Morse-Novikov cohomology of closed one-forms of rank 1

TL;DR

This paper investigates the Morse-Novikov cohomology related to closed one-forms of rank 1 on compact manifolds, with explicit calculations for certain locally conformally symplectic manifolds like the Inoue surface.

Contribution

It provides explicit computations of Morse-Novikov cohomology for closed one-forms of rank 1, especially on locally conformally symplectic manifolds such as the Inoue surface.

Findings

Explicit cohomology calculations for the Inoue surface

Characterization of Morse-Novikov cohomology in rank 1 cases

Insights into the structure of locally conformally symplectic manifolds

Abstract

We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods $\langle \int_\gamma \eta \mid [\gamma] \in \pi_1(M)\rangle$ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for the Inoue surface $\mathcal{S}^0$.