Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 3 (2016), 1403-1426.
We give a generalization of the concept of near-symplectic structures to dimensions. According to our definition, a closed –form on a –manifold is near-symplectic if it is symplectic outside a submanifold of codimension where vanishes. We depict how this notion relates to near-symplectic –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 –manifold over a symplectic base of codimension , 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- singular locus . We describe a splitting property of the normal bundle that is also present in dimension four. A tubular neighbourhood theorem for is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary.
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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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