Dehn Surgery and Satellite Knots

For certain kinds of $3$-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in S³ is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S³, a fake S³, a fake S³ with nonzero Rohlin invariant, S¹ × S², a fake S¹ × S², S¹ × S² # M with M nonsimply-connected, or a fake lens space.