Locally finite approximation of Lie groups, I

Invent. math. 83, 425-436 (t986) Inventio~le$ mathematicae �9 Springer-Verlag 1986

Locally finite approximation of Lie groups, I

Eric M. Friedlander* and Guido Mislin

1 Department of Mathematics, Northwestern University, Evanston, Illinois 60201, USA 2 Department of Mathematics, ETH-Zentrum, CH-8092 Ziirich, Switzerland

For a compact, connected Lie Group G, we earlier demonstrated that the classifying space BG was "cohomologically approximated" by the classifying spaces of (finite) groups of points of an associated algebraic group with values in finite fields [5]. In Theorem 1.3, we generalize this result to arbitrary Lie groups with only finitely many components. Namely, we prove that the classifying space of such a group can be "cohomologically approximated" by the classifying space of a group expressible as a countable union of finite groups (i. e. a locally finite group).

The usefulness of such an approximation results from the opportunity it affords to extend known theorems for finite groups to arbitrary compact Lie groups. With this philosophy, we present a family of applications of H. Miller's proof of the Sullivan Conjecture. Namely, our approximation theorem in conjunction with H. Miller's theorem enables us to provide a reasonably explicit description of the homotopy groups of mapping complexes with source the classifying space of a Lie group with only finitely many components (see Theorem 3.3 below). In [6], we prove further properties of our approximation which enable applications to the generalized cohomology of classifying spaces.

It is our pleasure to acknowledge the great benefit we have derived from many patient explanations of A. Borel as well as useful conversations with P. Gabriel and H. Miller and constructive suggestions of the referee. The first author gratefully acknowledges the warm hospitality of ETH Zurich during the development of this paper.

1. The approximation theorem

In Theorem 1.3, we demonstrate that any Lie group G with finitely many components admits a "locally finite approximation away from p". In the special case in which G is connected, this approximation theorem is a sharp- ened form of [5, Theorem 1.4].

* Partially supported by the N.S.F.

426 E.M. Friedlander and G. Mislin

As motivation, we shall recall the well known approximation of the Lie group SI~-I t /Z. The inclusion i: II~/Z~S 1 with tI~/Z given the discrete topology is a homomorphism of topological groups. The induced map on classifying spaces Bi: B~/Z- - ,BS 1 satisfies a) Bi*: H*(BS1; Z/nZ) ~H*(BII~/Z,Z/nZ) is an isomorphism for any n>0 ; and b) Homgrps(lI~/TZ, F ) = {1} for any finite group F.

As introduced below, the approximation q~: Bg--*BG differs in two important respects from the simple example Bi: B t ~ / Z ~ B S 1. Namely, we do not require q~ to be induced by a homomorphism of groups and �9 will fail in general to be a suitable cohomological approximation at the designated prime p since, for G connected, our conditions on ~b imply that g be p-acyclic. Replacing II~/Z by its subgroup II~/Z[1/p] of elements of order prime to p, we obtain an example

cI): BII~/TI [1/p]~ B S ~

of a locally finite approximation away from p.

1.1 Definition. Let G be a Lie group with finitely many components and let p be a prime number not dividing I~zo(G)], the order of the group of connected components of G. A map 4~: B g ~ B G from the classifying space of a countable locally finite group g to the classifying space of G is said to be a locally finite approximation away from p (occasionally abbreviated "l.f. approx/p") if 45 satisfies the following three conditions:

a) For any finite rco(G)-module A, whose order is not divisible by p, induces an isomorphism

~* : H* (BG, A) ~, H* (B g, A) = H* (g, A).

b) cb induces a surjection

~b,: g~-~l(Bg)~nl(BG)=no(G).

c) Let n = k e r ( ~ , : ,q~n0(G)). Then /4*(~z, 7Z/p)=0 and Homgrps(n, F)={I} for any finite group F.

