A classifying space for commutativity in Lie groups

Adem, Alejandro; Gómez, José

doi:10.2140/agt.2015.15.493

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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

DOI: 10.2140/agt.2015.15.493

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 $ℤ \times B_{com} U$ is a loop space and define a notion of commutative K–theory for bundles over a finite complex $X$ , which is isomorphic to $[X, ℤ \times B_{com} U]$ . We compute the rational cohomology of $B_{com} G$ for $G$ equal to any of the classical groups $SU (r)$ , $U (q)$ and $Sp (k)$ , 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

Mathematical Subject Classification 2010

Primary: 22E99

Secondary: 55R35

References

Publication

Received: 9 June 2014

Revised: 10 July 2014

Accepted: 11 July 2014

Published: 23 March 2015

Authors

Alejandro Adem
Department of Mathematics University of British Columbia 1984 Mathematics Road, Room 121 Vancouver BC V6T 1Z2 Canada

José Manuel Gómez
Departmento de Matemáticas Universidad Nacional de Colombia Calle 59A No 63 – 20, Bloque 43 Medellín Colombia

Volume 15, issue 1 (2015)