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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.

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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.

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  • $\begingroup$ Thank you very much, I did not know this theorem. But anyway, I interested mostly in the case when m=2, n=4. $\endgroup$ – nikitamarkarian Sep 14 '13 at 13:40
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    $\begingroup$ That particular embedding is actually of no serious topological interest, since it is not compatible with suspension. If you look at the embedding of loop n X in loop m of the (m-n)th suspension of X you will see what I mean; see Section 3 of math.uchicago.edu/~may/PAPERS/Final1.pdf. Discs are too round. Little cubes are square enough. The Steiner operads are both round and square. The point is that for very many real applications, not all E_n operads are created equal. $\endgroup$ – Peter May Sep 14 '13 at 14:13
  • $\begingroup$ @PeterMay Dear Peter, thank you for your comment. My question is not only about the embedding, but about interaction between this embedding and the formality theorem. And the formality theorem is quite irrelevant to topology, isn't it? $\endgroup$ – nikitamarkarian Sep 19 '13 at 19:09

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