The Fundamental Group on Algebraic Topology

Abstract

Algebraic topology is mostly about finding invariants for topological spaces. The fundamental group is the simplest, in some ways, and the most difficult in others. This project studies the fundamental group, its basic properties, some elementary computations, and a resulting theorem. In this paper, we will examine the construction and nature of the first homotopy group, which is more commonly known as the fundamental group of a topological space. We will first briefly cover the basics of point-set topology, then use these concepts to facilitate a rigorous study of the construction of the fundamental group.