The topology of Stein fillable manifolds in high dimensions I

TL;DR

This paper characterizes which high-dimensional almost contact manifolds admit Stein fillable structures using bordism theory and modified surgery, providing new insights into Stein fillability in high dimensions.

Contribution

It introduces a bordism-theoretic criterion for Stein fillability of high-dimensional almost contact manifolds, extending previous results with a novel application of Eliashberg's h-principle.

Findings

Any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable.

Provides conditions for Stein fillability of exotic spheres.

Discusses the Stein fillability of subcritical Stein manifolds.

Abstract

We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.