On the intersection form of fillings

TL;DR

This paper proves the uniqueness of the intersection form for exact fillings of certain contact manifolds with vanishing rational first Chern class, using specialized Reeb dynamics and developing a broader approach for ADC manifolds.

Contribution

It introduces an ad hoc method to establish the uniqueness of intersection forms for exact fillings and extends the approach to more general ADC manifolds.

Findings

Unique integral intersection forms for exact fillings with vanishing rational first Chern class.

Diffeomorphism type classification of certain flexibly fillable contact manifolds.

Obstructions to contact embeddings beyond exact fillings.

Abstract

We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC \`a la Lazarev) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.