2022/02/09 by Md. Shariful Islam, Islam, Md. Shariful
Computer Science · Mathematics · #57R30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2202.04508
openalex publication_date 2022/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 dω=d+ω\wedge, where ω 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 Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology.