Manifold and orbifold in the surface classification theorem

Thesis Bachelor of Science in Mathematics University of Rizal System-Morong. 2019

EXECUTIVE SUMMARY: The Orbifold (short term for orbit manifold) is a space locally modelled on Rn modulo finite group actions. It was first introduced by William P. Thurston. His orbifold theorem also known as surface classification theorem states that while not all three-manifolds have a geometric structure, any three-manifold can be split up into geometric pieces. Any geometric object is made up of either good or bad orbifolds. The study entitled Manifold and Orbifold in the Surface Classification Theorem was conducted at the University of Rizal System located at Sumulong Street, Morong Rizal. The researchers used the qualitative method because it is primarily explanatory research. It is used to gain an understanding of underlying reasons, opinions, and motivations. During the conduct of this study, the researchers were able to gathered data for analysis rationale. Based from the analysis, the researchers concluded that the orbifolds has local structure that is trivial if and only if figure certain has a x which does not lie in any map and the map is homeomorphism near X ̃. If this certain figure is a cyclic group generated by rotations, a group generated by reflections in a single line and a dihedral group generated by reflection in a line and by a rotation, it is non-trivial. Orbifolds which has a manifold covering are called good and orbfiolds without a manifold covering are called bad. It can also be classified as good orbifold elliptic, Euclidean or hyperbolic if their corresponding manifold covering is either sphere, Euclidean or hyperbolic by computing its Euler characteristics with the use of Conway's notation. The researchers recommended with the virtue of the preceding summary of findings and conclusions, the following recommendations are hereby stated: A follow up study on the structure of Klein bottle, Torus, Moebius band, Annulus and Projective plane of the surface classification theorem may be conducted. The concepts generated may be integrated in the lessons on Geometric Topology.