Algebraic & Geometric Topology

Near-symplectic $2n$–manifolds

Ramón Vera

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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 ωn1 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

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

Received: 12 August 2014
Revised: 19 August 2015
Accepted: 3 October 2015
First available in Project Euclid: 28 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 57R17: Symplectic and contact topology
Secondary: 57R45: Singularities of differentiable mappings

near-symplectic forms broken Lefschetz fibrations stable Hamiltonian structures singular symplectic forms folds singularities


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

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