# Embedding e_n -> e_m

Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to \mathbb{R}^m$. By the formality theorem $e_n$ is quasiisomorphic to its cohomology $H^*(e_n)$, which is an operad resembling the Poisson or the Gerstenhaber operad, that is it is generated by the commutative multiplication and the bracket.

The question is: can we write down an explicit $\infty$-morphism $H^*(e_n)\to H^*(e_m)$ that corresponds to the embedding under the quasiisomorphism mentioned above.

Do you know Theorem 1.4 ("relative formality") of Lambrechts-Volic paper on formality of the small disks? It says that when $m \geq 2n+1$ the $\infty$-morphism is in fact a morphism and it is the obvious one (forget the bracket to get a commutative algebra, then give it bracket zero to get a Gerstenhaber/Poisson algebra with bracket in different degree).
Update: The relative formality theorem mentioned above has since been improved by Tourtchine and Willwacher http://arxiv.org/abs/1409.0163 : the same conclusion holds for any $m \geq n+2$. So this covers in particular the case you care(d) about. Moreover, their result is sharp, in the sense that relative formality fails when $m=n+1$.
To my knowledge the only nontrivial case where all the higher components of this $\infty$-morphism have been worked out is for $e_1$ mapping into the framed little disks, which is due to Alm http://arxiv.org/abs/1501.02916. The "correction terms" end up being expressed in terms of period integrals on the moduli spaces $M_{0,n}$, and thus (using the work of Brown) in terms of multiple zeta values.