Stein fillings of homology 3-spheres and mapping class groups

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 with page a 4-holed sphere admits a unique Stein filling up to symplectic deformation. Furthermore, according to a property of deforming symplectic fillings of rational homology 3-spheres into strong symplectic fillings, we also show that a symplectically fillable integral homology 3-sphere supported by an open book with page a 4-holed sphere admits a unique symplectic filling up to symplectic deformation and blow-up.