Stabilization of the homotopy groups of the self equivalences of linear spheres

Assaf Libman

paper

Stabilization of the homotopy groups of the self equivalences of linear spheres

arXiv:1903.12550

Abstract

Let $G$ be a finite group. Let $U_{1}, U_{2}, \dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_{n}$ has infinite multiplicity. Let $V_{n} = \oplus_{i = 1}^{n} U_{n}$ and $S (V_{n})$ denote the linear sphere of unit vectors. Then for any $i \geq 0$ the sequence of group $\dots \to π_{i} map^{G} (S (V_{n}), S (V_{n})) \to π_{i} map^{G} (S (V_{n + 1}), S (V_{n + 1})) \to \dots$ stabilizes with the stable group $\oplus_{H} ω_{i} (B W_{G} H)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S (\oplus_{n} U_{n})$ .