Equivariant Obstruction Theory

II.I

Chapter II. Equivariant Obstruction Theory

In this chapter we shall assume that the reader is reasonably familiar with obstruction theory on CW-complexes.

We shall attempt to strike a reasonable balance between giving no details on the one hand and developing the theory from

scratch on the other by making use of the results, without proof,

of the classical theory.

1. The obst!uction coczcle

In this section n > I will be an integer, fixed throughout

the discussion. Let K be a G-complex and L a G-subcomplex. Let

Y be a G-space. We shall assume, for simplicity, that the set

yH of stationary points of H on Y is non-empty, arcwise connected

and n-simple for each subgroup H C G, (We note here that the

theory could be generalized to relative CW-complexes (K,L)

with no trouble.)

Assume that we are given an equivariant map

9: KnUL § Y. Let o be an (n+l)-ce11 of K and let fo: Sn § Kn

be a characteristic map for a (note that the characteristic maps

may be chosen equivariantly).

The subgroup G leaves K(o), and hence Im f , stationary. 0 O It follows that Go leaves Im(~ofo) stationary. That is, G __f ~ n) = y a. G Thus ~~ defines an element c~(o)~ ~n(Y o), and clearly

c~ (o) = 0 if o is in L. But, with ~n(u defined as in example

(3) of Chap. I, w this defines a cochain

c~ E cn+l(l,L;~n(Y)), II .2

-1 Gg o gGog Now c~ (go) is represented by ~Ofgo: Sn § Y = Y and

~ = ~~176 = go,of o so that c~ (g~) = g#(c~ (a)). This means that c~ is an equivariant cochain (by the defintion of

~n (Y))' that iS

c~c~+l(x,L'',~n(Y)).

It is called the obstruction cochain.

(1.1) Proposition. 6c~ = O.

Proof. Let t be an (n+2)-cell and consider the computation of (6c~)(T). To calculate this, one "pushes" the coeffi- G cients to those on t; that is to ~ (y t), and calculates the n classical coboundary. But c~ restricted to K(z) and with G coefficients pushed to ~ n (y T ) is just the obstruction cochain,

in the classical sense, to extending ~IK n n K(T) to Kn+l~ K(z).

Thus (6c~)(t) = 0 is a fact from the classical theory.

(1.2) Proposition. c~ = 0 s ~ can be extended eaui- variantly to K n+l UL.

Proof. If c cp(O) = 0 then clearly we may extend ~ to G KnL) LDo in such a way that 7(o) c Y o. Define, for geG and xGo, -1 G gGog G cpCgx) = g ~,(x)e gCY o) = y = y go

If gx = g'x then g, = gh for some he G so that g' ~Cx) = g ~C x) U (since ~(x) e YG~ which shows that this definition is valid.

The proof is completed by taking an (n+l)-cell from each orbit of G on the (n+1)-cells and following the procedure above. II.3

Now suppose that ~ and O are equivariant maps Kn UL § Y and let F: (Kn-lu L)xI § Y he an equivariant homotopy between

Ix n-lU L and O[K n-IUL. Define an equivariant map ?#FO: (KxI) n

U(LzI) § Y by

(~o# FO)(x,O) = ~(x)

FO)(x,1) = ~(x) f FO)(x,t) = F(x,t), If ~r = OIK n-I UL and F is the constant homotopy # will denote #F"

The deformation cochain d n ~F,O r (Y)) is defined by

d~,F,@(o) = C~#F@(Oxl) o

It is clear that

(1.3) 6d ~,F,O = c 9 = c~ .

If #F = #' that is if F is constant, then we put d 0 = d F, O.

(1.4) Proposition. Let 9~: KntJL § Y be equivariant and let d SCG(K,L;~n(Y)).n Then there is an equivariant map

O: KnUL § Y, coinciding with cf on Kn-Iu L, such that d~, 0 = d.

Proof. Let ~ be an n-cell of K, not in L, and choose a characteristic map f : (Bn,S n-l) § (Kn,K n-l) for a. Let o G jn = Bnx(o) usn-lxI C Bnxl and define u jn § y o by u =

~(fo(x)). As shown in non-equivariant obstruction theory, u may G be extended to a map u ~(Bn• § y o representing the element G (or any element) d(o)~ ~ (Y o). It is clear that such extensions n may be chosen equivariantly, since d is an equivariant cochain. II .4

Now 0 can be defined by gIK n-1UL ffi I Kn-1 UL and, for an n-cell o and xa o,

O(x) ffi T'(fS 1 (x) ,1).

