  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 <
g. This result 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 base spaces of the Seifert pieces has Euler characteristic X and
there are a total of f exceptional fibers in the Seifert pieces, then f - X is
no greater than 3g - 3 - p.
