Fox pairings of Poincar\'{e} duality groups

TL;DR

This paper explores Fox pairings in Poincaré duality groups using group cohomology, computing relevant cohomology groups and proposing fundamental and higher Fox pairings.

Contribution

It introduces a cohomological approach to Fox pairings in Poincaré duality groups and defines fundamental and higher pairings.

Findings

Computed cohomology groups of Fox pairings in Poincaré duality groups

Proposed fundamental Fox pairings and higher Fox pairings

Established connections between Fox pairings and group cohomology

Abstract

This paper develops the study of Fox pairings of a group $G$ from the viewpoint of group cohomology. We compute some cohomology groups of Fox pairings of $G$, where $G$ admits a Poincar\'{e} duality group pair. We also suggest fundamental Fox pairings and higher Fox pairings.