Complements of hyperplane arrangements as posets of spaces

TL;DR

This paper explores the structure of hyperplane arrangement complements as posets of spaces, providing tools to correct spectral sequence arguments in prior cohomology computations for both affine and toric arrangements.

Contribution

It introduces a poset of spaces framework for arrangement complements, enabling the correction of spectral sequence arguments in previous cohomology studies.

Findings

Poset of spaces structure for arrangement complements

Application to repair spectral sequence arguments

Extension to toric hyperplane arrangements

Abstract

The complement of an arrangement A of a finite number of affine hyperplanes in complex n-space has the structure of a poset of spaces indexed by the intersection poset, L(A). The space corresponding to G in L(A) is homotopy equivalent to the complement of the hyperplanes in the central arrangement A_G normal to G. This poset of spaces structure can be used to repair a spectral sequence argument in two earlier papers of Davis, Januszkiewicz, Leary and Okun for computing certain cohomology groups of arrangement complements. Similarly, toric hyperplane arrangements have the structure of a diagram of spaces and this structure can be used fix a spectral sequence argument in an earlier paper of Davis and Settepanella.