CLASSIFICATION of SURFACES Contents 1. Introduction 1 2. Torus
CLASSIFICATION OF SURFACES YUJIE ZHANG Abstract. The sphere, M¨obiusstrip, torus, real projective plane and Klein bottle are all important ex- amples of surfaces (topological 2-manifolds). In fact, via the connected sum operation, all surfaces can be constructed out of these. After introducing these topological spaces, we define and study important prop- erties and invariants of surfaces, such as orientability, Euler characteristic, and genus. Last, we give a proof of the classification theorem for surfaces, which states that all compact surfaces are homeomorphic to the sphere, the connected sum of tori, or the connected sum of projective planes. Contents 1. Introduction 1 2. Torus, projective plane and Klein bottle 1 2.1. First definitions 2 2.2. Basic surfaces 2 3. Orientability, Genus and Euler characteristic 4 3.1. Computations 5 3.2. Connected Sum 5 4. The Classification Theorem 7 Acknowledgments 10 References 10 1. Introduction The main goal of this paper is to develop a geometric intuition for topological subjects. The structure of the paper is as follows. In x2, we make some basic definitions and describe the construction of basic surfaces such as the sphere, torus, real1 projective plane, and Klein bottle. In x3, we define some fundamental invariants of surfaces: orientability, genus, and Euler characteristic. Our definitions, in keeping with an intuitionistic spirit and a geometric preference, will occasionally be less- than-entirely rigorous, but we hope they will capture the imagination of the reader for whom this account may provide a first introduction. In x4, we proceed to the proof of the classification theorem for surfaces.
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