  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.
