Homology of the $\mathbb {E}_d$ operadOperad and cooperad structures

We now describe an operad structure on $\mathrm{Pois}^d$ and the corresponding dual cooperad structure on $\mathrm{Siop}^d$.

Given an $S$-tree, we can produce a bracket expression. For example, we send

We can then send forests to products of bracket expressions, e.g.

The Jacobi identity translates to the usual Jacobi identity of the Lie/Poisson bracket. With the bracket expressions, we can interpret $\mathrm{Pois}^d$ as operad in the usual way by imposing the Leibniz rule

$\{ X, Y \cdot Z\} = \{ X, Y\} \cdot Z + (-1)^{|X| |Y|} Y \cdot \{ X, Z\}$

Under the configuration pairing, one checks that this gives the following cooperad structure on $\mathrm{Siop}^d$. To give an operad structure is to give a map $\circ _a: \mathcal{O}(m) \otimes \mathcal{O}(n) \to \mathcal{O}(m + n - 1)$ for every tree of the form

where the grafting vertex is the $a$th vertex. Label the two vertices $A$ and $B$. To make $\mathrm{Siop}^d$ a cooperad, we need a map $\mathrm{Siop}^d(m + n - 1) \to \mathrm{Siop}^d(m) \otimes \mathrm{Siop}^d(n)$. This map sends $G$ to $G_A \otimes G_B$, where the edges of $G_1$ and $G_2$ are specified by the following procedure:

• For any edge $ij$ of $G$, consider the leaves $i$ and $j$ in the tree above. Let $v$ be the meet of $i$ and $j$ (so that $v = A$ or $B$), and let $J_v(i) = J_v(j)$ be the branches of $v$ over which $i$ and $j$ lie (in this case, one of $J_v(i)$ and $J_v(j)$ will be $i$ or $j$). Then add an edge to $G_v$ from $J_v(i)$ to $J_v(j)$.

Theorem 5.1

The map $\mathrm{Siop}^d(n) \to H^*(\operatorname{Conf}_n(\mathbb {R}^d)) = H^*(\mathbb {E}_d(n))$ is an isomorphism of cooperads.

Corollary 5.2

The map $\mathrm{Pois}^d(n) \to H_*(\operatorname{Conf}_n(\mathbb {R}^d)) = H_*(\mathbb {E}_d(n))$ is an isomorphism of operads.

Proof
Since the cooperad structure is compatible with the product structure, it suffices to show that it preserves $\circ _a$ on $a_{ij}$.

Consider the composite

$\mathbb {E}_d(m) \times \mathbb {E}_d(n) \overset {\circ _a}{\longrightarrow } \mathbb {E}_d(m + n - 1) \overset {\alpha _{ij}}{\longrightarrow } S^{d - 1}.$

Consider the homotopy where at time $t$, the disks in the first factor are scaled by $t$ and the disks in the second factor are scaled by $t^2$. As $t \to 0$, we see that this approaches the projection onto the $v$th factor followed by $\alpha _{J_v(i) J_v(j)}$, as promised.

Proof