Algebraic Topology, WS 2021/22
Prof. Dr. C. Löh
News

The time/location changed! (In order to resolve the
time conflict with Differential Geometry I.)

If you are interested in taking this course,
please register in HIS/LSF at the end of SS 2021.

The seminar
Proof Lab: Simplicial Topology
in WS 2021/22 accompanies this course; no previous knowledge of algebraic topology or programming is
required.

If you plan to write a bachelor thesis under my supervision in SS 2022 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2022.
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:

What is algebraic topology?

The fundamental group and covering theory

The EilenbergSteenrod axioms

Singular homology

Cellular homology

Classical applications of (co)homology.
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.
Time/Location
Tuesday, 8:3010:00, M 101,
Friday, 8:3010:00, M 101.
Exercise classes
tba
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!
Literature
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.
Prerequisites
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.
Exams
tba
Last change: July 22, 2021