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De Rham model for string topology. (English) Zbl 1066.55008
Let \(M\) be a simply connected closed oriented \(n\)-dimensional manifold and \(LM\) the associated free loop space. M. Chas and D. Sullivan have defined a commutative algebra structure on the shifted ordinary homology \(\mathbb{H}_*(LM)=H_{*+n}(LM)\), that is a generalization of the intersection product on \(H_*(M)\). Using ring spectra, R. Cohen and J. D. S. Jones have shown that the algebra \(\mathbb{H}_*(LM)\) is isomorphic to the Hochschild algebra \(HH^*(C^*(M), C^*(M))\).
In this paper the author gives a proof of this isomorphism over the real numbers. His proof is based on the theory of Chen iterated integrals. The above isomorphism appears as the corresponding holonomy map. The author’s approach gives new algorithms for computing the homology of free loop spaces dans Hochschild cohomologies.

MSC:
55P62 Rational homotopy theory
55N35 Other homology theories in algebraic topology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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