Symplectic 4-manifolds with fixed point free circle actions

We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.Comment: 5 pages (to appear in Proc. Amer. Math. Soc.

The topology of Stein fillable manifolds in high dimensions II

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.Comment: We 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. Kellne

Contact open books with flexible pages

We give an elementary topological obstruction for a $(2q{+}1)$-manifold $M$ to admit a contact open book with flexible Weinstein pages: if the torsion subgroup of the $q$-th integral homology group is non-zero, then no such contact open book exists. We achieve this by proving that a symplectomorphism of a flexible Weinstein manifold acts trivially on cohomology. We also produce examples of non-trivial loops of flexible contact structures using related ideas.Comment: 9 page