Homology decompositions and groups inducing fusion systems

TL;DR

This paper explores the relationship between group constructions that realize fusion systems and homology decompositions of p-local finite groups, showing that certain cohomology rings are closely related.

Contribution

It establishes a connection between group realizations of fusion systems and homology decompositions, providing new insights into their cohomology rings.

Findings

Cohomology ring of Robinson's construction closely approximates that of the realized fusion system

Links between group constructions and homology decompositions are clarified

Enhances understanding of the algebraic structure of p-local finite groups

Abstract

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.