Classification of surfaces

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This is a report that I made over the summer of 2019. This was under the Summer of Science initiative by the Maths and Physics Club. This was made under the mentorship of a senior student of the Mathematics Department.

Abstract

We start with an introduction to topology and then we move on to classify connected, compact 2-dimensional manifolds into three categories- the sphere, connected sum of tori, and connected sum of projective planes.

Introduction about the paper

The aim of this paper is to introduce some basic concepts of topology and prove the classification theorem using them.
The paper begins with Section 2 by introducing concepts from topology that will be required. This section assumes no prior knowledge of topology. In this section, we also prove the lemmas and theorems that are useful in building the required theory.
Before moving on to Section 4, we prove two basic theorems about continuous functions in Section 3 along with one other result about \(\mathbb{R}.\) In Section 4, we introduce the concept of manifolds which will shall be explored for the rest of the paper. After finishing the proof of our desired theorem in Section 11, we briefly talk about the Euler characteristic in Section 12.