TITLE: Jones' theorem for non-simply-connected spaces?
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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.