CHAPTER I
Equivariant cellula r an d homolog y theor y
1. Som e basi c definition s an d adjunction s
The object s o f stud y i n equivarian t algebrai c topolog y ar e space s equippe d
with an action by a topological group G. Tha t is , the subject concern s spaces X
together wit h continuou s action s G x X X suc h tha t ex x and g{g lx) =
(ggf)x. Map s / : X Y ar e equivarian t i f f{gx) = gf(x). W e the n sa y
that / i s a G-map. Th e usua l construction s o n space s appl y equall y wel l in
the categor y G^o f G-space s an d G-maps . I n particula r G acts diagonall y o n
Cartesian product s o f space s an d act s b y conjugatio n o n th e spac e Map(X , Y)
of maps fro m X t o Y. Tha t is , we define g / b y (g f)(x) gf{g~
1x).
As usual , w e tak e al l space s t o b e compactl y generate d (whic h mean s tha t
a subspac e i s closed if its intersectio n wit h eac h compac t Hausdorf f subspac e i s
closed) an d wea k Hausdorf f (whic h mean s tha t th e diagona l I c l x l i s a
closed subset , wher e th e produc t i s given th e compactl y generate d topology) .
Among other things , thi s ensure s that w e have a G-homeomorphism
(1.1) Map( X x Y,Z) * Map(X,Map(Y,Z) )
for an y G-space s X , F , an d Z.
For us , subgroup s o f G are assume d t o be closed . Fo r H C G, w e writ e
XH
= {x\hx = x for h G H}. Fo r x G X, Gx = {h\hx = x} is called th e
isotropy grou p o f x. Thu s x G X
H
i f H is contained i n Gx. A good dea l
of th e forma l homotop y theor y o f G-space s reduce s t o th e ordinar y homotop y
theory o f fixed point spaces . W e le t NH be th e normalize r o f H in G and
let WH = NH/H. (W e sometime s writ e N
G
H and W
G
H.) Thes e "Wey l
groups" appea r ubiquitousl y i n th e theory . Not e tha t X H i s a ^.ff-space. I n
equivariant theory , orbit s G/H play th e rol e o f points , an d th e se t o f G-map s
G/H G/H can b e identifie d wit h th e grou p WH. W e als o hav e th e orbi t
spaces X/H obtaine d b y identifyin g point s o f X i n th e sam e orbit , an d thes e
too ar e Wif-spaces . Fo r a space K regarded a s a G-space wit h trivial G-action ,
we have
(1.2) GW{K,X)^W(K,X G)
l i
http://dx.doi.org/10.1090/cbms/091/02
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