The space of commuting elements in an exceptional Lie group and maps between classifying spaces

TL;DR

This paper investigates the surjectivity of a map between spaces of homomorphisms and maps from classifying spaces for free abelian groups and compact Lie groups, extending previous work by Atiyah and Bott.

Contribution

It provides new conditions determining when the map is surjective in rational cohomology for free abelian groups of rank at least 3 and compact Lie groups.

Findings

Identifies conditions for surjectivity in rational cohomology

Extends Atiyah and Bott's results to higher rank free abelian groups

Analyzes the structure of homomorphism and mapping spaces in this context

Abstract

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(\pi,G)_0$ denotes the null-component of the space of homomorphisms from $\pi$ to $G$, and $\mathrm{map}_*(B\pi,BG)_0$ denotes the null-component of the space of maps from $B\pi$ to $BG$. Since the classifying space functor is continuous, there is a continuous map $\Theta\colon\mathrm{Hom}(\pi,G)_0\to\mathrm{map}_*(B\pi,BG)_0$. Atiyah and Bott studied this map when $\pi$ is a surface group, and proved surjectivity in rational cohomology. In this paper, we obtain the condition that the map $\Theta$ is surjective or not in rational cohomology when $\pi$ is $\mathbf{Z}^m$ for $m\geq 3$ and $G$ is a compact connected Lie group.