The transfer maps
determined by an n-dimensional symmetric representation
(C,~,U) of a ring with involution R in ~n(~)=Dn(S) are
morphisms of quadratic L-groups
(C,a,U) ! : Lm(R ) ...... , Lm+n(A) = Lm+n(S) (m,n~0)
[]
Example 4.3 Given a Hurewicz fibration F ,E PIB with J
the fibre F a finite n-dimensional geometric Poincare
complex we shall define in ~5 below a symmetric
representation (C(F),~,U) of Z[~I(B) ] in Dn(Z[~I!E) ]),
with F the pullback to F of the universal cover E of E I
and ~=([F]~-)-I :C(F) ,C(F) n-* the Poincare duality
chain equivalence. The algebraic surgery transfer maps
will be defined in ~5 to be
v II
f~
e~
m
• rl
f't
0 gl
v 0 p,
I
v v
;vl
/A
~..
~-~,
~
~
~,, <
~ o
~
4 e.
N
~,
Pt
[]
~ 0
0 '~
0
C
N r'
,t.~,
°[°
~-
0 0
II
II
0
-,
0
0 Cl 0
0 0
II 1 0
r~
II
~.
f'r 1 rt
v O"
,<
"<
I~
~-
0 0
rt
~
0 rt
0 El
O~
~h
~,
+
~'~
\V
El
~ 0
'~
0
0
191
k C O = ES $
1
k : C O = ~S
l
C = 0 for r ~ 0 , r
k . , c0 = (rS) ;
1
(s I , S 2 , . • . ,S k)
((tl ,t 2 ..... t k) : tlSl+t2s2+...+tkSk ) ,
U = I : R = Mk(S) , H0(Homs(C.C)) °p = Mk(S)
The generalized Morita maps ~:L.(R) JL.(S) in this
case are just the usual Morita maps, which are
isomorphisms for the projective and round L-groups. See
Hambleton, Taylor and Williams [5] and Hambleton,
Ranicki and Taylor [4] for Morita maps in quadratic
L-theory.
£3
Example 4.6 Let F=V{*) ,E P:B be a k-sheeted finite
covering, so that K~E) is a subgroup of KI(B) of index
k. There are evident identifications of spaces
and also of Z-module chain complexes
c ( ~ ) = z [ x 1 ( s > ] = S Z [ ~ I ( E > ] k
The symmetric representation (C(F).~,U) of Z[KI(B) ] in
~ o ( Z [ K 1 ( E ) ] ) = ~ ( Z [ K 1 ( E ) ] ) associated to p:E ,B (as in
4.3) is given by
U : Z [ K I ( B ) ] = H o ( H o m z i K I ( B ) ] ( C ( F ) , C ( F ) ) ) ° P
192
restriction
H0(Homz[~I(E)](C(F),C(F)))°P = Mk(Z[~l(E)]) ,
a = e l : C(F) = eZ[~I(E)] k k
H°mz[~rl (E)](C(~)'Z[~I(E)]) = @Z[~I(E)Ik "
The algebraic transfer maps i n this case are the
composites
Palg : L m ( Z [ ~ I ( B ) ] ) U!
Lm(Mk(Z[~] (E) ])) , Lm(Z[~] (E) ])
with U! induced by U as in 2.5 and ~ the Morita maps of
4.5. In this case Palg can be described more directly
by the restrictions of Z[~l(B) ]-module actions to
Z[~l (E) ]-module actions, and it is clear that ! !
Palg=Pgeo "
[]
Example 4.7 The algebraic sl-bundle transfer maps of
Munkholm and Pedersen [ I0] and Ranicki [16,%7.8] !
Palg:Lm(R) JLm+I(S) are defined for any ring with
involution S, with R=S/(t-I) for a central element t6S
such that [=t -I (We are only dealing with the
orientable case here). From our point of view these are ! t
the quadratic L-theory transfer maps Palg=(C,a,U) ' of
4.1 with (C,~,U) the symmetric representation of R in
~I(S) given by
d = 1-t : Ct = S ~ C O -- S ,
193
CI = -t : C I = S ) C O = S
1 : C O = S , C 1 = S
For an sl-bund]e S I ~E P*B
S=Z[KI(E) ] , t=fibre~Kl (E).
orle takes R=Z[~I(B) ] ,
[3
~5. The al~ebraic surgery transfer
A map p:E ~B of connected spaces with homotopy
fibre of the homotopy type of a finite (or finitely
dominated) CW complex F determines a representation of
Z[~I(B) ] in ~(Z[~I (E) ] )
(C(F),U:Z[~I(S)] 'Ho(Homz[~I(E) ] (C(F),C(F)))°P)
as in 1.7. We shall now show that if F is a finite
n-dimensional geometric Poincare complex then for any
choice of orientation map w(B) :~i (B) )Z 2 in the base
there is defined a symmetric representation (C(F),G,U)
of Z[~I(B) ] in ~n(Z[~I(E) ]), and hence obtain from ~4
quadratic L-theory transfer maps
' ~) ' • = (c( ,=,u)" : Palg
Lm(Z[~I(B)]) , Lm+n(Z[~I(E)]) (m~O)
In ~8 below we shall identify these algebraic surgery
transfer maps with the geometric surgery transfer
maps.
There is no loss of generality in assuming that
F JE P JB is a Hurewicz fibration with the fibre
F=p-l(*) a finite CW complex F. If F is disconnected
then p:E ,B is the composite of a Hurewicz fibration
194
p' :E ~B' with connected fibre p,-1 (,) and a finite
covering B' ,B. Since transfer theory is we11-known
for finite covers (cf. 4.6) there is no loss of
generality in taking F to be connected. In fact, the
algebraic transfer maps are defined in exactly the same
way for disconnected F, and only the geometric
treatment of the orientation maps has to be modified by
using groupoids instead of groups.
Transport of the fibre along paths in the base
space gives a map QB ,F F which on KO induces a group
morphism U:KI (B) ,[F,F] to the monoid of homotopy
classes of self-maps of F (Whitehead [24,p. 186]).
Analogously, one has the pointed transport of the
pointed fibre along paths in E, defining a morphism
U+:~I(E) ~[F,F] + to the monoid of pointed homotopy
classes of pointed self-maps of F. Homotopy along a
path defines a morphism ~i (F) J IF,F] + (Whitehead
[24,p.98ff ]) .
Proposition 5.1 The transport maps define a morphism
from an exact sequence of groups to an exact sequence
of pointed sets
P . ~I(F) , ~l(E) , ~I(B) , {I}
I0 ii , , ~ ' I ( F ) , [ F . F ] + , [ F , F ] , ( 1 }
[]
We shall now use 5.1 in the case when F is a J
geometric Poincare complex to lift an orientation map
w(B) for ~| (B) to an orientation map w(E) for ~I (E).
195
Definition 5.2 An orientation map for a group K is a
morphism w:~ IZ2={+l }. Let Z[~] w denote the ring Z[~]
with the w-twisted involution
- : Z[~l ' Z[~] ; E n g , ~ n w(g)g -I gEff g gE~ g
Given a chain complex C in ~(Z[K]) let WCn-* denote the
n-dual chain complex C n-* in ~(Z[K]) defined using the
w-twisted involution on Z[~] • If w is trivial WCn-* is
written as C n-* . Let Z w denote the right Z[K]-module
with additive group Z and
Z w X Z[~] ' Z w ; (m,gE~E ngg) , m(gE~ w(g)ng)
Let Wz denote the left Z[~]-module defined in the same
way.
[]
When w is clear we abbreviate Z[~] w to Z[~] •
S
An n-dimensional geometric Poincare complex X is a
(connected) finite CW complex together with an
orientation map w(X) :~i (X) ,Z 2 and a fundamental
class
[X] 6 Hn(X;Z w(X) ) = Hn(ZW~z[~l(X) ]C(X))
such that the Z[~l (X) ]-module chain map
)c(~ [X]~_:w(X )n-* ,C(X) is a chain equivalence, with
the universal cover. See Wall [21] for the general
theory.
The orientation map w=w(X) :K=~I(X) ~Z 2 of an
196
I
n-dimensional geometric Poincare complex X is
determined by the topology of X. since the cap product
with a fundamental class [X]EHn(X;Z w) defines an
isomorphism of Z[~]-modules
[X]n- : Ho(WC(~)n-*) ........ , Xo(X) = Z
If Hn(x) is defined to be H0(C(X)n-*) using the
untwisted involution (~=g-t) on Z[~] then we get
Hn(x)mWz.
Definition 5.3 Let X be an n-dimensional geometric I
Poincare complex.
i) The degree of a pointed self-map f :X JX is the
number d(f)EZ such that
f
with f:X
cover X.
ii) The homotopy orientation
morphism
: Hn(x) , Hn(x) ; l " d(f) ,
,X a lift of f to a self map of the universal
of X is the monoid
^ zx w ~(x) : IX,El + .... = ; f b d(f) ,
w~th Z X the monoid defined by Z and multiplication.
[]
Let f :X ,X be a pointed self homotopy
equivalence, inducing an automorphism f.:~ ~ of the
fundamental group ~=~I(X). A lift f:X-----*X of f to the
universal cover X induces a Z-module chain equivalence
f:C(X) ,C(X) which is f.-equivariant
197
~(gx) : f,(g)(x) 6 C(X) (g6~,x6C(X))
The induced isomorphJ sm of additive groups
~ * H n w z : (X~): ,Hn(X~)=Wz is also f,-equivariant. Hence we
have
f. W
w : wf . : K J ~ ' Z 2
and f, defines an automorphism f,:Z[~] w ~Z[~] w of the
ring with involution Z[~] w The Z-module automorphism
f,:Hn(X;Zw)=z .Hn(x;zW)=z is such that
^ = Z 2 CZ x f,([X])=d(f)[X], with d(f)=w(f)6<2l} . In
particular, it follows that the orientation map w and A
the homotopy orientation w are related by a commutative
diagram of monoid morphisms
~](x) , [x,x]
I t^ wl I ( _ + l > , Z x
Proposition 5.4 For any pointed self homotopy
equivalence f :X ~X there is defined a chain homotopy
commutative diagram of Z-module chain complexes and
chain equivalences
d ( f ) ( f - 1 ) * Wc(~)n-* w c n-*
1 I 'XJ~l ~ I ExJ~-
c ( ~ ) ........... , c ( ~ )
with the horizontal chain maps f,-equivariant, and the
198
vertical chain maps ~i (X)-equivariant.
[3
s
Definition 5.5 An n-dimensional Poincare fibration
F *E P JB is a Hurewicz fibration with the fibre F an I
n-dimensional geometric Poincare complex, together with
an orientation map w(B) :~i (B) JZ 2 . The lift of w(B) is
the orientation map
!
p ' w ( B ) = w ( m ) : K I ( E ) , Z 2 ;
A
g ......... ~ w(B) (p, (g)) .w(F) (U+(g))
+ "%
with U as in 5.1 and w as in 5.3.
[]
i
Proposition 5.6 An n-dimensional Poincare fibration
F ,E P ~B determines a symmetric representation
(C({),~,U) of Z[~I(B) ] w(B) ~n ~n(Z[~(E) ] w(E)) with
~=([F]~-)-l :C(F) ,C(F) n-* the Poincare duality chain
equivalence and (C(F),U) the representation of Z[ffl(B) ]
in ~n(Z[ffl (E) ]) associated to p.
Proof: We have to show that
U : Z [ N I ( B ) ] w ( B ) , Ho(Homz[~I(E)])(C(F),C(F)))°P
is a morphism of rings with involution, or equivalently
that for every gE~I(B ) there is defined a chain
homotopy commutative diagram of Z[~I(E) ]-module chain
complexes
n 0
i-I-
r~
0 m
rr
0 r"
• f'r
0
rl"
n rl"
0 0 ~h
rt'
I -=
+ W 0
v
TI
O~ r'
v v r-,
p.. v
v
v
rr
0 El
,-~
0 ~.
rT
rt
i ~,
t~
0 rr
:Y
eo
O"
r)
,9
rl.
B)
13
N
v v 1 + N
r~
v v W
O
v
h~
0~ H
v %J
"J
• I
~-,,
I ~
rr
r~
0 ~
" ~
:::7
:3
""
0 I%
1 r~
(~
'~
~-~
1~
.
~ ~
O"
• ../
~ I~
,'I
/
~"
:~
0 0
< ~
t~
m
~'
~ 0 rt
rr
0 rt
:3
" 0
~'~
0 ~
[]
r~
,o
r~
~"
o
~-h
~1
rt
"f
0 ~
" 0
"O
~'h
El
,~
'
1=
"M
N
,..,
0 ~
v
t
~o
g =
t
~ v
! I*
~o
~o
200
# p = -®(c(v),~,u) : B(Z[~I(B)] ) J ~)n(~[~[(E) ])
and the generalized Morita maps ~ of %3.
%6. The geometric st_ Irgery transfer
Wall [22] defined the rel~ surgery obstruction
o,(f,b)~Lm(Z[~l(X) ]) for a normal map
(f,b) : (M,~M) ~(X,~X) from a compact m-manifold with
boundary (M,~M) to a finite m-dimensional geometric /
Poincare pair (X,~X) with ~f=fl :~M ~X a homotopy
equivalence, and b:~M----~ X a map from the stable normal
bundle of M to a topological reduction of the Spivak
normal fibration ~X of
involution on Z[~] (X) ]. The
property that o,(f,b)=0 if
X, with the w(X)-twisted
surgery obstruction has the
(and for m~5 only if) (f,b)
a homotopy equivalence of
B with finitely
orientation map
to realize every
~s normal bordant rel~ to
pairs. Given a connected space
presented ~I(B) , and given an
w(B) :~I(B) ~Z2, it is possible
element XELm(Z[~I(B) ] ) (m)5) as the surgery obstruction
of an m-dimensional normal map (f,b):(M,~M) J(X,~X)
with a ~l-isomorphism reference map X )B and
orientation map w(X) :~i (X) ,,,~l (B) w~B~ ~Z 2
x = ~.(f.b) E Lm(Z[NI(B) ])
s
The total space E of an n-dimensional Poincare
fibration F ,E P,B over an m-dimensional geometric s
Poincare complex B is homotopy equivalent to an
(m+n)-dimensional geometric Poincare complex, with the !
orientation map the lift w(E)=p'w(B) :~I(E) ,Z 2 in the
sense of 5.5 of the orientation map w(B) :~i (B) JZ 2
(Ouinn [ 12] , Gottlieb [2]).
Quinn [II] used the realization theorem for
201
surgery obstructions to define geometric transfer maps
in the quadratic L-groups for a fibre bundle (or even a
block fibration) F ,E P JB with the fibre F a compact
n-manifold
! Pgeo : Lm(Z[~fl(B)]) , Lm+n(Z[KI(E)]) ;
O,((f,b) : (M,0M) ,(X,~X))
, ~ , ( ( g . c ) : ( N . D N ) ' ( Y . D ¥ ) )
Here, (g,c) : (N,~N) ,(Y,~Y) the (m+n)-dimensional
normal map equipped with a reference map Y~E obtained
from the n-dimensional normal map (f,b) :M -~X by the
pullback of p along a reference map X ,B.
The surgery obstruction of Wall [22] was defined
using geometric intersection numbers on the homology
remaining after surgery below the middle dimension. The
theory of Ranicki [ 14] , [ 15 ] associates an invariant in
Lm(Z[~I(X) ]) to a normal map (f,b):(M,DM) :(X,DX) of p
m-dimensional geometric Poincare pairs, with b:~ M ~PX
a map of the Spivak normal fibrations and ~f:~M ,OX a
homotopy equivalence. The quadratic kernel of (f,b) is I !
an m-dimensional quadratic Poincare complex (C(f) ,~) !
over Z[~l (X) ] . Here, C(f" ) is the algebraic mapping
cone of the Umkehr Z[~l(X)]-module chain map
-I ([x]~-) m - * f
f : c ( ~ , o ~ ) , c ( ~ ) , c ( ~ ) m - *
[M]~- ,
with X the universal cover of X, f:M .X a
~l(X)-equivariant lift of f to the pullback cover
M=f X of M. The Poincare duality chain equivalence is
given up to chain homotopy by the composite
202
!
(I+T)~o : C(f" ) m-*
[M]~- , c ( ~ )
e ,~ m- ~
, C(~,~M)
C(~,O ~ )MM e , c(f') J
with e :C(M,DM) ~C(f ! ) the inclusion. The quadratic
signature of (f,b) is the cobordism class
!
o,(f,b) = (C(f"),~) 6 Lm(Z[~l (X) ])
J
A normal map from a manifold to a geometric Poincare i
complex determines a normal map of geometric Poincare
complexes with quadratic signature the surgery
obstruction.
Definition 6.1 The seometric surgery transfer maps of t
an n-dimensional Poincare fibration F ~E P,B with
finitely presented ~i (B)
Pgeo : Lm(Z[~I(B) ]) Lm+n(Z[~1(E) ]) ;
a.((f,b) :M iX) ' a.((g,c):N ~Y) (m~5)
are defined using the quadratic signature of normal I
maps of geometric Poincare complexes. Here, (g,c) :N ...... ,Y
is the (m+n)-dimensional normal map obtained from an
m-dimensional normal map (f,b) :M ,X by the pullback of
p along a reference map X ,B.
[]
Theorem 6.2 The geometric surgery transfer maps of an i
n-dimensional Poincare fibration F ,E P ~B coincide
with the algebraic surgery transfer maps
203
! ! Pgeo = Palg :
Lm(Z[~I(B) 1) Lm+n(Z[ffl(E) ]) (m~5)
[3
The proof of 6.2 is
proof would express the
pullback normal map of s
geometric Poincare
tensor product of
map of the base
complexes (f,b) :M
complex (C(F) ,¢) . proof of the
the untwisted
deferred to %8. The ideal
quadratic kernel of the
the total (m+n)-dimensional
complexes (g~c) :N :¥ as a twisted
the quadratic kernel of the normal J
m-dimensional geometric Poincare w
,X and the symmetric Poincare
This would generalize the chain level
surgery product formula in RanickJ [ I 5 ] in
case p=projection:E=BXF ,B
o.((f,b)Xl :MXF ,XXF) = a.(f.b)~o (F)
6 Lm+n(Z[KI(B)XKI(F) ])
which expressed the quadratic signature of a product
(f,b)Xl as the tensor product of the quadratic
signature of (f,b) and the symmetric signature
(F)=(C(F),#)6Ln(Z[~I(F) ]). However, this would
require the development of a fair amount of new
technology, translating the homotopy action of QB on I
the geometric Poincare complex F into a chain homotopy l
action of C(QB) on the symmetric Poincare complex
(C(F) ,#) over Z[~l(E) ] . For the purpose at hand we can
assume by the realization theorem that the
m-dimensional normal map (f,b) :M ~X is
[ (m-2)/2]-connected. In the highly-connected case we
can give a chain level geometric interpretation of both
the element U!o.(f,b)~Lm(~n(Z[~l(E) ])) and its image
204
under the generalized Morita map
~:Lm(~n(Z[~l(E) ])) ~Lm+n(Z[~I(E) ]). For a fibre bundle
F ~E P ,B ~ t is possible to dispense with some of the
algebra, using instead the fibred intersection theory
of Hatcher and Quinn [6] as o,tlined in Appendix I
below.
%7. Ultraquadrat~c L-theory
Ultraquadratic L-theory was developed in %7.8 of
RanickJ []6] in connection with the algebraic theory of
codimension 2 surgery. We use it here to recognize
quadratic Poineare complexes in the image of the
generalized Morita maps ~:Lm(~n(~)) JLm+n(~) of %3,
providing a tool for the identification in %8 below of
the algebraic and geometric surgery transfer maps.
Let A be an additive category with involution. As
in Ranicki [ 15] , [ 19] define for any finite chain
complex C in A and c=+1 the Z-module chain complex
W%C = WOZ[z2]HomA(C ,C) ,
.
with the generator TEZ 2 acting on HomA(C ,C) by the
E-transposition involution T =ET and W the standard K
free Z[Z2 ]-module resolution of Z
I-T w : . . . : Z ( Z 2 ] , Z [ Z 2 ]
I + T , Z ( Z 2 ]
I-T , Z ( Z 2 ]
An m-chain ~E(W%C)m is a collection of morphisms
@ = <~sEHomA(C ,C)m_s ~s~0}
such that for a cycle there is defined a chain map
205
(I+TE)~0:cm-* JC. An m-dimensional E-quadratic i
(Poincare) complex (C,~) ~n ~ is an m-dimensional chain
complex C in ~ together with an element
~EQm(C,~)=Hm(W%C) (such that (l+T¢)~O:Cm-* ~C is a
chain equivalence). The skew-suspension isomorphisms
: Qm(C,E) ' Qm+2(SC, -E) ; ~ , S~
are defined by (S~) s=+~ s (s~0), for any finite chain
complex C in ~. The skew-suspension maps
S:Lm(~,E) ~Lm+2(A,-¢) (m~0) ~n the +~-quadratic
L-groups are also isomorphisms, so that
Lm(A,E) = Lm+2(A,-¢) = Lm+4(A,E) (m~>0)
For E=I we write Qm(C, l)=Qm(C), Lm(~, 1)=Lm(~) ,
l-quadratic = quadratic.
and
Ultraquadratic complexes are E-quadratic complexes
(C,~) with ~s=0 for s~l.
For any finite chain complex C in ~ define the
abelian group
(Hom~ * m(C) ~ H m (C ,C)) = H 0 (Hom~(C m-* ,C))
C m-* of chain homotopy classes of chain maps : iC.
Definition 7. 1 An m-dimensional £-ultraquadratic
(Poincare) complex i__nn ~ (C, ) is an m-dimensional chain
complex C in ~ together with an element ~EQm(C) (such
that (I+Tc)~:cm-* ,C is a chain equivalence).
D
There is a corresponding notion of cobordism of
206
I
c-ultraquadratic Poincare complexes in A, with the
group denoted by ~m(A,c), and m-dimensional cobordism
by Lm(~) for c=+l. The c-ultraquadratic L-groups are
4-periodic, with
~m(A,E) = Lm+2(A,-E) = Lm+4(A,¢) (m~>0)
by skew-suspension isomorphisms, just like for the
c-quadratic L-groups L,. If A=B(R) for a ring with
involution R we write ~m(A) as Lm(R).
Define a map Qm(C) ,Qm(C,c); ,~ by ~0=~, ~s~0
(s~l). An m-dimensional E-ultraquadratic (Poincare)
complex (C,~) determines an m-dimensional quadratic J
(Poincare) complex (C,#). The forgetful maps in the
cobordism groups
Lm(A,¢) , Um(A,c) ; (C,,) , (C,,) (m~>0)
are surjective for even m and injective for odd m.
A
The ultraquadratic L-group Lm(~ ) was identified ~n
~ 7 . 8 of [16] with the cobordism group Cm_ 1 of knots
k:sm-lcsm+l (m~4). A Seifert surface for a knot
k:Sm-Icsm+l is a codimension 1 framed submanifold
Mmcs m+I with boundary ~M=k(sm-l). Inclusion defines an
m-dimensional normal map (f,b) : (M,~M) )(Dm+2,S m-I )
with quadratic kernel o,(f,b)=(C,~) such that
(Dm+2,M)=H,(M) The framing determines a map H,(C)=H,+I M jsm+I-M which induces a chain map ~:C m-* JC,
i
defining an m-dimensional ultraquadratic Poincare
complex (C,$) over Z. The knot complement u=sm+l-(open
nbhd. of k(sm-l)) has boundary Ou=sm-IxsI , and there is
defined an (m+l)-dimensional normal map
(U,~U) ~(Dm+2 sm-I 1 , )XS which is a F-homology
207
equivalence. Let (L m+l M m ; ,zM m) be the fundamental
domain for the infinite cyclic cover U of U obtained by
cutting U along M, and let
((e;f,zf),(a;b,zb)) :
(L m+l ;M m,zM m) , Dm+2X([0,1];<0},{l>)
be the corresponding (m+l)-dimensional normal map of
triads. The inclusions j:M ,L, k:zM ,L induce ! !
