On the homotopy groups of the self equivalences of linear spheres

Abstract

Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V{)}^{\ast n}$ denote the $n$-fold join of $S(V)$ with itself and let $\aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V{)}^{\ast n}$. We show that for any $k\ge 1$ there exists $M>0$ which depends only on $V$ such that $|π_k \aut_G(S(V)^{*n})| \leq M$ is for all $n\gg 0$.