Title: Subgroups of the Group of Self-Homotopy Equivalences
Authors: Martin Arkowitz, Gregory Lupton and Aniceto Murillo.
Classification Nos. (1991): Primary 55P10; Secondary 55P62, 55Q05.
Addresses: Department of Mathematics, Dartmouth College, Hanover
NH 03755 U.S.A.
Department of Mathematics, Cleveland State University, Cleveland
OH 44115 U.S.A.
Departmento de Algebra, Geometria y Topologia, Universidad de
Malaga, Ap. 59, 29080 Malaga, Spain
e-mail Addresses: Martin.Arkowitz@Dartmouth.edu
Lupton@math.csuohio.edu
Aniceto@agt.cie.uma.es
Abstract: Denote by $\mathcal{E}(Y)$ the group of homotopy classes
of self-homotopy equivalences of a finite-dimensional complex $Y$.
We give a selection of results about certain subgroups of
$\mathcal{E}(Y)$. We establish a connection between the Gottlieb
groups of $Y$ and the subgroup of $\mathcal{E}(Y)$ consisting of
homotopy classes of self-homotopy equivalences that fix homotopy
groups through the dimension of $Y$, denoted by
$\mathcal{E}_{\#}(Y)$. We give an upper bound for the solvability
class of $\mathcal{E}_{\#}(Y)$ in terms of a cone decomposition of
$Y$. We dualize the latter result to obtain an upper bound for
the solvability class of the subgroup of $\mathcal{E}(Y)$
consisting of homotopy classes of self-homotopy equivalences that
fix cohomology groups with various coefficients. We also show that
with integer coefficients, the latter group is nilpotent.