Equivariant homology theories indexed by representations

R O C Z N IK I PO LSK T EG O TO W A RZY STW A M A TEM A TYCZNEGO Séria I : P E A C E M A TEM A TY CZN E X I X (1977)

|WOJCIECH PUblKOWSKll

Equivariant homology theories indexed by representations

CONTENTS

Introduction... 343

0. Notations and introductory re m a rk s... 343

1. Index-systems for (?-th.eories... 344

2. Unreduced U-theories... 346

3. Spectra and related ^ -th e o rie s... 348

4. Transformations between U-theories... 350

5. Some rem arks... 352

Introduction. This paper is a slightly modified version of Chapter I of my dissertation written under prof. K. Gçba. I am greatful to him and to prof. A. Jankowski for helpful conversations.

In literature there are many examples of equivariant analogues of “generalized homology and cohomology theories” , hut there was no general definition which covered all of them. Such a definition is given in Section 1 — and next some properties of “equivariant generalized homology and cohomology theories” (called shortly “G-theories” ) are proved — with special attention to differences between equivariant and non-equivariant case. This paper has thus rather unifying and ideol­

ogical character — no hard and deep theorems are proved.

0. Notations and introductory remarks. In the sequel G will denote a compact Lie group, BG — additive group of the Grothendieck ring of real representations of G (denoted usually by BO(G)), BG+ — additive semi­

group of representations (BG+ <= BG), neBG+ — ^-dimensional trivial representation of G.

E em ark . We do not distinguish between representation V and corresponding element in BG+. To be precise — we must choose repre­

sentatives Va for all aeBG+ and fix isomorphisms

such that diagrams

У a + Vfi+y

Ш + у р X \ ^ a , P + y

/ ^

r .+ V p +Уу v

<Pa,/S+^\

а+Р+У

у / fa+PiV

a+P

JrV v

\

commute.

We can do it (for example) as follows:

1. Let {У0 = 1 , Vlf V2, ...} be a set of irreducible representations of G (one for each isomorphism class).

For any aeRO(G)+ there exists exactly one representative Va = V0+ . . . + F 0 + F x + . . . . + 7 ! + ... + V 3+ ... + 7 S

-<-*

tim es-* times-»-

2. Define cpa>ft to be a permutation of summands, which sends F^-sum- mands of Vp after 7 fc-summands of Va (k = 0 , 1 , . . . ) .

The category T(G) of G-spaces in which we work is not very impor­

tant. It must satisfy certain usual conditions. The best such category seems to be the category of О-spaces of O-homotopy type of finite G-GW- complexes in the sense of Matumoto [12]. T. (G) denotes corresponding category of pointed О-spaces, and T2(0) — of О-pairs. hT(G), hT.(G) and hTz(G) are suitable homotopy categories (i.e., morphisms areO-homo- topy classes of O-maps).

To be reasonably short — we use some standard conventions and notations from algebraic topology (for example writting X instead of (X , 0) in T 2(0) or using the same letter for a О-map and its O-homotopy class).

Notice one useful generalization of “suspension functor” to the equivariant case :

Let VeRG+. Denote by 8 v eT.(G) its one-point compactification (with distinguished point “at infinity” ), and by

2JV: JiT.{G)->hT.{G)

a functor defined on objects Z V{X) — 8 V

X (smash product with 8 V),

called V-suspension.

1. Index-systems for 0-theories. In all known examples of G-theories —

homology functors are indexed either by integers (as various (?-bordism

theories [8], [9], [20], [21], [23] and singular ^-homology [2], [5], [10],

[11], [13]), or by cyclic group (H#-theories [18]), or by representations of G (stable (r-homotopy [19], Ебг-bordism [14], [15], [16]). These examples are either theories on T2{G) — as geometric (r-bordism [20], [21], [8], [9], [14], [15], [23] and singular G-homology [2], [5], [10], [11], [13], or reduced theories (i.e., theories on T.(G )) — and in this case suspen­

sions are indexed by (all or some) representations of G — as homotopical (r-bordism [7], [8], [16], stable (x-homotopy [19], K G-theories [18].

In Section 1 we restict ourselves to the case of reduced G-theories.

^-theories on T 2(G) will be considered in Section 2.

