A genus zero Lefschetz fibration on the Akbulut cork

TL;DR

This paper constructs a genus zero positive allowable Lefschetz fibration on the Akbulut cork and uses it to distinguish exotic Stein surfaces via monodromy factorizations.

Contribution

It introduces a genus zero PALF on the Akbulut cork and applies it to differentiate exotic Stein surfaces through mapping class group factorizations.

Findings

Constructed a genus zero PALF on the Akbulut cork.

Established a correspondence between smooth structure differences and monodromy factorizations.

Extended the construction to infinitely many exotic Stein surface pairs.

Abstract

We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinitely many exotic pairs of compact Stein surfaces such that one is a cork twist of the other along an Akbulut cork. The difference of smooth structures on each of exotic pairs of compact Stein surface is interpreted as the difference of the corresponding positive factorizations in the mapping class group of a common surface of genus zero.