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DECOMPOSITIONS AND CONTACT STRUCTURES

This theorem plays a pivotal role in studying cobordisms of contact structures and understanding filling properties of contact structures, see [2, 6, 13, 14, 16, 19]. This better understanding of fillings leads to various topological applications of contact geometry. Specifically, the much studied property P for knots was established by P. Kronheimer and T. Mrowka in [28]. A non-trivial knot has property P if non-trivial surgery on it never gives a homotopy sphere. In addition P. Ozsváth and Z. Szabó in [34] gave an alternate proof of a characterization of the unknot via surgery which was originally established in [29]. This characterization says that the unknot is the only knot on which p-surgery yields −L(p, 1). Moreover, in [34] it is shown that the Thurston norm is determined by Heegaard Floer Homology. Ideally the reader should be familiar with low-dimensional topology at the level of, say [36]. In particular, we will assume familiarity with Dehn surgery, mapping tori and basic algebraic topology. At various points we also discuss branch coverings, Heegaard splittings and other notions; however, the reader unfamiliar with these notions should be able to skim these parts of the paper without missing much, if any, of the main line of the arguments. Since diffeomorphisms of surfaces play a central role in much of the paper and specific conventions are important we have included an Appendix discussing basic facts about this. We also assume the reader has some familiarity with contact geometry. Having read [15] should be sufficient background for this paper. In order to accommodate the reader with little background in contact geometry we have included brief discussions, scattered throughout the paper, of all the necessary facts. Other good introductions to contact geometry are [1, 20], though a basic understanding of convex surfaces is also useful but is not covered in these sources.