1.2 Remark. Observe that / 4 , ( ~ , 7 / p ~ ) = ~ / 4 , ( ~ , Z / p " ) = 0 if /4*(~,Z/p)=0,

where ~p~=l imZ/p" . Because ~ = k e r ( O , : g--*~0(G)) is locally finite, the

g roups / t , (n , ~) and/4*(n, ~) are trivial and, since 7Zp~,-~/7Z(p~, (1.1 c) implies therefore that /4,(~,Z~p~)=0=/4*(n,Z~p)). It follows that /4"(~,7/p~)=0, a property of ~ which will be used later. Moreover, because no(G ) is a finite group of order prime to p, we conclude that /4*(g, B)=0 for any 7l~p~[~0(G) ]- module B.

We proceed to state and then to prove the existence of locally finite approximations away from p.

1.3 Theorem. Let G be a Lie group with finitely many components and let p be a prime number not dividing the order of ~zo(G ). Then there exists a locally .finite approximation away from p (1.f. approx/p), ~: Bg-*BG.

Locally finite approximation of Lie groups, I 427

In our proof of Theorem 1.3, we consider G ~ G / R with R a complex torus and G / R ~ G surjective with finite kernel. The following two lemmas reduce the proof of Theorem 1.3 to a consideration of (~ which we analyze using tech- niques of [5].

1.4 Lemma. Let f : G ~ G be a surjective homomorphism of Lie groups with finitely many components and let p be a prime not dividing I~zo(G)l. Assume ker ( f ) has the homotopy type of the torus (S~) ~. I f cb: Bg-~BG is a 1.f. approx/p, then there is a map of fibration sequences

B ( ~ / Z D/p])" --, B ? - --, Bg

l: 1o B ker ( f ) , B G - - - * B G

with each vertical map a t.j~ approx/p.

1.5 Lemma. As in (1.4), let f : G-~G be a surjective homomorphism of Lie groups with finitely many components and let p be a prime not dividing Ino(G)[. Assume that ker ( f ) is finite and let P c k e r ( f ) be a p-Sylow subgroup. I f ~: B g ~ B G is a l.f. approx/p, then so is ~: B~=pul l - back ( B g ~ B G * - - B d / P ) ~ B d .

Proof of Theorem 1,3. By a well-known theorem [8, Chap. XV], the quotient of G by a maximal compact subgroup K is contractible so that B K o B G is a homotopy equivalence. The complex form K(~) of K has reductive connected component and BK~BK(I I ; ) is also a homotopy equivalence. Thus, it suffices to assume G = K(C), which we shall do for notational convenience.

Let G denote the image of Ad: G~Aut(G~ Then G is the group of complex points of a smooth integral group scheme Gz which fits in an extension

where G) ~ is the Chevalley integral group scheme associated to the adjoint form of G o and ~- is the constant integral group scheme associated to the image of rro(G ) in 7ro(Aut (G~ [14; Chap. XXIV - 1.3]. The identity component R of the center of G is a complex torus normal in G. Moreover, G / R ~ G is surjective with finite kernel. Consequently, Lemmas (1.4) and (1.5) imply that it suffices to verify the existence of a 1.f. approx/p for (~.

Define ~=Gz0Fv) and define ~: B ~ B G as in the proof of [5; 1.4]. Namely, using etate homotopy theory we obtain a map B~-~[ I (Z /1 )~BG ,

l:gp

where (Z / l ) ~ ( - ) is the Bousfield-Kan Z/l-completion functor, which induces an isomorphism in cohomology with any finite (2g[1/p])[~-module as coefficients [4]. Then ~ is the unique lift for the lifting problem

B(~

Y+ l:l-p

428 E.M. Friedlander and G. Mis|in

whose existence and uniqueness is given by the vanishing of/~*(Bg, t~) (since is locally finite) and/~*(Bg, Z/p) (by [13; remark on p. 582] applied to G~ used in conjunction with Sullivan's "arithmetic fibre square" technique [15]. So constructed, ~ satisfies (1.1 a) and (1.1 b). We have already observed that /4*(B~,Z/p)=0; because each simple factor of G~ d has the property that its group of lFp-rational points is a simple (discrete) group, and because

G a~t@ ~ Homg~p~(~,F)={1} for any finite group F, =ke r (~ : g~no(G)) equals 7 t~pJ, as required by (1.1 c).

