Critical Heegaard surfaces

Author:
David Bachman

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

MSC (2000):
Primary 57M99

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

Published electronically:
June 6, 2002

MathSciNet review:
1926863

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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 ( )*, 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 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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M99

Retrieve articles in all journals with MSC (2000): 57M99

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