Comparing Heegaard and JSJ structures of orientable 3-manifolds

Authors:
Martin Scharlemann and Jennifer Schultens

Journal:
Trans. Amer. Math. Soc. **353** (2001), 557-584

MSC (2000):
Primary 57M50

Published electronically:
September 15, 2000

MathSciNet review:
1804508

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Heegaard genus of an irreducible closed orientable -manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if of the complementary components are not Seifert fibered, then . This generalizes work of Kobayashi. The Heegaard genus also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space and exceptional fibers, then .

**[CG]**A. J. Casson and C. McA. Gordon,*Reducing Heegaard splittings*, Topology Appl.**27**(1987), no. 3, 275–283. MR**918537**, 10.1016/0166-8641(87)90092-7**[Ha]**Wolfgang Haken,*Some results on surfaces in 3-manifolds*, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39–98. MR**0224071****[He]**P. Heegaard, Forstudier til en Topologiskteori for de Algebraiske Aladers Sammenhaeng,*Ph. D. thesis*, Copenhagen, 1898.**[Ja]**William Jaco,*Lectures on three-manifold topology*, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR**565450****[Ko]**Tsuyoshi Kobayashi,*Structures of full Haken manifolds*, Osaka J. Math.**24**(1987), no. 1, 173–215. MR**881754****[MR]**Yoav Moriah and Hyam Rubinstein,*Heegaard structures of negatively curved 3-manifolds*, Comm. Anal. Geom.**5**(1997), no. 3, 375–412. MR**1487722**, 10.4310/CAG.1997.v5.n3.a1**[Prz]**A. Przybyszewska, in*Knot theory from Vandermonde to Jones*by J. Przytycki, Mathematics Institute, Odense University, Preprint 43, 1993.**[RS]**Hyam Rubinstein and Martin Scharlemann,*Comparing Heegaard splittings—the bounded case*, Trans. Amer. Math. Soc.**350**(1998), no. 2, 689–715. MR**1401528**, 10.1090/S0002-9947-98-01824-8**[Sc]**M. Scharlemann, Heegaard splittings of compact -manifolds, to appear in*Handbook of Geometric Topology*, ed by R. Daverman and R. Sherr, Elsevier Press.**[Sc2]**M. Scharlemann, Local detection of strongly irreducible Heegaard splittings,*Topology and its applications***90**(1998), 135-147. MR**99h:37040****[Sc3]**M. Scharlemann and J. Schultens, The tunnel number of the sum of knots is at least ,*Topology***38**(1999), 265-270. CMP**98:05****[Sc4]**J. Schultens, Additivity of tunnel number, to appear.**[Sco]**Peter Scott,*The geometries of 3-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, 10.1112/blms/15.5.401**[ST]**Martin Scharlemann and Abigail Thompson,*Thin position for 3-manifolds*, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231–238. MR**1282766**, 10.1090/conm/164/01596**[W]**Friedhelm Waldhausen,*Heegaard-Zerlegungen der 3-Sphäre*, Topology**7**(1968), 195–203 (German). MR**0227992**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M50

Retrieve articles in all journals with MSC (2000): 57M50

Additional Information

**Martin Scharlemann**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106

Email:
mgscharl@math.ucsb.edu

**Jennifer Schultens**

Affiliation:
Department of Mathematics, Emory University, Atlanta, Georgia 30322

Email:
jcs@mathcs.emory.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02654-4

Received by editor(s):
March 22, 1999

Published electronically:
September 15, 2000

Additional Notes:
Research supported in part by NSF grants and MSRI

Article copyright:
© Copyright 2000
American Mathematical Society