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

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.

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