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# 19223811 Forschungsmodul: Topologie “Homology and Cohomology of Groups”

• Students at FU should register via Campus Management and all students should register via MyCampus/Whiteboard.
• The course will be held remotely using WebEx.

### Summer Term 2020

>>> Please note that the start of the Summer Term has been postponed one week. <<<

• Time and place:  Tuesday, 14:00-16:00h, online

• 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.

## Talks

DateTitleSpeaker
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)