Contact structures on product five-manifolds and fibre sums along circles

Abstract

Two constructions of contact manifolds are presented: (i) products of S¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that CP² × S¹ admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group Z₂.