Remarks on generalized homology theories

A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES M ATHEM ATICSE X V (1971)

A.

Ja n k o w s k i

(Warszawa)

Remarks on generalized homology theories

Introduction. This paper is devoted to the following question:

(i) Bepresentability of the generalized homology theory by the cospectrum.

(ii) The existence and properties of Hurewicz homomorphism for generalized homology theory.

The notions of spectrum and cospectrum were introduced by Lima [5].

Given a cospectrum F we define a sequence of functors Yn ( • ) and natural transformations an: Fn( • ) Yn_,rlS( •); if F is convergent, then the se­

quence (Fw, an) is generalized homology theory. For a spectrum X the generalized homology theory X* ( • ) was defined by Whitehead [8].

In the first part we will introduce a notion of duality between co­

spectrum and spectrum, which is less restrictive than that given by Lima ; we also show that if the cospectrum F is dual to the spectrum X , then the generalized homology theories F* ( • ) and X* ( • ) are naturally isomor­

phic. This is a slight generalization of results of Spanier [7].

It follows from the results of Lima that there exists a dual co­

spectrum for spectrum consisting of finite complexes.

The second part is devoted to the Hurewicz homomorphism X - K ( ‘) ^ Нп(-Ло)

where hn is a generalized homology theory such that % = 0 for i < 0.

This homomorphism was considered in [2] p. 14 for bordism theories and was called the Steenrod representation.

PART I. COSPECTRA AND GENERALIZED HOMOLOGY

1. Preliminaries. All spaces and maps considered in this paper belong

to the category «9>1 of based (7TF-complexes and based maps. The only

exception are function spaces (with compact-open topology) which belong

to the category <9% of spaces of the homotopy type of (717-complexes^

186 A. J a n k o w s k i

The full subcategory of ■9>1 consisting of finite OIF-complexes will be denoted by . The reduced suspension and function space functors we will denote by 8 and F( -, •) respectively. We recall some definitions.

De f in it io n

1.1. A spectrum in the category i = 0 , 1 , 2, is a se­

quence { Xk, gk} of spaces and maps gk: 8 X k -> X k+1 in

The spectrum is convergent if the following conditions are satisfied:

(1.1.a) there exist Q kuch that nqJrk{Xk) =

0

for all Jc and q ^ Q . (l.l.b ) for any integer q there exist N q such that for h > N q

(<?*)* Hq+*+i ( SXk) Hq+k+1(Xk+1) is an isomorphism C1).

The definition of a map of spectra is obvious. The category of spectra in will be denoted by ^ and the objects of ^ by X , F, ...

Following Whitehead [

8

] we define, for any spectrum X, the genera­

lized homology and cohomology groups

Xn(A) = lim 7in+k(A # X k) , X n(A) = lim [8kA , X n+k],

к к

where denotes the smash product. The homology and homotopy groups of a spectrum X are defined as follows:

—v —^

Hn{X) = limHn+k[ Xk), 7in(X) = lim жп+к(Хк).

~k ~k

For details we refer to [

8

]. In the sequel we will use the notation and definitions of Spanier [7] without detailed references.

2. Cospectra.

—

De f in it io n

2.1. A cospectrum F in «5^, % = 0 ,1 is a sequence { Y k, ek}

of spacesУ and maps ek : Y k+1- > 8 Y k in

A cospectrum F = {F fc, ek} is convergent if the following conditions are satisfied:

(

2

.

1

.a) there exist Q such that dim Y k ^ h A Q for all 1c, (2.1.b) for each q there exist N q such that for h > Nq

ek. : Hk+^ ( 8 Y k) - Hk+«+1( Yk+1) is an isomorphism.

(]) This convergence conditions are adopted from [7], p. 349. After the prep­

aration of this paper it was pointed to me by C. Wall and D. Burghelea that the second one is inessential. The same remark applies to the cospectra.

The definition of a map of cospectra is obvious. We will denote the category of cospectra in by Sfi and the objects of ^ by Y, ...

