Fundamental group and covering spaces

• Grant number: 25/04471-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2025
• End date: December 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Eduardo Rosinato Longa
• Grantee: Renan Barreto Sampaio
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

We will study two classical topics that are at the origin of Algebraic Topology: the fundamental group of a topological space and covering spaces. We will begin with the formal description of the fundamental group of a space and examine some of its properties. Then, we will describe this group in the case of the circle, which, together with the Van Kampen theorem, will allow us to compute the fundamental group of various spaces, including that of compact surfaces. In the second part, we will explore the concept of a covering space of a given topological space. After establishing an initial foundation, we will analyze how different coverings can be related, which will enable us to define the covering automorphism group of a given space and connect it to our earlier exploration of the fundamental group. (AU)