Algebraic & Geometric Topology

Heegaard splittings and the pants complex

Jesse Johnson

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We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We then use a similar method to compare different manifolds, defining a distance which converges under stabilization to an integer related to Dehn surgeries between the two manifolds.

Article information

Algebr. Geom. Topol., Volume 6, Number 2 (2006), 853-874.

Received: 5 May 2006
Accepted: 11 May 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M27: Invariants of knots and 3-manifolds 57M99: None of the above, but in this section

Heegaard splitting curve complex pants complex


Johnson, Jesse. Heegaard splittings and the pants complex. Algebr. Geom. Topol. 6 (2006), no. 2, 853--874. doi:10.2140/agt.2006.6.853.

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  • J F Brock, The Weil–Petersson metric and volumes of 3–dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495–535
  • F Costantino, D Thurston, 3–manifolds efficiently bound 4–manifolds, preprint (2005)
  • A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221–237
  • J Hempel, 3–manifolds as viewed from the curve complex, Topology 40 (2001) 631–657
  • T Kobayashi, A construction of $3$–manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka J. Math. 29 (1992) 653–674
  • M Lackenby, The asymptotic behaviour of Heegaard genus, Math. Res. Lett. 11 (2004) 139–149
  • H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non-Haken $3$–manifolds, Topology 35 (1996) 1005–1026
  • E Sedgwick, An infinite collection of Heegaard splittings that are equivalent after one stabilization, Math. Ann. 308 (1997) 65–72
  • J Souto, personal correspondence (2005)
  • F Waldhausen, On mappings of handlebodies and of Heegaard splittings, from: “Topology of Manifolds (Georgia, 1969)”, Markham, Chicago, Ill. (1970) 205–211