Compact Stein surfaces with boundary as branched covers of B 4

Abstract

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B 4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.

Families of contact 3-manifolds with arbitrarily large Stein fillings

“…. ∪ h m , where X 0 consists of 0-and 1-handles and each h i , 1 ≤ i ≤ m, is a 2-handle attached to [25], also see Akbulut-Ozbagci [1]). An oriented compact 4-manifold with boundary is a Stein surface, up to orientation preserving diffeomorphisms, if and only if it admits an allowable Lefschetz fibration over the 2-disk, a.k.a "PALF".…”

“…Following the works of Eliashberg and Gompf on handle decompositions of compact Stein manifolds, Loi and Piergallini, proved that any Stein domain admits a Lefschetz fibration structure [25] (and an alternative proof was later given by Akbulut and Ozbagci [1]). Moreover, the Stein structure on an allowable Lefschetz fibration can be chosen so that the contact structure it induces on the boundary agrees with the one that the Thurston-Winkelnkemper construction would hand when applied to the natural open book induced by the Lefschetz fibration on the boundary.…”

Fillings of Genus–1 Open Books and 4–Braids

“…Building on Eliashberg's topological characterization of Stein fillings and the work of Rudolph on braided surfaces, they showed that every Stein filling come from an allowable Lefschetz fibration on X [26] (also see [1], and [8] for a further generalization to Lefschetz fibrations over arbitrary compact surfaces with non-empty boundaries), which arise as a branched covering of the Stein 4-ball along a braided surface S. On the other hand, the works of Rudolph [32] and Boileau and Orevkov [10] established that quasipositive links are precisely those which are oriented boundaries of smooth pieces of complex analytic curves in the unit 4-ball, realized as braided surfaces.…”

“…By the pioneering works of Rudolph [32] and Boileau and Orevkov [10], these links are characterized as oriented boundaries of smooth pieces of complex analytic curves in the unit 4-ball -which can be realized as braided surfaces. An intimate connection between fillings of contact 3-manifolds and quasipositive braids is provided by Loi and Piergallini [26], who showed that every Stein filling of a contact 3-manifold is a branched cover of the Stein 4-ball along a braided surface.…”

Stein fillings of homology 3-spheres and mapping class groups

“…If the monodromy (t α1 , t α2 , · · · , t αm ) transforms another m-tuple (t α ′ 1 , t α ′ 2 , · · · , t α ′ m ) by Hurwitz moves and total conjugations, we write Loi and Piergallini [17], and Akbulut and Ozbagci [1] showed that every open book decomposition with monodromy which has a positive factorization supports a Stein fillable contact structure, and every Stein fillable contact structure is supported by an open book decomposition with monodromy which has a positive factorization. It can be easily check that, for any PALF f : X → D 2 with fiber Σ, we obtain the open book decomposition with page Σ and monodromy the total monodromy of f .…”