Contemporary Mathematics

Volume 316, 2002

Lusternik-Schnirelmann Category in Homotopy Theory

Peter Hilton

Dedicated to the memory of Israel Berstein and Tudor Ganea

1.

Introduction

The notion of Lusternik-Schnirelmann category (which we will usually abbrevi-

ate here as LS-cat) first appeared in a paper by Lusternik and Schnirelmann

[LS]

in 1934. It was a property of smooth manifolds related to the minimum number of

singularities of certain special functions defined on those manifolds. It was certainly

not difficult to see that the LS-category could be extended to arbitrary spaces (so

surely to CW-complexes) and was a homotopy invariant. However, its nature and

its purpose were rather far removed from the usual interests of homotopy theorists.

In particular, it was a numerical invariant, not a functor (thus very different from,

say, the homology groups and homotopy groups), and so did not seem likely to be

responsive to the methods of study which had proved so powerful in connection

with the algebraic invariants of homotopy type.

A connection with the familiar concerns of homotopy theorists is, however,

furnished by the observation that spaces X with LS-cat X

=

2 (we are here nor-

malizing so that spaces of LS-category 1 are simply the contractible spaces) include

all the suspension spaces

~Y.

Thus, if

Y

is a CW-complex with base point

yo,

then

(1.1)

~y

=

Y

x

I/ (Y

x

j) U (Yo

x

I),

with the obvious base point. Then we may enlarge the 'northern and southern

hemispheres'

Y

x [0,

~]

and

Y

x

[~,

1] to open sets which are, indeed, not merely

contractible in

~y

but contractible in themselves. Moreover, suspension spaces are

provided with a crucial structure map

f-l:

~y

-+

~y

V

~Y,

given by

0::::; t::::;

~

~

::::; t::::;

1,

(1.2)

( )

{

((y,2t),*)

f-ly

t -

' - (*,(y,2t-1))

here* stands for the base point and

~YV~Y

is the one-point union of two copies of

~y,

regarded as embedded in

~y

x

~y

in the obvious way. The map

f-l

satisfies, up

to homotopy, 3 axioms dual (in the sense

1

of Eckmann-Hilton) to the group axioms

2000

Mathematics Subject Classification.

Primary 55M30; Secondary 55P30, 55P99.

1

This is the sense in which we will talk of dual concepts throughout this paper.

@2002 American Mathematical Society

http://dx.doi.org/10.1090/conm/316/05491