ON FACTORIZATIONS OF THE SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES

Title & Authors
ON FACTORIZATIONS OF THE SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES
Shi, Yi-Yun; Zhao, Hao;

Abstract
For a pointed space X, the subgroups of self-homotopy equivalences $\small{Aut_{\sharp}_N(X)}$, $\small{Aut_{\Omega}(X)}$, $\small{Aut_*(X)}$ and $\small{Aut_{\Sigma}(X)}$ are considered, where $\small{Aut_{\sharp}_N(X)}$ is the group of all self-homotopy classes f of X such that $\small{f_{\sharp}=id\;:\;{\pi_i}(X){\rightarrow}{\pi_i}(X)}$ for all $\small{i{\leq}N{\leq}{\infty}}$, $\small{Aut_{\Omega}(X)}$ is the group of all the above f such that $\small{{\Omega}f=id;\;Aut_*(X)}$ is the group of all self-homotopy classes g of X such that $\small{g_*=id\;:\;H_i(X){\rightarrow}H_i(X)}$ for all $\small{i{\leq}{\infty}}$, $\small{Aut_{\Sigma}(X)}$ is the group of all the above g such that $\small{{\Sigma}g=id}$. We will prove that $\small{Aut_{\Omega}(X_1{\times}\cdots{\times}X_n)}$ has two factorizations similar to those of $\small{Aut_{\sharp}_N(X_1{\times}\cdots{\times}\;X_n)}$ in reference [10], and that $\small{Aut_{\Sigma}(X_1{\vee}\cdots{\vee}X_n)}$, $\small{Aut_*(X_1{\vee}\cdots{\vee}X_n)}$ also have factorizations being dual to the former two cases respectively.
Keywords
self-homotopy equivalences;wedge spaces;product spaces;loop spaces;suspension;
Language
English
Cited by
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