Uma introdução ao Grupo Fundamental e aplicações

Abstract

This work presents some of the basic notions of algebraic topology, focusing on the concept and basic results related to the Fundamental Group. We begin with the basic definitions of paths and loops. Then, we introduce the definition of the Fundamental Group, covering, and lifting of paths, proving some of its fundamental properties. Subsequently, we deal with the problem of topological classification from the perspective of techniques accessible to undergraduate students or beginning graduate students. We discuss the di⇤culty of classifying certain surfaces that commonly arise during a mathematics undergraduate degree using ideas typically encountered in a course on general topology or topology of metric spaces. Then, we demonstrate that with some basic results about the Fundamental Group, it is possible to distinguish some of these surfaces. Finally, we present several applications of the Fundamental Group in Algebra and Topology, primarily proving the following classical results: Fundamental Theorem of Algebra, Brouwer’s Fixed Point Theorem, Jordan Curve Theorem, and the Invariance of Domain.