In this course we develop two important ways to study a topological space:
(1) via its coverings and its fundamental group,
(2) via its singular homology.
We shall try to emphasise both formal/abstract properties and concrete examples. Part (1) will eventually focus on the equivalence (under some mild conditions on the topological space) between coverings, on the one hand, and representations of the fundamental group, on the other hand.
Part (2) will end with a discussion of the famous Brouwer fixpoint theorem in arbitrary dimension, and the hairy ball theorem.
The course will use on a modest level the tools and language of category theory and homological algebra. All required notions are introduced during the course.
Hours of class per week
Homework and final exam (oral or written, depending on the number of students).
Basic theory of groups as in Algebra 1, basic theory of vector spaces as in Lineaire algebra 1 – 2, basic theory of topological spaces as in Topologie.
We will treat parts of
J. M. Lee: Introduction to topological manifolds, Springer GTM 202.
W. Fulton: Algebraic Topology: A First Course, Springer GTM 153.
Both books are available as e-book through the university network.
See this webpage