Rational maps and string topology

Kallel, Sadok¹ ; Salvatore, Paolo²

¹ Laboratoire Painlevé, Université de Lille I, Villeneuve d’Ascq, France
² Dipartimento di matematica, Università di Roma “Tor Vergata”, Roma, Italy

Geometry & topology, Tome 10 (2006) no. 3, pp. 1579-1606.

Résumé

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.

Détail

DOI : 10.2140/gt.2006.10.1579

Keywords: mapping space, rational map, string product

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