Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups

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: ε ∗(X) , the subgroup of homotopy classes which induce the identity on homology groups, ε ∗(X) , the subgroup of homotopy classes which induce the identity on cohomology groups and ε # dim + r ( X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ⩽ 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 ε ∗(X) = ε ∗(X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine ε # dim + r ( X) for certain spaces X and completely determine the group when X = S m × S n and X = CP n ∨ S 2 n .