A new construction of symplectic manifolds

Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.

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Lectures on Symplectic Geometry

Motivation: Mechanics and optics Billiard in the circle and the square Billiard ball map and integral geometry Billiards inside conics and quadrics Existence and non-existence of caustics Periodic trajectories Billiards in polygons Chaotic billiards Dual billiards Bibliography Index.

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Geometry and billiards

Intrigue Holonomic approximation: Jets and holonomy Thom transversality theorem Holonomic approximation Applications Differential relations and Gromov's $h$-principle: Differential relations Homotopy principle Open Diff $V$-invariant differential relations Applications to closed manifolds The homotopy principle in symplectic geometry: Symplectic and contact basics Symplectic and contact structures on open manifolds Symplectic and contact structures on closed manifolds Embeddings into symplectic and contact manifolds Microflexibility and holonomic $\mathcal{R}$-approximation First applications of microflexibility Microflexible $\mathfrak{U}$-invariant differential relations Further applications to symplectic geometry Convex integration: One-dimensional convex integration Homotopy principle for ample differential relations Directed immersions and embeddings First order linear differential operators Nash-Kuiper theorem Bibliography Index.

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Introduction to the h-Principle

On Seifert-manifolds

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

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An introduction to contact topology

We prove that every closed, connected contact 3-manifold can be obtained from S 3 with its standard contact structure by contact (±1)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

A Legendrian surgery presentation of contact 3-manifolds

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact (\pm 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S3 by a sequence of such contact (\pm 1)-surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into (\pm 1)-surgeries. We use this to give explicit surgery diagrams for all contact structures on S3 and S1 \times S2, as well as all overtwisted contact structures on arbitrary closed, orientable 3-manifolds. This amounts to a new proof of the Lutz-Martinet theorem that each homotopy class of 2-plane fields on such a manifold is represented by a contact structure.

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Surgery diagrams for contact 3-manifolds

We prove that every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact surgery of coefficient plus or minus 1 along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in slightly stronger form).

1. IntroductionIt has been known for some time that contact structures show a high degree of topological flexibility in the sense that many topological operations can be performed on contact manifolds while preserving the contact property. For instance, Martinet [14] used a surgery description of 3-manifolds to show that every closed, oriented 3-manifold admits a contact structure, and alternative proofs of this result were given later by Thurston and Winkelnkemper [18], who based their proof on an open book decomposition, and Gonzalo [8], who used branched covers. These, however, are all strictly 3-dimensional constructions.

Hansjörg Geiges

Papers

An introduction to contact topology

A Legendrian surgery presentation of contact 3-manifolds

Surgery diagrams for contact 3-manifolds

A Legendrian surgery presentation of contact 3-manifolds

Constructions of contact manifolds