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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

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 BG BH 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).

classifying spaces, mapping spaces, Lie groups
Mathematical Subject Classification 2010
Primary: 55R35
Secondary: 55R37, 55R40
Received: 2 January 2019
Revised: 21 June 2019
Accepted: 27 July 2019
Published: 23 April 2020
Cihan Okay
Department of Physics & Astronomy
University of British Columbia
Vancouver, BC
Ben Williams
Department of Mathematics
University of British Columbia
Vancouver, BC