De f in it io n 2 .2 .

The cohomology groups of a cospectrum Y are defined as

Hn{Y) = lim Hn+k(Y k).

~k

For a cospectrnm F in ■9>l we define the sequence of functors on the category SPX

F ft( - ) = И т [ Г я+* ,Я * ]

~k

and we get natural equivalences of functors

We have the following obvious proposition:

Pr o p o s it io n

2.3. I f Y is a convergent cospectrum in £fx, then the sequence {Yn, an) is a generalized homology theory on the category

3. Functional dual of a cospectrnm. Let F { -, •) be the function space with the compact-open topology. The maps

Я : S F (A ,B ) -> F { A , SB), y : F (A, B) -> F ( S A ,S B )

are defined in an obvious way. Let / : F {A , B) F(G, B) be a map induced by / : C -> A.

De f in it io n

3.1. A functional dual (2) F (Y) of a convergent co- spectrum F in is the spectrum F(Y) = {Fk(Y), gk} defined as follows:

Fk(Y) =

*

for к > Q, for к < ф ,

and r]k : SFk(Y) Fk+1{Y) is the composition of maps SFk(Y) = S F (Y k, S*k) ± F ( Y k, S ™ + ' ) ± F ( 8 Y k, S“ +2) % F ( Y t+l, S“ +2) = Fk+l( H Note that the spectrum F(Y) is in <9% and the weak homotopy type of F(Y) does not depend on Q.

(2) This is a slight generalization of the functional dual of a space in the sense of [7].

188 A. J a n k o w s k i

It is well known that there exists a natural equivalence

It is easy to check that in the diagram

s I

[SB,

8

F ( Y k, S24)]

a J,

[

8

B , F ( Y k, S 2k+1)-[

В I

[SB, F ( Y k,

8

2k+2)]

[ S B , F ( Y k+

1

,

8

2 M )]

I I s

I

V

II I I I

V

[

8

BJjfYk,

8

2k^

J,s [S B M Y k, S lk+1-\

[ S B % Y t + *,

8 2k+ 4

the middle square II is commutative. Commutativity of I is pr ovee in [6], p. 350 and the commutativity of lower square is obvious. Thud the diagram (1) is commutative.

Th e o r e m 3 .2 .

I f Y is a convergent cospectrum in then the spectrum F(Y) is convergent.

P roof. It follows from Theorem 1 of [5] that Tiq+n{Fn(Y)) = 0 for q < —Q. This proves condition (2.1.a). For the proof of (2.1.b) let us consider a part of diagram (1) with В =

8

q.

[

8

q, F ( Y k,

8

2k)]

s I

[S*+1, S F (Y k, £ 2*)]

4

[

8

Q+

1

, F ( Y k, S2k+1)]

В I

[S3+S - î W t ,S 2*+2)]

s 2*]

s

[

8 Q + 1

Yk, £2fc+1]

[£9+2r fc,

8

2k+2]

Since nr(F[Yk, S2i)) = 0 for r ^ к Qy the left-hand suspension map is an isomorphism for q < 2(& — Q) —

1

. The right-hand suspen­

sion is an isomorphism for q < 3 k — Q—1. Hence

M U : nQ+

1

(SF(Yk,

8 2

k))->jzq+

1

(F(SYk,

8

2k+2)) is an isomorphism for q <

2

{k —-Q)— 1 and

M U : -®fc+g+l (

8

F ( Y k, S2*)) ^k+ S+l (F(

8

Y k,

8

2k+2))

is ail isomorphism for к > -/ +

2

Q + 2.

