Equivariant Classical Cohomology

Chapter I. Equivariant Classical Cohomology

1. G-complexes

Let G be a finite group. By a G-complex we mean a CW complex K together with a given action of G on K by cellular maps such that

(*) For each ge G, {x6K]g(x) = x} is a subcomplex of K.

-1 Note that for each g~ G, the fact that g: K § K and g : K ~ K are assumed to be cellular implies that, in fact, each g: K § K in an automorphism of the given CW structure of K. Also it follows from the condition (*) that if g~ G leaves any point x6 K fixed then g must leave K(x) pointwise fixed. (K(A), for any subset ACK, denotes the smallest subcomplex of K containing

A. It is a finite subcomplex iff A has compact closure.)

Let K be a G-complex and L a subcomplex invariant under G.

Then an easy inductive argument on the skeletons of K shows that K has the equivariant homotopy extension property with respect to L. That is, if f: K ~ X is an equivariant map

into any space X with a given G-action and if F': L• § X is

any equivariant homotopy then there exists an equivariant homotopy F: K• § X extending F'.

Taking the case in which X -- LxIO Kx{O} with f and F' the obvious maps we obtain the fact that L• U K• is an equivariant retract of KxI, the retraction being F: KxI § LxlU

Kx{O}. Let B C X be the set of points x such that F(x,1)

L• Then B is a neighborhood of L in K and the composition ~-2

F BxI ~ LxIU Kx{O} § K is an equivariant strong deformation retraction of B onto L.

Now apply these facts to the G-complex KxI and the subcomplex A = LxI U Kx{O}. Let U be a neighborhood of A possessing an equivariant strong deformation retraction onto A. Let f: K § I be a continuous function such that f(x) = 0 on some neighborhood of L and f(x) = 1 unless xxI C U. By taking x § inf{f(g(x))Ig~G} we can assume that f(g(x)) = f(x) for all g ~ G. Define

P t : KxI § Kxl by Ft(x,s ) = (x,s(l-tf(x))). This forms a deformation of Kxl into U which is equivariant and leaves A stationary. Following this by the deformation of U into A we see that A = LxI O Kx{0} is an equivariant strong deformation retract of Kxl.

Now identify L~ to a point, so that K• becomes the mapping cylinder M- KxI/Lx{l} of the collapsing map K § K/L.

Now our deformation becomes a deformation retraction of M onto

K•215 KUC L (K with the cone C L on L attached). On the other hand M can be deformed equivariantly into the face

Kx~}/Lx{l~ K/L. This shows that for any pair (K,L) of G-complexes, the G-complex K/L is of the same equivariant homotopy type as

K UC L .

Let us recall a construction central to the cohomology theory of CW complexes. Let K be a CW complex and pick an orientation for each cell of K. (If K is a G-complex it may be assumed that the operations of G preserve these orientations, because of (*), but this is not important.) Let Cn(K) be the I-3 free abelian group generated by the n-cells of K. Cn(K ) is isomorphic to the singular homology group Hn(Kn/Kn-1;Z), or

to H (K n p Kn-I;z) n Suppose that o is an n-cell of K and let fa: Sn'l § Kn-1 be a characteristic (attaching) map for a. Collapsing K n-2 to a point, we obtain an induced map

(1.1) S n-1 -+ Kn-1 § Kn-1/K n-2 = VT/~ where z ranges over the (n-l)-cells of K (T/~ is an oriented

(n-l)-sphere and V denotes the one point union). For each

T there is a projection VT/~ § T/~ (collapsing all other spheres).

Let fTa denote the composed map fT : sn-I + O The map (1.1) provides a singular homology class

Bo e Cn_l(K) = Hn_l(Kn-1/Kn-2 ) and we clearly have that

ao = XT[T: olT where [z: o} = 0 unless T is an (n-l)-cell and, for an (n-l)- cell z in K, [T: o} = deg iT: sn-1 § T/~ O (for fixed o this is non-zero for only a finite number of cells T T, in fact s is a trivial map except for a finite number of cells T). The correspondence o + ~o generates a homomorphism

a: Cn(K) .+ Cn_I(K) which, in fact, is just the singular homology connecting homomorphism of the triple Kn, Kn-l, Kn'2. That is, ~ is equivalent to the composition I-4

J* Hn(Kn K n-l) ~* ~ Hn-I ( Kn- I) ' Hn-1 (Kn-I Kn-2 )

