Volume 12 (2007)

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DOI: 10.2140/gtm.2007.12

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.

List of participants