Nonacyclic Reidemeister torsions of manifolds of odd dimension

TL;DR

This paper extends Reidemeister torsion to odd-dimensional manifolds with non-acyclic cohomology, introducing new topological invariants and analyzing specific 3-manifold representations.

Contribution

It defines nonacyclic Reidemeister torsions for odd-dimensional manifolds, introduces related invariants, and proposes a volume form for SU(n)-character varieties.

Findings

Defined nonacyclic Reidemeister torsion for odd-dimensional manifolds

Introduced topological invariants including nonacyclic abelian torsions and Alexander polynomials

Computed torsions for certain 3-manifold representations and compared with existing work

Abstract

Given an oriented closed manifold $M$ of odd dimension and a unitary representation $\rho : \pi_1(M) \ra \GL_n(\F)$, we define a Reidemeister torsion, even if the cohomology associated with $\rho$ is not acyclic. As corollaries, we introduce some topological invariants of $M$, which include the nonacyclic extensions of abelian torsions and the Alexander polynomials of links. Further, we propose a volume form of the $\SU(n)$-character varieties of $M$. Moreover, we compute the Reidemeister torsions of some representations of 3-manifolds and compare the works of Farber--Turaev.