In this course we will introduce two ways to study a topological space:

(1) Via coverings and the fundamental group

(2) Via singular homology.

The first part will discuss the Seifert-Van Kampen theorem and Galois theory of covering spaces.

The second part will treat singular homology and the little homological algebra needed. As applications, we will discuss classical theorems in geometry and topology such as the Brouwer fixed point theorem, the hairy ball theorem, invariance of dimension…

**Hours of class per week**

2

**Final grade**

Homework and oral

**Prerequisites**

Algebra 1—3, Lineaire algebra 1—2, Topologie

**Further information:**

Course webpage to be announed.

If time permits, we will elaborate on the tight relation between the category of covering spaces of a given path connected space X, on the one hand, and the category of representations of the fundamental group of X, on the other.