#### Volume 15, issue 1 (2015)

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A classifying space for commutativity in Lie groups

### Alejandro Adem and José Manuel Gómez

Algebraic & Geometric Topology 15 (2015) 493–535
##### 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 $ℤ×{B}_{com}U$ is a loop space and define a notion of commutative K–theory for bundles over a finite complex $X$, which is isomorphic to $\left[X,ℤ×{B}_{com}U\right]$. 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.

##### Keywords
commuting elements, Lie groups, classifying spaces
Primary: 22E99
Secondary: 55R35