Surgery is the most effective way of constructing manifolds. This is especially true in dimensions 3 and 4, where Kirby calculus provides a method for manipulating surgery diagrams. The groundbreaking results of Donaldson (on Lefschetz fibrations) and Giroux (on open book decompositions) now allow one to incorporate analytic structures into these diagrams: symplectic or Stein structures in the 4-dimensional case, contact structures in the 3-dimensional situation. This volume gives an introduction to the surgery techniques adapted to these additional structures. The necessary topological background on Lefschetz fibrations and open book decompositions is developed in the book. Also included are rapid introductions to the basics and applications of Seiberg--Witten and Heegaard Floer theories.

Klappentext

This book is about an investigation of recent developments in the field of sympletic and contact structures on four- and three-dimensional manifolds from a topologist's point of view. In it, two main issues are addressed: what kind of sympletic and contact structures we can construct via surgery theory and what kind of sympletic and contact structures are not allowed via gauge theory and the newly invented Heegaard-Floer theory.

Inhalt

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.