Rational string topology. (English) Zbl 1200.55015

Félix, Yves; Thomas, Jean-Claude; Vigué-Poirrier, Micheline

J. Eur. Math. Soc. (JEMS) 9, No. 1, 123-156 (2007).

The basic construction of string topology is to provide a (graded) commutative product on the homology of the free loop space \(LM\) of a closed oriented manifold \(M\), and a graded Lie bracket on its \(S^1\)-equivariant homology. Understanding the underlying structures at the chain level of this “loop product” and “string bracket” is a much more subtle story not fully uncovered. The purpose of this paper is to clarify the situation for simply connected \(M\) and when working with rational field coefficients. It is possible in this case to describe explicitly in terms of Sullivan models (for spaces and maps) the duals in cohomology of the loop product and the string bracket (Theorems A and B of the paper). This implies for example that the loop product on \(H_*(LM)\) is invariant under orientation preserving maps which are quasi-isomorphisms.
A Sullivan model for the composition of free loops \(LM\times_MLM\rightarrow LM\) is also given. Using “cap-homomorphisms”, the authors manage to describe the dual of the loop product in terms of chains of a differential graded Lie algebra. In so doing, they prove that there is a natural isomorphism of graded algebras
\[ J: {\mathbb H}_*(LM)\longrightarrow HH^*(C^*(M), C^*(M)) \]
where the term on the left is \(H_{*+\dim M}(M)\) and the term on the right is Hochschild cohomology with its Gerstenhaber product,; \(C^*(M)\) being the algebra of singular cochains. A similar result has been obtained by S. A. Merkulov [Int. Math. Res. Not. 2004, No. 55, 2955–2981 (2004; Zbl 1066.55008)] when working with real coefficients. In its final section, the paper shows that the isomorphism \(J\) is compatible with the Adams-type algebra isomorphism \(H_*(\Omega M)\rightarrow HH^*(C^*(M),{\mathbb Q})\) via a homomorphism \({\mathbb H}_*(LM)\rightarrow H_*(\Omega M)\) which corresponds to the Gysin map of the embedding of the based loop space \(\Omega M\hookrightarrow LM\), and which then takes loop product to Pontryagin product.

Reviewer: Sadok Kallel (Villeneuve d’Asq)

MSC:

55P50	String topology
55P35	Loop spaces
55N45	Products and intersections in homology and cohomology
17A65	Radical theory (nonassociative rings and algebras)

Keywords:

rational homotopy; Hochschild cohomology; free loop space; string homology

Citations:

Zbl 1066.55008

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