Geometry & Topology

Rational maps and string topology

Sadok Kallel and Paolo Salvatore

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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–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.

Article information

Geom. Topol., Volume 10, Number 3 (2006), 1579-1606.

Received: 23 September 2003
Revised: 28 August 2006
Accepted: 11 September 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58D15: Manifolds of mappings [See also 46T10, 54C35]
Secondary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 26C15: Rational functions [See also 14Pxx]

mapping space rational map string product


Kallel, Sadok; Salvatore, Paolo. Rational maps and string topology. Geom. Topol. 10 (2006), no. 3, 1579--1606. doi:10.2140/gt.2006.10.1579.

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