# The Complex of Affinely Commutative Sets

## TL;DR

This paper investigates the homotopy properties of the classifying space for commutativity in certain groups, establishing conditions under which it is contractible and analyzing associated maps for Lie and discrete groups.

## Contribution

It introduces a new map $rak{c}$ linking $E_{com} G$ to $B[G,G]$ and proves its non-nullhomotopic nature for non-abelian groups, also analyzing its connectivity.

## Key findings

- $E_{com} G$ is contractible iff $G$ is abelian for compact Lie and discrete groups.
- The map $rak{c}$ is not nullhomotopic for non-abelian groups.
- $rak{c}$ is 3-connected for $G=O(n)$ with $n \\ge 3$.

## Abstract

We show that for some classes of groups $G$, the homotopy fiber $E_{\mathrm{com}} G$ of the inclusion of the classifying space for commutativity $E_{\mathrm{com}} G$ into the classifying space $BG$, is contractible if and only if $G$ is abelian. We show this both for compact connected Lie groups and for discrete groups. To prove those results, we define an interesting map $\mathfrak{c} \colon E_{\mathrm{com}} G \to B[G,G]$ and show it is not nullhomotopic for the non-abelian groups in those classes. Additionally, we show that $\mathfrak{c}$ is 3-connected for $G=O(n)$ when $n \ge 3$.