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2014 | Buch

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Contact and Symplectic Topology

herausgegeben von: Frédéric Bourgeois, Vincent Colin, András Stipsicz

Verlag: Springer International Publishing

Buchreihe : Bolyai Society Mathematical Studies

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Symplectic and contact geometry naturally emerged from the
mathematical description of classical physics. The discovery of new
rigidity phenomena and properties satisfied by these geometric
structures launched a new research field worldwide. The intense
activity of many European research groups in this field is reflected
by the ESF Research Networking Programme "Contact And Symplectic Topology" (CAST). The lectures of the Summer School in Nantes (June 2011) and of the CAST Summer School in Budapest (July 2012) provide a nice panorama of many aspects of the present status of contact and symplectic topology. The notes of the minicourses offer a gentle introduction to topics which have developed in an amazing speed in the recent past. These topics include 3-dimensional and higher dimensional contact topology, Fukaya categories, asymptotically holomorphic methods in contact topology, bordered Floer homology, embedded contact homology, and flexibility results for Stein manifolds.

Inhaltsverzeichnis

Frontmatter

Vladimir Igorevich Arnold and the Invention of Symplectic Topology

Abstract

In 1965, with a Comptes rendus note of Vladimir Arnold, a new discipline, symplectic topology, was born. In 1986, its (remarkable) first steps were reported by Vladimir Arnold himself. In the meantime…

Michèle Audin

Topological Methods in 3-Dimensional Contact Geometry

An Illustrated Introduction to Giroux’s Convex Surfaces Theory

Abstract

These notes provide an introduction to Giroux’s theory of convex surfaces in contact 3-manifolds and its simplest applications. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surface is encoded by finitely many curves on the surface. Then we will describe the bifurcations that happen in generic families of surfaces (with one or sometimes two parameters). We sketch a proof of the fact that the standard contact structure on S ³ is tight (due to Bennequin) and that all tight contact structures on S ³ are isotopic to it (due to Eliashberg).

Patrick Massot

A Beginner’s Introduction to Fukaya Categories

Abstract

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology.

Denis Auroux

Geometric Decompositions of Almost Contact Manifolds

Abstract

These notes are intended to be an introduction to the use of approximately holomorphic techniques in almost contact and contact geometry. We develop the setup of the approximately holomorphic geometry. Once done, we sketch the existence of the two main geometric decompositions available for an almost contact or contact manifold: open books and Lefschetz pencils. The use of the two decompositions for the problem of existence of contact structures is mentioned.

Francisco Presas

Higher Dimensional Contact Topology via Holomorphic Disks

Abstract

The aim of this notes is to explain some non-fillability results in higher dimensional contact topology, which are closely related to the question of how to define overtwistedness. We start with an overview of some basic examples and theorems known so far, comparing them with analogous results in dimension three. We will also describe an easy construction of non-fillable manifolds by Fran Presas. Then we will explain how to use holomorphic curves with boundary to prove the non-fillability results stated earlier. No a priori knowledge of holomorphic curves will be required; though many properties will only be quoted.

Klaus Niederkrüger

Contact Invariants in Floer Homology

Abstract

In a pair of seminal papers Peter Ozsváth and Zoltan Szabó defined a collection of homology groups associated to a 3-manifold they named Heegaard-Floer homologies. Soon after, they associated to a contact structure ξ on a 3-manifold, an element of its Heegaard-Floer homology, the contact invariant c(ξ). This invariant has been used to prove a plethora of results in contact topology of 3-manifolds. In this series of lectures we introduce and review some basic facts about Heegaard Floer Homology and its generalization to manifolds with boundary due to Andras Juhász, the Sutured Floer Homology. We use the open book decompositions in the case of closed manifolds, and partial open book decompositions in the case of contact manifolds with convex boundary to define contact invariants in both settings, and show some applications to fillability questions.

Gordana Matić

Notes on Bordered Floer Homology

Abstract

Bordered Heegaard Floer homology is an extension of Ozsváth-Szabós Heegaard Floer homology to 3-manifolds with boundary, enjoying good properties with respect to gluings. In these notes we will introduce the key features of bordered Heegaard Floer homology: its formal structure, a precise definition of the invariants of surfaces, a sketch of the definitions of the 3-manifold invariants, and some hints at the analysis underlying the theory. We also talk about bordered Heegaard Floer homology as a computational tool, both in theory and practice.

Robert Lipshitz, Peter Ozsváth, Dylan P. Thurston

Stein Structures: Existence and Flexibility

Abstract

This survey on the topology of Stein manifolds is an extract from the book of Cieliebak and Eliashberg (From Stein to Weinstein and Back—Symplectic Geometry of Affine Complex Manifolds, Colloquium Publications, vol. 59, 2012). It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfréd Rényi Institute of Mathematics, Budapest.

Kai Cieliebak, Yakov Eliashberg

Lecture Notes on Embedded Contact Homology

Abstract

These notes give an introduction to embedded contact homology (ECH) of contact three-manifolds, gathering together many basic notions which are scattered across a number of papers. We also discuss the origins of ECH, including various remarks and examples which have not been previously published. Finally, we review the recent application to four-dimensional symplectic embedding problems. This article is based on lectures given in Budapest and Munich in the summer of 2012, a series of accompanying blog postings at floerhomology.wordpress.com , and related lectures at UC Berkeley in Fall 2012. There is already a brief introduction to ECH in the article of M. Hutchings (in Proceedings of the 2010 ICM, vol. II, pp. 1022–1041, 2010), but the present notes give much more background and detail.

Michael Hutchings

A Topological Introduction to Knot Contact Homology

Abstract

Knot contact homology is a Floer-theoretic knot invariant derived from counting holomorphic curves in the cotangent bundle of \(\mathbb{R}^{3}\) with Lagrangian boundary condition on the conormal bundle to the knot. Among other things, this can be used to produce a three-variable polynomial that detects the unknot and conjecturally contains many known knot invariants; a different part of the package yields an effective invariant of transverse knots in \(\mathbb{R}^{3}\).

In these notes we will describe knot contact homology and the geometry and algebra behind it, as well as connections to other knot invariants, transverse knot theory, and physics. Topics to be treated along the way include: Legendrian contact homology for Legendrian knots in R ³; the conormal construction and Legendrian contact homology in five dimensions; a combinatorial formulation of knot contact homology in terms of braids; transverse homology, a filtered version associated to transverse knots; and relations to the HOMFLY polynomial (and knot homologies) and to recent work of Vafa and others in string theory.

Lenhard Ng

Titel: Contact and Symplectic Topology
herausgegeben von: Frédéric Bourgeois
Vincent Colin
András Stipsicz
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-02036-5
Print ISBN: 978-3-319-02035-8
DOI: https://doi.org/10.1007/978-3-319-02036-5

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Vladimir Igorevich Arnold and the Invention of Symplectic Topology

Topological Methods in 3-Dimensional Contact Geometry

A Beginner’s Introduction to Fukaya Categories

Geometric Decompositions of Almost Contact Manifolds

Higher Dimensional Contact Topology via Holomorphic Disks

Contact Invariants in Floer Homology

Notes on Bordered Floer Homology

Stein Structures: Existence and Flexibility

Lecture Notes on Embedded Contact Homology

A Topological Introduction to Knot Contact Homology

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