分野別セミナー

The ordered configurations of k points in a manifold M, F(M,k), is a well studied invariant of the manifold. After giving a short introduction to the topic and some classical results, we move on to two questions:

1) Do homotopy equivalent manifolds have homotopy equivalent configuration spaces?
2) Can you find a formula for the rational homotopy type of a configuration space given the rational homotopy type of the manifold?

Longani-Salvatore showed that two homotopy equivalent closed manifolds can have non-homotopy equivalent configuration spaces. On the other hand whem M is a complex projective manifold work of Fulton-Macpherson and Kriz gives a formula for the rational homotopy type of F(M,K) using only the cohomology algebra of M. Under some connectivity hypotheses we give a similar formula for the rational homotopy type of F(M,k) for a general closed manifold M.