The notion of a Heegaard
splitting of a 3-manifold is as old as 3-dimensional
topology itself; we may recall, for example, that
Poincaré described his dodecahedral space by
means of a Heegaard diagram. It also seems to be the
case that one of the early motivations for the study
of automorphisms of surfaces was the desire to
understand 3-manifolds through their Heegaard
splittings. Nevertheless, for many years knowledge
about Heegaard splittings was limited; the following
is a more or less complete list of things that were
known up to 1970: 3-manifolds are triangulable, and
hence possess Heegaard splittings (Moise, 1952); any
two Heegaard splittings of a given 3-manifold become
isotopic afer some number of stabilizations
(Reidemeister, Singer, 1933); S3 has a
unique splitting of any genus up to isotopy
(Waldhausen, 1968); Heegaard genus is additive under
connected sum (Haken, 1968); the algebraic
characterization of Heegaard splittings in terms of
splitting homomorphisms (Stallings, 1966).
Starting in the 1980's, progress
in the subject began to accelerate and it entered
more and more into the mainstream of 3-dimensional
topology, with developments coming from several
different directions. There are now enough general
results and techniques established to justify
speaking of the theory of Heegaard splittings. A
(certainly incomplete) list of recent advances in the
subject is the following: the classification of
Heegaard splittings of Seifert fiber spaces; the
notion of strong irreducibility; the introduction of
the curve complex into the study of Heegaard
splittings; the use of normal and almost normal
surfaces; results obtained using Cerf theory
(sweep-outs); the application of the theory of
minimal surfaces; geometric topological methods,
including the theory of laminations; results relating
Heegaard splittings to hyperbolic structures, for
instance hyperbolic volume; results on the tunnel
number of knots; the use of Heegaard splittings to
define Heegaard Floer homology.
It was against this background
that the Technion Workshop on Heegaard Splittings was
held in the summer of 2005. The goal was to gather
people with a specific interest in Heegaard
splittings in one workshop where the state of the art
could be exposed and discussed.
It was decided by the
participants to publish a proceedings of the workshop
with the hope of making available to interested
people a concentrated source of information about the
current state of research on Heegaard splittings.
Some papers were solicited from non-participants
whose interests are close to the field. We wish to
thank all those who contributed to this volume for
their efforts.
The volume also contains a list
of problems about Heegaard splittings, contributed by
some of the workshop participants.
