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

TL;DR

This paper establishes a link between the existence of certain closed one-forms on most 3-manifolds and properties of twisted Alexander polynomials, providing new insights into the topology of these manifolds.

Contribution

It demonstrates an equivalence between the existence of non-singular closed one-forms and the unitary minimality of terms in twisted Alexander polynomials for most 3-manifolds.

Findings

Most closed 3-manifolds admit non-singular closed one-forms if and only if their twisted Alexander polynomials have unitary u-minimal terms.

The work connects cohomology classes with algebraic invariants, enriching the understanding of 3-manifold topology.

Provides criteria to determine the existence of certain forms based on polynomial properties.

Abstract

We prove that for "most" closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$ is equivalent to the fact that every twisted Alexander polynomial $\Delta^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup with finite index $H < \pi_1(M)$ has a unitary $u$-minimal term.