Contact 3-manifolds with infinitely many rational homology ball Stein fillings

Abstract

We construct the first examples of contact $3$-manifolds admitting infinitely many pairwise non-homeomorphic Stein fillings which are rational homology balls. Our examples are planar. They are obtained by a simple modification of Baykur's recent planar contact $3$-manifold $(Y,ξ)$ with infinitely many pairwise non-homeomorphic Stein fillings, and the fillings are distinguished by their fundamental groups. For one of these rational homology balls we give an explicit Kirby diagram, with a single $1$-handle and a single $2$-handle attached along a trefoil, which exhibits its boundary as $(0,-8)$-surgery on a two-component link. We also identify the smooth topology of Baykur's example: $Y$ is obtained by $(0,-8)$-surgery on the link $7^2_4$, and it is Seifert fibered over the Klein bottle with one exceptional fiber. Finally, we construct further Stein fillings of $(Y,ξ)$, which realize all the groups $\mathbb Z/m\rtimes_{-1}\mathbb Z$, $m\ge0$, as their fundamental groups.