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

Chris Wendl
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics
[Reviewed by
Chris Seaton
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This book is an introduction to the intersection theory of punctured J-holomorphic curves in symplectic 4-manifolds with boundary and applications to symplectic cobordisms and fillings of contact 3-manifolds.  The main theme of the text is the use of the moduli space and intersection theory of punctured J-holomorphic curves to understand the structure of a symplectic 4-manifold, locally as a foliation and globally as a bordered Lefschetz fibration along with the induced open book decomposition at the boundary. After summarizing the situation for closed J-holomorphic curves, the main body of the text is a careful exposition of the asymptotic behavior and intersection theory of punctured J-holomorphic curves. The text is a well-written introduction to the subject that is appropriate for advanced graduate students with a background in algebraic topology and some differential geometry. As a researcher who works in nearby fields but was only passingly familiar with the topics covered in the text, I found the book an inviting exposition of the subject, with the core material well explained and including many brief discussions of related topics as invitations to further reading.
A notable feature of the style of the book is the effort the author puts into motivating the subject and each result along the way. The introductory “Motivation” section illustrates the topic of the text by stating and sketching proofs of two results of Gromov and McDuff, one in the closed case and in the open case, about recovering the global structure of a symplectic 4-manifold from information about its submanifolds or behavior at infinity, respectively. Thereafter, the author is careful to explain the goals and obstacles of each step and then remind the reader of them when stating new definitions or technical results. This is the case both locally and globally. Globally, the first two lectures cover the intersection theory of J-holomorphic curves in symplectic 4-manifolds and applications in the closed case, then introduce contact manifolds, symplectic cobordisms, and punctured J-holomorphic curves, using the closed case as a careful set-up for the investigations of these latter topics in the remaining three lectures. Locally, many results are stated in order to motivate the technology needed to prove them and then proven later, and the proofs of some of the more technical results are left to the appendices. This leads to a non-linear structure but makes the lectures a very pleasant and understandable read.
The book focuses on the geometric picture and usually avoids analysis. Much of the necessary background is surveyed or covered in the appendices, which make up half of the book and include summaries of the relevant properties of closed and punctured J-holomorphic curves and relevant analytic results. An additional appendix summarizes Siefring’s intersection theory of punctured J-holomorphic curves in a more linear manner than the lectures to serve as a reference. The lectures and some of the appendices include a handful of exercises as well as well-chosen figures that illustrate the key constructions.
Chris Seaton is Professor of Mathematics at Rhodes College.