Contemporary Mathematics

Volume 181, 1995

On Finiteness of Subgroups of Self-Homotopy Equivalences

Martin Arkowitz and Gregory Lupton

Section 1 - Introduction

If

X is a topological space, we denote by £(X) the set of homotopy classes

of self-homotopy equivalences of X. Then £(X) is a group with group operation

given by composition of homotopy classes. In this paper we investigate the fol-

lowing natural subgroups of £(X) : (i) £.(X), the subgroup of homotopy classes

which induce the identity on integral homology H.(X), (ii) £#(X), the subgroup

of homotopy classes which induce the identity on homotopy groups 1r#(X), (iii)

£.#(X)= £.(X)

n£#(X).

We establish some basic general results which give con-

ditions under which one or other of these subgroups is finite or infinite. We also

give some specific finiteness results when

X

is a homogeneous space of the form

X

=

U(n)/(U(n

1

)

x · · · x U(nk)) and when X is a product of spheres. Our ap-

proach uses methods from rational homotopy theory. We first prove results about

the minimal model counterpart of £(X) and then translate these into corresponding

statements about the subgroups of £(X).

For a space X, the groups £.(X) and £#(X) have been studied extensively.

For £.(X) we cite the work of

[Wi], [Ok],

[Sa1],

[Sa2]

and

[Za].

For £#(X) we cite

[A-C], [D-Z], [Ts1],[Ts2], [Ts3]

and

[Di].

In particular, Maruyama has obtained

1991 Mathematics Subject Classification. Primary 55P10, 55P62, 55835. Sec-

ondary 55P60.

©

1995 American Mathematical Society

0271-4132/95 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/181/02026

Volume 181, 1995

On Finiteness of Subgroups of Self-Homotopy Equivalences

Martin Arkowitz and Gregory Lupton

Section 1 - Introduction

If

X is a topological space, we denote by £(X) the set of homotopy classes

of self-homotopy equivalences of X. Then £(X) is a group with group operation

given by composition of homotopy classes. In this paper we investigate the fol-

lowing natural subgroups of £(X) : (i) £.(X), the subgroup of homotopy classes

which induce the identity on integral homology H.(X), (ii) £#(X), the subgroup

of homotopy classes which induce the identity on homotopy groups 1r#(X), (iii)

£.#(X)= £.(X)

n£#(X).

We establish some basic general results which give con-

ditions under which one or other of these subgroups is finite or infinite. We also

give some specific finiteness results when

X

is a homogeneous space of the form

X

=

U(n)/(U(n

1

)

x · · · x U(nk)) and when X is a product of spheres. Our ap-

proach uses methods from rational homotopy theory. We first prove results about

the minimal model counterpart of £(X) and then translate these into corresponding

statements about the subgroups of £(X).

For a space X, the groups £.(X) and £#(X) have been studied extensively.

For £.(X) we cite the work of

[Wi], [Ok],

[Sa1],

[Sa2]

and

[Za].

For £#(X) we cite

[A-C], [D-Z], [Ts1],[Ts2], [Ts3]

and

[Di].

In particular, Maruyama has obtained

1991 Mathematics Subject Classification. Primary 55P10, 55P62, 55835. Sec-

ondary 55P60.

©

1995 American Mathematical Society

0271-4132/95 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/181/02026