F I B E R E D M O D U L E S A N D C O B O R D I S M S . I I *

V. R . K i r e i t o v UDC 513.836

w 5. Le t ~ be the c a t e g o r y of finite C W - c o m p l e x e s with d i s t inguished points , the mappings of which p r e s e r v e the d i s t inguished points. The objects of the c a t e g o r y 9r will be cal led s imp ly complexes for b rev i ty . If X1, X 2 . . . . , X n a r e complexes f r o m /X p , then we shal l cal l the i r r educed union the complex

X, A X ~ A . . . AX~=X, XX~X... XXd(x,VX~V... V X , ) u . . . u ( x , v . . . VX~- ,Vx . ) ,

where x i E X i , i = 1, 2, . . . , n, are the dist inguished points, and the symbol V denotes the operat ion of taking the bouquet of spa c e s .

One cal ls a s p e c t r u m of d e g r e e k a sequence X = {{Xn, fr0}n =1,2 . . . . . of complexes X n and maps fn: Sk /~ Xn ~ Xn +I, whe re S k is the k - d i m e n s i o n a l unit sphe re of a r i t h m e t i c space R k +1, where the c o m - plex X n m u s t be ( k n - 1 ) - c o r m e c t e d and the g roup rrkn{Xr0 * 0, and the map fn, cal led the suspens ion map, m u s t be a h o m o t o p y equ iva lence up to the 2 k n - l - s k e l e t o n , n = 1, 2, . . . .

By a map ~: X ~ Y of s p e c t r a X and Y of d e g r e e k i s mean t a sequence of maps { ~ 0 n } n = l , 2 , . . . , ~~ : Xn ~ Yn such that the d i a g r a m

Sk A X ~n~_~ SkA Y~

Xn+l q~n+l ~ y,,+,

is h o m o t o p y comm ut a t i ve fo r each n = 1, 2, . . . . The map ~ is ca l led an equiva lence of the s p e c t r a X and y if ~0n is a homotopy equiva lence up to the 2 k n - l - s k e l e t o n fo r each n = 1, 2, . . . . Spec t r a a r e equ iva len t if the re ex i s t s an equ iva lence of one s p e c t r u m into the o ther .

A s p e c t r u m X = { (Xn, fn)}n =1, 2 . . . . of d e g r e e k is cal led mul t ip l ica t ive ff fo r each na tu ra l number n, m_> 1 the re is g i v e n a m a p f n , m : X n /~ X m ~ X n + m s u c h t h a t t h e m a p s f n + t o ( f n A 1)oX, f n + m o(1 A fn, m), fn, m +t ~ (1 A fm) ~ #, a r i s i n g f r o m the d i a g r a m

k .fnA1 tn+l, ra (s/~,XOAX~--- x,,+IAX., /

[~ tl

1k i' m [

fn, m + l

r e p r e s e n t , r e s p e c t i v e l y , e l emen t s d ' , d, ~,, of the g roup [S k A Xn A Xm, X n + m ] , connected by the r e l a - t ions a ' = a = ( -1 )ka" ; in the d i a g r a m k and g a r e the na tu ra l h o m e o m o r p h i s m s . A f ami ly {fn, m } n , m =*,a . . . . of maps with the indicated p r o p e r t y is cal led a muI t ip l i ca t ive s t r u c t u r e on the s p e c t r u m X. To s p e c t r a of d e g r e e k _> 2 one c a r r i e s over the concep ts of a s soe ia t iv i ty , commuta t iv i ty , and un i t a r ines s of m u l t i p l i c a - t ive s t r u c t u r e s in t roduced in [1].

* The p r e s e n t paper is a cont inuat ion of the pape r " F i b e r e d modules and c o b o r d i s m s . I" published in S ib i r sk . Matem. Zh., 14, No. 5, 1006-1024 (1973).

T r a n s l a t e d f r o m Sib i r sk i i Ma tema t i chesk i i Zhurna l , Vo[. 14, No. 6, pp. 1247- i258 , N o v e m b e r - D e - c e m b e r , 1973. Or ig ina l a r t i c l e submi t t ed Apr i l 23, 1970.