It is clear that d~, 0 = d.

The cocycle c~ ~ -Gc n+l (K,L;~n(Y)) represents a cohomology

class

[c~]~ H n+l(K,L;~n(y)G ) which depends, by (1,3), only on the equivariant homotopy class

of ~IK n-1U L. Moreover, if [c~ ] ffi 0, then by (1.4) q[K n-lu L

extends to 0: Kn UL + Y such that c 0 = 0 (choose d with 6d = -c~).

Hence, by (1.2), we have the following result:

(1.5) Theorem. Let ~: Kn UL § Y be equivariant. Then

~[ K n-I UL can be extended to an equivariant ma r Kn+lu L § Y

iff [c~] - o.

Remark. Suppose that ~,0: K § Y are equivariant and

that F: (Kn-Iu L)xI § Y is an equivariant homotopy between the

restrictions of ~ and 0 to Kn-Iu L. As above we obtain an

equivariant map ~#FO= (Kn-l• U Q § u where Q ffi (LxI) U (K•

Then the obstruction to extending ~#F 0 to (KnxI) U q is

^n§ C~#F@ ~c G (KxI,LxIU Kx~l;~n(Y)) ~

This group is isomorphic to Cn(K,L;~n(Y))G and this isomorphism

takes C~#FO into d~,F, @ (now a cocycle). II .5

2. Primary obstructions

At various points in this section we shall make one or more of the following assumptions:

(I) yH is r-simple, non-empty and arcwise connected for all r and

H c G (e.g. ~0 (Y) = 0 = ~1 (Y))"

(2) H~+I(K,L;~ (Y)) = 0 for all r < n. r

(3) HG(K,L;~r(Y)) = 0 for all r < n.

r-I (4) H G (K,L;~ (Y)) = 0 for all r < n. r

Numbers appearing in each statement indicate which of these assumptions are used. The results in this section are all easy applications of w to the study of extensions of equivariant maps and homotopies. The proofs will be omitted since they offer no difficulties.

Suppose first that we are given an equivariant map f: L § Y.

(2. I) Lemma. (1,2) There exists an equivariant extension f of f to K n UL. n

(2.2) Lemma. (I,3) If fn and gn are equivariant extensions of f to K nUL then [Cfn ] = [c ]. gn

(Hint : Use (2.1) to find a homotopy fn-i ~ gn-I relative to L.)

(2.3) Definition. (1,2,3) Let y n+l (f) e-GNn+l (K'L;~n(Y)) be the (unique) cohomology class [cf ] for any equivariant n+l n extension fn of f to K nUL. y (f) is called the primary obstruction to. extendin# f and is an invariant of the equivariant homotopy class of f. II .6

(2.4) Proposition. If k: K' § K is cellular and equivariant then ~n+l(f-k) = k *(Tn§ (f)) when this is defined.

(This is also true without cellularity but we have not yet defined k in the general case.)

(2.5) Theorem (Extension). (1,2,3) If we also have

that -GHr§ ffi 0 for n r dim(K-L) then an equivariant

map f: L § Y has an equivariant extension to K iff 7n+l(f) ffi 0.

Now suppose that we are given two equivariant maps

f,g: K § Y such that f[L ffi g[L. These induce an equivariant map

f#g: Q § Y where Q = (KX~I) U (LXI).

There is a natural isomorphism

(2.6) ~: HGn(K,L;~n(Y)) ~-~ NG'n+l(lxI,Q'~n(Y)),

(induced by the obvious isomorphism on the cochain level). We

define, under conditions (1,5,4):

C2.7) n(f,g) = ~-l(Tn+l(f#g))

and note that

n n n C2.8) (f,g) + ~ (g,h) = ~ (f,h)

and

(2.9) ~n(fok,gok) ffi k (n(f,g))

(where k: (K',L') § (K,L) is cellular and equivariant) when this

is defined.

