A note on the symplectic topology of $b$-manifolds

A Poisson manifold $(M^{2n} , \pi)$ is $b$-symplectic if $\bigwedge^n \pi$ is transverse to the zero section. We prove an $h$-principle for open, $b$-symplectic manifolds, which shows that an open, orientable manifold $M$ is $b$-symplectic if and only if $M \times \mathbb{C}$ has an almost-complex structure. For closed, oriented manifolds, we observe that a cosymplectic manifold is the singular locus of a $b$-symplectic manifold if and only if it is symplectically fillable. We use this observation to prove that every $3$-dimensional, closed, orientable cosymplectic manifold is the singular locus of a closed, orientable $4$-manifold. We also discuss extensions of this result to higher dimensions.

Full Text (PDF format)

P.F. has been supported by NWO-Vrije competitie grant “Flexibility and Rigidity of Geometric Structures” 612.001.101, and by IMPA (CAPES-FORTAL project). D.M.T. has been partially supported by FCT Portugal (Programa Ciˆencia) and ERC Starting Grant no. 279729. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize, a Chaire d’Excellence de la Fondation Sciences Mathématiques de Paris and is partially supported by grants with reference MTM2015-69135-P (MINECO-FEDER) and 2014SGR634 (AGAUR). This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098).

The authors have been partially supported by European Science Foundation network CAST, Contact and Symplectic Topology. We would like to thank the CRM-Barcelona for their hospitality during the Research Programme Geometry and Dynamics of Integrable Systems.

Received 16 January 2014

Published 8 September 2017