1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g/est I7th SLreet, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrievaI system, or transmitted, in any form or by any means, | electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A t copy of this article is available from the publisher for $15.00. [

J

875

Let X, Y be s p e c t r a with degrees k, l , r espec t ive ly . By the product X A Y of the s p e c t r a X and Y is meant the spe c t rum defined by the conditions X A Y = {(x A Y)n, hn}n =1,2 . . . . . (X A Y)n = Xn A Yn, the map h n : S k + l A (x A Y)n-'* (x A Y)n+l is defined as the composi t ion-of all maps of the d i ag ram

S k+~ "X ~ Y" A ( / \ ),, = sk+zAX,,AY,~ X---ALSk,/kS~AX,,AY,~ --+ SkAX,~As~AY,~ *,,4g,~ X,~+~Ay,~+t=(X/ky),~+>

The number k + l, in general~ is not the degree of the spec t rum X A Y; however the s p e c t r a we consider in what follows a re such that the degree of thei r product is always equal to the sum of the degrees of the s p e c t r a multiplied; hence eve rywhe re in what follows we r e s t r i c t ou r se lves to s p e c t r a of this kind.

In the obvious way one defines the product of s e v e r a l spec t r a , and one can note that the product of s p e c t r a is assoc ia t ive up to equivalence.

Let X l, X 2, . . . , X k be mult ipl icat iye s p e c t r a with degrees l l , /2 , �9 �9 �9 , lk , respec t ive ly , and X 1 A x 2 A �9 A x k be their product, let {f~n, n} be a mul t ip l icat ive s t ruc tu re on the spec t rum Xi, i = 1, 2,

�9 . . k. We define on the s pec t rum X 1 A x 2 A . . . A x k a sequence of maps {hm, n} , called the product of the mul t ip l ica t ive s t ruc tu r e s {f~n,n}" The map

h~. ~: ( X ' A X ~ A . . . A x ~) ~ A (X'AX~A �9 �9 �9 A x ~) o-~ ( X ' A X ' A . �9 �9 A x ~) ~+~

we define as the composi t ion of all maps of the d i a g r a m

(X~AX' A . . A X % A ( X ~ A X ~ A . . .AX~)~ ~ X,, . /kZmA. k , 2 . ~ ~ . . A X ~ A X , A X ~ A . . . A X ~

- ~ ( X d A X , * ) A -. .A(X~kAX,~) ~ , ,^l~. , i . . .m ~

- + X ~ + , A X ~ + , A . . . A X e + , = (X~A X ' A . . . Ax~)~+,,

w h e r e X, g a re the canon ica l h o m e o m o r p h i s m s , i .e . , hm, n = ( f~ ,n A f2m,n A . . . A fkm,n) o ~ o x.

The proof of the following l e m m a reduces to the ver i f ica t ion of commuta t iv i ty of the corresponding d iag ram.

LEMMA 5.1. The sequence {hm, n} is a mult ipl icat ive s t ruc tu re on the spec t rum X 1 A x 2 A .- . A x k. If x l, x 2, . . . , x k a re mul t ip l icat ive , assoc ia t ive s p e c t r a with unit, then the spec t rum X 1 A x 2 A . . . A X k is assoc ia t ive , commutat ive , and has a unit.

A spec t rum X = {(Xn, fn)}n =l, ~ . . . . of degree k defines on the ca tegory ;Yt an ex t r ao rd ina ry homology theory X, given by the fo rmulas :

X ( K ) = s X~(K),

X~ (K) = lira ind{n~+~ ( X J ~ K ) , ([~/~l) .}.

If X, Y a re two s p e c t r a with degrees k, l, r e spec t ive ly , then the group X(Y) (the X-homology of the s p e c t r u m Y) is defined by the conditions

X ( Y ) = ~ , X , ( Y ) ,

X, (Y) = lira ind{X~n+~(Y.), (g~).},

where {gn} is the suspens ion map of the s p e c t r u m Y.

If X, Y are mul t ip l icat ive s p e c t r a and {fn, m}, {gn, m} are thei r r e spec t ive mut t ip l icat ive s t ruc tu res , then the d i a g r a m

Xzn+~(Y~) | (g~'~)~ xz(~+m+~+~(Y~+~)

reduces to a pair ing of groups

X,~+~( Y~) | X~+~(Y~) -+X~+,~+~+~ (Y~+,~).

One ve r i f i e s that this pair ing, up to mul t ip l icat ion by the coefficient (-1) i /m , is consis tent with the homo- m o r p h i s m s induced by the suspens ion mappings of the spec t ra , and hence, we pass to the l imit as n - - ~,

876

m - - ,% and we get a pa i r ing

X,(Y) | Xj(Y)-+ X~+iY).

