On a Lichnerowicz type cohomology attached to a function

TL;DR

This paper introduces a new cohomology theory for smooth manifolds, called Lichnerowicz type cohomology attached to a function, exploring its fundamental properties and relations to existing cohomologies.

Contribution

It defines and analyzes a novel cohomology linked to a function, extending classical concepts with new properties and invariance results.

Findings

Establishes a de Rham type isomorphism for the new cohomology

Shows dependence on the chosen function and explores singular forms

Demonstrates homotopy invariance and Mayer-Vietoris sequence for the cohomology

Abstract

In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invariance and next, a regular case is considered. The notions are introduced using techniques from the study of two cohomologies of a smooth manifold: the Lichnerowicz cohomology and the cohomology attached to a function.