# Vector bundles over classifying spaces of p-local finite groups and Benson-Carlson duality

## TL;DR

This paper characterizes vector bundles over p-local finite groups' classifying spaces, establishes a stable elements formula for their invariants, and explores Gorenstein properties of their cochain augmentation maps.

## Contribution

It provides a new description of the Grothendieck group of vector bundles and proves a stable elements formula for cohomological invariants of p-local finite groups.

## Key findings

- Grothendieck group described via subgroup representation rings
- Stable elements formula for generalized cohomological invariants
- Augmentation map is Gorenstein, impacting cohomology ring structure

## 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 in terms of representation rings of subgroups of its Sylow. 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 map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the cohomology ring of these classifying spaces.