Periodic Surface Homeomorphisms and Contact Structures

TL;DR

This paper explores the relationship between periodic surface homeomorphisms, rational open books, and contact structures, providing combinatorial criteria for Stein fillability and constructing explicit symplectic fillings.

Contribution

It introduces a method to associate contact structures to periodic maps via marked data sets and establishes combinatorial conditions for Stein fillability.

Findings

Certain combinatorial data sets yield Stein fillable contact structures.

An analogue of Mori's construction for symplectic fillings is developed.

Sufficient conditions for Stein fillability based on monodromy positivity are proven.

Abstract

Periodic surface homemorphisms (diffeomorphisms) play a significant role in the the Nielsen-Thurston classification of surface homeomorphisms. Periodic surface homeomorphisms can be described (up to conjugacy) by using data sets which are combinatorial objects. In this article, we start by associating a rational open book to a slight modification of a given data set, called marked data set. It is known that every rational open book supports a contact structure. Thus, we can associate a contact structure to a periodic map and study the properties of it in terms combinatorial conditions on marked data sets. In particular, we prove that a class of data sets, satisfying easy-to-check combinatorial hypothesis, gives rise to Stein fillable contact structures. In addition to the above, we prove an analogue of Mori's construction of explicit symplectic filling for rational open books. We also prove a sufficient condition for Stein fillability of rational open books analogous to the positivity of monodromy in honest open books as in the result of Giroux and Loi-Piergallini.