Algebraic topology is a branch of mathematics which developed slowly during the 20th century, with the almost parallel development of the two main categories-homology and homotopy. In this study we will concentrate on the homotopical aspects of this subject in our investigation of fundamental group theory and its applications in classifying topological spaces. We will begin by discussing some basic topological and algebraic constructs, which will serve as a foundation for the development of fundamental group theory. Included in this development of fundamental group theory is a discussion of path homotopies, basic group theory, and covering spaces.; Since there is no solitary method for calculating the fundamental group of a topological space, we will investigate several techniques used to calculate the fundamental group of some simple, familiar topologicals spaces like that of the circle. We will then ascertain several concepts which will allow us to extend the simple examples worked to compute the fundamental group of more complex topological structures like the Mobius Strip.
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