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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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  • [AM90] S. Akbulut and J. McCarthy, Casson's Invariant for Oriented Homology 3-spheres, An exposition, Mathematical Notes, vol. 36, Princeton University Press, Princeton, NJ, 1990. MR 90k:57017
  • [Bac] D. Bachman, A normal form for minimal genus common stabilizations., in preparation.
  • [BO83] F. Bonahon and J. P. Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Scient. École Norm. Sup. (4) 16 (1983), 451-466. MR 85c:57010
  • [Bon83] F. Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), 305-314. MR 85d:57008
  • [Cer68] J. Cerf, Les difféomorphismes de la sphére de dimension trois ( $\gamma(4)=0$), Springer-Verlag, 1968, Lecture Notes in Mathematics #53. MR 37:4824
  • [CG87] A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology and its Applications 27 (1987), 275-283. MR 89c:57020
  • [FHS] M. Freedman, J. Hass and P. Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983), 609-642. MR 85e:57012
  • [Gab87] D Gabai, Foliations and the topology of three-manifolds. III, J. Diff. Geom. 26 (1987), 479-536. MR 89a:57014b
  • [Hem] J. Hempel, 3-manifolds as viewed from the curve complex, Topology, 40 (2001), 631-657.
  • [PR87] J. Pitts and J. H. Rubinstein, Applications of minimax to minimal surfaces and the topology of 3-manifolds, Miniconference on geometry and partial differential equations, 2 (Canberra 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., 12, Austral. Nat. Univ., Canberra, 1987, pp. 137-170. MR 89a:57001
  • [RS96] H. Rubinstein and M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (1996), 1005-1026. MR 97j:57021
  • [Rub96] J. H. Rubinstein, Lecture notes from conference at UC Davis, 1996.
  • [Sch98] M. Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology and its Applications 90 (1998), 135-147. MR 99h:57040
  • [ST93] M. Scharlemann and A. Thompson, Heegaard splittings of ${(surface)} \times {I}$ are standard, Math. Ann. 295 (1993), 549-564. MR 94b:57020
  • [ST94] M. Scharlemann and A. Thompson, Thin position for 3-manifolds, A.M.S. Contemporary Math. 164 (1994), 231-238. MR 95e:57032
  • [Wal68] F. Waldhausen, Heegaard Zerlegungen der 3-sphäre, Topology 7 (1968), 195-203. MR 37:3576

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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

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

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