## Algebraic & Geometric Topology

### Near-symplectic $2n$–manifolds

Ramón Vera

#### 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 $ωn−1$ 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 $NZ$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1403-1426.

Dates
Revised: 19 August 2015
Accepted: 3 October 2015
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895851

Digital Object Identifier
doi:10.2140/agt.2016.16.1403

Mathematical Reviews number (MathSciNet)
MR3523044

Zentralblatt MATH identifier
1342.53110

#### Citation

Vera, Ramón. Near-symplectic $2n$–manifolds. Algebr. Geom. Topol. 16 (2016), no. 3, 1403--1426. doi:10.2140/agt.2016.16.1403. https://projecteuclid.org/euclid.agt/1511895851

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