**Prerequisites**

- Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, and ideally a first encounter with the fundamental group

- Knowledge about basic constructions with vector spaces and abelian groups.

- Familiarity with the definition of a differentiable manifold or at least a submanifold of R^n.

- Familiarity with the definition of a category and a functor is also helpful. For those who haven't seen this before, the "Intensive course on Categories and Modules" http://www.rolandvdv.nl/AT18/#

**Aim of the course**

The aim of the course is to introduce the basic tools of algebraic topology: homotopy and (co)homology groups. Homotopy groups are higher-dimensional generalizations of the fundamental group and classify maps from spheres to our chosen space up to homotopy. We will use differential topology to show that they can vanish in low enough degrees and to construct the mapping degree. As computations with homotopy groups are very difficult, we will introduce (co)homology groups of spaces, which are much more computable. On the other hand, they are more difficult to define and we will present a route via Eilenberg--Mac Lane spaces. Using (co)homology, we have tools to show that many spaces are not homeomorphic (and not even homotopy equivalent). We will also present other applications.

**Rules about Homework/Exam**

There will be regular hand-in homework sets throughout the course and there will be a written exam at the end.

The score from the homework assignments will count as a bonus for the final examination: if the grade from the written exam is at least 5.0 and the average grade from the homework assignments is higher, then the written exam counts 75% and the homework assignments count 25%. For the retake, no homework bonus will be given (unless you are unable to attend the final exam because of a scheduling conflict with another exam or other serious circumstances; in this case contact us as early as possible and you will be able to take the retake with your usual homework bonus). In case you take the retake, whether the better grade or automatically the retake grade counts depends on your university policy.

**Lecture notes/Literature**

Hatcher: Algebraic Topology

Milnor: Topology from a differentiable viewpoint

**Lecturers**

Inbar Klang

Lennart Meier

- Docent: Inbar Klang
- Docent: Lennart Meier
- Docent: Marco Nervo
- Docent: Sven van Nigtevecht