Stable splittings, spaces of representations and almost commuting elements in Lie groups

TL;DR

This paper investigates the topology of spaces of almost commuting elements in Lie groups, providing a stable splitting after suspension and explicit descriptions for certain representation spaces using symmetric products.

Contribution

It introduces a homotopical approach to analyze these spaces, deriving a stable splitting and explicitly describing stable factors for rank one compact Lie groups.

Findings

Stable splitting after one suspension for spaces of almost commuting elements.

Explicit descriptions of representation spaces as products of Eilenberg-MacLane spaces.

Complete characterization of stable factors for rank one compact Lie groups.

Abstract

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.