  We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher
loop homology of a manifold in order to compute the homology of the spaces of
continuous and holomorphic maps of the Riemann sphere into a complex projective
space. This product makes sense on the homology of maps from a co-H space to a
manifold, and comes from a ring spectrum. We also build a holomorphic version
of the product for maps of the Riemann sphere into homogeneous spaces. In the
continuous case we define a related module structure on the homology of maps
from a mapping cone into a manifold, and then describe a spectral sequence that
can compute it. As a consequence we deduce a periodicity and dichotomy theorem
when the source is a compact Riemann surface and the target is a complex
projective space.
