Stein fillings of homology $3$-spheres and mapping class groups

TL;DR

This paper proves the uniqueness of Stein and symplectic fillings for certain homology 3-spheres supported by open books with 4-holed sphere pages, using combinatorial mapping class group techniques.

Contribution

It establishes the uniqueness of Stein and symplectic fillings for specific homology 3-spheres via combinatorial methods, a new approach in the field.

Findings

Unique Stein filling up to diffeomorphism for the specified 3-spheres.

Unique symplectic filling up to diffeomorphism and blow-up for the specified 3-spheres.

Application of combinatorial techniques of mapping class groups to fillability problems.

Abstract

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillings of a rational homology $3$-spheres into strongly symplectic fillings, we also show that a symplectically fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique symplectic filling up to diffeomorphism and blow-up.