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# Forschungsseminar Geometrie und Topologie

## Winter Term 2011/12

• 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:

# 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

DateTitleSpeaker

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

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)