The diffeomorphism type of canonical integrations of Poisson tensors on surfaces

TL;DR

This paper characterizes the diffeomorphism type of the canonical integration of Poisson tensors on surfaces, showing it is usually diffeomorphic to the cotangent bundle unless the surface is a 2-sphere with an area form.

Contribution

It extends previous results by identifying the diffeomorphism type of the canonical integration for most Poisson structures on surfaces.

Findings

Canonical integration is diffeomorphic to the cotangent bundle for non-spherical cases.

Special case: when the Poisson tensor is an area form on the 2-sphere, the diffeomorphism type differs.

Generalizes previous results on Poisson integrations on surfaces.

Abstract

A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) groupoid structure. In this short note we show that when $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$, this extending results in \cite{Ma09} and \cite{BCST12}.