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).