Z-module chain maps j,k:C=C(f" ) ~D=C(g" ) such that
j-k:C JD is a chain equivalence. The ultraquadratic
structure -~~eQm(C ) is determined by the symmetric
structure (I+T)~:C m-* ,C and j,k, since up to chain
homotopy
(j-k)-I j = ( ( I + T ) ¢ ) I : C J C ,
^ 3- (j-k)-Ik---T~((I+T) ) 1 : C C
More generally:
Proposition 7.2 Let (C,~) be an m-dimensional I
E-quadrat Jc Poincare complex in ~%. A cobordism
((3 k) :CeC ~D,(5@,@@-~)) with 3-k:C ;D a chain
equivalence determines an E-ultraquadratic structure
~EQm(C) with image ~EQm(C,E) , such that
(C,~) = D(cm-*,~) 6 im(~:L0(Dm(A),E)--"~Lm(A,E))
with (C m-* ^ ,#) a nonsingular E-quadratic form in Dm(A).
Proof : Define a morphism in ~m(~)
-I -I. J (j-k)
h = (j-k) 3 : C ..... , D J C .
By the chain homotopy invariance of the Q-groups we can
replace ((j k),(5~,~-~)) by a homotopy equivalent
208
c o b o r d i s m ( ( h h - l ) : C ~ C : C , ( 5 ~ , ~ - @ ) 6 O m + l ( ( h h - 1 ) , e ) ) .
On the chain level
h%(¢) - (h-l)%(~) = d(B~) £ (W%C) m ,
so that there J s d e f i n e d a chain homotopy
( I + T E ) S q J 0 : h ( I+TE)%b 0 ~- ( [ + T e ) q J 0 ( 1 - h ) :
C m-* ~ C
The m-dimensional ¢-ultraquadrat~ c
(C,~) in ~ defined by the chain map
I
Poincare complex
= h ( I + T c ) ~ 0 : cm_. ( I + T E ) ~ 0 h ~ C , C
^ ^* Cm- * is such that ~+~ ~-(I+TE)~O : ,C. Define a chain
x~(W%C)m+I such that ~-~b=d(x+~)~(W%C) m by
0 if s=0 cm+l_r_s Xs = : . • C
hTE#s_ 1 if s~>1 r
Thus ~--~EQm(C,E) and
A
(C,#) = (C, ) = ~(cm-*,~) 6 Lm(A,¢)
[3
Corollary 7.3 Let (f,b) :M ,X be an (i-l)-connected i
normal map of (n+2i)-dimensional geometric Poincare
complexes, and let
((e;f,zf),(a;b,zb)) : (L;M,zM) , XX([0, | ] ;{0},{I})
be an (i-l)-connected normal bordism between (f,b) and
a disjoint copy (zf,zb). If the (i-l)-connected normal
I m
C
u I ~
- m
I 0
? I=~.
Il
l
~ m
f~
0
r/i
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0 ~ 0
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210
Z[~I(X) ] satisfying the hypothesis of 7.2. It follows
that @6On+2i(C) $6H0(Hom~(cn+2i-*
map
is the image of the element
,C)) defined by the composite chain
-I j ( j - k ) : c n + 2 i - * ¢ 0 , C , ,,, D ~ C
with ¢ 0 [M]~-:cn+2i-* = ,C the Poincare duality chain
equivalence. The nonsingular (-)i-quadratic form
(s-ic n+2i-* ; , ) in ~n(Z[~I(X) ]) is such that
o , ( f , b ) ( C , @ ) I I ( S - J C n + 2 i - *
6 i m ( u : t 0 ( D n ( Z [ z ~ ] ( X ) ] ) , ( - ) j ) , Un (Z[ ,~ 1 ( x ) l , ( _ ) i ) )
= im(~:L2i(~n(Z[Kl (X) ] )) 'Ln+2i(Z[~l (X) ]))
rl
Proposition 7.4 Let ((3 j' ) :C~C' ,D, (5~,~-~')) be a l
cobordism of m-dimensional E-quadratic Poincare
complexes in ~, such that D, C(3) and C(~' ) are the
suspensions of (m-l)-dimensional chain complexes (up to
chain equivalence), with m~l . The chain homotopy
classes of the chain maps
-I 3" = inclusion : G = S D , S-Ic(j ' ) = F
= inclusion :
G = S-ID , S-]C(j) ~- C(j') m-* = F m-l-*
are the components of a morphism of c-symmetric forms
in Dm_I(A)
211
[ : l o.o, , HE(F) = (F~Fm_I_ , , [ 0 c 0
such that y /~=(I+T_E)HO:C__ m-l-* ,G for a certain
element @EQm_I (G m-l-* -e) determined by (B~,@~-@)
A
in ~)m_l (~) and if 0Eim(Qm_l (G m-l-*) ,Qm_l (Gm-I-*,-E))
then G is a lagrangian of the hyperbolic c-quadratic
form
HE(F)=(FeFm-I-*, [0 I0})
and (F,G) is a nonsingular c-quadratic formation in
Dm_l(~) such that
(C,~) = ~(F•G) E im(~:Ll(Dm_l(A),E) 'Lm(A,c))
Proof: Let (Dm+l-*,H) be the (m+l)-dimensiona] l
c-quadratic complex in ~ (not in general Poincare)
defined by the algebraic Thom construction• the image
of (S~/~-~')EQm+I (C(j j') •E) under the isomorphism
((1+Te)(~0,~0~-~)~) -I :
Qm+l (C(j j' ) ,E)
(Dm+l-* Qm+l '¢) = Qm-I (Gm-l-* , -¢)
Up to chain homotopy
_i D inclusion ~( /~ : G = S
S-Ic(j j' ) ~_ D m-* = G m-l-* ,
212
so that there exists a chain homotopy
Y /~ ~- (I+T_E)0 0 : G J G m-l-*
and
. * . * G i n - 1 - *
as required for G to be a lagrangian in HE(F). If
O~Om ( G m - l - * ( G m - l - * -I ,-E) is the image of OEQm_ 1 ) then
(G,0) Js the hessian (-¢)-quadratic form in Dm_l(~)
required for G to be a lagrangian in HE(F). The
algebraic Thom construction defines a one-one
correspondence between the homotopy equivalence classes J
of (m+l)-dimensional E-quadratic Poincare pairs in
and (m+l)-dimensional E-quadratic complexes in
(Proposition 3.4 of Ranicki [14]). Thus
((j j') :C~C' 'D, (~,~')) is homotopy equivalent to I
the (m+l)-dimensional E-quadratic Poincare pair
((0 +I) :OD ,D,(O,O~)) defined by
. I 1 d~D 0 (-)rd D
~D r = Dm-r~D r , ~D r - I = D m - r + l ~ D r-I '
- (o o 1 D O 0 = 1 0
~D m - r _- D r ~ D m - r • OD r = o m - r ~ D r ,
213
(_)m-r+s$ o ] =
~$I o o
~D m - r - 1 = D r + l ~ ) D m - r - 1
85 = 0 : ~D m-r-s : OD s
J ~D r =
( s ~ 2 )
Dm-reDr
Up to chain homotopy
* - 1 inclusion : F = S C(j')
S-IC(j j') - Dm-* Gm-l-*
so that there is defined a chain
f :C ,C(~ ). Choosing a representative
O:D )D m+1-* and a chain
* ~ ) ~ . ,D r e + l - * X : Y ~ ( I + T .D ......... d e f i n e a
g : D D , C ( ~ ) b y
equivalence
chain map
homotopy
chain map
g = 1 × ] :
0 W
r~D r * Gm-r-I = ~F = D m- ) C(~ )r r-I
such that
f%(#) -- g%(~O) 6 Om(C(I/ ),e)
Now (C(~ ),g%(~@)) is the m-dimensional E-quadratic i
Poincare complex in ~ constructed in 3.2 from the
nonsingular E-quadratic formation (F,G) in Dm_l (~), so
that
(C,~) = (C(M), f%(~)) = (C(~),g%(~O))
214
= I/(F,G) 6 im(/l:Ll(Dm_l(A),E) 'Lm(A,£))
[]
%8. The connection
We now connect the algebra and the geometry,
verifying the claim of Theorem 6.2 that the geometric J
surgery transfer maps for an n-dimensional Poincare
fibration F JE Y,B coincide with the algebraic surgery
transfer maps
! ! Pgeo = Pa]g :
Lm(Z[~I(B) ] ) , Lm+n(Z[~I(E) ]) (m~0)
We know from 1.9 how a CW complex structure behaves
under transfer on the cellular chain level. The
strategy is to encode the L-theory data in CW complex
structures, and to decode the lifted L-theory data from
the CW lifts using the ultraquadratic L-theory of %7.
We consider first the case m=2i. By Chapter 5 of
Wall [22] every e]ement x6L2i(Z[~I(B) ] ) (i~3) is the
Witt class of the nonsingular (-)i-quadratic form in
B(Z[~I(s)])
(Ki(M) , k:K
: Ki(M)
i(M)XKi(M) ' Z [ K I ( B ) ] ,
, Z[~I (B) ] / < a - ( - ) i a [ a ~ Z [ ~ l (B) ]} )
on t he kernel Z[~l(B)]-module
Ki(M) = ~i+1 (f) = Hi(f ! ) = ker(f, :Hi(M)
of an (i-l)-connected normal map (f,b):(M,~M) J(X,aX)
215
f r o m a 2i-dimensional manifold with boundary (M,OM) to s
a 2J-dimensional geometric Poincare pair (X,~X) , with
~f :~M, ,~X a homotopy equivalence, and with a
~l-isomorphism reference map X ,B such that
w(X) :KI (X) ~] (B) w~B~ ~ 2 . The adjoint of k defines an
isomorphism in ~(~[KI (B) ] )
k : K i (M) , K i (M) ;
u -~ ( v k(u,v) )
(Ki(M) ,k, ~) can be viewed as a nonsingular
(-)i-quadratic form (Ki(M)*,~) over Z[~I (B) ], with ~ an
equivalence class of ~[~I(B) I-module morphisms
~:Ki(M)* :Ki(M) such that
^ "^* -1 * ~0+(-)i~ = k : K i (M) , K i (M)
~(k(v))(k(v)) = ~(v) (vEKi(M)) ,
^ i+l * with equivalent to ~+X+( - ) X for any
Z [ ~ I ( B ) ] - m o d u l e morphism x:Ki(M ) ~Ki(M). The surgery
obstruction is thus given by
x = o.(f,b) = (Ki(M),k,I~) = (Ki(M) . )
E L2i(Z[~I(B)])
We shall be regardin E modules as O-dimensional
chain complexes, and for any qEZ we write sqc for the
q-fold suspension of a chain complex C, with
dsq C = d C : (sqc) = C r r-q ( s q C ) r _ I = Cr_q_ 1
! The quadratic kernel o.(f,b)=(C(f" ),~) of the
216
(i-l)-connected 2i-dimensional normal map (f,b) :M ~X
is an (i-l)-connected 2i-dimensional quadratic i
Poincare complex over Z[KI(B) ] which is homotopy ; ^
equivalent to (SiKi(M), ). Thus we can identify ~0=~,
and up to chain homotopy
( I+T)~ 0 ~+( ) i ; * X-1
v)2i-* i * t i (M) C(f" = S Ki(M) , C(f" ) = S K i
!
The quadratic structure ~EQ2i(C(f" )) is the equivalence *
class of Z[KI (B) ]-module morphisms :Ki (M) ~Ki(M)
described above . A choice of representative ~ is a
choice of ultraquadratic structure ~EQ2i(C(f" )) for the
quadratic structure ~Q2i(C(f !)). We now fix a choice
of ~o
Let {Vl,V2, ... ,Vk) be a base for the f.g. free
Z[Kl(B)]-module Ki(M) , and use the dual to define a
base for Ki(M)=Ki(M)* The functor of additive
categories with involution
p = -~(C ),~,U)
B(ZI~I(B)]) ' ~n(Z[KI(E) ])
sends t he morphisms in ~(Z[KI (B) ] )
Ki(M ) = ~Z[KI(B) ] , Ki(M) = ~Z[~I(B) ] k k
to chain homotopy classes of Z[KI (E) I-module chain
maps
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.,. t_
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m
(3
u.~.
~.
r~
IA
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"
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,~
o"
O v
Cl
v II
5~
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ffl
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+ +.+,
I :+
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v v
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It I x*
220
v .:S i Jint(M2iXDq) with nullhomotopies in X. Let V . be J 3
a regular neighbourhood of vj(S i) in MXD q, and let
Pj=closure(MXDq-Vj), so that
MXD q = VjVDvjPj ~P = OV V~(MXD q) , J J
(Vj ,~Vj) = vj (Si)X(Di+q,s i+q-I )
The intersection number
Xj,j, = X(vj,vj,) ~ Z[~I(B)] (l~j,j'~k)
is the image of I ~ Z [ K t ( B ) ]
Z[KI (B) ]-module morphism
under the composite
Hi(Si ) = Z[~I(B) ]
vj,
, , Hi (M) ~ Hi+ q (MXDq,MXS q-I )
' Hi+q(MXDq,P j) ~ Hi+q(Vj,OV j)
= Ho(Si ) = ZtnI(B) I
which can also be expressed as ~J
vj,
Hi(S i) = Z[~I(B) ] • Hi(M,~M )
([M]~-) v .
, H i ( ~ i ) = z I ~ I ( B ) j
i
The pullbacks from the n-dimensional Poincare fibration I
F ~E P ~B define framed Poincare immersions
w .:FnX Si ~N n+2i with nullhomotopies in Y, and with 3
~ i ~i (E)-equivariant lifts wj :FXS :N (l~j~k). Let
221
Wj,QjCNXD q be the total spaces o f the fibrations over
Vj,PjCMXD q, so that
NXD q = WjVDwjQ j , ~Oj = DWjQD(NXD q) ,
(Wj,DW~) = w3(FXSi)X(Di+q,s i+q-I )
For any embedding D 2i+qcint(V )CM2iXDq J
the pair
( (MXDq-int(D2i+q))Uvj,xlDi+ixDq ' P3 VDi+Ixsq-I )
has a relative CW structure with one (i+q)-cell and one
(i+q+l)-cell, such that the cellular chain complex in
8(Z[~I(B) ]) is kj,j, :Z[~I(B) ] ,Z[~I(B) ]. By 1.9 the
chain homotopy class of the Z[Kl(E)]-module chain map #
p (k3, j,) :C(F) :C(F) coincides with the composite
w3, c ( { ) , s - i c ( { x s i ) ,
s-iC(N,~N) .,. s-i-qc(NXDq 0(NXDq))
, s-i-qc(NXDq Q'j) __ s-i-qc(wj,0Wj) ~ C(F~XS i)
, c ( ~ ) ,
and hence also with the composite
W . !
3 c(~) , s-ic({xs i) , s - i c ( ~ , O~)
- 1 ( I N ] A - )
. . . . . . . . . , s i c ( ~ ) n - *
r .~ 4¢
w .
3 ,, s i c ( ~ x s i ) n - *
[F]A- , C ( F ) n - * , C ( F )
222
The (j, j' )-component
equivalence
of the Z[~I(E) ]-module chain
((I+T)~0)-I :
i - - - ~ n+2i-* C(g" ) = aS C(F) , C(g" ) ~ $siC(F) n-* k k
is thus the composite
sic(F)
# p ( X j , j , )
, s i c ( F )
( [ F ] ~ - ) - 1
, siC(F) n-*
and up to chain homotopy
t n+2i_~ (I+T)~0 : C(g')
# -I
~sic(~ ) p (X k
= ~sic(F) n-* k
~[F]~-
! i C , C(g ) = as (F) k
[]
We extend the description of the symmetric
structure of ~(g,c) given by 8. 1 to the quadratic
structure, using the ultraquadratic L-theory of ~7. A
choice of ultraquadratic structure ~:Ki(M) ~ ,Ki(M) for
o,(f,b) is used to construct a normal bordism between
(f,b) :M~X and a copy (zf,zb) :zM IzX which encodes
the quadratic self-intersection form ~ in the CW
structure. The quadratic structure of ~,(g,c) is then
decoded from the CW structure of the pullback normal
bordism between (g,c):N ~Y and a copy (zg,zc):zN ~zY,
using 1.9 and 7.3. The construction of the bordism is
motivated by the way in which the infinite cyclic cover
223
of a knot complement can be obtained hy cutting along a
Seifert surface.
Lemma 8.2 A choice of ultraquadratic structure $ for
(Ki(M),X,~) can be realized by an (i-l)-connected
(2i+])-dimens~onal normal bordism
((e; f,zf), (a;b,zb)) : (L;M,zM) xx([0,1];<0>,<l>)
between (f,b) :M ~X and
(zf,zb) :zM ,zX, such that
Z[~I(B) I-module morphisms j,k:Ki(M) ,K
the inclusions 3 :M ~L, k:zM ~L J s
j-k:K~ (M) ,K i (L) with
(j-k)-13 = ~)~ ; Ki(M) ~ Ki(M) ,
(j-k) lk = (_)i+l~ k : K~ (M) , Ki (M)
a
the
disjoint copy
difference of the
~(L) induced by
an isomorph:ism
The (i-l)-connected (2i+l)-dimensional normal map
(e/(f=zf),a/(b=zb)) : (L/(M=zM),DMXS I )
, ( x , D x ) x ( [ o , t ] / o = l ) = ( x , O x ) x s 1
is a Z[~t(B) ]-homology equivalence, with the homotopy
equivalence ~fXl :~MXS 1 ,~XXS 1 on the boundary.
Proof: Every based f.g. free lagrangian of the
(-)l-quadratic form (Ki(M),X,~)~(Ki(M),-X,-~) can be
realized by disjoint framed embeddings of S i in
MVaMX[0, I ]zM with nullhomotopies in X, such that the
trace of the surgeries on these framed embedded
i-spheres defines a normal bordism between (f,b) and
(zf,zb). The realization of the lagrangian
°.
r-'
f~
~t
v w v et
rt
0
÷ I I 1 °.
W
v
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v v N
v
0
,°
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01
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l.a.
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th
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n-
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~ l.a
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r- I
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v v
2 2 5
((h;g,zg),(d;c,zc)) :
(P;N,zN) , YX([0,1 ] ;{0},{I})
be the (n+i-l)-connected (n+2i+l)-dimensional normal
bordism between (g,c) :N ,Y and a disjoint copy
(zg,zc) :zN ~zY obtained from ((e;f,zf), (a;b,zb)) by
pullback from F ....... ~E P~B along the reference map X ~B.
The (n+i-l)-connected (n+2i+I)-dimensional normal map
(h/(g=zg),d/(c=zc))
P/(N=zN) Yx( [0, I I/0=I) = Y×S
is a Z[NI(E) l-homology equivalence. By 7.3 the
quadratic kernel ~,(g,c) is determined by the chain
homotopy classes of the Z[NI(E) l-module chain maps t ! ! I
C(g" ) *C(h" ), C(zg" ) ~C(h" ) and the Poincare duality !)n+2i-* t
chain equivalence C(g ,C(g" ). We shall now
arrange CW structures for (e,a) Jn such a way that only
cells in dimensions i,i+1 occur in the relevant pairs,
and 1.9 appl~es to obtain the Z[~I(E) l-module chain
homotopy data in the total spaces of the pullbacks from
F ,E P ~B as the algebraic transfers of Z[NI(B) l-module
data.
L is the trace of surgeries on (i-1)- and
i-spheres in M, so that (L,M) has a relative CW
structure with i- and (i+l)-cells, with the cellular
chain complex in B(Z[~I (B) ]) given by
d = j : C(L,M) i+ 1 = Ki(M) '~ C(L,M) i = Ki(e)
Replacing e:L ,XX[O, 1 ] by the inclusion of L in the
mapping cylinder it may be assumed that L is a
subcomplex of X, such that (X,L) and (X,M) have
cellular chain complexes in ~(Z[NI (B) ])
i+IK i (e)
: C(X,M)i+ I = Ki(M)~Ki(L)
e ~
c(x,~) -- s
d = ( 3 I )
i ~.
P~
~'
°
0 0
0
<
r~
r"
15"
~.
0
r~
II
v I v
! v !
t,,~,
v I!
r-
rt
::r
rt
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rr
0 3 0 0 rI"
0
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=r
0 r~
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I%
1 0
v ~
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m ',-
,. =l
II ~.
~ 0
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1
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°
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~r~,
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XZ
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@
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~
X~
v U
v
227
(n+2i)-dimensional quadratic t
Poincare
( ~ S i C ( F ) , O ) o v e r Z [ f f I ( E ) ] w i t h k
t n+2i-* ( I + T ) O o : C(g') = ~)S ic (FJ) n-*
k
e[F]~-
# -I esic(~ ) p (k ) , i , C(g) = ~s C(F) , k k
complex
~0 = P # ( $ X ) ( I + T ) / 7 0 :
' n+2i-* ~9[ F ]~- C(g') = ~SiC(F~) n-* , , , eSiC(F °) k k
p ( ' i , C ( g " ) = ~ S C ( > ) ,
k
' n+2i-r-s ! ~s = 0 : C(g') , C(g )r (s~l)
[]
This completes the proof of Theorem 6.2 in the
case m=2i, and we proceed to the case m=2i+l .
By Chapter 6 of Wall [22] every element
xEL2i+I(Z[~I(B) ]) (i~2) is the Witt class of the kernel
nonsingular (-)i-quadratic formation over Z[KI(B) ]
(F,G) = (Ki+l (U,~U),Ki+I (Mo,DU))
of an (i-l)-connected (2i+l)-dimensional normal map
(f,b) :(M,DM) l(X,~X) with ~f :DM ~DX a homotopy
equivalence, and with a ~l-isomorphism reference map
X .......... ~B such that w(X) :~I (X) ,KI (B) w~B~ ,Z 2 . Here, U is
the connected sum of a sufficiently large number k~0 of
framed embeddings siQint(M) with nullhomotopies in X to
generate the f.g. Z[KI (B) l-module Ki (M), and
M0=closure(M-U). Thus F=Ki+I (U,~U) is a based f.g. free
Z[~I(B) l-module , and G=Ki+I(Mo,~U) is a based f.g. free
~.
t~
~ ~
0 ~
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~.
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El
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t~
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~ ÷
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U t~
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~ m
~" 0
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-v~
e~ll ~, ~
...,,+I
'~.
~ ~.m
"~DQI
'-'- 0 nl
~ nl~
v
°m ~,r~
~v~
~ ~.+t== ~0
0 ~.0
m
~"
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m 0
-
rr
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o"
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,s q~ t~ co
,I
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tD v
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v ~
x tD ~,~
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+ v
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II
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:=
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231
%9. Change of K-theory
We now extend the definition of the algebraic !
surgery transfer maps (C,a,U) " :Lm(R) JLm+n(S) to the
intermediate L-groups, and show that they are
compatible with the Rothenberg exact sequences.
An involution R ~R;r ,r on a ring R determines a
duality involution *:P(R) ,~(R);P ,P =HOmR(P,R) on
the additive category ~(R) of f.g. projective R-modules
by
R X P , e ; (r,f) , (x , f(x).r) ,
e(P) : P ~ P ; x , (f ~ f(x))
The duality involution on P(R) determines involutions
on the algebraic K-groups
* : K 0 ( R ) ' K 0 ( R ) ; [ P I ' [ P ] ,
* : K1 (R) , K1 (R) ;
T(f :P 'Q) , T(f :Q 'P )
and also on the reduced K-groups
Ki(R) = coker(Ki(Z) ,K.(R)) (i--0 I)
The intermediate quadratic L-groups L~(R) of a
ring with involution R are defined for *-invariant
subgroups X~Ki(R) (i=0, I), such that x EX for all xEX.