We have that ~2 = 0 since the composition

H n 1 (K n-I ) J* ~ H (K n- I Kn-2 ) a* ~ H (K n -2) - n-1 ' n-2

(part of the homology sequence of the pair (Kn'l,Kn-2)) is

zero. Note that 82 = 0 is equivalent to the equation

~[~l~] [x s~] = 0 for given ~,o. T

2. Equivariant cohomolog7 theories

Let G be a finite group and let ~ denote the category

of G-complexes and (continuous) equivariant maps. Let /~0

denote the category of G-complexes with base point and base

point preserving equivariant maps (base points are always

assumed to be left fixed by each element of G and, in the

case of G-complexes, to be a vertex) Let 4 2 be the category

of pairs (K,L), L C K a subcomplex, of G-complexes.

We use the abbreviation "Abel" to stand for the category

of abelian groups.

An e~uivariant (generalized) cohomolo~y theory on the

category ~ is a sequence of contravariant functors n: ~2 § Abel (n~ Z)

together with natural transformations 6n: ~n(L,~) ~ ~n+I(K,L),

such that the following three axioms are satisfied (we put ~n(L) = ~n(L,~)) :

(1) If f0" fl are equivariantly homotopic maps (in ~2)

then ~n(fo) = ~n(fl). I-5

(2) The inclusion (K,KN L) C (KUL,L) induces an isomorphism

~n(KUL,L ) ~ ~ ~n(K,KnL)

(3) If (K,L)~ ~2 then the sequence

(K,L) j * ~n(K) i ~ ~n(L ) ~ ~n+l (K, L) § . . o is exact.

Remark. If G is abelian then the operations by elements of G are morphisms ~ § ~ (i.e. they are equivariant). Thus, in this case, each ~n(K,L) has a natural G-module structure.

There are functors ~2 § ~0 and D0 § ~2 defined by

(K,L) § K/L and K § (K,x0) where x 0 is the base point of Ko

L/L is the base point of K/L (taken to be a disjoint point if

L ffi ~, in which case K+ denotes K/~). Standard arguments can be used to translate the above axioms into an equivalent set of axioms for a "single space" theory on ~0" (See, for example,

G. W. Whitehead, Generalized homology theories, Trans. A. M. S.

102 (1962), pp. 227-285.)

In fact for K G~ 0 let SK = SA K (with the obvious G action, trivial on the "circle factor" S) denote the reduced suspension of X. Then an equivariant cohomology theory on ~0 is a sequence of contravariant functors

n: /0~0 § Abel together with a sequence of natural transformations of functors n on(K): ~n(K) -~n+I(sK) satisfying the following three axioms

(1') If fo' fl are equivariantly homotopic (in ~0 ) then

= (fl) . ~-6

(2') on(K) is an isomorphism for each n and K.

(3') The sequence

is exact.

Most of the material of Chapter I of Eilenberg-Steenrod goes over directly to these generalized theories. Later on in these notes we shall show how to construct such theories using rather standard methods and shall consider some special cases of interest. We shall not concern ourselves with these matters at present, but shall confine ourselves to a discussion of

"coefficient groups".

In non-equivariant theories the "coefficients" of the theory are defined to be ~ (pt) (or ~ (pt+)) and these (graded) groups are the primary distinguishing feature between different cohomology theories. In fact for (non-equivariant) "classical" theory (= r theory + dimension axiom) the knowledge of the coefficient group (40(P t) in this case) allows computation of the r of any finite simplicial complex. Essen- tially this is true because homotopy points (i.e. contractible objects such as simplexes) form the basic building blocks of all complexes.

For equivariant theory the situation is slightly more complicated, for now the "building blocks" are essentially the orbits (in an appropriate sense) of G. That is, the coset spaces G/H, where H ranges over the subgroups of G (not necessarily normal), form a representative set of building blocks. Thus a "coefficient system" should contain all the ~* ~, + groups CG/H) Cor ~ ((G/H)))). But this is not enough,

for we must specify how the building blocks "fit together".

That is, we must consider the equivariant maps G/H § G/K and a "coefficient system" must incorporate the induced homomorphisms

(G/K) * (G/H) in its structure.

In the following sectiom we define precisely what we mean by a coefficient system.

Terminology: A cohomology theory on ~ or ~0 will be called "classical" (="equivariant classical cohomolo gy" but

~lassical equivariant cohomology"as defined, for example, in

Steenrod and Epstein, Cohomology Operations) if it satisfies the additional "dimension" axiom:

(4) O~ n(G/H) = 0 for n f 0 and all H,

or, for a single space theory,

(4') ~ n((G/H} +) = 0 for n ~ 0 and all H.

