Burak Ozbagci

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.

LEFSCHETZ FIBRATIONS ON COMPACT STEIN SURFACES

Let M be a compact Stein surface with boundary. We show that M admits infinitely many nonequivalent positive allowable Lefschetz fibrations over D 2 with bounded fibers.

Invariants of contact structures from open books

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

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].)

ON THE TOPOLOGY OF COMPACT STEIN SURFACES

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over S 2 . (2) There exists a minimal elliptic fibration over D 2 , which is not Stein. (3) The circle bundle over a genus n � 2 surface with euler number e = 1 admits at least n+ 1 mutually non-homeomorphic simply-connected Stein fillings. (4) Any surface bundle over S 1 , whose fiber is a closed surface of genus n � 1 can be embedded into a closed symplectic 4-manifold, splitting the symplectic 4-manifold into two pieces both of which have positive b + . (5) Every closed, oriented connected 3-manifold has a weakly symplectically fillable double cover, branched along a 2-component link.

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g ≥ 3 and h ≥ 2. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

Contact 3-manifolds with infinitely many Stein fillings

Infinitely many contact 3-manifolds each admitting infinitely many, pairwise non-diffeomorphic Stein fillings are constructed. We use Lefschetz fibrations in our constructions and compute their first homologies to distinguish the fillings.

Commutators, Lefschetz fibrations and the signatures of surface bundles ☆

Abstract We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with nonzero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.

Exotic Stein fillings with arbitrary fundamental group

Let G be a finitely presentable group. We provide an infinite family of homeomorphic but pairwise non-diffeomorphic, symplectic but non-complex closed 4-manifolds with fundamental group G such that each member of the family admits a Lefschetz fibration of the same genus over the two-sphere. As a corollary, we also show the existence of a contact 3-manifold which admits infinitely many homeomorphic but pairwise non-diffeomorphic Stein fillings such that the fundamental group of each filling is isomorphic to G. Moreover, we observe that the contact 3-manifold above is contactomorphic to the link of some isolated complex surface singularity equipped with its canonical contact structure.