Weinstein Handlebodies for the Painlev\'e Betti Spaces

TL;DR

This paper demonstrates that the generic fiber of Painlevé Betti moduli spaces can be described using Weinstein handle attachments along Stokes Legendrians, providing explicit handlebody diagrams and extending handle presentation methods.

Contribution

It introduces a novel Weinstein handlebody description for Painlevé Betti spaces and extends handle presentation techniques to surfaces with boundary.

Findings

Weinstein handlebody diagrams for all Painlevé equations' Betti spaces.

Extension of handle presentation methods to surfaces with boundary.

Identification of the generic fiber as a Weinstein handle attachment along Stokes Legendrians.

Abstract

We prove that the generic fibre of the Betti moduli space associated to any of the ten Painlev\'e equations coincides with the result of attaching Weinstein handles along the Stokes Legendrian, and provide Weinstein handlebody diagrams for each of them in the process. We moreover extend the procedure for obtaining presentations in Gompf normal form of Weinstein manifolds obtained by attaching handles to Legendrian lifts of cooriented curves in surfaces to their unit cotangent bundle to the case where the surface may have nonempty boundary.