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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?

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  • $\begingroup$ 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 $\endgroup$ – Ryan Apr 12 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. $\endgroup$ – user2520938 Apr 12 at 21:09
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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.

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  • $\begingroup$ Thanks for this answer! Can you explain a bit on why cochains lose information compared to chains? $\endgroup$ – user2520938 Apr 12 at 21:38
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    $\begingroup$ Essentially the quasi-isomorphism 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 quasi-isomorphisms after applying the cobar functor. I could explain more through email, if needed! $\endgroup$ – Manuel Rivera Apr 12 at 21:52
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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.

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    $\begingroup$ 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. $\endgroup$ – Manuel Rivera Apr 13 at 17:49
  • $\begingroup$ @Manuel: maybe if one puts $A=\mathbb{R}$ and supposes that $X$ is simply-connected. There is a forgetful map $$\check H^{k-1}_{lf}(LM,\mathbb{R})\to \check H^{k-1}(LM,\mathbb{R}),$$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! $\endgroup$ – Konrad Waldorf Apr 14 at 6:59

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