12 I . EQUIVARIANT CELLULA R AN D HOMOLOG Y THEOR Y

and

(1.3) G%{X,K) £ * &(X/G,K).

If Y is an if-space , ther e i s an induce d G-spac e G x

H

Y. I t i s obtained fro m

G x 7 b y identifyin g (gh, y) wit h (# , hy) fo r g € G, ft € if, and y £ F . A bi t

less obviously, w e also have the "coinduced " G-spac e Map # (G,y), whic h i s th e

space o f if-map s G — Y with lef t actio n b y G induce d b y th e righ t actio n o f

G o n itself , (g • f){g') = f{gfg) Fo r G-space s X an d if-space s y , w e have th e

adjunctions

(1.4) GW{G x

H

Y,X) * H&{Y,X)

and

(1.5) H&{X,Y) s G«T(X,Map H(G,y)).

Moreover, fo r G-space s X , w e have G-homeomorphism s

(1.6) Gx HX^(G/H)xX

and

(1.7) Map/ f (G,X) £ * Map(G/if,X).

For th e first, th e uniqu e G-ma p G xH X — (G/H) x X tha t send s x G l t o

(eH,x) ha s inverse that send s (gH,x) t o th e equivalence class of (g^g~

1x).

A homotopy between G-maps X — y is a homotopy ft : X xl — y that i s

a G-map , where G acts trivially on J. Ther e results a homotopy categor y hGW.

Recall that a map o f spaces i s a weak equivalenc e i f it induce s a n isomorphis m

of all homotopy groups . A G-map / : X —* Y is said to b e a weak equivalenc e

if fH : XH — • YH i s a weak equivalenc e fo r al l if C G . W e let ftG^T denot e

the categor y constructe d fro m hGW b y adjoinin g forma l inverse s t o th e wea k

equivalences. W e shall b e mor e precis e shortly . Th e algebrai c invariant s o f G -

spaces that w e shall b e interested i n will be defined o n the categor y hG^f.

General Reference s

G. E . Bredon . Introductio n t o compac t transformatio n groups . Academi c Press . 1972.

T. tor n Dieck . Transformatio n groups . Walte r d e Gruyter . 1987.

(This referenc e contain s a n extensiv e Bibliography. )

2. Analogue s fo r base d G-space s

It will often b e more convenient to work with based G-spaces . Basepoint s ar e

G-fixed an d are generically denoted by *. W e write X+ fo r the union of a G-spac e

X an d a disjoint basepoint . Th e wedge , o r 1-point union , o f based G-space s i s

denoted by XvY. Th e smas h produc t i s define d b y X A Y = X x Y/X V Y.

We write F(X iY) fo r th e base d G-spac e o f based map s X —• Y. The n

(2.1) F(X A Y, Z) * F(X, F(Y, Z)).

We write GST for th e categor y o f based G-spaces , an d w e have

(2.2) G3T{K,X) s 2T{K,X G)