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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– 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.
Kallel, Sadok 1 ; Salvatore, Paolo 2
@article{GT_2006_10_3_a9, author = {Kallel, Sadok and Salvatore, Paolo}, title = {Rational maps and string topology}, journal = {Geometry & topology}, pages = {1579--1606}, publisher = {mathdoc}, volume = {10}, number = {3}, year = {2006}, doi = {10.2140/gt.2006.10.1579}, url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2006.10.1579/} }
Kallel, Sadok; Salvatore, Paolo. Rational maps and string topology. Geometry & topology, Tome 10 (2006) no. 3, pp. 1579-1606. doi : 10.2140/gt.2006.10.1579. http://geodesic.mathdoc.fr/articles/10.2140/gt.2006.10.1579/
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