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.
We will elaborate on the tight relation between coverings of a given topological space and representations of its fundamental group. This will be done in several level of generality, culminating in Grothendieck’s functorial formulation of Galois theory. We will highlight the similarities with the Galois theory of field extensions.
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
Homework and half an hour oral exam
Algebra 1 – 3, Lineaire algebra 1 – 2, Topologie
The course will make use of the language of categories and functors. The basic definitions will be recalled. However a little familiarity with these notions, at level of [this page](https://en.wikipedia.org/wiki/Category_(mathematics) and this page is desirable.