Fox-Neuwirth cell structures and the cohomology of symmetric groups

TL;DR

This paper employs Fox-Neuwirth cell structures to analyze the mod-two cohomology of symmetric groups, providing new proofs of classical results and enhancing understanding of configuration space topology.

Contribution

It introduces a novel approach using Fox-Neuwirth structures to compute symmetric group cohomology and offers simplified proofs of established theorems.

Findings

Computed mod-two cohomology of symmetric groups

Provided simplified proofs of Nakaoka and Madsen's results

Enhanced understanding of configuration space topology

Abstract

We use the Fox-Neuwirth cell structure for one-point compactifications of configuration spaces as the starting point for understanding our recent calculation of the mod-two cohomology of symmetric groups. We then use that calculation to give short proofs of classical results on this cohomology due to Nakaoka and to Madsen. v2. Added references, expanded exposition on the cochain model and a mathematical correction in the final section. v3. Added references.