Zeroth Poisson homology, foliated cohomology and perfect Poisson manifolds

TL;DR

This paper establishes a deep connection between zeroth Poisson homology and foliated cohomology in regular Poisson manifolds, introduces the concept of perfect Poisson manifolds, and provides new examples through homology computations.

Contribution

It proves isomorphisms between Poisson homology and foliated cohomology for regular unimodular Poisson manifolds and introduces the notion of perfect Poisson manifolds with new examples.

Findings

Zeroth Poisson homology is isomorphic to top foliated cohomology in regular Poisson manifolds.

For regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic.

New families of perfect Poisson manifolds are constructed using homology computations.

Abstract

We prove that for regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group and we give some applications. In particular, we show that for regular unimodular Poisson manifolds top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what is a perfect Poisson manifold. We use these Poisson homology computations to provide families of perfect Poisson manifolds.