# Group Cohomology, SS 2019

Prof. Dr. C. Löh / Dr. D. Fauser / J. Witzig

News

• The lecture notes are updated (version of July 25).
(25.07.) Sign error on p. 126 (in the computation of the homology of S_3) corrected.
(22.07.) Proposition 4.2.2: Statement about detection via free modules corrected (indeed, this works only in the case of finite cohomological dimension; the other descriptions also work in the infinite-cd case).
• The new etudes are online: Etudes 13 (sheet of July 25, no submission).
• The final exercise sheet is online: Sheet 13 (sheet of July 22, optional submission before July 29, 10:00).
This sheet contains optional exercises (bonus points!). Please make sure that you reach 50% of the points (of the normal exercises).
• The new exercise sheet is online: Sheet 12 (sheet of July 15, submission before July 22, 10:00).
This is the last sheet that will count into the Studienleistung. In addition, there will be Sheet 13 with lots of bonus points! Please make sure that you reach 50% of the points (of the normal exercises).
• Quasi-sequel to this course: Bounded cohomology and simplicial\ volume Arbeitsgruppe Löh: Dr. Marco Moraschini; course homepage
• information on oral exams: see below
• Organisational matters
• If you plan to write a bachelor thesis under my supervision in WS 2019/20 (in Topology/Geometry), you should participate in a seminar in the Global Analysis and Geometry group before WS 2019/20.

Group Cohomology

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 of applications. 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 dynamical systems,
• 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.

Lecture notes: pdf. Topics covered so far:
• Literature
• Introduction
• What is group cohomology?
• Why group cohomology?
• Overview of this course
• The basic view
• Foundations: The group ring
[The group ring, Modules over the group ring, The domain categories for group (co)homology]
• The basic definition of group (co)homology
[The simplicial and the bar resolution, Group (co)homology]
• Degree 0: (Co)Invariants
• Degree 1: Abelianisations and homomorphisms
[Homology in degree 1: Abelianisation, Cohomology in degree 1: Homomorphisms, Application: Hilbert 90]
• Degree 2: Presentations and extensions
[Homology in degree 2: Hopf's formula, Cohomology in degree 2: Extensions]
• Changing the resolution
[Projective resolutions, The fundamental theorem of group (co)homology, Example: Finite cyclic groups, Example: Free groups]
• (Co)Homology and subgroups
[Restriction and (co)induction, The Shapiro lemma, Transfer]
• The geometric view
• Foundations: Geometric group theory
[Quasi-Isometry, Amenability]
• Uniformly finite homology
[Uniformly finite homology of spaces, Uniformly finite homology of groups, Application: Ponzi schemes and amenability]
• Bounded cohomology
[Bounded cohomology of groups, Application: A characterisation of amenability, Application: Quasi-morphisms, Application: Stable commutator length]
• The derived view
• Derived functors
[Axiomatic description, A construction, The two sides of Tor, Group homology as derived functor, Group cohomology as derived functor, The derived category]
• The Hochschild-Serre spectral sequence
[Terminology for spectral sequences, Classical spectral sequences, The spectral sequence of an extension, A proof of Hopf's formula, Universal coefficients and products]
• The topological view
• Classifying spaces
[The standard simplicial model, Changing the classifying space, Examples of classifying spaces, Group (co)homology via classifying spaces]
• Finiteness conditions
[Cohomological dimension, Finite type]
• Application: Free actions on spheres
[From sphere actions to periodic resolutions, From periodic resolutions to Sylow subgroups]
• Appendix
• Amalgamated free products
• Some homological algebra
• Homotopy theory of CW-complexes

comments:
• (02.05.) When defining the bar resolution in the lecture, I wrote [g_2|...|g_{n}] as last summand in the definition of the boundary operator. It should have been [g_1|...|g_{n-1}] (as in the lecture notes).
• (02.05.) The computation of H_1 today was unnecessarily obfuscated; in the lecture notes, I provide a more transparent, algebraic, argument (the geometric observation is still helpful, but now exiled into a separate remark).
• (02.05.) Diagram on p. 19 (computation of the bar cocomplex): In the lecture, I wrote "g^{-1} a" instead of "g a"; as we do not have to flip between left and right, "g a" is better (but both descriptions lead to the same object because in the category of \Z-modules the two descriptions are isomorphic).
• (20.05) Proof of Theorem 1.6.15: proof of step 1: It is easy to reduce to the case of 1 + p^{n-1} (see the diagram in the lecture notes; the right arrow is not the identity; thus, these extensions are not equivalent, but they are "isomorphic"). When discussing this during the lecture, I was confused about something that is not relevant to this comparison.
• (06.06.2019) The floored chain in the proof of Theorem 2.2.15 is corrected (in the lectures the scoping of the floor was wrong).
• (06.06.2019) Corrected the averaging in the cohomological transfer (and some other typos here and there).
• (04.07.) Proofs of Corollary 1.5.3 and 1.5.4 are corrected: of course, we only have an upper bound for rk_\Z H_1(N;\Z)_G by |R| (and not for rk_\Z H_1(N;\Z), because N is only normally generated by R; taking coinvariants ensures that H_1(N;\Z)_G is generated by R).
• (22.07.) Proposition 4.2.2: Statement about detection via free modules corrected (indeed, this works only in the case of finite cohomological dimension; the other descriptions also work in the infinite-cd case).
• (25.07.) Sign error on p. 126 (in the computation of the homology of S_3) corrected.

