Critical Heegaard surfaces

Bachman, David

doi:10.1090/S0002-9947-02-03018-0

Critical Heegaard surfaces
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by David Bachman

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

DOI: https://doi.org/10.1090/S0002-9947-02-03018-0

Published electronically: 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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