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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asked Apr 12, 2021 at 12:45
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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 $\endgroup$– RyanApr 12, 2021 at 18:08
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$\begingroup$ @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
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$\begingroup$ One way of proving the Jones statement is via the Eilenberg-Moore "spectral sequence" - you get more general than simply-connected then. Namely, for the EMsseq, you only need the two-sided action of $\pi_1(X)$ on $H^*(\Omega X)$ to be nilpotent (for every $*$ - here I'm assuming you already know that the cohomology groups are finitely generated). $\endgroup$– Maxime RamziApr 9, 2023 at 20:43
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3 Answers
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Corollary 9.5 in the paper Rivera and Zeinalian - Cubical rigidification, the cobar construction, and the based loop space 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: Rivera and Saneblidze - A combinatorial model for the free loop fibration.
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 co-Hochschild homology is not invariant under quasi-isomorphisms of coalgebras.
answered Apr 12, 2021 at 21:28
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$\begingroup$ Thanks for this answer! Can you explain a bit on why cochains lose information compared to chains? $\endgroup$ Apr 12, 2021 at 21:38
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1$\begingroup$ 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! $\endgroup$ Apr 12, 2021 at 21:52
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The more natural statement is already in Goodwillie's Cyclic homology, derivations, and the free loopspace, namely that if $X$ is connected, then choose a base point $x \in X$ and consider $C_{*}(\Omega_{x}X)$ as a dg algebra with respect to the Pontryagin product. Then
$$HH_{*}(C_{*}(\Omega_{x}X)) \simeq H_{*}(LX)$$
and the identification is circle-equivariant.
The Jones statement quoted by the OP then follows from Koszul duality between $C_{*}(\Omega_{x}X)$ and $C^{*}(X)$ in the simply connected case.
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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 Loop-fusion cohomology and transgression an isomorphism $$\smash{\check H}^k(X,A) \cong \smash{\check H}^{k-1}_\text{lf}(LX,A), $$ where the index "lf" denotes a version of Čech cohomology adapted to loop spaces by including fusion.
answered Apr 13, 2021 at 9:29
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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
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$\begingroup$ @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! $\endgroup$ Apr 14, 2021 at 6:59
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$\begingroup$ TeX note: the superscript $k$ in $\check H^k$
\check H^kis too high. You can bring it back to earth by\smashing: $\smash{\check H}^k$\smash{\check H}^k. (Side by side: $H^k\check H^k\smash{\check H}^k$.) I have edited accordingly. $\endgroup$– LSpiceApr 9, 2023 at 18:22