Classification of surfaces

• Grant number: 19/16142-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2019
• End date: November 30, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ivan Struchiner
• Grantee: Sophia Lopes Ribeiro Fiorotto
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis, AP.TEM

Abstract

The main goal of this project is to introduce basic concepts of general and algebraic topology. To achieve that, we will use the study of classification of surfaces as motivation to bring up new concepts. Our approach to the problem is to first give a list of compact connected surfaces. All of which will be constructed from a polygonal region in the plane by identifying its edges using the labelling scheme, developing concepts like quotient topology. From this, we will show that any compact connected surface is homeomorphic to one in the list and that any two surfaces in the list are not homeomorphic to each other, developing concepts from algebraic topology. (AU)