# Cohomology of Groups and AlgebrasWS 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.

## Files:

 No. Thema Datum file 1 Syllabus 01.10.2022 Contents.pdf 2 My Lecture Notes 01.01.2023 Cohomology of Groups and Algebras 3 Exercises 01.10.2022 exercises_coho.pdf

Topics in the Exam:

1. Split exact sequences and group extensions
2. Factor systems and equivalent group extensions
3. G-modules and low-degree cohomology groups
4. Functors, resolutions, homology, cohomology
5. Lie algebras and Lie algebra cohomology

Dietrich Burde