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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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Abstract: The Heegaard genus $g$ of an irreducible closed orientable $3$-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 $p$ of the complementary components are not Seifert fibered, then $p \leq g-1$. This generalizes work of Kobayashi. The Heegaard genus $g$ also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space $P$ and $f$ exceptional fibers, then $f - \chi(P) \leq 3g - 3 - p$.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02654-4
PII: S 0002-9947(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