The Subgroups of the Group of Self-homotopy Equivalences of Product Spaces

The subgroups $\mathscr{E}_\#(X\times Y)$ and $\mathscr{E}_\Omega(X\times Y)$ of the group of self-homotopy equivalences $\mathscr{E}(X\times Y)$ as well as their $p-$localization are studied, and some corresponding short split exact sequences are obtained. where $p$ is a prime.