Fillings of unit cotangent bundles of nonorientable surfaces

TL;DR

This paper classifies minimal weak symplectic fillings of the canonical contact structure on unit cotangent bundles of nonorientable surfaces, showing they are s-cobordant to disk cotangent bundles, with uniqueness results for the Klein bottle.

Contribution

It provides a classification of minimal weak symplectic fillings for nonorientable surfaces, extending understanding beyond the real projective plane.

Findings

Fillings are s-cobordant to disk cotangent bundles for all nonorientable surfaces except the real projective plane.

The minimal weak symplectic filling of the Klein bottle's unit cotangent bundle is unique up to homeomorphism.

The results generalize known classifications for orientable surfaces to nonorientable cases.

Abstract

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.