It follows from. Proposition 1 of [

6

] that we have a commutative diagram

tf*+a+

1

H s n , s “ +2)) B k- ^ ' ( S Y k) ff*+a+i H n + i > s 2*+2)) i я ‘-«+1( Ч +1)

where

⁹⁹, 90

are induced by slant product. By Theorem 3 of [

6

],

⁹⁹

and cp' are isomorphisms for к > q + 2Q and by the convergence of Y there exists an X _ q such that 4 is an isomorphism for к > N_q. If к > max(iV_e, q-\- +

2

Q +

2

), then the map rjk* = Q kM)* is an isomorphism and the proof of Theorem 3.2 is concluded.

4. Duality.

De f in it io n

4.1. A

pairing of a cospectrum

F

=

{Yfc, sk} and a spec­

trum

X

= {.

X k

, £>4 is a

sequence of maps uk : X k% Y k -,S™

such that the diagram

-Y jc + l

Y k+1 Ж № к

jlK-e*

u k+ 1

S X kJfS Y k

I (uk ) l , l

gt2k+2 is commutative.

Starting with a paring и = {%} we construct homomorphism

<pn : H n ( X ) - > H - n ( Y )

as follows. Let us denote

—1 for £ = 1 , 2 (mod 4), 1 for £ = 0 , 3 (mod 4).

For all n, к the map uk : X k$ Y k -> 82k induces the slant product

<Puk = < & kW : H k+n(Xk) - Hk- n{Y k) and let

1

tyriyk ^n+ktyuj,’

Consider the diagram Hn+k(Xk)

I Hn+k+A SX k)

'J- ^k*

■®n+fc+l (-^-fc+l)

Vn.k

dn + k V [u k )1

V n . k + l

■> H k~n(Y k) ,

- H k ~ n + 1 ( S Y k )

1 ek

-> н ‘" п+1( П +1)

19 0 A. J a n k o w s k i

we have and

ек(Р(г

1

к)1Л — 9uk+

1

Qk*-

The upper square is commutative for n-\-k = 1 ,3 (mod 4) and in this case ôn+k = ôn+k+l. The same square is anticommutative for n-f-fc = 0 ,2 (mod 4) and in this case ôn+k = — <5и+Л;+1. Therefore that for all n, к the square

Hn+k(Xk) H k~n(Y k)

I ek*a » . I eka Я»+*+1(**+1) Hk~n+

1

(Y k+l) is commutative and we define

<Pn lim (pn^k • Thus

< * 4 =

<?n : H J X ) - + I I - n(Y).

De f in it io n

4.2. A paring и = {uk} is called a duality if and only if <pn are isomorphisms.

4— < -

We are going to prove that for Y in the duality и induce the weak homotopy equivalence / : X -> F(Y).

De f in it io n

4.3. Define the sequence of maps f k : X k ^ F t (¥) = _FCFt ,S “ ) by the formula

(fki.æ))(y) = UAX, У) • It can be easily checked that the diagram S X k — k ~> S F (Y k, S2k)

I I

I **

x h+x— + F (Y k+

1

, S 2k+2)

is commutative and thus / = {fk} is a map of the spectra.

Note that the sequence e = {ek} of evaluation maps ek : F { Y k,S*k) # Y k -+S*k

is a pairing of the spectrum F(Y) and cospectrum F and is in fact a duality.

- > —>• <—

Th e o r e m

4.4. The map / : X ->F(Y) is a weak homotopy equivalence.

Proof. We have the commutative diagram

-X- к Ж ~ ^k

! ^ s a

F ( Y k, S“ ) Ж ГкА and we infer that the diagram

Ff-n+k^^k) h

Hn+k(F{Yk, &*)) /

/ V n . k

\Фп,к

\ /

Hk~n( Y k)

where ipnk is defined by the duality e = {e^ is commutative. Thus we have the commutative diagram

H J X ) /* Hn(F(Y))

V » y'Vn

\ <- S H~n(Y)

By Theorem 3 of [6], y>n is an isomorphism for each n and yn is as­

sumed to be an isomorphism for each n. Therefore /* is an isomorphism for each n. Consequently by Theorem 3.5 of [7], / is a weak homotopy

equivalence. ^

Let A be a complex in and Y = { Y k1 ek} a cospectrum in Define a cospectrum A % Y = {A J* Y k, ek}, where ek is the composition A # ^4-1 - A # 8 Y k - 8 (A Г #*). _____

It can be easily seen that the cospectrum A % Y is convergent if F is convergent.

