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Critical Heegaard surfaces
Author(s):
David
Bachman
Journal:
Trans. Amer. Math. Soc.
354
(2002),
4015-4042.
MSC (2000):
Primary 57M99
Posted:
June 6, 2002
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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:
10.1090/S0002-9947-02-03018-0
PII:
S 0002-9947(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
Posted:
June 6, 2002
Copyright of article:
Copyright
2002,
American Mathematical Society
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