19223811 Forschungsmodul: Topologie “Homology and Cohomology of Groups”
- FU-Students should register via Campus Management.
- Non-FU-students should register via MyCampus/Whiteboard.
- The course will be held remotely using WebEx.
Summer Term 2020
Lecturer: Dr. Gabriel Angelini-Knoll, Prof. Dr. Holger Reich, Prof. Dr. Elmar Vogt
>>> Please note that the start of the Summer Term has been postponed one week. <<<
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Time and place: Tuesday, 14:00-16:00h, online
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Leistungsnachweis/criteria for proof of performance:
Grade and credit points will be awarded based on a presentation and written summary.
Prerequisites: We assume basic knowledge of topology as taught in Topology I and II.
Content: Homology and Cohomology of Groups
The subject of this seminar is a wonderful mix of algebra, in the form of homological algebra, and topology. In Topology II you have encountered the homology and cohomology groups of a group G with coefficients in a G-module M as the tor groups Tor^{ZG}(Z,M) respectively the Ext groups Ext_{ZG}(Z,M) where ZG is the integral group ring of G. For topologists it is the homology and cohomology of any K(G,1), i. e. of a connected CW-complex with fundamental group G and vanishing higher homotopy groups. We will see that it easy to show that the two desrciptions lead to naturally isomorphic groups.
This leads to an attractive interplay between homological algebra and topology where one can use topology to obtain information about the structure of the group G, or use homological algebra to obtain insight in topological questions. The seminar is also an ideal opportunity to apply what you have learned in Topology II to some concrete questions.
The seminar will be based on portions of textbooks, both classics in this field, by Kenneth Brown and Alejandro Adem and James Milgram.
Please Read I.0-1.8 Brown on your own.
Talks
| Date | Title | Speaker |
|---|---|---|
| 21.04. | Talk 1: Motivation and some homological algebra (I.4,I.6, II.1-II.3 Brown) | Gabriel Angelini-Knoll |
| 28.04. | Talk 2: Homology of a group and topology (II.4-II.7 Brown, II.1 Adem-Milgram) | Elmar Vogt |
| 05.05. | Talk 3: Group homology and cohomology with coefficients (III.1,III.5-III.6, Brown, II.3 Adem-Milgram) | Georg Lehner |
| 12.05. | Talk 4: Transfer maps (III.8-III.10 Brown, II.5-II.6 Adem-Milgram) | Daniel Krupa |
| 19.05. | Talk 5: Low dimensional cohomology and group extensions (IV.2-IV.4,IV.41,IV.51 Brown, I.1,I.6,I.81 Adem-Milgram) | Elisabeth A.d. Siepen |
| 26.05. | Talk 6: Products (V.1,V21,V.31,V.4-V.5,V.61 Brown, II.4 Adem-Milgram) | Joshua Egger |
| 02.06. | Talk 7: Tate cohomology: definitions (VI.2-VI.4 Brown, II.7 Adem- Milgram) | Ferry Saavedra |
| 09.06. | Talk 8: Tate cohomology: basic properties (VI.5-VI.6 Brown, II.7 cont. Adem-Milgram) | Colin Rothgang |
| 16.06. | Talk 9: Duality and Cohomologically trivial modules (VI.7-VI.8 Brown) | Elmar Vogt |
| 23.06. | Talk 10: Classification theorems for cohomology of finite groups (VI.9 Brown, IV.5-IV.6 Adem-Milgram) | Aldo Kiem |
| 30.06. | Talk 11: Spectral sequences: definitions (VII.2-VII.4 Brown) | Gabriel Angelini-Knoll |
| 07.07. | Talk 12: Spectral sequences: examples (VII.5-VII.6 Brown, IV.1-IV.2 Adem-Milgram) | Evgeniya Lagoda |
| 14.07. | Talk 13: Equivariant homology (VII.7-VII.10 Brown, V.0-V.1 Adem-Milgram) | Elmar Vogt |
1This section is optional because it either covers more advanced material or it is a review of material covered in the course Topology II held in the Winter Semester 2019/2020
Literature:
- K. Brown: Cohomology of groups. Springer Verlag Graduate Texts in Mathematics 87 (1982)
- A. Adem and J.Milgram: Cohomology of Finite Groups. Springer Verlag Grundlehren der mathematischen Wissenschaften 309 (1994)