Later on, we shall prove existence and uniqueness theorems (of the Eilenberg-Steenrod type) for such "classical" theories.

3. The categor 7 of canonical ~rbits.

The category of canonical orbits of G, denoted by ~G' is defined to be the category whose objects are the left coset spaces G/H and whose morphisms are the equivariant (with respect to left translation) maps G/H § G/K. I-8

For future reference we shall classify the equivariant maps G/H § G/K. Suppose f is any map

f: G/H § G/K and put

f(H) = aK where aeG.

Then f is equivariant ifs f(gH) = gaK for all ge G. Conversely, the formula f(gH) = gaK defines a map (which must be equivariant) provided that

f(ghH) = f(gH) for a11he H. That is, we must have ghaK = gaK for all ha H.

This is equivalent to haK = aK and hence to

-1 (3.1) a Ha C K .

Thus we have the following result: Let a@ G be such that a-lHa CK. Define

~: G/H § G/K by ~(gH) ffi gaK.

Then ~ is equivariant, that is, ~ Ghom(G/H,G/K) and every equivariant map has this form. Also, clearly, ~ = ~ iff aK -- bK, that is, iff a-lb~ K.

Suppose that (5.1) is satisfied. Then the inclusion a-lHa c K induces a natural projection G/a-IHa § G/K (equivariant) -I -I and, similarly, the inclusion H C aKa induces G/H § G/aKa .

Now right translation by a induces an equivariant map Ra: G/H §

G/a-IHa (given by gH § gHa = ga(a-IHa)) and also Ra: G/aKa -1 §

G/K. Clearly the diagram I-9

-1 G/H * G/aKa

(3.2) It a

G/a- itta o G/K

commutes. Thus equivariant maps are precisely those maps induced by inclusions of subgroups and by right translations.

In particular hom(G/H,G/H) consists of the right trans-

lations by elements of the normalizer N(H) of H (i.e. ae N(H)

yields gH § gHa = gaH). Since RaR b = Rba, and generally ab = ba, -I the correspondence a § R a yields an isomorphism

(3.3) N(H)/H ~ hom(G/H,G/H).

For example, let G = Zp, where p is prime. Then ~G

consists of the objects G/G and G/(e} (that is essentially of

a point P and of G) together with the following morphisms

P § P

G § P

~: G § G for each a~ G

(where here ~ = R takes ~ into ga). a

4. Generic coefficient systems

(4.1) Definition. A (g?neric) qoefficient system (for G)

is defined to be a contravariant functor 0 G ~ Abel.

If M,N: ~G ~Abel are coefficient systems, a morphism

T: M ~ N is a natural transformation of functors. With this

definition, the (generic) coefficient systems for G form T-IO

an abelian category C G = Dgram(~G,Abel ) . (~G denotes the

dual category to ~G and the fact that ~G is an abelian cate-

gory is a special case of a result of Grothendieck; see Maclane,

Homology, IX, 3.1, p. 258.)

Examples:

(i) Let be an equivariant cohomology theory and let

q be an integer. Define

hq: d~ G § Abel by hq(G/H) = ~q(G/H) and if f: G/H § G/K is equivariant, let hq(f) = ~q(f): ~'q(G/K) § ~&q(G/H).

(2) Let A be a G-module. Define

M: ~G § Abel as follows: Let M(G/H) = A H (the set of stationary points of

H in A). For g e G with H ~ gKg "I note that the operation by

g: A § A takes A K into A H, (for aG A K implies that

Hga C gKg-lga = gKa = ga). Denote this map A K § A H by gH "~ g-I If ~ = g' so that g' ~K, then clearly gH,K = gH,K" Thus for ~: G/H + G/K we let : AK § AH . M(~) = gH,K

(3) Let Y be a G-space with a base point Y0" Define

~q(Y) eG, that is ~q(u ~G -~Abel as follows:

~q(Y)(G/H) = ~q(yH,y0)

~q(y)(~) = g#: *q(yK y0) § *q(yH'yo} where ge G satisfies H C gKg "I so that g maps yK § yH (see

example 2). (In this example we assume each ~I(yH ) to be

abelian when q = 1.) I-ii

Remark. Since hom(G/H,G/H) ~ N(H)/H we have that, for any coefficient system Me e M(G/ll) possess a natural N(H)/H- G' module structure.

