Critical Heegaard Surfaces

Details Critical Heegaard Surfaces Critical Heegaard Surfaces

2002-01-22

arxiv.org/abs/math/0201203v1

Transactions of the American Mathematical Society 354 (2002), 4015-4042.

Mathematics

Geometric Topology

28 pages, 8 figures, to appear in Transactions of the AMS

Scientific paper

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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