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

An accessible introduction to the intersection theory of punctured holomorphic curves and its applications in topology.

Chris Wendl (Author)

9781108497404, Cambridge University Press

Hardback, published 26 March 2020

194 pages, 20 b/w illus. 40 exercises
23.5 x 15.6 x 1.5 cm, 0.39 kg

'This is a comprehensive and outstanding research book on the Verlinde formula, conformal blocks, generalized theta functions and related topics.' Zhenbo Qin, MathSciNet

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.

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.

Subject Areas: Algebraic topology [PBPD], Differential & Riemannian geometry [PBMP], Complex analysis, complex variables [PBKD]