Some Properties of Aut$_∗(X)$ and the Subgroup Aut$_Σ(X)$ (Communications in Mathematical Research, 2021)
Let Aut$_∗(X)$ denote the group of homotopy classes of self-homotopyequivalences of $X$, which induce identity automorphisms of homology group. Wedescribe a decomposition of Aut$_∗(X_1∨· · ·∨X_n)$ as a product of its simpler subgroups.We consider the subgroup Aut$_Σ(X)$ of all self homotopy classes α of $X$ such that $Σα = 1_{ΣX} : ΣX → ΣX$, and also give some properties of Aut$_Σ(X)$.