The Classification of Surfaces
Weizhe Jiao · 2021
This study aims to prove the existence of classification of compact, connected surfaces (without boundaries) up to homeomorphisms. In the first part of the paper, we prepare the basic knowledge that we should know before we start to consider the top theorem, such as compactness and connectedness. Then, this paper enumerates some typical surfaces such as the Möbius strip, torus, and Klein bottle to set up the standard that what is a surface. Triangulation is applied to analyze the topological graph, helping us calculate the Euler characteristics of abstract surfaces that are difficult to count faces. Moreover, we define the topological invariants of surfaces such as orientability to better compare the different characters of each surface. Finally, we prove the Homeomorphism between the combinatorial sphere and its corresponding surfaces. In other words, we can discriminate the surfaces with the genus (How many handles or Möbius strips are on the protosphere).