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 Ghomology theory is the analogue of a homology theory in the equivariant context. It is a functor, which instead of spaces digests spaces with a Gaction.
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
Date  Title  Speaker  

Guest talk:  
16.03. 
On the EilenbergMoore 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  JanDavid 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. 
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. 
QuillenLichtenbaum 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 LTheory, KTheory 15, No.3, p 201252
[4] T. tom Dieck, Transformation groups, de Gruyter
[5] W. Lück, Chern characters for proper equivariant homology theories and applications to K and Ltheory, Crelle 543, p 193234
[6] W. Lück, Transformation groups and algebraic Ktheory, LNM 1408