Geometry & Topology

Rational maps and string topology

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.

Article information

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

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

https://projecteuclid.org/euclid.gt/1513799775

Digital Object Identifier
doi:10.2140/gt.2006.10.1579

Mathematical Reviews number (MathSciNet)
MR2284046

Zentralblatt MATH identifier
1204.58006

Citation

Kallel, Sadok; Salvatore, Paolo. Rational maps and string topology. Geom. Topol. 10 (2006), no. 3, 1579--1606. doi:10.2140/gt.2006.10.1579. https://projecteuclid.org/euclid.gt/1513799775

References

• C P Boyer, B M Mann, J C Hurtubise, R J Milgram, The topology of the space of rational maps into generalized flag manifolds, Acta Math. 173 (1994) 61–101
• M Chas, D Sullivan, String topology
• D Chataur, A bordism approach to string topology, Int. Math. Res. Not. (2005) 2829–2875
• F R Cohen, R L Cohen, B M Mann, R J Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991) 163–221
• R L Cohen, Multiplicative properties of Atiyah duality, Homology Homotopy Appl. 6 (2004) 269–281
• R L Cohen, V Godin, A polarized view of string topology, from: “Topology, geometry and quantum field theory”, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cambridge (2004) 127–154
• R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773–798
• R L Cohen, J D S Jones, J Yan, The loop homology algebra of spheres and projective spaces, from: “Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001)”, Progr. Math. 215, Birkhäuser, Basel (2004) 77–92
• Y Félix, L Menichi, J-C Thomas, Gerstenhaber duality in Hochschild cohomology, J. Pure Appl. Algebra 199 (2005) 43–59
• J Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989) 247–286
• K Gruher, P Salvatore, String topology in a general fiberwise setting and applications
• J W Havlicek, The cohomology of holomorphic self-maps of the Riemann sphere, Math. Z. 218 (1995) 179–190
• P Hu, Higher string topology on general spaces
• J C Hurtubise, Configurations de particules et espaces de modules, Canad. Math. Bull. 38 (1995) 66–79
• S Kallel, Configuration spaces and the topology of curves in projective space, from: “Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999)”, Contemp. Math. 279, Amer. Math. Soc., Providence, RI (2001) 151–175
• S Kallel, P Salvatore, work in progress
• J R Klein, Fiber products, Poincaré duality and $A\sb \infty$-ring spectra, Proc. Amer. Math. Soc. 134 (2006) 1825–1833
• A W Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser, Boston (1996)
• J Leborgne, String Spectral Sequences
• G Segal, The topology of spaces of rational functions, Acta Math. 143 (1979) 39–72
• R M Switzer, Algebraic topology–-homotopy and homology, Classics in Mathematics, Springer, Berlin (2002) Reprint of the 1975 original