Algebraic Topology 1

The aim of the course is to show how basic geometric structures may be studied by transforming them into algebraic questions. Studying geometric objects by associating algebraic invariants to them is a powerful idea which influenced many areas of mathematics. For example, deciding about the existence of a map between spaces (often a difficult task) may be translated into deciding whether an algebraic equation has a solution (which is often quite simple). The goal of the course is to introduce the most important examples of such invariants, such as singular homology and cohomology groups, and to calculate them for fundamental examples and constructions of topological spaces. Basic notions in Category Theory and Homological Algebra will be reviewed according to the knowledge of the participants.

1. Knowledge. The student has knowledge of fundamental concepts and methods in algebraic topology, in particular singular homology.

2. Skills. The student is able to apply his or her knowledge of algebraic topology to formulate and solve problems of a geometrical and topological nature in mathematics.

The learning methods and activities depend on the course instructor, but will in general consist of lectures and exercises.