Algebraic Topology, WS 2018/19

Prof. Dr. C. Löh / D. Fauser / J. Witzig


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 and (non-)embeddability results. 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 "Group Cohomology" in the summer 2019, where (co)homology of groups will be studied. The course in SS 2019 can also be attended independently of the present course on Algebraic Topology.

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: pdf. Topics covered so far:


Monday, 10--12, M 102,
Thursday, 10--12, M 104.

Exercise classes

Do 14--16, H31
Fr 8--10, M009

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!

Sheet 1, of October 15, 2018, submission before October 22, 2018 (10:00) will be discussed in the exercise classes on October 25/26
Sheet 2, of October 22, 2018, submission before October 29, 2018 (10:00) will be discussed in the exercise classes on November 2/?


These etudes help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded.

Sheet 0, of October 15, 2018, no submission, will be discussed in the exercise classes on October 18/19.
Sheet 1, of October 19, 2018, no submission,


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, October 22, 2018