An application of (2.5) to this situation yields: II .7

(2.10) Theorem (Homotop~v~. (1,S,4) If we also have that

H Gr(K,L;~r(Y)) = 0 for n < r < dim(K-L) and if f,g: K § Y are

equivariant with f[L = g[L, then f and g are equivariantly homo-

topic (relative to L) iff n(f,g) = 0.

A standard argument now proves the following result:

(2.11) Theorem (Classification). Assume that (1) holds

and also that ~ ~ r-1 H (K,L;&r(Y)) = 0 = H G (K,L;~r(Y)) fox" r < n

r+l ~Pl (K,L;~r(u 0 H G (K,L;~r(Y)) for r > n,

Let f: K § Y be an equivariant map. Then the equivariant homotopy

classes (relative to L) of maps g: K § Y (with g[L = f[L) are in

one-one correspondence with the elements of

fIG (K, L;~n (Y))

and g § n(g,f) is such a correspondence.

As a matter of notation, we shall use double brackets:

[[X~Y]], where X and Y are G-spaces, to denote the equivariant

homotopy classes of (equivariant)maps X § Y. Thus, for L = ~,

the conclusion of (2.11) states that [[g]] ~§ n(g,f) is a

one-one correspondence

[ [K; Y] ] = H n(K;~n(Y))G II .8

3. The characteristic class of a map

In this section we assume that Y is a G-space with base point YO eYG such that

~q(Y,y0 ) = 0 for q < n,

for a given integer n _> 1. If n = I, we assume that ~l(Y,y0)

(that is, each ~I(YH,y0 )) is abelian.

Let K be a G-complex and let 0 denote the constant

(equivariant) map 0: K § Y0 eY. For any equivariant map

f: K ~ Y we define the characteristic class of f to be

(3.1) X n (f) = ~ n (f,0)~H n(K;~n(y)G )

If k: K' § K is cellular and equivariant then by (2.9)

X n (fok) = k * (xn(f)).

(The cellularity condition is unnecessary as will follow from

later facts.)

The following four results are standard and immediate

consequences of the definitions and of w We shall omit their proofs :

r (3.2) Proposition. If HG(K;~r(Y)) = 0 for r > n then

two maps f,g: K § Y are homotopic iff xn(f) = •

(3.3) Theorem. .If (K,L) is a G-complex pair and f: L § Y

is given with characteristic class • HG(L;~n(Y)),n then the primary obstruction to extending f to K equivariantly is

n+l * n v Cf) = ~ Cx Cf))

where 6 : HG(L;~n(Y))n § Hn+I(K,L;~n(Y))G is the coboundary. II .9

(5.4) Corollary. I_~f H r+I(K,L;G ~r(Y) ) - 0 for r > n then

an equivariant map s L § Y has an equivariant extension to K is

Xn (f)~Im[i * : H G (K;~n(Y)) § H n(L;~n(Y))]G

(3.5) Theorem. If f,g: K § Y are equivariant and if fl L = g l L, then

xn(s xn(g) = j * ( n (f,g)).

(Here xn(f) and xn(g) are in HG(K;~n(Y)),n ms (s is in

* H (K,L;~n(Y)) and j is induced by (K,r § (K,L).)

We conclude this section with some remarks on the case in which Y is, itself, a G-complex. These remarks will not be used in any essential way elsewhere in these notes. The identity

1: Y § Y yields a class

xn(y) = xn(1) = mn(1,0) e H~(Y;~n(Y)), which is the primary obstruction to equivariantly contracting u and is called the characteristic class of Y.

For any f: K § Y we obviously have

n * n C3.6) x Cf) = f Cx CY)).

By Chap. I, (10.5) we have that

n ~ HG(Y;~n(Y)) ~ Hom(~n(Y),~n(Y)) and it can be shown that under this isomorphism Xn(y) corresponds to the identity homomorphism. (Perhaps the easiest way to prove this is to note that Y has the equivariant homotopy type of a

G-complex which has no cells in dimensions between 0 and n, and then to prove the result in this case. See w This is, os course, an important result since it allows the computation of the characteristic class. II.i0

4. Hopf G-spaces

Let Y be a G-space with base point YO" Let G act diago- nally on yxy, that is g(y = (gy Such a space Y together with a base point preserving equivariant map 0: yxy § y

is said to be a Hopf G-space if the restriction Yv Y § Y of 0

is equivariantly homotopic to 1V 1. This obviously implies that yH for H CG is a Hopf-space

For example, if Y is any G-space with base point, then the loop space ~Y is a Hopf G-space where the action of G on

a loop, or generally on a path f: I + Y, is defined by g(f)(t) =

gCfCt)).

