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19223811 Forschungsmodul: Topologie “Homology and Cohomology of Groups”

Summer Term 2020

Dozenten: 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. <<<

  • 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.

Please Read I.0-1.8 Brown on your own.

Talks

DatumTitelSprecher
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) N.N.
05.05.  Talk 3: Group homology and cohomology with coefficients (III.1,III.5-III.6, Brown, II.3 Adem-Milgram) N.N.
12.05.  Talk 4: Transfer maps (III.8-III.10 Brown, II.5-II.6 Adem-Milgram) N.N.
19.05.  Talk 5: Low dimensional cohomology and group extensions (IV.2-IV.6 Brown, I.1,I.6,II.8 Adem-Milgram) N.N.
26.05.  Talk 6: Products (V.1-V.6 Brown, II.4 Adem-Milgram) N.N.
02.06.  Talk 7: Tate cohomology and basic properties (VI.2-VI.6 Brown, II.7 Adem- Milgram) N.N.
09.06.  Talk 8: Cohomology theory of finite groups, duality and classification theorems (VI.7-VI.9 Brown, IV.5,IV.6 Adem-Milgram) N.N.
16.06.  Talk 9: Spectral sequences and examples (VII.2-VII.6 Brown, IV.1-IV.2 Adem- Milgram) N.N.
23.06.  Talk 10: Equivariant homology (VII.7-VII.10 Brown, V.0-V.1 Adem-Milgram) N.N.
30.06.   TBA N.N.
07.07.   TBA N.N.
14.07.   TBA N.N.

Literatur:

  • 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)