Let us introduce the definition:

D e f in it io n 1.1. Index-system (for (x-theories) is a diagram / = ( M - ^ B B - ^ B ) ,

where M, В are abelian groups, and <p, rp — group-homomorphisms.

We are going to index (x-homology functors by elements of В and suspensions by elements of M. To be more precise:

D e f in it io n 1.2. Let # — (M -£-* BG-X> B) be an index-system.

A (reduced) — (x-theory (indexed by / ) consists of:

(a) a family of functors

hr: hT.(G)->Ab, reB , satisfying :

hr(Y)->hr{X)-+hr(XfY ) exact for any (x-cofibration Y-+X, (b) a family of natural isomorphisms

oa: hr->hr+y)(Pal > , ae(p~1(BG+), satisfying

oaob = oa+b under natural isomorphisms of functors

ZVajrVb = 2 7

«+

=

<0+Ь).

R em ark . “Inverting the arrows” in Definition 1.2 we get a defi­

nition of a (reduced) (x-cohomology theory. Remarks of this type are usually dropped in the rest of this paper. Cohomological versions are used with no comments in some places.

E x a m p l e s . 1.1. Homotopical (x-bordism of T. Tom Dieck (a) unoriented [9], with index-system

B G ^ -> B G — >Z ,

where dim denotes virtual dimension.

(b) unitary [8], with index-system

Z ® BU (G )-^> BG — >Z , where BU(G) is the unitary representation ring,

<p\RU( 0 ) ~ “forgetting a complex structure”, <p\z — imbedding onto trivial real representations.

1.2. Stable (x-homotopy (and (x-cohomotopy) of G. Segal [19] and homotopical E(x-bordism of [16] are (x-theories with index-system

B G ~ > B G — >BG .

1.3. ÜL^-theory (complex) [18] has as its index-system Z ® B U ( G ) - ^ B G ^ ^ Z 2,

where y — as in example 1.1(b), dim2 — virtual dimension modulo 2.

1.4. For other examples see Section 2, where we show how to get a reduced (x-theory from the unreduced one — and give examples of unreduced (x-theories.

2. Unreduced (x-theories. Unreduced ^G- theory is defined on the category J i T 2(G). It has boundary homomorphisms instead of suspensions.

To index the homology functors we use — as previous — a homo­

morphism

ip: BG~>B.

D e f in it io n 2.1. An unreduced (x-homology theory consists of:

(a) a family of functors

hr : hT2(G)->Ab, reB , (b) a family of natural transformations

d: hr-+hr_vl-Q, r e B ,

where Q: hT2(G)->JiT2(G) is a restriction, i.e., Q (X , Y) = Y subjected to usual axioms:

I. E x c isio n . Let U = Int( U) c JJ c Int(A), (X , A), ( X —U, A —

— U)eT2{G). к: ( X — U, A — U) c —> (X , A) induces isomorphisms к*: hr(X — U , A — U)->hr(X ,A ), reB .

II. E x a c t seq u e n ce s. For any reB , ( X , A )eT2{G) a sequence hr{A)— > hr(X )— > hr{ X ,A ) — * h r_vl(A )— + ...

is exact (non-named arrows are induced by inclusions).

R em ark. As in non-equivariant case — one can deduce from these axioms exact sequence of a triple, Mayer-Yietoris sequence and similar general properties.

E x a m p l e s . 2.1. Singular (or G-cellular) G-homology and ^-coho­

mology theory [2], [5], [10], [11], [13]. In this case ip = dim: RG-+Z.

2.2. Geometric G-bordism: unoriented [9], [20], [21], oriented [23]

and unitary [8] — are (unreduced) G-theories with ip — dim.

2.3. Geometric jRG-bordism theory [14], [15], [16] is a G-theory on hT2{G) indexed by id: RG-+RG.

Now we are going to explain a relationship between reduced and unreduced G-theories.

In the non-equivariant case — reduced and unreduced theories are in one-to-one correspondence [22], but in equivariant case it is not so.

Not every reduced G-theory can be obtained from some unreduced one.

T h e o r e m 2.1. There is one-to-one correspondence between unreduced G-theories indexed by ip: RG-^R and reduced G-theories with index-system Z ^ R G - ^ R , i{l) = 1 .

P roof. We give a short description of the correspondence men­

tioned in Theorem, omitting the verifications, which are as in [22].

