Twisted Blanchfield pairings and symmetric chain complexes

TL;DR

This paper introduces a new twisted Blanchfield pairing for symmetric chain complexes over group rings, establishing its sesquilinearity, hermitian property, and nonsingularity under specific conditions, with applications to 3-manifolds.

Contribution

It defines the twisted Blanchfield pairing for symmetric triads over group rings and proves key properties, extending classical invariants to more general settings.

Findings

The pairing is sesquilinear.

The pairing is hermitian under certain conditions.

The pairing is nonsingular under certain conditions.

Abstract

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted Blanchfield pairing is then associated to a 3-manifold together with a decomposition of its boundary into two pieces and a unitary representation of its fundamental group.