The topology of Stein fillable manifolds in high dimensions II

TL;DR

This paper advances the understanding of Stein fillability in high-dimensional contact manifolds, introducing new maximal structures, fillability conditions for products, and obstructions based on number theory.

Contribution

It identifies maximal almost contact manifolds in high dimensions, explores fillability of product manifolds, and establishes obstructions to Stein fillability using Bernoulli numbers.

Findings

Existence of maximal almost contact manifolds in each dimension

Product M x S^2 admits weakly fillable contact structures under certain conditions

Certain high-dimensional spheres and manifolds are not Stein fillable

Abstract

We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not. Concerning obstructions to Stein fillings, we show that the (8k-1)-dimensional sphere has an almost contact structure which is not Stein fillable once k > 1. As a consequence we deduce that any highly connected almost contact (8k-1)-manifold (with k > 1) admits an almost contact structure which is not Stein fillable. The proofs rely on a new number-theoretic result about Bernoulli numbers.