Proof of Lemma 1.4. The fibration ~: B d ~ B G is classified by an element ~'eH3(BG, Z_ ") where Z" is the 7to(G)-module given by the action of rco(G ) on r~(ker(f)). Let ~*~: X-~Bg be the induced fibration classified by 4~* ~' e H a(g, Z"). Because Z" is a trivial rt = ker (~ . " 9~rto(G))-module, (1.1 c)

_ _ t~3, ~n- implies (cf. Remarkl .2) that /-I*(~,Z")-~/-)*(~,Z[1/p]") so that t g , _ ) "--/~3(9 , 2g [1/p]") which is isomorphic to H2(9, I~/~ [1/p]") because g is locally finite. We conclude the existence of a map of fibration sequences

B(Q/Z[1/p])" , Y ,B 9

B ker ( f ) , X , B 9

in which 2 satisfies (1.1 a), (1.1 b), and (1.1 c). Write Y=B~, 9=li_~mg~ with each

g~ finite, ~=li_~m~ with ~ c ~ the inverse image of g c.g, and A=II~/Z[1/p]".

Because HZ(g,, A)= lim H z(9,, m A) where ,,A = k e r ( A - ~ A ) , we conclude that

each ~ and thus also ~ are locally finite. By construction, X - . B d and therefore c I ) : B ~ X - - , B d satisfy (1.1a) and (lAb)_ Since ~ = k e r ( ~ 5 . : ~ l t o ( d ) =no(G)) is an extension of A =l~/Z[1/p]" by re, ~ also satisfies (1.1 c).

Proof of Lemma 1.5. Let C denote ker ( f )c~d ~ Because C is a finite normal subgroup of d ~ it is central in d ~ Because IPI and Ir~o(d)] are relatively prime, P must be the subgroup of k e r r consisting of all p-torsion elements and therefore P is normal in G. By definition of ~, we have a map of fibration sequences

B(ker(f)/P) , B-g , Bg

B(ker (f)/P) , B(G/P) , BG

and a central extension

(1.3.1)

l ~C/P---,fr-orc--* l (1.3.2)

where ~ = k e r ( ~ . : ~no(d /P)=no(d ) ) . Using (1.3.1), we easily conclude that (1.1 a) for 4~ implies (1.1 a) for ~. Observe that the obstructions for existence


and uniqueness for the lifting problem

8~

lie in /t*(~, P)=H~ H*(~, e)). By (1.1 c) for 45 and (1.3.2), /t*(~, Z /p )=0 so that a unique lifting 45: B ~ B d , of 45 exists and automatically satisfies (1.1 a). Because �9 satisfies (1.1 b), the fibre of �9 and thereby of 45 is connected, thus, 45 and thus 45 satisfy (1.1 b). As for (1.1 c), we have already observed the vanishing of H*(~,Z/p). Finally, consider any map 0: ~ F with F finite. By (1.1c) for ~, image (0 )cF equals image(C/P) so that we may assume F is abelian of order prime to p. Applying (1.1 a) for 45, we conclude that

Horn (~, F)=H~(~, F) = H~ (~, Z [n0(G)] |

is isomorphic to HI(BG, Z[n0(G) ] | ~ F) which is 0 because ~o is connected.

2. A direct construction

A well known theorem of Chevalley asserts that a connected reductive complex linear algebraic group G is obtained by base change from a reductive group scheme G z (i.e. an affine smooth group scheme over specZ with connected, reductive geometric fibres c.f. [14; p. 15]). In Proposition 2.1, we extend this result to any linear algebraic group over the complex numbers, whose connected component is reductive. Such a group G is obtained by base change from a group scheme G A smooth over specA for some ring A of S-integers in a number field. This enables us to provide in Theorem 2.2 a more explicit construction of a locally finite approximation away from p whose locally finite group arises as the discrete group of lFv-valued points of such a group scheme G A. One important consequence of this alternate construction is that the resulting locally finite approximation is induced by a chain of homomorphism and base change maps. The disadvantage of this construction is that it only provides locally finite approximations away from p for all sufficiently large primes p and does not apply to small primes.

