Heegaard Splittings and Surgery on 2- and 3-Manifolds

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 S 3, S ² x S ¹ , and T ³ 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 S ³ . Finally we introduce Dehn surgery, a method by which any compact, closed, orientable 3-manifold can be obtained. Dehn surgery on S ³ 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 S ³ . 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.