Homology of the $\mathbb {E}_d$ operadDefinition of the $\mathbb {E}_d$ operad

# 1 Definition of the $\mathbb {E}_d$ operad

Definition 1.1

$\mathbb {E}_d(n)$ is the space of all $n$ disjoint balls in $D^d$. This is an operad in the usual way.

As a homotopy type, we have $\mathbb {E}_d(n) \simeq \operatorname{Conf}_n(D^d) \cong \operatorname{Conf}_n(\mathbb {R}^d)$, with the first homotopy equivalence given by shrinking the balls. Recall that

Definition 1.2

For any topological space $X$, the configuration space $\operatorname{Conf}_n(X) \subseteq X^n$ is the space of $n$ distinct points in $X$, labelled $x_1, \ldots , x_n$.

The homology of $\operatorname{Conf}_n(\mathbb {R}^d)$ as a group turns out to be pretty easy to compute.

Example 1.3

For $n = 2$, it is easy to see that $\operatorname{Conf}_2(\mathbb {R}^d) \simeq S^{d - 1}$. Indeed, by translation, we can assume $x_1 = 0$, and $x_2$ can be any point in $\mathbb {R}^d \setminus \{ 0\} \simeq S^{d - 1}$.

A more symmetric way of seeing this is to consider the subspace of $\operatorname{Conf}_2(\mathbb {R}^d)$ consisting of points such that $x_1 = - x_2$ and $\| x_1\| = \| x_2\| = 1$. This is homeomorphic to $S^{d - 1}$ and is a deformation retract of $\operatorname{Conf}_2(\mathbb {R}^d)$ by translation and scaling.

We focus on the case $d > 2$, so that $S^{d - 1}$ is simply connected. We have fiber sequences

The vertical map forgets the $n$th point and is a fibration. The fiber over a point is $\mathbb {R}^d \setminus \{ x_1, \ldots , x_{n - 1}\} \simeq \bigvee _{n - 1} S^{d - 1}$. Inductively, by the Serre spectral sequence, we see that the (co)homology of $\operatorname{Conf}_n(\mathbb {R}^d)$ is free, concentrated in degrees that are multiples of $d - 1$, and the Serre spectral sequence degenerates at $E^2$ for degree reasons.

This tells us the homology and cohomology groups completely as groups, but we lack geometric understanding of all these classes, which is needed to compute the operad structure. What we shall do is to produce explicit (co)homology classes, and this preliminary calculation will tell us when we have found all the classes. For this purpose, all we need are the following two properties:

1. $H_{d - 1}(\operatorname{Conf}_n(\mathbb {R}^d))$ and hence $H^{d - 1}(\operatorname{Conf}_n(\mathbb {R}^d))$ is free of rank $\binom {n}{2}$.

2. The cohomology of $\operatorname{Conf}_n(\mathbb {R}^d)$ is generated as a ring by elements in degree $d - 1$.

In fact, what we will do is to compute the cooperad structure on the cohomology, which we only have to verify in degree $d - 1$, and then dualize to get the operad structure on homology.