#### Volume 14, issue 1 (2014)

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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
##### Abstract

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.

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