(2.1) Proposition. Let G C be a linear algebraic group over ~ with reductive connected component. There exists a number field L c (E, a finite set of primes of L with associated ring A of S-integers in L, and a group scheme G a smooth over A, with reductive connected component G ~ , such that Gr G A X spec(A) spec (~).

Proof. We claim that there is a ~-group G~ such that

G~ x spec(~) spec (C) _~ G C.


From the classification of reductive groups we know that there is a group G~ with Gg? X spec~O ) spec(ll2)~G ~ By adapting the classification of nonabelian extensions of abstract groups [9, Chap. IV.8] to the case of algebraic groups over

and I1~ (using Hochschild-cohomology as described in [2]), one proceeds as follows. From the short exact sequence of group schemes 1-*G~Gc-*nr n c the constant group scheme on the abstract group n=Ge(~)/G~ we obtain an abstract kernel (in the sense of Mac Lane)

GO �9 (no, e, ~br ne~(AutgrG~176162

Note that Er215 ) where E 0 is the constant group scheme AutgrG~/IngrG ~. In follows that ~9r215162 ) for a unique ~90: n0~E0 , and we obtain an "abstract kernel" (no, G~,~pO), where n 0 =Go/G ~. We denote the associated obstruction 3-cocycle by c O: n 0 • n0-* C 0, C o the center of G~. Observe that [co] ~ H3(n0, Co) is 0, since by extension of scalars one has an isomorphism a: H*(nO, CO)_-~H*(nc, CO, CecG ~ the center, and therefore a[co]=O, since the corresponding abstract kernel is realized by Gr As a consequence, the ~-extensions of G O with quotient n O and action ~O 0 are classified by H2(nO, CO) which is a finite group, isomorphic to H2(n~, C 0 under or, and we conclude that there is a unique G o with G o • ~p~0~ spec ( ~ ) = Gr The affine algebraic group G o is obviously of the form Ga• ), where A is the ring of S-integers in a number field L, S a finite set of primes in L, and we may assume G A to be smooth over A with reductive connected component by choosing S large enough.

We now present our alternate construction of a locally finite approximation q~: Bg~BG away from p. This construction is used systematically in [6] and is there denoted by BFa~BG.

Let G A be a group scheme over specA. Let s p e c B ~ s p e c A be a base change morphism. Denote by GA(B ) the group of B-points of G a and, as in [-5, w write GA(B)~ for the constant group scheme over specB with group of components GA(B ). The natural B-morphism p: GA(B)B-*GB=GA• induces a morphism of simplicial schemes Bp: BGA(B)B~BG B and of associated etale homotopy types (BGA(B)B)~t-*(BGB),t (cf. [4]). In case (specB)~ t is contractible, there is a natural homotopy equivalence BGa(B)~-(BGA(B)8)~t where GA(B) is given the discrete topology. Hence one obtains in this case a natural homotopy class

(2.2) BG A(B)-*(BGB)et.

We will make use of this map in the following special case. Assume A ~ W~v ) = W, the Witt vectors of IF v, and B = I F z with A o B the obvious map. Assume furthermore given an embedding q): WOFv)-*~. We have naturally associated a chain of maps

(2.3) (B GFp)e t - * (B G w)r ~ (B Gr -* (B Ge)s.et--~ Sing B Gr (C)

in the notation of [4; 8.8]. We assume now that G ~ is reductive. Then, according to [4; 8.8] the corresponding chain of maps with G replaced by G O determines homotopy equivalences between the Sullivan completions ()^/P

Locally finite approximation of Lie groups, 1 431

away from p. If we assume moreover that Gw/G~ is a constant group scheme over W, then we have a chain of homotopy equivalences

(B ~ p )et ~ (B n W)et ~ (B 7~C)et --~ (B ne)s.et ~ Sing B ~c(~).

