Fillings of unit cotangent bundles

TL;DR

This paper investigates the topology of Stein and exact fillings of the canonical contact structure on the unit cotangent bundle of a closed surface, establishing uniqueness and homological constraints.

Contribution

It proves a uniqueness theorem for Stein fillings up to s-cobordism and characterizes the homology of exact fillings, linking them to the disk cotangent bundle.

Findings

Any Stein filling is s-cobordant rel boundary to the disk cotangent bundle.

The rational homology of exact fillings matches that of the disk cotangent bundle.

For square-free $g-1$, all exact fillings share the same integral homology and intersection form.

Abstract

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $\Sigma_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $\Sigma_g$. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if $g-1$ is square-free, then any exact filling has the same integral homology and intersection form as $DT^*\Sigma_g$.