UC Berkeley

The main result of this dissertation shows that every orientable closed 3-manifold admits a nonzero degree map onto at most finitely many homeomorphically distinct non-geometric prime 3-manifolds. Furthermore, for any integer d > 0, every orientable closed 3-manifold admits a map of degree d onto only finitely many homeomorphically distinct 3-manifolds. This answers a question of Yongwu Rong. The finiteness of JSJ piece of the targets under nonzero degree maps was known earlier by the results of Soma and Boileau–Rubinstein–Wang, and a new proof is provided is this dissertation. We also prove analogous results for dominations rela-tive to boundary. As an application, we describe the degree set of dominations onto integral homology 3-spheres.