Algebraic & Geometric Topology

Non-isotopic Heegaard splittings of Seifert fibered spaces

David Bachman and Ryan Derby-Talbot

Full-text: Open access


We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3–manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if M has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen’s conjecture.

Article information

Algebr. Geom. Topol., Volume 6, Number 1 (2006), 351-372.

Received: 2 May 2005
Revised: 6 December 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M60: Group actions in low dimensions

Heegaard Splitting essential Surface


Bachman, David; Derby-Talbot, Ryan. Non-isotopic Heegaard splittings of Seifert fibered spaces. Algebr. Geom. Topol. 6 (2006), no. 1, 351--372. doi:10.2140/agt.2006.6.351.

Export citation


  • D Bachman, S Schleimer, Surface bundles versus Heegaard splittings, Comm. Anal. Geom. to appear
  • J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma \sb{4}=0)$, Lecture Notes in Mathematics, No. 53, Springer, Berlin (1968)
  • A Hatcher, Notes on Basic 3-Manifold Topology, available at\char'176hatcher/3M/3Mdownloads.html
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society, Providence, R.I. (1980)
  • W Jaco, J H Rubinstein, $1$-efficient triangulations of 3-manifolds, In preparation
  • W Jaco, J H Rubinstein, $0$-efficient triangulations of 3-manifolds, J. Differential Geom. 65 (2003) 61–168
  • T Li, Heegaard surfaces and measured laminations, I: the Waldhausen conjecture, preprint
  • M Lustig, Y Moriah, Nielsen equivalence in Fuchsian groups and Seifert fibered spaces, Topology 30 (1991) 191–204
  • Y Moriah, J Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998) 1089–1112
  • K Morimoto, M Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143–167
  • M Sakuma, Manifolds with infinitely many non-isotopic Heegaard splittings of minimal genus, preliminary report, (unofficial) proceedings of the conference on various structures on knots and their applications (Osaka City University) (1988) 172–179
  • J Schultens, The stabilization problem for Heegaard splittings of Seifert fibered spaces, Topology Appl. 73 (1996) 133–139
  • E Sedgwick, The irreducibility of Heegaard splittings of Seifert fibered spaces, Pacific J. Math. 190 (1999) 173–199
  • H Zieschang, Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam, Invent. Math. 10 (1970) 4–37