Let M~ e Since ~G contains, in particular, the G" objects G = G/{e} and P = G/G with the morphisms

1: P § P

r: G § P

~: G § G we have that M "contains" the abelian groups M(P) and M(G) with the homomorphisms M(1) = i and

e = M(r): M(P) § M(G)

a. = H(a): M(G) § M(G) which satisfy M(~) = M(~) = M(~)M(8) and M(~)M(r) = M(r~) =

M(r); that is,

f (ab). = a.b.

a.E = E .

Thus we may consider M(G) to have a G-module structure defined by (a,m) § a.(m) and M(P) to have a trivial G-module structure and e : M(P) § M(G) to be an equivariant homomorphism (i.e. r M(P) § M(G)G).

Of course, if G = Z where p is prime, then this is all P of the structure of an M~ ~G" That is, in this case, a coefficient system consists of an abelian group MO, an abelian group G M1 with a G-module structure and an homomorphism e : M0 § M1.

Moreover, a morphism between two such systems M and M' is a commutative diagram of G-module homomorphisms: Z-12

C M0 + M 1

t ! E ! i M0 ) M 1'

For example, when G = Z and Y is a G-space with base point, P (Y) consists of the group i (yG), the group ~ (Y) on which G q q q acts by the induced homomorphisms g#: ~ (Y) § ~ (Y), and the q q homomorphism r : ~ (yG) § ~ (y)G C ~ (Y). induced by inclusion q q q yG C Y.

5. Coefficient systems on a G-complex.

Let K be a G-complex. From K we form a category ~F~ whose objects are the finite subcomplexes of K and whose morphisms are as follows: If L and L' are finite subcomplexes of K, then hom(L,L') consists of a11 maps g: L § gLCL' for ge G

(hom(L,L') may be empty). Note that we do not distinguish between maps induced by different elements of G if they are the same map.

Clearly the morphisms of ~ are just the inclusion maps

L C L', the maps a: L § aL induced by operations by elements of

G, and the compositions of these.

We should note that for most purposes only the objects

K(o) of ~ for cells o of K are of importance, but for some constructions one needs the more general subcomplexes.

We define a canonical contravariant functor 0:~+ ~ G as follows: For L C K a finite subcomplex, let G L =

{g Gig leaves L pointwise fixed). We put

0 (L) = G/G L o

If gL ~ L' and f denotes the map L § L' induced by operation by ga G, then we see that -1 g GL,g C G L

and we put OCf) = g: OCL') § OCL), that is OCf) is g: G/G L' G/G L which takes g'GL, into g'gG L-

In other words, if L C L' then GL, C G L and O(inclusion)

is the natural map G/G L + G/GL, while if g: L § gL then

Gg L = gGLg-1 and O(g: L § gL): O(gL) = G/gGLg-I ~ G/G~ O(L) is

right multiplication by g.

Now if M ~ ~G is a generic coefficient system, that is,

if M: ~G § Abel is a contravariant functor, then

MO: ~ § Abel

is a covariant functor and is called a (simple) coefficient

system, on K. We generalize this as follows:

A local coefficient system on K is a covariant functor

: 3< § Abel o

Again by Grothendieck's result, the local coefficient systems

on K form an abelian category ~K = Dgram(~, Abel).

The coefficient systems MO: 7( § Abel, for M~ G' clearly form a subcategory ~K of ~ ~K.

Notation. If ~@ ~C K and o is a cell we let Jr(o) -

~(K(o)) and for K(~)C KC~ ) we let ~(z § o) denote

~(inclusion: K(T) § K(o)). Note that if [z: o] ~ 0 then

K(T) ~ K(o) so that T § o is "in" ~. 1-14

6. Cohomology

Let ~: ?C§ Abel be in ;~C K. Orient the cells of K

in such a way that G preserves the orientations and define

Cq(K;Z) to be the group of all functions f on the q-cells of K with

fCo) s ;r Define 6 : Cq(K;Z) § cq+I(K;Z) by

(6.13 (6f)(o) = T [T: o].2~(z § o)s T (which makes sense since K(x)C K(o) whenever [T: o] ~ 03. In other words (6f3(o3 is defined by "pushing" all coefficients to

~(o3 and then taking the usual coboundary. This remark shows that 66 = 0 since to compute (66f3(~3 we push coefficients to

~(~3 and then compute (classica~ coboundaries twice which necessarily gives zero. Of course)66 = 0 also follows by direct

computation.

