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Critical Heegaard Surfaces

David Bachman
Transactions of the American Mathematical Society
Vol. 354, No. 10 (Oct., 2002), pp. 4015-4042
Stable URL: http://www.jstor.org/stable/3072994
Page Count: 28
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Critical Heegaard Surfaces
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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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