LUSTERNIKSCHNIRELMANN CATEGORY IN HOMOTOPY THEORY
5
is homotopy commutative, (Tn
/)f..Lx
~
f..LY
f.
We note that, in the case n = 2, this definition precisely dualizes, in the sense
of EckmannHilton, the usual notion of an Hspace homomorphism.
Of course, we do not abandon Definition 2.1 (or Definition 2.2) with the disco
very of the Whitehead criterion. Thus, for example, the earlier definition is certainly
the most convenient in observing what happens when we attach a cone to a space.
This we now discuss.
Let f: X
t
Y be a map of CWcomplexes, and let ex be the cone on
X, that is, the space obtained from X xI by identifying (X x 0)
U
(*
xI) to
the basepoint. We then form the mapping cone of
f,
also described as the cofibre
off, by taking the disjoint union of ex and y and identifying (x, 1) with f(x),
x E X. We write
e
f
or Z for the mapping cone of
f
and k: Y
"+
Z for the natural
embedding of Y in the mapping cone. Note that (i) iff is a cofibration, so that we
may regard
X
as a subcomplex of
Y
then, by identifying ex to a point, we obtain
a homotopy equivalence h: Z  Y /X giving rise to a commutative diagram
y
(2.3)
k

q",.
z
!h
Y/X
where
q
is the obvious projection onto the quotient space; and
(ii)
k: Y
"+
Z is
always a cofibration, so
(2.4)
ek
~
Z/Y
=EX, the suspension of
X.
We now prove
THEOREM 2.4. Let f: X  Y be a map of eWcomplexes with mapping cone
ef. Then
LScat
e
f ::;
LScat Y
+
1.
PROOF. In the mapping cone ef, slice the cone ex at the level t =
~'
t E I.
Then the part of eX given by t
~
is a contractible open set of e f; and the part of
e f, consisting of Y and the part of eX given by t
~,
may be flattened down onto
Y,
to which it is therefore homotopyequivalent.
If
we assume that LScat = n, so
that Y may be covered by n open sets contractible in Y, then
e
f
may be covered
by n open sets contractible in
e
f,
together with the given contractible open set of
ex given by t
~·
Thus LScat ef::; n
+
1, so that
LScat
e
f ::;
LScat Y
+
1.
D
COROLLARY 2.5. If X is a connected eWcomplex then LScat X ::; dim X+
1.
This corollary, whose statement was essentially known to LusternikSchnirel
mann, may be generalized to
THEOREM 2.6. If X is an (r I)connected CWcomplex, r;:::: 1, then
cat X
~dimX +
1.
r