Contact manifolds with flexible fillings

TL;DR

This paper demonstrates that flexible Weinstein fillings of a contact manifold with vanishing first Chern class have isomorphic cohomology, and in some cases, are symplectomorphic, revealing rich structures in high-dimensional contact topology.

Contribution

It establishes cohomology invariance for flexible fillings and constructs new examples of Weinstein domains with distinct contact boundaries, advancing understanding of contact and symplectic topology.

Findings

Flexible Weinstein fillings have isomorphic integral cohomology.

In certain cases, all flexible fillings are symplectomorphic.

There exist infinitely many contact structures in high dimensions with almost Weinstein fillings.

Abstract

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology; in certain cases, we prove that all flexible fillings are symplectomorphic. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Similar methods are used to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. We also prove relative analogs of our results, which we apply to Lagrangians in cotangent bundles.