The intermediate L-groups for X=<0>,Ki(R) are written
as
K0 ( R ) ( 0 ) ~ K I ( R ) s L . ( R ) = L ~ ( R ) , L . ( R ) = L . ( R )
( 0 } ~ K 0 ( R ) K I ( R ) I . , ( R ) = L , ( R ) = L ~ ( R ) = L , ( R )
232
For *-invarJant subgroups XC_X'C_Ki(R) there is defined a
Rothenberg exact sequence
with
, L X ( R ) ~ LX' ( R ) n n
, ,,, H n ( z 2 ; X ' / X )
............ : L X (R) n-I
) o , °
Hn(z2 ;X'/X ) :
{aEX' /Xla =(-)na}/(b+(-)nb IBEX' /X}
See RanickJ [ ] 3] , [ ]4] for further details.
We consider first the torsion case XC_K] (R).
A representation (C,U) of R in D(S) determines a
transfer map in the abso]ute torsion groups
(C,U) :KI(R) 'KI(S) (Example 1.8), and also in the
' K (R) (S) By reduced torsion groups (C,U)' : 1 'KI "
definition, D(S) is the homotopy category of finite
chain complexes of based f.g. free S-modules. We sha] 1
now make use of the bases.
Proposition 9.1 Let (C,(I,U) be a symmetric
representation of R in Dn(S) , for some rings with
involution R,S.
i) For any *-invariant subgroups XC_K 1 (R) , YC_K 1 (S) such
:C n-* that (C,U) "(X)C_Y and T(~:C )6Y there are defined
transfer maps in the intermediate torsion L-groups
' LX(R) : L Y (S) (n>0) (C,~,U)" : m m+n
ii) For any *-invariant subgroups X~X'~KI (R) ,
Y~Y'GKI(S) such that (C,U) !(X)~Y, (C,U) !(X')~Y ' , T(~)6Y
there is defined a morphism of Rothenberg exact
sequences
233
LX, ^ Z2 ' LX(R) , (R) ......... ~ Hm( ;X'/X) ~ ... m m
( c , a , u ) " ( c , ~ , u ) " ( c , u ) '
Y ' ~m+n ' LYm+n(S) --~ Lm+ n (S) --~ (~o ;Y'/Y)~ ~ ....
Proof : The transfer map in the reduced torsion groups
(C,U)" :KI (R) ~KI (S) is such that
*(C,U) ! = (-)n(c,u)!* : K] (R)
Let m=2i. For any nonsingular (-)i-quadratic form (M,~)
on a based f.g. free R-module M=R k the n-dimensional p
(-)i-quadratic Poincare complex (~C,@) representing k
! (C,U) "(M,%b) has reduced torsion
T ( ( I + T ) e o : ~ g C n - ,@C) k k
' i * * -- (C,U)' T(~+(-) ~ :S :M ) 6 K1 (S)
image of T(~+(-) i@*)6KI(R).~ Similarly for the m=2i+l
and formations.
[]
Next, we consider the projective case X~Ko(R) . It
is more convenient to work with the preimage of X in
Ko(R) , so we regard X as a *-invariant subgroup of
Ko(R) such that [R]EX.
Given a ring S let E(S)=D(~(S)), the homotopy
category of finite-dimensional f.g. projective S-module
chain complexes. A representation (C,U) of a ring R in
E(S) determines transfer maps in the algebraic
K-groups
! (C,U)" : Ki(R ) = Ki(•(R))
234
Ki(S ) = Ki(?(S)) (i=O, I)
(Example 1.8). For n~0 let En(S)=Dn(D(S)), the full
subcategory of E(S) with objects n-dimensional f.g.
projective S-module chain complexes. An involution on S
determines the n-duality involution C :C n- on ~n(S).
Proposition 9.2 Let (C,~,U) be a symmetric
representation of R in ~n(S), for some r~ngs with
involution R,S.
i) For any *-invariant subgroups X~K0(R) , Y~K0(S) such !
that [R]6X, [S]6Y, (C,U)(X)~Y~K0(S) there are defined
transfer maps in the intermediate class L-groups
' LX(R) , L Y (S) (n~O) (C,~,U)" : m m+n
ii) For any *-invariant subgroups X~X'~K0(R) , !
Y~Y'~K0(S) such that [R]EX, [S]EY, (C,U) " (X)~Y,
(C,U) " (X')~Y' there is defined a morphism of Rothenberg
exact sequences
, LX(R) , LX' (R) m m
, H m ( z 2 ; X ' / X )
I I l (c,a,u) (c,~,u) (
!
C,U) '
y, ^ + -- L +n(S) ) Lm+n(S) ---* I! m n(Z2;Y'/Y) I . o , °
[]
The proof of 9.2 is somewhat more involved than
that of 9.1.
A splitting (B,r,i) in ~ of an object (A,p) in the
idempotent completion ~ is an object B in ~ together
with morphisms r:A ~B, i:B ~A in ~ such that
ri = 1 : B ......... ~ B , Jr = p : A , A
Lemma 9.3 A functor of additive categories F:~ ~
235
extends to a functor F: ,~ if and only if for each
object (A,p) in ~ the object (F(A),F(p)) in B has a
splitting in ~. Any two such extensions of F are
naturally equivalent.
Proof: It is clear that the splitting condition is
necessary for F to extend to {, so we need only prove
that it is sufficient. For each object (A,p) in
choose a splitting (B,r,i) of the object (F(A),F(p)) in ^ ^
~, and set F(A,p)=B, with (B,r,i)=(F(A), I, ]) for
p=l :A JA. For a morphism f : (A,p) ~(A' ,p' ) let
i f r' ~(f) : {(A) = B , A , A' , {(A') = B'
[]
An additive category A is idempotent complete if
the functor ~ ,~;A ,(A, I) is an equivalence of
categories. Applying 9.3 to I :A 'A we have that A is
idempotent complete if and only if every object (A,p)
in ~ splits in A. If ~ is idempotent complete every
functor F:~ :~ extends to a functor F:A ........ ,~, namely
the composite of and an equivalence S
For any ring S the additive category ~(S) of f.g.
projective S-modules is idempotent complete, with every
object (A,p) in ~(S) split by the triple (B,r,i)
defined by
r : A ~ B = im(p) ; x , p(x) ,
i = inclusion : B , A
This is the special case n=0 of:
Lemma 9.4 For any ring S and any n~O the homotopy
category En(S ) of n-dimensional f.g. projective
S-module chain complexes is idempotent comp]ete. 2
Proof: For every chain homotopy projection p=p :D ~D
of an object D in En(S) there exists by Lemma 3.4 of
LUck [7] an (n+l)-dimensional infinitely generated
0 ~
. :~
~
. !~
rT
~ 0
~
rt
rt
0 ~
~
v ~
R
~ D
0
v f/
)
~'1
t't
0 ~
, k~
~ 0
'~
0 II
',,ID
I ,~
-
v v
+ r~
v
Cl
f:I
oo
t ~
v II v v
rt
t~
0 0 r,t
°° I
v • .
~
=1,
~I
v
~ o
~
+
~ ~
v
~ 0
v ~
N
0
0 ~
rT
0 v
< 0
0 ~,
0 lf
f' ~
' ~
. 0
~
~u
0 ~
~ ~
~h
.
0 0
X- 1 >
v
~e
v
0 < 0
='0
~.,
0
~ 0
rr
f3
N
rt
0
ffl
0 <
rt
0
0 ~
[]
0 0 ~
"
0 ~
" ~
" I,
~
0 ..
0 "~
"~
/ 1
0
'~
0 ".-
4 G
' ~
"
n I~
'
0 ~
r
0"
G
~ v
m
~ v
°.
0 I 0 D
:r
0 0
0 r-r
'~
0
mr
237
F = -@(C,~,U) : B(R) ' ~[n(S) ; R ) C
[]
The proof of 9.2 is now completed by observing
that the transfer map in the projective class groups !
( C , U ) " : K o ( R ) ,K0(S) is such that
' )n ' *(C,U)" = (- (C,U)" * : K0(R) : K0(S)
Remark 9.6 Our methods also apply to construct
algebraic surgery transfer maps in the round L-groups
L~X(R) of Hambleton, Ranicki and Taylor [4], which are
defined for *-invariant subgroups X~KI(R). For any
symmetric representation (C,~,U) of R in En(S) and any
*-invariant subgroup X~KI (R), Y~KI (S) such that t
(C,U) "(X)~Y there are defined round L-theory transfer
maps
' LrX(R) , L rY (S) (m~0) (C,~,U)" : m m+n
which are compatible with the round L-theory Rothenberg
exact sequences.
D
Remark 9.7 The connection established in %8 between the
algebraic and geometric surgery transfer maps extends
to the intermediate cases, and also to round L-theory.
[]
Remark 9.8 Our algebraic constructions apply also to
the E-quadratic L-groups L,(R,E), which are defined for
a ring with involution R and a central unit £ER such
that EE=I. L2i(R,E ) (resp. L2i+l(R,e)) is the Witt
group of nonsingular (-)iE-quadratic forms (resp.
formations) over R. A symmetric representation (C,~,U)
of R in Dn(S) such that U(E)=~:C ....... ,C for a central
unit
238
~6S with ~=I induces transfer maps
!
(C,a,U) " : L m ( R , ¢ ) , Lm+n(S,~ ) (m~O)
Hitherto we considered the
L,(R, I)=L,(R) , with 0=I£S.
c a s e c = 1 6 R f o r which
[3
Appendix l . Fibred intersections
! ! The proof of Pgeo=P~Ig_
algebraic properties of the
bundle F ,E P :B with the
n-dimensional manifold it is
the algebra:ic and geometric
coincide more directly, using
theory of Hatcher a n d Quinn
in %8 makes heavy use of
the L-groups. For a fibre
fibre F a compact
possible to verify that
surgery transfer maps
the bordism intersection
[6] to obtain fibred
versions of the geometric intersection forms (resp.
formations) used by Wall [22] to define the surgery
obstruction of a highly-connected even (resp. odd-)
dimensional normal map. The quadratic kernel of the
pullback normal map is the fibred intersection form
(resp. formation) both algebraically and geometrically.
We now sketch the argument for the intersection pairing
in the even-dimensional case, leaving the
sel f-intersection function ~ and the odd-dimensional
case to the interested reader.
Given two maps vi :Qi )M (i=l ,2) let E(Vl ,v 2 ) be
the pointed space of triples (Xl ,x2,w) defined by
points xiEQi and a path ~:[0,[] )M from ~(0)=Vl(X 1 ) to
~(1)=v2(x2) , so that there is defined a homotopy fibre
square
E il'v2) ' i 1 t vl
v 2 Q 2 ...... ~ M
239
Given a stable vector bundle ~ over a space M let fr
~n (M,~) be the bordism group of n-manifolds N equipped
with a map N ~M and a compatible stable bundle map
~N ~" For trivial ~ this is the usual framed fr S
cobordism group Qn (M)=~n(MV{*})" For
vl =v2 :Q] =Q2={ *} ,M the homotopy pullback is the loop
space, E(* *)=QM
Now suppose that M is an m-manifold, and that
vi:Q i ,M is an immersion of a qi-manifold Qi (i=1,2)
such that vl (QI) intersects v2(Q2) in general position.
Let QI~Q2 denote the corresponding
(ql+q2-m)-dimensional submanifold of M. The bordism
Jnvariant of the intersection ([6,2. I ]) is the bordism
class
)%(Vl,V2) = [QI~Q2]
60 fr m(E(v v ) ~Q2~TM) ql+q2 - 1' 2 '~QI
If Q1 and Q2 are (q]+q2-m+I)-connected the map
E(*,*)=QM JE(Vl ,v 2) induces an isomorphism ([6,3. I ])
fr (QM) O~rl+q2 -m(E(*'*)) = Oql+q2_ m
fr _m(E(v 1 v2 ) ~Q2~TM) Oql+q2 ' '~QI
which is used as an identification.
Let (f,b) :M ~X be an (i-1)-connected
2J-dimensional normal map with a ~l-isomorphism
reference map X ~B, with the surgery obstruction
~,(f,b)=(Ki(M),k,~)6L2i(Z[~|(B) ]) defined as in Chapter
5 of Wall [22]. Let Vl,V2, ... ,Vk be a base of the
kernel f.g. free Z[~I(B) ]-module Ki(M)=~i+l(f).
Represent each vj6Ki(M ) by a pointed framed immersion i v :S ~M with a nullhomotopy in X. The values taken by
3 i the (-) -symmetric form (Ki(M),X) on the base elements
are just the bordism intersections
240
k(vj,vj,) f r 6 n 0 (E(vj,vj,),p i~ i~fM)
S S
fr = n O ( Q M ) = H o ( Q M ) = Z [ K [ ( B ) ]
(l~j, j'~k)
Now let (g,c) :N ,Y be the (i-l)-connected
(n+2i)-dimensional normal map with a ~l-isomorphism
reference map Y ,E obtained from (f,b) :M ",X by the
pullback of the fibre bundle F ,E ~ ,B along X :B. The • S i pointed framed immersions v : ~M (I~j~k) with 3
nullhomotopies in X lift to pointed framed immersions i w .:S XF ,N with nullhomotopies in Y. On the chain
3 level this corresponds to lJ fting the kernel
i i Z[~I(B) I-module chain complex C(f" )=S Ki (M)=~S Z[KI (B) ]
k
to the kernel Z[K1 (E) ] - m o d u l e chain complex
' ic( C(g')=~)S F~). The bordism intersections k
fr(E(wj,w3, ),~ ~ ) k(wj,wj, ) E n n SIXFe~S XF~TN
fr = n n (QMXF,~F) (l~j, j'~k)
are the images of the bordism intersections )~(vj,vj,)
under the geometric bordism transfer map
fr p" = -XF : •0 (riM)
f r Qn ( QMXF, ~F ) ;
X J XXF
The Poincare duality isomorphism of based f.g. free
Z[~I(B) ]-modules
(k(vj,vj,)) :
* t si ' 2i-* SiK (M) ~ C(f' ) (M C(f') = - - = K ) 1 i
i
is lifted to the Poincare duality chain equivalence of
241
chain complexes of based f.g. free Z[KI(E) l-modules
' n+2i-* (X(wj,wj,)) : C(g') = ¢siC(F) n-* k
!
", C ( g ' ) : e s i c ( ~ ) k
Using the Poincare duality Z[~I(E) l-module chain
equivalence [F]~-:C(F) n-* ~C(F), the action of ~M on
the ~] (E)-equivariant homotopy type of F and Hurewicz
maps there is defined a commutative diagram
flfr(f~M ) ~ , H0(QM) ~- , Z[~I(B)]
I l Iu
I [ F I ~ -
f r ( n M X F :o F ) ~ It ( n M X F ) ~ H n ( C ( F " ) ® Z [ ~ I ( E ) l C ( F O ) ) • •n ' n
The anticlockwise composition gives the geometric
surgery transfer Pgeo on the level of intersections,
while the clockwise composition gives the algebraic
surgery transfer P a l g
Appendix 2. A counterexample in symmetric L-theory
An n-dimensional Poincare fibration F .... ~E P ,B does
not in general induce transfer maps in the symmetric !
l,-groups p" : L m ( Z [ K I ( B ) ] ) ~Lm+n(z[~I(E) ]), either
algebraically or geometrically. It is not possible to
define p geometrically since the symmetric L-groups
are not geometrically realizable (Ranicki [ 16,7.6.8]).
There are two obstructions to an algebraic definition !
of p , which requi~es the lifting of an m-dimensional
symmetric Poincare complex (C,~) over Z[~I(B) ]
representing an element (C,#)6Lm(Z[KI(B) ]) I
(m+n)-dimensional symmetric Poincare complex
over, Z[KI(E),] representing the putative
p" (C,#)=(C" ,#" )ELm+n(z[KI(E) ]). The symmetric
are not so it cannot be assumed 4-periodic,
to an ! !
( c " , ~ ' )
transfer
L-groups
that (C,#)
242
is highly-connected as in the quadratic case. In the
following discussion we assume that the fibre F is
flnJ te, and that the chain complex C consists of based
f.g. free ~[~i (B) ]-modules. The two obstructions to ! !
] ifting (C,#) to ( C ' , ~ " ) are given by:
i) it may not be possible to lift C to a based !
f.g. free Z[K l (E) ]-module chain complex C" with a ! ! ! !
filtration FoC'C_FIC'C...C_FmC" =C" such that the
connecting chain maps between successive filtration
quotients are given up to chain homotopy by
' ' r #(Or) = p#(d C) : FrC'/Fr_IC" = S p
! !
S(Fr_IC" /Fr_2C' = srp#(Cr_l ) (l~r~m)
r # where S denotes the r-fold dimension shift and p
the functor of %1
is
# p = -®(C(F),I:) : 3(Z[~I(B> 1) , Dn(Z[~I(E)]) ,
!
ii) even if C' exists, it may not be possible to ¢
lift the m-dimensional symmetric Poincare structure # ¢
on C to an (m+n)-dimensional symmetric Poincare ! !
structure #' on C" .
If C can be assembled over B in the sense of Ranicki !
and Weiss [2(3] then it can be lifted to C ° , but in
general it is not possible to assemble Z[~I(B) I-module
chain complexes, so already i) presents a non-trivial
obstruction to the existence of transfer in symmetric
L-theory. Even if the obstruction of i) vanishes (e.g.
if B is an Eilenberg-MacLane space K(~I(B),I)) then ii)
may present a non-trivial obstruction. This is
illustrated by the following example, which exhibits
the failure of a projection of rings with involution
p:S ,R=S/(I-t) (t = central unit E S, ~=t-IEs) to 1
induce an S -bundle symmetric L-theory transfer map
' O(R p" :L ) ........... ,LI(S) analogous to the SI-bundle quadratic !
L-theory transfer map p" :L0(R ) ~L[(S) (cf. 4.7). The
243
! ! ! transfer p" (C,¢)=(C" ,¢" ) of a 0-dimensional symmetric
Poincare complex (= nonsingular symmetric form) (C,#) k
over R with C0=R is defined if the symmetric kXk
matrix
#0 = (@0) 6 Mk(R)
!
can be lifted to a kXk matrix ~06Mk(S) !
p(#~)=#0~Mk(R) and
t ! * !
t#; - (#6) = ( Z-t)# i 6 Mk(S)
such that
! t * for some symmetric kXk matrix #i=(#i) EMk(S) , so that
! ! •
( C ' , ¢ ' ) : i s a 1 - d i m e n s i o n a l s y m m e t r i c P o i n c a r e c o m p l e x t
over S w:ith C'=C(l-t :S k :sk). In particular, for
2 2 S = Z2 [Z2XZ2 ] = Z2 [t,u]/(t -l,u -I)
= t , u = t+u+l ,
p : S ' R = Z2[Z2 I = Z2[u]/(u2-1) ;
t ~ ] ~ U J U
the transfer is not defined for the 0-dimensional J
symmetric Poincare complex (C,#)=(R,u) over R, for !
although C can be lifted to C" and #0 can be lifted to ! !
#0 there does not ex:ist a symmetric #~. Both the
obstruct:ions to i) and ii) vanish for the visible
symmetric L-groups VL (Z[~]) of Weiss [23] provided
that B is an Eilenberg-MacLane space K(~I (B), I) , in
which case there are defined transfer maps
! L m VL m+n p : v ( Z [ ~ I ( B ) 1) , ( Z [ ~ I ( E ) I ) .
REFERENCES
[ i ] W.Browder and F.Quinn
surgery theory for G-manifolds and
stratified sets
Proceedings 1973 Tokyo Conference on
244
Man~ folds, Tokyo Univ. Press, 27-36 (1974)
[2] D.Gottlieb S
Poincare duality and fibrations
Proc. A.M.S. 76, 148-150 (1979)
[3] I.Hambleton, J.Milgram, L.Taylor and B.Williams
Surgery with finite fundamental group
Proc. Lond. Math. Soc. (3) 56, 349-379 (1988)
[4] l.Hambleton, A.Ranicki and L.Taylor
Round L-theory
J. Pure and Appl. Alg. 47, 131-154 (1987)
[5] I.Hambleton, L.Taylor and B.Williams
Maps between surgery obstruction groups
Proc. 1982 Arhus Topology Conf.,
Springer Lecture Notes 1051, 149-227 (1984)
[6] A.Hatcher and F.Quinn
Bordism invariants of intersections of
submanifolds
Trans. A.M.S. 200, 326-344 (1974)
[7] W.L~ck
The transfer maps induced in the
algebraic K 0- and Kl-~r0ups by ~ fibration I.
Math. Scand. 59, 93-121 (1986)
[B] W.L~ck and I.Madsen
Equivariant L-theory II.
to appear
[9] W.L;Jck and A.Ranicki
Chain homotop~ prooections
to appear in J. of Algebra
[I0] H.Munkholm and E.Pedersen
The sl-transfer i___nn surgery theory
Trans. A.M.S. 280, 277-302 (1983)
245
[ 1 1 ] F.Quinn
geometric formulation of surgery
Princeton Ph.D.thesis (1969)
[121 S
Surgery on Poincare and normal spaces
Bull. A.M.S. 78, 262-267 (1972)
[ 1 3 ] A.Ranicki
Algebraic L-theory I__~. Foundations
Proc. Lond. Math. Soc. (3) 27, 101-125 (1973)
[14] The algebraic theory of surgery I. Foundations
Proc. Lond. Math. Soc. (3) 40, 87-192 (1980)
[15] The algebraic theory of surgery II.
Applications to topology
Proc. Lond. Math. Soc. (3) 40, 193-283 (1980)
[161 Exact sequences i__nn the algebraic theory o f surgery
Mathematical Notes 26, Princeton (1981)
[171 The algebraic theory of finiteness obstruction
Math. Scand. 57, 105-126 (1985)
[181 The algebraic theory of torsion I. Foundations
Algebraic and Geometric Topology,
Springer Lecture Notes 1126, 199-237 (1985)
[191 Additive L-theory
Mathematica Gottingensis 12 (1988)
[20] A.Ranicki and M.Weiss
Chain complexes and assembly
Mathematica Gottingensis 28 (1987)
[21] C.T.C.Wall d
Poincare complexes
Ann. of Maths. 86, 213-245 (1970)
246
[ 2 2 ] Surgery on compact manifolds
Academic Press (1970)
[23] M.Weiss
On the definition
preprint
of the symmetric
[24] G.W.Whitehead
Elements of homotopy
Springer (1978)
t h eory
W.L~ck: Mathematisches Institut,
Georg-August Universit~t,
Bunsenstr. 3-5,
34 G~ttingen,
Bundesrepublik Deutschland.
A.Ranicki : Mathematics Department,
Edinburgh University,
Edinburgh EH9 3JZ,
Scotland, UK.
SOME REMARKS ON THE KIRBY-SIEBENMANN CLASS R. J. Milgram
In this note we study the relations that hold between the Kirby-Siebenmann class { K S } • H4(BsToP; Z/2) and the first Pontrajagin class.
The first result is that that the natural map p0 : BSTOP ~ B s e does not detect { K S } no mat ter what coefficients might be used. However, the homology dual of { K S } is in the image of the Hurewicz map
lr4(BsToP) ~ H4(BsToP; Z/2).
