Burak Ozbagci – författare

This book is about an investigation of recent developments in the field of symplectic and contact structures on four and three dimensional manifolds, respectively, from a topologist�s point of view. The level of the book is appropriate for advanced graduate students. There is no doubt that symplectic and contact structures are in the center of attention nowadays for low-dimensional geometers and topologists. In this volume there are two main issues that are addressed: what kind of symplectic and contact structures we can construct via surgery theory and what kind of symplectic and contact structures are not allowed via gauge theory and newly-invented Heegaard-Floer theory. It turns out that interesting results about contact structures can be obtained for example when the "classical" surgery theory is coupled with the Heegaard-Floer theory. The close relationship between symplectic and contact structures is another theme in the volume which naturally arises when one wants to perform symplectic cut and paste operation.The material in the volume is based on two groundbreaking results of the near past Donaldson's result on the existence of Lefschetz pencils on symplectic four manifolds and Giroux' correspondence between contact structures and open book decompositions on three manifolds. The volume makes an attempt to illustrate some consequences of these results and incorporate them with the new developments in the Heegaard-Floer theory, especially the Ozsvath-Szabo contact invariants.

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.