## Homology of (S^n, X)

Goodwillie, Burghelea-Fiedorowicz and others proved that homology of free loop space of a topological space is isomorphic to Hochschild homology of chains considered as an algerbra under the Pontryagin product. I already discussed it here, because this fact “explains” multiplication on the Hochschild homology of an algebra over the operad of chains of little discs.

Below I cite a generalization of this statement that was announced by A. Voronov in section I.5.4 of “String topology and cyclic homology.

**UPD:** As I learned from V. Turchin, in P. Salvatore, “Configuration spaces with summable labels” this statement is proved for mapping space from any manifolds, not only spheres.

Let be the operad of little n-spheres and be the operad of *framed* little n-spheres. For a topological space on the space of pointed maps from n-sphere to , that is , the natural action of is defined. (**Remark:** By the May recognition theorem, spaces acted by are exactly n-fold loops. It means that we have a functor from category of -spaces to -spaces, namely -space goes to the n-fold loop space, and the latter is acted by . In P. Salvatore, N. Wahl “Framed discs operads and the equivariant recognition principle” it is proved that spaces are n-fold loop spaces of a space with -action. The mentioned functor is the result of this recognition map being applied to with trivial -action.)

Anyway, given a -space , Voronov produces a model for as follows. Let be the space of framed discs on n-sphere. Then the model is factoriszed by relations presented by the picture

where . This model obviously allows to get homology of from the one of . For we have usual Hochschild homology. It’s amazing that for one needs action of *framed* little n-spheres operad, action of just little spheres is not enough.

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