Vector bundles over classifying spaces of p‐local finite groups and Benson–Carlson duality

Abstract

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p‐local finite group (S,F,L) in terms of representation rings of subgroups of S. We also prove a stable elements formula for generalized cohomological invariants of p‐local finite groups, which is used to show the existence of unitary embeddings of p‐local finite groups. Finally, we show that the augmentation C∗(|L|p∧;double-struckFpfalse)→double-struckFp is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of false|scriptLfalse|p∧.

A completion theorem for fusion systems

“…where the first isomorphism holds by Theorem 4.2 in [11]. This isomorphism is induced by restriction to K(BS) and so the isomorphism from R(F ) ∧ IF to K(|L| ∧ p ) is induced by the restriction of the ring isomorphism R(S) ∧ IF → K(BS) to R(F ) ∧ IF .…”

“…In this section we determine the prime ideals of the representation ring of a saturated fusion system F over a finite p-group S. The argument is inspired by Section 6 of [1]. We use this information to compare different topologies on representation rings of subgroups of S. We start by recalling the definition of F -invariant representations from [11]. Definition 2.1.…”

“…A representation ρ is F -invariant if and only if its character χ ρ is F -invariant in the sense that χ ρ (s) = χ ρ (f (s)) for any morphism f in F (See Remark 3.3 in [11]). In particular, the regular representation of S and the trivial representation are F -invariant for any saturated fusion system F over S. Note that the set Rep(F ) of isomorphism classes of unitary representations of S which are F -invariant is a monoid under the operation induced by direct sum of representations.…”

“…It is closed under tensor product as well. By Proposition 5.7 in [11], it can be described by stable elements:…”

“…from Theorem 4.2 of [11] with the projection to the factor K(BS). In particular, it is injective and we identify K(|L| ∧ p ) as the subring of stable elements of K(BS).…”