Cohomology of hyperplane complements with group ring coefficients

TL;DR

This paper calculates the cohomology of hyperplane complement spaces with group ring coefficients, revealing it is nonzero only in the degree equal to the arrangement's rank, thus providing precise algebraic topological insights.

Contribution

It provides an explicit computation of the cohomology with group ring coefficients for hyperplane complements, a novel result in algebraic topology.

Findings

Cohomology is nonzero only in the degree equal to the rank of the arrangement.

Provides explicit formulas for the cohomology groups.

Enhances understanding of hyperplane arrangement topology.

Abstract

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.