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The groundbreaking results of the near past - Donaldson's result on Lef schetz pencils on symplectic manifolds and Giroux's correspondence be tween contact structures and open book decompositions - brought a top ological flavor to global symplectic and contact geometry. This topological aspect is strengthened by the existing results of Weinstein and Eliashberg (and Gompf in dimension 4) on handle attachment in the symplectic and Stein category, and by Giroux's theory of convex surfaces, enabling us to perform surgeries on contact 3-manifolds. The main objective of these notes is to provide a self-contained introduction to the theory of surgeries one can perform on contact 3-manifolds and Stein surfaces. We will adopt a very topological point of view based on handlebody theory, in particular, on Kirby calculus for 3- and 4-dimensionalmanifolds. Surgery is a constructive method by its very nature. Applying it in an intricate way one can see what can be done. These results are nicely com plemented by the results relying on gauge theory - a theory designed to prove that certain things cannot be done. We will freely apply recent results of gauge theory without a detailed introduction to these topics; we will be content with a short introduction to some forms of Seiberg-Witten theory and some discussions regarding Heegaard Floer theory in two Appendices.
List of contents
1. Introduction.- 2. Topological Surgeries.- 3. Symplectic 4-Manifolds.- 4. Contact 3-Manifolds.- 5. Convex Surfaces in Contact 3-Manifolds.- 6. Spinc Structures on 3- and 4-Manifolds.- 7. Symplectic Surgery.- 8. Stein Manifolds.- 9. Open Books and Contact Structures.- 10. Lefschetz Fibrations on 4-Manifolds.- 11. Contact Dehn Surgery.- 12. Fillings of Contact 3-Manifolds.- 13. Appendix: Seiberg-Witten Invariants.- 14. Appendix: Heegaard Floer Theory.- 15. Appendix: Mapping Class Groups.
Summary
The groundbreaking results of the near past - Donaldson's result on Lef schetz pencils on symplectic manifolds and Giroux's correspondence be tween contact structures and open book decompositions - brought a top ological flavor to global symplectic and contact geometry. This topological aspect is strengthened by the existing results of Weinstein and Eliashberg (and Gompf in dimension 4) on handle attachment in the symplectic and Stein category, and by Giroux's theory of convex surfaces, enabling us to perform surgeries on contact 3-manifolds. The main objective of these notes is to provide a self-contained introduction to the theory of surgeries one can perform on contact 3-manifolds and Stein surfaces. We will adopt a very topological point of view based on handlebody theory, in particular, on Kirby calculus for 3- and 4-dimensionalmanifolds. Surgery is a constructive method by its very nature. Applying it in an intricate way one can see what can be done. These results are nicely com plemented by the results relying on gauge theory - a theory designed to prove that certain things cannot be done. We will freely apply recent results of gauge theory without a detailed introduction to these topics; we will be content with a short introduction to some forms of Seiberg-Witten theory and some discussions regarding Heegaard Floer theory in two Appendices.
Additional text
From the reviews:
"This book introduces the reader to modern trends in low-dimensional contact and symplectic geometry. … They also touch on and hint at many other interesting topics. This book is a highly recommended introduction to this exciting circle of ideas." (John B. Etnyre, Mathematical Reviews, Issue 2005 k)
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From the reviews:
"This book introduces the reader to modern trends in low-dimensional contact and symplectic geometry. ... They also touch on and hint at many other interesting topics. This book is a highly recommended introduction to this exciting circle of ideas." (John B. Etnyre, Mathematical Reviews, Issue 2005 k)
