The homology of an $E\_2$-algebra is a Gerstenhaber algebra.


How precisely is the Gerstenhaber structure related to the $E\_2$-structure?


Obviously, the Gerstenhaber product is the commutative product that the $E\_2$-product induces in homology.


But what precisely is the interpretation of the Gerstenhaber bracket? It cannot be the commutator of the $E\_2$-product because that would be zero in homology, right?


Thanks for any hints.



What is the interpretation of the Gerstenhaber bracket?
I think the most transparent interpretation is by identifying $E\_2$ algebras with brace algebras (which was proved by [McClure and Smith](https://arxiv.org/abs/math/9910126)).


Namely, a brace algebra satisfies the relation
$$ab - (-1)^{|a||b|}ba = (-1)^{|a|} d(a\{b\}) - (-1)^{|a|}(da)\{b\} + a\{db\},$$
where $a\{b\}$ is one of the brace operations (so, it witnesses the first-order commutativity of the multiplication). The Gerstenhaber bracket is the antisymmetrization of the first brace:
$$[a, b] = a\{b\} - (-1)^{(|a|+1)(|b|+1)} b\{a\}.$$
Note that from the first equation you can see that $[-, -]$ is a $d$-closed operation, so it makes sense on homology.


So, the Gerstenhaber bracket is a secondary operation: the product is commutative on homology, but the homotopy might not be (anti)symmetric.


