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Refined Kirby calculus for integral homology spheres

Kazuo Habiro

Geometry & Topology 10 (2006) 1285–1317

arXiv: math.GT/0509039

Abstract

A theorem of Kirby states that two framed links in the 3–sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on framed links to be able to be replaced with a sequences of algebraically canceling pairs of handle-slides. Then, using the first result, we obtain a refinement of Kirby’s calculus for integral homology spheres which involves only ± 1–framed links with zero linking numbers.

This paper is dedicated to Professor Yukio Matsumoto on the occasion of his sixtieth birthday.

Keywords
Kirby calculus, framed link, surgery, handle-slide, integral homology sphere, band-slide, Hoste move
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
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Publication
Received: 20 December 2005
Accepted: 20 June 2006
Published: 18 September 2006
Proposed: Colin Rourke
Seconded: Peter Teichner, Rob Kirby
Authors
Kazuo Habiro
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606–8502
Japan