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

Okay, Cihan¹ ; Williams, Ben²

¹Department of Physics & Astronomy, University of British Columbia, Vancouver, BC, Canada
²Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 883-923

Résumé

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 BG when G is a compact connected Lie group, in the sense that a mod-ℓ homology isomorphism BG → BH for such groups induces a mod-ℓ homology isomorphism B(ℤ,G) → 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(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).

Détail

DOI : 10.2140/agt.2020.20.883

Classification : 55R35, 55R37, 55R40
Keywords: classifying spaces, mapping spaces, Lie groups

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