## Spherical homology, the geometric case

Let be a topological space and be its double loop space. As we already discussed, is acted by the operad of framed little discs. It follows that the complex of chains is acted by the chains of framed little discs, that is it is a f2-algebra. Here we consider spherical homology of this f2-algebra and show that two definitions (this and that) give the same result. It follows, by the way, that they coincide being applied to free algebras.

So, consider the spherical homology of f2-algebra . As it is follows from this post, the first definition of the spherical homology gives the chain complex of the space of maps from 2-sphere to : . Let us see that the second definition gives the same.

Firstly, by the Goodwillie’s result (and of others) , where means the free loop space. One may see that and the product on the Hochschild complex is induced by the Pontryagin product on . Secondly, left and right module structures provided by the statement is induced by maps presented by the picture:

Finally, consider the fiber bundle , where the last arrow send a sphere to its equator. If one presents as the classifying space and takes the standard simplicial realisation of it, then the homological spectral sequence of the fiber bundle appears to be the standard complex calculating . This reasoning resembles the one from the mentioned Goodwillie’s paper.

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