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

TL;DR

This paper investigates the surjectivity of a map between homomorphism spaces and classifying space maps for Lie groups and various groups, revealing conditions under which the map is surjective or not in rational cohomology.

Contribution

It extends previous work by analyzing the surjectivity of the map for free abelian groups and classical Lie groups, providing new results and counterexamples.

Findings

Surjectivity in rational cohomology for $Z^m$ and classical groups except $SO(2n)$.

Non-surjectivity for $Z^m$ with $m extgreater 2$ and $SO(2n)$ with $n extgreater 3$.

Analysis of the cokernel dimension in rational homotopy groups for these cases.

Abstract

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. Then there is a map $\Theta\colon\mathrm{Hom}(\pi,G)_0\to\mathrm{map}_*(B\pi,BG)_0$ between the null-components of the spaces of homomorphism and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi=\mathbb{Z}^m$ and the classical group $G$ except for $SO(2n)$, and that it is not surjective for $\pi=\mathbb{Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$. As an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi=\mathbb{Z}^m$ and the classical groups $G$ except for $SO(2n)$.