**Colloquium Publications**

Volume: 52;
2012;
726 pp;
Hardcover

MSC: Primary 53; 57; 37; 32;
Secondary 58; 14

Print ISBN: 978-0-8218-8746-2

Product Code: COLL/52.R

List Price: $109.00

AMS Member Price: $87.20

MAA Member Price: $98.10

**Electronic ISBN: 978-0-8218-9096-7
Product Code: COLL/52.R.E**

List Price: $109.00

AMS Member Price: $87.20

MAA Member Price: $98.10

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#### Supplemental Materials

# \(J\)-holomorphic Curves and Symplectic Topology: Second Edition

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*Dusa McDuff; Dietmar Salamon*

The theory of \(J\)-holomorphic curves has been of great importance
since its introduction by Gromov in 1985. In mathematics, its
applications include many key results in symplectic topology. It was
also one of the main inspirations for the creation of Floer
homology. In mathematical physics, it provides a natural context in
which to define Gromov–Witten invariants and quantum cohomology, two
important ingredients of the mirror symmetry conjecture.

The main goal of this book is to establish the fundamental theorems
of the subject in full and rigorous detail. In particular, the book
contains complete proofs of Gromov's compactness theorem for spheres,
of the gluing theorem for spheres, and of the associativity of quantum
multiplication in the semipositive case. The book can also serve as an
introduction to current work in symplectic topology: there are two
long chapters on applications, one concentrating on classical results
in symplectic topology and the other concerned with quantum
cohomology. The last chapter sketches some recent developments in
Floer theory. The five appendices of the book provide necessary
background related to the classical theory of linear elliptic
operators, Fredholm theory, Sobolev spaces, as well as a discussion of
the moduli space of genus zero stable curves and a proof of the
positivity of intersections of \(J\)-holomorphic curves in
four-dimensional manifolds. The second edition clarifies various
arguments, corrects several mistakes in the first edition, includes
some additional results in Chapter 10 and Appendices C and D, and
updates the references to recent developments.

#### Readership

Graduate students and research mathematicians interested in symplectic topology and geometry.

#### Table of Contents

# Table of Contents

## $J$-holomorphic Curves and Symplectic Topology: Second Edition

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the second edition ix10
- Preface xi12
- Introduction 116
- 𝐽-holomorphic curves 1732
- Moduli spaces and transversality 3954
- Compactness 7590
- Stable maps 115130
- Moduli spaces of stable maps 153168
- Gromov–Witten invariants 201216
- Hamiltonian perturbations 257272
- Applications in symplectic topology 295310
- Gluing 369384
- Quantum cohomology 417432
- Floer homology 487502
- Fredholm theory 531546
- Elliptic regularity 549564
- The Riemann–Roch theorem 579594
- Stable curves of genus zero 619634
- Singularities and intersections 653668
- Bibliography 695710
- List of symbols 711726
- Index 715730
- Back Cover Back Cover1744