Ozsva?th-Szabo? invariants of contact surgeries
In this thesis, I am going to deal with contact manifolds of dimension three: these are orientable manifolds with a plane field that is nowhere tangent to a surface. Contact manifolds split into two families, the overtwisted ones and the tight ones, according to the presence or absence of a certain embedded disc. While the overtwisted ones are classified by homotopy data only, tight contact structures are much harder to study, and only a handful of classification results are known. I am going to study what happens when doing contact surgeries along Legendrian links in S 3 : more specifically, I am going to look at contact manifolds through the eyes of Heegaard Floer homology, by computing their Ozsvath-Szab6 invariants. It's a classical result that doing negative contact surgeries along Legendrian links in ( S3, tst) yields back tight (in fact, Stein fillable) contact structures, so I am going to discuss positive contact surgeries. The main result gives necessary and sufficient conditions for the nonvanishing of the contact invariant, in terms of computable, integer-valued invariants of the knot and the surgery coefficient. On one hand, this gives many new examples of tight contact manifolds, and on the other hand it gives obstructions to fillability of contact manifolds in most cases. Some of the techniques developed in the proof also allow me to refine the result to deal with some positive rational surgeries. This work has been inspired by earlier works of Lisca and Stipsicz, and the Floertheoretic part is mostly a refinement of their results. In recent years, they dealt with the problem of computing the invariant for positive contact surgeries using "classical" Heegaard Floer homology: here I push further and us~ sutured and bordered Floer homology to get more control on some of their intermediate lemmas. I also turn some of their topological statements into algebraic statements about certain gluing maps associated to surgeries, acting on sutured Floer homology. On the topological side, the bulk of the thesis is a detailed study of the interactions between the Legendrian cabling construction and contact surgeries.
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