A completion theorem for fusion systems

TL;DR

This paper establishes a completion theorem linking twisted K-theory of p-local finite groups to twisted representations of fusion systems, enabling explicit calculations of K-theory for exotic 7-local finite groups.

Contribution

It proves a new completion theorem for fusion systems' twisted K-theory, connecting it to representation theory and applying it to exotic 7-local finite groups.

Findings

Isomorphism between twisted K-theory and completed representation Grothendieck group

Explicit computation of K-theory for Ruiz-Viruel exotic 7-local groups

New theoretical framework for understanding p-local finite groups' K-theory

Abstract

We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.