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

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.