2 PETER HILTON
(existence of two-sided identity, existence of two-sided inverses, associativity), which
may be regarded as being responsible for inducing a natural group structure, for
any space Z, in the set of homotopy classes [EY, Z].
Now a very significant result of George Whitehead [Wh] showed that, for con-
nected CW-complexes X, the property LS-cat X :::; n is equivalent to the assertion
that X admits a structure map of a certain kind. One furnishes X with a base point
*•
and considers the subspace Tn X of the nth Cartesian power xn, consisting of
those points
(x1,x2,· · ·,xn), Xi
EX, for which
Xi=*
for some
i,
1:::;
i:::;
n.
We follow Ganea in calling Tn X the fat n-fold wedge of n copies of X, and let
j :
Tn X
.......+
xn be the inclusion. Then Whitehead showed that cat X :::; n if, and
only if, the diagonal map
~:
X ---+ xn can be deformed down
2
into Tn X. An
equivalent formulation which suits our purposes better is to say that cat X :::; n if
and only if X admits a structure map
JL:
X ---+ Tn X such that
j
JL
~ ~:
X ---+ xn;
and even to regard the statement 'cat X :::; n' as furnishing X with such a struc-
ture map JL, so that a space X with LS-catX:::; n is really a pair (X,JL). Notice
that the structure map JL generalizes the structure map JL for a suspension space
EY in the sense that the condition jJL
~ ~
reduces, in the case
n =
2, to the
first condition satisfied by
JL:
EY --+ EY V EY, which guarantees that the induced
multiplication on [EY, Z] has a two-sided identity. Moreover, it should be borne in
mind that the properties of
JL:
EY
~
EY V EY are dual to those of the structure
map (loop-composition)
JL:
OZxOZ--+ OZ for the loop-space OZ. Indeed, [EY, Z]
and [Y, OZ] are naturally isomorphic groups, the isomorphism being induced by the
adjunction
(1.3)
ry:
[EY,Z]---+ [Y,OZ].
Thus it seems reasonable to speculate that the surge of interest in LS-category
among homotopy theorists in the 1960's sprang from the Whitehead characteriza-
tion. Indeed, among the first questions to receive attention from algebraic topolo-
gists at that time were (a) whether all spaces X with LS-cat X :::; 2 were suspensions
and (b) whether all homomorphisms between suspensions were suspension maps.
Here a map
f
between two spaces X, Y with LS-cat :::;
n
would be called a homo-
morphism if the diagram
(1.4)
homotopy commutes. Notice that, prior to Whitehead's contribution, the notion
of a homomorphism was effectively inconceivable. We will, in fact, give counterex-
amples to the propositions enunciated above, in Section 5.
In Section 2 we give the original definition of LS-category and describe the
passage to George Whitehead's definition. We also show that, if
f:
X ---+ Y is a
map with mapping cone C
f,
then
(1.5) LS-cat C
f :::;
LS-cat Y
+
1.
2
The paper by Berstein and Ganea
[BG],
which appeared in 1961, is often quoted for this
result, and for many interesting applications; but there is no doubt that the key idea was to be
found in Whitehead's 1956 paper
[Wh].
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