On the Construction of the Fundamental Group and Its Topological Invariance

This paper investigates the algebraic topological topic of the fundamental group in efforts to show its topological invariance. This is done first by providing background both historical and practical, and then by developing basic homotopy theory, leading to path homotopy. A path product is defined, which motivates a product for homotopy classes of paths. The product on homotopy classes of paths is shown to have desired properties, namely associativity, identity, and inverses. The fundamental group is then defined with homotopy classes of loops paired with the product. Its point dependence is shown to be irrelevant to the calculation of the fundamental group on path connected spaces, and then the topological invariance of the fundamental group is shown in generality. Some examples of fundamental groups are given, and from this, spaces are shown to not be homeomorphic. Future research would involve the calculational techniques of the fundamental group, including covering spaces and deformations. Another direction is to research the implication of the fundamental group on other areas of mathematics, including its implication of the Fundamental Theorem of Algebra.