Weinstein Handlebodies for the Painlevé Betti Spaces

TL;DR

The paper establishes a precise Weinstein handlebody description for the Betti moduli spaces associated to the Painlevé equations, showing that for each type $X\in\mathfrak{P}$ the generic fibre $M_X$ is Weinstein deformation equivalent to the handlebody obtained by attaching along the Stokes Legendrian $\Lambda_X$. It develops a boundary-aware extension of Gompf's normal-form construction, together with a practical framework (matching paths, Hurwitz moves, Dehn twists) to realize skeleta from Lefschetz fibrations and branched covers. Explicit handlebody diagrams are provided for all Painlevé spaces (PI–PVI, including FN and deg variants), and the main theorem is verified via a case-by-case analysis of the Stokes Legendrians, including a computation of $H_1$ for $M_{III(D8)}$. The results connect symplectic topology, microlocal sheaf theory, and the irregular Riemann-Hilbert/Betti moduli framework, with implications for geometric Langlands programs.

Abstract

We prove that the generic fibre of the Betti moduli space associated to any of the ten Painlevé 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.

Figures (42)

• Figure 1: Legendrian attaching links for the Painlevé moduli spaces.
• Figure 2: Stokes data associated to the Painlevé spaces. Each diagram should be interpreted as lying on $\mathbb{P}^1$, and the coorientation is taken to point away from the puncture.
• Figure 3: Legendrian Reidemeister and Gompf moves
• Figure 4: Handle slides over a $(\pm 1)$-framed Legendrian sphere $\Sigma$ in the front, using the Reeb chord in red. We reiterate that for Weinstein handlebody diagrams, the surgery coefficient is always $-1$; therefore, the diagram on the left describes a move which may always be applied to any surgery diagram of a Weinstein manifold. The move on the right will become useful in dehn_fronts.
• Figure 5: Cancelling the blue $2$-handle with a $1$-handle; $\Lambda$ refers to a Legendrian tangle, possibly involving further $1$-handles and the blue attaching sphere, and $\Lambda'$ is the Legendrian tangle obtained from $\Lambda$ by removing the blue attaching sphere.

Theorems & Definitions (63)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8