We present a new method for simplifying the topology of a 3D shape. Unlike existing methods that either remove all topological features or offer indirect control over the target topology, our method allows the user to prescribe the number of topological features of each type (e.g., components, handles, and voids). Our method is based on persistent homology, and its core is a novel algorithm that simplifies a filtration (a sequence of evolving shapes) to remove topological features with low persistence (duration in the filtration) in a controllable manner. Experiments demonstrate that our method is highly successful in practice in realizing the prescribed topology while making minimal changes to the shape.