Braided Multisections and Aspherical Stein Fillings

Abstract

We construct Stein fillings diffeomorphic to complements of a connected symplectic unbranched multisection and a regular sphere fiber in the ruled surface $Σ_g\times S^2$. For every $g\geq2$ and every multisection degree $p\geq4$, a fixed contact three-manifold admits infinitely many pairwise nonhomeomorphic aspherical fillings of this form. Their first and second integral homology groups are $\Z^{2g}\oplus\Z/p$ and $\Z^{(2g-1)(p-1)}$, respectively, and their intersection pairings vanish. We vary the braid monodromy while preserving the marked contact boundary. Unbounded torsion orders in the first homology of regular unbranched covers of degree $p2^{p-1}$ detect infinitely many isomorphism types of fundamental groups. We also give an equivariant Weinstein handle construction for the cyclic filling $W_{g,4}$. The contact boundaries are supported by spinal open books with planar pages and have positive ordinary support genus.