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

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

2009-06-29

arxiv.org/abs/0906.5242v2

Math. Ann. 348 (2010), 195-210

Mathematics

Symplectic Geometry

15 pages, 4 figures; v2: We have incorporated a result of Baykur on Stein decompositions of 4-manifolds. This gives a much str

Scientific paper

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.

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