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RESEARCH BLOG 8/13/04 Recently, Moriah, Schleimer, and Sedgwick posted a paper which is
 

Summary: RESEARCH BLOG 8/13/04
Recently, Moriah, Schleimer, and Sedgwick posted a paper which is
aimed at resolving a conjecture of Sedgwick's about Heegaard split-
tings. Namely, Sedgwick conjectured that an irreducible 3-manifold
with infinitely many distinct (non-isotopic) strongly irreducible Hee-
gaard splittings is Haken. The motivation for this conjecture is that
the known examples of manifolds with infinitely many strongly irre-
ducible Heegaard splittings (first discovered by Casson and Gordon)
are Haken. The splittings are obtained from a minimal genus splitting
by "adding" copies of a particular incompressible surface. If one takes
two (orientable, coorientable) surfaces in a 3-manifold which are trans-
verse, there is no canonical way to add them, by cutting and pasting, to
get another embedded surface, since for each intersection curve, there
are two ways to resolve the intersection. But once one has made one
such choice for each intersection curve, one may add arbitrarily many
copies of the pair of surfaces together so that the choice of resolution
is always the same. Another motivation for Sedgwick's conjecture, is
that if one had an infinite collection of strongly irreducible Heegaard
splittings, one could put them in almost normal form with respect to
a fixed triangulation, by a result of Rubinstein and Stocking. Taking

  

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

 

Collections: Mathematics