Thesis advisor: Ian Biringer / We are interested in proving the following statement: Given a 3-manifold M with boundary and a homeomorphism of the boundary f : ∂M → ∂M such that there is some power that extends to M, there is some k depending only on the genus g(∂M) and some l < k such that ƒᶩ extends to M. We will prove that the power needed to extend is not uniformly bounded with some examples, we will prove the statement is true if M is boundary incompressible and we will show that the general statement reduces to effectivising some technical results about pure homeomorphisms extending to compression bodies. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.