In fact there is a unique non-zero element [KS] • z d B s T o P ) of order 2, and po([KS]) # 0 • 7q(Bsa) . In particular this implies that w4 + { K S } is a mod(24) fiber-homotopy invariant of SPIN-TOP bundles. However, it is interesting to ask what happens when w2 is non-zero. To understand this we introduce an intermediate classifying space, BTSG for which we have a factorization
f P~ po = p" f , BSTOP ' BTSG BSG.
BTSG is univeral for the vanishing of transversality obstructions through dimension 5. Additionally, BTSa is built out of finite groups (Z/2-extensions of the symmetric groups S,~) in the same way that BSG is constructed from the S , . As a result, explicit construction of homotopy classes of maps into BTSG is often possible.
We show that H4(BTSG; Z/2) = Z /2 (~ Z/48 and that the homology dual of the Kirby-Siebenmann class maps to 24 times the second generator. Thus, this transversality theory does detect {KS} . But note also the Z/48. Our main question is the extent to which it gives rise to a fiber homotopy invariant of topological R"-bundles. The general result is
T h e o r e m I: Let ~, ¢ be two stabte R'~-bundles over X , and suppose they are fiber homotopy equivaient. Then there is a E H2(X; Z/2) and
24a 2 + PI(~) + 24{KS(~)} = P1(¢) + 24{KS(%b)}
in H4(X; Z/4S) where PI(() is the Z/48 reduction o[ the t~rst Pontraja~in c ~ s .
In other words, there is an element A • H4(BTSG; Z/48) with f*(A) = PI + 24{KS}, and (I) gives the effect of different liftings of a map po "g : X ----* BSTOP , BSG on A.
H2(BsToP; Z/2) = Z/2 with generator w2, so the possible factorizations of P0 through BTSG differ in their effect on A only by 24w~. In particular this gives
C o r o l l a r y : I f M 4 is a compact dosed topological manifold with even index, and u is its stable normal bundle, then wg = 0 • H2(M; Z/2) and
v ' f * (A) = PI(u) + 24{KS(~)}
is independent of the choice o f f factoring po.
This note came about in answer to a question of Frank Quinn. He pointed out that in [M-M l the exact structure of BSTOP, and the various surgery maps in dimension 4 were never worked out. But currently it appears very useful to understand them. Of course, we do not a t t empt to work out explicit geometric methods for evaluating the new invariants. But knowing what they are and how they fit together should make that fairly direct.
248
T h e h o m o t o p y t y p e s o f Bso, Bsa in d i m e n s i o n <_ 7
A Postnikov system for Bso through dimension 7 is given by
(1) Bso ,K(Z/2, "2.) ,K(Z, 5)
withK- invar ian t2{Sq2Sql ( t2)+t2 .Sql ( t2)} . (Note that HS(K(Z/2, 2); Z) = Z / 4 w i t h generator having mod(1) reduction "f and
(2) "7 = Sq2Sql( t2)+t2"Sq ' (12)-
Moreover, fl4(t~) = 7-) The stable homotopy of spheres is given in the first 6 dimensions by
(3) 7r,~(S °) =
Z i = 0 Z/2 i = 1, generator 77 Z/2 i = 2, generator '~1 Z/24 i = 3, generator v 0 i = 4 , 5 Z/2 i = 6, generator '~2 = v 2
and we will use the same names for the corresponding elements in ~ri+1(Bsa) ~ Try(S°). One relation that should be kept in mind is r}~l = 12v, since it also holds in rc.(BsG), though the relation q2 = ~1 which holds stably does not hold in ~r.(Bsa).
L e m m a (4): A Postnikov system for Bsc through 7 is given by
K ( Z / 2 , 2) × K(Z / 2 , 3) × K(Z/2, 7) ,K(Z/24 , 5)
where the K-invariant is 2{Sq2Sqa(t2) + t2 " Sql(t2)} + 4{,-,C'q2(ta)}.
P r o o f : With Z/24-coefficients the K-invariant for Bso maps back to the image of the cor- risponding K-invariant for Bso. Hence, the class in (2) must appear in the K-invariant. Also, the kernel of the map HS(K(Z/2, 2,3); Z/24) ~ HS(K(Z/2, 3); Z/24) is gen- erated by 4Sq2(ts). It follows that 4Sq2(ts) is the only term which can be added to the K-invariant. But, in fact, this term must be involved in the K-invariant because there is the homotopy relation which we have already noted qt;1 = 12v, since 7/is detected by Sq 2.
In order to understand the integral homology of Bso, BSTOP, and the intermediate space BTSG which we will introduce shortly, we need a method for obtaining Bochstein information from K-invariants. The following result will suffice.
,,,¢
L e m m a (5): Let K ( Z / 2 ' , j ) × K ( Z / 2 , j + 1 ) - - K ( Z / 2 " , j + 1) be given with
= 2",~(~j) + 2"-~(~i+~),
then tile fiber E of the map ~: is K ( Z / 2 i + ' - u ' - I × Z/2~').
P r o o f : The homotopy exact sequence of the fibration in dimensions j , j + 1 is
(6) 0 ,.,+~(E) ,Z/2 ,,Z/2' , . , (E) ,Z/2~. ,0
249
But the term 2"-ILj+~ in a*(~j+1) implies that g. is injective in (6). Thus E is a K 0 r , j ) and r is given as an extension in the sequence
o , z / 2 "-~ ,~j(E) ,Z/2'----,0.
The type of this extension is determined by the term 2~(~(, j)) in ~*(Lj+~). From this (5) follows.
(4) and (5) imply that there is a mod(8) Bochstein
/3s(~) = {Sq2(~a)} in H*(Bsc; Z/2).
Additionally, the Hurewicz image of z, is {w~} + 2{tg"} since this is already true in Bso, where it is well known. As a consequence H4(Bsa; Z) = Z/2 ~ Z/24 with generators {w~}, {w~'*} respectively, and 12u is in the kernel of the Hurewicz map.
T h e s t r u c t u r e of BSTOP t h r o u g h d i m e n s i o n 7
From the fiberings
(7) G/O , Bso , BsG
i l i G/TOP , BSTOP ' B s o
and the well known result of Kirby-Siebenmann that 7r4(G/TOP) = x4(G/O) = Z, but that the map between them is multiplication by 2, we get the diagram of extensions in a-4,
(8)
0 , Z
0 , Z
.24 Z
, ~ ( B s r o v )
Z/24 ,,, 0
:t , Z/24 ~ 0
The only way this diagram can commute is if ~r4(BsToP) = Z/2 @ Z with the element of order 2 mapping to 12 • v, and the generator of the Z-summand mapping to v.
Z/2 i = 2 L e m m a (9): zG(BSTOP) = Z (9 Z/2 i = 4
0 4 < i < 8 . BSTOP through this range is given by
Moreover, a Postnikov system/'or
(10) K(Z/2 , 2 ) x K(Z/2, 4) , K ( Z , 5)
with K-jnvariant 2{Sq2Sq~(~_) + ~2" SqI(L2)}.
(This is clear.)
In particular, the class {KS*} 6 H4(BsToP; Z) which is in the Hurewicz image of the element of order 2, must go to zero in H4(BsG; Z), since, in homotopy, it goes to 12v. This shows that {KS*} has no homology (or cohomology) relations implied by the
250
map into Bsa . However, in homotopy, the fact that it maps to 12u should have some consequeences.
The s p a c e BTSG
The failure to detect the l{{rby-Siebenmann class in H.(Bsa; Z) is the influence of the first exotic class L3. In fact, the term 4Sq2(ta) in the 5-dimensional K-invariant (4) is exactly the difficulty. (For example, if we kill u,2 but leave t3 in H*(Bs6; Z/2) the resulting space has only Z/4-torsion in H4( ; Z).) Hence it is natural to consider the classifying space BTSG obtained from B s c by killing the exotic class t3- For definiteness, recall that t3 is detected with 0-indeterminacy in the Thom-complex M S G by applying the twisted secondary operation associated to the relation (w2 + Sq2)(u,2 + Sq 2) to the Thorn class, and using the Thorn isomorphism to bring the class back to Bse . For details see [R].
We have the fibration sequence
(11). K ( Z / 2 , 2) ,BTsG ' BsG , K (Z /2 , 3)
with K-invariant L3. This is the universal space for fiber homotopy transversality to hold in the Thom space, at least through dimension 5 (Compare [B-M]). Indeed, a fiber homotopy sphere bundle ( ~ X and reduction to BTSG is equivalent to the condition ~3(() = 0 6 H3(X; Z/2), together with a specific choice of 2-dimensional cochain c so
where f : X - -~ B s a classifies (. This situation is very close, but certainly not the same as the situation studied in [F-K]. Also, there is a factorization of the canonical map BSTOP--*BsG as
BSTOP ~ BTSG , BSG.
Precisely, there are exactly two such factorizations differing by a map
BSTOP --~ K ( Z / 2 , 2).
Now, we look at the 6-skeleton of BTSG. This is the 6-skeleton of the 2-stage Postnikov system
K(Z/2 , 2 ) x K ( Z / 3 , 4) ,K(Z/8 , 5)
with K-invariant 2{Sq2Sq ~ (~2)+ ~" Sq~(~2)}. From (5) the resulting space has 4 ~h integral homology group given as
H4(BTsG; Z) : Z/2 (9 Z/48
with generators (w4)*, (w~)* respectively. Here, w~ can be identified with t2. Note that this implies that the Kirby-Siebenmann class maps non-triviMty to 24((w~)').
The proof of theorem (I)
f Lemma (12): Let X ~ B T s c be given and suppose f ' is the composite
(~,f) X , K(Z / 2 , 2) x BTS G ' BTSG
251
where # is the principal bundle map K(Z/2 , 2) × BTSG ~ BTSG, then
f"*{w~} = f'{w~.} +24a 2 6 H4(X; Z/48).
P roo f : H4(K(Z, 2) x BTSG; Z/16) = (Z/2) 2 $ Z/4 ~ Z/16 with generators
8(t2 ® w2), 8(1 ® w4) of order 2, (4t~ ® 1) of order 4, and (1 ®w~) of order 16.
We will show that > (w~) = 8(tr, ® 1) + 1 ® wa. We first note, by nat.urality and the primitivity of w~ in H4(Bso; Z) that 8(L.~ ® u,~.)" is not in this image. Next, we look at the cohomology Serre spectral sequence of the fibering
K(Z /2 , 2) :BTsG ~BsG
with Z/16-coefficients. E ° 4 = H4(K(Z/2,~; Z/16) = Z/4, with generator 4t~. Also, E~ o = H4(BsG; Z/16) = Z/2 • Z/8 with generators 8w4, 2(w2), and
E~ o = Hh(BsG; Z/16) = (Z/2) 3 + Z/8.
Here, only the Z/8 is of interest. It has generator Sq2(~), so d~(4~) = 4Sq2(t3), and at Eoo' ~, i + j = 4, only E ° 4 = Z/2, E 4 0 = Z/8 ~ Z/2 are non-zero. Thus there is a non-trivial extension for H4(BTSa; Z/16)
0 ,Z/8 (generator 2w 2) ,Z/16 ,Z/2 (generator 8~) ~.0.
But this forces the result.
Theorem (I) is direct from (12). The corollary follows, also, since the assumption of even index implies that w2(M4) ~ = 0 (mod 2). Hence, either lifting gives the same map in cohomology with Z/48-coefficients.
C o n c l u d i n g r e m a r k s
From Quillen's work we know that BsG®Z2 can be identified with B(B+(SO(Fa))) in dimensions ~ 6, and as B(B+(Soo)) in all dimensions. Here, Sod is the infinite symmetric group. Similarly we can describe BTSG as B(B+(SO(F3))) in this same range. Moreover, BTSG can be given as B(B + (Sod)) in all dimensions. Here, these new groups are described by central extensions
Z/2 ,SO(F3) *SO(F3) ,0
z/2 , ~ ~s~ ,o
where, for Sod the extension is the (unique) non-trivial one for which the transposition (1, 2) continues to have order 2. This might be very useful in understanding Casson's recent results on the Rochtin invariant.
It seems direct to use the description above of BTSG by finite models to calculate the order of the classes which carry the remaining Pontrajagin classes. I hope to return to this later.
Also, there is a second factorizing space for the map BSTOP ~ Bsa, namely the space where we kill all the exotic classes cr(e 2,_1,2~_1). The precise structure of these classes is not entirely known, but there is considerable information in [R]. So it should
252
be possible to understand the higher torsion in the cohomology and homology of this intermediate classifying space. Moreover, it is likely that it is the universal space for the vanishing of transversality obstructions.
B i b l i o g r a p h y
[B-M] G. Brumfiel-J.Morgan, Homotopy theoretic consequences of N. Levitts obstruction theory to transver- sality for sphericad fibrations, Pac. J. Math (1976) 1-100
[F-K] M. Freedman-R. Kirby, A geometric proof of Rochlin's theorem, Algebraic and Geometric Topology, A.M.S. Proceedings of Symposia in Pure Mathematics, Vol. XXXII(1) (1978) 85-98
[M-M] Ib Madsen-R.J.Milgram, Classifying Spaces for Surgery and Cobordism of Manifolds, Ann. of Math Studies #92, Princeton U. Press (1979)
JR] Doug Ravenal, Thesis, Brandeis University (1970)
November, t987 Sonderforschungsbereich 170 G/Sttingen Universit~t
The Fixed-Point Conjecture for p-Toral Groups
by
Dietrich Notbohm
i. Introduction
Suppose that X is a space with an action of the topological
group G. Let X G and X hG denote the fixed-point set respectively the homotopy fixed-point set of this action. We define
X hG := maPG(EG,X)
as the space of G-maps in the category Top of topological spaces and maps. As model for EG any acyclic G-complex is possible.
(Here complex always means CW-complex.) X hG is then unique up to homotopy.
The definition is not given in the category S of semisimplicial sets, as it happens in [DZ] and [M] for finite groups. For topological groups the space EG, constructed as nerve over a category, is not a simplicial set, but a semisimplicial object over the category Top . Therefore the same is true for the space
maPG(EG,X) ,
where X is interpreted as the singular chain complex of the topological space X. For finite groups both definitions agree up to weak homotopy [BK; chapter VIII ].
There are two other interpretations of the homotopy fixed-point set. The first one is as section space
U(EGXGG~BG)
of the fibration EGXGX ~ BG ,
the second one is as fixed-point set
map(EG,X) G ,
where G operates canonically on map(EG,X) . Let p be a prime, for all time fixed. X~ denotes the
E/p-completion in the sense of Bousfield and Kan [BK]. X is called E/p-good, if X~ is p-complete [BK; 1,5].
Especially nilpotent and other "nice" spaces are ~/p-good. Look at [BK;VII].
254
The unique G-map EG -~ * , where * is the one point set with a trivial G-action, induces a map
X G = maPG(*,X) , maPG(EG,X)
Functoriality of the composition gives a composite map
X G^ __~ X *G ~ X ~hG P P P
which fits into a commutative diagram
X G , X hG
1 l X G^ ~ X ~G
P P
Definition: A topological group N is called a p-toral group, iff there exists an exact sequence
1 --~ T --~ N --~ P --~ 1 ,
where T is a Torus and P a finite p-group.
Theorem: If N is a p-toral group and X a E/p-good connected finite N-complex, then the map
X N^ ~ X AhN P P
is a weak homotopy equivalence.
Remark: The analogue theorem for finite p-groups, but without the technical condition E/p-good, is proved by H. Miller in [M]. It is the foundation of the rest of the paper. For this result J. Lannes found another proof.
It is a pleasure to thank J. McClure for valuable discussions about the book of Bousfield and Kan.
2. Proof of the Theorem
We need some remarks: 2.1 Remark: Let
1 ~K ~G ,H ,I
be an exact sequence of topological spaces and assume, that H is finite. Let X be a G-space. H acts on the
255
fixed-point set X K canonically. We have
(X K) H = X G
As H is finite, EG is an acyclic K-complex of finite type. We get
X hK = map(EG,X) K ,
where map(EG,X) is a G-space. Hence using the above equation and the exponential law for mapping spaces, we get the analogue:
X hG ~ (xhK) hH
2.2 Remark: Let f:X 1 * X 2 be a weak homotopy equivalence
and a G-map between two G-spaces X I, X 2 The horizontal
map in the diagram
EGXGX 1 , EGXGX 2
\ Y BG
is a weak homotopy equivalence. Because BG is a complex, the two spaces
map(BG,EGXGX i) , i = 1 , 2
are weak homotopy equivalent as well.
We denote with map(BG,BG)
id
the connected component of the identity and with
map(BG,EGxGXi) s
the space of all maps, which are homotopic to a section. If we look at the two fibrations
hG • ~ map~BG,EGXGXi) s ' " X 1 b map(BG,BG)id f
it is easy to see, that the two homotopy fixed-point sets
X. hG are weak homotopy equivalent. 1
256
Proof of the theorem: i) reduction to the case of a torus. Let
1 tT ~N ~P tl
be the exact sequence belonging to th~ p-toral group N.
X T is a finite P-complex. It is proved in [M] that
X N^p = (xT) PAp ' (xTp) hP
is a weak homotopy equivalence. Setting
X~ hT = maPT(EN,X ~) ,
remark 2.1 implies a weak homotopy equivalence
x^hNp "w (Xp hT)hN
The map
X T^ ~ X ̂ h T P P
is P-equivariant. Together with (2.21 we can reduce therefore the problem to the case of a torus.
ii) Let n be the dimension of T. We can think of ~/pk c S 1
k as the group of the roots of unity with order p and define
Ok := (~/pk)n , O. := ~ O k
The homomorphism o. ---~ T induces a mod p-equivalence
Bc~ --~ T ,
which is the same as to say that the map
Hj(Bc.;~/p) , Hj(BT;~/p)
is an isomorphism.
Now let X be a ~/p-good connected finite T-complex. Then there are the following maps
hT ho. x; --, x;
T __ xO~ X
--~ lira X ~hOk , p
--~ tim X ck
As T is a finite Uk-complex for all k, X has also the
structure of a finite Ok-complex. Using Miller's Theorem
[M] and the following three propositions, the proof will be finished in a straightforward way.
257
2.3 Proposition: Let X be a finite T complex. Then it is
X T = i~XUk
and the sequence of the fixed-point sets is a finite sequence.
2.4 Proposition: Let X be a E/p-good finite T-complex. Then the map
X~ hT , X; ha-
is a weak homotopy equivalence.
2.5 Proposition: Let X be a finite T-complex. Then it is
h a . ha . ~n(X ~ k) "n (X~ ) ~i,
3. Proofs of the Propositions 2.3 - 2.5
Proof of 2.3: If X is a finite T-complex, it consists of a finite number of cells of the form T/A x e , where AcT is a closed subgroup, n
T/Axe n belongs to X T if A=T and it belongs to X uk
if UkCA . Because a~ is dense in T and because there is
only a finite number of orbit types T/A, we get
X ok = X T
for k big enough.
3.1 Lemma: Let Y be a p-complete space. Let
Y • E ~ B
be a fibration, such that the action of ~i B on
Hn(Y;E/p) is nilpotent. Then the E/p-completion
induces a homotopy equivalence
between the section spaces. Proof: Under the above assumption the mod R fibre lemma [BK; II, 5] is applicable. We get a Eibre square
258
E I E *
B ~ B A P
w h i c h i n d u c e s a c o m m u t a t i v e d i a g r a m
map(B,E) s P map(B,B)id
map B~,E~) s , map(B~,B~)id
where the rows are fibrations and the columns are given by the completion. With the universal property of pullback diagrams, which fibre squares are, you can prove, that (**) is up to homotopy a fibre square too. The fibres of the rows in (**) are exactly the section spaces. This implies the Lemma.
3.2 Lemma: Let E. p B. , i=0,1 , be fibrations with 1 1
p-complete fibre, in such a way that the diagram
i 0 --~ il
B 0 --* B 1
is a fibre square. Assume that the operation of ~I(BI)
on Hj(EI;~/p) is nilpotent and that the map B 0 ~ B 1
is a mod p equivalence. Then the two section spaces
F(E 1 -~ B I) , F(E 0 -~ B 0)
are weak homotopy equivalent. Proof: The assumptions of the mod R fibre square lemma [BK; II, 5.3 ] are satified. We get up to homotopy a fibre square
E0p P Elp
1 1 % ; '
with homotopy equivalences in the rows. This implies that the associated section spaces of the fibre squares are weak homotopy equivalent. If you use 3.1, the proof will be finished.
Proof of 2.4: The diagram
259
Ec~xa X p --~ ETXTX p
BC~ --~ BT
is a fibre square with a p-complete fibre in the columns. Moreover BT is 1-connected. Lemma 3.2 applies.
3.3 Lemma: Let GlCG2c... be a ascending sequence of groups
and define G. := ~ G k . Let X be a G.-space. Then the map
X hG- ~ hl~!~m xhGk
is a weak homotopy equivalence.
Proof: For the definition of holim see [BK; XI].
We choose the Milnor model for the spaces EGw and EG k. Then EG. is exactly the union of the spaces EG k or
EG~ = ~ EG k .
This implies that
hG~ = xhGk X = maPG (EG ,X) = ~ maPGk(EGk,X)
xhGk On the other hand the maps ~ X hGk-I are fibrations. According to [BK; XI] there is a weak homotopy equivalence
EG. = lim X hGk hG k , holim X e----
Proof of 2.5: Because of Lemma 3.3 there is an exact sequence
0 , lira I ~n+l(X~ hak) , ~n(X~ ha') , lim ~n(X~ h~k)
for all base points [BK; XI, 7.4] By [M] we get
, 0
~n(X~ hak ) s ~n(X°kp)
Proposition 2.3 implies that the lim ~term must vanish. q
References:
260
[BK]
[DZ]
[M]
A.K. Bousfield and D.M. Kan: Homotopy Limits, Completion, and Localisation; Lecture Notes in Math. 304, Springer 1972.
W.G. Dwyer and A. Zabrodsky: Maps between Classifying Spaces; preprint.
H. Miller: The Fixed-Point Conjecture; to appear.
Dietrich Notbohm Mathematisches Institut der Georg-August-Universit~t Bunsenstr. 3-5
D-3400 G6ttingen Bundesrepublik Deutschland
Simply connected manifolds without S~-symmetry
V. Puppe
Several authors have studied the question of existence of manifolds
with little or no symmetry (s.[I],[2],[6],[7],[16],[17]), e.g.E. BloQmberg has
shown (s.[2]) that there exiSt closed manifolds which do not admit any
effective topological (continuous) action of a compact Lie group. For
his argument the presence of a rather complicated fundamental group is
essential.
From a completely different point of view M. Atiyah and F.Hirzebruch
had proved earlier (s.[1]) that a compact spin manifold M can not admit
an effective differentiable St-action if the ~-genus ~(M) is different
from zero. It has been shown, though, that the differentiability as-
sumption in their result is crucial, i.e. there exist examples of topo-
logical effective SI-actions on spin manifolds with ~(M) • O (s.[3] VI.
9.6 and [4]).
Here we prove, using the connection between P.A. Smith-theory and
deformation of algebras (s.[12],[13],[14]), that there exist simply con-
nected, closed, oriented, differentiable manifolds M such that any
closed, orientable manifold M with H~(M;~) ~ H*(M;~) (as algebras over
~) has no topological St-symmetry, i.e. does not admit any non trivial
topological St-action; in fact, there exist examples which admit non
trivial topological ~-actions only for (at most) finitely many primes
p (compare [11]).
S.Kwasik and R.Schultz have studied topological St-actions on 4-mani-
folds and - among other results - they show, by completely different
methods, that there exist many closed simply connected 4-manifolds with-
out topological St-symmetries (s.[19]).
I want to thank R. Buchweitz, J. Damon and A. larrobino for illumi-
nating conversations on the deformation theory of Artin algebras. In
fact, what is described in this note is more or less an interpretation
of certain algebraic results about deformations of algebras (s.[9],[IO])
in the context of SI-action from the view point of [12],[13],[14].
