Winter Term 2011/12

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

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

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

Configuration spaces

The configuration space $F_k(M)$ is the space of configurations of $k$ distinguishable particles in the manifold $M$. It is topologized  as a subspace of the $k$-fold product $M^{\times k}$. Dividing out the action of the symmetric group leads to the space $C_k(M)$ of $k$ indistinguishable particles. There are also variants where the particles are charged or have labels in an auxiliary space.

Forgetting some of the points leads to fibrations between these spaces for varying $k$. The fundamental group  $F_k( \IR^2 )$ is the braid group and it turns out that $F_k( \IR^2)$ is in fact a classifying space for the braid group.

Another point of view interprets $F_k(M)$ as the complement of the "thick diagonal" in the product $M^{\times k}$, which is an arrangement of subspaces homeomorphic to $M^{\times (k-1)}$, and whose combinatorics is given by the partition lattice. This leads to computations of cohomology and (stable) homotopy types of configuration spaces.

Combining the spaces $C_k(\IR^n)$ for varying $k$ leads to the space $C(\IR^n)$ of finite subsets of $\IR^n$. Up to a group completion this space is homotopy equivalent to $\Omega^n S^n$. Configuration space models are also used to obtain stable splitting results.

Schedule

Date	Title	Speaker
19.10.	Program discussion
2.11.	1. Basic fibrations	Dimitrios Patronas
9.11.	2. The two dimensional case and the braid group	Salvador Sierra Murillo
16.11.	Nonpositively curved cube complexes	Sebastian Meinert
	Guest talk:
23.11. 15:45h	Configuration spaces of graphs	Tadeusz Januszkiewicz
Tuesday	Guest talk:
29.11. 14h c.t.	A Controlled Vietoris-like theorem for simplicial complexes	Spiros Adams--Florou
30.11.	3. Subspace arrangements I	Pavle Blagojevic
7.12.	4. Subspace arrangements II	Pavle Blagojevic
14.12.	5. Compactifications I	Holger Reich
04.01.	6. Compactifications II	Holger Reich
	Guest talk:
11.01.	On Thompson's group T and applications to algebraic K-theory	Marco Varisco
18.01.	7. Compactifications III	Moritz Schmitt
1.02.	8. Epstein-Glaser renormalization I	Erik Panzer
8.02.	9. Epstein-Glaser renormalization II	Marko Berghoff

Literature

[B87]	Bödigheimer, C.-F.: Stable splittings of mapping spaces; Lecture Notes in Math. 1286
[FH01]	Fadell, Edward R. and Husseini, Sufian Y.: Geometry and topology of configuration spaces; Springer Mongraphs in Mathematics (2001)
[FM94]	Fulton, William and MacPherson, Robert: A compactification of configuration spaces; Ann. of Math. (2) 139 (1994)
[McD75]	McDuff, Dusa: Configuration spaces of positive and negative particles; Topology 14 (1975)
[OT92]	Orlik, Peter and Terao, Hiroak: Arrangements of hyperplanes; Grundlehren der Mathematischen Wissenschaften 300; Springer
[S73]	Segal, Graeme: Configuration-spaces and iterated loop-spaces; Invent. Math. 21 (1973)
[ZZ93]	Ziegler, Günter M. and Zivaljevic, Rade T.:Homotopy types of subspace arrangements via diagrams of spaces; Math. Ann. 295 (1993)