Configuration spaces on the sphere and higher loop spaces

We show that the homology over a field of the space Lambda(n)Sigma(n)X of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space C(S-n,X) on the n-sphere with labels in X and of its completion, that depend only on the homology of X. In many but not all cases the homology of C(S-n,X) coincides with the homology of Lambda(n)Sigma(n)X. In particular we obtain the homology of the unordered configuration spaces on a sphere.