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RESEARCH BLOG 8/18/04 CONJECTURES OF WALDHAUSEN AND SEDGWICK
 

Summary: RESEARCH BLOG 8/18/04
CONJECTURES OF WALDHAUSEN AND SEDGWICK
Tao Li posted two papers on the ArXiv which claim to solve some
conjectures about Heegaard splittings. The first paper solves the Wald-
hausen conjecture. This conjecture states that for any g N, an
atoroidal 3-manifold M has only finitely many Heegaard splittings of
genus g. A solution was claimed a while ago by Jaco and Rubin-
stein, using their tool of 1-efficient triangulations, but has yet to ap-
pear. They show that given a closed irreducible 3-manifold, one may
find a triangulation which has only one normal 2-sphere (linking the
vertex), and only finitely many normal tori. Then the proof of the
Waldhausen conjecture follows by a standard argument using almost
normal branched surfaces (which I alluded to last blog). First, one may
assume that the Heegaard splittings are strongly irreducible by detele-
scoping. If there is an infinite number of strongly irreducible Heegaard
splittings of genus g, then one may take a subsequence of them
which are almost normal and fully carried by a fixed almost normal
branched surface. Taking differences between the normal coordinates,
one obtains a normal surface of Euler characteristic zero, which there-
fore must consist of copies of the finitely many normal tori. Then by

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics