LUSTERNIK-SCHNIRELMANN 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 Eckmann-Hilton, the usual notion of an H-space 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 CW-complexes, 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 eW-complexes with mapping cone
ef. Then
LS-cat
e
f ::;
LS-cat 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 homotopy-equivalent.
If
we assume that LS-cat = 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 LS-cat ef::; n
+
1, so that
LS-cat
e
f ::;
LS-cat Y
+
1.
D
COROLLARY 2.5. If X is a connected eW-complex then LS-cat X ::; dim X+
1.
This corollary, whose statement was essentially known to Lusternik-Schnirel-
mann, may be generalized to
THEOREM 2.6. If X is an (r- I)-connected CW-complex, r;:::: 1, then
cat X
~dimX +
1.
-r
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