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.

Lemma 4.1

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.

This pairing of forests and graphs is called the configuration pairing.

Proof

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The pairing is the degree of the map

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

The lemma is then clear from the previous identification.

Proof

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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:

Lemma 4.2

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

$\begin{tikzpicture} [scale=0.3] \draw (0, 0) -- (-3, 3); \draw (-2.5, 2.5) -- (-2, 3); \draw (-2, 2) -- (-1, 3); \draw (-1.5, 1.5) -- (0, 3); \draw (0, 0) -- (3, 3); \node at (0.7, 2) {$\cdots$}; \end{tikzpicture}$

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

$\begin{tikzpicture} [scale=0.7] \node (1) at (0, 0) {$i_1$}; \node (2) at (1, 1) {$i_2$}; \node (3) at (2, 0) {$i_3$}; \node (4) at (3, 1) {$i_4$}; \node (5) at (4, 0) {$i_5$}; \draw [->] (1) -- (2); \draw [->] (2) -- (3); \draw [->] (3) -- (4); \draw [->] (4) -- (5); \end{tikzpicture}$

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

Proof

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[Hint] The first follows form the observation that tall trees are exactly trees where the leftmost vertex is as far away from the root vertex as possible.

Proof

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

Lemma 4.3

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

Corollary 4.4

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