Compact Stein surfaces with boundary as branched covers of $\B^4$

TL;DR

This paper demonstrates that Stein surfaces with boundary can be represented as simple branched covers of the 4-ball with positive braided branch sets, linking Stein fillability of 3-manifolds to positive open-book decompositions.

Contribution

It establishes a correspondence between Stein surfaces with boundary and positive braided branched covers of the 4-ball, providing a new characterization of Stein fillable 3-manifolds.

Findings

Stein surfaces with boundary are equivalent to certain branched covers of ^4.

A 3-manifold is Stein fillable iff it admits a positive open-book decomposition.

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.