Therefore, the maps in (2.3) induce all homotopy equivalences between the associated Sullivan completions away from p. Using (2.2) we thus obtain for such a G w a natural homotopy class (depending on ~p: W ~ )

h q~: BGA(IFp) = BGwOFp)-~(BGc(ff~)) ̂ /p

(2.4) Theorem. Let G be a Lie group with finite component group. Let K c G denote a maximal compact subgroup and Gr the linear algebraic group over whose Lie group of complex points is a complex from for K. Let A be a ring of S-integers in a number field as in (2.1) such that Gr A • Then for almost all primes p, there is a locally finite approximation away from p

�9 : BGAOPp)~BG.

Furthermore, if G is connected, we may choose A =7, and p arbitrary.

Proof. Let p be a prime not lying under a prime of S and assume that p does not divide lTz0(G)l. Let W denote the Witt vectors of IFp and choose an embedding 99: W ~ r with A c W. Consider the natural homotopy class discussed above

h ~p : B G A OFp) ~(BGr ^/p.

We claim that this map lifts to a unique homotopy class

~': BG AOFp)~BGr162 ).

As in the proof of (1.3) it suffices to verify that fI*(BGAOFp),~)=O, which clearly holds, and that ffI*(BGA(IFp),Z/p~)=O. For the latter it suffices to see that G~ is p-acyclic, since p does not divide LGAOFp)/G~ Since the radical R of G ~ has a p-acyclic group of lFp-points (isomorphic to OF*)") we may assume G ~ semisimple in which case we have already observed that G~ is p-acyclic (cf. proof of (1.3)). If we construct ~3~ B.G~176 in a similar way we obtain a homotopy commutative diagram of fibration sequences

BG~ , BG A~p) ~ BG A~:p)/G~

ooj oi BG~162 - , BGr162 > BGr162176162

where ~ is constructed as ~, using the constant group scheme GA/G~ instead of G A. Obviously, 7 ~ is a homotopy equivalence. By [5; 2.3], ~0 is a H*( ,Z/nZ)- isomorphism for n prime to p. An easy spectral sequence argu- ment thus shows that ~ satisfies (1.1 a). Clearly, (1.1b) holds since 7' is an equivalence. We have already observed that G~ is p-acyclic. Moreover,


every map G~ to a finite group factors through G~ where R c G ~ denotes the radical. But G~ is a product of (abstract) simple groups. Thus Homgrps(G~ F)= {1} which completes the proof that �9 : BGA(IPp)-*BGc(~2) is a 1.f. approx/p. Since the inclusions Gc(C),--K~G are homotopy equivalences, we obtain an associated 1.f. approx/p ~ by composing with BGr ~_ B K ~_ BG.

(2.5) Remark. The map 4~: BGAOFp)~BG constructed in (2.4) is induced by group homomorphisms and base change maps in the following natural way. There is a chain of group homomorphisms

G AOPv)~G A(W)~G A(r ~G

with GA(IFv) , GA(W ) discrete and GA(•),K , G given their usual Lie group topology, inducing a homotopy commutative diagram

BG A~p) , BGA(W ) ,BGA(r ), BK - , BG

(BG;p~{ p ~ - (B!w):{ v ~ - (B!r p ' ' (B!) -/p ~ _ (B!)~/p

The resulting map BGA(IFp)-o(BGf/p lifts, as we have seen, in a unique way up to homotopy to BG, and this lift ~: BGAOFp)-oBG is our locally finite approximation away from p.

3. Applications to mapping complexes

H. Miller has shown that map,(Bn, X) is weakly contractible, where 7t is a locally finite group, X a finite dimensional, connected complex, and m a p , ( - , - ) the mapping complex of pointed maps between pointed spaces [11]. In Theorem3.1, we provide a natural extensions of Miller's Theorem to Lie groups with finitely many components. (As first observed by B.I. Gray [7] and then systematically studied by W. Meier [10] and A. Zabrodsky [16], there do exist examples of non-trivial maps from the classifying space of a compact Lie group to a finite complex.) We follow the "vanishing" result of Theorem 3.1 with an explicit description in Theorem 3.3 of the homotopy groups of map,(BG, X) for somewhat restricted X. Finally, in Theorem 3.5 we identify map, (BG, X) for X restricted as in (3.3) and further restricted by the condition that its universal cover be rationally an H-space.

