A homotopy theoretic realization of string topology. (English) Zbl 1025.55005

Cohen, Ralph L.; Jones, John D. S.

Math. Ann. 324, No. 4, 773-798 (2002).

For an oriented, closed smooth manifold \(M^d\), Chas and Sullivan used intersection theory on singular chains to equip the homology of the free loop space \(H_*(LM)\) with a degree \(-d\) product, called the loop product. This paper is an attempt to show how this product arises from an \(A_\infty\)-ring structure on the Thom spectrum constructed from the pull-back of the stable normal bundle along the evaluation map \(LM\to M\). Thus the loop homology \(H_*(LM)\) is isomorphic to the Hochschild/Gerstenhaber homology. The authors (and some others) have realized that the written proof is not complete. Indeed they have assumed that Atiyah duality carried all necessary structure. There is something to prove here. Moreover, in the statement of Theorem 1 one should have that \(M\) is simply connected. If it is not simply connected there may be convergence problems in the cosimplicial spectrum. As written in a recent report: “Among the experts it is not doubted that the loop homology is isomorphic to Hochschild/Gerstenhaber homology in the simply connected case. However, the isomorphism itself may depend on the manifold structure and may not be homotopy invariant. Existence and uniqueness of this isomorphism remain quite interesting questions to resolve”.

Reviewer: J.C.Thomas (Angers)

MSC:

55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P48	Loop space machines and operads in algebraic topology
57R19	Algebraic topology on manifolds and differential topology
55N35	Other homology theories in algebraic topology
55U99	Applied homological algebra and category theory in algebraic topology

Keywords:

string topology; Hochschild homology; loop product; spectrum; loop homology