\(J\)-holomorphic curves and symplectic topology (Q2897327)

This book is the second edition of [Zbl 1064.53051] written by two top specialists on this field. As said in the preface to this edition, the main purpose of it is to correct various errors in the first edition and to update the discussions of current work in the field.NEWLINENEWLINEThe main change of Chapter 1 is that two passages (``Applications'' and ``Quantum cohomology'') from Section 1.4 of the first edition were extended into the new Section 1.5: Application and further developments. Here, some new references and recent relevant results are mentioned.NEWLINENEWLINEChapter 2 has no big changes. Minor errors were corrected; a few sentences, and Exercise 2.5.3 and Remark 2.5.6 were newly added.NEWLINENEWLINEIn Chapter 3, Remark 3.12 was newly added, and Section 3.4 was rewritten. Other changes are minor.NEWLINENEWLINEAs to Chapter 4, Proposition 4.1.5 was newly added, and may lead to Proposition 4.1.5 even if the almost complex structure \(J\) is only \(\omega\)-tame. Remarks 4.3.2 and 4.3.6 were newly added. But a main change is a revised form of Section 4.4, where the statement and proof of the isoparametric inequality was corrected.NEWLINENEWLINEIn Chapter 5, the proofs of Theorem 5.2.2 and Theorem 5.3.1 became clearer and more rigorous. Remark 5.2.3 and Exercise 5.3.3 were newly added.NEWLINENEWLINEThe main changes of Chapter 6 is to make the proof of Theorem 6.2.6(II) clearer and more rigorous.NEWLINENEWLINEIn Chapter 7, the proofs of Theorem 7.2.3 and Proposition 7.4.8 were rewritten. In addition, Exercise 7.2.4 was newly added.NEWLINENEWLINEIn Chapter 8, the authors added Example 8.4.3 and Remark 8.6.13, where some relevant developments and work were mentioned.NEWLINENEWLINEChapter 9 reviews many applications in symplectic topology. Some of them are proved in detail, but most of the important results were only outlined or the references of these results are pointed out.NEWLINENEWLINEIn Chapter 10, Section 10.9 was newly added. It is shown for an almost complex structure-\(J\)-independent \(z\in S^2\) that a new geometric formulation of the gluing theorem may easily be obtained. Such a result is enough for one to complete the proof of associativity of quantum multiplication.NEWLINENEWLINEIn Chapter 11, more discussions about coefficient rings for quantum cohomology rings and more examples of the quantum cohomology are given.NEWLINENEWLINEChapter 12 increases some new references in Floer homology, such as functoriality of Donaldson category under Lagrangian correspondence and Heegaard-Floer theory.NEWLINENEWLINEThe changes of the appendices mainly appear in Appendices C, D.NEWLINENEWLINEThe discussions on Serre duality in the final passage of Section C.1 largely expands the original one. The proof of the sum formula for the Fredholm index of Theorem C.4.2 was rewritten. In Section C.5.2, the authors added the proof of integrablity of almost complex structures in dimension two and a second proof of Theorem C.5.1.NEWLINENEWLINETheorem D.2.6 and its proof are a complete rewriting of the passage below (D.2.4) in the first edition. The final two passages on deleting marked points and canonical sections are deleted because they are given in other form in the new Section D.6 on the cohomology of the moduli space of stable curves of genus zero. The material of Sections D.4 and D.5 from the first edition was extended into the present Section D.4.NEWLINENEWLINETo sum up, the authors put a lot of effort into the new edition to make it as clear and readable as possible. This book will be very useful to anyone wanting to study symplectic geometry and topology.NEWLINENEWLINEFinally, there also exist several other important books published in recent years. Their contents are complementary to this book to some extent, where new ideas and techniques in this field were developed [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. II. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53003); \textit{M. Audin} and \textit{M. Damian}, Théorie de Morse et homologie de Floer. Les Ulis: EDP Sciences; Paris: CNRS Éditions (2010; Zbl 1217.57001); \textit{K. Cieliebak} and \textit{Y. Eliashberg}, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1262.32026)].