Let us denote the product gCy,y') by y~ly' in a given

Hopf G-space Y. Let (K,L) be a pair of G-complexes and let

c: KnUL § y be equivariant, where Y is (also) as in w

We have the map

~: Kn•L § Y

defined by (W ~ ~)(x) = c:(x)~ ~(x). Since addition in the

homotopy groups of a Hopf-space is induced by the Hopf-space G O operation, as is well-known and since each Y is a Hopf-space

it follows immediately that

cctn~ = cc:+ co in CG+I (K,L;~n(Y)).

It follows immediately that in the situation of (3.1),

with Y a Hopf G-space and f,f': K § Y equivariant, we have

(4.1) X n (fro f, ) = X n (f) + x n( f')- II.11

5. Equivariant deformations and homotopy type

In this section we shall prove some elementary facts concerning equivariant deformation. These results could be encompassed in an obstruction theory of deformation (which con- tains the obstruction theory of extensions) but we have chosen not to do so.

Let Y~ B be a pair of G-spaces and assume that

(5.11 ~q(Y,B) = 0 for all 0 ~ q ~ n,

in the sense that, for every subgroup H C G, every map

(Bq,Sq-l) § (yH,BH) is deformable, through such maps, to a map into B H. (We allow the case n = |

(5.2) Lemma. Le_t_t (K,L) be a pair of G-complexes..with dim(K-L) < n and let ~: K,L § Y,B be an equivariant map. Then

is equivariantl~ homoto~s relative to L to a map into B.

Proof. Consider KXl. We wish to extend the map

ILxI ~ ~x(0) on LxI UKx{O) to Kxl such that Kx{l} goes into B.

The extension is defined inductively on the KnxI and proceeds much as in the proof of (1.21. The details are omitted.

As above, double brackets [[XIY]] denote the set of equivariant homotopy classes of equivariant maps X § Y, where X and Y are G-spaces.

(5.3) Corollary. Inclusion i: B § Y induces a one-one correspondence

i#: [[K;B]] ~ [[K;Y]] for every G-complex K with dim K < n. II.12

Proof. i# is onto by (5.2). If f: K § B can be equivariantly deformed, through Y, to g: K § B, then by (5.2) the homotopy may be deformed, relative to the ends, into B. This shows that i# is one-one.

(5.4) Theorem. Let f: Y § Y' be an equivariant map of

G-spaces such that f#: ~q(Y) ~ ~q(Y') for all q ~ 0. Then

f#: [[K~Y]] § [[K;Y']] is a one-one correspondence for every G-complex K.

Proof. Let M be the mapping cylinder of f, with the f natural G-action. M and Y' have the same equivariant homotopy f type so that f may be replaced by the inclusion i: Y ~ Mf. The hypothesis implies easily that fiq(Mf,Y) = 0 for all q ~ O. Thus the result follows from (5.5).

(5.5) Corollary. If of: K § K' is an equivariant map between two G-complexes such that. ~#: ~q(K) ~ ~q (K') for all q ~ 0 then ~ is an equivariant homotopy equivalence.

Proof. ~#: [[K';K]] -L [[K';K']] by (5.4}. Let -i ~: K' § K represent ~# (1). That is, ~ ~: K' § K' is equi- -I variantly homotopic to the identity. Clearly ~# = ~# is bijective so that there is (similarly) a O: K ~ K' with ~8 ~ I

(equivariantly). Then 0 ~ ~0 ~ ~ so that ~ ~ ~ ~0 ~ I as was to be shown.

(5.6) Proposition. Every equivariant map f: K 1 § K 2 between two G-complexes is equivariantly homotopic to a cellular map. An equivariant homotopy between cellular maps may be II,15

deformed equivariantl~, relative to the ends, into a. cellular

homotopy.

