Kirby calculus in manifolds with boundary
Abstract
Suppose there are two framed links in a compact, connected 3-manifold (possibly with boundary, or non-orientable) such that the associated 3-manifolds obtained by surgery are homeomorphic (relative to their common boundary, if there is one.) How are the links related? Kirby's theorem gives the answer when the manifold is S3, and Fenn and Rourke extended it to the case of any closed orientable 3-manifold, or twisted S1 cross S2. The purpose of this note is to give the answer in the general case, using only minor modifications of Kirby's original proof.
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