Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds

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A famous example of Casson and Gordon shows that a Haken 3–manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3–manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3–manifold M M , there is a number N N such that any two Heegaard splittings of M M are equivalent after at most N N stabilizations.