## Introduction to Symplectic TopologySymplectic structures underlie the equations of classical mechanics and their properties are reflected in the behaviour of a wide range of physical systems. Over the last few years powerful new methods in analysis and topology have led to the development of the modern global theory ofsymplectic topology, including several striking and important results. At its publication in 1995, Introduction to Symplectic Topology was the first comprehensive introduction to the subject, and has since become an established text in this fast-developing area of mathematics. This second editionhas been significantly revised and expanded, with new references and examples added and theorems included or revised. A section has been included on new developments in the subject, and there is a more extensive discussion of Taubes and Donaldson's recent contributions to the subject.From reviews of the first edition: '...an authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.' Book Reviews, AMS |

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### Contents

Introduction | 1 |

From classical to modern | 11 |

Linear symplectic geometry | 37 |

Symplectic manifolds | 81 |

Almost complex structures | 117 |

SYMPLECTIC MANIFOLDS | 146 |

Symplectic Fibrations | 197 |

Constructing Symplectic Manifolds | 233 |

Generating functions | 280 |

The group of symplectomorphisms | 311 |

The Arnold conjecture | 339 |

Symplectic capacities | 371 |

New directions | 417 |

458 | |

473 | |

Areapreserving diffeomorphisms | 265 |

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### Common terms and phrases

4-manifold Arnold conjecture ball blow-up boundary Chapter Chern class choose closed cohomology class coisotropic compact symplectic compactly supported complex structure condition connected sum consider construction corresponding cotangent bundle critical points defined denote diffeomorphism differential dimension ellipsoids equation equivalent exact example Exercise exists fibration fibre finite fixed points follows function H geometry given Gromov Ham(M Hamiltonian flow Hamiltonian function Hamiltonian isotopy Hamiltonian symplectomorphisms Hamiltonian vector field Hence Hofer homology homomorphism homotopy hypersurface identity implies integral intersection invariant isomorphism J-holomorphic curves Kahler Lagrangian submanifold Lagrangian subspace Lemma Lie algebra loop matrix McDuff metric Moreover neighbourhood nondegenerate nonsqueezing theorem nontrivial orbits oriented phism plectic Proposition prove quotient result Riemann surface satisfies shows solution Symp(M symplectic action symplectic embedding symplectic fibration symplectic form symplectic manifold symplectic structure symplectic submanifold tangent topology torus transverse trivial unique vanishes vector space zero section