If X is a paracompact, finitistic St-space which is totally non
homologous to zero (TNHZ) in the Borel construction X G := EG x X (G=S I) G
with respect to ~ech cohomology with rational coefficients and if
dim~ H~(X;~) < ~, then the cohomology algebra B := H~(xG;~) of the fix
point set X G can be viewed as a deformation of the algebra A := H~(X;~).
262
A one parameter family of deformations A[t] = A ~ ~[t] (where "N" indi-
cates the twisting of the multiplication)with Ao=A and AI~B (disregard-
ing the grading is given by the cohomology H~(XG;~) of X G considered
as an algebra over H~(BG;~) = ~[t], deg(t) = 2 (s.[12] for details).
The property (TNHZ) is automatically fulfilled if H°dd(x;~) = O be-
cause then the Leray-Serre spectral sequence degenerates already for de-
gree reasons.
This suggests the following program to exhibit manifolds without S l-
symmetry:
I. Find a rigid graded algebra A* over @ with A °dd = 0 and dim~ A ~ < ~,
which fulfils Poincar6 duality ("rigid" means roughly that all alge-
bras obtained from A ~ by deformation are isomorphic to A~; s. [20]
for a discussion of different notions of rigidity and how they relate
to each other).
2. Realize A ~ as the rational cohomology algebra of a manifold M.
3. Check that H~(M;~) Z H~(MG;~) implies M = M G.
Yet there are several obstacles:
a) First of all it is not known (to me) and seems to be a very difficult
question to decide whether there exist non trivial rigid algebras of
the desired form (s. [20] for examples of non-commutative finite di-
mensional rigid algebras, in particular over ~/p ) .
b) To realize a Poincar@ algebra as the cohomology of a (simply connect-
ed) compact manifold one needs certain extra conditions to be satis-
fied if the dimension is divisible by 4 (s.[ 18]).
c) Even if one would find a non trivial rigid graded algebra A, the
isomorphism between A and some deformation of A need not respect the
grading (supposing the algebra obtained by deformation has an a pri-
ori grading).
But not every deformation of an algebra A ~ ~ H*(X;~) which is pos-
sible algebraically can be realized by an St-action on X. The one pa-
rameter families A ~ ~[t] which correspond to St-actions on X have cer-
tain special properties (s.[12],[13]) :
Thereexists a grading on A I (namely the one given by the isomorphism
A I ~ H~(xG;~)) such that A ~ ~[t] embeds into the trivial family
A I ® ~[t] as a graded algebra over ~[t] and the cokernel of this em-
bedding is ~[t]-torsion. In fact, this is just a reformulation of the
theorem, which says that the morphism H~(XG;~)~H~((XG)G;~) localization
induced by the inclusion X G --~ X becomes an isomorphism after localiza-
tion at (O). The property (TNHZ) gives that H~(XG;~) ~ H*((XG) G;@) is
injective° In particular A ~ ~[t] is then a jump deformation (s.[8]) in
the sense that all A := (A ~ ~[t] ® ~£ where ~e is ~ considered as a 6
~[t] ~[t]-module via ~[t] ~ ~, t ~ e, are isomorphic for e # O; moreover
263
A I has a filtration such that the associated graded algebra is iso-
morphic to A O (s.[ 13]).
Hence for the first part of our program we only need to know that A
is rigid with respect to "g-deformations", i.e. deformations of the
special kind described above. If A has that property we will say that A
is g-rigid. It is shown in [5] and [8] that a non trivial g-deformation
of A lowers the dimension of the second Hochscbild cohomology of A with
coefficients in A (the space of infinitesimal deformations), i.e.
dim~ H2 (Ao,Ao) > dim~ H2 (AI,AI). On the other hand there exist non
smoothable graded Artin algebras (already defined over ~) (s.[9],[I0]),
i.e. algebra which do not admit deformations to ~ x...x ~, in fact not
even to ~ x...x ~ if one extends the ground field to 6. Therefore, if
one starts with a non smoothable graded Artin algebra and considers all
algebras which can be obtained from A by (iterated) g-deformations,
there must exist non trivial g-rigid graded algebras (among the compo-
nents, i.e. direct factors of the algebras obtained). By Quillen's re-
sults in rational homotopy (s.[15] and also [18])one can realize such
an algebra as the rational cohomology of a simply connected finite CW-
complex and hence obtains:
Proposition!:There exist simply connected finite CW complex X such that
the strictly commutative graded algebra H*(X;~) is g-rigid (H°dd(x;~)=O).
For every St-action on such a space X the rational cohomology of X G
is isomorphic as a filtered algebra to H*(X;~) (s.[5]) . Within the
("filtered") isomorphism type of H*(X;~) one can choose a graded alge-
( ~ Ai) sh°uld be maximal f°r bra A* with "minimal degree", i.e. dim~ i o
each q. Let Y be a simply connected finite CW-complex with H*(Y;~)=A*,
then one gets the following:
Corollarv.1: For any St-action on Y the inclusion yG ~ y induces an iso-
morphism in rational cohomology.
Proof: By the choice of Y one has that H*(Y;~) and H*(yG;~) are iso-
morphic as graded vector spaces (and as filtered algebras). Therefore
the morphism H*(Y;~) ~ ~[t] ~ H~(yG;~) ® ~[t] induced by yG --~ y, can
only become an isomorphism after localization, if the evalution at t=O
(i.e. applying - ® ~o), which gives H*(Y,~) ~ H*(yG,~), is an iso- ~[t]
morphism, too.
Remark: Of course Corollary I does not imply that any St-action on Y
284
must be trivial. Instead of Y one could as well take YxD 2 ~ Y (D 2 :=
{x 6 ~2; IXl ~ I}) which clearly admits a non trivial St-action with
(YxDZ) G = y x {O}.
TO exhibit simply connected manifolds without Sl-symmetry we are look-
ing for connected Poincar~ algebras over ~ which are g-rigid. It is
shown in [9] and [10] that there are non smoothable connected Poincar~
algebras A (graded Gorenstein algebras) with A °dd = O and formal dimen-
sion of A equal to 6. This leads to the following
Proposition 2: If A is a non smoothable connected Poincar~ algebra of
formal dimension 6 with A °dd = O, then A is g-rigid.
Proof: Let B be an algebra obtained from A by a g-deformation. Then B = k U B. and B. is a connected Poincar~ algebra of even formal dimension
i=1 i i
< 6 with B °dd = 0 for i = I ,k It is easy to see that any connected -- 1 i . . . .
Poincar~ algebra C of formal dimension fd(C) = 2 or 4 and with C °dd = 0
is smoothable. (For fd(C) = 2 this is obvious since C ~ ~[x]/(x2) ; for
fd(C) = 4 a somewhat round about but simple argument is to observe that
C®¢ ~ H*(M;¢) where M is a connected sum of a number of copies of ~P2s
and therefore admits an St-action with isolated fix points.) Hence there
must be a component, say BI, in B with fd(B1) = 6. By the inequalities
A i > dim ~ B i there is precisely one component with formal dimen- i=q i=q
sion 6.
The top dimensional generator a 6 A 6 is mapped to a non zero element
in B1 ® 1 c BI ® ~[t] by the morphism A ~ ~[t] ~ B ® ~[t] ~ BI ® ~[t], k
= ~ BI . where the last map is induced by the projection B N B i i=I
(Otherwise the first map in the composition would not become an iso-
morphism after localization.)
Evaluated at t = O one obtains a morphism A ~ BI of connected
Poincar~ algebras which has non zero degree and hence is an isomorphism.
This implies, of course, that A ~ B ~ Bl, since dim~ A = dim~ B.
Theorem I: There exist simply connected, closed, oriented differentiable
6-dimensional manifolds M such that no closed, orientable manifold with
the same rational cohomology algebra as M admits any non trivial S l-
action.
Proof: Choose a non smoothable connected Poincar~ algebra A* of formal
dimension 6 with A °dd = O. By Sullivan's results (s.[18]) there exists
265
a simply connected, closed, oriented 6-dimensional differentiable mani-
fold M with Hm(M,~) ~ A m. By Proposition 2 the algebra A m is g-rigid;
in fact the proof of Proposition 2 shows that for any S~-action on a
manifold M with Hm(M;~) = A m the inclusion of the fix point set M ~G ~
induces an isomorphism in rational cohomology. Since M is closed and
orientable this implies M ~G = M, i.e. the action is trivial.
Remarks: a) For the above argument it is, of course, essential to assume
to be closed. The manifold with boundary M x D 2 and the open manifold 0
M x D 2 are homotopy equivalent to M and clearly admit non trivial S z-
actions.
b) The following example of a graded Gorenstein algebra is due
to A. Iarrobino (s.[9] Ex.7,[IO]). It waschecked on a computer to admit on-
ly deformations to algebras of the same "type", in particular it is not
smoothable.
A := R/J with R = ~[a,b,c,d,e,f] and the ideal J is generated by
{3ab-4ac-3bd,ad,ae,b~-af,12bc-9af-16bd-12ce,be,bf,3c2-4ac,3cd-3ac-4ce -
3df,cf,3d2-4bd,12de-12bd-16df-9a2,3e2-4ce,ef-ce-a2,3f2-4df}.
If one assigns the degree 2 to all the generators a,b,c,d,e,f then
J is a homogeneous ideal and A is a connected Poincar~ algebra of for-
mal dimension 6 with A °dd = O. Iarrabino remarks that the above example
should not be considered as a rare exception but one of many similarly
constructed.
By [18] the algebra A can be realized as the rational cohomology of
a closed oriented 6-dimensional differentiable manifold. Such a mani-
fold does not admit any non trivial SZ-action.
We now discuss some implication of the above method to the non ex-
istence of non trivial cyclic group actions. If suffices to consider
groups G = ~ of prime order p.
The following difficulties occur:
a) An analogue of Sullivan's result about realizing a rational Poincar~
algebra as the cohomology of a manifold seems completely out of reach
in the ~-case.
b) Even if one has a simply connected, closed, oriented manifold M with
H°dd(M;~%) = O and such that H*(M;~) is g-rigid in an appropriate
sense (note that H*(B ~ ; ~ ) is not just the polynomial ring over
~ in one variable in case p is an odd prime), M need not be TNHZ
in EG x M with respect to Hm(-,~) for all actions of G = ~ on M. G
There could be actions such that the Leray-Serre spectral sequence
of the fibration M ~ EG x M ~ BG does not collapse from the E2-term G
266
on and the action of G on H~(M;~) may be non trivial either.
Hence to prove an analogous result to Theorem I above for a given
fixed prime p (G = ~%) by just immitating the proof does not seem to
work. But it is possible to use similar arguments in order to show that
there are simply connected, closed, oriented differentiable manifolds
that do not admit any non trivial ~%-action, for almost all (i.e. all
but finitely many) primes p. This answers a question of P. L~ffler and
M. RauBen (s. [11]) to the neqative.
I am grateful to T. Petrie for suggesting this possibility.
We use Example 7 of [9] is. Remark b) above) to show the following:
Lemma: There exists a connected Poincar~ algebra A • of fd(A ~) = 6 with
A °dd = O, defined over ~ , such that A • := A ~ ® ~/ is g-rigid for al- P
most all primes.
By g-rigid we mean that any embedding (as graded algebras) of an one
parameter family A ~p O ~[t] ~ B * ® ~[t] (deg(t) = 2) into a trivial
family with cokernel being ~[t]-torsion must actually be an iso-
morphism.
Proof: The algebra given as Example 7 in [9] can be written as A • ® ~,
where A * is a connected graded Poincar6 algebra of fd(A ~) = 6 with
A Odd = O, defined over ~ and free as a Z~-module4 To prove that A • is g- P
rigid we show that the part of the Hochschild cohomology H2'-(Ap;Ap ) c
which classifies the infinitesimal deformations of negative weight (of
commutative algebras) is zero for almost all primes p. An element in
H2,-(A~.A~ c , p, p, is represented by a 2-cycle ~p: A~®A~ ~ A~, i.e. a symmetric
bilinear form of negativ degree (as a map of graded vector spaces) with
@Hp = O. Let ~: A~®A * ~ A • be a symmetric bilinear form such that
® id~ ~p. Then ~ = P'6 for some ~ 6 C 3 (A*;A ~) and 66 = O since
A * is torsion free,and P[6] = O in H3(A*;A*). If p is a prime which does
not occur in the torsion of H3 (A*;A ~) then [6] must be zero and there
exists an ~ 6 C2 (A~,A ~) such that ~ = ~ and therefore @(~-p~) = O, i.e.
:= ~-po 6 Z 2 (A~;A ~) and ~ ® id~ ~p. We can "symmetrize" ~ to get
~ =2Hp. = (~+~t) 6 Z2(A~'A~)'c where ~t(a1,a2) := ~(a2,al) , with ~ ® id~
(More conceptually one might use Harrison or Andr~-Quillen cohomology
of commutative algebras instead of Hochschild cohomology. But since we
have to exclude a finite number of primes anyway we may as well stick
to odd primes p, making 2 a unit in ~.) By [9] and [10] [~] = O in
H2'-(A~®~'A~®~)C , i.e. there exists a morphism ~: A~®~ ~ A~®~ such that
267
6~ = U ® id@. Let ~(p) denote the localization of ~ at p .
For almost all primes p the morphism aQ is already defined over Z~(p),
A* ® ZZ( such that ~(p) = i.e. there is a map ~(p) : A* ® ~(p) p)
® id~Z(p) . It follows that 6(~(p) ® id2Z/p) = 2~p, i.e. [2,~p] and hence
[~p] = O in H2'-c (Ap,~p* *) for almost all primes. Since H 2'-c (A*p,A*p) is fi-
nitely generated on gets: H 2'- c (A*p,A*p) = 0 for almost all primes p
The argument now proceeds as in the rational case before to give
that A* is g-rigid for almost all primes. P
As before we can realize A* ® (~ as the rational cohomology of a
closed, oriented, differentiable 6-dimensional manifold M. For almost
all primes p one has H*(M;2Z/. ) ~ H*(M;ZZ ) ® ZZ/p. It follows from the ~P
isomorphism A*®~ ~ H*(M;~) = H*(M;ZZ) ® ~, that A* ® Z~(p) ~ H* (M; ~ZZ(p)
and hence A ~p ~ H*(M;ZZ) ®Zg/p for almost all primes p, since the denomi-
nators of the rational coefficient matrix which gives the isomorphism
between A* ® ~ and H*(M;Z~) ® ~, and the torsion of H*(M;~) involve on-
ly finitely many primes. Let P denote the set of primes, such that A* P
is g-rigid, A*p = H*(M;~/p) ~ H*(M;Z~)® ZZ/p for p6P and 2Z/p must act
trivially on H*(M;ZZ(p)) (which - for a given M - is the case if p is large enough).
Assume Z~/p acts on M for some p 6 P.
The localization theorem for the equivariant cohomology given by the
Borel construction works as well if we use coefficients 2Z(p) instead of
H*(EG MG;ZZ(p)) induced by the in- ~/p, i.e. the map H*(EG ~ M; ZZ(p))
clusion M G ~ M becomes an isomorphism after inverting the multiplication
with the polynomial generator t 6 H*(BG;2Z(p)) ~ ~(p)[t]/p. (t)"
The group ~Z/p acts trivially on H*(M;2Z(p)); hence the E2-term of the
Leray-Serre spectral sequence of the fibration M ~ EG × M ~ BG is given G
by E2 = H*(BG; H*(M;ZZ(_))) ~ H*(BG;ZZ(p)) ® H*(M;2Z(p)).
Since H°dd(BG;ZZ(p)) ~= O = H°dd(M;~(p)) the spectral sequence col-
lapses already for degree reasons (as in the Sl-case with rational coef-
ficients). One therefore gets a morphism
H*(MG;2Z(p ) ) ~ H*(BG;2Z(p) )@ H*(M;2Z(p) ) ---~ H~(MG)G,2Z(p) ) ~ H*(BG;ZZ(p))® H*(MG;ZZ(p) )
which becomes an isomorphism after localization.
(Note that H*(MG;~(p)) can not have p-torsion since H°dd(MG;2Z/p) : 0
s. e.g. [3], VII (2.2))
Tensoring this morphism with ZZ/p gives an embedding
H~(M;~Z/p) @ ZZ/p[t] --~ H*(MG;ZZ/p) ® Zg/p[t] such that the cokernel is
ZZ/p [ t l-torsion.
Since A*p ~ H*(M;ZZ/p) is g-rigid it now follows, that M G ~ M induces
268
isomorphism H~(M;~) ~ H~(MG;~) and hence an we get:
Theorem 2: There exist simply-connected, orientable, closed 6-dimension-
al differentiable manifolds M such that for any closed orientable mani-
fold ~ with H*(M;~) ~ M*(M;~) a non-trivial action of ~ on M is only
possible for at most a finite number of primes p.
References
[1 ] ATIYAH, M.F. and HIRZEBRUCH, F.: Spin manifolds and group actions, Essays on Topology and Related Topics (M@moires d6di~s ~ G. de Rham), 18-28. Berlin-Heidelberg-New York: Springer 1969
[2] BLOOMBERG, E.M. : Manifolds with no periodic homeomorphism. Trans. Amer. Math. Soc. 202, 67-78 (1975)
[3] BREDON, G.: Introduction to compact transformation groups. New York-London: Academic Press 1972
[4] BURGHELEA, D.: Free differentiable S ~ and S 3 actions on homotopy spheres, Ann. Sci. Ecole Norm. Sup. (4) 5, 183-215 (1972)
[5] COFFEE, J.P.: Filtered and associated graded rings, Bull.Amer. Math. Soc. 78, 584-587 (1972)
[6] CONNER, P.E. and RAYMOND, F.: Manifolds with few periodic homeomor- phisms, Proceedings of the Second Conference on Compact Trans- formation Groups (Univ. of Massachusetts, Amherst 1971) Part II. Springer Lecture Notes in Math. 299, 1-75 (1972)
[7] CONNER, P.E., RAYMOND, F. and WEINBERGER, P.: Manifolds with no periodic maps, Proceedings of the Second Conference on Compact Transformation Groups (Univ. of Massachusetts, Amherst 1971) Part II. Springer Lecture Notes in Math. 299, 81-108 (1972)
[8] GERSTENHABER, M.: On the deformation of rings and algebras IV. Ann. of Math. 99, 257-276 (1974)
[9] IARROBINO, A.: Compressed algebras and components of the punctual Hilbert scheme, Algebraic Geometry, Sitges 1983, Proceedings. Spinger Lecture Notes in Math. 1124, 146-166 (1985)
[10] IARROBINO, A. and EMSALEM, J.: Some zero-dimensional generic singularities; finite algebras having small tangent space, Compositio Math. 36, 145-188 (1978)
[II] LOFFLER, P. und RAUSSEN, M.: Symmetrien von Mannigfaltigkeiten und rationale Homotopietheo~ie. Math.Ann. 271, 549-576 (1985)
[12] PUPPE, V.: Cohomology of fixed point sets and deformation of alge- bras, Manuscripta Math. 23, 343-354 (1978)
[13] PUPPE, V.: Deformation of algebras and cohomology of fixed point sets, Manuscripta Math. 30, 119-136 (1979)
[14] PUPPE, V.: P.A. Smith theory via deformations. Homotopie alg~brique et alg~bre locale, Luminy, 1982, Ast~risque 113-114, Soc.Math. de France, 278-287 (1984)
[15] QUILLEN, D.: Rational homotopy theory, Ann. of Math. 90, 205-295 (1969)
6] SCHULTZ, R.: Group actions on hypertoral manifolds. I. Topology Symposium Siegen 1979, Proceedings Springer Lecture Notes in Math. 788, 364-377 (1980)
[17] SCHULTZ, R.: Group actions on hypertoral manifolds. II. J. Reine Angew. Math. 325, 75-86 (1981)
[18] SULLIVAN, D.: Infinitesimal computations in topology. Publ. I.H.E.S. 47, 269-331 (1977)
[19] KWASIK,S. and SCHULTZ,R.: Topological circle actions on 4-manifolds. Preprint (1987)
[20] GERSTENHABER, M. and SCHACK, S.D.: Relative Hochschild cohomology, rigid algebras, and the Bockstein. J.Pure Appl.Algebra 43,
53-74 (1986)
2 × 2 - MATRICES AND APPLICATION
TO LINK THEORY
b y
P i e r r e VOGEL
In m a n y s u b j e c t s in t opo logy , p a r t i c u l a r l y in low d i m e n s i o n n a l t o p o l o g y , a
g r e a t dea l of t h e d i f f i cu l ty of t h e t h e o r y c o m e f r o m t h e p r e s e n c e of v e r y big g r o u p s
l ike : f r e e g r o u p s , b r a i d g r o u p s , m a p p i n g c l a s s g r o u p s , f u n d a m e n t a l g r o u p s of
s u r f a c e s or 3 - d i m e n s i o n n a l m a n i f o l d s . . . . It is v e r y d i f f i c u l t to m a k e d i r e c t
c o m p u t a t i o n s in s u c h a g r o u p G . A poss ib l e w a y to s t u d y it is to c o n s i d e r h o m o l o g y
g r o u p s Hn(G). T h e s e f u n c t o r s H n a re d e r i v e d f u n c t o r s of t h e a b e l i a n i z a t i o n f u n c t o r H z
and t h e m o r p h i s m f r o m Z[G] to Z[HI(G)] is t h e u n i v e r s e l r e p r e s e n t a t i o n of the a l g e b r a
7,[G] to a c o m m u t a t i v e a l g e b r a . A p o s s i b l e w a y to c o n s t r u c t o t h e r i n v a r i a n t s is to
c o n s i d e r r e p r e s e n t a t i o n s in t h e a l g e b r a of 2 × 2 - m a t r i c e s w i t h e n t r i e s in a
c o m m u t a t i v e r ing . Th i s m e t h o d w a s a l r e a d y u s e d in s o m e p a r t i c u l a r c a s e s . In [2]
Culler a n d S h a l e n c o n s i d e r r e p r e s e n t a t i o n s of t h e f u n d a m e n t a l g r o u p of a s u r f a c e or a
3 - d i m e n s i o n n a l m a n i f o l d in SL2(£) a n d o b t a i n s m a n y i n t e r e s t i n g r e s u l t s a b o u t
3 - d i m e n s i o n n a l m a n i f o l d s . In [ I ] Casson c o n s i d e r s r e p r e s e n t a t i o n s of t h e f u n d a m e n t a l
g r o u p of a s u r f a c e in SU z and c o n s t r u c t s an i n v a r i a n t in 7. for h o m o l o g y 3 - s p h e r e s .
In th i s p a p e r w e c o n s i d e r r e p r e s e n t a t i o n s f r o m an a l g e b r a R in a n a l g e b r a
M2(A) , w h e r e A is a c o m m u t a t i v e r ing , and w e c o n s t r u c t f u n c t o r s o~L a n d C s a t i s f y i n g
t h e fo l l owing p r o p e r t i e s : if R is an a lgeb ra , C(R) is a c o m m u t a t i v e r ing a n d ~ ( R ) is a
C ( R ) - a l g e b r a . M o r e o v e r w e h a v e a n a t u r a l r e p r e s e n t a t i o n f r o m R to ~L(R) an th i s
r e p r e s e n t a t i o n is in s o m e s e n s e t h e u n i v e r s e l r e p r e s e n t a t i o n f r o m R to t h e a l g e b r a of
2 x 2 - m a t r i c e s w i t h e n t r i e s in a c o m m u t a t i v e r ing ,The a l g e b r a J~L(R) is no t e x a c t l y an
270
a l g e b r a of 2 x 2 - m a t r i c e s b u t w e h a v e a t r ace m a p t and a d e t e r m i n a n t m a p 8 f r o m
J~L(R) to C(R) a n d if K is a C(R)- a l g e b r a w h i c h is an a l g e b r a i c l y c losed f ie ld , ~L(R)®K is
, i n a l m o s t all cases , i s o m o r p h i c t o M 2 ( K ) .
