  Kirby proved that two framed links in S^3 give orientation-preserving
homeomorphic results of surgery if and only if these two links are related by a
sequence of two kinds of moves called stabilizations and handle-slides. Fenn
and Rourke gave a necessary and sufficient condition for two framed links in a
closed, oriented 3-manifold to be related by a finite sequence of these moves.
The purpose of this paper is twofold. We first give a generalization of Fenn
and Rourke's result to 3-manifolds with boundary. Then we apply this result to
the case of framed links whose components are null-homotopic in the 3-manifold.
