Tópicos de topologia algébrica e o grupo fundamental do círculo

Abstract

In this work, we present the introductory study to the fundamental group, which can be an important tool to answer one of the main questions of topology: if two spaces (or surfaces) are homeomorphic or not. Initially, we approach basics concepts of topological spaces necessary for the following study: homotopy between applications and, more precisely, between paths. Subsequently, we verify that the equivalence classes formed by homotopic closed paths provide a group-specific structure for each space. In a second moment, we calculate the fundamental group of the unit circle. We also present, using the mentioned tools, a demonstration for the Fundamental Theorem of Algebra.