Non-contractible configuration spaces

Let $F(M,k)$ be the configuration space of ordered $k-$tuples of distinct points in the manifold M. Using the Fadell-Neuwirth fibration, we prove that the configuration spaces $F(M,k)$ are never contractible, for k≥ 2. As applications of our results, we will calculate the LS category and topological complexity for its loop space and suspension.