A classifying space for commutativity in Lie groups

Abstract

In this article we consider a space $B_{com}G$ assembled from commuting elements in a Lie group $G$ first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that $\mathbb{Z}\times B_{com}U$ is a loop space and define a notion of commutative K–theory for bundles over a finite complex $X$, which is isomorphic to $\left[X,\mathbb{Z}\times B_{com}U\right]$. We compute the rational cohomology of $B_{com}G$ for $G$ equal to any of the classical groups $SU\left(r\right)$, $U\left(q\right)$ and $Sp\left(k\right)$, and exhibit the rational cohomologies of $B_{com}U$, $B_{com}SU$ and $B_{com}Sp$ as explicit polynomial rings.