Summary: LECTURE NOTES ON GENERALIZED HEEGAARD
TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS
These notes grew out of a lecture series given at RIMS in the summer of 2001.
The authors were visiting RIMS in conjunction with the Research Project on Low-
Dimensional Topology in the Twenty-First Century. They had been invited by
Professor Tsuyoshi Kobayashi. The lecture series was first suggested by Professor
The lecture series was aimed at a broad audience that included many graduate
students. Its purpose lay in familiarizing the audience with the basics of 3-
manifold theory and introducing some topics of current research. The first portion
of the lecture series was devoted to standard topics in the theory of 3-manifolds.
The middle portion was devoted to a brief study of Heegaaard splittings and
generalized Heegaard splittings. The latter portion touched on a brand new
topic: fork complexes.
During this time Professor Tsuyoshi Kobayashi had raised some interesting
questions about the connectivity properties of generalized Heegaard splittings.
The latter portion of the lecture series was motivated by these questions. And
fork complexes were invented in an effort to illuminate some of the more subtle