Symplectic topology of \(b\)-symplectic manifolds

Abstract

A Poisson manifold \((M^{2n}, \pi)\) is \(b\)-symplectic if \(\bigwedge^{n}\pi\) is transverse to the zero section. In this paper we apply techniques of Symplectic Topology to address global questions pertaining to \(b\)-symplectic manifolds. The main results provide constructions of: \(b\)-symplectic submanifolds à la Donaldson, \(b\)-symplectic structures on open manifolds by Gromov’s \(h\)-principle, and of \(b\)-symplectic manifolds with a prescribed singular locus, by means of surgeries.