# 4 The homology–cohomology pairing

To compute the homology–cohomology pairing, since $H^*(\operatorname{Conf}_n(\mathbb {R}^d))$ is generated by the $a_{ij}$, it suffices to compute the composites

$\prod _{v \in F} S^{d - 1} \overset {P_F}{\longrightarrow } \operatorname{Conf}_n(\mathbb {R}^d) \overset {\alpha _{ij}}{\longrightarrow } S^{d - 1}$for forests $F$.

If $i$ and $j$ are in different trees, by taking the limit $\varepsilon \to 0$, we see that this map is homotopic to the constant map $(\pm 1, 0, \ldots , 0)$, hence is nullhomotopic. Otherwise, if they meet at $v$, then it is easy to see that up to a sign, this is projection onto the $v$^{th} factor.

Let $F$ be an $n$-forest and $G$ an $n$-graph. Then under the homology–cohomology pairing, $\langle G, P_F\rangle$ is $\pm 1$ iff

For every edge $(i, j)$ in $G$, the corresponding leaves in $F$ are in the same component

The map sending an edge $(i, j)$ to the meet of leaves $\{ i, j\}$ gives a bijection between edges of $G$ and vertices of $F$.

Otherwise, it is zero.

*configuration pairing*.

The lemma is then clear from the previous identification.

Using this explicit description of the pairing, let us show that the maps $\mathrm{Pois}^d(n) \to H_*(\operatorname{Conf}_n(\mathbb {R}^d))$ and $\mathrm{Siop}^d(n) \to H^*(\operatorname{Conf}_n(\mathbb {R}^d))$ are isomorphisms. We already know that the second map is surjective. Our plan is as follows:

Write down a spanning set of $\mathrm{Pois}^d(n)$ and $\mathrm{Siop}^d(n)$.

Show that the homology–cohomology pairing pairs these elements perfectly.

Deduce the maps must be injections and $\dim \mathrm{Pois}^d(n) = \dim \mathrm{Siop}^d(n)$, so they are both isomorphisms.

The following lemma follows from applying the Jacobi and Arnold identities:

$\mathrm{Pois}^d(n)$ is spanned by “tall” forests, i.e. forests whose trees look like

where the leftmost vertex has minimal label amongst the leaves. $\mathrm{Siop}^d(n)$ is spanned by “long graphs” whose components look like

where $i_1 < i_2, \ldots , i_5$.

Each of these elements are specified by partitions of $\mathbf{n}$ together with some ordering data.

The pairing of a tall forest and a long graph is $1$ if they correspond to the same partition, and $0$ otherwise.

$\mathrm{Pois}^d(n) \to H_*(\operatorname{Conf}_n(\mathbb {R}^d))$ and $\mathrm{Siop}^d(n) \to H^*(\operatorname{Conf}_n(\mathbb {R}^d))$ are isomorphisms.