Operads of moduli spaces of points in C^d

We compute the structure of the homology of an operad built from the spaces TH_{d,n} of configurations of points in C^d, modulo translation and homothety. We find that it is a mild generalization of Getzler's gravity operad, which occurs in dimension d = 1.


Introduction
In a wealth of papers, e.g., [Get94,Get95,GK94,GK98,Vor00,KSV95], a number of connections between moduli spaces of curves and operads have been firmly established. In this note, we explore an operad built out of moduli spaces of points in higher-dimensional objects.
In [CGK09], Chen, Gibney, and Krashen study a variety TH d,n of configurations of n points in affine d-space modulo the action of the affine group, and define a compactification T d,n of this variety. In dimension d = 1, these varieties return the familiar moduli spaces of points in P 1 : TH 1,n = M 0,n+1 and T 1,n is the Deligne-Mumford compactification M 0,n+1 .
Just as in dimension 1, TH d,n and T d,n (or, for our purposes, their complex points) give rise to operads. In the case of T d,n , as for M 0,n+1 , this structure arises via grafting of trees of projective spaces (as in a free operad). The operadic structure on TH d,n may be derived from this via a form of transfer, though this is not quite the approach we take here. Write H * (TH d ) for the operad whose n th term is ΣH * (TH d,n ) (here Σ indicates a shift of degree by 1).
Definition 1.1. Let Grav d be the operad of graded Z-modules generated by k-ary operations {a 1 , . . . , a k } ∈ Grav d (k) of dimension 2d − 1, and c ∈ Grav d (1), of dimension −2, subject to the relations If d = 1, this is precisely the gravity operad introduced by Getzler in [Get94,Get95], where it was shown to be isomorphic to the operad ΣH * (M 0,n+1 ). It is the purpose of this note to extend this result to the higher-dimensional setting: Theorem 1.2. There is an isomorphism of operads H * (TH d ) ∼ = Grav d in "arity" n > 1.
We expect this computation to be useful in determining the structure of the homology of the operad T d . One concrete application of this result is as follows (derived from Theorem 1.2 and [Wes08]): Corollary 1.3. Let X = Ω 2d Y for an S 1 -space Y (more generally, let X be an algebra over the (2d)-dimensional framed little disks operad). Then the shifted equivariant homology ΣH S 1 * (X) is an algebra over the suboperad (Grav d ) >1 of arity > 1.
It is a pleasure to thank Michael Ching and Danny Krashen for helpful conversations about this material. The author was partially supported by NSF grant DMS-0705428 and ARC grant DP1095831.

The cohomology of TH d,n
Recall that the ordered configuration space of n points in C d , Conf n (C d ), is the space This space is acted upon (component-wise) by the affine group Aff(C d ) ∼ = C × ⋊ C d . If n > 1, the action is free.
The affine group is homotopy equivalent to its subgroup . This is a fibration, and is equivariant for the S 1 -action. Therefore, there is a commutative diagram of fibrations is homotopy equivalent to S 2d−1 , so for degree reasons the Serre spectral sequence for p collapses at E 2 . This allowed [CLM76] to prove that there is a ring isomorphism [GJ94]). Consequently, one can identify H * (Conf n−2 (C d \ {a, b})) as a quotient of H * (Conf n (C d )): Now, since Conf 2 (C d ) is S 1 -equivariantly homotopy equivalent to S 2d−1 , Conf 2 (C d )/S 1 ≃ CP d−1 . Therefore the Serre spectral sequence for p is of the form Again, the spectral sequence collapses because all differentials are determined on the fibre, and there are no possible targets for generators of the cohomology of the fibre for degree reasons. We conclude: Proposition 2.2. There is a ring isomorphism It is worth remarking that in dimension d = 1, this is a reflection of the wholly unsurprising fact that there is a homeomorphism M 0,n+1 ∼ = Conf n−2 (C \ {0, 1}).

