Nonacyclic Reidemeister torsions of manifolds of odd dimension
Given an oriented closed manifold $M$ of odd dimension and a unitary representation $ρ: π_1(M) \ra \GL_n(\F)$, we define a Reidemeister torsion, even if the cohomology associated with $ρ$ 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.
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