Geometry & Topology

On the stable equivalence of open books in three-manifolds

Emmanuel Giroux and Noah Goodman

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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.

Article information

Geom. Topol., Volume 10, Number 1 (2006), 97-114.

Received: 19 September 2005
Accepted: 28 October 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57R52: Isotopy

open books fibered links plumbing plane fields contact structures


Giroux, Emmanuel; Goodman, Noah. On the stable equivalence of open books in three-manifolds. Geom. Topol. 10 (2006), no. 1, 97--114. doi:10.2140/gt.2006.10.97.

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