Pr o p o s it io n

4.5. There in an isomorphism, Л» : F(Y)n(A) - {A ~jfŸ)n(_S2n) natural with respect to maps in £f0.

Proof. Fix n and consider the diagram [8kA , F ( Y n+k, 8 2n+2k)]

I 4’*

[8kA # r n+fc, S2»*2*]

[A % Y n+k, 82n+k]

[8k+1A , F( Yn+k+1, S2n+2k+2)]

к 2 j V *

[8k+1A ^ Y n+k+1, S2n+2k+2]

Î s'fc+i г - IA # **+*+» ^ +fc+1]

En + k oS

1 9 2 A . J a n k o w s k i

The upper square is commutative because of commutativity of dia­

gram (1) for В =

8

k A and the lower square is also commutative. Since all yjk are isomorphisms and Sk is isomorphism for h sufficiently large, we can define

Лп — (lim$fc)_1 lim, y k

~k ~k

and Лп is an isomorphism. The naturality is obvious and the proposition follows.

Let A' be the p-dual of A. By Lemma 5.8 of [7] we have the iso­

morphism

r n : ¥ J

8

^ A ' ) -> ( Г У Ь „ ( « 2”)-

Corollary

4.0. I f Y is a cospectrum in X is the dual spectrum and A, A' are p-dual, then there is an isomorphism

Dn - . b ( A ) ^ Y n{S™-*A').

Now we are going to state our main result.

Th eo r em

4.7. I f Y is the cospectrum in dual to the spectrum X, then there is a natural isomorphism Фп : Xn(A) -> Yn(A) of the generalized homology theories on ^ 0.

P roof. Let A' be the p-dual of A for some p. It follows from the results of [8] that there is an isomorphism

DntP: X * - n( A ')- > X n(A).

Let Фп = a2n~vо Dp_no D~^v . It can be shown that Фп is independent of the choises and is natural. This completes the proof.

PART II. THE HUREWICZ HOMOMORPHISM

5. Preliminaries. In this part we consider spectra and drop the upper arrow.

Let A, В be (n—1)-connected and let C be (m—1)-connected. Let

а < 1 Р ( А , л п{А)),

/5сЯ“ (Б ,я „(В » ,

~ ^ Н ^

1

(

8

В,т,п+

1

(

8

В)), y zH m(C, x m(G)), d*K'±~(A J<С,жм (А MO))

denote the fundamental classes of A , B, SB, G, A

(7,

respectively.

The following lemmas concerning fundamental classes are known.

Lem m a

5.1. I f f : A В is a map and f # : H n[Af Ttn(A)) -> R n[A, nn(B)) is the coefficient homomorphism induced by /* : лп(А)

лп(В), then f*p = f * a .

Lem m a

5.2. I f a* : Hn+1(8B, лп(В ))-> Hn+1(8B, лп+1(8В)) is the coefficient homomorphism induced by the homotopy suspension

8 : жJ B ) -+nn+l{8B) and

<T : E " ( B , жп(В)) H " + l(S B , nn(B)) is the cohomology suspension, then p = a# ap.

Lem m a

5.3. I f a J^yeHn+m(A G, лп(А)®лт{0)] is the external cup product and

x* . Hn+m{A | C , » <( i ) 8 ^ ( ( ! ) ) ^ p « ( i Ж С , я п+т(А # 0 ) ) is the coefficient homomorphism induced by natural pairing

x : лп{АЩлт(С) ^п+т и Ж 0 ), then и# (а % y) — ô.

Let X be а ( —1)-connected spectrum. By a result of [8] we can as­

sume X k to be (fc—l)-connected. лк(Xk) will shortly be denoted by nk.

