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Journal ArticleDOI

A proof of a theorem of Luttinger and Simpson about the number of vanishing circles of a near-symplectic form on a 4-dimensional manifold

01 Jan 2006-Mathematical Research Letters (International Press of Boston)-Vol. 13, Iss: 4, pp 557-570
TL;DR: In this paper, a proof is given of a theorem announced some years ago by Luttinger and Simpson to the effect that a compact 4-manifold that has a near-symplectic form in a given cohomology class admits one in the same class whose zero locus consists of any given, but strictly positive number of disjoint, embedded circles.
Abstract: A proof is given of a theorem announced some years ago by Luttinger and Simpson to the effect that a compact 4-manifold that has a near-symplectic form in a given cohomology class admits one in the same class whose zero locus consists of any given, but strictly positive number of disjoint, embedded circles.
Citations
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TL;DR: In this paper, it was shown that every closed oriented smooth 4-manifold X admits a singular Lefschetz fibration over the 2-sphere, and that the fibration can be chosen so that F is a fiber component.
Abstract: We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber component. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus, we get alternative proofs of previous existence results. Also shown is the existence of broken Lefschetz pencils with connected fibers on any near-symplectic 4-manifold.

37 citations

Posted Content
TL;DR: In this article, it was shown that every closed oriented smooth 4-manifold X admits a singular Lefschetz fibration over the 2-sphere, and that the fibration can be chosen so that X is a fiber.
Abstract: We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus we get alternative proofs of previous existence results. Also shown is the existence of broken Lefschetz pencils with connected fibers on any near-symplectic 4-manifold.

37 citations


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Journal ArticleDOI
TL;DR: In this paper, the topology of broken Lefschetz fibrations is studied by means of handle decompositions, and a generalization of the symplectic fiber sum operation to the near-symplectic setting is given.
Abstract: The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations and pencils, showing that all near-symplectic closed 4-manifolds can be supported by such objects, paralleling a result of Auroux, Donaldson and Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg�Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire.

36 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that any near-symplectic form on four-manifolds can be modified by an evolutionary process taking place within a finite set of balls, so as to have a prescribed positive number of zero-circles.
Abstract: We give elementary proofs of two `folklore' assertions about near-symplectic forms on four-manifolds: that any such form can be modified, by an evolutionary process taking place within a finite set of balls, so as to have a prescribed positive number of zero-circles; and that, on a closed manifold, the number of zero-circles for which the splitting of the normal bundle is trivial has the same parity as 1+b_1+b_2^+.

26 citations

Posted Content
TL;DR: Gerig and Gerig as mentioned in this paper extended Taubes' SW=Gr theorem to non-symplectic 4-manifolds and showed that the Seiberg-Witten invariants are well-defined.
Abstract: Author(s): Gerig, Chris | Advisor(s): Hutchings, Michael | Abstract: For a closed oriented smooth 4-manifold X with $b^2_+(X)g0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds.The main results are the following. Given a suitable near-symplectic form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the "chamber" for defining the latter invariants when $b^2_+(X)=1$.In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces.

16 citations

References
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Journal ArticleDOI
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

2,482 citations

Journal ArticleDOI
TL;DR: In this paper, a special class of contac t s t ructures on 3-manifolds, called overtwisted contact s tructures, is defined, which can be defined by a 1-form 7 with 7/x (d~) nowhere 0.
Abstract: A contac t s t ructure on a (2n + l ) -d imensional manifold is a cod imens ion 1 tangent d is t r ibut ion which can be defined (at least locally) by a 1-form 7 with 7/x (d~)\" nowhere 0. In this paper we will s tudy a special class: overtwisted contact s tructures on 3-manifolds. Examples of overtwisted structures were s tudied by Lutz [11], Gonza l o and Varela [10], Er landsson [5], and Bennequin [1]. We give here the complete classification of overtwisted structures. Fo r example, it follows from the classification that all known examples of nons t anda rd (but homotop ica l ly standard) contac t s tructures on S 3 are equivalent. The paper has the following organizat ion. In Sect. 1 we give basic definit ions and formulate main results. In Sect. 2 we prove miscel laneous lemmas needed for the main theorem. In Sect. 3 we prove the main theorem, and in Sect. 4 discuss open quest ions a round the subject.

490 citations

Journal ArticleDOI
TL;DR: In this paper, it is shown that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1-form ca of maximal rank, and then some further equivalent conditions are derived.
Abstract: transformations in general and to the study of global contact transformations in the special case of euclidean space. In attempting to generalize Lie's results to more general manifolds, it becomes clear that there are intrinsic global differences between the even and odd dimensional cases. In this paper, only the odd dimensional case will be discussed. Intuitively, a manifold carries a contact structure if the coordinate transformations can be chosen to preserve the 1-form dz - y'dx' up to a non-zero, multiplicative factor. We first show that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1form ca of maximal rank, and then we derive some further equivalent conditions. It is well known that the existence of such a 1-form implies that the structure group of the tangent bundle can be reduced to the unitary group. (See, e.g., Chern, [7]). If this can be done, we say that the manifold is an almost-contact manifold. The obstructions to the existence of such a structure are investigated and it is shown that the primary obstruction is the third Stiefel-Whitney class. This solves completely the question of the existence of U(2) structures on five dimensional manifolds. We turn then to a discussion of global contact transformations, i.e., transformations which preserve ca up to a non-zero, multiplicative factor, T. Lie's results are shown to be valid in general by providing intrinsic proofs of his theorems. It should be noted that in this context, in general, analysis occurs only in the definitions, while the proofs consist simply of algebraic manipulations. Sheaves are employed at this point only because they provide a convenient language. Finally, we show that the factors T which can occur in contact transformations are not arbitrary. These results are then applied to the study of deformations (in the

397 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a near-symplectic 4-manifold can be decomposed into two symplectic Lefschetz fibrations over discs, and a fibre bundle over S 1, which relates the boundaries of the fibration to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form.
Abstract: We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4–manifold equipped with a “near-symplectic” structure (ie, a closed 2–form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4–manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2–form. Conversely, from such a decomposition one can recover a near-symplectic structure.

119 citations

Journal ArticleDOI

85 citations