## Over deze cursus

This course is optional for mathematics students. The course is recommended to students interested in pure mathematics, such as differential geometry, algebraic geometry, algebraic topology, algebra, logic. Please find more information about the study advisory paths in the bachelor at the.

**Leerdoelen:**

We will start with some questions outside and inside of topology that we will solve during the course using methods of homotopy theory and algebraic topology. In particular, we will define homotopies, homotopy equivalences and the fundamental groups. The fundamental group of a topological space will occupy a major part of this course: The first computation (with already some applications) is the fundamental group of the circle. The key computational tool for more complicated spaces is the van Kampen theorem. We will show that the fundamental group is closely related to free group actions and covering space theory.

An important test ground for these ideas are two-dimensional manifolds, i.e. surfaces. We will give a classification of these, both in the oriented and non-oriented case.

The last part of the course will deal with homology. Homology is an algebraic invariant that is much more suitable to high dimensional phenomena than the fundamental group. While it takes a bit to prove the key computational tools (like homotopy invariance and the Mayer-Vietoris sequence), there will be a significant payoff.

At the end of this course, a student is able to:

- Define homotopies between maps, homotopy equivalences between topological spaces and illustrate these notions by some examples.
- Define the fundamental group and compute it for a large class of spaces.
- State the classification of surfaces.
- Explain the terms "connected sum", "orientable" and "Euler characteristic".
- Define (universal) covers and know the standard theorems and examples of them
- Define the notion of homology, and compute it for some small familiar examples.
- Decide for many pairs of spaces whether they are homotopy equivalent (or even homeomorphic) or not.
- Provide applications (like the Brouwer fixed point theorem).

**Onderwijsvormen:**

Two times per week two hours of lectures and two times per week tutorials.

**Toetsing:**

There are two exams for this class, the midterm will count for 30% and the final exam will count for 50% of the final grade.Furthermore, there is homework which counts for 20% of the final grade.

In case of a retake exam, the retake exam counts for 80% and the homework for 20% of the final grade, while the original grades of midterm and final exam are deleted.

**Herkansing en inspanningsverplichting:**

Students with a final grade lower than 4 are eligible to do the retake exam only if they have handed in solutions to all hand-in homework problems, either before or after the final exam.

**Taal van het vak:**

The language of instruction is English.

## Leerresultaten

Zie onder vakinhoud.

## Voorkennis

WISB124 Inleiding Groepen en Ringen en WISB243 Inleiding Topologie. Zie de cursusplanner (cursusplanner.uu.nl) voor de inhoud van deze vakken: selecteer Faculteit Betawetenschappen en vervolgens het programma van de bachelor Wiskunde van het meest recente jaar.

- Code
**WISB341** - Studiepunten
**ECTS 7.5** - Email contactpersoon