The group of stable self-equivalences

Abstract

THE GROUP of homotopy classes of homotopy equivalences of a topological space with itself, denoted here as G(X), has been studied by various authors [3, 6, 111. It is of course a homotopy invariant, although not a functor from the homotopy category to the category of groups. The determination of the groups G(X) bears on the question of the choices for k-invariants in a Postnikov decomposition. Nevertheless, our knowledge of the groups G(X) is scanty. In general, we know that if X is a l-connected, finite complex, G(X) is finitely-generated (see remark in [3]). Unfortunately, it is shown [lo] that there are uncountably many, nonisomorphic groups with two generators. Thus, the mere fact that G(X) is finitely-generated does little to help in finding what the actual possibilities for G(X) are.