Certain numbers on the groups of self-homotopy equivalences

Abstract

For a connected based space X, let [X, X] be the set of all based homotopy classes of base point preserving self map of X and let 6(X) be the group of self-homotopy equivalences of X. We denote by A(#)(k) (X) the set of homotopy classes of self-maps of X that induce an automorphism of pi(i) (X) for i = 0,1, ..., k. That is, [f] is an element of A(#)(k) (X) if and only if pi(i)(f) : pi(i)(X) -> pi(i)(X) is an isomorphism for i = 0,1, ..., k. Then, E(X) subset of A(#)(k)(X) subset of [X, X] for a nonnegative integer k. Moreover, for a connected CW-complex X, we have E(X) = A(#)(X). In this paper, we study the properties of A(X) and discuss the conditions under which E(X) = A(#)(k) (X) and the minimum value of such k. Furthermore, we determine the value of k for various spaces, including spheres, products of spaces, and Moore spaces. (C) 2014 Elsevier B.V. All rights reserved.