LUSTERNIK-SCHNIRELMANN CATEGORY IN HOMOTOPY THEORY
3
In Section 3 we show that if
f :
X
------
Y is actually a homomorphism (for
LS-cat:::; n) in the sense of (1.4), then
(1.6) LS-cat C 1 :::; LS-cat Y,
and we may, in fact, furnish C 1 with a structure map such that the canonical
map k: Y
------
C 1 is a homomorphism. In fact, the conclusion follows if we weaken
the hypotheses by merely requiring that
f
be a quasi-homomorphism; this puts no
restriction on LS-cat X.
In Section 4 we give some important properties of spaces
X
with LS-cat
X :::;
n;
and in Section 5 we discuss two modifications of the notion of category, namely,
weak category and strong category. Finally, in Section 6, we consider notions dual
to category and its modifications.
Throughout this article we are much concerned to find accessible examples to
demonstrate the significance of the ideas introduced. Proving the facts about these
examples is usually not easy; and it is, perhaps, the pervading difficulty of carrying
out calculations involving LS-category that led to some loss of popularity for the
notion among homotopy theorists (until recent years). The problems that remain
open are really hard!
It is a pleasure to have this opportunity to commend the survey article pub-
lished by loan James in 1978
[Ja].
It remains the best source of information about
what is known- although James makes plain that his survey is not comprehen-
sive- thus, for example, it does not contain the quasi-homomorphism theorem of
Section 3. The bibliography accompanying [Ja] is especially valuable.
Finally, we adopt, in this paper, the custom of not always distinguishing nota-
tionally between a map and its homotopy class.
2. Definitions of the LS-category
Let us place ourselves firmly in the category of CW-complexes. We first repeat
the original definition of LS-category (abbreviated, as we have said, to LS-cat), as
applied to objects of this category.
DEFINITION
2.1. Let
X
be a connected CW-complex. Then we say that
LS-cat X :::; n
if X may be covered by
n
open sets
U1, U2, · · ·,
Un such that, for each i, 1 :::; i:::;
n,
the embedding map
ji : Ui
~
X
is nullhomotopic.
Our first step is to observe that, using the CW-topology, we may show that we
may replace the covering of X by n open sets by a covering by n subcomplexes.
Moreover, we may furnish X with a base point* EX and we may work henceforth
in the pointed category of CW -complexes (and pointed maps). Thus we have the
equivalent definition
DEFINITION
2.2. Let
X
be a connected pointed CW-complex. Then we say
that
LS-cat
X :::; n
(sometimes even abbreviated to cat
X:::;
n)
if
X
may be covered by n (pointed) subcomplexes
K
1 ,
K2, · · ·, Kn
such that, for
each i, 1 :::; i :::;
n,
the embedding
ji:
Ki
~
X
is nullhomotopic (in the pointed
sense).
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