TITLE: What is the interpretation of the Gerstenhaber bracket?
QUESTION [8 upvotes]: 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.

REPLY [13 votes]: I think the most transparent interpretation is by identifying $E_2$ algebras with brace algebras (which was proved by McClure and Smith).
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.