Zero-sets of near-sympletic forms

Perutz, Tim

doi:10.4310/jsg.2006.v4.n3.a1

Cited by 24 publications

(49 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, by deforming ! , one can show that on any near-symplectic manifold, one can reduce the number of circles to 1; this was proved by Perutz [12]. We give a new proof of this result in Theorem 6.1 as an application of the techniques developed in this paper.…”

Section: Near-symplectic Manifoldsmentioning

confidence: 70%

“…In particular, this allows us to divide the zero-set into two pieces, the even circles where the line bundle L is orientable, and the odd circles where L is not orientable. This definition is motivated by the following result of Gompf that the number of even circles is equal to 1 b 1 C b C 2 modulo 2 [12]. In particular, observe that if there is only one zero circle which is even, the manifold X cannot be symplectic.…”

Section: Near-symplectic Manifoldsmentioning

confidence: 99%

“…An important such deformation is provided by the Luttinger-Simpson model given on D 4 R 4 where the birth (or death) of a circle can be observed explicitly [12]:…”

Section: Near-symplectic Manifoldsmentioning

confidence: 99%

“…Furthermore, we will provide one-parameter families for the deformations corresponding to the 4 moves given in Section 3. These will be near-symplectic cobordisms in the sense of the following definition given by Perutz [12]. The strategy will be the same as the construction of the local model around a cusp singularity.…”

Section: Generic Deformations Of Wrinkled Fibrations and 1; 1/-stabimentioning

confidence: 99%

“…The Figure 18 drawn in the most symmetric fashion presents a different choice of three vanishing cycles D 1 , D 2 and D 3 on the twice punctured torus, and through similar arguments as above one can see that this picture stands for a broken Lefschetz fibration that can be used to replace a positive Lefschetz singularity locally. This contains the nontrivial part of Perutz's example of a broken Lefschetz fibration (Example 1.3 in [12]; also see Example 3.2 in [4]), and corresponds to the broken Lefschetz fibration that Lekili obtains after perturbing a positive Lefschetz singularity in his paper. One can then draw the curves C 1 , C 2 and C 3 by joining the vertices of the hexagon formed by D 1 , D 2 , D 3 in the center as in Figure 18, and get the picture we had above (Figure 16, on the left) up to isotopy.…”

mentioning

confidence: 98%

See 4 more Smart Citations

Wrinkled fibrations on near-symplectic manifolds

Lekili

2009

Geom. Topol.

View full text Add to dashboard Cite

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, ie, manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them; see Auroux, Donaldson and Katzarkov [3] and Gay and Kirby [8]. We present a set of four moves which allow us to pass from any given broken fibration to any other which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.

show abstract

Section: Near-symplectic Manifoldsmentioning

confidence: 70%

Section: Near-symplectic Manifoldsmentioning

confidence: 99%

“…An important such deformation is provided by the Luttinger-Simpson model given on D 4 R 4 where the birth (or death) of a circle can be observed explicitly [12]:…”

Section: Near-symplectic Manifoldsmentioning

confidence: 99%

Section: Generic Deformations Of Wrinkled Fibrations and 1; 1/-stabimentioning

confidence: 99%

mentioning

confidence: 98%

See 3 more Smart Citations

Wrinkled fibrations on near-symplectic manifolds

Lekili

2009

Geom. Topol.

View full text Add to dashboard Cite

show abstract

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

2019

View full text Add to dashboard Cite

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.2010 Mathematics Subject Classification. 57R17, 53D17. after combining the definition of generalized bLf [19] together with the results of [6] and [10] on Poisson structures. Theorem 1.1. Let M and X be closed oriented smooth manifolds with dim(M ) = 2n, dim(X) = 2n − 2, and f : M → X a generalized broken Lefschetz fibration. Then there is a complete singular Poisson structure of rank 2 whose associated bivector vanishes on the singularity set of f . If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The proof of this theorem is a direct application of Theorem 2.11 and the definition of completeness. The integrability condition is verified in the relevant cases, as explained in 2.4.In section 3 we focus on the Poisson structure on the total space of a generalized wrinkled fibration in dimension 6. We give the general steps for buidling Poisson bivectors around all types of singularities of corank 1, which can be applied on any given dimension. This idea allows us to show the following. Theorem 1.2. Let M be a closed, orientable, smooth 6-manifold equipped with a generalized wrinkled fibration f : M → X over a smooth 4-manifold X. Then there exists a complete Poisson structure whose symplectic leaves correspond to the fibres of the given fibration structure, and the singularities of both the fibration and the Poisson structures coincide. Moreover, for each singularity, the Poisson bivector and induced symplectic form on the leaves are given by the following equations:Folds: 13) If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The existence of a Poisson structure with the stated properties follows from Theorem 2.11, previously shown by the first and third named authors together with García-Naranjo [10]. The proof of this theorem follows from an application of Theorem 2.11 and the definition of completeness.These results allow us to present in section 2.5.1 countably many examples of Poisson structures on the same underlying smooth manifold that are Morita inequivalent. In our examples the leaves of the symplectic foliations change topology, as the fibrations involved undergo deformations.In section 3, the local models for the bivectors are shown to hold true. Then in section 4 we prove that the local models for the symplectic forms on the leaves are also the claimed ones.Finally, in section 5 we turn to near-symplectic geometry. Explicit models of near-symplectic forms have appeared in previous work [11,1,14,19]. Here we show a further construction of local near-symplectic forms that follows a...

show abstract