Introduction to Topological Manifolds

• Grant number: 25/20367-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2025
• End date: October 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ivan Struchiner
• Grantee: Carlos Filipe Flor de Carvalho
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 22/16097-2 - Modern methods in differential geometry and geometric analysis, AP.TEM

Abstract

This undergraduate research project aims to introduce the student to the study of topological manifolds, structures that are crucial for advanced mathematical research. The project begins with an introduction to General Topology, allowing the student to gain familiarity and expertise in this field and understand the definition of a topological manifold. Afterward, the project will continue with the construction of key examples of manifolds and the tools needed to demonstrate the first part of the Classification Theorem for Compact Manifolds, the first central theorem of the research. Finally, Homotopy Theory and the study of the Fundamental Group of topological spaces will be approached, along with a brief exploration on Free Groups, culminating in the understanding and application of the second part of the Classification Theorem for Compact Surfaces and the Seifert-Van Kampen Theorem.Part I - Content:- Topological spaces and examples, convergence, continuity, homeomorphisms, Hausdorff spaces - Definition of topological manifolds- Product and quotient spaces- Connectedness and compactness - Main examples of manifolds (the torus, Klein s bottle, Moebius strip, projective plane, etc.)Part II - Content:- Cell complexes e CW-complexes- Classification of 1-manifolds- Surfaces, connected sum, poligonal presentation, part 1 of compact surfaces theorem- Euler characteristic and orientabilityPart III - Content:- Paths, homotopy and fundamental group- The fundamental os spheres Sn, homotopy equivalence- The fundamental group of the circlePart IV - Content:- Free products, free groups, groups presentation and free abelian groups- Part 2 of Compact Surfaces Classification Theorem- Theorem of Seifert-van Kampen (AU)