Nonacyclic Reidemeister torsions of manifolds of odd dimension

Given an oriented closed manifold $M$ of odd dimension and a unitary representation $\rho :{\pi }_1(M)\text{ra}{\text{GL}}_n(\text{F})$, 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 $\text{SU}(n)$-character varieties of $M$. Moreover, we compute the Reidemeister torsions of some representations of 3-manifolds and compare the works of Farber–Turaev.