2
GEORGE W . WHITEHEA D
right formulatio n i s no t ye t clear . On e suc h theore m ha s bee n foun d b y D . S . Kah n i n
his thesi s (unpublished) . Fo r a genera l discussio n o f thes e problems , th e reade r i s re -
ferred t o Adams ' Seattl e lecture s [7] .
1. Preliminarie s
In wha t follows , al l space s wil l b e assume d t o b e provide d wit h base-points , an d
all map s an d homotopie s t o preserv e th e base-point ; th e base-poin t o f an y spac e wil l
be denote d indiscriminatel y b y a n asterisk . Moreover , base-point s o f CT-complexe s
are assume d to b e vertices . Categorie s o f space s an d map s withou t base-poin t (her e
termed "fre e s p a c e s " an d "fre e maps" ) wil l b e assume d t o b e imbedde d i n th e corre -
sponding base d categorie s b y th e devic e o f adjoinin g a n externa l base=point .
Let S
Q
b e th e categor y o f compactl y generate d Hausdorf f space s [68 ] havin g th e
homotopy typ e o f a CJF-complex . I f X, A ar e object s o f S
Q
wit h A a close d subspac e
of X, th e pai r {X, A) i s sai d t o b e admissible i f an d onl y i f th e inclusio n ma p o f A
into X is in th e categor y S
Q
(i.e. , X an d A hav e th e sam e base-point ) an d i s a
cofibration. I t i s easil y see n tha t
(1.1) / / A is the base-point of X, then (X, A) is admissible.
(1.2) If (Z , A) is an admissible pair in S
Q
, then there is a CW-pair {K
7
L) such
that {X, A) and {K, L) have the same homotopy type.
Let {Xj A) b e a n admissibl e pai r i n S
Q
. The n th e quotien t spac e X/A i s th e
identification spac e obtaine d fro m X b y collapsin g A t o a point , th e base-poin t o f
X/A; X/A belong s t o §
0
.
Recall tha t th e categor y S ha s (finite ) product s an d sums ; i f X, Y € S
Q
, the n
their categorica l produc t X x Y i s th e cartesia n produc t suitabl y retopologized . More -
over thei r su m X V Y = X x {* ! \J {* ! x Y i s a subspac e o f X x Y , an d th e pai r
(X x Y , X V Y ) i s admissible . Henc e thei r reduced join X A Y = X x Y/X V Y i s de -
fined an d belong s t o S
Q
. Moreover , th e reduce d joi n functor i s defined ; map s
/: X -+X'
9
g: Y -+Y' induc e a m a p / A g : X A X ' _ Y A Y ' wit h th e usua l
properties. On e o f th e advantage s o f workin g i n th e categor y S i s
(1.3) The reduced join functor is commutative and associative, up to natural
homeomorphism.
Let X, Y , Z € S
0
, an d le t i^ X - * X V Y , i^ Y - X V Y b e th e natura l inclu -
sions. The n i
x
A 1: X A Z - » (X V Y ) A Z an d %
2
A 1: Y A Z — {X V Y ) A Z induc e
a ma p
h-. U A Z ) V ( y A Z ) - , ( i v y ) A Z
and i t i s easil y verifie d tha t h i s a homeomorphism . Henc e
(1.4) TA e operation of reduced join is distributive over that of addition, up to
