Rational maps and string topology

Kallel, Sadok; Salvatore, Paolo

doi:10.2140/gt.2006.10.1579

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Rational maps and string topology

Sadok Kallel and Paolo Salvatore

Geometry & Topology 10 (2006) 1579–1606

DOI: 10.2140/gt.2006.10.1579

arXiv: math.AT/0309137

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.

Keywords

mapping space, rational map, string product

Mathematical Subject Classification 2000

Primary: 58D15

Secondary: 55R20, 26C15

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Publication

Received: 23 September 2003

Revised: 28 August 2006

Accepted: 11 September 2006

Published: 28 October 2006

Proposed: Ralph Cohen

Seconded: Haynes Miller, Bill Dwyer

Authors

Sadok Kallel
Laboratoire Painlevé Université de Lille I Villeneuve d’Ascq France

Paolo Salvatore
Dipartimento di matematica Università di Roma “Tor Vergata” Roma Italy

Volume 10, issue 3 (2006)