Heegaard Diagrams Through Morse Theory

This thesis aims to study the concept of Heegaard diagrams of 3-manifolds using onlyelementary concepts. This is done by describing Heegard splittings using the handledecompositions detailed by Morse functions, concepts that only require a backgroundin basic and differential topology. First, we define several core concepts in topology,such as smooth manifolds, homotopy, and retraction. Then we describe important aspects of Morse theory, like Morse functions and Morse lemma. Using Morse lemma,one can find that a manifold can be decomposed into a union of handles, each associated with a critical point of the Morse function. Restricting our study to 3-manifolds,we find that we can separate the manifold into two similar handlebodies, consistingof a solid sphere with several 3-dimensional 1-handles attached, called a Heegaardsplitting. It is then shown how one can construct the 3-manifold by detailing how theboundaries of the two handlebodies attach to the abstract Heegaard surface. The result is a complete description of the topological properties of the 3-manifold using acomparably simpler 2-manifold (Heegaard surface) and a collection of curves on thatsurface. Finally, the thesis ends with a set of examples, where we find the Heegaarddiagrams of product 3-manifolds of a circle and a genus g surface.