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.