Homotopy Gerstenhaber algebras: description via operads vs derivations

There are at least a couple of definitions in the literature for an $$E_2$$-algebra, also known as a homotopy Gerstenhaber algebra, also known as $$G_{\infty}$$-algebra.

Suppose $$V$$ is a graded vector space. Let $$S(V), \text{Lie}(v)$$ denote the graded symmetric algebra and the graded free Lie algebra of $$V$$ respectively and $$S_+(V)$$ be the symmetric algebra in strictly positive degree.

$$\textbf{Definition (Tamarkin)}$$ A $$G_{\infty}$$-algebra structure on $$V$$ is a degree $$+1$$ map $$\delta: S_+(\text{Lie}(V^*[1])[1]) \rightarrow S_+(\text{Lie}(V^*[1])[1])$$ which behaves as a derivation of both $$\cdot$$ and $$[\,,]$$ such that $$\delta^2 = 0$$.

Unpacking this leads to saying that we have a collection of maps $$m_{k_1, \dots, k_n}: V^{\otimes k_1} \otimes \dots \otimes V^{\otimes k_n} \rightarrow V$$ of degree $$3-(k_1 + \dots k_n + n)$$ obeying appropriate symmetry and associativity relations.

The second definition pertains to algebras over the little disk operad. Let $$D_2(k)$$ denote the configuration space of $$k$$ little disks inside a big disk and let $$\text{Chains}_{\bullet}(D_2(k))$$ be the singular chain complex. Letting $$\mathcal{P}(k) = \text{Chains}_{\bullet}(D_2(k))$$, one can define an operadic structure on this collection of vector spaces. This leads one to the

$$\textbf{Definition (Getzler-Jones?)}$$ An $$E_2$$-algebra is an algebra over the operad $$\text{Chains}(D_2)$$.

How can one show that these two definitions are equivalent?

Short of a full proof of equivalence, it would be nice to understand a description of the cycle in $$D_2(k_1 + \dots +k_n)$$ which corresponds to the map $$m_{k_1, \dots, k_n}$$. For example: if one were working in $$H_{\bullet}(D_2)$$, the homology operad, one associates to the point class in configuration space of two disks the operation $$\cdot = m_2$$, and to the cycle involving one little disk going around the other the bracket $$[\,,] = m_{1,1}$$ in the Gerstenhaber algebra. Is there an explicit description of the cycle corresponding to $$m_{k_1, \dots, k_n}$$?

The equivalence between these two notions is nontrivial, since it amounts to a choice of formality isomorphism for the operad of little disks.

Let $$D_2$$ be the little disks operad. The easy part of the equivalence is to show that $$G_\infty$$ algebras are equivalent to algebras over the operad $$H_*(D_2)$$. If you take the operad of graded $$\mathbb Q$$ vector spaces, $$H_*(D_2)$$, its algebras are Gerstenhaber algebras-- this can be seen by computing the homology of the configuration spaces. Then Koszul self-duality for the Gerstenhaber operad gives the relationship between $$G_\infty$$ algebras and Gerstenhaber algebras. I.e. every $$G_\infty$$ algebra may be rectified to an algebra over $$H_*(D_2)$$.

Let $$C_*(D_2)$$ be chains on the little disk operad. To obtain an equivalence between $$C_*(D_2)$$ algebras and $$H_*(D_2)$$ algebras, one needs to choose a zig-zag of quasi-isomorphisms between the two operads. These formality isomorphisms are difficult to construct, and there are many possible choices, because these operads admit a large automorphism group. See Kontsevich's paper, https://arxiv.org/abs/math/9904055, for a discussion.

• Thanks for the answer. Please take a look at the edits I made. – dayar Feb 19 '20 at 19:50

Here are the operads that are involved in that game:

• the operad $$D_2$$ of little disks, which is a topological operad.

• its chain operad $$C_{-*}(D_2,\mathbb{k})$$, which is an operad in cochain complexes (of $$\mathbb{k}$$-modules).

• its homology operad $$H_{-*}(D_2,\mathbb{k})$$, which is known to be isomorphic to the Gerstenhaber operad $$G^{\mathbb{k}}$$, which is itself a binary quadratic operad satisfying the Koszul property.

• the minimal resolution of $$G^{\mathbb{k}}$$ is the operad $$G^{\mathbb{k}}_\infty$$ governing ($$\mathbb{k}$$-linear) $$G_\infty$$-algebras.

Being a resolution, $$G^{\mathbb{k}}_\infty$$ is obviously quasi-isomorphic to $$G^{\mathbb{k}}$$. As Phil Tosteson mentions in his answer, the difficult part relies on proving that $$C_{-*}(D_2,\mathbb{k})$$ is formal. It's only proven over a field $$\mathbb{k}$$ of characteristic zero, and it's actually not formal for $$\mathbb{k}=\mathbb{F}_p$$ (see e.g. the introduction of https://arxiv.org/pdf/1903.09191.pdf). To my knowledge, there are essentially two different proofs:

Note that the formality quasi-isomorphisms from these two proofs happen to coincide, after the correct "choice of associator" has been made. See https://arxiv.org/abs/0905.1789.

Finally, I don't think it is meaningful to ask which cycle corresponds to the map $$m_{k_1,\dots,k_n}$$. The reason is that $$m_{k_1,\dots,k_n}$$ is not closed. You may ask if they are represented by nice chains... I don't know the answer to that question (and I suspect that it is close to be as hard as proving the formality itself), but for the $$m_{1,\dots,1}$$ the answer is known:

• first observe that $$D_2$$ is weakly homotopy equivalent to the operad of compactified configuration spaces of points in the plane.

• $$m_{1,\dots,1}$$ (with $$n$$ "$$1$$"s) can be represented by the top-dimensional/fundamental cell of the compactified configuration space $$\overline{C}_n\simeq D_2(n)$$ of $$n$$-points in the plane.

• Thanks! In $\overline{C}_n$ is the group of dilations and overall translations implicitly modded out? So that the fundamental cell of $\overline{C}_n$ is $2n-3$ dimensional? – dayar Feb 21 '20 at 1:58
• Yes, the group we use to mod out is the semi-direct product of translations of (real, positive) dilations. hence, yes, the dimension of the top dimensional cell is 2n-3. – DamienC Feb 21 '20 at 13:48
• Great. It might be worth asking this as a separate question, but what is the argument that there is a morphism of operads from the $L_{\infty}$-operad to the suboperad of top dimensional cells in the compactified configuration space? If you know any references I'd appreciate it. – dayar Feb 21 '20 at 15:43
• I don't have a reference in mind, but I think the argument is not so difficult. As a graded operad (ie without the differential), the $L_\infty$-operad is free. So, you define a morphism of graded operads sending $m_{1,\dots,1}$ to the top dimensional cell in $\overline{C}_n$. Then you check that it is compatible with the differentials (intuitively, the boundary of the top dimensional cell is a union of products of top-dimensional cells of $\overline{C}_m$'s for $m<n$. – DamienC Feb 22 '20 at 9:58