## Algebraic & Geometric Topology

### A classifying space for commutativity in Lie groups

#### Abstract

In this article we consider a space $BcomG$ 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 $ℤ × BcomU$ is a loop space and define a notion of commutative K–theory for bundles over a finite complex $X$, which is isomorphic to $[X, ℤ × BcomU]$. We compute the rational cohomology of $BcomG$ for $G$ equal to any of the classical groups $SU(r)$, $U(q)$ and $Sp(k)$, and exhibit the rational cohomologies of $BcomU$, $Bcom SU$ and $Bcom Sp$ as explicit polynomial rings.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 493-535.

Dates
Revised: 10 July 2014
Accepted: 11 July 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510840921

Digital Object Identifier
doi:10.2140/agt.2015.15.493

Mathematical Reviews number (MathSciNet)
MR3325746

Zentralblatt MATH identifier
06425412

Subjects
Primary: 22E99: None of the above, but in this section
Secondary: 55R35: Classifying spaces of groups and $H$-spaces

#### Citation

Adem, Alejandro; Gómez, José. A classifying space for commutativity in Lie groups. Algebr. Geom. Topol. 15 (2015), no. 1, 493--535. doi:10.2140/agt.2015.15.493. https://projecteuclid.org/euclid.agt/1510840921

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