I t is w e l l k n o w n t h a t a b r a i d w i t h n c o m p o n e n t s ac t s o n t h e f r e e g r o u p F n on
n l e t t e r s . But th i s ac t ion on F 2 = F(x,y) is no t v e r y i n t e r e s t i n g if t h e b r a i d is p u r e and
h a s o n l y 2 c o m p o n e n t s . On the o t h e r h a n d if w e r e p l a c e t h e b r a i d b y an e m b e d d i n g L
of 2 i n t e r v a l s in I x IR E w h i c h is s t a n d a r d on t h e b o u n d a r y , L d o e s n ' t act on F 2 n e i t h e r
on t h e a l g e b r a Z [ F 2] e x c e p t if L is a b r a i d .
In th i s p a p e r w e wi l l p r o v e t h a t t h e r e ex i s t s a r i ng A , a l g e b r a i c e x t e n s i o n of
a p o l y n o m i a l r i n g of 5 v a r i a b l e s , a n d a m o r p h i s m f r o m t h e r i n g C(7.[F2]) to A , s u c h
t h a t L acts on ~L(7.[F2])®A b y c o n j u g a t i o n b y an e l e m e n t on t h e f o r m u + v x y , w h e r e u
a n d v b e l o n g s to A . The pa i r (u ,v ) in A 2 is w e l l d e f i n e d up to a s c a l a r a n d d e p e n d s
o n l y on t h e c o n c o r d a n c e c lass of L . This i n v a r i a n t is e x p l i c i t l y c o m p u t a b l e , as it is
s h o w n in an e x a m p l e , and it is a b s o l u t e l y n o t t r i v i a l .
I - F u n c t o r s JvL and C
Le t R be t h e a l g e b r a of 2 x 2 m a t r i c e s w i t h e n t r i e s in a c o m m u t a t i v e r ing A.
The t r a c e t r and t h e d e t e r m i n a n t de t a r e m a p s f r o m R to A s a t i s f y i n g t h e fo l lowing :
i) t r is A - l i n e a r and de t is A - q u a d r a t i c .
ii) de t is mu l t i p l i c a t i ve .
iii) fo r e v e r y x,y in R:
t r (xy) - t r (x) t r (y) + de t ( x + y ) - de t (x) - d e t ( y ) = 0
M o r e o v e r , fo r e v e r y m a t r i x in R, w e h a v e the C a y l e y - H a m i l t o n f o r m u l a :
iv) x 2 - t r (x) x + de t (x) = 0
On t h e o t h e r h a n d , w e h a v e a m a p : x ~ x- f r o m R to R d e f i n e d by:
x = t r ( x ) - x
The map: x -~x is an ( a n t i - ) i n v o l u t i o n of R and s a t i s f i e s t h e fo l lowing :
V x ~ R x + x = t r ( x )
x x = x x = d e t ( x )
271
D e f i n i t i o n 1.1 A q u a s i 2 x 2
c o m m u t a t i v e r ing A e q u i p p e d w i t h an invo lu t ion
sa t i s fy ing the following:
P0
m a t r i x a l g e b r a is an a l g e b r a R o v e r a
and maps t and 8 f rom R to A
V x ~ R t(x) ~ x + x
8(x) = x x = x x
PI t is A - l i n e a r and 8 is A - q u a d r a t i c
P2 8 is mul t ip l ica t ive
P3 v x , y ~ R t(xy) - t(x) t (y) = ~(x) + 8(y) ~ 8(x+y)
P4 V x ~ R x z - t ( x ) x + 8(x) = 0
P5 Vx ,yER xy + yx - x t ( x ) - t ( x ) y - t (xy) + t(x) t (y) = 0
R e m a r k 1.2 P r o p e r t i e s P4 and P5 are o b v i o u s c o n s e q u e n c e s of p r o p e r t y
P0 . M o r e o v e r , if A is inc luded in R, PI , P2 , P3 are also c o n s e q u e n c e s of P0 .
R~mark J.~ If A is a c o m m u t a t i v e r ing, t h e n M2(A) is a quas i 2 x 2 mat r ix
a lgeb ra ove r R. But a quas i 2 x 2 mat r ix a lgebra ove r A is g e n e r a l l y not i s o m o r p h i c to
M2(A). For ins tance , if A is t he field ~ and R is the q u a t e r n i o n i c s k e w field 14 e n d o w e d
w i t h the s t a n d a r d involu t ion , it is e a s y to check t h a t R is a quas i 2 x 2 mat r ix a lgebra
not i s o m o r p h i c to M2(~).
Let us d e n o t e by ~ ( resp. ~ 2 ) the c a t e g o r y of a l g e b r a s ( resp . quas i 2 x 2
mat r ix a lgebras) . If R and S are a lgebras over c o m m u t a t i v e r ings A and B, a m o r p h i s m
into ~ f r o m (R,A) to (S,B) is a couple of c o m p a t i b l e h o m o m o r p h i s m s f r o m R to S and A
to B. A m o r p h i s m is a m o r p h i s m in ~ 2 if it r e s p e c t s t r a c e s , d e t e r m i n a n t s and
involu t ions .
T h e o r e m 1.4 There ex is t s a func to r (o~,C) f rom ~ t o ~ 2 and a m o r p h i s m
f rom the i d e n t i t y func to r of ~ to (~L,C) sa t i s fy ing the fol lowing:
For each A - a l g e b r a R , e ach m o r p h i s m ~ f r o m R to a quas i 2 x 2 mat r ix
a lgebra M ove r B fac tor izes un ique ly t h r o u g h the C(R)- a lgebra ~L(R) .
272
Proof S u p p o s e t h a t R is an a l g e b r a o v e r a c o m m u t a t i v e r ing A. Denote b y A"
t h e r ing A[R R]. If x be l ongs to R, t he two c o r r e s p o n d i n g e l e m e n t s in R Rc A[R R] will
b e d e n o t e d b y t (x) a n d 8(x) r e s p e c t i v e l y .
So, w e ge t m a p s t and 8 f r o m R to A'. If w e force t and 8 to s a t i s fy p r o p e r t i e s
P I , P2 and P3, we ge t a q u o t i e n t A" of A'.
Now w e set:
R' = R ® A " A
We h a v e a A"- l i n e a r map - - f r o m R' to i t se l f d e f i n e d by:
V x ~ R x®1 = I ® t ( x ) - x ® I
Le t R" b e t h e q u o t i e n t of R' b y t he t w o - s i d e d idea l g e n e r a t e d b y t h e fo l l owing
e l e m e n t s :
x y = y x , x ~ R ' , y ~ R "
x ~ - - 8(x) , x ~ R '
The A " - a l g e b r a R" is c l ea r ly a quas i 2 × 2 m a t r i x a lgebra . M o r e o v e r , i t is t h e u n i v e r s a l
one. Now w e set:
C(R) = A"
,P'L(R) = R"
and (,YL,C) is a f u n c t o r f r o m ,~ to ~g'2 s a t i s fy ing t he d e s i r e d p r o p e r t y .
Le t us c o n s i d e r t he fo l lowing e x a m p l e :
R is t h e g r o u p r ing 3~[F(x,y)l, w h e r e F(x,y) is t he f r e e g roup g e n e r a t e d b y x
and y; R is an a l g e b r a o v e r 7..
T h e o r e m I.~ In th i s case, we have :
C(R) = Z ( t ( x ) , t (y) , t (xy) , 8(x), 8(y), 8(x)- ' , 6(y) -I ]
a n d ,Pa,.(R) is a f r ee C(R)- m o d u l e w i t h bas i s I, x, y and xy.
P r o o f Denote b y CI(R) t h e s u b r i n g of C(R) g e n e r a t e d b y t(x), t (y) , t (xy) , 6(x),
8(y), 8(x)- ' a n d 8(y) -I.
Claim I for e v e r y u ~ F(x,y), t (u ) l ies in Ct(R).
273
The p roo f is b y i n d u c t i o n on t he l e n g t h l(u) of t h e w o r d u in F(x,y).
S u p p o s e t h a t t (u ) l ies in Ct(R) for e v e r y u in F(x,y) of l e n g t h less t h a t n, and
le t u be a w o r d in F(x,y) of l e n g t h n.
If u c o n t a i n s a p o w e r x p, w i t h p=O, l :
u = v x P w
b y t he C a y l e y - H a m i l t o n f o r m u l a ( p r o p e r t y P4), x p be longs to:
Ct(R) • x Ct(R)
If p is l ess t h a n - 1 , v w a n d v x w h a v e l e n g t h less t h a n n a n d t (u ) l ies in Ct(R) b y
induc t ion . If p is - 1, v w h a s l e n g t h less t h a n n, so:
t ( u ) E Ct(R) ~ t ( v x w ) E Ct(R)
The s a m e ho lds if u c o n t a i n s a n o n t r iv i a l p o w e r of y. Thus it is e n o u g h to cons ide r the
case w h e r e u does no t con t a i n xPor yP (p~ 0 ou 1 ). Hence, t he w o r d u has t h e fo l lowing
fo rm:
u = xyxy...
u = yxyx ...
If n is b i g g e r t h a n 3, u c o n t a i n s (xy) 2 or (yx) 2 and t (u ) b e l o n g s to CI(R). In t h e o t h e r
case, we have :
n.< I ~ t (u ) E C~(R)
t (xy) = t (yx) ~ C,(R)
t (xyx) = t (x 2 y ) E Ct(R)
t ( y x y ) = t (y 2x) E Ct(R)
a n d t he c la im is p r o v e d .
Claim 2 For e v e r y u in R, t (u ) and 8(u) be long to Ct(R).
Le t u be an e l e m e n t of R . S ince t is l i nea r , t (u ) l ies in CI(R). S ince 8 is
q u a d r a t i c , 8(u) b e l o n g s to Cj(R) for e v e r y u in R if and o n l y if:
V u ~ F(x,y) 8(u) E CI(R)
Vu,v ~ F(x,y) 8 ( u + v ) - 8(u) - 8 ( v ) ~C)(R)
But t h a t is e a s y to check b e c a u s e of p r o p e r t y P3.
Le t ovLl(R) b e t h e s u b r i n g of d~L(R) g e n e r a t e d b y R a n d Ct(R). An e a s y
274
c o n s e q u e n c e of c l a i m s I a n d 2 is:
Claim 3 ,~LI(R) is a q u a s i 2 × 2 m a t r i x a l g e b r a o v e r Cj(R).
By t h e u n i v e r s a l p r o p e r t y of dVL(R), w e h a v e :
C(R) = Ct(R)
d"L(R) = d%)(R)
On t h e o t h e r h a n d , it is e a s y to c h e c k tha t :
Cj(R) + x C(R) + y C(R) + x y C ( R )
is an a l g e b r a . Then :
dCL(R) = C(R) + x C(R) + y C(R) + xy C(R)
Now, c o n s i d e r t h e r e p r e s e n t a t i o n p f r o m R to M2(£) d e f i n e d by :
w h e r e u, v, a, b, c a r e c o m p l e x n u m b e r s , and:
u * 0 , v * 0 , ac - b . 0
By u n i v e r s a l p r o p e r t y , w e h a v e m a p s :
p . : dVL(R) - . M2(C)
p . : C(R) ~ £
a n d w e c h e c k :
p . ( t ( x ) ) = u + v p . (6(x)) = uv
p . ( t ( y ) ) = a + c p . (6 (y ) ) = a c - b
9 . ( t ( x y ) ) = au + cv
If a,b, c, u, v a r e c h o s e n to b e a l g e b r a i c a l l y i n d e p e n d a n t , p . ( t ( x ) ) , p . ( t ( y ) ) ,
p . ( t ( x y ) ) , p . (8(x)) , p . ( 8 (y ) ) a r e a l g e b r a i c a l l y i n d e p e n d a n t too a n d C(R) is t h e p o l y n o m i a l
r i n g Z [ t (x) , t (y ) , t (xy) , 8(x), 8(y), 8(x) -1, 8(y)-* ] . M o r e o v e r , 1, p . (x) , P.(Y), p . ( x y ) a r e
l i n e a r l y i n d e p e n d a n t , so:
,JVL(R) = C ( R ) ~ x C ( R ) ~ yC(R)(D x y C ( R )
§2 - Re la t ion w i t h r e p r e s e n t a t i o n s .
275
Def in i t ion 2,1 le t K be a f ie ld and R be a r ing. Two r e p r e s e n t a t i o n s p and p'
f r o m R to M2(K) a re ca l led a l m o s t c o n j u g a t e if e i t h e r p and p' a re c o n j u g a t e or p ( resp
p') is e x t e n s i o n of I - d i m e n s i o n a l r e p r e s e n t a t i o n s ~ and [3 ( r e sp cx' and i3') and:
= cz' and ~ = [3'
or: ~ = [3' and 13 = ~'
The se t of r e p r e s e n t a t i o n s f rom R t o M 2 ( K ) m o d u l o a lmos t con juga t i on will be d e n o t e d
b y R2(R).
P r o p o s i t i o n 2 - 2 Let K be a field. Two a l m o s t c o n j u g a t e r e p r e s e n t a t i o n s f rom
a r ing R to M2(K) i n d u c e t h e s ame m o r p h i s m f rom C(R) to K.
P r o o f Let p and p' b e a l m o s t c o n j u g a t e r e p r e s e n t a t i o n s f r o m R to M2(K). If p
and p' a re con juga te , we h a v e a c o m m u t a t i v e d i ag ram:
M2(K)
/ 1 M2(K)
and b y t he u n i v e r s a l p r o p e r t y , w e h a v e d i a g r a m s :
M,(K) K
Y ,FL(K) . C(R) =
M2(K) K
T h e n m o r p h i s m s P. and p ' . f r o m C(R) to K are t he same .
If p and p' a re no t con juga t e , P and p' a re c o n j u g a t e to r e p r e s e n t a t i o n s p)
a n d p'~ f r o m R to t h e s u b r i n g M'2(K) of u p p e r t r i a n g u l a r m a t r i c e s in M2(K). M o r e o v e r ,
t h e d i a g o n a l e v a l u a t i o n g ives a m a p f r o m M'2(K) to K 2 a n d we ge t a c o m m u t a t i v e
276
d iag ram:
M'2(K)
R p , i ~ K2
M'2(K)
On t h e o t h e r h a n d , M'2(K) and K 2 are quas i 2 x 2 m a t r i x a l g e b r a s o v e r K and
f, f' and t he i nc lu s ion M'2(K) in M2(K) are m o r p h i s m s in ~ 2 " Then , if w e a p p l y t he
f u n c t o r C, we ge t : P. - Pl. = P'l. = P'.
T h e o r e m 2 - 3 le t R b e a r ing and K be a f ield. Le t f be a m o r p h i s m f r o m
C(R) to K. T h e n t h e r e ex i s t s an e x t e n s i o n L of K and a r e p r e s e n t a t i o n f r o m R to M2(L)
i n d u c i n g f. M o r e o v e r , L can be c h o s e n to be K or a q u a d r a t i c e x t e n s i o n of K or, if K has
c h a r a c t e r i s t i c 2, a sub f i e ld of J K
T h e o r e m 2- 4 Let R be a r ing and K a field. Let p and p' b e r e p r e s e n t a t i o n s
f r o m R to M2(K). Then , p and p' a re quas i c o n j u g a t e if and on ly if p and p' i n d u c e t he
s a m e m o r p h i s m f r o m C(R) to K.
P roo f If f is a c h a r a c t e r f rom C(R) to K, le t us d e n o t e b y R2(R,f) t he a l m o s t
con jugacy c lasses of r e p r e s e n t a t i o n s f rom R to M2(K) i n d u c i n g f f r om C(R) to K.
Let A be t he fo l lowing K - a l g e b r a :
A = d%(R) ® K C(R)
w h e r e t he C(R)- a l g e b r a s t r u c t u r e of K is g i v e n b y f.
The a l g e b r a A is a q u a s i 2 × 2 m a t r i x a l g e b r a o v e r K. b y t h e u n i v e r s a l
p r o p e r t y of dVL(R) we have :
R2(R,f) = R2(A,Id)
Then , if w e w a n t to p r o v e t h e o r e m 2 - 3 , it is e n o u g h to s h o w t h a t R2(A®L , Id) is no t
e m p t y for some a l g e b r a i c e x t e n s i o n L of K. T h e o r e m 2- 4 is e q u i v a l e n t to t h e fac t t h a t
R2(A, Id) h a s at mos t one e l e m e n t .
277
Case I S u p p o s e t h a t t he c h a r a c t e r i s t i c of K is d i f f e r e n t f r o m 2 and:
VXEA t(X) 2 = 4G(X)
In this case, d e n o t e by f the map ( { /2 ) t f rom A tO K. Since t is l inear , f is l inear too.
On the o the r hand, f2 is e q u a l to 8 and f2 is mul t ip l ica t ive . Then, for e v e r y
x ,yE K t h e r e exists ~=±I such that:
f(xy) = c f(x) f(y)
Since f(xy) and f(x) f(y) a re bi l inear , it is e a s y to see tha t z doesn ' t d e p e n d
on x and y, and f is mul t ip l ica t ive . The morph i sm:
x ---, f(x)
be longs to R2(A, Id) and t h e o r e m 2- 3 is p roved in this case (wi th L = K)
Let p be an e l e m e n t of R2(A, Id). Since t(x) z is equa l to 4 8(x) for e v e r y x
in A , p(x) is e i t he r the scalar matr ix f(x) or this matr ix plus some n i lpo t en t m a t r i x .
Then , if p is not the scalar r e p r e s e n t a t i o n f t he r e exis ts some e l e m e n t x o in A such
tha t p(x o) is the matr ix :
p(x 0) = If(0 x0)
in some basis in K z.
L e t x E A and ( :
f(x 0)
b l be the matr ix p(x). We h a v e : d Y
a + d = t(x) = 2 f ( x )
f(x o) a + c + f(x o) d = t(x o x ) = 2 f(x o x ) = 2 f ( x o) f(x)
Then c is zero, and p is t he fol lowing morph i sm:
for some map g f rom A tO K, and p is a lmos t con juga te to:
I 0:1 Therefore , t h e o r e m 2 - 4 is p r o v e d in this case.
Case 2 Suppose tha t K is of charac te r i s t i c 2 and the t race t is nul l on A.
In this case, deno te by f the map 4~- f rom A to 4 ~ a n d by L the image of f.
It is ea sy to see tha t f is an a lgebra ic h o m o m o r p h i m and the scalar r e p r e s e n t a t i o n f
is an e l e m e n t of R2(A®L, I d ) . H e n c e , t h e o r e m 2 - 3 is p r o v e d in this c a s e .
278
Let p be an e l e m e n t of R2(A, Id). If p(x) is a scalar matr ix , for e v e r y x
in A, L is equa l to K and p is the scalar r e p r e s e n t a t i o n f .
If L is equa l to K and p(x o) i s n o t a scalar matr ix for some XoEA, we can s h o w , a s i n
the f i rs t c a s e , tha t t he r e exists a map g f rom A to K such tha t p is con juga te to the
r e p r e s e n t a t i o n (f0 g l a n d t h e n p i s a l m ° s t c ° n j u g a t e t ° ( f 0 ~ I
Suppose tha t L is d i f f e r e n t f rom K. Let Xoe A such tha t f(x o) is not in K.
Then p(x o) is not a scalar matr ix and , as a b o v e , p®L is conjugate to a r e p r e s e n t a t i o n p':
0-1o :I such tha t g(x o) is not zero.
If x is an e l e m e n t of A, p'(x) is a l inear c o m b i n a t i o n of 1 and p'(Xo), and
t he r e exis t a, b E K such that:
p (x )= a + b p ( x o)
f(x) = a + b f(x o)
T h e r e f o r e L is the ex tens ion K[f(Xo)] of K and the re exist un ique funct ions ~ and [~ f rom
L to K such that:
V u e L u = ~ ( u ) + [3(u)f(x o)
and we have:
V X e A p(X) = ~(f(X)) + ~(f(x)) p(x o)
The conjugacy class of p is the conjugacy class of p(x o) which is the conjugacy class of:
(,,°0, :I So R2(A, [d) has at most 1 e l emen t .
Case 3 We suppose tha t we are not in cases I or 2 and that:
~'x,yE A t(xy) 2 - t (xy) t(x) t(y) + 8(X) t ( y ) 2 + 8 ( y ) t (X) 2 - 4 6(X) 8(y) = 0
Since we are not in cases I or 2, t he r e exists Xoe A such that :
t(Xo ) 2 - 4 6(x o) = 0
Let ~. and It be two e l e m e n t s of a quadra t i c ex tens ion L of K such that:
7, + )~ = t(x o) ~.It = 8(x o)
Since t(Xo )2 = 4 8(x o) , ?, is d i f f e r e n t f rom 11. Then, for e v e r y XeA, the fo l lowing
equa t ions :
2 7 9
a + b : t ( x )
~L a + I t b : t ( x o x )
h a v e a u n i q u e s o l u t i o n . D e f i n e m a p s f a n d g f r o m A t o L b y :
f ( x ) = a
g (x ) ~ b
S i n c e w e h a v e :
t (x o x) 2 - t (x o x) t (x o) t ( x ) + 8(x o) t ( x ) 2 + 8(x) [t(Xo )2 - 4 8(Xo)] : 0
i t is n o t d i f f i c u l t t o c o m p u t e 8(x). A f t e r c o m p u t a t i o n , w e ge t :
~(x) : f ( x ) g ( x )
C l e a r l y , f a n d g a r e K - l i n e a r .
L e t x a n d y b e t w o e l e m e n t s in A. W e se t :
a = f (x ) b = g (x )
I f a = b t h e r e e x i s t u n i q u e e l e m e n t s ce a n d [3 in L s u c h t h a t :
ce + [5 : t ( y ) a n d ace + b [3 ~ t ( x y )
a n d w e h a v e a s a b o v e : 8 ( y ) = ce [3
C o n s e q u e n t l y , w e h a v e :
ce : f ( y ) a n d [3 : g ( y )
or : ce : g ( y ) a n d [3 = f ( y )
S u p p o s e t h a t : ce : g ( y ) a n d [3 = f ( y )
L e t u E L s u c h t h a t : u~ . + a ~ uFt + b
T h e n t ( u x o + x) 2 is d i f f e r e n t f r o m 4 8(u x o + u) a n d , a s a b o v e , t h e r e e x i s t u n i q u e
e l e m e n t s ce' a n d 9' i n L s u c h t h a t :
ce '+ [3'= t ( y )
(u~ . + a) ce'+ ( u ~ . + b ) [ 3 ' = t ( ( u x o
~' [3' = 8 ( y )
A n d w e h a v e :
t( ( u x o + x ) y ) = (u~ . + a) ce + ( u l t + b ) [3
or: t ( ( u x o + x ) y ) = (u~ . + a)[3 + (u It + b )ce
I n o t h e r w o r d s :
or ;
+ x) y )
U( ~L[3+itce ) + ace + b ~ = u( ~,ce + It[3 ) + ace + b [3
u( %[3+itce ) + ace + b[3 = u( ~L[3 + ~tce) + a[3 + b c e
280
i.e. u(t~-13)(~.-I t) = 0 or (~ -13 ) (a -b ) = 0
a n d th i s is i m p o s s i b l e .
h e n c e , if f (x) = g(x) , w e h a v e :
t ( x y ) = f (xy ) + g (xy) = f ( x ) f ( y ) + g(x) g (y)
Of c o u r s e , t h e s a m e h o l d s if f (y) ~ g(y) .
