The Classification of Surfaces is one of the problems which gave rise to the modern topology. It has become one of the signature theorems of the area, which now is called algebraic topology. It states that any closed connected surface is homeomorphic to the sphere, the connected sum of tori, or the connected sum of projective planes. In this thesis we are going to go over the geometric, topological, and algebraic tools necessary for understanding, proving and using the theorem together with some useful examples of surfaces.
Thesis itself consists of three chapters. The first part talks about homotopy theory and defines the fundamental group, which is an algebraic invariant between topological spaces. In addition, we learn some basic ways of calculating the fundamental group for some easy- to-imagine examples. The second chapter introduces free groups and free products, which altogether let us calculate the fundamental group in more complex cases. The third and final chapter introduces the geometric ideas behind the classification theorem, which includes polygonal regions and labelling schemes together with operations on them. As a result, we overview the construction of any two-dimensional compact surface and classification theorem as a main goal.
