SELF-HOMOTOPY EQUIVALENCES RELATED TO COHOMOTOPY GROUPS

Abstract

Given a topological space X and a non -negative integer k, we study the self-homotopy equivalences of X that do not change maps from X to n -sphere S-n homotopically by the composition for all n >= k. We denote by epsilon(K)(X) the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of epsilon(k) (X), which has been studied by several authors. We prove that if X is a finite CW complex, there are at most a finite number of distinguishing homotopy classes epsilon(k) (X), whereas epsilon(k) (X) may not be finite. Moreover, we obtain concrete computations of epsilon(k) (X) to show that the cardinal of epsilon(k)(X) is finite when X is either a Moore space or co-Moore space by using the self -closeness numbers.