Course Details for A.Y. 2013/2014

Course Objectives

The goal of this course is to provide  the motivations, definitions and techniques for the translation of  topological problems into algebraic ones, hopefully easier to deal with. On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.

Course Content

• General Topology: Topological spaces and continuous maps, induced, quotient and product topology,  Metric spaces, Hausdorff spaces, Compact spaces, connected spaces, Paths and path connected spaces
• Manifolds and surfaces: The pancake problems, n-dimensional manifolds, surfaces  and classification of surfaces. 
• Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.  
• Covering spaces: The fundamental group of a covering space, the fundamental group of a orbit space,  lifting theory and existence theorems, the Borsuk-Ulam theorem, the Seifert-Van Kampen theorem, the fundamental group of a surface
• Introduction to singular homology : standard and simplicial simplexes. Introduction to singular homology : standard and simplicial simplexes.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

The student must know the basic notions of General Topology and Geometry contained in the exams Geometry A, B and Algebra

Assessment Methods and Criteria

Oral Exam

Textbooks

• Czes Kosniowski, A first course in algebraic topology , Cambridge University Press. 1980.    
• Czes Kosniowski, Introduzione alla Topologia Algebrica , Zanichelli Ed.. 2010.    

Course page updates

• A.Y. 2012/2013
• A.Y. 2013/2014
• A.Y. 2014/2015
• A.Y. 2015/2016
• A.Y. 2016/2017
• A.Y. 2017/2018
• A.Y. 2018/2019