The Complex of Affinely Commutative Sets

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 $\mathfrak{c}:E_{\mathrm{com}}G\to B[G,G]$ 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 $n\ge 3$.

Various minor additions and improvements to wording and exposition