Mr Mark Bott
Doctor of Philosophy supervision
The classification of surfaces
The goal of this study is to classify all compact two-dimensional manifolds up to homeomorphism, and in doing so, attempt to develop and understand some of the techniques used in modern topology. We begin in chapter one by looking at applications of manifolds in mathematics and physics in order to motivate the problem, and then thoroughly defining a topological manifold. The idea of complexes is then introduced, and used to give a complete classification of 1-manifolds and prove Tibor Rado's celebrated triangulation theorem for 2-manifolds. Using the triangulation theorem, a complete list of possible compact surfaces is given. Time is put aside to develop the concept of the fundamental group. The theory developed is then used to show that the possible compact surfaces listed prior are all topologically distinct. This gives the complete classification theorem, namely that every compact, connected 2-manifold is homeomorphic to exactly one of the sphere, a connected sum of tori, or a connected sum of projective planes. The project concludes with a brief look at the idea of classifying smooth manifolds up to diffeomorphism, and observing how classification problems become far more difficult (and interesting) in higher dimensions. An attempt has been made to make the study self contained, and keep assumed knowledge to a minimum. All that is required is an understanding of basic point set topology, (an excellent explanation of which is given in chapters two and three of [Mun74]), elementary group theory (as can be found in chapter one of [Fra02]), and basic multivariable calculus (see chapter one of [Spi65]).