Group Cohomology, SS 2019
Prof. Dr. C. Löh
If you are interested in taking this course in SS 2019, please register as a course participant in HIS/LSF, so that the number of teaching assistants can be determined. In case you
do not have yet an RZ-account, you can also send an email to email@example.com to register.
If you plan to write a bachelor thesis under my supervision in SS 2019 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2019.
Group cohomology is an invariant that connects algebraic and geometric
properties of groups in several ways. For example, group cohomology
admits descriptions in terms of homological algebra and also in terms
of topology. Different choices of coefficients for group cohomology
leads to different invariance properties, whence to different types
Group cohomology naturally comes up in algebra, topology, and geometry.
For example, group cohomology allows to
generalise the Hilbert 90 theorem in Galois theory,
classify group extensions with given Abelian kernel,
generalise the classical group-theoretic transfer,
generalise finiteness properties of groups
(such as finiteness, finite generation, finite presentability, ...),
characterise the class of amenable groups (which
are important in large-scale geometry),
study which finite groups admit free actions on spheres,
characterise which groups admit non-trivial quasi-morphisms,
prove that certain groups admit significantly different
prove rigidity results in topology and geometry
In this course, we will introduce the basics of group homology and
cohomology, starting with elementary descriptions and calculations.
Depending on the background of the audience, we will then either
focus on a more algebraic perspective or on a more topological one.
If all participants agree, this course can be held in German; solutions to the
exercises can be handed in in German or English.
Monday, 10--12, M 102,
Thursday, 10--12, M 104.
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
is not necessary, but helpful.
Knowledge on algebraic topology (as in the course
in WS 18/19
) is not necessary, but helpful.
Last change: January 12, 2019