András I. Stipsicz

Surgery on Contact 3-Manifolds and Stein Surfaces

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.

Ozsváth–Szábo invariants and tight contact three-manifolds I

Let S 3 r (K) be the oriented 3-manifold obtained by rational r-surgery on a knot K ⊂ S 3 . Using the contact Ozsvath-SzabO invariants we prove, for a class of knots K containing all the algebraic knots, that S 3 r (K)carries positive, tight contact structures for every r ≠ 2g s (K) - 1, where g s (K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Σ(2,3,4) and -Σ(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m ∈ N there exists a Seifert fibered rational homology 3-sphere M m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.

Heegard Floer invariants of Legendrian knots in contact three-manifolds

We define invariants of null–homologous Legendrian and transverse knots in contact 3–manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non–loose knots in overtwisted 3–spheres. Moreover, we apply the invariants to find transversely non–simple knot types in many overtwisted contact 3–manifolds.

Concordance homomorphisms from knot Floer homology

Abstract We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3 . These invariants can be used to give homomorphisms from the smooth concordance group C to Z , giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus2 Lefschetz fibrations over S2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures. In this paper we will prove the following theorem. Theorem 1.1. There are infinitely many (pairwise nonhomeomorphic) 4-manifolds which admit genus-2 Lefschetz fibrations but do not carry complex structure with either orientation. Matsumoto [6] showed that S×T #4CP 2 admits a genus-2 Lefschetz fibration over S with global monodromy (β1, ..., β4), where β1, ..., β4 are the curves indicated by Figure 1. (For definitions and details regarding Lefschetz fibrations see [6], [5].)

Contact surgeries and the transverse invariant in knot Floer homology

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3.

An exotic smooth structure on CP²#6CP²

We construct smooth 4–manifolds homeomorphic but not diffeomorphic to CP#6CP2 . AMS Classification numbers Primary: 53D05, 14J26 Secondary: 57R55, 57R57

On the existence of tight contact structures on Seifert fibered 3-manifolds

We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.

Planar open books and Floer homology

Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre.

Seifert fibered contact three-manifolds via surgery

Using contact surgery we dene families of contact structures on certain Seifert bered three{manifolds. We prove that all these contact structures are tight using contact Ozsv ath{Szab o invariants. We use these examples to show that, given a natural number n, there exists a Seifert bered three{manifold carrying at least n pairwise non{isomorphic tight, not llable contact structures.