LOCALIZATION AND COMPLETION THEOREMS FOR MU-MODULE SPECTRA

J.P.C. GREENLEES AND J.P. MAY

Abstract. Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU; examples include all of the standard homotopical MU-modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M_{∗}(BG) and M^{∗}(BG). The G-spectrum MU_{G} that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum S_{G}, and there is an MU_{G}-module M_{G} whose underlying MU-module is M. This allows the use of topological analogues of constructions in commutative algebra. The computation of M_{∗}(BG) and M^{∗}(BG) is expressed in terms of spectral sequences whose respective E_{2} terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU_{∗}^{G} and its module M_{∗}^{G}. The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global ℐ_{∗}-functor with smash product.

Contents

1. Introduction and statements of results 1

2. The strategy of proof 7

3. Constructing sufficiently large finitely generated ideals 9

4. The idea and properties of norm maps 13

5. Global I_{∗}-functors with smash product 15

6. The passage to spectra 18

7. Wreath products and the definition of the norm map 20

8. The proof of the double coset formula 23

9. The norm map on sums and its double coset formula 26

10. The Thom classes of Thom spectra 29

References 31

The authors are grateful to Gustavo Comezaña, Jim McClure, and Neil Strickland for helpful conversations. The first author thanks the University of Chicago for its hospitality and the Nuffield Foundation for its support. The second author acknowledges support from the NSF.

1. Introduction and statements of results

Completion theorems relate the nonequivariant cohomology of classifying spaces to algebraic completions of associated equivariant cohomology theories. They are at the heart of equivariant stable homotopy theory and its nonequivariant applications. The most celebrated such result is the Atiyah-Segal completion theorem [1]. For any compact Lie group G, it computes K(BG) as the completion of the representation ring R(G) at its augmentation ideal. A more recent such result is the Segal conjecture [2]. For any finite group G, it computes the cohomotopy π^{∗}(BG) as the completion of the equivariant cohomotopy π_{G}^{∗} at the augmentation ideal of the Burnside ring A(G). Unlike the Atiyah-Segal completion theorem, in which the representation ring is under good algebraic control, the Segal conjecture relates two sequences of groups that are largely unknown and difficult to compute.

Shortly after the Atiyah-Segal completion theorem appeared, Landweber [19] and others raised the problem of whether an analog might hold for complex cobordism. It was seen almost immediately that the appropriate equivariant form of complex cobordism to consider was the stabilized version, MU_{G}^{∗}, introduced by tom Dieck [6]. Shortly after the question was raised, Löffler [23] proved the following result.

Theorem 1.1 (Löffler). If G is a compact Abelian Lie group, then

(MU_{G}^{∗})_{Ĵ_{G}} ≅ MU^{∗}(BG),

where J_{G} is the augmentation ideal of MU_{G}^{∗}.

Here MU^{∗}(BG) is completely understood [20, 22, 23], and the result is not difficult because the Euler classes of the irreducible complex representations of G, which of course are all 1-dimensional, are under good control. There has been no further progress in the last twenty years. In fact, in his 1992 survey of equivariant stable homotopy theory [3], Carlsson stated the problem as follows:

“Formulate a conjecture about MU^{∗}(BG), for G a finite group.” In [12], the first author introduced a new approach to the Atiyah-Segal completion

theorem (for finite groups), in which he deduced it from what we now understand to be a kind of localization theorem giving a computation of K_{∗}(BG) in terms of local cohomology. When such a localization theorem holds in homology, it is a considerably stronger result than the implied completion theorem in cohomology. For example, the localization theorem for stable homotopy theory is false, although the completion theorem for stable cohomotopy is true. We refer the reader to [17, §§6-8] for a general discussion of localization theorems in equivariant homology and completion theorems in equivariant cohomology. As promised there and in the announcement [7], we shall here prove such theorems for stabilized equivariant complex cobordism. Landweber [19] had noted that the problem of studying MU_{∗}(BG) seemed to be even harder than the problem of studying MU^{∗}(BG).

To make sense of the approach of [12], one must work in a sufficiently precise context of highly structured ring and module spectra that one can mimic constructions in commutative algebra topologically. The theory developed by Elmendorf, Kriz, Mandell, and the second author [9] provides these essential foundations. That paper was written nonequivariantly, but, as stated in a metatheorem in its introduction, all of its theory applies verbatim to G-spectra for any compact Lie group G; see also [8, 17, 10]. In the language of that paper, stabilized equivariant cobordism is represented by an algebra MU_{G} over the equivariant sphere G-spectrum S_{G}. The underlying nonequivariant S-algebra of MU_{G} is MU. In earlier language, this means that MU_{G} is an E_{∞} ring G-spectrum with underlying nonequivariant E_{∞} ring spectrum MU.

