Oberseminar Algebra und Geometrie

Sommersemester 2020: Cohomology of Groups

Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.

All talks take place Fridays at 12:30 in

Up-date (Coronavirus pandemic) - 9 April 2020

Due to the Coronavirus pandemic the activities of the Oberseminar are currently suspended. We look forward to returning to normal seminar activities in the future. The original schedule of talks can be found below.

Short description

The aim of the seminar is to (re)visit fundamental concepts and results that are relevant to understanding the cohomology of groups. In the first half we will mainly follow lecture notes of Kenneth S. Brown [Br10]. More details can be found in his book [Br82], which develops the subject on the basis of elements from algebra and algebraic topology. After a brief look at Galois cohomology, we turn to selected cohomological topics concerning profinite groups. For this we use introductory notes of Grégory Berhuy [Be18] and selected sections from John S. Wilson's book [Wi98]. Additional other sources, including the classic books by Leonard Evens [Ev91] and by Karl W. Gruenberg [Gr70], are listed below. Due to time considerations, we will not prove all, but only selected results. It is up to the speakers to make a judicious choice of what they emphasise and treat in more detail. The maxim is: "Each talk should have a clear pitch - do not assume that your audience already knows why they should care about your particular topic".


24.04.20 1.  The Homology of a Group (Max Lindh)

Main source: Sec. 0-1 of [Br10]; Ch. II of [Br82]

Keywords: basic homological algebra, co-invariants, homology, examples: free groups and cyclic groups, topological interpretation, Hopf's theorems.

01.05.20 no seminar (Maifeiertag)
08.05.20 2.  Homology and Cohomology with Coefficients (Johannes Fischer)

Main source: Sec. 3 of [Br10]; Ch. III of [Br82]

Keywords: homology and cohomology groups, Tor and Ext, extension and co-extension of scalars, induced and co-induced modules, Eckmann-Shapiro Lemma, dimension shifting, transfer.

15.05.20 3.  Products (Djurre Tijsma)

Main source: Ch. V of [Br82]

Keywords: tensor product of resolutions, cross-products, cup and cap products, example: integral cohomology of Sym(3).

22.05.20 4.  Finiteness Conditions I (Margherita Piccolo)

Main source: Sec. 2, 4 of [Br10]; Ch. VIII, Sec. 1-5 of [Br82]

Keywords: cohomological dimension, Serre's theorem, resolutions of finite type, groups of type FPn.

29.05.20 5.  Finiteness Conditions II (N.N.)

Main source: Sec. 4 of [Br10]; Ch. VIII, Sec. 6-11 of [Br82]

Keywords: groups of type FP and FL, topological interpretation, examples: arithmetic groups, duality groups, virtual notions.

05.06.20 6.  Equivariant Homology (Tuan Thuong Dang)

Main source: Sec. 5 of [Br10]; Ch VII of [Br82]

Keywords: double complexes, spectral sequences, equivariant homology, Hochschild-Serre spectral sequence, example: amalgamations.

12.06.20 7.  Cohomology of Finite Groups (Heng Xie)

Main source: Sec. 6 of [Br10]; Ch. VI of [Br82]

Keywords: functoriality, local computation, complete cohomology (Tate cohomology), duality, periodic cohomology.

19.06.20 8.  Galois Cohomology - an Introduction (Bruno Laurent)

Main source: [Be18]

Keywords: cohomology sets, Hilbert's Theorem 90, Galois descent, example: conjugacy of matrices.

26.06.20 9.  Cohomology of Profinite Groups and a Lemma of Serre (N.N.)

Main source: Ch. 9, 10, 11 of [Wi98], in particular Sec. 11.5

Keywords: continuous cohomology of profinite groups, comparison with abstract cohomology, a lemma of Serre (Lemma 11.5.4 in [Wi98]).

03.07.20 10.   Cohomology of Uniform Pro-p Groups - a Theorem of Lazard (Luis Augusto de Mendonca)

Main source: Ch. 8, 11 of [Wi98], in particular Sec. 8.7 and 11.6

Keywords: uniform pro-p groups, graded algebras for pro-p groups, Poincaré duality groups, a theorem of Lazard (Theorem 11.6.1 in [Wi98]).

10.07.20 11.  Cohomological Goodness (Martina Conte)

Main source: Ch. I, Sec. 2.6 of [Se97] and [GJZ08]

Keywords: basic concepts and results from [Se97], selected results from [GJZ08].

17.07.20 Programme discussion for the next semester (chaired by Stefan Schröer)
from 13.00 Norddeutsches Gruppentheorie-Kolloquium 2020 at HHU
(organised by Benjamin Klopsch and Moritz Petschick)


[Be18]  Berhuy, G.: An introduction to Galois cohomology and its applications, notes, 2018; available here.
[Br82]  Brown, K. S.: Cohomology of Groups, Springer-Verlag, New York, 1982.
[Br10]  Brown, K. S.: Lectures on the Cohomology of Groups, in: Cohomology of Groups and Algebraic K-Theory (eds. Ji, Lizhen e.a.), Beijing: Higher Education Press, Advanced Lectures in Mathematics 12, 2010, pp. 131-166; available here.
[Ev91]  Evens, L.: The cohomology of groups, Claredon Press, Oxford, 1991.
[Gr71] Gruenberg, K. W.: Cohomological topics in group theory, LNM 143, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
[GJZ08] Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A.: Cohomological goodness and the profinite completion of Bianchi groups, Duke Math. J. 144 (2008), 53-72.
[Se97] Serre, J.-P.: Galois Cohomology, Springer-Verlag, Berlin, 1997.
[Wi98] Wilson, J. S.: Profinite Groups, Clarendon Press, Oxford, 1998.


WS 2019/20: Intersection theory

SS 2019: Knots and primes

WS 2018/19: The Grothendieck group of varieties and stacks

SS 2018: Arithmetic Groups - Basics and Selected Applications

WS 2017/18: Algebraic K-theory

SS 2017: Berkovich spaces

WS 16/17: Resolution of singularities and alterations

SS 2016: Modular Representation Theory

WS 15/16: The Milnor Conjectures

SS 2015: Rationality

WS 14/15: Essential Dimension

SS 2014: Varieties of Representations

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