The two basic ingredients in the proof of the following theorem are Miller's Theorem and the existence of locally finite approximations (Theorem 1.3). More effort than might be expected is required to prove Theorem 3.1 because the space X is not assumed to be either finite or simply connected.

(3.1) Theorem. Let G be a Lie group with finitely many components, let X be a finite dimensional, connected complex and let X--*Jg be the Sullivan profinite completion of X. Then map, (BG, X) is weakly contractible.


Proof Write X=limmX ~ where each X= is a connected complex with finite

homotopy groups, the limit being taken in the homotopy category. Because ni(map,(BG, X))=[XiBG, X] ~-li+mm[XiBG, X=] for any i>0, it suffices to prove

�9 ^

that any composition of the form Z'BG~X~X= is null homotopic. If d denotes the dimension of X, then )(+X~, being induced from X~X=, factors through (skaX=) ~, where sk~X, is a finite connected complex with finite fundamental group (isomorphic to nt(X~)). We conclude that it suffices to prove that map, (BG, ~') is weakly contractible, whenever Y is a connected finite complex with finite fundamental group. We first verify that map,(BG,(~)-)~map,(BG, ~') is a weak equivalenc where (Y)" denotes the universal covering space of Y. Because SIBG is simply connected for i> 1, it suffices to prove that any map f : BG~Y factors through (~')'. Choose any prime p and any 1.f. approx/p q': Bo+BG as in (1.3). By (1.1b), f factors through (Y)" if fo~b: B o ~ Y is null-homotopic. Let F denote the homotopy fibre of Y-~Y; F is connected and n,(F)"~n,+l(Y)| is a ~[nl (Y)]- module. By (1.2), /~*(g, n , (F))=0, so that foq , lifts uniquely to a map Bg+Y which is necessarily null-homotopic by Miller's Theorem. Because Y--*Y induces an isomorphism on fundamental groups, the universal covering (Y)" of ~" is equivalent to (~')* which is equivalent to I-I(Z/I)~(Y), where the product is

l

indexed by all primes l and where (Z/l)oo(Y) is the Bousfield-Kan Z/I-completion. Since (Z/I)o~(Y) is H*( ,7.//)-local in the sense of Bousfield [-1], ~I,*" map,(BG,(Z/1)~o(Y))~map,(Bg,(TI/l)~o(Y)) is an equivalence whenever p 4:1. By (1.3), p can be chosen different from any specified prime l, so that the equivalences

map, (BG, Y)~map, (BG, (Y)')~H map, (BG, (Z/l)~ (Y))

reduce the proof of the theorem to the proof that map,(Bg,(Z/l)o~(f')) is weakly contractible whenever p+l. This is given by [11; Theorems C,D] , because (7//l)oo(f') is a nilpotent space with bounded 7//q-homology for every prime q.

In order to be able to relate map, (BG, X) to map, (BG, X) we will need the following auxiliary result.

(3.2) Lemma. Let G be a Lie group with finitely many components and let X be a finite dimensional connected complex with universal covering f (~X. Then the induced map

map, (BG, X)--,map, (BG, X) is a weak equivalence.

Proof. It obviously suffices to prove that 2~-~ X induces a bijection of connected components

u , : n0(map,(BG, )())= [BG, X] ~ n0(map,(BG, X))= [BG, X].

Clearly, u , is monic; it remains to show that every f : BG~X lifts to )~ or, equivalently, that f , : nl(BG)~gx(X) is trivial. Because a locally finite approximation 4: Bo~BG induces a surjection on fundamental groups, it suffices to


prove that ( f o ~ ) , : n~(Bg)-,nl(X ) is trivial. This is indeed the case, since X is finite dimensional and therefore, by [11, Theorem 10.1], foq~ is equivalent to the constant map.

