Topological complexity of symplectic 4-manifolds and Stein fillings

TL;DR

This paper demonstrates that the Euler characteristic of symplectic 4-manifolds and Stein fillings cannot be bounded solely by the genus of compatible Lefschetz pencils or open books, respectively, revealing unbounded topological complexity.

Contribution

It constructs the first examples of arbitrarily long positive Dehn twist factorizations, solving open problems about bounds related to Lefschetz pencils and Stein fillings.

Findings

No a priori bound on Euler characteristic from genus of Lefschetz pencils.

No similar bound for Stein fillings from open book genus.

Existence of arbitrarily long positive Dehn twist factorizations.

Abstract

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by Auroux, Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci and Stipsicz, independently.