Stein-fillability and positivity in the mapping class group

TL;DR

This paper constructs non-positive open books supporting Stein-fillable contact structures and proves that for non-planar surfaces, the monoid of Stein-fillable monodromies equals the positive monodromies, answering a longstanding question.

Contribution

It provides a complete characterization of monodromies supporting Stein-fillable structures in relation to positivity, especially for non-planar surfaces.

Findings

Constructed infinite family of non-positive open books supporting Stein-fillable structures.

Proved monoid of Stein-fillable monodromies equals positive monodromies only for planar surfaces.

Resolved a long-standing question about the mapping class group and Stein-fillability.

Abstract

We construct an infinite family of non-positive open books with once-punctured torus pages that support Stein-fillable contact structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.