4

PETER HILTON

We next observe that, since the pair

(X, Ki)

satisfies the homotopy extension

property, we may replace the condition

ji

~

0:

Ki

---+

X

by the condition that

the identity map Id:

X

---+

X

be homotopic to a map

ki : X

---+

X

such that

ki(Ki)

=

*·

We are now in a position to state, and justify, George Whitehead's characteri-

zation of LS-cat (see [Wh]). Given the CW-complex

3 X,

let

xn

be the topological

product of n copies of

X

and let

Tn X

be the subcomplex of

xn

consisting of those

points (x1, x2, · · ·,

Xn)

such that

Xi=*

for some i, 1 ::; i::; n. We will follow Ganea

in referring to

Tn X

as the fat wedge of n copies of

X.

Then Whitehead showed

THEOREM 2.3.

Let X be a pointed, connected CW-complex. Then LS-catX ::;

n if and only if the diagonal map

~

: X

---+

xn may be deformed to a map J.L

sending X into Tn X.

PROOF. Let

Pi: xn

---+

X

project

xn

onto its ith factor

X.

Then

if~ ~

J.L: X

---+

xn,

ld=

Pi~ ~

PiJ.L: X

---+

X.

We may assume

J.L

cellular and define

Ki

= J.L-l

xi,

where

xi

is the subcomplex of

xn

given by

Xi

=

*·

Since

J.L

maps

X

into the fat wedge

TnX,

it follows that

X=

U~=l

Ki;

moreover,

PiJ.L

maps

Ki

to

*·

Thus LS-cat X ::;

n

according to Definition 2.2.

Conversely, let there be given homotopies

Id~

ki: X

---+

X

with

ki(Ki)

=

*•

where

X

=

U~=l

Ki·

Then these homotopies combine to yield a homotopy

~ ~

J.L: X---+ xn;

and, if

x

E

Ki,

then

J.L(x)

E

Xi

~

Tn X,

so that

J.L(x)

E

Tn X

for

all x

EX. 0

It was surely Whitehead's Theorem 2.3 which stimulated the interest of ho-

motopy theorists in LS-cat. Of course, it is an easy consequence of Definition 2.1

that LS-cat is a homotopy invariant, but Whitehead's Theorem placed the notion

of LS-cat firmly in the domain of the theory of

spaces with structure maps.

For, if

we now regard

J.L

in Theorem 2.3 as a map

J.L=

X

---+

Tn

X, then LS-cat X ::;

n

if

there exists such a map

J.L

giving rise to a homotopy commutative diagram

X

xn

(2.1)

jj '

rnx

and we think of spaces

X

with LS-cat

X ::; n

as pairs

(X, J.L).

Viewed in this way,

spaces with LS-cat X::; n are direct generalizations of co-H-spaces [HMR], which

are given by diagram (2.1) with n = 2; if X is a co-H-space, it carries a structure

map

J.L: X---+ XV X,

the wedge of two copies

X,

such that

j J.L

~ ~

: X

---+

X

x

X.

Thus non-contractible

co-H-spaces are precisely spaces of LS-category 2.

We may now introduce the idea of a (homotopy-)homomorphism of spaces

with LS-cat::;

n.

If

(X, J.Lx)

and (Y,

J.LY)

are two such spaces, then

f: X

---+

Y is

a homomorphism if the diagram

(2.2)

X

J.Lx!

rnx

3

Henceforth we will always assume

X

to be connected and pointed.