**Cohomology of Groups and Algebras**

**WS 2022/2023**

**Dietrich Burde**

**Lectures:** Monday 13:15-14:00 and Wednesday 13:15-14:45 in SR 8

**Exercises:** Monday 11:30-12:15 in SR 8

This page contains informations and pdf-files for this lecture and its exercise class.

Homology and cohomology has its origins in topology, starting
with the work of Riemann (1857), Betti (1871) and Poincare (1895)
on *homology numbers* of manifolds. Although Emmy Noether observed
in 1925 that homology was an abelian group rather than just Betti numbers,
homology remained a part of the realm of topology until about 1945.
During the period of 1940-1955 came the rise of algebraic methods. The homology and
cohomology of several algebraic systems were defined and explored: Tor and Ext
for abelian groups, homology and cohomology of groups and Lie algebras,
the cohomology of associative algebras, sheaves, sheaf cohomology and spectral
sequences. At this point the book of Cartan and Eilenberg (1956)
crystallized and redirected
the field completely. Their systematic use of derived functors, defined by projective
and injective resolutions of modules, united all the previously disparate homology
theories. Several new fields grew out of this: homological algebra,
K-theory, Galois theory, etale cohomology of schemes and so on.

We start the lecture by giving an elementary definition of group cohomology,
along with group extensions and factor systems. We give interpretations of
the n-th cohomology group for small n.
Then we will study the functorial definition of cohomology. In the last part we will treat
Lie algebra homology and cohomology, along with some applications.

The chapters are as follows:

Chapter 1: Introduction.

Chapter 2: Group extensions.

Chapter 3: Cohomology of Groups.

Chapter 4: Cohomology of Lie algebras.

Here is a syllabus and a bibliography available.

**Exercises:** There will be exercises for this lecture. Participation is possible for
those present in the first meeting at the beginning of March. In general, presence is
mandatory for the exercise class. Students being absent more than once will be signed out.
This will not apply in case of illness with a medical certificate. The rules may also change
due to the present COVID-19 situation.

There are 36 exercises in total. For each week everyone is supposed to prepare three exercises,
i.e., we start with 1,2,3 for the second week, then 4,5,6 for the third week, then 7,8,9, etc.

VO Exam 2023:

Wednesday, January 25-th, 13:15 - 14:30, Written exam, Seminarraum 8. No documents allowed.

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Files:

**Topics in the Exam**:

- Split exact sequences and group extensions
- Factor systems and equivalent group extensions
- G-modules and low-degree cohomology groups
- Functors, resolutions, homology, cohomology
- Lie algebras and Lie algebra cohomology

Dietrich Burde
Last modified: Di Jan 10 14:13:28 CET 2023