# 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.