**I-** Is the following statement still a conjecture [see this article](http://arxiv.org/pdf/0711.1946.pdf) ?


**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 ?



Gerstenhaber conjecture for free loop space
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.


