Algebraic & Geometric Topology

Near-symplectic $2n$–manifolds

Ramón Vera

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 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

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

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

Permanent link to this document
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

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

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

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


Export citation

References

  • S Akbulut, Ç Karakurt, Every $4$–manifold is BLF, J. Gökova Geom. Topol. GGT 2 (2008) 83–106
  • D Auroux, S,K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043–1114
  • R,\.I Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. 2008 (2008) Art. ID rnn 101
  • R,\.I Baykur, Topology of broken Lefschetz fibrations and near-symplectic four-manifolds, Pacific J. Math. 240 (2009) 201–230
  • K Cieliebak, E Volkov, First steps in stable Hamiltonian topology, J. Eur. Math. Soc. $($JEMS$)$ 17 (2015) 321–404
  • S,K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205–236
  • D,T Gay, R Kirby, Constructing symplectic forms on $4$–manifolds which vanish on circles, Geom. Topol. 8 (2004) 743–777
  • D,T Gay, R Kirby, Constructing Lefschetz-type fibrations on four-manifolds, Geom. Topol. 11 (2007) 2075–2115
  • M Golubitsky, V Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer, New York (1973)
  • R,E Gompf, A,I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)
  • K Honda, Local properties of self-dual harmonic $2$–forms on a $4$–manifold, J. Reine Angew. Math. 577 (2004) 105–116
  • K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629–664
  • Y Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. 13 (2009) 277–318
  • T Perutz, Zero-sets of near-symplectic forms, J. Symplectic Geom. 4 (2006) 237–257
  • T Perutz, Lagrangian matching invariants for fibred four-manifolds, I, Geom. Topol. 11 (2007) 759–828
  • C,H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S\sp 1\times B\sp 3$, Geom. Topol. 2 (1998) 221–332
  • K Wehrheim, C Woodward, Pseudoholomorphic quilts, preprint (2015)