Comparing Heegaard and JSJ structures of orientable 3-manifolds

Scharlemann, Martin; Schultens, Jennifer

doi:10.1090/S0002-9947-00-02654-4

Comparing Heegaard and JSJ structures of orientable 3-manifolds
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by Martin Scharlemann and Jennifer Schultens PDF

Trans. Amer. Math. Soc. 353 (2001), 557-584 Request permission

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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