Nikita Markarian’s mathblog

Homology of (S^n, X)

Posted in Mathematics by nikitamarkarian on October 30, 2009

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

Anyway, given a fE_n-space A=\Omega^n X, Voronov produces a model for (S^n,X) as follows. Let fDS^n(k) be the space of k framed discs on n-sphere. Then the model is \coprod fDS^n(i)\times A^i factoriszed by relations presented by the picture

where C\in fE_n. This model obviously allows to get homology of (S^n, X) from the one of \Omega^n X. For n=1 we have usual Hochschild homology. It’s amazing that for n>1 one needs action of framed little n-spheres operad, action of just little spheres is not enough.

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