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Critical Heegaard surfaces


Author: David Bachman
Journal: Trans. Amer. Math. Soc. 354 (2002), 4015-4042
MSC (2000): Primary 57M99
DOI: https://doi.org/10.1090/S0002-9947-02-03018-0
Published electronically: June 6, 2002
MathSciNet review: 1926863
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Abstract: In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.


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Additional Information

David Bachman
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
Address at time of publication: Mathematics Department, Cal Poly State University, San Luis Obispo, CA 93407
Email: bachman@math.uic.edu, dbachman@calpoly.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03018-0
Keywords: Incompressible surface, Heegaard splitting, stabilization, curve complex
Received by editor(s): December 22, 2000
Received by editor(s) in revised form: January 10, 2002
Published electronically: June 6, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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