SELF-HOMOTOPY EQUIVALENCES RELATED TO COHOMOTOPY GROUPS
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초록
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 $n\geq k$. We denote by $\E_{k}^{\sharp}(X)$ the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of $\E_\sharp^k(X)$, 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 $\E_{k}^{\sharp}(X)$, whereas $\E_{\sharp}^{k}(X)$ may not be finite. Moreover, we obtain concrete computations of $\E_{k}^{\sharp}(X)$ to show that the cardinal of $\E_{k}^{\sharp}(X)$ is finite when $X$ is either a Moore space or co-Moore space by using the self-closeness numbers.
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 $n\geq k$. We denote by $\E_{k}^{\sharp}(X)$ the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of $\E_\sharp^k(X)$, 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 $\E_{k}^{\sharp}(X)$, whereas $\E_{\sharp}^{k}(X)$ may not be finite. Moreover, we obtain concrete computations of $\E_{k}^{\sharp}(X)$ to show that the cardinal of $\E_{k}^{\sharp}(X)$ is finite when $X$ is either a Moore space or co-Moore space by using the self-closeness numbers.
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