(3.3) Theorem. Let G be a Lie group with finitely many components and let X be a finite dimensional connected complex. Then all homotopy groups nj(map,(BG, X)), j> l, are rational vector spaces. Moreover, if ni(X ) is finitely generated for each i> 1, then, for j> 1

n~(map, (BG, X))_~ Iq Hk-J( BG, nk+ ~(X)| k > j

and each element in nj(map,(BG, X)) is represented by a phantom map ZJBG--*X for any j>O.

Proof. Let Y ~ X be the universal covering. Then map, (BG, Y ) ~ m a p , (BG, X) by (3.2). Denote by Yn the Bousfield H , ( ; R)-localization of Y (cf. [1]). Since Y is an arithmetic space in the terminology of [3], there is a pullback diagram

Y ,l-[Yz/~ 1

l

We note that map,(BG, F[ Yz/,)~-I-Imap,(BG, Yz/~)is weakly contractible, since by the universal property of Yz/l, any locally finite approximation Bg---,BG away from a prime different from l gives rise to a weak equivalence map, (BG, Yz/l)-~map, (B g, Yz/~), and the latter is weakly contractible according to Miller's Theorems C and D [11]. Thus map,(BG, Y)~-map,(BG, F), F the homotopy fibre of Y--"I-I Yz/t, or equivalently of Y,~(1-I Y~,/l)~). The homotopy groups of F are all rational vector spaces, and as a consequence, OF is a product of Eilenberg-MacLane spaces. Therefore for j > 1,

hi(map, (BG, Y)) ~- n j_ 1 (map, (BG, OF)) = [Z j- 1 BG, OF]

= ~] Hk-I+I(BG; n k + l ( F ) ) ' k > j

which is a r space. Assume now that hi(Y) is finitely generated for i__>2. Then nk(F)~--nk+ I(Y)| [15; Theorem 3.1], so that

hi(map. (BG, X))~ I-I Hk-J( BG, nk+l (X) |

for j=>l. Let f : Z i B G ~ X represent an element of nj(map.(BG, X)) for any j>=0. By

(3.2) f lifts to f : ZiBG~Y , Y the universal cover of X; by (3.1) fi determines the null-homotopic map(f )^ : ZIBG-~Y. Moreover, [Z, Y ] ~ [ Z , Y] is injective whenever Z is a finite complex because Y is simply connected with finitely generated homotopy groups (cf. [15]). We conclude that f and therefore f is null-homotopic when restricted to any finite skeleton of ZJBG.


We require the following (presumable well known) lemma in order to determine the weak homotopy type of map, (BG, X) for X with 3? rationally an H-space.

3.4 Lemma. Suppose Y is a simple connected space of finite type, whose rationalization Yr is an H-space. Then the homotopy fibre F of the Sullivan completion Y--, Y is a connected H-space, homotopy equivalent to l-I K(V,, n), where V is a rational vector space isomorphic to ~,+I(Y)| ,>__1

Proof. As observed in (3.3), rc i(F) ~ gi + 1 (Y) | :~/Z ~ gi + l ((fz)~)/~i + 1 (Y~) and F is the homotopy fibre of Y~-~(f')~. Since Yo is an H-space and Y is of finite type, all k-invariants of Y and Y are torsion classes. Consequently, Yo~(f')~ is the product of maps K(~,(YQ), n)---,K(rc,((f')~), n), so that F is homotopy equivalent to 1-1 K(~~174 n).

n>l

We can now prove the following structure theorem on map, (BG, X). We point out to the reader that ~./TZ. is a ~-vector space of the same cardinality as the real numbers.

3.5 Theorem. Let G be a Lie group with finitely many components and let X be a connected finite dimensional complex with gi(X) finitely generated for all i>2. Suppose that the rationalization (X)Q of the universal cover X of X is an H-space. Then there is a weak equivalence

map, (BG, X) ~ - 1~ K(Wj, j) j>=O

where Wj~-- [ I Hk-J(BG; gk+ I(X)| In particular, there are no essential k > j

maps BG--*X if and only if for every m> 1 either H,,(BG) or ~,,+ I(X) is a finite group.

