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].