On the stable equivalence of open books in three-manifolds

Giroux, Emmanuel; Goodman, Noah

doi:10.2140/gt.2006.10.97

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On the stable equivalence of open books in three-manifolds

Emmanuel Giroux and Noah Goodman

Geometry & Topology 10 (2006) 97–114

DOI: 10.2140/gt.2006.10.97

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

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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

Volume 10, issue 1 (2006)