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On Kirby calculus for null-homotopic framed links in $3$–manifolds

Kazuo Habiro and Tamara Widmer

Algebraic & Geometric Topology 14 (2014) 115–134

Kirby proved that two framed links in S3 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.

$3$–manifold, framed link, surgery, Kirby calculus, null-homotopic link
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Received: 23 February 2013
Revised: 19 June 2013
Accepted: 21 June 2013
Preview posted: 21 November 2013
Published: 9 January 2014
Kazuo Habiro
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
Tamara Widmer
Institut für Mathematik
Universität Zürich
Winterthurerstr. 190
CH-8057 Zürich