The topology of Stein fillable manifolds in high dimensions II

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.

arXiv commentWe corrected mistakes in the proofs of Lemma 2.9 and Corollary 2.10. This lead to an assumption being removed from the statement of Theorem 1.3. The paper is now published in Geometry and Topology. The appendix was written by Bernd C. Kellner

References in corpus (10)

• Geometry of contact transformations and domains: orderability versus squeezing
• Weak and strong fillability of higher dimensional contact manifolds
• The topology of Stein fillable manifolds in high dimensions I
• How to recognise a 4-ball when you see one
• Contact structures on 5-manifolds
• Loose Legendrians and the plastikstufe
• Almost contact 5-folds are contact
• Constructions of contact forms on products and piecewise fibered manifolds
• Contact structures on M \times S^2
• Almost Complex Structures on (n-1)-connected 2n-manifolds

Cited by in corpus (8)

• The topology of Stein fillable manifolds in high dimensions I
• Maximal contact and symplectic structures
• Subcritical contact surgeries and the topology of symplectic fillings
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Contact surgery and symplectic caps
• Contact structures on M \times S^2
• Disconnecting the moduli space of G_2-metrics via U(4)-coboundary defects
• A landscape of contact manifolds via rational SFT