Thus, the group X(Y) is provided with the s t ruc tu re of a graded ring, which, if the con t ra ry is not s ta ted, we always have in mind.

Let X, Y be mul t ip l ica t ive s p e c t r a with degrees k, l , r e spec t ive ly ; let S O be the ze ro -d imens iona l sphere . Since S O is a group, Z (S ~ is a r ing for any mult ipl icaf ive spec t rum Z.

THEOREM 12. The r ing (X A Y)(S~ is mul t ip l ica t ive ly i somorph ic to the r ing X(Y).

Proof . We consider the d i rec t s p e c t r u m of groups A i = { r r k n + l m + i 0 f n AYn) , (fn A gn).} over the se t I x I, where I is the se t of natural numbers with the i r natural order ing relat ion. The l imit l imA i coin-

~ , m ~ c o

cides with the l imi t of the cofinal par t of this spec t rum defined over the se t A =- I x [, consis t ing of pa i r s (n, n), n s I. N o w the l imit of this cofinal pa r t is , as is not hard to see , the group (X A Y)i(S~ On the other hand, the l imit l imA i co inc ides with the repea ted l imit l i m l i m A I. But l imAi is the s p e c t r u m V(m )

= {X/m +i(Ym), gin).} and l i m V ( m ) in i ts own r ight is the group Xi(Y ). Consequently, the additive group m ~ c c

(X A Y) (S ~ is graded i somorph ic to the group x(Y) . . . .

Now we shal l show that in fac t the i somorp.hism is mult ipi icat ive. F o r this we define a pair ing of s p e c t r a A i and AJ with values in the s p e c t r u m A ~ + ]. We se t

(1A~A1),

(fn, n')A(gm, m') " - -> gh(n+n')+l(m+m')4.i+] ((X,J~X,v)/~(Y~AY,,.~,)) . . . . ak(~+~,)+~( . . . . ')+~+j (Xn+n,AYm+m,),

( l ) " " " + " ' ( ( . f ..... , ) A ( g . . . . ) ) o ( t A ~ A ' 9 , o ~ . (Dn, m ; n ' , m " ~ = -

One ver i f i e s that the maps g0n, m; n ' , m ' a re cons is tent with the r e s t r i c t i on h o m o m o r p h i s m s of the s p e c t r a A i, A J, A i + J, and thus define a pair ing l im A i | l im AJ --- i im A i + J. Moreover , fo r the s p e c t r a over the se t

A ~-- I • I, the indicated pair ing is consis tent with the pair ing defined by the mul t ip l icat ive s t ruc tu re on the s p e c t r u m {((X A Y)n, f n / \ g n ) } n = l , 2 . . . . . Finally, if one passes in the s p e c t r u m A i to the l imit as n ~ ~, and then as m ~ ~, the indicated pair ing of s p e c t r a A i induces a mui t ip i icat ive s t ruc tu re in the group X(Y), that which is defined in the s tandard way fo r this group. Since the repeated [imit of the spec t rum A i coin- cides with its double l imit, the t heo rem is proved.

Let X be a reduced muit ip l icat ive homology theory on the ca tegory ~ . Let A be a f in i te -d imens iona l a ssoc ia t ive a lgebra with unit over the field K of r ea l or complex numbers . We assume , that for the theory X( ) there ex i s t s a Thorn i s o m o r p h i s m in the ca tegory of f ibered A-modules , i .e . , if ~A is a f ibered A-module , then there ex i s t s an i s o m o r p h i s m of graded groups which is functor ia l with r e spec t to A - h o m o m o r p h i s m s of f ibered A-modules

(p::X (X:Up) -~ X (T~),

where deg~p~ = d imR~; X~ U p is the disconnected union of the base X~ and the point p; the point p is d i s - t inguished; T~ is the Thorn space of the bundle ~.

