Rational maps and string topology

Abstract

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.