Dehn surgery and satellite knots

For certain kinds of 3 3 -manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in S 3 S³ is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S 3 S³, a fake S 3 S³, a fake S 3 S³ with nonzero Rohlin invariant, S 1 × S 2 S¹ S², a fake S 1 × S 2, S 1 × S 2 # M S¹ S², S¹ S² \# M with M M nonsimply-connected, or a fake lens space.