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Rational maps and string topology

Sadok Kallel and Paolo Salvatore

Geometry & Topology 10 (2006) 1579–1606

arXiv: math.AT/0309137


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.

mapping space, rational map, string product
Mathematical Subject Classification 2000
Primary: 58D15
Secondary: 55R20, 26C15
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Received: 23 September 2003
Revised: 28 August 2006
Accepted: 11 September 2006
Published: 28 October 2006
Proposed: Ralph Cohen
Seconded: Haynes Miller, Bill Dwyer
Sadok Kallel
Laboratoire Painlevé
Université de Lille I
Villeneuve d’Ascq
Paolo Salvatore
Dipartimento di matematica
Università di Roma “Tor Vergata”