We study the mod- homotopy type of classifying spaces for commutativity, , at a prime . We show that the mod- homology of depends on the mod- homotopy type of when is a compact connected Lie group, in the sense that a mod- homology isomorphism for such groups induces a mod- homology isomorphism . In order to prove this result, we study a presentation of 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 and the locally finite group , where is a Chevalley group. We see that the naïve analogue for of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a action on .
"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