Forschungsseminar Geometrie und Topologie

Winter Term 2011/12

 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:

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

Literature