The action of U (d)
Notice that there is an action of U (d) on Conf n (C d ), of which the S 1 = U (1)-action is but a part. The Pontrjagin ring of U (d) is where ∆ k is a generator of dimension 2k−1, obtained iteratively from fibrations over odd-dimensional spheres (see, e.g., [MT91,SW03]). These classes induce natural maps via the group action.
Proof. We use the dual action, in cohomology. That is, H * (U (d) acts on H * (Conf n (C d )) via dual maps ∆ * k which decrease degree by 2k − 1. Because each ∆ k is primitive, ∆ * k is a derivation. It is easy to see for degree reasons that the action of ∆ * k on H * (Conf n (C d )) is null except when k = d, and there, ∆ * d (x ij ) = 1, ∀ij. If we define y ij := x ij − x 12 , then H * (Conf n (C d )) is generated multiplicatively by y ij , ij = 12 along with x 12 . Write Y for the subalgebra generated by {y ij | ij = 12}. By the computations above, i * carries Y isomorphically onto H * (Conf n−2 (C d \ {a, b})).
Note that That is, H * (Conf n (C d )) is a free Λ[x 12 ]-module, generated by Y . Clearly ∆ * d (y ij ) = 0, and since ∆ * d is a derivation, this implies that Y ⊆ ker ∆ * d . For a general element y + y ′ x 12 , we see that Remark 3.2. This implies that the subspace ker ∆ d = im ∆ d is isomorphic to the shifted copy {a, b})).

TH d as an operad
Proposition 4.1. For each d > 0, there is an operad T H d in the category of S-modules whose n th term T H d (n) is weakly equivalent to the (shifted) suspension spectrum ΣΣ ∞ (TH d,n ) + for n > 1.
The category of S-modules, introduced in [EKMM97] is a rigidification of the stable homotopy category of spectra to admit a symmetric monoidal smash product. For those with little background or patience for the stable homotopy category, this proposition has the immediate (and down-toearth) consequence: Corollary 4.2. The collection H * (TH d )(n) := ΣH * (TH d,n ), n > 1, form a (non-unital) operad in the category of graded abelian groups.
We note that the shift by 1 is important; it accounts for a degree shifting S 1 -transfer map inherent in this construction. On T H d , this transfer exists as an actual map between the spectra forming the operad. For H * (TH d ), it comes from a homological transfer map: for an S 1 -bundle E → B, the transfer sends an element of H q (B) to the (q + 1)-dimensional cycle lying over it in H q+1 (E).
Proof. Let D 2d denote the operad of 2d-dimensional little disks, after [May72]. In [SW03], this was shown to be an SO(2d)-operad. Consider the group homomorphism S 1 → SO(2d) where z ∈ S 1 acts on C d = R 2d by z · (z 1 , . . . , z d ) = (z · z 1 , . . . , z · z d ). By restriction, this makes D 2d into an S 1 -operad. Using the machinery of [Wes08], we define T H d as the homotopy fixed point operad T H d := D hS 1 2d . Now D 2d (n) is S 1 -equivariantly homotopy equivalent to Conf n (C d ). Moreover, since the action of S 1 on the latter space is free (n > 1), and its quotient TH d,n is equivalent to a finite CW complex, D 2d (n) is S 1 -equivariantly finitely dominated. Thus by Theorem D of [Kle01], the norm map gives a homotopy equivalence Although this result does not apply to the unary part of the homotopy fixed point operad (i.e., the Spanier-Whitehead dual D hS 1 2d (1) = F (BS 1 + , S 0 )), it will play a role in the section below in studying the interaction of the Chern class with the rest of the operad.
A low-technology proof of Corollary 4.2 is given in section 3.2 of [Wes08].

The proof of Theorem 1.2
Since D 2d is an SO(2d)-operad (and hence U (d)-operad), H * (D 2d ) is a H * (U (d))-operad. Moreover, the primitivity of ∆ d implies that it is a derivation for the operad composition on H * (D 2d (n)) = H * (Conf n (C d )). That is, the operad compositions • i satisfy See [SW03]. A consequence of this fact is therefore that ker ∆ d ⊆ H * (D 2d ) = H * (Conf * (C d )) is a suboperad. One can now copy the proof of Theorem 4.5 of [Get94] to get