A method for integral cohomology of posets

TL;DR

This paper introduces a novel method for computing the integral cohomology of posets using homological algebra and spectral sequences, applicable to simplicial complexes and related structures, with applications to fusion systems and Coxeter groups.

Contribution

It develops a new computational approach for poset cohomology based on spectral sequences and homological algebra, extending previous methods.

Findings

Provides an alternative proof of Webb's conjecture for saturated fusion systems.

Computes cohomology of Coxeter complexes for finite and infinite Coxeter groups.

Establishes a connection between the method and discrete Morse theory.

Abstract

We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups.