Morse-Novikov cohomology on foliated manifolds

TL;DR

This paper extends Morse-Novikov cohomology to foliated manifolds, exploring its properties, invariance, and Hodge theory, revealing new dualities and structures in the leafwise context.

Contribution

It introduces a novel study of Morse-Novikov cohomology relative to foliations, including homotopy invariance, Hodge decomposition, and duality results for Riemannian foliations.

Findings

Proves homotopy invariance of leafwise Morse-Novikov cohomology.

Establishes Hodge theorem and Poincaré duality for Riemannian foliations.

Demonstrates a Hodge diamond structure for leafwise Morse-Novikov cohomology.

Abstract

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_\omega=d+\omega\wedge$, where $\omega$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincar{\'e} duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology.