On the mod-ℓ homology of the classifying space for commutativity

Okay, Cihan; Williams, Ben

doi:10.2140/agt.2020.20.883

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On the mod-$\ell$ homology of the classifying space for commutativity

Cihan Okay and Ben Williams

Algebraic & Geometric Topology 20 (2020) 883–923

DOI: 10.2140/agt.2020.20.883

Abstract

We study the mod- $ℓ$ homotopy type of classifying spaces for commutativity, $B (ℤ, G)$ , at a prime $ℓ$ . We show that the mod- $ℓ$ homology of $B (ℤ, G)$ depends on the mod- $ℓ$ homotopy type of $B G$ when $G$ is a compact connected Lie group, in the sense that a mod- $ℓ$ homology isomorphism $B G \to B H$ for such groups induces a mod- $ℓ$ homology isomorphism $B (ℤ, G) \to B (ℤ, H)$ . In order to prove this result, we study a presentation of $B (ℤ, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gómez. We also study the relationship between the mod- $ℓ$ type of a Lie group $G (ℂ)$ and the locally finite group $G ({\bar{𝔽}}_{p})$ , where $G$ is a Chevalley group. We see that the naïve analogue for $B (ℤ, G)$ of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a $G$ action on $B (ℤ, G)$ .

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Keywords

classifying spaces, mapping spaces, Lie groups

Mathematical Subject Classification 2010

Primary: 55R35

Secondary: 55R37, 55R40

References

Publication

Received: 2 January 2019

Revised: 21 June 2019

Accepted: 27 July 2019

Published: 23 April 2020

Authors

Cihan Okay
Department of Physics & Astronomy University of British Columbia Vancouver, BC Canada

Ben Williams
Department of Mathematics University of British Columbia Vancouver, BC Canada

Volume 20, issue 2 (2020)