Time/Location

Monday, 10--12, M 102,
Thursday, 10--12, M 104.

Exercise classes

group 1: Thursday, 12--14, M 102
group 2: Friday, 8:30--10, H 32
• The exercise classes start in the second week; in this first session, some basics material will be discussed (as on the sheet Etudes 0).
• Organisational matters

Exercise sheets

Solutions can be submitted in English or German and in teams of up to two people. Please do not forget to add your name to all your submissions!

 Sheet 1, of April 29, 2019, submission before May 6, 2019 (10:00) will be discussed in the exercise classes on May 9/10 Sheet 2, of May 6, 2019, submission before May 13, 2019 (10:00) will be discussed in the exercise classes on May 16/17 Sheet 3, of May 13, 2019, submission before May 20, 2019 (10:00) will be discussed in the exercise classes on May 23/24 Sheet 4, of May 20, 2019, submission before May 27, 2019 (10:00) will be discussed in the exercise classes on May 29(!)/31 Sheet 5, of May 27, 2019, submission before June 3, 2019 (10:00) will be discussed in the exercise classes on June 6/7 Sheet 6, of June 3, 2019, submission before June 11(!), 2019 (10:00) will be discussed in the exercise classes on June 13/14 Sheet 7, of June 10, 2019, submission before June 17, 2019 (10:00) will be discussed in the exercise classes on June 19/21 Sheet 8, of June 17, 2019, submission before June 24, 2019 (10:00) will be discussed in the exercise classes on June 27/28 Sheet 9, of June 24, 2019, submission before July 1, 2019 (10:00) will be discussed in the exercise classes on July 4/5 Sheet 10, of July 1, 2019, submission before July 8, 2019 (10:00) will be discussed in the exercise classes on July 11/12 Sheet 11, of July 8, 2019, submission before July 15, 2019 (10:00) will be discussed in the exercise classes on July 18/19 Sheet 12, of July 15, 2019, submission before July 22, 2019 (10:00) will be discussed in the exercise classes on July 25/26 Sheet 13, of July 22, 2019, optional submission before July 22, 2019 (10:00)

Etudes

These etudes help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded.

 Sheet 0, of April 25, 2019, no submission, will be discussed in the exercise classes on May 2/3. Sheet 1, of May 2, 2019 no submission Sheet 2, of May 9, 2019 no submission Sheet 3, of May 16, 2019 no submission Sheet 4, of May 23, 2019 no submission Sheet 5, of May 30, 2019 no submission Sheet 6, of June 6, 2019 no submission Sheet 7, of June 13, 2019 no submission Sheet 8, of June 20, 2019 no submission Sheet 9, of June 27, 2019 no submission Sheet 10, of July 4, 2019 no submission Sheet 11, of July 11, 2019 no submission Sheet 12, of July 18, 2019 no submission Sheet 13, of July 25, 2019 no submission

Literature

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.

Prerequisites

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 of Kommutative Algebra) is not necessary, but helpful.
Knowledge on algebraic topology (as in the course in WS 18/19) is not necessary, but helpful.

Exams

Please read the information on organisation and formalities of this course.

Dates for the oral exams (25 minutes):
• Wed, July 31
• Wed, September 25
Registration details:
• Please register in FlexNow for the oral exam. Registration deadline: Two weeks before the exam. (You can then de-register until one week before the exam).
• Moreover, for the oral exam, you also need to register with Ms. Bonn (M 217) for a time slot!
• If applicable: For the Studienleistung (successful participation in the exercise classes), please register in FlexNow before July 25.
• If none of the dates for the oral exams fits (e.g., because you are going abroad), please contact me directly.
. If you prefer to take a combined oral exam on Algebraic Topology and Group Cohomology, please contact me.

Last change: July 25, 2019