A considerable virtue of the kind of localization theorem that we have in mind is that, when it applies to an S_{G}-algebra R_{G} with underlying nonequivariant S-algebra R, it automatically implies localization and completion theorems for the computation of M_{∗}(BG) and M^{∗}(BG) for the underlying R-module M of any split R_{G}-module M_{G}. (The notion of a split G-spectrum is defined and discussed in [21, II.84], [15, §0], and [17, §3].) This is an especially happy feature of our work since MU_{G} is split and, as will be proven elsewhere [25], every MU-module M is the underlying nonequivariant spectrum of a split MU_{G}-module M_{G}. Therefore, by [9, V§4], our work applies to all of the standard MU-modules that are constructed from MU by quotienting out the ideal generated by a regular sequence of elements of MU_{∗} and localizing by inverting some other elements. In particular, it applies to the Brown-Peterson and Morava K-theory spectra. There is a long and extensive history of explicit calculations of groups M_{∗}(BG) and M^{∗}(BG) in special cases. Our theorem gives a general conceptual framework into which all such computations must fit.

As we shall make precise shortly, the theorem shows that these nonequivariant homology and cohomology groups are isomorphic to the equivariant homotopy groups of certain homotopical J_{G}-power torsion MU_{G}-modules Γ_{J_{G}}(M_{G}) and homotopical completion MU_{G}-modules (M_{G})_{J_{G}}^{∧}, where J_{G} is the augmentation ideal of MU_{∗}^{G}. There result spectral sequences for the computation of these homotopy groups in terms of “local cohomology groups” and “local homology groups” that can be computed from knowledge of the ring MU_{∗}^{G} and its module M_{∗}^{G}. Thus the theorem establishes a close connection between the geometrically defined equivariant cobordism groups and the homology and cohomology of classifying spaces with coefficients in MU-modules.

This is entirely satisfactory on a conceptual level. However, like the Segal conjecture, our theorem relates two sequences of groups that are largely unknown and difficult to compute. Thus, on the computational level, it merely points the direction towards further study. Explicit computations will require better understanding of MU_{∗}^{G} than is now available. We recall an old and probably false conjecture.

Conjecture 1.2. MU_{∗}^{G} is MU_{∗}-free on generators of even degree.

This was claimed by Löffler [22, 23] when G is Abelian, but his proof has not appeared. Comezaña [4] has recently worked out a complete proof in this case. Little is known for non-Abelian groups.

Since our work is based on the importation of techniques of commutative algebra into equivariant stable homotopy theory, we briefly recall the relevant algebraic constructions; see [16] for details and discussion. Let R be a graded commutative ring and let I = (α_{1}, …, α_{n}) be a finitely generated ideal in R. Define K^{•}(I) to be the tensor product of the graded cochain complexes

K^{•}(α_{i}) = (R → R[1/α_{i}]),

where R and R[1/α_{i}] lie in homological degrees 0 and 1. Up to quasi-isomorphism, K^{•}(I) depends only on the radical of I. For a graded R-module M, define

H_{I}^{s,t}(R; M) = H^{s,t}(K^{•}(I) ⊗ M),

where s indicates the homological degree and t the internal grading. Such “local cohomology groups” were first defined by Grothendieck [18]. It is easy to see that H_{I}^{0}(R; M) is the submodule

Γ_{I}(M) = {m ∈ M |I^{N}m = 0 for some N}

of I-power torsion elements of M. If R is Noetherian it is not hard to prove directly that the functor H_{I}^{∗}(R; ?) is effaceable and hence that local cohomology calculates the right derived functors of Γ_{I}(?) [18]. It is clear that the local cohomology groups vanish above degree n, but in the Noetherian case Grothendieck’s vanishing theorem shows the powerful fact that they are zero above the Krull dimension of R. The other fact we shall use is that if β ∈ I then H_{I}^{∗}(R; M)[1/β] = 0; this is a restatement of the easily proven fact that K^{•}(I)[1/β] is exact [16, 1.1]. We abbreviate

H_{I}^{∗}(R) = H_{I}^{∗}(R; R).

These algebraic local cohomology groups are relevant to topological homology groups.

