#### Volume 14, issue 1 (2014)

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A spectral sequence for fusion systems

### Antonio Díaz Ramos

Algebraic & Geometric Topology 14 (2014) 349–378
##### 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.

##### Keywords
Lyndon–Hochschild–Serre spectral sequence, fusion system, strongly closed subgroup, Tate's nilpotency criterion
##### Mathematical Subject Classification 2010
Primary: 55T10
Secondary: 55R35, 20D20