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
Previous Page Next Page