Weinstein Handlebodies for the Painlev\'e Betti Spaces

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.