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I gave Jones’ theorem (long requested at Hochschild homology) a quick statement and references. Copied this also to the entries free loop space, cyclic loop space and cyclic homology and Sullivan models of free loop spaces:
Let be a simply connected topological space.
The ordinary cohomology of its free loop space is the Hochschild homology of its singular chains :
Moreover the -equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space is the cyclic homology of the singular chains:
(Loday 11)
If the coefficients are rational, and 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 carries the structure of a smooth manifold, then the singular cochains on 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.
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