Proof. This is an easy consequence of (5.2) using (Y,B) =

(K2,K~).

This result can be used to extend the definition of the

induced cohomology homomorphism of an equivariant map K1 § K2 to arbitrary (non-cellular) maps. Another method of doing this

is given in w

6. Eilenberg-HacLane G-complexes

Let ~ be any element of the abelian category e G . A

G-space of type (~,n) is defined to be a G-space Y with

for q # n ~ =q(Y'Yo) I0~ for q = n, where YO ~ yG ~ ~.

For such a space (2.11) provides a one-one correspondence

(6.1) [[K;Y]] ~ HG(K;~n ) for all G-complexes K, given by [[f]] = xn(f)

(where 0 denotes the constant map K § YO and the notation on the right is from w Moreover, if ~: K * K' is cellular and equivariant then, by (2.9),

[[K'iY]] ~ , H Gn(K';~)

'# (6.2)

[ [K;Y] ] = , H n(K;~)G 11-14 commutes, where ~#([[f]]) = [[fo~]]. Thus (6.1) is a natural equivalence of functors.

Note that if Y is a G-space of type (~,n) then the loop space ~Y (see w has type (~,n-1). This is an immediate consequence of the obvious fact that (~y)H = ~(yH).

If Y is a Hopf G-space then we can define an addition in

[ [K,Y] ] by

(6.3) [[f]] + [[f,]] = [[fur,]].

Then, by (4.1), the correspondence (6.1) preserves addition.

Thus, in this case, if we are given an equivariant map ~: K § K'

(not necessarily cellular) we can define ~ * : HG(K';~)n § Hn(K;~ G ) # by commutativity of (6.2), since W is always defined. The

obvious additivity of ~# implies that ~ is a homomorphism.

Thus, in this way, we can dispense with the definitions of

in Chap. I, w as well as Proposition (5.6) (used to extend the

definition of ~ to non-cellular maps).

We shall now show how to construct a G-complex K of type

(~,n) for any ~e ~ G and n ~ i. We shall restrict our attention,

for convenience onl~ to the case n > i. This is not much loss

of generality since ~K has type (~,i) when K has type (5,2).

The construction is based on the following two lemmas which use

the notation of Chap. I, w 10.

First we shall introduce some further notation. If T is

a G-set, T + is T together with a disjoint base point, SqT + is the th q reduced suspension of T + (that is, the one point union of

q-spheres, one for each member of T), and CSqT + is the reduced II.15 cone of this (that is, the one point union of (q+l)-cells, one for each member of T). Note that there are natural isomorphisms

(for q > 1)

(6.4) F T ~ ~q(SqT +) ~ -qC (SqT+;Z) ~ It--q (SqT+;X) of elements of e G"

(6.5) Lemma. Let q > 1 and let Y be a G-space with base point Y0 and with ~0(Y,y0) = 0 = ~l(Y,y0). Then for any G-set T, the assignment to an equivariant homotopy class [[f]] (of a map f: SqT + § Y) of the induced morphism f#: F T ~ ~q(SqT +) § ~q(Y) in

G is a one-one correspondence

[[SqT+,Y]] ~ Horn (FT,~q(Y))"

In particular t every morphism ~: F T § ~q(Y) in e G is represented by an equivariant map f: SqT + § Y and f is equivariantly extendible to CSqT § § Y iff a = f# is trivial.

Proof. A direct proof of this should be fairly obvious.

However, we note that it is, in fact, a special case of the equivariant homotopy classification theorem (2.11). That is, take

K = SqT + , let L be the base point, and let O: K § Y be the constant map into Y0" The conditions of (2.11) are satisfied for n = q since, in fact, K has no cells in dimensions other than 0 and q.

The classification assigns to an equivariant map f: K § Y the class mq(f,0) in

H~(SqT+;~q(Y)) ~ Hom(~(SqT+;Z),~q(Y))

Hom(~q(SqT +) ,~q(Y)) II.16

(see (9.5) of Chap. I). It is obvious from the definition of n (f,0) that the corresponding homomorphism ~ (SqT § § ~ (Y) is q q precisely the induced map f#.