Let

ef : Нг{- ,щ ) НЦ •, лк+1)

be the coefficient homomorphism induced by the composition of maps

S ek*

Щ = ^(Xft.) — > лк+1(8Х к) — > лк+1(Хк+1) — л:л+1.

Let X k be the Eilenberg-MacLane space К (л к, Tc) and let

£кеНп{Хк, л к), akeHk(Xk, лк) be the fundamental classes of X k and X k respectively.

De f in it io n

5.4. Let X be a (—l)-connected spectrum and let X k be (Tc—l)-connected. Maps ek : 8 X k -> X k+1 of a spectrum X — {X k, ~ek}

are defined uniquely up to homotopy by the condition

£k a k+ 1 = G s1c a k '

Maps f k : X k -> X k are defined uniquely up to homotopy by the condition

f k a k — $ k ‘

R o c z n ik i P T M — P r a c e M a t e m a t y c z n e X V < 13

194 A . J a n k o w s k i

It can be easily checked (using Lemmas 5.1 and 5.2) that the diagram o' -y sk „ y

---> 1

_ \<h+1 s x k^ i - x

k + 1

is homotopy commutative and thus

/ = { f k }

is a map of spectra. It is obvious that nn{X) — 0 for n # 0 and

/* : n0(X) -> 7z0{X) is an isomorphism.

A multiplicative spectrum is a spectrum X — {X k, sk} and a sequence of maps

(P n, m • -^~n Ж ^-r, -n+m satisfying some obvious relations with sk.

Denote by £n>m the fundamental class of X n % X m and let cp*m be the coefficient homomorphism induced by

Vn,m* • ^n+mi-^-n Ж -^-m) ^ '^'n+mi^-n+m) ‘

De f in it io n

5.6. The

maps

Pn,m • ^-n Ж n+m

are defined uniquely up to homotopy by the conditions

^Pn.m^n+m Фщт^п Ж

•

Proposition

5.6. I f the sequence <pn>m is a multiplication on a spec­

trum X, then the sequence cpn>m is a multiplication on the spectrum X and f : X -> X is the multiplicative map.

Proof. The first part is obvious. In order to prove the second one we have to show that

(.fn+mPn.m) an+m

(PPn.mfn Ж fm) an+m •

We have, by Lemmas 5.1 and 5.3

/ /• 4* _ * Г* _ * t _ t

\Jn+m^Pn,m) ®n+m Pn,mJn+m^n+m Vn.m^n+m Ч^п,т^п,т

= <PÏ,mX*Çn Ж = <p

Z

i

n ^ * (fn an) Ж (fm am) (fn Ж fm) !п,т ^ ®n Ж ®m (fn Ж fm) *Pn,m®n,?

(Vn,mfn Ж fm) ®n+m'

It is obvious that if X is a ring-like spectrum and g : S -> X is the

unit map, then X is a ring-like spectrum, the composition g : S X X

being a unit, and / : X -> X is a homomorphism of the ring-like spectra.

6. Definitions of Hurewicz and Hopf homomorphisms. Becall that the map of spectra h : X F induces the natural transformations

h* : X*(-) Denote as usual

X 9 = X 9(8°) = л__й{Х) and X q = X q(8°) = ng(X).

Since Xq = X й = 0 for and X0 = X° = X0, there exist natural equivalences

T* • -F* ( ■ ) ^ -Й"* ( ’ J -X"o) ?

t*

:* * ( • ) -> H* ( •, X0) .

De f i n i t i o n

6.1. Natural transformations

Z* = т */*:ЛГ,(-)->Я*(-,Х0),

X = T * f : X * ( - ) - * H * ( - , X 0) ,

are called Hurewicz and Hopf homomorphisms, respectively.

The following proposition is due to Whitehead [8]:

Pr o p o s it io n 6.2.

