Nikita Markarian’s mathblog

Spherical homology, the geometric case

Posted in Mathematics by nikitamarkarian on November 15, 2009

Let X be a topological space and \Omega^2 X be its double loop space. As we already discussed, \Omega^2 X is acted by the operad of framed little discs. It follows that the complex of chains C_*(\Omega^2 X) 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 C_*(\Omega^2 X). 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 X: HS_*(C_*(\Omega^2 X))=C_*(S^2, X). Let us see that the second definition gives the same.

Firstly, by the Goodwillie’s result (and of others) HH(C_*(\Omega^2 X))=C_*(L\Omega X), where L means the free loop space. One may see that L\Omega X=\Omega LX and the product on the Hochschild complex is induced by the Pontryagin product on \Omega LX. Secondly, left and right module structures provided by the statement is induced by maps \Omega LX \times \Omega^2 X \to \Omega^2 X presented by the picture:


Finally, consider the fiber bundle \Omega^2 X\times \Omega^2 X \to (S^2, X) \to LX, where the last arrow send a sphere to its equator. If one presents LX as the classifying space B(\Omega LX) and takes the standard simplicial realisation of it, then the homological spectral sequence of the fiber bundle appears to be the standard complex calculating \mathop{Tor}_{C_*(\Omega LX)}^*(C_*(\Omega^2 X),C_*(\Omega^2 X)). This reasoning resembles the one from the mentioned Goodwillie’s paper.


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