Heegaard Splittings and Surgery on 2- and 3-Manifolds

Abstract A Heegaard splitting of a 3-manifold is a representation of it as the union of two handlebodies with the same boundary. Each splitting is defined by an attaching homeomorphism between the boundaries of these two handle-bodies. We discuss surgery on surfaces (2-manifolds) to explain why specifying precisely g pairs of curves suffices to define the attaching homeomorphism of two genus-g surfaces. Then we demonstrate certain Heegaard splittings of man­ifolds such as S3, S² x S¹, and T³ and offer techniques to visualize them. We observe a simple classification of compact, closed, orientable 3-manifolds by Heegaard genus. Manifolds that admit genus-1 splittings are also lens spaces, which can be defined as particular quotient spaces of the 3-sphere S³. Finally we introduce Dehn surgery, a method by which any compact, closed, orientable 3-manifold can be obtained. Dehn surgery on S³ along some knot or link K entails removing an open tubular neighborhood N(K), defining a homeomor-phism of δ ¯N(K), and attaching the new neighborhood ¯N′(K) to the boundary of the complement of N(K) in S³. In general, Dehn surgery along the un-knot produces a lens space. Thus we discuss methods of obtaining, visualizing, and classifying 3-manifolds and the connections between surgery and Heegaard splittings.