A spectral sequence for fusion systems

TL;DR

This paper introduces a new spectral sequence for fusion systems with strongly closed subgroups, extending classical group cohomology tools to more general and exotic fusion system contexts.

Contribution

It constructs a spectral sequence applicable to fusion systems, generalizing the Lyndon-Hochschild-Serre spectral sequence for broader algebraic structures.

Findings

The spectral sequence converges to the cohomology of fusion systems with strongly closed subgroups.

It applies to finite simple groups and exotic fusion systems, beyond traditional group extensions.

Proves an analogue of Stallings' result and derives Tate's p-nilpotency criterion.

Abstract

We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove an analogue of a result of Stallings in the context of fusion preserving homomorphisms and deduce Tate's p-nilpotency criterion as a corollary.