Now we define an operation of G on Cq(K;Z3 as follows:

If g&G and feCq(K;j~) we put

(6.23 gCf)(o) -- ZCg)CfCg-lo)).

-1 Here &~(g) refers to ~(g: K(g o3 + K(o33. Let us abbreviate

(g3 = g..

Replacing o by g(o) in (6.23 we obtain

C6.33 gCf) Cgo) = g.CfCa33

It is clear that the automorphism s § g(s of C (K;~3 defines an action of G on C (K;;~) by chain mappings, Thus the

fixed point set 1-15

Cq(K;~) G = (fe cq[g(f) = f for all geG} is a subcomplex. It is also denoted by Cq(K; ~). By (6.3)

C * (K; ~ )G consists precisely of the equivariant cochains f

(i.e. such that f(go) = g,(f(o))).

We define the equivariant cohomology group

(6.4) Hq(K;.~) = HQ(c*(K; :~)G).

If M~ e G (so that Me ~ e K C ~* eK ) we use the abbreviation

(6.s) HqcK;M) - XqCK;MO).

If L is a subcomplex of K, invariant under G, then there

is a restriction map C (K;~) § C (L; ~) whose kernel is the

# relative cochain group C (K,L; ~). There is a splitting homomorphism C (L;~~') § C (K;~') defined by extension of a cochain by zero (not a chain map). This clearly commutes with operations by G so that the sequence

* * * G 0 § C (K~L; ~') G § C (K;:~') G § C (L; ~) § 0

is exact. With the obvious definitions we obtain an induced

cohomology exact sequence n n ... § n(K,L;;~)G § HG(K ; ~) § HG(L;~ ) § H +1 (K,L; ~) * ...

7. Equivariant maps.

This section is not necessary to our main line of thought

and it is included merely for the sake of completeness.

Let G and G' be finite groups and let ~ : G § G' be a homomorphism. Let K be a G-complex, K' a G'-complex and let

: K § K' be a cellular map which is equivariant (i.e. 1-16

CgCx)) = Cg) The map ~ (together with 9) induces a functor

(between the categories associated with K and K' respectively) as follows: If L C K, let ~(L) = K'(~(L)) and if f is the composition L-K~ gL C L 1 then ~(f) is the obvious composition

K'C~CL)) § V'Cg)K'C~CL)) = K'C~ Cg)~CL)) = K'C~CgL)) C K'C~CL1)) .

(By abuse of notation we might define u on morphisms by writing

(g) = ~,(g).)

Let ~': 7~' ~ Abel be a local coefficient system on K'.

Then ~ 'u ~ § Abel is a local coefficient system on K. Sup- pose that ~: ~ § Abel is any local coefficient system on K.

Then we define a u A from ~' to ~ to be a natural transformation

of functors on ~C. Now there is an obvious chain map C (K;:~'u §

* C (K; :~) induced by A and this is clearly equivariant with respect to the actions by G. Thus A induces a homomorphism

C7.1) ~ : HGCK;~'u ) § HGCK;~ ).

We shall define a canonical homomorphism

C7.2) u : HG,(K';~" ) § HG(K;~'u ) so that together with (7.1) we will obtain a homomorphism

u : HG,(K';;~'' ) § HGCK;;~ ) (also denoted merely by A ).

In fact note that the cellularity of ~ implies that induces a map Kn/K n-I § K'n/K 'n-I and hence induces a chain mPp 1-17

~.: Cn(K) § Cn(K' ).

Define

(7.3) u : C (K'; ~'') § C CK; :~'~P) by

(f) Co) = fCq,.Co)) where the right hand side is shorthand for

na~'(K'CTa) § KtC,Co)))fC"ra)& ~'(K'(*Co))) = ~'~/Ca) where ~.(o) = Zn "~a fC n (K').

Now we compute -1 v C~*Cg)Cf))Ca) = CS~(g)Cf))C~.(o)) = ~' C s* Cg)) Cf(s- (g) ~.Ca)) -I = C~'v)Cg)CfC~.cg-lo))) = C~'V) Cg)Cv*Cf)Cg a))

= gC~ Cf))Co).

Thus, if yCg)(f) = f for all geG, then

gCu (f)) = u (SPCg) Cf)) = V (f). * )~(G) * G Therefore (7.3) takes C (K'; :;~' into C CK;~'~) Since c * (K';~') G' c c * (K';~') ~*(G) we obtain a chain map * G' * G C (K';~') § C (K;~'u which induces our promised map (7.2) upon passage to homology.

