Oberseminar Algebra und Geometrie

Sommersemester 2021: Cohomology of Groups

Chaired by I. Halupczok, B. Klopsch and M. Zibrowius.

Organised by P. Cubides Kovacsics and B. Klopsch.

All talks take place Tuesdays at 10:30 in 25.21 HS 5F and/or online (via Cisco Webex). In persona talks should be kept to 90 min, pure online talks should be kept shorter, up to 70 min (including questions).
Please ensure to be on the mailing list to be notified about short-term changes.

Due to the Coronavirus pandemic the activities of the Oberseminar were suspended. We aim to return to half-way normal seminar activities this semester. The schedule of talks below has been adapted from the original plan for the summer semester 2020, taking into consideration that talks may very well take place online and will be kept shorter.

Aims and Content - a short description

The aim of the seminar is to (re)visit fundamental concepts and results that are relevant to understanding the cohomology of groups. 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. We also give a brief look at Galois cohomology; for this we use introductory notes of Grégory Berhuy [Be18]. 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".
Selected cohomological topics concerning profinite groups, which were part of the original programme for 2020, are possibly covered in the Advanced Seminar on Group Theory; see John S. Wilson's book [Wi98] or Jean-Pierre Serre's classic [Se97] for the development of the theory in this direction.

Schedule

13.04.21 get ready - no seminar yet
20.04.21 1.  The Homology of a Group I (Dominic Witt)

Main source: Sec. 0-1 of [Br10]; Ch. (I and) II, Sec. 1-5 of [Br82]

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

27.04.21 2.  The Homology of a Group II (Max Lindh)

Main source: Ch. II, selected Ex. from Sec. 5, Sec. 6-7 of [Br82]

Keywords: some application (e.g. one-relator groups or group extensions), functoriality, Mayer-Vietoris sequence for the homology of amalgamated free products.

04.05.21 3.  Homology and Cohomology with Coefficients I (Margherita Piccolo)

Main source: Sec. 3 of [Br10]; Ch. III, Sec. 0-5 of [Br82]

Keywords: homology and cohomology groups, Tor and Ext, extension and co-extension of scalars, induced and co-induced modules.

11.05.21 4.  Homology and Cohomology with Coefficients II (Luca Maria Di Gravina)

Main source: Sec. 3 of [Br10]; Ch. III, Sec. 6-10 of [Br82]

Keywords: Eckmann-Shapiro Lemma, dimension shifting, transfer.

18.05.21 5.  Products (Iker De las Heras Kerejeta)

Main source: Ch. V of [Br82]

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

25.05.21 6.  Finiteness Conditions I (Luis Augusto Mendoça )

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

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

01.06.21 7.  Finiteness Conditions II (NN)

Main source: Sec. 4 of [Br10]; Ch. VIII, Sec. 5-9 of [Br82]

Keywords: groups of type FPn, groups of type FP and FL, topological interpretation, examples: arithmetic groups.

08.06.21 8.  Finiteness Conditions III (David Bradly-Williams)

Main source: Sec. 4 of [Br10]; Ch. VIII, Sec. 9-11 of [Br82]

Keywords: examples, duality groups, virtual notions.

15.06.21 9.  Spectral Sequences (Florian Felix)

Main source: Sec. 5 of [Br10]; Ch VII, Sec. 1-6 of [Br82]

Keywords: double complexes, spectral sequences, homology of a union, Hochschild-Serre spectral sequence.

22.06.21 10.  Equivariant Homology (NN)

Main source: Sec. 5 of [Br10]; Ch VII, Sec. 7-10 of [Br82]

Keywords: equivariant homology, example: amalgamations.

29.06.21 11.  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.

06.07.21 12.  Galois cohomology I (Djurre Tijsma)

Main source: Sec. 1-5 of [Be18]

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

13.07.21 13.  Galois cohomology II (Martina Conte)

Main source: Sec. 6-12 of [Be18]

Keywords: functorial group actions, twisted forms, Galois descent lemma, example: conjugacy of matrices again

Literature

[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.
[Se97] Serre, J.-P.: Galois Cohomology, Springer-Verlag, Berlin, 1997.
[Wi98] Wilson, J. S.: Profinite Groups, Clarendon Press, Oxford, 1998.

Archive

SS 2020 and WS 20/21: cancelled due to pandemic

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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