#### Volume 16, issue 3 (2016)

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Near-symplectic $2n$–manifolds

### Ramón Vera

Algebraic & Geometric Topology 16 (2016) 1403–1426
##### Abstract

We give a generalization of the concept of near-symplectic structures to $2n$ dimensions. According to our definition, a closed $2$–form on a $2n$–manifold $M$ is near-symplectic if it is symplectic outside a submanifold $Z$ of codimension $3$ where ${\omega }^{n-1}$ vanishes. We depict how this notion relates to near-symplectic $4$–manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration as a singular map with indefinite folds and Lefschetz-type singularities. We show that, given such a map on a $2n$–manifold over a symplectic base of codimension $2$, the total space carries such a near-symplectic structure whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension-$3$ singular locus $Z$. We describe a splitting property of the normal bundle ${N}_{Z}$ that is also present in dimension four. A tubular neighbourhood theorem for $Z$ is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary.

##### Keywords
near-symplectic forms, broken Lefschetz fibrations, stable Hamiltonian structures, singular symplectic forms, folds, singularities
##### Mathematical Subject Classification 2010
Primary: 53D35, 57R17
Secondary: 57R45
##### Publication
Received: 12 August 2014
Revised: 19 August 2015
Accepted: 3 October 2015
Published: 1 July 2016
##### Authors
 Ramón Vera Department of Mathematics The Pennsylvania State University University Park State College, PA 16802 United States Department of Mathematical Sciences Durham University Science Laboratories South Rd Durham DH1 3LE United Kingdom