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.