Refined Kirby calculus for integral homology spheres

Habiro, Kazuo

doi:10.2140/gt.2006.10.1285

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Refined Kirby calculus for integral homology spheres

Kazuo Habiro

Geometry & Topology 10 (2006) 1285–1317

DOI: 10.2140/gt.2006.10.1285

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 $\pm 1$ –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

Volume 10, issue 3 (2006)