Algebraic Topology, WS 2021/22

Prof. Dr. C. Löh


Algebraic Topology

Algebraic topology studies topological spaces via algebraic invariants -- by modelling certain aspects of topological spaces in the realm of algebra, e.g., by groups and group homomorphisms. Classical examples include homotopy groups and (co)homology theories.


Algebraic topology has various applications, both in theoretical and in applied mathematics, for instance, through fixed point theorems, (non-)embeddability results, topological data analysis, and many more. For example, Nash's proof of existence of certain equilibria in game theory is based on a topological argument. Topics covered in this course include: This course will be complemented with the course "Geometric Group Theory" in the summer 2022. The course in SS 2019 can also be attended independently of the present course on Algebraic Topology. Moreover, there probably will also be a continuation of the Algebraic Topology Series.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Lecture notes: will be made available.


Tuesday, 8:30--10:00, M 101,
Friday, 8:30--10:00, M 101.

Exercise classes


Exercise sheets

Solutions can be submitted in English or German and in teams of up to two people. Please do not forget to add your name to all your submissions!


This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books.
A list of suitable books can be found in the lecture notes.


All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Knowledge about manifolds as in Analysis IV is not necessary, but helpful.
Knowledge about basic homological algebra (as in the last two weeks of Kommutative Algebra) is not necessary, but helpful.



Last change: July 22, 2021