We shall say that the i s o m o r p h i s m ~ is mul t ip l ica t ive , if for any two f lbered A-modules ~, ~? the d ia - g r a m

X (T~)CDX (T~l) --~ X (T~AT~) -+ X (T (~ • h)) ?

x (x~ U p) | (x~ U q) ~ x ((x~ U p)A(X~ U p))-~ x ((x~• x~) U r)

is commuta t ive ; here X is the pair ing map. Let ~ = (~r0n=l 2 be a fami ly of f ibered A-modules . We shall say that this fami ly is a 2/-family, if the following con'di't[ons a re sat isf ied:

877

1. T h e r e ex i s t s a A - m o d u l e A such that fo r each na tu ra l n u m b e r n _> 1 the re is defined a map of f ibe red A - m o d u l e s :

F . :~ .e t (A) -+ ~.+4

2. F o r each n, m _> 1 the re is defined a map of f ibe red A - m o d u l e s : Y- n , m : ~n x ~m ~ ~n +m;

3. The f ami ly {(T~n, T Y ' n ) } n = l , 2 . . . . is a mul t ip i ica t ive s p e c t r u m with mul t ip l ica t ive s t r u c t u r e

{T Y" n ,m}.

Here T~ n is the Thorn complex , and T Y" n, T 9- n ,m a re the maps of Thorn complexes induced by the m a p s ~" n and ~r- n ,m, r e s p e c t i v e l y . I f {Xn} n =~, ~ , . . . is the f ami ly of bases fo r the f ibered modules of the f a m i l y g (X n = Xgn), then the m a p Y" n induces a map fn: Xn ~ Xn+l" We denote by the symbol B

the induct ive s y s t e m { (Xn, fn)}n =l, a, . . . �9 We set , us ing a mul t ip l ica t ive homology t h e o r y Y ( ) ,

Y(B)---- 2 Y,(B),

Y,(B) = lira indtY,(X.Op), (I,O.}. n ~ c ~

The pa i r ing defined (fn,m). : Yi(Xn U p) | Yj(X m U q) - - Yi +j(X U r )n +m, is cons i s t en t with pa s sage to the l im i t of the induct ive s y s t e m and def ines o r / the g roup Y(B) the s t r u c t u r e of a g raded r ing, which will a l so a lways be unders tood .

Le t go be a mui t ip i ica t ive Thorn i s o m o r p h i s m of the t h e o r y Y in the c a t e g o r y of f ibered A - m o d u l e s , = {~n)n =1, 2 , . . . jus t as be fo re , a 7 - f a m i l y of f ibe red A - m o d u l e s .

THEOREM 13. The re ex i s t s an i s o m o r p h i s m 5 : Y(T~) --* Y(B) of g raded r ings .

P roof . We r e c a l l that if d i m R A = l, then

Y,(T~) = lira ind{Y,.+,(T~.), (T$-,~).}

and

T h e r e ex i s t s a T h e m i s o m o r p h i s m

and the diagram

Y~ (B) = l im ind{Y,(X,~Up) }.

ep.':Y~(X.Up)--,-Y,,~+,(T~,O, n=i, 2 . . . . .

q~ t

] (t~). I (v~'#. t

r

is commuta t i ve fo r each n = 1, 2, by v i r tue of the func to r i a l i t y of the i s o m o r p h i s m (pin. P a s s i n g to the l im i t of the induct ive s y s t e m as n ~ ,% we get an i s o m o r p h i s m of g roups

O,:Y~(B)--,-Y~(T~),

induced by the i s o m o r p h i s m { (pin} n = l, 2 . . . . of induct ive s y s t e m s .

We se t ~ = (~i)i --1,2,. , ~ : Y(B) ~ Y(T~). We shal l show that the i s o m o r p h i s m ~ is mul t ip l ica t ive . In fact , mul t ip l i ca t ion in the ' r ings Y(B) and Y(T~) is defined, r e spec t i ve ly , by the pa i r ings a) and b):

x (& ,0. y X a) Yi(X,~Up)(~Y~(X,,,UP)---~Yi+i((X,~Up)/~(X''@p)) " ~ i+~(T~++~UP)

X b) Y~+~ (T~,)@Y~+j(T~) ~ Yz(,+~)+~+; (T~./~T~,~) ...... ~ Yz(,+,,+~+J) (T~+~,).

878

The d iagram a) can, with the help of the Thom isomorphism, as was shown, be mapped into the dia- g ram b). The total d iagram obtained is commutative, since the f i rs t square on the left is commutative be- cause (p is a multiplicative Thorn i somorphism, and the second square is commutative by virtue of the functorial i ty of the Thorn i somorphism. Passing to the limit of the inductive sys tem as n - - ~, m -- ~, we get the multiplicativity of the i somorph i sm ~. We shall also formulate here the following asser t ion, the proof of which is t r ivial and is hence omitted.

i LEMMA 5.2. If ~i = ( ~ n ) n = l , 2 , . . . , i = 1, 2, + . . , k is a finite number of Y-famil ies of fibered A- modules, then the family

I+

is, in a natural way, a y-family of fiberedA-modules. The symbol II denotes the operation of direct product of fibered A'modules.

w 6. In this paragraph the results obtained earlier are applied to compute some cobordism rings. In what follows, the following conventions are used.

