The Spherical Manifold Realization Problem

The Lickorish-Wallace theorem states that any closed, orientable, connected 3-manifold can be obtained by integral Dehn surgery on a link in S^3. The spherical manifold realization problem asks which spherical manifolds (i.e., those with finite fundamental groups) can be obtained through integral surgery on a knot in S^3. The problem has previously been solved by Greene and Ballinger et al. for lens space and prism manifolds, respectively. In this project, we determine which of the remaining three types of spherical manifolds (tetrahedral, octahedral, and icosahedral) can be obtained by positive integral surgery on a knot in S^3. We follow methods inspired by those presented by Greene.