The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type.

There is an equivalence of Gerstenhaber algebras

$HH^*(C_*(\Omega X,\mathbb{Q}), C_*(\Omega X,\mathbb{Q}) \cong HH^*(C^*(X,\mathbb{Q}),C^*(X,\mathbb{Q})$

On the left hand side we have Pontryagin product on the based loop space and on the right hand side rational cochains. $HH^*$ denotes Hochschild cohomology.

I have never seen anyone speak to the following enhanced statement, which makes me wonder if there is a counterexample or if I am simply missing some literature.

$HCH^*(C_*(\Omega X), C_*(\Omega X) \cong HCH^*(C^*(X),C^*(X))$

The question is: Is this statement true, false or unknown?

Here we are looking at Hochschild cochains in the homotopy category of $B(\infty)$ algebras. For the background police, a $B(\infty)$ algebra is a type of dg-Gerstenhaber structure, that naturally gives rise to a Gerstenhaber structure by passing to homology. For more info, see the paper of Keller mentioned below.

It is possible to prove this theorem when $C^*(X)$ is equivalent to a graded simply connected Koszul algebra( i.e. X is both formal and coformal). I believe this is due to Keller in a paper called the "Derived Invariance of Higher Structures of the Hochschild complex".


Looking closer at Keller's paper, the result seems to be in there. Namely, in his main theorem in section 3.3, he proves that fully faithful dg-functors $per(A) \to D(B)$ induced by an $A\otimes B^{op}$ module X induce $B(\infty)$ morphisms $\phi_X: HCH^*(B,B) \to HCH^*(A,A)$. Additionally, in the same theorem, he proves that if the map $per(B^{op}) \to D(A^{op})$ induced by X is also fully faithful, then $\phi_X$ is invertible.

These criterion all apply to M a simply connected space, $X= \mathbb{Q}$ the trivial local system, $A=C_*(\Omega(M))$, and $B= C^*(M)$.


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