I f f ( x ) = g(x) and f (y) = g ( y ) , w e h a v e :
t ( x y ) 2 - 4 t (xy ) f ( x ) f ( y ) + 4 f ( x ) a f ( y ) a= 0
[ t ( x y ) - 2 f(x) f ( y ) ] 2 = 0
t ( x y ) - 2 f ( x ) f ( y ) = f ( x ) f ( y ) + g ( x ) g ( y )
T h e r e f o r e fo r e v e r y x a n d y in A , w e h a v e :
f ( x y ) + g (xy ) = f ( x ) f ( y ) + g ( x ) g ( y )
a n d : f ( x y ) g ( x y ) = f(x) g(x) f (y) g (y )
H e n c e w e h a v e t w o p o s s i b i l i t i e s :
f ( x y ) = f(x) f (y ) a n d g (xy ) = g(x) g (y)
or : f ( x y ) = g(x) g (y ) a n d g (xy ) = f(x) f (y )
S u p p o s e t h a t f(x) and g(x) a r e d i f f e r e n t . L e t y ~ A . If f (xy ) is e q u a l to g(x)
g (y) , w e h a v e :
f (x (y+ I ) ) = g(x) g(y) + f(x)
a n d : f ( x ( y + l ) ) = f ( x ) ( f ( y ) + l ) o r f ( x ( y + l ) ) = g ( x ) ( g ( y ) + l )
S ince f (x) a n d g(x) a r e d i f f e r e n t , w e g e t :
g(x) g (y) = f(x) f (y)
T h e s a m e h o l d s if f (y ) = g (y ) a n d t h e n in a n y c a s e .
F ina l ly w e h a v e :
~ ' x , y E A f (xy) : f(x) f (y ) g ( x y ) = g ( x ) g ( y )
a n d ( j ~ I i s a r e p r e s e n t a t i o n i n R z ( A ® L , I d ) .
N o w s u p p o s e t h a t p is a r e p r e s e n t a t i o n in R2(A®L , Id) . The r e p r e s e n t a t i o n
p®L is c o n j u g a t e t o a r e p r e s e n t a t i o n p' s u c h t h a t :
S u p p o s e t h a t x a n d y a r e t w o e l e m e n t s in ^ a n d tha t :
0,,>=(: :I :I
281
W e h a v e :
a + d = t (x ) = f (x) + g(x)
~,a + I t d = t (x o x ) = f (x o x) + g(x o x)
a = f (x) d = g (x )
a n d t h i s i m p l i e s :
8(x) = ad - b c = ad ~ b c = 0
T h e n p'x) a n d p ' (y) a r e t r i a n g u l a r .
S u p p o s e t h a t : c = 0 b ~ 0 13 = 0 Y = 0
t h e n w e h a v e :
f ( x y ) = a c t + b Y = f (x) f ( y ) = a
w h i c h is i m p o s s i b l e . H e n c e , p' h a s t h e f o l l o w i n g f o r m :
fo r s o m e m a p ~ f r o m A t o L.
if w e c h a n g e f a n d g, w e m a y as w e l l s u p p o s e t h a t :
S u p p o s e t h a t ~ is ze ro . T h e n , for e v e r y x E A, p'(x) is a l i n e a r c o m b i n a t i o n o f 1 a n d p'(x o)
a n d t h e r e e x i s t t w o f u n c t i o n s c~ a n d 13 f r o m A to K s u c h t h a t :
p(x) = ~(x) + p(x o) ~(x)
M o r e o v e r , w e h a v e :
t (x ) = 2 ~ ( x ) + ~(x) t (x o)
t (x o x) = ct(x) t (x o) + O(x) t(X2o )
a n d ~ (x ) a n d ~(x) d e p e n d o n l y o n t (x) , t(Xo), 8(x), 8(x o) a n d t (x o x)-
S i n c e p(x o) is c o n j u g a t e to:
t (x )
8(x0 ) -~] p is c o n j u g a t e to:
I: :I I">° + 13 8 ( x ) 0
If 0 is n o n z e r o , t h e r e e x i s t s x~E A s u c h t h a t 0(x I) = O. W e h a v e :
282
Since p(x o) is c o n j u g a t e to:
t(Xo) -1
8(x ° ) 0
t h e r e e x i s t s a m a t r i x :
(: :) in GLz(L) s u c h tha t :
I: :)l: and :
I: :)(o A f t e r c o m p u t a t i o n w e get :
a ~ 0
s Z - pit
},-IZ
:) ,',o> -' (: ,) 8(x o) 0 d
- I
If: :I ,M,K>2 b = 0
aug ~-13 au - - ~ K E K b(~.- g) ~.-)t b(Z-g)
2 auit Zi3 - ~p. aulz
~ K + - - E K b(~.- ~) ~.-~. b (~.-).t )
S u p p o s e t h a t L is d i f f e r e n t f r o m K. T h e n L is a q u a d r a t i c e x t e n s i o n of K a n d w e h a v e a
Galois a c t i o n - on L:
-~ : Z ~ - : p
So w e get :
aull au ~ K a n d ~ K ~ lz ~ K
b(~.- g) b(~,- g)
a n d t h e n L is e q u a l to K.
So L is e q u a l to K a n d p (= p') is q u a s i c o n j u g a t e to:
a n d t h e o r e m s 2 - 3 a n d 2 - 4 a r e p r o v e d in th i s case .
C ~ 4 W e s u p p o s e t h a t w e a re n o t in ca se I o r 2 o r 3.
For a n y x a n d y in a q u a s i 2 x 2 m a t r i x a l g e b r a , s e t :
A(x.y) = t ( x y ) 2 - t ( x y ) t ( x ) t ( y ) + t (x )ZS(y) ÷ t (y )ZS(x) - 4 8 ( x ) 8 ( y )
203
I n t h i s c a s e t h e r e e x i s t x a n d y in A S u c h t h a t ~ ( x , y ) is n o t z e r o . L e t A) b e t h e
s u b a l g e b r a o f A g e n e r a t e d b y x a n d y . C l e a r l y Aj is g e n e r a t e d as a K - v e c t o r s p a c e b y
I, x, y , x y . S u p p o s e w e h a v e a r e l a t i o n :
a + b x + c y + d x y = 0 a , b , c , d ~ K
T h e n w e ge t :
2 a + b t ( x ) + c t ( y ) ÷ d t ( x y ) ~ 0
a t (x ) ÷ b t (x 2 ) + c t ( x y ) + d t ( x 2 y ) = 0
a t ( y ) + b t ( x y ) + c t ( y 2 ) + d t ( x y 2 ) = 0
a t ( x y ) + b t ( x 2 y ) + c t ( x y 2 ) + d t ( x 2 y 2 ) - - 0
I t is n o t d i f f i c u l t t o c h e c k t h e f o l l o w i n g :
t (x 2 ) = t (x ) 2 - 2 8(x)
t ( y 2 ) = t ( y ) 2 - 2 8(y)
t ( x 2 y ) = t ( x ) t ( x y ) - 8(x) t ( y )
t ( x y 2 ) = t ( y ) t ( x y ) - 8 (y ) t (x )
t ( x 2 y a ) : t ( x y ) t (x ) t ( y ) - t (x 2 ) 8(y) - t ( y a ) 8(x) + 2 8(x) 8 (y )
a n d t h e d e t e r m i n a n t o f t h i s s y s t e m is ~,(x,y) 2 w h i c h is n o t z e r o . T h e r e f o r e ,
( 1, x , y , x y ) is a b a s i s o f A i -
L e t a + b x + c y + d x y b e a n e l e m e n t o f t h e c e n t e r o f ^v W e h a v e :
x ( a + b x + c y + d x y ) = ( a + bx + c y + d x y ) x
( c + d x ) ( x y - y x ) = 0
B u t w e h a v e t h e f o l l o w i n g f o r m u l a :
8 ( x y - y x ) -- ( x y - y x ) ( x y - y x ) ~ 8 ( x y ) + ~ (yx) + t ( x y 2 x) - t ( x y ) t ( y x )
= 2 8 ( x ) 8 ( y ) + t ( x 2 y 2 ) - t ( x y ) 2 = - &(x,y)
T h e n w e ge t :
( c + d x ) ~ ( x , y ) ~ 0 ~ c = d ~ 0
a n d a ÷ b x w h i c h c o m m u t e s w i t h y is a m u l t i p l e o f I. T h e r e f o r e , t h e c e n t e r o f Aj is K.
On t h e o t h e r h a n d , it is n o t d i f f i c u l t to s e e t h e r e is n o c h a r a c t e r f r o m Aj to
K. T h e n Aj is s i m p l e a n d A t ® L is i s o m o r p h i c to M2(L) fo r s o m e q u a d r a t i c e x t e n s i o n L
of K. C o n s e q u e n t l y t h e r e e x i s t e l e m e n t s ei} in A) ® L , i : 1,2 j= ! ,2 s u c h t h a t :
e i j e l , j, : 0 i f j = i'
= e l i , if j = i'
284
Le t us d e f i n e t h e f o l l o w i n g m a p s fij f r o m A tO L:
Vi.j V X E A fii(X) = t ( x e i j )
Cla im For e v e r y x,y in A and e v e r y I,l in {1,2} w e h a v e :
f i j (xy) -- f i l (x) f l j (Y) + f i2(x)f2j(Y)
P roo f of t h e c la im: For e v e r y u,v in A w e h a v e :
8(u e . + v e . ) = 8(u ez)) + 8(v el1) + t(u eli) t ( v e~l) - t (u e . v e . )
b u t 8 is m u l t i p l i c a t i v e and 8(e . ) is zero . T h u s w e h a v e :
t (u e . ) t ( v e . ) = t (u e . v e . )
and th i s imp l i e s :
t ( x y e . ) = t( x ( e . + e 2 2 ) y e . ) = t(x e . y eat) + t(x e22 y e . )
= t(x e,,) t ( y e . ) + t( x e~, e . e~2 y e . )
= t(x e . ) t ( y e H) + t(x e m ell) t(el2 y ell)
= t(X e . ) t ( y ell) + t(X e2j) t (y e12)
So w e h a v e
f i j (xy) = t ( x y eji) = t(eli x y eil e . )
= t(eli x exl) t (y ell ell) + t(el i x e21) t (y eil el2)
= f i l (x) f l j (Y) + f i2(x)f2j(Y)
and t h e c l a im is p r o v e d .
As a c o n s e q u e n c e of t he c la im, w e h a v e a m o r p h i s m f f r o m A tO M2(L):
\ 21 22/
and it is n o t d i f f i cu l t to see t h a t f is a m o r p h i s m in t h e c a t e g o r y s~ 2 (i.e. it p r e s e r v e s
t r a c e and d e t e r m i n a n t ) .
N o w l e t u a n d v be t w o e l e m e n t s of A s u c h t h a t f (u) = 0. W e h a v e :
t (u ) = t ( f (u) ) = 0
t ( u v ) = t ( f ( u v ) ) = 0
~ u + U = 0
UV+ VU= 0
and th i s imp l i e s : uv = v u
Le t a,b in A. W e h a v e :
285
u ( a b - b a ) = a u b - b ~ u = a b u a - b ~ u = 0
In p a r t i c u l a r :
u ( e l ~ ez t - e2t e l 2 ) = u (eft - e 2 2 ) = 0
But e , - e 2 2 is i n v e r t i b l e . T h e n u is t r i v i a l a n d f i n d u c e s a m o n o m o r p h i s m f r o m A tO
M2(L).
T h a t p r o v e s t h e o r e m 2= 3.
If w e h a v e a r e p r e s e n t a t i o n f r o m A to M2(K), A is n o t a s k e w f i e ld a n d A is
i s o m o r p h i c to M2(K). H e n c e t w o r e p r e s e n t a t i o n s f r o m A to M2(K) a r e c o n j u g a t e a n d
t h e o r e m 2 - 4 is p r o v e d ,
§ 3 - A n i n v a r i a n t for l inks .
D e f i n i t i o n 3 - 1
A l ink of n i n t e r v a l s is an e m b e d d i n g of I x { l , 2 . . . . . n} to I x ~ 2 w h i c h is
s t a n d a r d o n t h e b o u n d a r y .
T w o l i nks a r e c o n c o r d a n t if t h e r e is an e m b e d d i n g F f r o m [2 x { l . . . . . n}
t o I x I x I R 2 s t a n d a r d o n a l x l x { 1 . . . . . n} a n d i n d u c i n g fi o n I × { i } x { l . . . . . n} f o r i = O,l.
T h e s e t o f c o n c o r d a n c e of l i n k s of n i n t e r v a l s is a s e t C n , w h i c h is a c t u a l l y a
g r o u p fo r t h e j u x t a p o s i t i o n l a w [3].
L e t L b e a l ink of n i n t e r v a l s . D e n o t e b y X t h e c o m p l e m e n t o f L a n d b y X o
a n d Xt the t o p p a r t a n d t h e b o t t o m p a r t o f OX. L e t x i ( r e s p x' i) b e t h e e l e m e n t of ~j(X o)
. t h ( r e s p a1(Xt)) w h i c h t u r n s a r o u n d t h e I c o m p o n e n t o f L in X o ( r e s p Xt). T h e
f u n d a m e n t a l g r o u p ~tx(X o) is a f r e e g r o u p w i t h b a s i s x i . . . . . x n. T h e s a m e h o l d s fo r
7tt(Xt). Bu t lit(X) is g e n e r a l l y n o t f r e e . We o n l y k n o w t h e f o l l o w i n g [3]:
T h e r e e x i s t s a u n i v e r s a l g r o u p G n d e p e n d i n g o n l y o n n a n d a m o r p h i s m
f r o m Jtt(X o) t o G n s u c h t h a t fo r a n y l ink L, ~ e x t e n d s u n i q u e l y o n xt(X). M o r e o v e r , t h e r e
e x i s t s a u n i q u e a u t o m o r p h i s m "c L d e p e n d i n g o n t h e c o n c o r d a n c e c l a s s o f a l ink L o n G n
s u c h tha t :
Vi= 1 ..... n 1;L(X i) = X' i
286
This a u t o m o r p h i s m sa t i s f ies the fol lowing:
for e v e r y i, "~L(Xi) is c o n j u g a t e to x i and ~L(X~X2 _. Xn) = X,X 2 ... X.
In fact, G. is t h e a l g e b r a i c c losu re of ~)(X o) in t he s e n s e of L e v i n e [4].
The p r o b l e m is t h a t G n is c o m p l e t e l y u n k n o w n and it is t h e r e f o r e d i f f icu l t
to g ive a d e s c r i p t i o n of some a u t o m o r p h i s m of G n.
F rom n o w on, we wil l s u p p o s e t h a t L is a l ink w i t h 2 c o m p o n e n t s . We set:
x I = x x 2 = y
T h e n )~i(Xo) is t h e f r ee g r o u p F(x,y).
No ta t ion 3 - 2 We set t h e fo l lowing in t he r ing C(Z[F(x,y)]):
a = t(x) b = t (y) c = t (xy)
= 8(x) ~ = 8(y)
A d e n o t e s t h e r i ng C(Z[F(x ,y ) ] ) = Z [ a , b, c, ~, - I , ~ , ~ - , ] and ~ is t h e a l g e b r a
JvL(Z[F(x,y)]). A is t he e l e m e n t of A de f ined by:
A = c 2 - abc + a2~ + b Z ~ - 4 ~
S is t he mu l t i p l i c a t i ve s u b s e t of A w h i c h cons i s t s of p o l y n o m i a l s
p(a2(~ -l , bZ~ -l) of Z [ a2ct -I , b a ~ -l] c A such t h a t P(4,4) = I.
is t h e c o m p l e t i o n of S-IA w i t h r e s p e c t to t he idea l g e n e r a t e d b y A:
= l im S -~ A I A n
Ais t h e s u b r i n g of A w h i c h cons i s t s of all e l e m e n t s of A a l g e b r a i c o v e r A.
T h e o r e m 3- ,~ le t L b e a l ink of 2 i n t e r v a l s . T h e n t h e m o r p h i s m f r o m
Z[Ttz(Xo)] to ~ e x t e n d s u n i q u e l y to a m o r p h i s m f r o m Z[)tj(X)] to ,J~L®A. M o r e o v e r
t h e r e ex i s t s a u n i q u e a u t o m o r p h i s m OL f rom ~L®A to i t se l f such tha t :
(DL(X) = x' ~OL(y) = y '
F u r t h e r m o r e t h e r e ex i s t e l e m e n t s u,v in A, u n i q u e up to m u l t i p l i c a t i o n b y a sca lar in A
such tha t :
VzEJ4~ ~L(Z) = (U+VXy) Z(U+VXy) -I
T h e a u t o m o r p h i s m OL d e p e n d s o n l y on t h e c o n c o r d a n c e c l a s s of L a n d t h e
c o r r e s p o n d a n c e L -~OL is a r e p r e s e n t a t i o n of t he g r o u p C 2 to Aut(J~L®A).
287
R e m a r k 3 - 4 In fact the m o r p h i s m f rom Z[Ttt(Xo)] to ~ e x t e n d s u n i q u e l y
to a morph i sm f rom Z[G 2] to J%®A, and we have a canonical r e p r e s e n t a t i o n f rom G 2 to
( ~ ® ^ ) *
The p roof of t h e o r e m 3 - 3 is qu i t e long and wi l l be d i v i d e d in s e v e r a l
l emmas .
L e m m a 3 - 5 Let (K, K o) be a pair of f in i te complexes . We suppose tha t K o is
h o m o t o p y e q u i v a l e n t to a b o u q u e t of t w o cerc les and tha t KIKo is con t rac t ib le . Let x
and y be t he g e n e r a t o r s of 7t~(Ko). Let c be the a u g m e n t a t i o n map:
~: Z[x, x -I, y, y-l] -=~ g ~(x) = ~(y) = 1
Then w e have :
-1 ) -I -1 Vi<~2 Hi(~l(K), ;[)(Ko); ~ (I Z[x, x , y, y-A] ) = 0
Proof: Let B be the ring: B : c-l( l )-I Z[x, x -l, y, y- l]
We h a v e an a u g m e n t a t i o n map: B - , Z
Since B is noe ther ian , and (K, K o) is f inite, H,(K, K o ; B) is f in i te ly gene ra t ed .
Let Hp(K, K o ; B) be t he f i rs t non t r iv ia l homology group of (K, K o) , if i t e x i s t s . Since
H.(K, K o ; Z ) van i shes , Hp(K, K o ; B) is ki l led by some e l e m e n t of B going to 1 in Z
T h e r e f o r e Hp(K, K ; B) v a n i s h e s too and (K, K o) is B- acyclic. But, for i ~ 2
HiOtl(K), ~tl(Ko); B) is a quo t i en t of Hi(K, K o ; B). That p r o v e s the l e m m a .
L e m m a 3 - 6 Let X be the c o m p l e m e n t of a link of 2 in t e rva l s . Le t M be a
Z[x, x-1 Y, y-l]_ module such tha t e v e r y e l e m e n t of c-I( I ) acts b i j ec t ive ly on M. Then:
V i<2 Hi(~I(X), 7tl(Xo); M) = Hi(G2 , F(x,y); M) = 0
Proof: The module M is a module o v e r the ring:
B = ( - l ( 1 ) - i g [ x , x - t Y,Y-I]
Then, by the un ive r sa l coef f ic ien t spec t ra l sequence , it is enough to p rove :
Vi<~2 Hi(itj(X), ~tj(Xo); B) = Hi(G 2 , F(x,y); M) = 0
288
The f i r s t p a r t of t h a t is p r o v e d in l e m m a 3 - 5.
T h e r e ex i s t s a s e q u e n c e of f in i te c o m p l e x e s [3]:
X o c Kj c K 2 c ...
such tha t :
Vi Ki/Xo is c o n t r a c t i b l e
G 2 = ~tj(u K n)
S o w e h a v e : Vi H i ( U K n , X o ; B ) = lim H i ( K n , X o ; B ) = 0
and t he l e m m a can be eas i ly deduced .
L e m m a 3 - 7 Let n ~ l be an i n t e g e r , a n d F n b e t h e g r o u p of u n i t s of t h e
a l g e b r a S - t d % / i n , w h e r e I is t he t w o - s i d e d idea l of S-IJ% g e n e r a t e d b y x y - y x . Let X
be t h e c o m p l e m e n t of a l ink of t w o i n t e r v a l s . T h e n t he m o r p h i s m f r o m F(x,y) to F n
f ac to r i ze s u n i q u e l y t r o u g h G 2 and ~ ( X ) .
ProQf: This l e m m a wil l be p r o v e d b y i n d u c t i o n on n.
For n= I , Fj is c o m m u t a t i v e and t he l e m m a is o b v i o u s s ince:
Hj(F(x,y); Z ) = HIOtI(X); Z ) = H~(G 2 ; Z )
On t he o t h e r h a n d , we h a v e an exac t s e q u e n c e (for n ~ I ):
1 - > 1 + I n / i n + t ~ Fn, I - ) F n - ) 1
Let G b e t h e g r o u p ~t1(X) or G 2 . By i n d u c t i o n w e h a v e a c o m m u t a t i v e d i a g r a m :
f% 1 "-)1+ I " / i n * l -o Fn. t "-) F n - ) 1
1' 1' F(x,y) ~ G
a n d w e w a n t to p r o v e t h a t t h e r e ex i s t s a u n i q u e m o r p h i s m f r o m G to Fn. j w h i c h
m a k e s t he d i a g r a m c o m m u t e :
F n + I ~ F n
Y F(x,y) -~ G
nl is c o m m u t a t i v e , it is e n o u g h to p r o v e t h a t Since t h e m u l t i p l i c a t i v e g r o u p I + I i n .1
13 I + I I i a . 1 is b y t he c o n j u g a t i o n a c t i o n a B - m o d u l e .
2 8 g
Let (o be the e l e m e n t x y - y x of S-Io~L. It is easy to check the fol lowing:
x~ ~ ( o x T o ) =~x
y ( o =(o y Y co : ( o y
Then I n is g e n e r a t e d by (o n and In/ in ÷ s is add i t ive ly i somorphic to
S-~vL/ ~ S-17.[x, x-I, Y, Y-I, ~ - ~ - -l, Y , Y -I] (o
On the o the r hand we have:
x ( l + (onu)x -1 = I + ( o n x ' u x - 1
y ( l + ( o n u ) x ~ = I + (on y ' u y -I
w h e r e x' and y ' a re x and y if n is even , and x- and y- i f n is odd.
Then I + In / i n÷1 is i somorph ic to S Z [ x , x , y , y , x , x , Y , Y - S ] a n d F ( x , y ) a n d G
acts on it (via Hj(F(x,y)) : HI(G) ) t r iv ia l ly if n is e v e n and in the fo l lowing w a y if n is
odd:
x(u) : x - x -su y(u) = y- y - l u
I f n i s e v e n , I + In / in ,1 i s a B - m o d u l e .
