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I- Is the following statement still a conjecture see this article ?

Conjecture (?) Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild cohomology of cochain complexes associated to $M$ is isomorphic as a Gerstenhaber algebra to $H_{\ast+d}(LM)$ the $d$-shifted homology of the free loop space on $M$.

II- What are the recent advances in solving the conjecture ?

III- Is there a short proof of the following fact: $HH^{\ast}(C^{\ast}(M))$ is isomorphic to $H_{\ast+d}(LM)$ as a graded vector space ?

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    $\begingroup$ I believe this is a known result, but I don't know if there is a direct reference (my knowledge of the literature is several years old). Namely in Malm's thesis, ericmalm.net/ac/research/string-topology/thesis.pdf, he proved the desired equivalence for $HH^*(C_*(\Omega(M))$ (actually a more refined statement at the BV level). Then for simply connected $M$, there is a general Koszul duality theorem due to Keller which gives the result you are asking about. See page 3 of mth.kcl.ac.uk/~lekili/koszul.pdf for the explicit statement or ... $\endgroup$
    – Daniel Pomerleano
    Commented Sep 27, 2015 at 11:12
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    $\begingroup$ mathoverflow.net/questions/70151/…. I was asking about Q coefficients because that was of interest to me, but the result of Keller webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf applies over any field. It should be said that the situation over Q is more direct because one can convert the topology into algebra using rational homotopy theory. $\endgroup$
    – Daniel Pomerleano
    Commented Sep 27, 2015 at 11:14

2 Answers 2

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I. To my knowledge over any field this conjecture is still open.

II. Over a field of characteristic zero, this conjecture is true, and we have more, we have a BV-isomorphism:

Félix, Yves; Thomas, Jean-Claude "Rational BV-algebra in string topology." Bull. Soc. Math. France 136 (2008), no. 2, 311–327.

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For III:

First show that the Hochschild homology of the DGA of cochains on $M$ is quasi-isomorphic to cochains on $LM$. To do this you can use Jones argument using a cosimplicial model for $LM$ or over the rationals one can construct an explicit map through Chen iterated integrals.

Now the "Poincaré duality step": relate Hochschild homology to Hochschild cohomology. To do this one may construct a lift of the iterated integrals map mentioned above by choosing a Thom form supported near the diagonal in $M \times M$. The details of this approach can be found on Merkulov's De Rham string topology. There are other ways of achieving this last Poincare duality step by using finite dimensional models for the cochains on $M$: either strict Frobenius algebras or cyclic $A_{\infty}$ algebras.

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    $\begingroup$ could you perhaps expand on your first paragraph? $\endgroup$
    – pro
    Commented Sep 27, 2015 at 14:04

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