TITLE: Self Homotopy Equivalences Which Induce the Identity on
Homology, Cohomology or Homotopy Groups
AUTHORS: Martin Arkowitz and Ken-ichi Maruyama
ABSTRACT: For a based, 1-connected, finite CW-complex $X$, we study
the following subgroups of the group of homotopy classes of self
homotopy equivalences of $X$: $\Cal E_*(X)$, the subgroup of
homotopy classes which induce the identity on homology groups,
$\Cal E^*(X)$, the subgroup of homotopy classes which induce the
identity on cohomology groups and $\Cal E_\#^{\roman{dim}+ r}(X)$,
the subgroup of homotopy classes which induce the identity on
homotopy groups in dimensions $\leq \roman{dim}\,X +r$. We
investigate these groups when $X$ is a Moore space and when $X$ is
a co-Moore space. We give the structure of the groups in these
cases and provide examples of spaces for which the groups differ. We
also consider conditions on $X$ such that $\Cal E_*(X) = \Cal E^*(X)$
and obtain a class of spaces (including compact, oriented manifolds
and $H$-spaces) for which this holds. Finally, we examine $\Cal
E_\#^{\roman{dim}+ r}(X)$ for certain spaces $X$ and completely
determine the group when $X = S^m \times S^n$ and $X = CP^n \vee S^{2n}$.