The situation with simple coefficient systems is slightly more complicated, and we shall now discuss this case. We define a functor

~: eG~ ~ G , by putting #(G/H) = G'/~(H) and, if a-lHa C K as in (3.1), so that ~Ca)-l~c H) ~Ca) {:::: ~CK) we put ~(~: G/H § G/K) =

~Ca): G'/~CH) § G'/~ CK). 1-18

The diagram

ol ro ~G + ~ ~G ' does not generally commute since

o'~CL) ffi @' (K' (~(L))) = G'/G~cL) while

@O(L) = ~(G/GL) = G'/? (G L) and ~(GL) C G~(L) are not generally equal. However the projection

G'/~ (GL) § G'/G~v(L)

is clearly functorial and provides a natural transformation

(7.4) @O § 0'u

of functors. Let M' e ~G' be a generic coefficient system for

G'. Since M' is a contravariant functor ~G' § Abellthe trans-

formation (7.4) induces a natural transformation

(7.5) MtQ'~ § M'~O

of functors ~§ Abel. In other words, (7.5) is a u

(7.6) M'O' ~ M'~O.

Thus we have an induced homomorphism

C7.7) HG,CK';M') § HGCK;M'@ )

(where the 0 and 0' have been dropped in accordance with our

notation conventions). 1-19

If H~ CG and M'e d G' we define a ~-morphism M' § M to be a natural transformation

M'# + H of functors ~ § Abel. Clearly, in combination with (7.7), G every ~-morphism M' § M induces a homomorphism

(7.8) HG,(K';H') + HG(K;H ).

8. Products

Suppose that K is a G-complex and K' is a G'-complex.

Then KxK' with the product cell-structure and the weak topology

is a GxG'-complex in the obvious way. If ~' and ~' are local

coefficient systems on K and K' respectively then define

KxK ' by (o~: ~ ~')CW) = ,,.~'(~1 w) /~ 5~" ('2 W) where ~1: KxK, § K and =2: KxK' § K' are the projections. The definition of

~ ~' on morphisms is obvious.

Suppose that f& cP(K; ~ *) and f'~ Cq(K';~'). Define

fxf, ~ CP+q(KxK, ; ,f. (T~ ~ ') by

Cfxf')CaxT) = f(o) Df'CT) where o and are Coriented) p and q-cells respectively (fxf, vanishes elsewhere). Cf,f') § fxf, is obviously bilinear.

If ge G and g'e G' then clearly

(gxg,)(fxf,) = g(f) x g,(f,).

It is also clear that 6(fxf,) ffi (6f)xs + (-1)Pfx6f ,. Thus x induces a chain map I"-20

~') § c~cK; x) | c~, oK'; -GxG'c p§ (K~K, ; ) and consequently, a "cross-product":

~ ~'). H~CK; ~) ~ H~,CK';~') § H GxG'p+q (KxK';

If ~ and ~' are simple then so is J~ ~' as the reader can check.

An internal product, the "cup-product" can be derived from the cross-product by means of equivariant diagonal approxi- mations. However, we have not given the necessary background for this since the definition of the cup product is more easily obtained as a consequence of general facts which we shall develop later in these notes.

t 9. Another description of_c_ochains.

We define an element

c (K;Z) e C ~n G by C_n(K;Z)(GIH ) = Cn(KH;z) together with the obvious values on morphisms of ~G" These objects, for n = 0,1,2,..., form a chain complex in the abelian category ~G" We can form the homology Hn(K;Z) = Hn(C,(K;Z))e_ ~G of this chain complex.

Clearly, this is just H (K;Z)(G/H) = H (KH;z) together, again, mn n with the obvious values on morphisms. Similar considerations apply to the relative case.

Let f e Cn(K;M) G where M~ G" Then for an n-cell o,

fCo) ~ MCG/G o). Suppose that oe KH. Then H C G so that we have O an element

M(G/H '+ G/Ger) f(o- ) M(G/H). 1-21

Denote this element by s This map clearly extends to

a homomorphism

(9.1) ~'(G/H): C (KH;z) -*-M(G/H). n

It is easily checked that (9.1) is natural with respect to the morphisms of ~G' so that f: C (K;Z) * M is a natural transfor- --n mation of functors. That is,

(9.2) s Hom(Cn(K;Z ),M) where Hom refers to the morphisms of the abelian category C G" Conversely, suppose we are given an element ~eHom(C__n(K;Z),M ).

