Journal of the
Korean Mathematical Society JKMS

For a pointed space $X$, the subgroups of self-homotopy equivalences ${\rm Aut}_{\sharp N}(X)$, ${\rm Aut}_{\Omega}(X)$, ${\rm Aut}_{\ast}(X)$ and ${\rm Aut}_{\Sigma}(X)$ are considered, where ${\rm Aut}_{\sharp N}(X)$ is the group of all self-homotopy classes $f$ of $X$ such that $f_{\sharp}=id:\pi_{i}(X)\to\pi_{i}(X)$ for all $i\leq N\leq \infty$, ${\rm Aut}_{\Omega}(X)$ is the group of all the above $f$ such that $\Omega f=id$; ${\rm Aut}_{\ast}(X)$ is the group of all self-homotopy classes $g$ of $X$ such that $g_{\ast}=id:H_{i}(X)\to H_{i}(X)$ for all $i\leq\infty$, ${\rm Aut}_{\Sigma}(X)$ is the group of all the above $g$ such that $\Sigma g=id$. We will prove that ${\rm Aut}_{\Omega}(X_{1}\times\cdots\times X_{n})$ has two factorizations similar to those of ${\rm Aut}_{\sharp N}(X_{1}\times\cdots\times X_{n})$ in reference [10], and that ${\rm Aut}_{\Sigma}(X_{1}\vee\cdots\vee X_{n})$, ${\rm Aut}_{\ast}(X_{1}\vee\cdots\vee X_{n})$ also have factorizations being dual to the former two cases respectively.

Keywords: self-homotopy equivalences, wedge spaces, product spaces, loop spaces, suspension

MSC numbers: 55P10