(6.6) Lemma. Let q > 1 and let Y be a G-space with base point Y0 and with ~0(Y,y0) = 0 = ~l(Y,y0). Let f: SqT + § Y be an equivariant base point preserving map and let Y' = Y LAf CSqT + be the (reduced) mapping cone of f with the obvious G action.

Let i: Y § Y' denote the inclusion. Then we have the following facts:

(1) i#: ~ (Y) § ~ (Y') is an isomorphism for r < q. r r .... (2) i#:~ (Y) § ~ (Y') is an epimorphism with Kernel i q q # Im{f#: F T ~ ~ (SqT +) ~ ~ (Y)}. q q

H Proof. For H C G it is clear that Y' is just the mapping yH § H cone of (SqT+) H § . But (SqT+) H = Sq(T H) Thus i#(G/H): ~(Y

§ nr(Y 'H) is induced by the inclusion of y}t in the mapping cone yH. of the restriction of f: Sq(TH) § § Similarly f#(G/H): ~r((SqT+) tt) (yH) is induced by the restriction of f ~r Since (1) and (2) are true iff the corresponding statements for the values on each G/H E ~ C are true, and since these corresponding statements are known results (see Hu, Homotopy Theory, p. 168) concerning (non-equivariant) attaching of cells, the lemma follows.

Using these two lemmas, the construction of K(~,n) complexes is now quite straightforward. Thus let T and R be G-sets such that there is an exact sequence

FR 0 in ~G (see Chap. I, w Let n > 1 and put Kn = SnT + Let 11.17

f: SnR + § SnT + be an equivariant map inducing ~ (via F T ~n (SnT+ ), etc.).

This exists by (6.5). Let K n+l - K n Uf CSnR +. By (6.6) we have

I ~n(Kn+l) ~

(K n+l) = 0 for r < n. r If K q has been constructed to be a G-complex of dimension q

(q > n + 1) such that

n(K q) ~ C6.7) ~ "'lKqj = 0 for r < n and n < r < q r let V be a G-set such that there is an epimorphism

FV 2C+ ~ (Kq). q

Let v: sqv + * K q be an equivariant map inducing y and let K q+l =

Kq~vsqV + . Then, by (6.6), K q+l satisfies (6.7) with q replaced by q + I. Let K = U K q . This is clearly a G-complex of type q

7. n-connected G-complexes

The method of killing the groups ~ used in the construction q of K(~,n) in the last section is, of course, an important tool.

We shall use it here in a rather straightforward way to prove the following result:

(7.1) Proposition. Let K be a G-complex with ~q(K) = 0 for all 0 ~ q < n. Then K has the same equivariant homotop~ type as a G-complex with no cells in dimensions q for 0 < q < n. II.18

Proof. Let L = Kn-1. Then the inclusion L * K is equivariantly homotopic to a constant map, by (2.10). That is, K is an equivariant retract of KU C But KUC L has the same equi- L. variant homotopy type as K/L. Thus there exist equivariant maps

K cr K/L ~ ~ K with ~ equivariantly homotopic to I. Clearly K/L has no q-cells for 0 < q < n so that ~ (K/L) = 0 for q < n. q Suppose that for some q ~ n we have constructed a G-complex

K ~ K/L and an equivariant map ,q: K § K with ,q~ = * W such q q that (*q)#: ~r(Kq) § ~r(K) is a monomorphism for r < q. Let T be a G-set and

a: F T + Ker{(~q)#: ~q(Kq) § ~q(K)} an epimorphism in C G. Let

f: SqT + § K q be an equivariant map inducing a. f may be assumed to be cellular by (5.6) (or merely because ~q(K~) § ~q(Kq) is an epimorphism).

Let Kq+l = Kq Uf CSqT +. By (6.5), ~q extends to ~q+l: Kq+ 1 § K and, by (6.6), (Oq+l)#: ~r(Kq+l) § ~r(K) is a monomorphism for r --< q . Let (K' ,, ' ) be the union of the (K q , ~q ).

Thus we obtain a G-complex K' ~ K/L with no q-cells for

0 < q < n and equivariant maps

K' with ~'~ = ~ ~ I. Also

* being a monomorphism with ~ ~# = 1, must be an isomorphism and it follows from (B.5) that K and K t have the same equivariant homotopy type.