Let a be an n-cell of K and regard o as an element of Cn(KGa;Z).

Define

f(o) = f(G/Go)(a) e M(G/Go) so that fe Cn(K;M). Let us check that f is equivariant. Apply- ing the fact that f is natural to the morphism g: G/G go G/gGog'l § G/G ~ of OG, we see that the diagram

G ~(G/Ga) Cn(K a;Z) 9 M(G/G ) O I g. Ig.: G ) Cn(K go Z ) ~o) M(G/Ggo) commutes. Thus fCgo) = ?~(G/Gg~)(go')fg, CfCG/G a)ca)) = g. CfCa)) as claimed.

We have demonstrated an isomorphism

(9.3) C Gn(K;M) ~ Hom(Cn(K;E),M) given by f ~ ~. It is clear that this isomorphism preserves the ~-22 coboundary operators. Thus we may pass to homology and obtain the isomorphism (9.4) H~(K;M) ~ Hn(Hom(~.(K;Z),M)).

Since Hom is left exact on ~G we obtain a canonical homomorphism

n (9 .s) HG(K;M ) § Hom(Hn(K;Z),M ).

It is also easy to check that if K has no (n-1)-cells, so that

C (K;Z) = 0, then (9 5) is an isomorphism (triviality of -n-I " H (K;Z) or even of H (K;Z) for 0 < q < n, is not sufficient -,-I ' --q for this).

Remark. If A is a G-module and M s ~G is the correspond- ing coefficient system as defined in w example 2, then an equivariant homomorphism Cn (K;Z) § A must take Cn (KH;Z)

Cn(K;z)H into AH = M(G/H). Thus it is clear that we have an isomorphism nomz(G) (Cn(K;Z),A) Hom(Cn(K;Z),M) CGn(K;M)

The left hand side is, by definition, the classical equivariant cochain group with coefficients in the G-module A.

10. A spectral sequence.

We shall show that the abelian category ~G contains sufficiently many projectfves and injectives. However, projective resolutions of length one (or even of finite length) do not generally exist, in contrast to the category Abel. Thus instead of a universal coefficient sequence linking homology and cohomology we obtain a spectral sequence. ]~-23

For a set S let F(S) denote the free abelian group based on S. Suppose that S is a G-set. Define an element

F S r C G by P S(G/H) = F(S H) together with the obvious values on morphisms of ~G (see w example 2). For example, if S is the set of n-cells of a

G-complex K which are not in the G-subcomplex L, then

F S = C_n(K,L;Z).

(10.1) Proposition. F S is pro~ective.

Proof. Let

IFs

Y *" a / / A ------~ B "~-----~ 0 8 be a diagram in e G with exact row and with 7 to be constructed.

Let S' c S be a subset containing exactly one element from each orbit of G on S. Given s e S' consider s as an element of G S F(S ) = Fs(G/Gs). Then a(s)~ B(G/Gs). Define y(s)e A(G/Gs) to be any element with 8(7(s)) = a(s). For ge G we let 7(gs) =

g.y(s)~ A(G/Ggs) (where g, = A(g: G/Gg s § G/Gs)). For H C G s

let j denote the projection G/H ~ G/G s. The ~ement s represents

an element of F(S H) = Fs(G/H), namely Fs(J)(s ). We define

u = A(j)u Now y has been defined on a set of

free generators of Fs(G/H ) for every H C G. Thus there is a unique extension to Fs(G/H ) for all H. This extension is clearly

a morphism F S § A with By = a, as claimed.

(10.2) Corollar X. C n(K,L;M)G ~ HOm(C_n(K,L;Z),M)

Proof. The exact sequence

0 -~ C (L;Z) + C (K;Z) + C (K,L;Z) ~ 0 1-24 of projective objects in ~G induces an exact sequence via the

functor Horn(. ,M) and the result follows.

(10.3) Corollary. C n(K,L;M)G is an exact functor of M.

Proof. This is immediate from (10.2).

It follows from (10.3) that an exact sequence 0 § M' §

M § M" § 0 in e G induces a long exact cohomology sequence of

(K,L) .

At the end of this section we shall show that e G con-

tains sufficiently many pro~ectives. In fact if S is the

disjoint union of all of the G-sets G/H for H C G then F S is

a (projective) generator of the category e G. Since e G

obviously satisfies Grothendieck's axiom AB5 (arbitrary direct

sums and exactness of the direct limit functor) it follows by

a result of Grothendieck that e G possesses sufficiently many

invectives (see Mitchell: Theory of Categories).

