Volume 10, issue 1 (2006)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On the stable equivalence of open books in three-manifolds

Emmanuel Giroux and Noah Goodman

Geometry & Topology 10 (2006) 97–114

arXiv: math/0509555

Abstract

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the theorem implies a conjecture of J Harer, namely, that any fibered link in the three-sphere can be obtained from the unknot by a sequence of plumbings and deplumbings of Hopf bands. The proof presented here involves contact geometry in an essential way.

Keywords
open books, fibered links, plumbing, plane fields, contact structures
Mathematical Subject Classification 2000
Primary: 57M50, 57R17
Secondary: 57M25, 57R52
References
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Publication
Received: 19 September 2005
Accepted: 28 October 2005
Published: 4 March 2006
Proposed: Rob Kirby
Seconded: David Gabai, Peter Ozsváth
Authors
Emmanuel Giroux
École Normale Supérieure de Lyon
69364 Lyon cedex 07
France
Noah Goodman
Massachusetts Institute of Technology
Cambridge
Massachusetts 02139
USA