Algebraic Topology

Welcome to the Algebraic Topology class (summer 2026)!

Lectures

We have lectures every Monday and Thursday:

Tutorials

Each week there will be a tutorial with Constantin Podelski:

Course description

In topology we study the shape of geometric objects: Which properties remain unchanged under continuous deformation? Unlike in other areas of geometry, topology ignores distances and angles and uses the much more flexible notion of a topological space. Topological spaces are made of rubber: We may bend, stretch, or compress them continuously, as long as we don't tear them apart and don't glue pieces together.

Thus a circle and an ellipse are topologically the same. Similarly, a coffee mug and a donut represent the same topological object, with the handle of the mug corresponding to the hole of the donut (illustration taken from wikipedia):

However, a circle cannot be continuously deformed into a figure eight: This would require us to identify two distinct points of the circle. One could also say that the circle has only one hole, while the figure eight has two.

Turning these intuitive observations into rigorous mathematical statements is surprisingly subtle. How can we define and detect features such as the “number of holes’’ of a space, and prove that they are invariant under continuous deformation?

Algebraic topology provides powerful tools to answer such questions. The basic idea is to attach algebraic objects (e.g. groups) to topological spaces in a way that captures their global structure. These algebraic invariants are designed to remain unchanged under continuous deformations and therefore allow us to distinguish topologically different spaces.

The course will give a self-contained introduction to two of the most important such invariants, homotopy groups and homology groups. We will develop their basic theory and see some striking applications:

• a circle cannot be continuously contracted to a point,

• the Brouwer fixed point theorem: Every continuous map from a disk to itself has a fixed point,

• the Jordan curve theorem: Every simple closed curve in the plane separates the plane into an inside and an outside region,

• the Borsuk–Ulam theorem: There are always two opposite points on Earth with the same temperature and air pressure.

Algebraic topology also plays a central role in modern mathematics. It is closely connected with geometry, analysis and group theory and has a wide range of applications from robotics and motion planning to topological data analysis. The course can be seen as a starting point for exploring these directions further.

Literature

to be added soon

Course material

Problem sheets, notes and other material will be uploaded on OPAL during the semester.