G n will denote, if there is no need to be more concrete, one of the groups O(n), SO (n), U(n), Sp(n).

G will denote the inductive limit limind G n with respect to the canonical inclusion G n -* G n +i BGm BG are n~c~

the classifying and stable classifying spaces, respectively, of the groups G n and G+ B-O n is the (in-1)- skeleton of the canonical cell division of the space BGn, where i : I, if G : O, SO; i = 2 , i f G = U; i : 4 , if G = Sp. y ~ is the universal vector bundle of the group Gn, acting in the standard wayin an n-dimensional vector space over the ground field. Thus, 7~ = (Eye, iry~, B~Gn). MG denotes the spectrum of Thorn complexes

of universal bundles of the series of groups {Gn)n:l,2, . . . . so that MG : {(Ty~, T ~7" n)}n:l,2, ~ ; •G

is the G-bordism ring ofapoint; ~G(K, L) is the G-bordism group of a finite pair of CW-complexes; ~G(K) is the reduced G-bordism group of the complex K with distinguished point,; one will also use for the group ~G(K) the notation MG(K).

We now define the spectrum whose homotopy groups will be computed presently. We set

X(k, l, m, n) = (kMO) /~ (IMSO) /~ (mMU) A (nMSp),

where k, l, m, n are natural numbers not less than 0; q. MG, as before, is the product of q copies of the spec t rum MG. Moreover , by definition, we set 0 �9 MG = S ~ the zero-d imens ional sphere with one dis t in- guished point. The following asser t ion about the spec t rum MG is well-known: the spec t rum MG is assoc ia - tive, commutative, and has a unit. The spec t rum X(k, l, m, n) is the product of spec t ra MG for different G and admits a multiplicative s t ruc ture which is also associat ive, commutative, and with a unit. :By the multiplicative s t ruc tu re on the spec t rum X(k, l, m, n.) witi always b9 meant just this s t ruc ture . Fur ther , i t is also known that the family of vec tor bundles {7~}~ -~ 9 {-/~ is the universal Gi-bundle ) is, in a natural way, a y - f a m i l y , and hence this is also true for the se r i e s

k l ra n

I i ~. ' ~+ I ] J i = 1 , 2 . . . .

We note that

k ~ m

.++<+, +, m, +)=+ (( H +o')• H ++~215 H • (If++') ' i i I i

i.e., the spectrum X(k, l, m, n) is the spectrum of Thom complexes of some y-series of vector bundles. The base of the i-th element of this series is the complex

k l m n

1 4. I

879

Let h i : B i --* B i +/, i = 1, 2 . . . . be the maps induced by the suspension maps of the spec t rum X(k, l, m, n); hij : B i x Bj --* B i + i be the maps induced by the mult ipl icat ion maps of the spec t rum X(k, l , m, n).

/ t l m n

The space n = ( H B o ) X ( I I B S O ) X ( 1 - I B u ) X ( I I B @ ) i s t h e i n d u c t i v e l i m i t o f t h e s y s t e m { ( B i , i i i I

hi)}i = i , 2 , . . . �9 Here one has on B the s t ruc tu re of an H-space and the H-space s t ruc tu res induced by the

maps {hij } on I imindB i = B coincide. The family of groups {(~2G(Bi) , (hi),)}t =1,2, is a spec t rum of ~ ~ �9 �9

abelian groups with pairing {(hm, n),} , hence the group ~G(B) is provided with a ring s t ruc ture . On the other hand, since B is an H-space, the group ~G(B) is a ring, called the Pontryagin G-bord i sm ring of the H-space B. It is known that the natural map lim ind { (~2G(Bi) , (hi),) } --* ~G(B) is an i somorph ism and

also multiplicative with r e spec t to the products defined in both groups. In o rde r to apply the resul ts ob- tained e a r l i e r to the calculation of the homotopy groups of the spec t rum X(k, l , m, n) it is n ece s sa ry to prove the following asser t ion .