Suppose now tha t n is odd. Let P(x,y) ~ 7.[x, x -s, y , y-l] such tha t P ( l ,1 ) is
I . Then P(x,y) acts on S-IZ[x, x-S Y, ,Y-I x , x-s , y-,y--S1 by mul t ip l ica t ion by
P( x-x -I, y-y-S). It is not diff icult to p rove the following:
2
-I)2 a - 2 c ~ _ -1 ( ~ x = ~ x x 1
-))2 b 2 = 2 ~ ] _ - i ( y y - - y y 1
The re fo re P has the fo l lowing form:
P = U + V x x -I y-~ y y - - + W y - + W x x - I - -'
2 -I U , V , W , T ~ Z [ a ~ , b 2 ~ -I]
and: U(4,4) + V(4,4) + W(4,4) + T(4,4) = 1
- - - - = I Set: P ' = U + Vx-x-1 + W y y -s+ T -X-x-ly Y
We have:
P P' =(U+V x-x -I )2+(W+T x-x -1 )2+(U+V x-x -I )(W+T x-x -s )(y ~- -i+ ~-y-Z) Thus we
have : P P ' = U' + V ' x - x -I U ' ,V ' E Z[aZc( -1,b 2~-1]
U'(4,4) + V'(4,4) = 1
In t he s a m e way , PP'(U'+V' x ~ - - i ) is a po lynomia l U" in Z[a2c~ -s, b2~ -I] such that :
S - l Z [ x , x
290
U"(4 ,4 ) = I
T h e r e f o r e , P P ' ( U ' + V ' x x - - J ) b e l o n g s to S a n d P is i n v e r t i b l e in
- - - - - I =I ] -1, y , y - t , x , x , y , y . T h u s 1 + I n / i n . 1 i s a B - m o d u l e .
L e t X b e t h e c o m p l e m e n t o f a l i nk o f t w o i n t e r v a l s . T h e n t h e
m o r p h i s m f r o m 7 . [F(x ,y) ] to JVL®A f a c t o r i z e s u n i q u e l y t h r o u g h Z [ G 2] a n d Z[nl(X)] .
P roof : W e h a v e :
2 ~) = ( x y - y x ) ( x y - y x ) = - 8 ( x y - y x )
= - 8 ( x y ) - 8 (yx) - t ( x y y x ) + t ( x y ) t ( y x ) = A
T h e r e f o r e w e h a v e :
l im (S-IJvL/in) = l i m ( S-IJ%/An ) = NL®~,
a n d t h e m a p f r o m 7 . [F(x ,y) ] to J~L®A f a c t o r i z e s u n i q u e l y t h r o u g h Z [ G 2] a n d Z[Ttl(X)].
L e t u b e a n e l e m e n t of G 2 or n,(X). If u l ies in G 2 , u is c o n t a i n e d in a f i n i t e l y
g e n e r a t e d s u b g r o u p G of G 2 s u c h t h a t F (x ,y ) c G is n o r m a l l y s u r j e c t i v e . If u is in 7tl(X),
s e t G = ~ti(X).
I n all c a s e s F(x ,y ) --)G is n o r m a l l y s u r j e c t i v e , a n d G is g e n e r a t e d b y x ,y a n d
e l e m e n t s z I . . . . . z n in [G,G], a n d w e h a v e :
Vi= 1 . . . . . n z i e [F(x,y), G]
So t h e r e e x i s t w o r d s W i ( z I . . . . . z n) in t h e s u b g r o u p [F(x ,y ) ,F (x ,y , z j ..... Zn)] o f t h e f r e e
g r o u p F(x ,y ,z I . . . . . z n) a n d w e h a v e :
Vi= 1 . . . . . n z i = W i ( z I . . . . . z n)
B u t w e h a v e a c a n o n i c a l m a p f r o m G to ,PL®h,. T h e n z, . . . . . z n c a n b e c o n s i d e r e d as
e l e m e n t s in ,.)vL@ A, a n d W i is a w o r d in ,PL@ h. w h i c h i n v o l v e s z I , . . . . . z n z -lj , ... Z-In . W e
c a n r e p l a c e z-li b y ~ 8(zi) -I, a n d b y m u l t i p l y i n g t h e r e l a t i o n a b o v e b y a p r o d u c t o f
6(zi), w e ge t :
V i = l . . . . . n z H ( z z ) % = w ' (z . . . . . z , z . . . . z ) i j j j i 1 n 1 n
B u t z i is c o n g r u e n t to l m u d (0 = x y - y x :
Z i = 1 + 6] U i
So w e g e t e q u a t i o n s :
(E i) @oi(U) + ca q:)li(u) . . . . . (o q Oq i (u ) = 0
291
w h e r e Opiis a p o l y n o m i a l f u n c t i o n of d e g r e e p d e p e n d i n g on u = (u~ . . . . . u n) w i t h v a l u e s
in J%®A a n d coef f i c ien t s in A.
On t h e o t h e r h a n d , t h e r e is a u n i q u e m o r p h i s m f r o m t h e g r o u p of
p r e s e n t a t i o n < x , y , z j . . . . . z n ; z i = Wi(z ~ ..... z n) > too~L®A w h i c h is s t a n d a r d on
F(x,y). T h e n e q u a t i o n s (E i) h a v e a u n i q u e so lu t ion Jn (~F~® ~)n a n d th i s so lu t i on has
a l g e b r a i c c o o r d i n a t e s ( o v e r A). T h e r e f o r e z i b e l o n g s to J%®A and t he i m a g e of G in
~I~®A is i n c l u d e d in ~ ® A . T h a t p r o v e s t he l e m m a .
We are n o w able to p r o v e t he f i r s t p a r t of t h e o r e m 3 = 3. Le t us cons ide r t he
fo l lowing d i a g r a m :
Z[F(x ,y) ]
Z [ ~ j ( X ) ] -~ ~ L ® A
Z[F(x ' ,y ' ) ]
The c o m p o s i t i o n map ): Z[F(x' ,y ' )] --) J%®A goes to a quas i 2 x 2 m a t r i x a lgebra . Then
~) i n d u c e s m o r p h i s m s ~ , ~ ' :
: J%(Z[F(x ' ,y ' ) ] ) - ) , ~ ,® A
~ ' : C(Z[F(x ' ,y ' )]) -) A
On t he o t h e r h a n d , we have :
~)(x') is c o n j u g a t e to x in J%®A
O(y') is c o n j u g a t e to y in (F~,®A
q)(x'y') = xy
T h e r e f o r e w e h a v e :
~)' ( t ( x ' ) ) = t (O(x ' ) )= t (x)
~' ( t ( y ' ) ) = t (O(y ' ) )= t (y)
~' ( 8 ( x ' ) ) = 8 ( ~ ( x ' ) ) = 8 ( x )
~' (8(y')) = 8(0(Y')) = 8(y)
~' ( t (x 'y ' )) = t(~)(x'y')) = 8(xy)
If we i d e n t i f y x' and x, y ' and y, 9 is a m a p f r o m ~ to J%®A i n d u c i n g t h e inc lus ion
A c A in t h e coe f f i c i en t r ing. T h u s 9 e x t e n d s to an e n d o m o r p h i s m ~)L of t h e A - a l g e b r a
~ , ® A .
S u p p o s e t h a t L and L' a re t w o l inks of 2 i n t e r v a l s s u c h t h a t L and L' are
c o n c o r d a n t . Le t X a n d X' b e t h e c o m p l e m e n t s of L a n d L' in I x R a a n d Y b e t h e
292
c o m p l e m e n t of the cobordism in I x I x IR a . We have a c o m m u t a t i v e d iagram:
~z[~, (x)] Z[F(x',y')] Z ~ T t , ( Y ) ] -~ ,3%®^
~ ~ Z n,(X')l
There fo re l inks L and L' induce the same m o r p h i s m from Z[F(x' ,y')] to ~L®A and ~L
and ~L' are the same. Hence, ~L depends only on the concordance class of L.
S u p p o s e tha t L and L' are two l inks of 2 i n t e r v a l s . Le t L" be the
juxtapos i t ion of L and L'. Let X, X', X" be the c o m p l e m e n t s of L, L', L" in I × IRa We have
the diagram:
Then we have:
Z[F(x,y)]
Z[~,(X)] c~
Z [ F ( x , y ) I " ~ g ~ [ ~ , ( X " ) ] fl--) J%®A
Z[n,(X')]
Z [ F ( x , y ) j ' ~ g'
~0LO (~ o~ of) =~ (~ g
and this implies:
~OL" = ~L' o ~)L
Thus ~L is an automorphism of J~L®Aand ~ is a representation of the group C 2 to
Aut(JV~®A) Now, the last th ing to do is to prove tha t ~L is the con juga t i on b y some
e l e m e n t in A(DAXy and tha t will be a consequence of the fol lowing lemma:
~ . 9 Let ~) be an a u t o m o r p h i s m of the a lgebra JVL®A which keeps
xy fixed. Then t he r e exists an e l e m e n t E in A~)AXy, u n i q u e up to a scalar, such tha t ~ is
the conjugat ion by E.
Proof: Let us denote b y A' the fol lowing algebra:
293
A ' = A[~.]/~z_ a~. +~
W e h a v e a Galois a c t i o n o f t h i s e x t e n s i o n :
~.~- , 7~ = a-7~
L e t K a n d K' b e t h e q u o t i e n t f i e l d s of A a n d A' . L e t $t a n d 0 b e t h e e l e m e n t s of K'
d e f i n e d by :
B
It+ It= b
ZI~+Z ~t=c
Actually 8 lies in K. Thus we have a representation p from d% to Mz(K'):
Let fl be the matrix: [0 :I in M2(K) c M2(K')
t t is e a s y to s e e t h e fo l lowing : V z E ~ l i p ( z ) = p ( z ) fl
w h e r e - - d e n o t e s t h e Galois a c t i o n o n K' e x t e n d e d to M2(K'). A n d t h e n , ~ 'L®K is
i s o m o r p h i c t o t h e s u b r i n g R o f m a t r i c e s A ~ Me(K') s u c h t h a t :
h A = A n
T h e a u t o m o r p h i s m $ i n d u c e s an a u t o m o r p h i s m $ o ° n R a n d on R®K' =
M2(K') a n d t h i s a u t o m o r p h i s m k e e p s t h e c e n t e r f i x ed . T h e r e f o r e t h e r e e x i s t s a m a t r i x
c o in GL2(K') s u c h t h a t : - !
V A E R (Po(A) = c o A C o
T h a t m e a n s :
V A E M2(K') f~ A = A n ~ f ] c o A C o -l = t--oAt-o -l f~
- - = - - C O I ~- - - -~VA~M2(K ' ) f l A = A f t ~ f l t o A C o -~ Coil Af~ - I - -
c o l f~ CoA = A f l - ' c o 1 t2 c o
T h e n , fo r a n y A in R, A c o m m u t e s w i t h f ~ ' ~ - - - c o 1 £I c o . Bu t R®K' is i s o m o r p h i c to
Mz(K'). H e n c e fl-1 ~o-i fi co is c e n t r a l a n d t h e r e e x i s t s k ~ K' s u c h t h a t :
f l t o = k t o fl
On t h e o t h e r h a n d , tPo(XY) is e q u a l to xy a n d t o c o m m u t e s w i t h xy . T h e n t h e r e e x i s t u ,v
K' s u c h t h a t : t o = u ÷ v p (xy) a n d t h i s i m p l i e s :
t i c o = f ~ ( u + v p ( x y ) ) = (u ÷ v p ( x y ) ) f ~
2 9 4
= (k u + k v p ( x y ) ) O
u = k u v = k v
S i n c e E o is i n v e r t i b l e , u (or v ) is n o t z e r o . B u t ~o is d e f i n e d u p to m u l t i p l i c a t i o n b y a
s c a l a r . T h e r e f o r e w e m a y as w e l l s u p p o s e t h a t u (or v ) is e q u a l to 1 a n d u a n d v b e l o n g
t o K. A f t e r m u l t i p l i c a t i o n b y s o m e e l e m e n t in A w e wi l l ge t :
£o = u + v p ( x y ) u , v E A
L e t u s se t : ~ = u + v x y ~ J~.®A
W e h a v e t h e f o l l o w i n g :
Vz e ,.)A.®A q)(z) c = c z
a n d t h e o r e m 3 - 3 is p r o v e d .
R e m a r k 3 - | 0 I t is n o t c l e a r t h a t u + v x y c a n b e c h o s e n to b e a u n i t in
JvL®A. B u t w e h a v e t h e f o l l o w i n g :
P r o p o s i t i o n 3 - I | L e t L b e a l ink of 2 i n t e r v a l s . T h e n t h e a u t o m o r p h i s m 0L
is t h e c o n j u g a t i o n b y a n e l e m e n t u + v x y ~ JA.®A s u c h t h a t u + v g o e s to I b y t h e
a u g m e n t a t i o n m a p f r o m A to Z s e n d i n g a, b, c to 2 a n d ~, [3 to I.
P roof : T h e a u t o m o r p h i s m 0L is t h e c o n j u g a t i o n b y a n e l e m e n t
= u + v x y E A O A X y
S i n c e ~-8(~) -I is t h e i n v e r s e of ~ in JVL®K , w h e r e K is t h e f r a c t i o n f i e l d o f A, w e h a v e
t h e f o l l o w i n g :
c y c - ~ 0 m o d S ( E ) ~ x ~ ~ 0 r o o d S ( c )
b u t w e h a v e :
y ~ - : E t(y ~--) - 2 f - : t(E f ) c - t(E) c ~- + a(~) ~- - - 2
E x E : e t( x ~ ) - c x : t ( x - ~ ) ~ - t (e) e x + 8 ( ~ ) x
T h i s i m p l i e s :
t ( e y - ) ( u + v x y ) - t ( ~ ) ( u y - + ~ v x ) ~ 0 m o d S ( E )
t ( x ~ ) ( u + v x y ) - t ( e ) ( u x + ~ v y ) ~ 0 m o d S ( E )
S i n c e ( I , x, y , x y ) a n d ( I , x, y , x y ) a r e A - b a s i s o f JvL®A, w e ge t :
u t(c) ~ 0 m o d S ( c ) v t ( E ) ~ 0 r o o d S ( c )
u t ( e y ) - 0 m o d S ( c ) v t ( ~ y ) ~ 0 m o d S ( ~ )
2 9 5
u t ( x E ) ~ 0 m o d 8(c) v t ( x - c ) ~ 0 m o d 8(E)
For x I . . . . . xp in A, d e n o t e b y < x) . . . . . x p > t h e i d e a l g e n e r a t e d b y x~ ..... x~. W e h a v e :
< u , v > . < t ( x ) , t ( c y ) , t ( x r ) > c < 8 ( ~ ) >
< u , v > . < 2 u + cv, bu + a[]v, au + b ~ v > c < 8 ( c ) >
I t is e a s y to c h e c k t h e f o l l o w i n g :
< 2 u + cv, bu + a l ]v , au + bc~v> ~ < b a ~ - a 2 1 3 , 2 a l ] - b c , 2 b ~ - a c > . < u , v >
T h e n w e h a v e :
< u , v > 2 < b 2 _ a 2 ~ , 2 a ~ _ b c , 2 b c ~ _ a c > c < 8 ( E ) >
l e t w b e an e l e m e n t of < u , v > z . T h e r e e x i s t X a n d Y in A s u c h t h a t :
w(bZc~ -a2l]) = X 8(~) w(2al ] - bc ) = Y 8(c)
T h e r e f o r e X(2a~ - bc ) is d i v i s i b l e b y b2~ - a2O.
L e t B b e t h e s u b r i n g of A d e f i n e d by :
B= S - S Z[a , b , ~, - i l], l I - l ] [[A]] ( s e e n o t a t i o n 3 - 2)
W e h a v e : A = B ~ c B . T h e n t h e r e e x i s t X o,X~ EB s u c h tha t : X = X o + Xjc
a n d w e d e d u c e :
2~I]X o - b X t (A - a2l] - b2(~ + 4~I]) ~ 0 m o d b2~ - 2 j ]
- b X o + 2a13X l - ab 2X) - 0 m o d b2~ - (~21]
X, b2A - 0 m o d b2~ - ~213
T h e n Xj is d i v i s i b l e b y b 2 _ z[~ ( in B) a n d X o a l so . T h e r e f o r e X is d i v i s i b l e b y
b2~ -- ~2l] in A a n d t h e n in A. Th i s i m p l i e s t h a t w i t s e l f is d i v i s i b l e b y 8(~), a n d w e
h a v e :
< u , v > 2 c < ~ ( ~ ) >
T h u s t h e r e e x i s t t h r e e e l e m e n t s r, s, t E A s u c h tha t :
2 u = r S ( c ) = r (u2+ c u v + ~ 0 v 2 )
u v = s 8(~) v 2 = t 8(~)
I t is e a s y to c h e c k t h a t :
2 s = r t r + c s + ~ t = I
M o r e o v e r , r, s, t d e p e n d o n l y on t h e h o m o t h e t y c lass of t a n d on t h e a u t o m o r p h i s m ~.
L e t us d e n o t e b y r o , s o , t o t h e i m a g e s o f r, s, t b y t h e a u g m e n t a t i o n
m o r p h i s m f r o m A to Z w h i c h s e n d s ix, ~ t o I a n d a, b , c t o 2. W e h a v e :
2 S o = S o t o
t o + 2 S o + t o = I
296
T h e r e f o r e it is e a s y to see t h a t t h e r e ex i s t s a u n i q u e i n t e g e r 0 s a t i s f y i n g t h e fol lowing:
= = 0 2 r o = ( I - 0 ) 2 s o 0 ( I - 0 ) to 2
Now, if w e c o n s i d e r a n o t h e r a u t o m o r p h i s m 0', we ge t a n o t h e r i n t e g e r 0' and it is e a s y
to check t h a t 0+0' is t h e i n t e g e r c o r r e s p o n d i n g to ~0 0' - On t h e o t h e r h a n d , if 0 is t he
c o n j u g a t i o n b y xy, w e h a v e u = 0, v~ I and t he c o r r e s p o n d i n g i n t e g e r is 0= I
T h e r e f o r e , t h e r e ex i s t s a n i n t e g e r n such t h a t t h e c o r r e s p o n d i n g i n t e g e r of c(xy) n is
zero , D e n o t e b y E'~ u' + v ' xy t h i s n e w e l e m e n t of ~ L ® A a n d b y r', s', t ' t h e
c o r r e s p o n d i n g e l e m e n t s in A c o n s t r u c t e d as above , We have :
r ' o = I s' o ~ 0
a n d r ' + s' xy goes to a un i t in ~ L ® Z . Hence c is a mu l t i p l e of:
(r ' * s' xy ) ( xy ) -n
w h i c h is i n v e r t i b l e in ~ 4 ® Z
8 4 - ~ .
Cons ide r t he l ink L g i v e n b y t he fo l lowing p ic tu re :
i , Z
The l ink is o r i e n t e d f r o m t h e top to t h e b o t t o m a n d x, y, x', y', z a re
e l e m e n t s of t h e f u n d a m e n t a l g r o u p It of t he c o m p l e m e n t of L c o r r e s p o n d i n g to p a t h s
t u r n i n g a r o u n d p a r t s "over" of L (see t he p i c t u r e )
Because L has t h r e e c ross ings , we h a v e t h e fo l lowing r e l a t i ons :
x z = ~ x x y ' = y ' z y z = z y '
and w e deduce :
x y ' = y' x - ) x' x
T h a n k s to t h e o r e m 3=4, t h e r e ex i s t s an e l e m e n t c in A~)Axy such tha t :
x' = c x £ - ' y ' = E y e - ' i n ~ . ® A
297
T h e r e f o r e w e h a v e in J~L®^:
- I = ! - ! - ! x ~ y c : c y E x c x e x
If w e m u l t i p l y on t h e lef t b y ~- 8(e) . w e get:
a n d th i s impl i e s :
~ - x E y E X = y E X ~ x ~ -
Let f be t h e a n t i i n v o l u t i o n of J%®A s e n d i n g
x t o x a n d y t o y ( a n d x y t o y x ) a n d l e t ~ ' = f(c). We h a v e :
Ex = x c ' y c = c ' y
~ - x = x ~- y ~ - = c~y
T h e n w e h a v e :
~ - x ~ y E x = x c ~ c y c x ~ x E ' ~ : E ' y x
y E X ~ X E - = y E XXC'~-~ x x y ~ r 'C
a n d w e get:
~ ' ~ ' y x = x y c C'c = f ( r ' c c ' y c )
We h a v e :
le t us set:
U = t ( c ~ ' ) V ~ 8(r') c ~ u + v x y
So w e h a v e :
C ' E E ' y x ~ U c ' y x - V ~ y x
= U ( u y x - + v x y y x ) - V ( u y x - + v y x y x - )
= U ( u y x - + [ ~ v x 2 ) - V ( u y x - + t ( y x ) v y x - v y x x y )
- - - - 2 - - - - 2 = U ( u y x * 1 3 v x ) - V ( u y x + t ( y x ) v y x - v ~ y )
and t h e e q u a t i o n :
E ' c E ' y x ~ f ( c ' e ~ ' y X)
g i v e s r i se to t h e fol lowing:
U u ( y x - x y ) - V u ( y x - x y ) - V v t ( y x ) ( y x - x y ) = 0
( U u - V u + V v t ( y x ) ) ( y x - x y ) = 0
a n d w e get:
U u - V u + V v ( a b - c ) = 0
On t h e o t h e r h a n d , w e h a v e :
U = t (cE ~ ) = t ( (u + v x y ) ( u + v x y ) )
298
2 u 2 2 = + 2 u v c + v t ( x y x y )
= 2 u 2 + 2 u v c + v 2 ( A + 2(Xi3)
( w i t h : zX = c 2 - a b c . a2ct + bZ~ - 4c~{}
V 8(c') u 2 z = = + U V C + O t ~ V
N o w it is e a s y to o b t a i n t h e f o l l o w i n g e q u a t i o n :
(u * ( a b - c ) v ) ( u 2 + c u r + c ( ~ v 2 ) + A u v 2 = 0
M o d u l o t h e a u g m e n t a t i o n i d e a l o f ^ , w e ge t :
(u + 2 v ) ( u 2 + 2 u v * v 2 ) = 0
B u t w e k n o w t h a t u a n d v c a n b e c h o s e n s u c h t h a t u + v is n o t c o n g r u e n t to z e ro . T h e n
u 2 + c u v + ~ [~ v 2 is n o t z e r o m o d u l o A, a n d w e h a v e :
u = ( c - a b ) v m o d
In t h i s e x a m p l e , w e c a n c h o o s e v to b e I, a n d u is t h e u n i q u e e l e m e n t in A, c o n g r u e n t
to c - a b m o d u l o ~ a n d s a t i s f y i n g t h e f o l l o w i n g e q u a t i o n :
(u + a h c)( u 2 - + c u + ~ ) + ~ u = 0
A c t u a l l y t h i s e q u a t i o n d o e s n ' t h a v e a n y s o l u t i o n in A. T h e e l e m e n t u b e l o n g s to a c u b i c
e x t e n s i o n o f A i n c l u d e d in A, a n d i t s e e m s to b e v e r y d i f f i c u l t to f i n d a s u b r i n g o f ^ ,
s m a l l e r t h a n A w h e r e w e c a n d o all t h i s c o n s t r u c t i o n fo r all l i n k s .
R e r e r e n c e s
[ I ] A . J . CASSON , o r a l c o m m u n i c a t i o n . See a l so :
A . M A R I N , L ' i n v a r i a n t d e C a s s o n , p r e p r i n t
[2] M . CULLER a n d P . B . SHALEN , V a r i e t i e s o f g r o u p r e p r e s e n t a t i o n s a n d s p i i t t i n g s
o f 3 - m a n i f o l d s . A n n . o f M a t h . 117 , n ° l ( 1 9 8 3 ) , pp . 1 0 9 - 1 4 6
[3] J . Y . LE D I M E T , C o b o r d i s m e d ' e n l a c e m e n t s d e d i s q u e s . To a p p e a r
[4] J . P . LEVINE , L i n k c o n c o r d a n c e a n d a l g e b r a i c c l o s u r e o f g r o u p s . P r e p r i n t
U n i v e r s i t ~ d e N a n t e s
D ~ p a r t e m e n t d e M a t h ~ m a t i q u e s
2 r u e d e la H o u s s i n i ~ r e
F - 4 4 0 7 2 NANTES C e d e x 03
