TITLE: Obstructions to $E_2$-algebra structure on $E_1$-algebra
QUESTION [9 upvotes]: Let $A$ be an $E_1$-algebra in chain complexes over $\mathbb Q$. 

Is there an easy way to check if $A$ admits the structure of an $E_2$-algebra (or $E_\infty$-algebra)?

REPLY [5 votes]: This isn't a complete answer, as it would fill a whole paper; and I haven't checked everything. But I think you can use a strategy similar to what is done in the paper The Intrinsic Formality of $E_n$-operads by Fresse and Willwacher. I'm not saying it's easy, but it's systematic.
The situation is the following. You have the diagram:
$$\require{AMScd}
\begin{CD}
E_1 @>>> \mathsf{End}_A \\
@V{\iota}VV \\
E_2
\end{CD}$$
and you want to know whether there exists a lift. (Of course there is nothing special about $\mathsf{End}_A$ here, it could be any operad.) This is a two-pronged question:

Does there exist a map $E_2 \to \mathsf{End}_A$ at all?
Can you find one that makes the diagram commute?

(The problem I see with this approach is that the second part depends on the first, but the first could have many different answers...)
You can use Bousfield's obstruction theory to answer these question. You want to find a path-component in the mapping space $\operatorname{Map}(E_2, \mathsf{End}_A)$, i.e. you're looking at $\pi_0$. Of course, the very first obstruction is to find a Poisson $2$-algebra structure on $H_*(A)$. If you deal with the resolutions correctly and everything, I believe that such a structure would correspond to a class in the first page of the Bousfield spectral sequence. You want to know whether it survives until the end. Bousfield's obstruction theory should tell you that obstructions live in the cohomotopy groups $\pi^{s+1} \pi_s X$ where $X$ is a certain bicosimplicial space. You want all these obstructions to vanish. Based on the work of Fresse–Willwacher, my (perhaps wild) guess is that you get obstructions indexed by classes in Kontsevich's graph complex $\mathrm{GC}_2$.
Now you have your morphism and you want to know if the diagram commutes, at least up to homotopy I guess. In other words, you want to know if the two maps $E_1 \to \mathsf{End}_A$ (the first one is given by the $E_1$-structure, the second by restricting the $E_2$-structure you found through $\iota$) are in the same path component, i.e. they are joined by a path. Relative Bousfield obstruction theory should tell you that obstructions to finding such a path live in $\pi^s \pi_s X$. If you check that they all vanish, this means that your two morphism are equal up to homotopy. Again, my guess is that obstructions should be indexed by the hairy graph complex $\mathrm{HGC}_{12}$.