There are dual “local homology groups” which, to the best of our knowledge, were first introduced in [13, 14]. Replacing K^{•}(I) by a quasi-isomorphic R-free chain complex K′^{•}(I), define

H_{s,t}^{I}(R; M) = H_{s,t}(Hom(K′^{•}(R), M)).

There is a tri-graded universal coefficient spectral sequence that converges to these groups; ignoring the internal grading t, which is unchanged by the differentials, it converges in total degree s = −(p + q) and satisfies

E_{2}^{p,q} = Ext_{R}^{p}(H_{I}^{−q}(R), M) and d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1}.

There is a natural epimorphism H_{0}^{I}(R; M) → M_{ Î} whose kernel is a certain lim^{1} group. It is not hard to check from the definition that if R is Noetherian and M is free or finitely generated then H_{0}^{I}(R; M) ≅ M_{I}^{∧}, and one may also prove that in this case the higher local homology groups are zero. It follows that, at least when R is Noetherian, H_{∗}^{I}(R; M) calculates the left derived functors of the (not necessarily right exact) I-adic completion functor. These algebraic local homology groups are relevant to topological cohomology groups.

Now let R_{G} be an S_{G}-algebra and M_{G} be an R_{G}-module; we always understand algebras and modules in the highly structured sense of [9]. For α ∈ R_{k}^{G}, let R_{G}[1/α] be the telescope of iterates

R_{G} → Σ^{−k}R_{G} → Σ^{−2k}R_{G} → ⋯

of multiplication by α and let K(α) be the fiber of the canonical map R_{G} → R_{G}[1/α]. For a finitely generated ideal I = (α_{1}, …, α_{n}) in R_{∗}^{G}, let K(I) be the smash product over R_{G} of the R_{G}-modules K(α_{i}). Up to equivalence of R_{G}-modules, K(I) depends only on the radical of I. Define

Γ_{I}(M_{G}) = K(I) ∧_{R_{G}} M_{G}

and

(M_{G})_{I}^{∧} = F_{R_{G}}(K(I), M_{G}).

There is a spectral sequence converging to Γ_{I}(M_{G})_{∗}^{G} (in total degree p + q), with

E_{p,q}^{2} = H_{I}^{−p,−q}(R_{∗}^{G}; M_{∗}^{G}) and d^{r} : E_{p,q}^{r} → E_{p−r,q+r−1}^{r},

and there is a spectral sequence converging to ((M_{G})_{I}^{∧})_{G}^{∗} (in total degree p + q) with

E_{2}^{p,q} = H_{−p,−q}^{I}(R_{G}^{∗}; M_{G}^{∗}) and d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1}.

Now take I to be a finitely generated ideal contained in the augmentation ideal J_{G} = Ker(R_{∗}^{G} → R_{∗}). Note that the ring R_{∗}^{G} need not be Noetherian and the augmentation ideal need not be finitely generated. In particular, MU_{∗}^{G} is not Noetherian and its augmentation ideal is not finitely generated, even when G is finite.

Since R[1/α] is nonequivariantly contractible for α ∈ J_{G}, the canonical map K(I) → R_{G} is an equivalence of underlying spectra and so induces an equivalence upon smashing with EG_{+}, where EG_{+} is the union of EG and a G-fixed disjoint basepoint. Inverting this equivalence and using the projection EG_{+} ⟶ S^{0}, we obtain a canonical map of R_{G}-modules

κ : EG_{+} ∧ R_{G} → K(I).

For an R_{G}-module M_{G}, κ induces maps of R_{G}-modules

EG_{+} ∧ M_{G} ≃ EG_{+} ∧ R_{G} ∧_{R_{G}} M_{G} → K(I) ∧_{R_{G}} M_{G} = Γ_{I}(M_{G})

and

(M_{G})_{G}^{∧} = F_{R_{G}}(K(I), M_{G}) → F_{R_{G}}(EG_{+} ∧ R_{G}, M_{G}) ≃ F (EG_{+}, M_{G}),

both of which will be equivalences for all R_{G}-module M_{G} if κ is an equivalence. We can now state our completion theorem for modules over MU_{G}.

Theorem 1.3. Let G be finite or a finite extension of a torus. Then, for any sufficiently large finitely generated ideal I ⊂ J_{G}, κ : EG_{+} ∧ MU_{G} → K(I) is an equivalence. Therefore,

EG_{+} ∧ M_{G} → Γ_{I}(M_{G}) and (M_{G})_{I}^{∧} → F (EG_{+}, M_{G})

are equivalences for any MU_{G}-module M_{G}.

