Holomorphic disks and three-manifolds invariants: Properties and applications

Ozsváth, Peter; Szabó, Zoltán

doi:10.4007/annals.2004.159.1159

Cited by 431 publications

(756 citation statements)

References 33 publications

(103 reference statements)

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“…There is also evidence that Heegaard-Floer homology is connected with orderability of the fundamental group of a closed 3-manifold [20,24,28]. In this paper we provide further evidence.…”

Section: Introductionsupporting

confidence: 65%

Ordered groups, eigenvalues, knots, surgery and L-spaces

Clay¹,

Rolfsen²

2011

Math. Proc. Camb. Phil. Soc.

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Abstract. We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a biordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard-Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

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“…There is also evidence that Heegaard-Floer homology is connected with orderability of the fundamental group of a closed 3-manifold [20,24,28]. In this paper we provide further evidence.…”

Section: Introductionsupporting

confidence: 65%

Ordered groups, eigenvalues, knots, surgery and L-spaces

Clay¹,

Rolfsen²

2011

Math. Proc. Camb. Phil. Soc.

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“…The above theorem is proved in Theorem 9.12 of [71]. A variant for Seiberg-Witten monopole Floer homology, with coefficients in Z/2Z is proved in [56].…”

Section: Basic Propertiesmentioning

confidence: 99%

“…It is proved in Section 10 of [71] that if Y is a rational homology threesphere and s is any Spin c structure over it, then HF ∞ (Y, s) ∼ = Z[U, U −1 ], thus, this invariant is not a very subtle invariant of three-manifolds. However, extra information can still be gleaned from the interplay between HF ∞ and HF + , with the help of some additional structure on Floer homology.…”

Section: 2mentioning

confidence: 99%

“…Broader summaries can be found in some of our other papers (c.f. [71], [79]). In Section 4 we describe in further detail the relationship between Heegaard Floer homology and knots, c.f.…”

Section: Some Background On Floer Homologymentioning

confidence: 99%

“…We have not attempted to summarize all of its properties; for additional properties, see [71], [79], [73].…”

Section: Basic Propertiesmentioning

confidence: 99%

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Heegaard Diagrams and Holomorphic Disks

Ozsváth

Szabó

International Mathematical Series

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3-Dimensional Methods in Contact Geometry

Honda

International Mathematical Series

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Introduction 1.1. First examples 1.2. Legendrian knots 1.3. Tight vs. overtwisted 1.4. Classification of contact structures 1.5. A criterion for tightness 1.6. Relationship with foliation theory 2. Convex surfaces 2.1. Characteristic foliations 2.2. Convexity 2.3. Convex decomposition theory 3. Bypasses 3.1. Definition and examples 3.2. Finding bypasses 4. Classification of tight contact structures on lens spaces 4.1. The standard neighborhood of a Legendrian curve 4.2. Lens spaces 4.3. Solid tori 4.4. Completion of the proof of Theorems 4.2 and 4.3 5. Gluing 5.1. Basic examples with trivial state transitions 5.2. More complicated example 5.3 . Tightness and fillability ReferencesA contact manifold (M, ξ) is a (2n + 1)-dimensional manifold M equipped with a smooth maximally nonintegrable hyperplane field ξ ⊂ T M, i.e., locally ξ = ker α, where α is a 1-form which satisfies α ∧ (dα) n = 0. Since dα is a nondegenerate 2-form when restricted to ξ, contact geometry is customarily viewed as the odd-dimensional sibling of symplectic geometry. Although Date: December 31, 2003. 1 2 KO HONDAcontact geometry in dimensions ≥ 5 is still in an incipient state, contact structures in dimension 3 are much better understood, largely due to the fact that symplectic geometry in two dimensions is just the study of area. The goal of this article is to explain some of the recent developments in 3-dimensional contact geometry, with an emphasis on methods from 3-dimensional topology. Basic references include [Ae, El2, Et1, Ge]. The article [Kz] is similar in spirit to ours.Three-dimensional contact geometry lies at the interface between 3-and 4-manifold geometries, and has been an essential part of the flurry in low-dimensional geometry and topology over the last 20 years. In dimension 3, it relates to foliation theory and knot theory; in dimension 4, there are rich interactions with symplectic geometry. In both dimensions, there are relations with gauge theories such as Seiberg-Witten theory and Heegaard Floer homology. Acknowledgements. This manuscript grew out of a lecture series given at the Winter School in Contact Geometry in München in February 2003 and a minicourse given at the Geometry and Foliations 2003 conference, held at Ryokoku University in Kyoto in September 2003. I would like to thank Kai Cieliebak and Dieter Kotschick for the former, and Takashi Tsuboi for the latter, as well as for his hospitality during my visit to the University of Tokyo and the Tokyo Institute of Technology during the summer and fall of 2003. Much of the actual writing took place during this visit.

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Holomorphic disks and three-manifolds invariants: Properties and applications

Cited by 431 publications

References 33 publications

Ordered groups, eigenvalues, knots, surgery and L-spaces

Ordered groups, eigenvalues, knots, surgery and L-spaces

Heegaard Diagrams and Holomorphic Disks

3-Dimensional Methods in Contact Geometry

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