Heegaard surfaces and measured laminations, I: The Waldhausen conjecture

Abstract

Abstract. We give a proof of the so-called generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods.

An algorithm to determine the Heegaard genus of a 3–manifold

“…In [13], the author proved a much stronger theorem, which says that there are only finitely many irreducible Heegaard splittings in a non-Haken 3-manifold, without the genus constraint. The proof in [13] also uses measured laminations, so one would hope the methods in this paper can be used to give an algorithmic proof of this theorem.…”

“…The proof in [13] also uses measured laminations, so one would hope the methods in this paper can be used to give an algorithmic proof of this theorem. Notation 1.4 Throughout this paper, for any topological space X , we use int.X /, jX j and x X to denote the interior, number of components and closure of X respectively.…”

“…The minimal genus of the Heegaard surface S among all Heegaard splittings is called the Heegaard genus of M . In [14], the author proved the so-called generalized Waldhausen conjecture. Theorem 1.1 [14] A closed orientable irreducible and atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy.…”

“…One goal in 3-manifold topology is to find algorithms to determine all the geometric and algebraic properties of any given 3-manifold. However, the proof of Theorem 1.1 in [14] is not algorithmic because of a compactness argument on the projective measured lamination space of a branched surface. In this paper, we give an algorithmic proof of Theorem 1.1.…”

Non-isotopic Heegaard splittings of Seifert fibered spaces

“…The recent proof of Waldhausen's conjecture (Li [7]) (see also work of Jaco and Rubinstein [6; 5]) establishes that a 3-manifold M admits infinitely many non-isotopic Heegaard splittings of some genus only if M contains an incompressible torus. We are interested in the converse of this statement.…”