Proof. The homotopy groups gj(map,(BG, X) for j > l are given by Theo- rem 3.3. Let F denote the homotopy fibre of the Sullivan completion Y--,Y where Y is the universal cover of X. Theorem 3.1 and Lemma 3.2 imply that the composition map,(BG, F ) ~ m a p , ( B G , Y)--*map,(BG, X) is a weak equivalence. Since F is a connected H-space by Lemma 3.4, F admits an H-inverse and therefore all connected components of map,(BG, F) are weakly equivalent:

map, (BG, F)~_map, (BG, F)o x K(Wo, 0),

where map, (BG, F)o denotes the connected component of the point map and

Wo-~ ~o(map, (BG, F))--[BG, F] ~= 1~ Hk( BG, 7~k+ l(X) | k>0

Clearly, map,(BG, F)o is a (weak) homotopy commutative and homotopy associative H-space since F is, and since all homotopy groups of map, (BG, F)o are ~-vector spaces, the k-invariants of map,(BG, F)o all vanish (cf. [12; Appendix]). Therefore, map, (BG, F)o ~_ 1-[ K (Wj,j).

j=>l


As a s imp le example , we see tha t m a p . (BS 3, $ 5 ) ~ K(Wo, 0), s ince

Wj~ I-I Hk-J( BS3, rtk+ ,($5) | = 0 for j > 0 . k>j

The re fo r e , every c o m p o n e n t o f m a p . ( B S 3, S 5) is c o n t r a c t i b l e a n d the set of

c o m p o n e n t s of map.(BS3, S 5) is in o n e - t o - o n e c o r r e s p o n d e n c e w i t h W 0 ~- H4(BS 3 , ~./Z) ~- ~../Z -~ [BS 3, $5].

References

1. Bousfield, A.K.: The localization of a space with respect to homology. Topology 14, 133-150 (1975)

2. Demazure, M., Gabriel, P.: Groupes alg6briques, 1; North-Holland, Amsterdam (1970) 3. Dror, E., Dwyer, W.G., Kan, D.: An arithmetic square for virtually nilpotent spaces. J. Math.

21, 242-254 (1977) 4. Friedlander, E.M.: Etale Homotopy of Simplicial Schemes; Ann. Math. Stud., Princeton (1982) 5. Friedlander, E.M., Mislin, G.: Cohomology of classifying spaces of complex Lie groups and

related discrete groups; Comment. Math. Helv. 59, 347-361 (1984) 6. Friedlander, E.M., Mislin, G.: Locally finite approximation of Lie groups, II, (To appear) 7. Gray, B.I.: Spaces of the same n-type for all n. Topology 5, 241-243 (1966) 8. Hochschild, G.: The Structure of Lie Groups. San Francisco: Holden-Day (1965) 9. MacLane, S.: Homology. Berlin, G~Sttingen, Heidelberg: Springer Verlag 1963

10. Meier, W.: Pullback theorems and phantom maps. Quart. J. Math. Oxford (2), 29 (1978) 11. Miller, H.: The Sullivan conjecture on maps from classifying spaces; Ann. Math. 120, 39-87

(1984) 12. Milnor, J.W., Moore, J.: On the structure of Hopf algebras; Ann. Math. 81, 211-264 (1965) 13. Quillen, D.: On the cohomology and K-theory of the general linear group over a finite field.

Ann. Math. 96, 552-586 (1972) 14. S6minaire de G6ometrie Alg6brique (SGA3), Sch6mas en groupes, III; Lecture Notes in

Mathematics Vol. 153. Berlin, Heidelberg, New York: Springer Verlag 1970 15. Sullivan, D.: Genetics of homotopy theory and the Adams conjecture. Ann. Math. 100, 1-79

(1974) 16. Zabrodsky, A.: On phantom maps and a theorem of H. Miller. Preprint 1983

Oblatum 29-I-1985/VIII 1985

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Locally finite approximation of Lie groups, I