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Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory

Part of Cambridge Tracts in Mathematics

  • Date Published: March 2020
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9781108759588

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  • Intersection theory has played a prominent role in the study of closed symplectic 4-manifolds since Gromov's famous 1985 paper on pseudoholomorphic curves, leading to myriad beautiful rigidity results that are either inaccessible or not true in higher dimensions. Siefring's recent extension of the theory to punctured holomorphic curves allowed similarly important results for contact 3-manifolds and their symplectic fillings. Based on a series of lectures for graduate students in topology, this book begins with an overview of the closed case, and then proceeds to explain the essentials of Siefring's intersection theory and how to use it, and gives some sample applications in low-dimensional symplectic and contact topology. The appendices provide valuable information for researchers, including a concise reference guide on Siefring's theory and a self-contained proof of a weak version of the Micallef–White theorem.

    • Presents an accessible introduction to the intersection theory of punctured holomorphic curves and its applications
    • Features self-contained proofs of the similarity principle and positivity of intersections, some of which have not appeared elsewhere in the literature
    • Includes a 'quick reference' appendix summarizing the main results needed to use the intersection theory in applications
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    Product details

    • Date Published: March 2020
    • format: Adobe eBook Reader
    • isbn: 9781108759588
    • contains: 20 b/w illus. 40 exercises
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • Table of Contents

    Introduction
    1. Closed holomorphic curves in symplectic 4-manifolds
    2. Intersections, ruled surfaces and contact boundaries
    3. Asymptotics of punctured holomorphic curves
    4. Intersection theory for punctured holomorphic curves
    5. Symplectic fillings of planar contact 3-manifolds
    Appendix A. Properties of pseudoholomorphic curves
    Appendix B. Local positivity of intersections
    Appendix C. A quick survey of Siefring's intersection theory
    References
    Index.

  • Author

    Chris Wendl, Humboldt-Universität zu Berlin
    Chris Wendl is Professor of Differential Geometry and Global Analysis at Humboldt University of Berlin. He is the author of Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds (2018), and a recent recipient of an ERC Consolidator Grant.

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