Critical Heegaard surfaces

Author:
David Bachman

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4015-4042

MSC (2000):
Primary 57M99

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