LEMMA 6~ canonical Thorn i somorph ism for G-bord ism theory ~G is multiplicative in the ca tegory of G-bundles.

Proof. We give i t for the case G = SO. In the other cases , the proof is analogous.

Let ~m be an SO-bundle of dimension m over the finite CW-complex X~. Let (M n, f) be a singular or ientable manifold, f: M n ~ X~. The canonical Thorn i somorph i sm with which the lemma is concerned is constructed in the following way. We choose on ~m any Riemannian met r ic and we consider the unit disk bundle in this Riemannian met r ic associated with the bundle ~m (without loss of genera l i ty one can assume that XL is an open smooth manifold), which wilt be assumed smooth. We denote this bundle by D(~m). Its boundary 0D(~m) is the unit sphere bundle associated with ~ m The pair (D(r 8D([m)) is a smooth mani- fold with boundary. Assuming the map f: M n ~ X~ smooth, we consider the induced bundles f! (D(~m)), f[ (aD(~m)). The total space of both bundles is a smooth manifold and Of! (D(~m)) = f! (~D(~m)). Hence the pair (f! (D(~m)), Of ! (D(~m)) is a manifold with boundary. Moreover , one has the natural map

F: (l' (D(~ ) ) , 0l, ( D ( ~ ) ) ) ~ ( D ( ~ ) , O D ( ~ ) )

of pairs of manifolds, so that we have the singular pair (f] (D(~m)), Of! (D(~m)), ~t-:), of the pair (D(~m), 0D(~m)). If the singular manifold (M~, f) is bordant to the singular manifold (M~, g), then the singular pairs corresponding to them are also bordant. There ar i se thus maps

~" " ~ + " ( D ( ~ ) , O D ( ~ ) ) , n = t , 2 , . . .

forming a s e r i e s { Cn}n=0, 1 . . . . �9 All the ~o n are additive, so that there a r i ses a homomorphism

~ : ~ s o ( X ) ~ M D ( U , OD(~) )

of degree m. Actually, g0 is an i somorphism, more exact ty the canonical Thorn i somorphism, since

~so(D(~), 0D(~)) ~ 5so(D([) /8D([) ) (cf. [2]).

Now let ~k ~91 be two SO-bundles; ~k x ~l be the i r d i rec t product. In o rde r to prove the mult ipl i - cativity of the i somorph i sm ~p, we must prove the commutat ivi ty of the diagram:

5so (T~) | 5so (T~l) ~ Dso (T~ A T~]) t

~ o (x~) | ~s0 (x~) A ~%o (x~ x x~)

or (in different notation) of the d iagram

g2so (D (~), OD (~))| (D 01) OD (TI)) x.~ ~so (D (~ x ~1), OD (~ x ~]))

~so (X~)@ ~so (X~) .7~so (X~ x X~).

Let (JG% ]), (JG m, g) be the singular manifolds of the complexes X[ and X~/, respect ively . Going along the bottom path of the diagram, we get the s ingular pair (f • g)! D(~ x ,?), (f x g)[ (0D(~ • ~1); 5 r x G) o f t h e p a i r (D(~ x ~), aD(~ x ~)). Nowgoing along the upper path, we get the pair (fl D(~), f! (~D(~)) x gI (aD(~))). However,

880

af ter "smoothing angles," the second pair coincides with the f i rs t . Passing to bord ism c lasses , we get the asse r t ion of the lemma.

Since the spec t rum X(k, l, m, n) is mult iplicative, the group ,'r(X(k, l, m, n)) = ~ ~ri(X(k , Z, m, n) i>~o

is provided with the s t ruc tu re of a graded ring; this s t ruc tu re one also has in mind always. The ring ~r(X(k, l, m, ~1)) by definition is the value of the theory X(k, l , m, n) on the sphere S ~

THEOREM14. a) TheringTrCX(k,l,m,n)),k> OisisomorphictothePontryaginring~20 ((HBO).

• ( H . s o ) • , 1 l 1

l - - I m

l

i i

n - - I

d) the ril~g 7r(X(0, 0, 0, n))is isomorphic to the Pontryagin ring eSp (HBSP). i

Proof . We shall prove, fo r example, case e). The other cases are proved analogously.

Since X(0, 0, m, n) = MU A X(0, 0, m - l , n), one has, by Theorem 12,

r~(X(O, O, m, n) )=X(O, O, m, n) (S~ O, m--i, n) (S")~MU(X(O, O, m--l, n) ). The spectrum X(0, 0, m-l, n) is the spectrum of Them complexes of the y-series of vector bundles

m - - i ~.

