A classifying space for commutativity in Lie groups

TL;DR

This paper studies the space B_{com}G formed from commuting elements in Lie groups, exploring its homotopy properties, classifying bundles, and computing rational cohomology for classical groups.

Contribution

It introduces homotopy-theoretic analysis of B_{com}G, establishes its role as a classifying space for commutative bundles, and explicitly computes their rational cohomology.

Findings

ZxB_{com}U is a loop space.

Defined a commutative K-theory for bundles.

Computed rational cohomology rings for B_{com}G.

Abstract

In this article we consider a space B_{com}G assembled from commuting elements in a Lie group G first defined in [Adem, Cohen, Torres-Giese 2012]. 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 ZxB_{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 [X,ZxB_{com}U]. We compute the rational cohomology of B_{com}G for G equal to any of the classical groups U(n), SU(n) and Sp(n), and exhibit the rational cohomologies of B_{com}U, B_{com}SU and B_{com}Sp as explicit polynomial rings.