From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: This week in the xxx mathematics archive (3 May - 7 May)
Date: 10 May 1999 11:56:20 -0700
Newsgroups: sci.math.research
Keywords: Heegaard splittings

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This week I'd like to discuss a line of research in 3-manifold topology
that has been picked up by the archive:  The structure of the set of
Heegaard splittings of a 3-manifold.  The archive articles on this
topic include math.GT/9903078, math.GT/9902010, math.GT/9803157,
math.GT/9803009, math.GT/9802101, and math.GT/9712262.  Although this
work involves a number of good people, I note that Marty Scharlemann is
a co-author of most of these articles.

3-manifold topologists have developed an intuitive distinction between
"deep geometric" structure on a 3-manifold and "superficial combinatorial"
structure.  For example, a triangulation of a 3-manifold is considered
superficial, because it is easy to find many of them and hard to know what
most of them mean.  A hyperbolic structure, on the other hand, is deep.
It is unique and immediately gives a lot of explicit information about
the topology of the manifold.

A Heegaard splitting of a closed 3-manifold is a decomposition of
the 3-manifold into two handlebodies; a handlebody being a standard
"doughnut" with one or more holes.  You get a Heegaard splitting from
a triangulation by taking a tubular neighborhood of the  vertices
and edges.  Given constructions such as these, Heegaard splittings
were long considered to be a type of superficial 3-manifold structure.
In the 1950's and 60's, when Papakyriakopoulos, Haken, Waldhausen, and
others made advances in combinatorial 3-manifold topology, there was
some faint hope that Heegaard splittings could lead to deep structure.
For example, Waldhausen proved that the only Heegaard splittings of the
3-sphere are the obvious ones. But even this tepid enthusiasm did not
last, since the whole combinatorial program was supplanted by Thurston's
geometrization program.

Surprisingly, Heegaard splittings are now interesting again.  Like other
cheap descriptions of 3-manifolds, there are moves that let you go
from  any Heegaard splitting to any other.  However, one difference is
that for Heegaard splittings there is only such move, stabilization,
and in one direction it is unique.  (The inverse, destabilization, is
not unique.) Thus the set of Heegaard splittings of a manifold form a
genealogical tree, with no ultimate ancestor. The classification theorem
for splittings of S^3 says that this tree is a stalk whose sole leaf is
the genus 0 splitting.

 From the recent work on splittings, it is known (at least for non-Haken
manifolds), that the tree is locally finite, indeed that it is finite
below any fixed genus g (Rubinstein-Scharlemann). It is known that the
tree is a stalk for any lens space (Bonahon) and for the 3-torus.  I'm not
sure, but I think that there is an example of a hyperbolic manifold for
which the tree has only finitely many leaves (irreducible splittings). On
the other hand, there is an example of Casson of a 3-manifold with
infinitely many irreducible splittings, necessarily of higher and
higher genus.  There is a theorem of Rubinstein and Scharlemann that for
non-Haken manifolds, a linear number of stabilizations suffice. That
is, any two splittings of genus g and g' have a common ancestor of
genus O(g+g').  There is a conjecture that one stabilization suffices.

Many of the arguments for these results consider interesting singular
foliations naturally associated to a Heegaard splittings. Namely, you
can thicken a Heegaard surface with many parallel copies of itself until
on each side the surface degenerates into an embedded graph.  This is a
generalization of decorating an n-sphere by constant-latitude n-1-spheres.
The starting point of the Rubinstein-Scharlemann argument, for example,
supposes that you have two such foliations, one for each splitting, and
that they are in the simplest possible relative position. The latitude
foliation also plays an important role in the Rubinstein-Thompson
algorithm to recognize the 3-sphere.  It is not clear if the foliation
has been fully exploited.  I have the feeling that the articles so far
are chipping away at a more general theory.

"This week in the xxx mathematics archive" may be freely redistributed
with attribution and without modification.

Titles in the xxx mathematics archive (3 May - 7 May)
-----------------------------------------------------
[set of new titles deleted, sorry -- djr]

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