{(II ) ( I I )} ~v ! • ~,' , which are U-bundles. The base of the i-th element of this family is the space i ~ t , 2 , . .

t 1

�9 ' m - - | n

(HB"'UI) X (1-[B'Sp,) : B i. Aeeording to Theorem t.3, there is a ring i somorph i sm t 1

MU (X (O, O, m - l , n) ) .~lim ind{~. (B~-), (h,) .},

and the multiplicative structure in the group limind {~2u(Bi) , (hi),} is assumed to be induced by the pairing i~oo

required in the definition of a T -family. Finally, according to our remark, made at the beginning of this paragraph, there is a ring isomorphism

lim ind{Qo- (B.~), (h0 . } ~ u (B), i-.)- ~

m - - I n

t t

In what follows , we use for the spae e BO)X(UBSO ) X ( E B U ) X ( ~ B S P ) the n o t a t i o n B ( k , l , l t i 1

m, n); here we assume that X is a one point space. l

As was shown earlier, the homotopy groups of the spectrum X(k,/, m, n) are the ring of a special type of eobordisms. In this paragraph the calculation of the ring 7r(X(k,/, m, n)) is reduced to the calcula- tion of the Pontryagin bordism ring of an H-space, which is a product of classifying spaces of classical groups, i.e., the Pontryagin bordism ring of the space B(k, l, m, n).

881

In the papers [2-4] this ring is ei ther computed, or methods are given for its computation. Leaving to the side the reformulat ion of the resul ts of these papers in general form, we note only, that all r ings of the form ~20(B(k , l, m, n)), ~SO(B(0, l, m, n)), ~u(B(0, 0, m, n)), ~2Sp(B(0 , 0, 0, n)) are f ree modules over the corresponding bordism rings. Moreover , if the case ~so(B(0, l, m, n))is excluded, then the remaining rings are polynomial rings in an infinite number of var iables over the corresponding bordism rings of a point.

We shall analyze in more detail one special case. Here, for clari ty, we shall begin with the original definitions.

Let K be the field of rea l or complex numbers. A s t ruc ture of type (m, ~) (m is a natural number, = =~1) on a K-vec tor bundle ~ is an automorphism ~r : ~ ~ ~, which is the identity on the base of the bundle [ , satisfying the following conditions: 2 ~ =~ and all s imi la r i ty invariants of the res t r ic t ion of the auto- morph ism ~r t o an a rb i t r a ry f iber of the bundle ~ are equal to one another and equal to the polynomial X m - e . If the base X~ is path connected, then to sat isfy this last condition, according to Theorem 5, it suf- f ices to ver i fy it on any one fiber of the bundle ~. An (m, e ) - s t ruc ture on a real vector bundle ~ for m = 2,

= - i is, obviously, a quasicompiex s t ruc ture . We note also that for m odd, a s t ruc ture of type (m, 1) by a change of sign becomes a s t ruc ture of type (In, -1 ) , and conversely, so that it is sufficient to r e s t r i c t ourselves to e i ther one of these cases . In the complex case it is sufficient to consider the case of a s t r u c - ture of type (m, 1). We shall say that a smooth manifold admits a s t ruc ture of type (m, e), if the stable normal bundle of this manifold admits a s t ruc ture of type (m, e). Considering the usual cobordism re l a - tion between pairs ( J//, 0, where J / is a closed smooth manifold, and L is a s t ructure of type (m, e) in

the stable normal bundle of J , we get a ring ~ (m, e) = ~ , ~(m, ~) of bordisms of a point of manifolds J .

i ~ 0

of this type. F r o m algebraic proper t ies of the polynomial x m - e and Theorem 8, it follows that the ring g] (m, e) is multiplicatively i somorphic to the ring of homotopy groups of the following spectra:

i f K = R , then

~(~, ')~n (X(2, 0, k - l , 0)), ~(~'. -')~:~ (X(0, 0, k, 0) ) , ~(~'+~)~ (X(l, 0, k~ 0));

i f K = C , then

e (~, ~)~= (X(0, 0, m, 0)).

