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

TL;DR

This paper introduces new methods for constructing contact structures on product five-manifolds and along circles, expanding the class of known contact manifolds and controlling their homotopy types.

Contribution

It presents two novel constructions of contact manifolds, including a new approach for 5-manifolds as products and fiber sums, with applications to CP^2×S^1 and manifolds with fundamental group Z_2.

Findings

All product 5-manifolds with certain decompositions admit contact structures.

CP^2×S^1 has contact structures in every homotopy class of almost contact structures.

Established contact structures for many 5-manifolds with fundamental group Z_2.

Abstract

Two constructions of contact manifolds are presented: (i) products of S^1 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 CP^2 \times S^1 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 Z_2.