Lecture Notes on Generalized Heegaard Splittings

The following sections are included:

• Preface
• Contents

The class of 3-manifolds provides a rich and beautiful world within the field of topology. Many great minds have contributed to our understanding of this world. One of the interesting classical features of 3-manifolds (to emerge towards the middle of the last century) is the fact that the categories DIFF (differentiable or smooth manifolds), TOP (topological manifolds) and PL (piecewise linear or PL manifolds) are equivalent. Standard references for the manifolds and the categories in which they are defined are [do Carmo (1992)], [Guillemin and Pollack (2010)], [Hatcher (1980's)], [Hempel (2004)], [Jaco (1980)], [Massey (1977)], [Milnor (1997)], or [Singer and Thorpe (1976)]. Each category offers a set of tools to study manifolds. In dimension 3, this allows us to choose the set of tools best equipped to solve a particular problem, to illustrate a certain phenomenon and to suit our particular taste. Our choice here will be to work in the TOP and PL categories.

In this chapter we discuss Heegaard splittings. Conceived by Poul Heegaard in his dissertation in 1898, inspired by a trip to Göttingen and the influence of Felix Klein, Heegaard splittings originally provided a framework in which to state a counterexample to Poincaré's original formulation of duality. (Thus forcing Poincaré back into topology.) The motivation for studying Heegaad splittings in recent decades lies in the natural decomposition of 3-manifolds they entail. This decomposition into “symmetric” pieces allows us to visualize many 3-manifolds. We provide several examples to illustrate the type of visualization they allow.

In the 1930's, Reidemeister and Singer independently proved theorems concerning a construction, see below, called stabilization. (See [Reidemeister (1935)] and [Singer (1933)].) Their motivation came from the lack of uniqueness of Heegaard splittings of 3-manifolds. In the 1960's, Haken and Waldhausen continued investigating the questions asked by Reidemeister and Singer. (See [Haken (1968)] and [Waldhausen (1968)].) In the late 1980's Casson and Gordon (see [Casson and Gordon (1987)]) further continued this study by introducing the notion of strong irreducibility and proving an extension of a key lemma due to Haken. In this chapter we discuss the concepts central to the work of Reidemeister, Singer, Haken, Waldhausen, Casson and Gordon. In the next chapter we will examine two well known theorems more closely.

Most of this chapter is devoted to a theorem of Waldhausen. For Waldhausen's original work, see [Waldhausen (1968)]. We also take a look at the theorem of Reidemeister and Singer.

In 1992, Scharlemann and Thompson defined the width of 3-manifolds and showed that manifold decompositions minimizing this width possessed many valuable properties. (See [Scharlemann and Thompson (1994b)].) The key insight is captured in Proposition 5.2.3. It tells us that a thin manifold decomposition provides a sequence of strongly irreducible Heegaard splittings meeting along incompressible surfaces. The notion thus refines the weak reductions inspired by the insights in [Casson and Gordon (1987)]. See also [Kobayashi (2003)]…

The following sections are included:

• Bibliography
• Index