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Given an oriented closed manifold $M$ of odd dimension and a unitaryrepresentation $\rho : \pi_1(M) \ra \GL_n(\F)$, we define a Reidemeistertorsion, even if the cohomology associated with $\rho$ is not acyclic. Ascorollaries, we introduce some topological invariants of $M$, which include thenonacyclic extensions of abelian torsions and the Alexander polynomials oflinks. Further, we propose a volume form of the $\SU(n)$-character varieties of$M$. Moreover, we compute the Reidemeister torsions of some representations of3-manifolds and compare the works of Farber--Turaev.

Nonacyclic Reidemeister torsions of manifolds of odd dimension