On the Strong Homotopy Lie-Rinehart Algebra of a Foliation

TL;DR

This paper explores the strong homotopy Lie-Rinehart algebra structure underlying the characteristic cohomology of a foliation, revealing its canonical nature and how it encodes the Lie bracket and action on cohomology.

Contribution

It introduces a strong homotopy Lie-Rinehart algebra framework for foliation cohomology, showing its canonical form and relation to characteristic structures.

Findings

The Lie bracket on characteristic cohomology arises from a strong homotopy structure.

The action on characteristic cohomology is derived from the same homotopy framework.

The strong homotopy structure is shown to be canonical up to isomorphism.

Abstract

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the space P of integral manifolds (of any dimension) of the characteristic distribution C of F. Similarly, characteristic cohomologies with local coefficients in the normal bundle TM/C of F may be interpreted as vector fields on P. In particular, they possess a (graded) Lie bracket and act on characteristic cohomology H. In this paper, I discuss how both the Lie bracket and the action on H come from a strong homotopy structure at the level of cochains. Finally, I show that such a strong homotopy structure is canonical up to isomorphisms.