It is reasonable to define K(J_{G}) to be K(I) for any sufficiently large I and to define Γ_{J_{G}}(M_{G}) and (M_{G})_{J_{G}}^{∧} similarly. The theorem implies that these MU_{G}-modules are independent of the choice of I.

Our main interest is in finite groups. However, the fact that the result holds for a finite extension of a torus and therefore for the normalizer of a maximal torus in an arbitrary compact Lie group suggests the following generalization. There should be an appropriate transfer argument, but we have not succeeded in finding one.

Conjecture 1.4. The theorem remains true for any compact Lie group G.

It is valuable to obtain a completion theorem about EG×_{G}X for a general G-space X, obtaining the motivating result about BG by taking X to be a point. For this purpose, we replace M_{G} by M_{G} ∧ X_{+} in the first equivalence and by F (X_{+}, M_{G}) in the second. If M_{G} is split with underlying nonequivariant MU-module M, then

π_{∗}^{G}((EG × X)_{+} ∧ Σ^{−Ad(G)}M_{G}) ≅ M_{∗}(EG ×_{G} X)

and

π_{∗}^{G}(F ((EG × X)_{+}, M_{G})) ≅ M^{∗}(EG ×_{G} X),

where Ad(G) is the adjoint representation of G [21, II.8.4]. Thus the theorem has the following immediate consequence.

Theorem 1.5. Assume the hypotheses of the theorem and assume that M_{G} is split. Then

Γ_{I}(Σ^{−Ad(G)}M_{G} ∧ X_{+})_{∗}^{G} ≅ M_{∗}(EG ×_{G} X)

and

(F (X_{+}, M_{G}))_{I}^{∧})_{G}^{∗} ≅ M^{∗}(EG ×_{G} X)

for any G-space X.

Implicitly replacing X by its suspension G-spectrum, we are entitled to the following spectral sequences.

Corollary 1.6. There is a homological spectral sequence that converges from

E_{p,q}^{2} = H_{I}^{−p,−q}(MU_{∗}^{G}; M_{∗}^{G}(Σ^{−Ad(G)}X_{+}))

to M_{∗}(EG ×_{G} X). There is a cohomological spectral sequence that converges from

E_{2}^{p,q} = H_{−p,−q}^{I}(MU_{G}^{∗}; M_{G}^{∗}(X_{+}))

to M^{∗}(EG ×_{G} X).

Combining Löffler’s theorem with ours, we see that the topology forces the following algebraic conclusion. A direct proof would be out of reach at present.

Corollary 1.7. If G is a compact Abelian Lie group and I is sufficiently large, then

H_{0}^{I}(MU_{G}^{∗}) ≅ ((MU_{G})_{Î})_{G}^{∗} ≅ (MU_{G}^{∗})_{Î}

and H_{p}^{I}(MU_{G}^{∗}) = 0 if p ≠ 0.

2. The strategy of proof

For clarity, we shall emphasize the general strategy of proof, focusing on MU_{G} only where necessary. Let G be a compact Lie group and let S_{G} be the sphere G-spectrum. We assume given a commutative S_{G}-algebra R_{G} with underlying nonequivariant commutative S-algebra R.

While our equivariant cohomology theories are RO(G)-graded, we are interested in their restrictions to integer degrees. For a (closed) subgroup H of G, let res_{H}^{G} denote the restriction

R_{G}^{∗} = R_{G}^{∗}(S^{0}) ⟶ R_{G}^{∗}(G/H_{+}) = R_{H}^{∗};

it is induced by the projection G/H_{+} ⟶ S^{0}. Let J_{H} denote the augmentation ideal in R_{H}^{∗}, namely the kernel of res_{1}^{H} : R_{H}^{∗} ⟶ R^{∗}. In [17, 7.5], we explained a general localization and completion theorem for the calculation of M_{∗}^{G}(EG_{+}∧X) and M_{G}^{∗}(EG_{+} ∧ X) for any split R_{G}-module M_{G} and G-spectrum X. In that theorem, we assumed that G is finite, each R_{H}^{∗} is Noetherian, and each theory R_{∗}^{H} has Thom isomorphisms. The Noetherian hypothesis ensures not only that each J_{H} is finitely generated but, much more deeply, that

\sqrt{res_{H}^{G}(J_{G})} = \sqrt{J_{H}}.

In fact, we did not discuss the verification of this last property in [17], and we remarked that its verification “can be the the main technical obstruction to the implementation of our strategy when we work more generally with compact Lie groups and non-Noetherian coefficient groups”. