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[[!include higher algebra - contents]]
Let $X$ be a simply connected topological space.
The ordinary cohomology $H^\bullet$ of its free loop space is the Hochschild homology $HH_\bullet$ of its singular chains $C^\bullet(X)$:
Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:
(Loday 11)
If the coefficients are rational, and $X$ is of finite type then this may be computed by the Sullivan model for free loop spaces, see there the section on Relation to Hochschild homology.
In the special case that the topological space $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that
This is known as Jones' theorem (Jones 87)
An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.
John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf)
Jean-Louis Loday, Free loop space and homology (arXiv:1110.0405)
Created on February 23, 2017 at 12:37:12. See the history of this page for a list of all contributions to it.