Smooth manifolds and their topological invariants

• Grant number: 14/22592-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2015
• End date: February 29, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ivan Struchiner
• Grantee: Pedro Mendes de Araújo
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The purpose of this project is to introduce the student to some fundamental aspects of algebraic topology (particularly the fundamental group), and also those of the geometry and topology of manifolds. Initially, the basic definitions and properties regarding differentiable manifods will be studied. Next, after the student has become familiarized with these properties and with many examples of manifolds, other topics such as the Brouwer fixed point theorem, and classical Lie groups will be discussed. In the second part of the project, emphasis will be given to algebraic topology, with focus on the fundamental group. The construction (and classification) of covering spaces, and the Seifert - van Kampen theorem will be studied. All of the concepts of algebraic topology will be exemplified to smooth manifolds. In particular, the process of constructing surfaces by cutting and glueing, and the calculus of invariants of these surfaces will be discussed (all with the main purpose of proving the classification theorem of compact surfaces). (AU)