SELF-CLOSENESS NUMBER AND WEAK HOMOTOPY DECOMPOSITION

Abstract

For any CW-complex X, there exists a weak homotopy decomposition X-(m) and a self-closeness number of X. In this paper, we study the self-closeness number of a weak homotopy decomposition of X. We prove that the self-closeness number of X-(m) is dominated by the self-closeness number of X. Moreover, we determine the set of self-homotopy classes and the group of self-homotopy equivalence classes of a weak homotopy decomposition of X, and herein we provide some examples of the self-closeness number for homotopy m-sections.