Vector bundles over classifying spaces of \(p\)-local finite groups and Benson-Carlson duality. (English) Zbl 1444.55007

Cantarero, José; Castellana, Natàlia; Morales, Lola

J. Lond. Math. Soc., II. Ser. 101, No. 1, 1-22 (2020).

This paper describes the Grothendieck group of complex vector bundles over the classifying space of a \(p\)-local finite group \((S,\mathcal{F},\mathcal{L})\) in terms of representation rings of the subgroups of \(S\). Here \(S\) is a finite \(p\)-group, \(\mathcal{F}\) is a saturated fusion system over \(S\), and \(\mathcal{L}\) is a centric linking system associated to \(\mathcal{F}\). The classifying space of \((S,\mathcal{F},\mathcal{L})\) is \(\vert\mathcal{L} \vert _{p}^{\wedge}\) and the main result of the paper shows the existence of an isomorphism between \( {\mathbb K}(\vert\mathcal{L} \vert _{p}^{\wedge})\) and an inverse limit of the representation rings of the subgroups of \(S\). The authors show also that the cohomology of the classifying space \(\vert\mathcal{L} \vert _{p}^{\wedge}\) is isomorphic to the inverse limit of the cohomology of the subgroups of \(S\), for any generalized cohomology theory. The authors also show that there exists a homotopy monomorphism \(\vert\mathcal{L} \vert _{p}^{\wedge}\rightarrow\mathrm{BSU}(n)_{p}^{\wedge}\) for some \(n\) whose homotopy fibre is an \({\mathbb F}_{p}\)-finite space with Poincaré duality.

Reviewer: Jonathan Hodgson (Swarthmore)

MSC:

55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology
20C20	Modular representations and characters
20D20	Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Keywords:

\(p\)-local finite group; fusion system; centric linking system; Cohen-Macaulay ring; Gorenstein ring