In [Another proof of M. Kontsevich formality theorem], Tamarkin provides a proof of the formality of the differential graded Lie algebra controlling the deformation of a polynomial associative algebra.

Tamarkin's proof makes a crucial use of the formality of the topological operad of little 2-disks.

Question: Is there any known use of the formality of the little $n$-disks operad for $n> 2$ in the context of deformation theory?

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    $\begingroup$ As shown by Tamarkin, one can use Koszul duality and $E_2$ formality to construct quantizations of Lie bialgebras. But the deformation complex of a bialgebra is an $E_3$-algebra and one can instead use $E_3$ formality to quantize Lie bialgebras as shown in arxiv.org/abs/1606.01504. Similarly, to quantize a quasi-triangular quasi-Lie bialgebra one can 1) apply $E_2$ formality (going back to Drinfeld), 2) use Koszul duality and $E_3$ formality or, conjecturally, 3) apply $E_4$ formality to its deformation complex. $\endgroup$ – Pavel Safronov Feb 11 '17 at 19:24

It sounds like you want to check out some papers of V. Turchin and W. Willwacher (with collaborators).

For one example, in the paper "Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots", Ann. Inst. Fourier 65 (2015) by Turchin and myself, we describe the complex of deformations of maps from chains on $e_m$ (operad of little disks in dimension $m$) to chains on $e_{m+k}$. The complex is related to the homology of the space of smooth embeddings with compact support of ${\mathbb R}^m$ into ${\mathbb R}^{m+k}$.

The calculation in that paper depends on the relative formality of the little disks operads. The relative formality holds for $k>1$. In the paper "Relative (non)-formality of the little cubes operad and the algebraic Cerf Lemma" by V. Turchin and T Willwacher they show that relative formality fails for $k=1$, and describe the deformation complex of operad maps $e_m\to e_{m+1}$.


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