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?

$\begingroup$ I’m not very familiar with Hochschild homology, but maybe this is reminiscent of work of Rivera and Zeinalian, see e.g. arxiv.org/abs/1612.04801 $\endgroup$– RyanApr 12, 2021 at 18:08

$\begingroup$ @Ryan I found that too, but that seems to only deal with the based loop space, which is a bit different from the free loop space. $\endgroup$– user2520938Apr 12, 2021 at 21:09
2 Answers
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 coHochschild 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 quasiisomorphism) 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 quasiisomorphisms of coalgebras.

$\begingroup$ Thanks for this answer! Can you explain a bit on why cochains lose information compared to chains? $\endgroup$ Apr 12, 2021 at 21:38

1$\begingroup$ Essentially the quasiisomorphism type of the singular chains coalgebra (or of the singular cochains algebra) only "remembers" the Malcev completion of the fundamental group. However, there is a stronger notion of weak equivalence between coalgebras that preserves the fundamental group in complete generality, namely, maps of coalgebras that become quasiisomorphisms after applying the cobar functor. I could explain more through email, if needed! $\endgroup$ Apr 12, 2021 at 21:52
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 $$\check H^k(X,A) \cong \check H^{k1}_{lf}(LX,A), $$ where the index "lf" denotes a version of Cech cohomology adapted to loop spaces by including fusion.

2$\begingroup$ Hi, this is interesting. Can you make an explicit claim relating this statement and Hochschild homology of the algebra of differential forms or cochains on M? It seems one should be able to make such a statement based on a quick look at that paper, since the path space $IM$ is being used as a resolution of $M$ in some sense. $\endgroup$ Apr 13, 2021 at 17:49

$\begingroup$ @Manuel: maybe if one puts $A=\mathbb{R}$ and supposes that $X$ is simplyconnected. There is a forgetful map $$\check H^{k1}_{lf}(LM,\mathbb{R})\to \check H^{k1}(LM,\mathbb{R}),$$so, using Jones' theorem, this gives a map $$\check H^k(M,\mathbb{R}) \to HH_k(\Omega_X).$$ But I haven't checked what this map is, actually! $\endgroup$ Apr 14, 2021 at 6:59