  We show that the homology over a field of the space 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 on the n-sphere with labels in X and of
its completion, that depends only on the homology of X. In many but not all
cases the homology of the configuration space coincides with the homology of
the mapping space. In particular we obtain the homology of the unordered
configuration spaces on a sphere.
