 
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 3manifold 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 1efficient triangulations, but has yet to ap
pear. They show that given a closed irreducible 3manifold, one may
find a triangulation which has only one normal 2sphere (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