Let {hr, d} be an unreduced G-theory for ip: RG— >R. The correspond­

ing reduced G-theory {hr, on} is defined as follows:

hr(X) = hr( X ,{ * } ) , where * is the distinguished point in X.

To define u’s we use exact sequence of the triple (GX, X , {*}), which gives an isomorphism

hr{CX, X)-^-> hr_vl(X , {*}) (G — reduced cone).

Combining d-1 with the isomorphism hr{GX, X)->hr{GXJX^ {X}) and G-homeomorphism C X /X — Z1X we obtain

a1: h ^ X ^ b ^ X ) . on are defined by iterations.

Now let us start with a reduced G-theory {hr1 an} indexed by Z — RG — > R .

Put hr(X ,A ) = hr(X/A).

To define d : hr(X , A)->hr_vl(A), look at the homomorphism p *: hr{X v C A )-*h r{X u C A IX u {*}),

X ^kj CA is regared as a pointed G space with distinguished point — the vertex * of the cone G A.

12 _ p r ace M a te m a tyczn e 19 z . 2

But X u (L 4 /X u {*} is O-homeomorphic to 2'1(JL u {*}) = U1(A /0), thus we can put

d = ( a 1) - 1 'p ^{* .}

Now it is clear why not every reduced О-theory can be obtained from some unreduced one: boundary homomorphisms allow us to define only trivial suspensions. (One can easily find an example of two different reduced О-theories which give the same unreduced O-theory.)

The next point in which there is a difference between equivariant and non-equivariant theories — are coefficients of the theory. In the non-equivariant case coefficients are simply the groups hr (point) (or hr(8°) for reduced theory). In the equivariant case the role of coefficients plays the restriction of the О-theory to the “category of canonical orbits”

0(G). 0(G) is a category with objects GjR for all closed subgroups R of G and with O-homotopy classes of О-maps as morphisms (see [2]).

D e f in it io n 2.2. Coefficients of an unreduced G-theory {hr, 0} are groups hr(GJH) and homomorphisms hr(GIR)-+Jir(G/K) induced by (r-maps G/R-^G/K.

For reduced О-theory we must use hr[(G/R)+) instead of Tir(GjR).

3. Spectra and related О-theories. In non-equivariant situation any (co-)homology theory is representable by suitable spectrum [4], [1].

We define a O-spectrum and show how it leads to G-theory — but we cannot prove the “representability theorem” for О-theory with general index-system. For index-system Z — > B G — > Z it is a simple consequence of E. H. Brown theory [4] and was proved in [13].

D e f in it io n 3.1. Let f = ( M — > JR G ~ > R ) be the index-system.

AG-spectrum A consists of:

(a) a family of pointed (z-spaces AreT.(G), reB, (b) a family of 6r-maps (in T.(G))

£“ : S*aAr-*A r+v9a, aç(p~1(BG+ ), reB , satisfying

pb a

a+b

cr + y>q>a ^{c r c r}

The definition of (z-theory with coefficients in 6r-spectrum is almost the same as in non-equivariant case — except one point about the group- structure. Namely the set of (z-homotopy classes of O-maps

Аг+у)<Ра\д.

usually lias no natural group-structure when <pa contains no trivial sum­

mand.

L em m a 3.1. Let X , Y, ZcT.(G), YeRG+.

(a) A set [Yv X , Y]G has natural group structure provided V has a trivial summand.

(b) Group structure from (a) is commutative provided V has two-di­

mensional trivial summand.

(c) Suppose V has a trivial summand. The mapping A lz : [ S VX , Y]g- * [ S r X A Z , Y ^a ZI ^s is then a group homomorphism.

P roof, (a) Let V = W +1. S vjS w is (т-homeomorphie to S r v S r which allows ns to define 6r-equivariant i?-cogroup structure on S v

V: S r ->Sr v S v .

The map V defines in usual way group structure in [L v X , Y"\a .

(b) Using two-dimensional trivial summand in V we can write a G- homotopy between V and rV, r: S v v S r ~>Sv v S v being “a transposition” , r{xvy) = (y v x).

Thus S v is a ^-equivariant H - commutative LT-cogroup. Commutati­

vity of [YVX , Y~\ ^q follows then in usual way.

(c) This part of the lemma follows directly from the definition of group structures in both sets. Q.E.D.

For X , YeT.(G) put {X , Y}e = lim ([X ,

n

By Lemma 3.1 it has natural abelian group structure. Now we are ready to define a .G-cohomology theory with coefficients in (х-spectrum A : D e f in it io n 3.2. (a) hr(X ; A) = lim {{X ^ X , Ar+ }Q, A“+&),

oeç)- 1( B G + )

where Л«+6 = ( ^ +We) . - i ^ : W i l d ,

(b) cr“ : hr{X\ A)-^hr+w4>a{E4>^X\ A) is the identity (both groups are the limit of the same direct system — but with translated indexes in the second case).

L e m m a 3.2. Functors hr, rcR and transformations ^{o a} , aep~1(RG+) form a reduced G-cohomology theory. *

P roof. Standard.

D e f in it io n 3.3. (a) hr{X , A) = lim ({/S'9’*1, X /\A _ r}Q, p%+b),

aeq>-l(№ +)

where

K +b = № * 4 ^ ) , - ^ : {№ , Х л Х „ а_л Ь { в * С + » ) ,Х л Х ^ .+м_,}в,

(b) oa: hr{X , A)-+hr+vv (XVaX } A) is induce dby maps HVa\ {S Vb, 1 л M ^a \ n - r}a-

L em m a 3.3. Functors hr,r e R and transformations oa, ae<p~1(RG+) form a reducèd G-homology theory.

P ro of. Standard.

B em ark . In Definitions’ 3.2 and 3.3 we can use (x-homotopy classes [ > ] ^g instead of { , }G provided <pM containes a representation with non-zero trivial summand.

E x a m p l e s . 3.1. Equivariant Thom spactra [8], [9], [16] define

“homotopical (x-bordism and (x-cobordism theories”: These spactra are constructed in usual way from “classifying (x-vector bundles” [7].

3.2. Stable (x-homotopy and (x-cohomotopy [19] are (x-theories de- fined by (x-spectrum of spheres 8 = ($ F, ow}, ow being canonical G- homeomorphism between E wS v and 8 r+w. (For v^RG+ we must put 8 V

— point.)

3.3. Eilenberg-Maclane (x-spectrum K (Z) = {K {Z , V), awj was de­

fined in [19].

Index-system for examples 3.2 and 3.3 is / = ( R G ^ 'R G ~ > R G ) .

4. Transformations between (x-theories. This paragraph is devoted to transformations between G-theories with different index-systems. To speak about such transformations — we first introduce a notion of map­

ping of index-system.

D e f in it io n 4.1. Let / = {M -Z+ R G -^> R), Ж = R G ~ > 8 ) be two index-systems.

Mapping (a, ft): #->Ж is a commutative diagram M--- ^— » RG---- ?----> R

a p

N ---- ^---- >RG---* S

where a and /? are homomorphisms of abeliam groups.

Now we are ready to define a transformation of (x-theories over some mapping of index-systems (in both directions — see Definitions 4.2 and 4.3). Let Ж, (a, /?) be as in Defefinition 4.1. Let {hr, oa), {gs, rb}

be (x-theories in dexed by f an d Ж respectively.

-

D e f in it io n 4.2. A tran sfo rm atio n ê : {hr, oa}->{gs, t 6} over the m ap p in g (a, /?): # is a fam ily of tran sfo rm atio n s of fun ctors

Kv~+g*v, VeRG,

such that following diagrams commute:

aab rb

Y Y

Ъ . . &

+Ф n . уФ ,vy>V+y>Mb

veEG, b€fi~1(BG+ ).

E x a m p l e s . 4.1. F o r an y in dex-system = ( M ~ ^ B G - ^ R) there exists a m ap p in g « / - > / , n am ely

+ R G --- - --- >BG

л

<р

и — ?— > д а ----*--- » E .

Moreover, stable 6r-homotopy [19] is universal among (r-theories with index-systems of the from (RG-^-> BG — » B). The same about G-co- homotopy. For precise formulation see Section 5.

4.2. ^-graded ^ -th e o ry has natural transformation into Z2-graded one — over the mapping of index-systems

B G ---->BG— — — >Z 4

id

BG --- -— > BG dim- - > Z%

This transformations ê is the identity for every veRG. Let J f ,

( a , ft), { \ , or*}, {g3, г6} be as in Definition 4.2.

