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

1 Definition of the Ed\mathbb {E}_d operad

Definition 1.1

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

As a homotopy type, we have Ed(n)Confn(Dd)Confn(Rd)\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 XX, the configuration space Confn(X)Xn\operatorname{Conf}_n(X) \subseteq X^n is the space of nn distinct points in XX, labelled x1,,xnx_1, \ldots , x_n.

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

Example 1.3

For n=2n = 2, it is easy to see that Conf2(Rd)Sd1\operatorname{Conf}_2(\mathbb {R}^d) \simeq S^{d - 1}. Indeed, by translation, we can assume x1=0x_1 = 0, and x2x_2 can be any point in Rd{0}Sd1\mathbb {R}^d \setminus \{ 0\} \simeq S^{d - 1}.

A more symmetric way of seeing this is to consider the subspace of Conf2(Rd)\operatorname{Conf}_2(\mathbb {R}^d) consisting of points such that x1=x2x_1 = - x_2 and x1=x2=1\| x_1\| = \| x_2\| = 1. This is homeomorphic to Sd1S^{d - 1} and is a deformation retract of Conf2(Rd)\operatorname{Conf}_2(\mathbb {R}^d) by translation and scaling.

We focus on the case d>2d > 2, so that Sd1S^{d - 1} is simply connected. We have fiber sequences

      \bigvee_{n - 1} S^{d - 1} \ar[r] & \Conf_n(\R^d) \ar[d]\\
      & \Conf_{n - 1}(\R^d)

The vertical map forgets the nnth point and is a fibration. The fiber over a point is Rd{x1,,xn1}n1Sd1\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 Confn(Rd)\operatorname{Conf}_n(\mathbb {R}^d) is free, concentrated in degrees that are multiples of d1d - 1, and the Serre spectral sequence degenerates at E2E^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. Hd1(Confn(Rd))H_{d - 1}(\operatorname{Conf}_n(\mathbb {R}^d)) and hence Hd1(Confn(Rd))H^{d - 1}(\operatorname{Conf}_n(\mathbb {R}^d)) is free of rank (n2)\binom {n}{2}.

  2. The cohomology of Confn(Rd)\operatorname{Conf}_n(\mathbb {R}^d) is generated as a ring by elements in degree d1d - 1.

In fact, what we will do is to compute the cooperad structure on the cohomology, which we only have to verify in degree d1d - 1, and then dualize to get the operad structure on homology.