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.
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