Algebraic & Geometric Topology

A classifying space for commutativity in Lie groups

Alejandro Adem and José Gómez

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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 × 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, × 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

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

Received: 9 June 2014
Revised: 10 July 2014
Accepted: 11 July 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

commuting elements Lie groups classifying spaces


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.

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