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 {S^3} is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S 3 {S^3} , a fake S 3 {S^3} , a fake S 3 {S^3} with nonzero Rohlin invariant, S 1 × S 2 {S^1} × {S^2} , a fake S 1 × S 2 , S 1 × S 2 # M {S^1} × {S^2}, {S^1} × {S^2} # M with M M nonsimply-connected, or a fake lens space.