There exist a spectral homology sequence {Erpq\

convergent to X* ( • ) and such that

K ,a = Hp( - , X q),

and a cohomology spectral sequence {Щ’9} convergent to X*(-) and such that

Ev,a = x q).

This spectral sequences are natural with respect to maps in ST q and ST x.

Consider the homology spectral sequence. We have the filtration

^n( ’ ) ’ ^n,0 ^n— 1,1 ' such that

^n.ol^n-i.i — Дn, 0 *

Thus we get an epimorphism Xn(-) -> Ef<0 and a monomorphism Е £ о - + К . о = В п( - , Х 0).

The edge homomorphism is the composition

X, M

n , 0 н п С

, X 0)

Pr o p o s it io n 6.3.

The Hurewicz homomorphism ip :

X n(*)

Hn(-, X 0)

coincides with the edge homomorphism defined above.

196 A . J a n k o w s k i

Proof. The proposition follows from naturality of spectral sequence of Proposition 6.2 and the triviality of spectral sequence for X.

Co r o l l a r y

6.5. I f X is MO or MSO, then the Hurewicz ho­

momorphism coincides with the Steenrod representation defined in [2], p. 14.

Proof. This is an immediate consequence of Proposition 6.3 and Proposition 7.1 of [2].

De f i n it i o n

6.5.

A

spectrum X is called a %-spectrum if for each space A the following condition is satisfied:

If Hr( A , X 0) = 0 for /’ = 0 , 1 , . . . , n —1, then Hr( A , X k) = 0 for r = 0 , 1 , . . . , n

— 1

and all 7c.

We are now going to state the main result of the second part of the paper.

Th e o r e m

6.6. Let X b e a %-spectrum. I f Hr{ A , X 0) — 0 for r = 0 , 1 , . . . , ..., n —1, then Xr{A) = 0 for r = 0 , 1 , . . . , n

— 1

and

I n : Х Л А ) ~ ^ H n ( A , X 0)

is an isomorphism.

« This can be proved by standard techniques of spectral sequences applied to homology spectral sequence of Proposition 6.2.

We shall now give some simple conditions for a spectrum to be a ^-spectrum.

Pr o p o s it io n

6.7. (a) I f X

0

is infinite cyclic, then X is a %-spectrum.

(b) I f X is a ring-lilce spectrum, then X is %-spectrum.

P roof. Part (a) is trivial. In order to prove (b) we note that A0 is a ring with unit and Xk is A0-module. There exist a spectral sequence Wp>q convergent to Hr( A , X k) with

Щл = Тотр{Нй{ А , Х

0

) , Х к).

If Hr{A , X Q) = 0 for r = 0 , 1 , ; . . , n —1, then = 0 for 0 < g < n.

Thus Ep q = 0 for p + g < n and Hr( A , X k) = 0 for r = 0, 1, ..., n

— 1

and all 7c.

Co r o l l a r y 6.8.

The spectra MO, MSO, M U, S are %-spectra.

7. Duality and Hurewicz homomorphism. Eecall that if A , A' are p-dual, then there exists a natural duality isomorphism

Dn>p: X n( A ) - * X p_n(A').

P

roposition

7.1. The diagram

X^(A) ^ Xp ( Af)

^,ХП ф Xp — П

H " ( A ,X 0) ^ H p_n( A ' , X 0) is commutative.

Let X be the multiplicative spectrum and let I be a X* - orientable vector bundle over A. Let UeXk(As) denote the Thom class and let

ü = xk ü.

P

roposition

7.2. The bundle £ is H*(-, X 0)-orientable, Ü is the cor­

responding Thom class, and the diagram

X n+k(A*) — -> Hn+k{AS X0)

t v t <p

X n( A ) - - * H n( A , X 0), where <p, ep are Thom isomorphisms, is commutative.

Proof. This follows from Proposition 3 of [3] since % is the multi­

plicative transformation of cohomology theories and %(i) = i where i, ~i aye units.

