5 Operad 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

$\begin{useimager} \[ \begin{tikzpicture}[scale=0.3, baseline={([yshift=-.5ex]current bounding box.center)}] \draw (0, 0) -- (-1, 1) node [above=-0.05] {\tiny i}; \draw (-0.5, 0.5) -- (0, 1) node [above=-0.05] {\tiny j}; \draw (0, 0) -- (1, 1) node [above=-0.05] {\tiny k}; \end{tikzpicture} \mapsto \{\{x_i, x_j\}, x_k\}. \] \end{useimager}$

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

$\begin{useimager} \[ \begin{tikzpicture}[scale=0.3, baseline={([yshift=-.5ex]current bounding box.center)}] \draw (0, 0) -- (-1, 1) node [above=-0.05] {\tiny $i$}; \draw (-0.5, 0.5) -- (0, 1) node [above=-0.05] {\tiny $j$}; \draw (0, 0) -- (1, 1) node [above=-0.05] {\tiny $k$}; \begin{scope}[shift={(3, 0)}] \draw (0, 0) -- (-0.5, 0.5) node [above=-0.05] {\tiny $\ell$}; \draw (0, 0) -- (0.5, 0.5) node [above=-0.05] {\tiny $m$}; \end{scope} \end{tikzpicture} \mapsto \{\{x_i, x_j\}, x_k\} \cdot \{x_\ell, x_m\}. \] \end{useimager}$

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

$\begin{tikzpicture} \foreach \x in {-1.5, -1, -0.5, 0, 0.5, 1, 1.5} { \draw (0, 0) -- (\x, 1); } \node [above] at (-1.5, 1) {\scriptsize $1$}; \node [above] at (1.5, 1) {\scriptsize $m\! +\! n\! -\! 1$}; \begin{scope}[shift={(0.5, 1)}] \foreach \x in {-0.75, -0.25, 0.25, 0.75} { \draw (0, 0) -- (\x, 1); } \node [above] at (-0.75, 1) {\scriptsize $a$}; \node [above] at (0.75, 1) {\scriptsize $a + n$}; \end{scope} \node [below] at (0, 0) {$A$}; \node [right] at (0.5, 1) {\tiny $B$}; \end{tikzpicture}$

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

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Since the cooperad structure is compatible with the product structure, it suffices to show that it preserves

\circ _a

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

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