#### Volume 20, issue 2 (2020)

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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
##### Abstract

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

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##### Keywords
classifying spaces, mapping spaces, Lie groups
##### Mathematical Subject Classification 2010
Primary: 55R35
Secondary: 55R37, 55R40