Geometry & Topology

On the stable equivalence of open books in three-manifolds

Emmanuel Giroux and Noah Goodman

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799704

Digital Object Identifier
doi:10.2140/gt.2006.10.97

Mathematical Reviews number (MathSciNet)
MR2207791

Zentralblatt MATH identifier
1100.57013

Subjects
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

Keywords
open books fibered links plumbing plane fields contact structures

Citation

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. https://projecteuclid.org/euclid.gt/1513799704


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References

  • J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma_4=0)$, Lecture Notes in Math. 53, Springer, Berlin (1968)
  • E Dufraine, Classes d'homotopie de champs de vecteurs Morse-Smale sans singularité sur les fibrés de Seifert, Enseign. Math. $(2)$ 51 (2005) 3–30
  • Y Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989) 623–637
  • D Gabai, The Murasugi sum is a natural geometric operation, from: “Low-dimensional topology (San Francisco, Calif., 1981)”, Contemp. Math. 20, Amer. Math. Soc., Providence, RI (1983) 131–143
  • D Gabai, The Murasugi sum is a natural geometric operation. II, from: “Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982)”, Contemp. Math. 44, Amer. Math. Soc., Providence, RI (1985) 93–100
  • E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: “Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)”, Higher Ed. Press, Beijing (2002) 405–414
  • R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $(2)$ 148 (1998) 619–693
  • J Harer, How to construct all fibered knots and links, Topology 21 (1982) 263–280
  • F Laudenbach, S Blank, Isotopie de formes fermées en dimension trois, Invent. Math. 54 (1979) 103–177
  • W D Neumann, L Rudolph, The enhanced Milnor number in higher dimensions, from: “Differential topology (Siegen, 1987)”, Lecture Notes in Math. 1350, Springer, Berlin (1988) 109–121
  • J R Stallings, Constructions of fibred knots and links, from: “Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford, Calif. 1976), Part 2”, Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, R.I. (1978) 55–60
  • W P Thurston, H E Winkelnkemper, 0375366
  • I Torisu, Convex contact structures and fibered links in 3-manifolds, Internat. Math. Res. Notices (2000) 441–454
  • V G Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 607–643, 672
  • F Waldhausen, }, Ann. of Math. $(2)$ 87 (1968) 56–88