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

Geiges, Hansjörg and Stipsicz, András I. (2010) Contact structures on product five-manifolds and fibre sums along circles. MATHEMATISCHE ANNALEN, 348 (1). pp. 195-210. ISSN 0025-5831

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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.

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