D e f i n i t i o n 4 .3. A transformation %: {gs, ^{r 6}} ->{hr, ^or®} over the mapping (a, ft): is a family of transformations of functors

Xv* 9 t v €RG)

such that following diagrams commute:

Xv yw

n°b

Qvv+v/лЬ

Y Y

y/ib xv+fib

г . у ф

" ^ Г1/у)К+у>ф **

VcRG, Ь€/г-1(да+).

E x a m p l e s . 4.3. “Stabilized (x-bordism” ) and transformation 9l*( )-^$R?’s( ) over the mapping

BO A

i

z l

- --- ->BG

г- ---- >BG

id dim

у --- > z was defined in [6].

4.4. Pontriagin-Thom construction gives a transformation from geo­

metric to homotopical (x-bordism [8] — over the mapping from Example 4.3.

E e m ark . We can also define maps of (x-spectra (over some map­

ping between their index-systems) and show that they lead to trans­

formations of corresponding ^-theories.

5. Some remarks.

A. Multiplicative structures. All ^-theories considered in examples of Sections 1 and 2 are in natural way “multiplicative ^-theories” . Gen­

eral definition of multiplicative (x-theory needs some assumption about index-system (to formulate anticommutativity of multiplication).

Namely, suppose that for index-system # — (M — > BG-^-> B) there exists a factorization

B G —4—> B

\ /

dim 2\ ^ }/в

D e f in it io n 5.1. Let {h r , a a] be a (reduced) (x-theory with index-sys­

tem £/ . Suppose there exists a factorization as above or all elements of groups hr(X) (re B , XeT.(G)) are of order 2.

Then multiplicative structure on the theory {hr, oa} is a family of pairings (for all X , YeT.(G))

hr(X) ® hs( Y)->hr+s{XA Y) satisfying usual axioms:

(i) naturality (with respect to induced maps), (ii) associativity,

(iii) identity (there exists an element l e h Q( S ° ) such that for every

a}ehr ( X ) it holds 1 "X = oc),

(iv) stability (suspension ca = multiplication by <ra(l)),

(v) anticommutativity:

For xehr(X), yehs (Y) it holds

Х ‘ У = ( — l f r'QSy ' X

(or os'y = y -os in the case of all elements of order 2).

R em ark . Evident reformalation gives a definition of multiplicative unreduced O-theory.

D e f i n i ti o n

5.2. Let # and ^q be as above. Multiplicative G-spectrum is a 6r-spe$trum {Ar, e®} with the additional structure:

(i) id e n tity : For any VeBG+ given a map i v : SlF->AvV,

(ii) m u ltip lic a tio n : For any r, seB given a map yr s : Ar л As->Ar+s such that usual (see [22]) diagrams commute.

R em ark . Standard procedure shows that multiplicative ^-spectra lead to multiplicative 6r-theories.

E x a m ple s.

Multiplicative structures in all the examples of O-theories mentioned in this paper are defined in the same way as in corresponding non-equivariant theories.

B. Universality of the stable G-homotopy. We shall prove that stable O-homotopy of = {nl, a v} is universal among O-homology theories with index systems of the form = (BG— > BG-X> B). Cohomological version (i.e., universality of the stable O-cohomotopy) is also true, with similar proof.

T h eo ee m

5.1. For any (reduced) G-homology theory h — {hr, or} in­

dexed by and for any aehQ(S°) there exists a transformation ya: ns->h which sends le n ^ S 0) onto a, transformation ya being over the mapping (id,v>): ° f index-systems (see Example 4.1).

P roof. (As in non-equivariant case.) Remember that я®(Х)

— ]imnv+v(Er X) (we use only V sufficiently large to be sure that v + A-VeBG+). Define maps v

yl- n ^ F (Xr X )- *h „(X ) 1>У

nv+F(XvX) = lS ’’+ r, Z r X ]e> f ^ ( a vr 1f , a ’ +r a<h,„(X), where

кф+Г)(8’ + г) - ± ^ h ^ +Fi(z r x ) ^ ^ h „(X ).

The maps y„ give a mapping of direct system, thus they define a map of its limit,

У a- n% (X)-+\v(X) .

Naturality of the transformation ya is clear because induced maps

■ commute with suspensions. Of course ya(l) = a. Q.E.D.

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