Homology of the $\mathbb {E}_d$ operadThe homology–cohomology pairing

# 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

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

2. 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
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

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:

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

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

3. 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 where the leftmost vertex has minimal label amongst the leaves. $\mathrm{Siop}^d(n)$ is spanned by “long graphs” whose components look like where $i_1 < i_2, \ldots , i_5$.

Proof
[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

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.