Homology Spheres Bounding Acyclic Smooth Manifolds and Symplectic Fillings

In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein embedding. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.