Families of contact 3-manifolds with arbitrarily large Stein fillings

TL;DR

This paper constructs large families of contact 3-manifolds with infinitely many Stein fillings that have unbounded Euler characteristics and small signatures, challenging previous conjectures in the field.

Contribution

It introduces a generalized framework for constructing Stein fillings on Lefschetz fibrations over various base surfaces and extends contact structure constructions to spinal open books.

Findings

Existence of contact 3-manifolds with infinitely many Stein fillings

Stein fillings with arbitrarily large Euler characteristics

Stein fillings with arbitrarily small signatures

Abstract

We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures ---which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].