Let M e e G and let M be an injective resolution of M.

Consider the double complex

Horn (C__, (K, L ; Z), M ).

Standard homological algebra applied to this double complex

yields a spectral sequence with

(10.4) E; 'q = ExtP(.H_q(K,L;Z),M)"-'-:-> HP+q(K,L;M).

(This notation means that E p'q converges to E p'q which is the r eo graded group associated with a filtration of HP+q(K,L;M). Also

Ext p refers to the pth right derived functor of Hom in the

category ~G" ) I-2S

By way of illustration we shall compute ExtP(A,M) in two rather elementary cases.

Example 1. Let A~ ~G be defined by A(G) = Z, with trivial G-operators, and A(G/H) ffi 0 for H ~ {e}. Let F. be a

Z(G)-free resolution of Z. Then F_., defined by F.(G) = F. and

~.(G/H) = 0 for H # {e}, is a projective resolution of A in ~G" Clearly Hom(F.;M)~ HOmz(G)(F.;M(G}) so that ExtP(A,M) ~ HP(G;M(G)), where the right hand side is the classical cohomology of G with

coefficients in the G-module M(G). If K is a connected G-complex

on which G acts freely and such that

H = 0 for 0 < q < N q (K;Z) then in (10 4) we have EP'q-~r M)~.~qHP (G ;M (G) ) for q < N. 2 u0 ' Consequently, we have an isomorphism

HG(K;M ) ~ Hn(G;M(G)) for n < N.

Example .2. Let B be an abelian group and let B_ e ~G

be defined by B(G/H) = B and B(j) ffi 1 for all morphisms j in

~G" Then, if M is an injective resolution of M, we have

Hom(B,M } ~ Hom(B(P),M (P)) = Hom(B,M (P))

where P is the point G/G. M (P) is clearly an injective reso-

lution of M(P) in Abel. Hence

ExtP(B,M) = ExtP(B,M(P))

where the right hand side is Ext in Abel. That is

Ext0(B,M) = Hom(B,M) ~-" Hom(B,M(P))

Ext I(B,M3 ~ ExtCB,M(P))

I ExtP(B_.M) = 0 for p > 1. "[- 2 6

In particular, if B is free abelian then ExtP(B,M) = 0 for p 0, that is, B is projective in ~G if B is projective in

Abel. (Of course, this also follows directly from (10.1) in the case in which G acts trivially on S.)

Let us return to the general discussion. There is an edge homomorphism n HG(K,L;M ) § Horn (H_n(K,L;Z},M) of (10.4} (coinciding with (9.5} when L = ~). Clearly this is

an isomorphism if each Hq(K,L;Z) is projective for q < n.

For example suppose that n 1, that K possesses

stationary points (e.g. k0) and that

q (K,k0) = 0 for q < n. The Hurewicz theorem, applied to each K H, shows that the (obvious}

Hurewicz homomorphism (in eG)

~q(K,k0) § --qH(K;Z)

is an isomorphism for 0 < q < n. Thus

(10.5) Hn(K;M}G ~ Hom(~ n (K,k0),M}

in this case.

We shall now justify our earlier contention that there

are enough projectives in e G. For any G-sets S and T let

E(S,T) denote the set of equivariant maps S § T. For K C G,

the assignment f + f(K) clearly yields a one-one correspondence

E(GIK,S) % SK.

(It is of interest to reconsider the material of w and the

examples of w in this light.) Thus FG/H(G/K ) = F((G/H) K) = F(E(G/K,G/H)). ]:-27

Now if a e H(G/H) the map s § M(s (a) os

E(G/K,G/H) ~ M(G/K)

induces a homomorphism F(E(G/K,G/H)) § M(GIK). This is clearly natural in G/K and hence is a morphism

~a: FG/H § M

in ~G" It is also clear that the generator H/H GFG/H(G/H )

corresponds to 1E E(G/H,G/H) and hence that ~a maps it into a e M(G/H).

We shall now explicitly exhibit a projective which maps onto a given H E e G. For a a M(G/H) let S be a copy os the

G-set G/H and let S(H) = ~ S be the disjoint union of these for all a e M(G/H) and all H C G. Then FS(M) = a~ F S a . The homomorphisms ~a: FS § M yield a homomorphism a (10.6) ~ = ~ ~o a : FS(H) § H which is clearly surjective.