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Non-isotopic Heegaard splittings of Seifert fibered spaces

David Bachman and Ryan Derby-Talbot

Appendix: R Weidmann

Algebraic & Geometric Topology 6 (2006) 351–372

arXiv: math.GT/0504605


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.

Heegaard Splitting, essential Surface
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57N10, 57M60
Forward citations
Received: 2 May 2005
Revised: 6 December 2005
Published: 12 March 2006
David Bachman
Mathematics Department
Pitzer College
1050 North Mills Avenue
Claremont CA 91711
Ryan Derby-Talbot
Mathematics Department
The University of Texas at Austin
Austin TX 78712-0257
R Weidmann
Fachbereich Informatik und Mathematik
Johann Wolfgang Goethe-Universität
60054 Frankfurt