Fibered Cohomology Classes in Dimension Three, Twisted Alexander Polynomials and Novikov Homology

We prove that for “most” closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1(M,\mathbf{R})$ is equivalent to the fact that every twisted Alexander polynomial ${\mathrm{\Delta }}^H(M,u)\in \mathbf{Z}[G\mathrm{/}\ker u]$ associated to a normal subgroup with finite index $H<{\pi }_1(M)$ has a unitary $u$-minimal term.