Translator Disclaimer
2020 On the mod-$\ell$ homology of the classifying space for commutativity
Cihan Okay, Ben Williams
Algebr. Geom. Topol. 20(2): 883-923 (2020). DOI: 10.2140/agt.2020.20.883

Abstract

We study the mod- homotopy type of classifying spaces for commutativity, B(,G), at a prime . We show that the mod- homology of B(,G) depends on the mod- homotopy type of BG when G is a compact connected Lie group, in the sense that a mod- homology isomorphism BGBH for such groups induces a mod- homology isomorphism B(,G)B(,H). In order to prove this result, we study a presentation of B(,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- type of a Lie group G() and the locally finite group G(𝔽 ̄p), where G is a Chevalley group. We see that the naïve analogue for B(,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(,G).

Citation

Download Citation

Cihan Okay. Ben Williams. "On the mod-$\ell$ homology of the classifying space for commutativity." Algebr. Geom. Topol. 20 (2) 883 - 923, 2020. https://doi.org/10.2140/agt.2020.20.883

Information

Received: 2 January 2019; Revised: 21 June 2019; Accepted: 27 July 2019; Published: 2020
First available in Project Euclid: 30 April 2020

zbMATH: 07195379
MathSciNet: MR4092314
Digital Object Identifier: 10.2140/agt.2020.20.883

Subjects:
Primary: 55R35
Secondary: 55R37, 55R40

Rights: Copyright © 2020 Mathematical Sciences Publishers

JOURNAL ARTICLE
41 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.20 • No. 2 • 2020
MSP
Back to Top