Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory (Cambridge Tracts in Mathematics, Series Number 220)

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.-- Publisher's description "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." -- Prové de l'editor Cover......Page 1 CAMBRIDGE TRACTS IN MATHEMATICS......Page 3 Lectures on Contact 3-Manifolds, Holomorphic Curvesand Intersection Theory......Page 5 Copyright......Page 6 Contents......Page 7 Preface......Page 9 Acknowledgments......Page 11 Introduction: Motivation......Page 13 Lecture 1.Closed Holomorphic Curves in Symplectic4-Manifolds......Page 23 Lecture 2 Intersections, Ruled Surfaces, and Contact Boundaries......Page 38 Lecture 3.Asymptotics of Punctured Holomorphic Curves......Page 56 Lecture 4.Intersection Theory for PuncturedHolomorphic Curves......Page 74 Lecture 5.Symplectic Fillings of Planar Contact3-Manifolds......Page 89 Appendix A.Properties of Pseudoholomorphic Curves......Page 106 Appendix B.Local Positivity of Intersections......Page 120 Appendix C.A Quick Survey of Siefring’s IntersectionTheory......Page 170 References......Page 189 Index......Page 194 Based on a series of lectures for students in topology, this book provides an entry point into the intersection theory of punctured holomorphic curves. Appendices featuring quick reference guides for applying the theory and self-contained proofs of key technical results also make it a valuable resource for researchers Motivation -- Closed holomorphic curves in symplectic 4-manifolds -- Intersections, ruled surfaces, and contact boundaries -- Asymptotics of punctured holomorphic curves -- Intersection theory for punctured holomorphic curves -- Symplectic fillings of planar contact 3-manifolds An accessible introduction to the intersection theory of punctured holomorphic curves and its applications in topology