On the homotopy groups of the self equivalences of linear spheres

TL;DR

This paper investigates the homotopy groups of the space of G-equivariant self-homotopy equivalences of iterated joins of complex linear spheres, showing their boundedness as the join parameter grows.

Contribution

It establishes a uniform bound on the homotopy groups of the automorphism space for large join iterations, revealing stability properties.

Findings

Homotopy groups of automorphism spaces are bounded independently of join number.

Stability of equivariant self-homotopy equivalences for complex linear spheres.

Boundedness depends only on the initial representation V.

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 $S(V)^{*n}$ 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 $S(V)^{*n}$. We show that for any $k \geq 1$ there exists $M>0$ which depends only on $V$ such that $|\pi_k \aut_G(S(V)^{*n})| \leq M$ is for all $n \gg 0$.