3. G-CW COMPLEXE S
13
and
(2.3) G2T{X, K) s 3T{X/G, K)
for a based spac e K an d a based G-spac e X. Similarly , fo r a base d G-spac e X
and a based if-spac e F , w e have
(2.4) G3T{G+ AH y , X) s ff^(Y, X)
and
(2.5) i f 9[Xy Y) * G^(X , i ^ ( C
+
, y)) ,
where
FH{G+,
y ) = Map#(G , F) wit h the trivial map as basepoint, an d we have
G-homeomorphisms
(2.6) G
+
AHX^(G/H)+AX
and
(2.7) F„(G
+
,X)*F(G/H+,X).
A base d homotop y betwee n base d G-map s X Y i s give n b y a base d G -
map X A1+ y . Her e the based cyhnde r X A 2+ i s obtained from X x / b y
collapsing the line through the basepoin t o f X t o the basepoint . Ther e results a
homotopy category hG&, an d we construct hG& b y formally inverting the weak
equivalences. W e writ e [X,y] o = hG^(X,Y); whe n X i s a G-C W complex ,
this i s the se t o f homotopy classe s of based G-map s X Y.
In eithe r th e base d o r th e unbase d context , cofibration s ar e define d b y th e
homotopy extensio n propert y an d fibrations ar e define d b y th e coverin g homo -
topy propert y exactl y a s in the nonequivarian t context , excep t tha t al l maps i n
sight ar e G-maps . Thei r theor y goe s through unchanged . A based G-spac e X i s
nondegenerately base d i f the inclusio n {* } X i s a cofibration . Fo r a base d
G-map / : X Y, w e write Ff an d Cf fo r th e homotop y fiber an d cofibe r o f
/ . Thu s Ff = {(u,x)\w E PY an d w(l) = f(x)} an d Cf = Y U / CX
f
wher e
CX X A I i s the reduce d con e on X.
3. G-C W complexe s
A G-C W comple x X i s th e unio n o f su b G-space s X n suc h tha t i s a
disjoint unio n o f orbit s G/H an d X
n+l
i s obtaine d fro m X
n
b y attachin g G -
cells G/H x
Dn + 1
alon g attaching G -map s G/H x
Sn

Xn.
Suc h an attachin g
map is determined by its restriction S n (X n ) H , an d this allows the inductiv e
analysis o f G-C W complexe s b y reductio n t o nonequivarian t homotop y theory .
Subcomplexes an d relativ e G-C W complexe s ar e define d i n th e obviou s way . I
will review my preferred wa y of developing the theory o f G-CW complexe s sinc e
this wil l serv e a s a mode l fo r othe r version s o f cellula r theor y tha t w e shal l
encounter.
We begi n wit h th e Homotop y Extensio n an d Liftin g Property . Recal l tha t
a ma p / : X Y i s a n n-equivalenc e i f 7r
q
(/) i s a bijectio n fo r q n an d
a surjectio n fo r q n (fo r an y choic e o f basepoint) . Le t v b e a functio n fro m
conjugacy classe s o f subgroups o f G t o th e integer s 1. W e sa y tha t a ma p
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