19243901 Group cohomology
- FU-Students should register via Campus Management.
- Non-FU-students should register via MyCampus/Whiteboard.
Summer Semester 2026
Lecturer: Kevin Li, PhD
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Time and place: Thursdays, 10-12h, SR 009, Arnimallee 6.
| Date | Topic |
| 16.04. | Course overview, review of homological algebra |
| 23.04. | Group rings, group cohomology as derived functor, cohomology of cyclic groups |
| 30.04. | Standard and bar resolutions, classification of central extensions |
| Tue 05.05. 08:15 h usual room |
Classifying spaces |
| 07.05. | No lecture (EGGG meeting) -- moved to 05.05. |
| Tue 12.05. 08:15 h usual room |
Künneth formula, Mayer--Vietoris |
| 14.05. | No lecture (public holiday) -- moved to 12.05. |
| 21.05. | Right-angled Coxeter and Artin groups, Shapiro's Lemma |
| 28.05. | Transfer map, geometric dimension |
| 04.06. | Cohomological dimension, Eilenberg--Ganea Theorem |
| 11.06. | Serre's Theorem, virtual cohomological dimension |
| 18.06. | G-CW-complexes, Lück's blow-up construction |
| 25.06. | No lecture -- moved to 07.07. |
| 02.07. | Geometric finiteness properties, Schanuel's Lemma |
| Tue 07.07. 08:15 h usual room |
Cohomological finiteness properties |
| 09.07. | Brown's criterion |
| 16.07. | Bestvina--Brady groups |
Prerequisits: Basic knowledge of topology (fundamental group, CW-complexes, homology), homological algebra, and group theory.
Course overview: Group cohomology is a beautiful interplay between algebraic topology and group theory.
A CW-complex can be small in different ways; it can be finite dimensional or of finite type. For a group G, one obtains two corresponding notions by requiring the existence of a small classifying space BG.
- The geometric dimension of G is the minimal dimension of a model for BG.
- We say that G is of type F_n if there exists a model for BG with finite n-skeleton.
Group cohomology is simply the cellular/singular cohomology of BG or, equivalently, the Ext groups of the trivial ZG-module Z. This will yield two algebraic counter-parts; the cohomological dimension and type FP_n.
A guiding theme of the lecture will be to compare the geometric/topological notions to the algebraic ones. It is a famous open problem of Eilenberg—Ganea from 1957 if the geometric and cohomological dimensions coincide for every group. The revolutionary cubical Morse theory of Bestvina—Brady from 1997 showed that the finiteness properties F_n and FP_n do not agree in general.
Group cohomology has many applications, e.g., to geometric group theory. The finiteness properties F_n and FP_n and the cohomological dimension are quasi-isometry invariants. Recurring examples of groups will be right-angled Artin and Coxeter groups.
References:
- Brown's book ``Cohomology of groups", 1982.
- Löh's lecture notes ``Group cohomology", 2019.
- Geoghegan's book ``Topological methods in group theory", 2008.
- Davis' book ``The geometry and topology of Coxeter groups", 2008.
- Wilton's lecture notes ``Non-positively curved cube complexes", 2011.