Applying Theorem 14 gives ring i somorphisms:

fl(~, ' ) ~ o ( B ( t , 0, k - L 0)), ~(2k, -')'~v(B(O, O, k - i , 0)),

~(2k+~, i)~Qo(B(O, 0, k, 0)), if K = R ; ~(~, ~)~v(B(0, 0, m - l , 0)), if K=C.

Let, for example, K =R, m = 3, ~ = 1. Then ~(3,1) ~ ~o(B(0, 0, 1, 0)) = ~o(BU). According to [2], ~o(BU) ~ ~Oz 2 | H.(BU; Z2) , where the i somorphism is of r ings, and H,(BU; Z2) is the Pontryagin homo-

logy ring of the H-space BU. According to [3], the ring

H.(BU; Z2igZ2[xi, x2,. . . , x . . . . . . ], deg x~=2i, i=1, 2 . . . . .

where Z2[xl, x 2 . . . . . Xn . . . . ] is the r ing of polynomials over the field Z 2 in the genera tors xl, x2, . . . , Xn, . . . . Thus, we have, finally:

~(~, ~)~qd[x~, x2 . . . . . x , , . . . ] , deg x,=2/.

Thus one can completely calculate all r ings of the form ~ (m, t).

The calculation of the cobordism ring of manifolds admitting the s t ruc ture of a fibered module over complete mat r ix rings, reduces to the calculation of the homotopy groups of the spec t ra M(kSO), M(kO), M(kU), M(kSp). The ring of homotopy groups of the spectrum M(2R) is also isomorphic to the cobordism ring of manifolds equipped with Af-module structure for f(X) = X 2 such that the fiber of the fibered module

is a Af-free module.

882

I t is e a s y to calcula te the ra t ional homotopy groups of these spec t r a . In fact , there are additive i s o - m o r p h i s m s of graded groups

H* (MkSO)..~Q[X~, Xz, . . . , X,,, . . .], deg X~=4i,; H*(MkU)~Q[X,, X2 . . . . . X . . . . . ], deg X~=2i; H* (MkSp) .~Q [X~, X~ . . . . . Z~, . . . ] , deg X~=4i; H* (MkO) ~0; Q is the field of rational numbers.

Whence in the s tandard way one gets the following asse r t ions :

rka,~(n(MkSO) | =rkQ[X~, X2 . . . . . X . . . . . ], deg X~=4i; rk:~ (n (MkU) | =rkQ[X~, X . . . . . , X . . . . . ], deg X,=2i; rh~,~(z~(MkSp) | =rkQ[X~ X2 . . . . . X . . . . . ]~ deg X~=4i;

rk~(:x(MkO)| here everywhere re=O, ~, 2 . . . . ;

i f i ~ 0 (mod4), then the groups zci(MkSO ) and 7ri(MkSp) a re finite; the groups 7ri(MkO ) a re finite for i ~ 0.

The calcula t ion of the per iodic pa r t of the homotopy groups of these s p e c t r a runs into the following obstruct ion. F o r s impl ic i ty we consider the spec t rum M(2U). The cohomoIogy with coefficient~ in the group Z 2 of the space T(0m(C ) e 0re(C)) is i somorph iea l ly mapped onto the ideal of the r ing H* (BU(m); Z2) , genera ted by the e l emen t c 2 , where c m is the highest Chern c lass of this r ing. There are re la t ions S~4c2

2 2 2 2 _ 2 2 = ClCm, Sq (elCm) - e~c m = Sq4c 2 . ~ m

On the other hand, the c lass clca m is not the value of any eohomology opera t ion on the e l emen t c ~ . This means that the cohomology of the s p e c t r u m M(2U), considered as a module over the Steenrod a lgebra , cannot be r ep re sen ted as the d i rec t sum of monogenic modules , as happens in the c l a s s i ca l case .

1Q 2. 3.

4.

L I T E R A T U R E C I T E D

G. Whitehead, "Genera l i sed homology t h e o r i e s , , Transo Amer . Math. Soc., 102, 227-283 (1962). P. Conner and E. Floyd, Smooth Per iod ic Maps [Russian t ranslat ion] , Mir, Moscow (1969). H. Caftan, "Demons t ra t ion homologique des t h e o r e m e s de per iodici te de Bolt, I, II, HI," Semina i re Henri Caf tan, Exposes 16-18 (1959/1960). P. Landweber , "On the s implec t i c b o r d i s m groups of the spaces Sp(n), HP(n), BSp(n)," Mich. Math. J . , 15, No. 2, 145-153 (1968).

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