C

orollary

7.3. I f Mn is X*-orientable manifold, then Mn is H* ( •, X0)- orientable and the diagram

Xp{Mn) — X n~p(Mn)

\ x \ x

Hp(Mn, X0) — Hn~p( Un, X0) is commutative.

This corollary is a generalization of Thom representation theorem.

8. An application. Let G be a finitely generated abelian group and let K '(G ,2) be the Moore space of G. Given a spectrum X we denote by X® the spectrum X G = {X k_2 Ж K'{G, 2), ef}, where ek = ek_2 % id.

The homology theory X!j?(-) will be denoted by X* (•,(?).

E x a m p le. If S is the sphere spectrum, then S ( - , G) are stable homotopy groups with coefficient group G (in the sense of Katuta). This follows immediately from Theorem 4.7.

Coefficients X G are related to coefficients X n as follows.

P

roposition

9.1. There is an exact sequence

0 e®X„ -> -Xj -> Tor(G, 0.

198 A. J a n k o w s k i

Proof. It can easily be seen that X G = яп{Х?) = Xn+

2

(K'(G, 2)).

By Proposition 6.2 we have a spectral seqnence convergent to X*{K{G, 2)) and such that

Щ,л = Н , { К \ в , 2 ),У ) =

G <S> X q Tor {G, X q)

0

for p =

2

, for p — 3, for p Ф 2 , 3 , and the standard arguments complete the proof.

Let X be a ( —1)-connected spectrum. Then, by the proposition 8.1, X G = 6r®X0 and there is the Hurewicz homomorphism

X* :X * (A ,G )-+ H * (A ,G ® X b ).

The following proposition generalizes the Hurewicz theorem for homotopy groups with coefficient group G in the sense of Katuta.

Pr o p o s it io n

8.2. I f X is

( —

1 )-connected spectrum, X

0

is infinite cyclic, Xq is finitely generated and G is finite, then X G is a %-spectrum.

Proof. We have Xff = G . Assume Hr(_A,G) = 0 for r = 0 , 1 , ..., ... , n —1. We have to prove that Hr{A , X*f) — 0 for r =

0

,

1

, ..., n —1.

It follows for Proposition 8.1 that there is an exact sequence

...

- * B r( A,

G ® K ) i ? ) Тог(в, X,_,)) -*

-+H r^ ( A , G ® X q)

First consider the cases g = 0 , 1 . Since Tor(6?, Х _ г) = Tor(G ,X 0)

= 0 we have

H M , X % ) = H r( A , G )

= 0

and

Hr( A ,X f ) = H r{A ,G

0

X l) = 0 by the general coefficient theorem. For q > 2 we have

Tor(6r, X3_1) ^ G 0T(A 3_1)

(here Т (Х а_г) denotes the torsion subgroup of Xg_x) and it follows from the general coefficient theorem that Hr(A, G®Xq) — Hr(A, Tor(G, Х9_г))

— 0. The coefficient exact sequence shows that Hr( A, X^) = 0 . This

completes the proof.

References

[1] J. M. Cohen, Some results on the stable homotopy groups of spheres, Bull. Amer.

Math. Soc. 72 (1966), p. 723-735.

[2] P. E. C onner and E.E. F lo y d , Differentiable periodic maps, Ergebenisse der Mathematik, Berlin 1964.

[3] A. D o ld , Delations between ordinary and extraordinary homology, Colloq. on Algebraio Topology, Aarhus 1962.

[4] D. W. K ah n , Induced maps for Tostnikov systems, Trans. Amer. Math. Soc.

107 (1963), p. 432-450.

[5] S. L im a , The Spanier-W hitehead duality in two new categories, Summa Math.

Bras. 4 (1953), p. 91-144.

[6] J. M oore, On the theorem of Borsuk, Fund. Math. 43 (1956), p. 195-201.

[7] E. H. S p a n ie r , Function spaces and duality, Ann. of Math. 70 (1959), p. 338-378.

[8] G. W. W h ite h e a d , Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), p. 227-283.