Representing Dehn twists with branched coverings

TL;DR

This paper demonstrates that all homologically non-trivial Dehn twists on surfaces with boundary can be represented as lifts of braid group half-twists via branched coverings, linking surface mapping classes to braid theory.

Contribution

It establishes a method to realize Dehn twists as lifts of braid group half-twists through branched coverings, extending to surfaces with disconnected boundaries.

Findings

Any homologically non-trivial Dehn twist is a lift of a braid half-twist.

Any Lefschetz fibration on B^2 is a branched cover of B^2 x B^2.

The approach applies to surfaces with disconnected boundary.

Abstract

We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B^2 is a branched covering of B^2 x B^2.