Jones' theorem for non-simply-connected spaces? Let $X$ be a smooth manifold. Jones' theorem says that $H^\bullet(\mathcal{L}X)\cong HH_\bullet(\Omega^\bullet_X)$, where $\mathcal{L}X$ is the free loop space of $X$. Is there a modification of this theorem that works for $X$ which is not simply connected? Maybe when $\pi_1(X)$ is abelian?
 A: Corollary 9.5 in the paper https://arxiv.org/abs/1612.04801 tells us the following:
For any pointed path connected space $(X,b)$ the co-Hochschild homology of (a pointed version of) the differential graded coalgebra of singular chains in $(X,b)$ is naturally isomorphic to the homology of the free loop space of $X$.
For another perspective on this isomorphism you may take a look at the following paper: https://arxiv.org/abs/1712.02644
If we use cochains or differential forms (considered as dg algebras under quasi-isomorphism) we lose information regarding the fundamental group. The dual of the above result does not hold under stronger conditions on the fundamental group. Also note that I am using singular chains on the underlying space as a coalgebra model and that coHochschild homology is not invariant under quasi-isomorphisms of coalgebras.
A: One possibility to understand the difference between the cohomologies of $X$ and $LX$ is to add "fusion"-conditions on the loop space side. This works for general manifolds $X$, and the key word here is "transgression".
For example, Kottke and Melrose have proved in https://arxiv.org/pdf/1309.7674.pdf an isomorphism
$$\check H^k(X,A) \cong \check H^{k-1}_{lf}(LX,A), $$
where the index "lf" denotes a version of Cech cohomology adapted to loop spaces by including fusion.
