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Forschungsseminar Algebraische und Geometrische Topologie

Summer Term 2011

 Prof. Dr. Holger Reich     -    Prof. Dr. Elmar Vogt     -    Prof. Dr. Günter M. Ziegler


  • Time and place:  Wednesday 15 -17 h,  SR Villa,  Arnimallee 2


 

Apart from several guest talks in this semester we would like to study the following topic:

Equivariant homology and Bredon homology


A G-homology theory is the analogue of a homology theory in the equivariant context. It is a functor, which instead of spaces digests spaces with a G-action.

It is well known that a homology theory is essentially determined by its value on a point and if this is concentrated in dimension 0, then we are dealing with ordinary homology with coefficients in an abelian group. In the equivariant context the smallest building blocks of a space are not points but orbits. Bredon homology is the analogue of an ordinary homology theory in the equivariant context. A classical theorem of Dold says that after rationalizing, i.e. tensoring with Q, every homology theory is a sum of shifted ordinary homology groups. The relevant isomorphism is usually called Chern character. A similar theorem is true but quite involved in the equivariant context and  was proven in 2002 by Wolfgang Lück, see [5]. The seminar starts with a review of classical group homology. Then we will develop basic concepts that are useful whenever one is talking about group actions. Finally we will formulate Lücks theorem about the equivariant Chern character and outline its proof.

Schedule

DateTitleSpeaker 
       
  Guest talk:    

16.03.
10h ct

On the Eilenberg-Moore spectral sequence
 
John McCleary
 
Abstract
 
       
  Guest talk:    

23.03.

 
Leafwise symplectic structures on Lawson's foliation on S^5  Yoshihiko Mitsumatsu
(Chuo Uhiversity, Japan)
Abstract
 
       

20.04.

1. Review of group homology from the algebraic perspective  Carsten Schultz   

27.04.

2. Classifying spaces and review of group homology from the topological perspective Jan-David Salchow   

4.05.

3. Explicit computations Pavle Blagojevic   

11.05.

4. Homological algebra of functors - Bredon homology from the algebraic perspective Dimitrios Patronas   

18.05.

5. Classifying spaces of categories and homotopy colimits Sebastian Meinert   
       
  Guest talk:    

24.05.
14h ct

Coassembly
 
Bruce Williams
(University of Notre Dame) 
 
       
  Guest talk:    

25.05.
 

Dualität und axiomatische Homologie
 
Tammo tom Dieck
(Universität Göttingen) 
Abstract

 
       

1.06.

6. Examples of classifying spaces  Mark Ullmann  

8.06.

7. Equivariant homology theories and examples  Benjamin Matschke   
       
  Guest talk:    

22.06.

Quillen-Lichtenbaum Phenomena in Stable Representation Theory  Dan Ramras
(New Mexico State Univ.) 
Abstract

       
  Guest talk:    

29.06.

Property (T)
 
Andrzej Zuk
(Université Paris 7) 
 
       

6.07.

8. The Chern character I  Fabian Lenhardt   

13.07.

9. The Chern character II  Elmar Vogt   

Literatur

Basic sources for this seminar include:

 

[1] G. E. Bredon, Equivariant Cohomology Theories, LNM 34, Springer
[2] K.S. Brown, Cohomology of groups, Springer
[3] J. Davis, W. Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15, No.3, p 201-252
[4] T. tom Dieck, Transformation groups, de Gruyter
[5] W. Lück, Chern characters for proper equivariant homology theories and applications to K- and L-theory, Crelle 543, p 193-234
[6] W. Lück, Transformation groups and algebraic K-theory, LNM 1408