On sections of genus two Lefschetz fibrations

Onaran, Sinem

doi:10.2140/pjm.2010.248.203

Cited by 9 publications

(7 citation statements)

References 6 publications

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“…In the case g = 2, Onaran [13] showed that, in the mapping class group of Σ 2,n , the boundary multitwist t δ 1 t δ 2 • • • t δn can be written as a product of positive Dehn twists about nonseparating simple closed curves for n ≤ 8. In [18], Tanaka improved this result to n = 4g + 4 for any g.…”

Section: A Questionmentioning

confidence: 99%

Arbitrarily Long Factorizations in Mapping Class Groups

Dalyan

Korkmaz

Pamuk

2014

Int Math Res Notices

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Abstract. On a compact oriented surface of genus g with n ≥ 1 boundary components, δ1, δ2, . . . , δn, we consider positive factorizations of the boundary multitwist t δ 1 t δ 2 · · · t δn , where t δ i is the positive Dehn twist about the boundary δi. We prove that for g ≥ 3, the boundary multitwist t δ 1 t δ 2 can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn-Morris, who proved this result for g ≥ 8. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of contact three manifolds.

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Section: A Questionmentioning

confidence: 99%

Arbitrarily Long Factorizations in Mapping Class Groups

Dalyan

Korkmaz

Pamuk

2014

Int Math Res Notices

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“…Note that the relations in Map(Σ k g ) obtained by [34], [59] and [71] give us many concrete examples of Stein fillable contact structures satisfying the above assumption. Furthermore, there are non-planar contact 3-manifolds satisfying the assumption (e.g.…”

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confidence: 98%

Calabi–Yau caps, uniruled caps and symplectic fillings

Mak

Yasui

2017

Proc. London Math. Soc.

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We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, they imply that any exact filling of the standard contact structure on the unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinite family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b 2 .Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness properties and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero.

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“…In the genus 2 case, Onaran [15] has given relators for Σ 2,b , b 8. Analogously to the previous case, there can be no such relator for b > 12; it is not known whether relators exist for the remaining cases 9 b 12.…”

Section: Curve Configurations As Obstructions To Planaritymentioning

confidence: 99%

Mapping class group relations, Stein fillings, and planar open book decompositions

Wand

2011

Journal of Topology

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The aim of this paper is to use mapping class group relations to approach the ‘geography’ problem for Stein fillings of a contact 3‐manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that, for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple obstruction to planarity, which we interpret in terms of the existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non‐planar structures that cannot be shown to be non‐planar by other existing methods.

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On sections of genus two Lefschetz fibrations

Cited by 9 publications

References 6 publications

Arbitrarily Long Factorizations in Mapping Class Groups

Arbitrarily Long Factorizations in Mapping Class Groups

Calabi–Yau caps, uniruled caps and symplectic fillings

Mapping class group relations, Stein fillings, and planar open book decompositions

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