Positive factorizations of mapping classes

TL;DR

This paper investigates the maximal length of positive Dehn twist factorizations in surface mapping classes and their implications for symplectic 4-manifolds and Lefschetz pencils, providing complete classifications and bounds.

Contribution

It completely characterizes which boundary multitwists admit arbitrarily long positive factorizations and determines the existence and maximal Euler characteristic of genus-g Lefschetz pencils with n base points.

Findings

Boundary multitwists with arbitrarily long positive factorizations are classified.

Existence of genus-g Lefschetz pencils with n base points is determined.

Only symplectic 4-manifolds of general type can have arbitrarily large topology.

Abstract

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.