# The homology class of a Poisson transversal

## TL;DR

This paper investigates the homology classes of compact Poisson transversals in Poisson manifolds, establishing conditions under which these transversals are non-trivial in homology, with extensions to Dirac geometry.

## Contribution

It generalizes known results from symplectic geometry to specific classes of Poisson structures and extends findings to Dirac geometry.

## Key findings

- All compact Poisson transversals are non-trivial in unimodular Poisson structures.
- The property holds for Poisson manifolds with closed leaves.
- Counterexamples exist where the property does not hold.

## Abstract

This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent non-trivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.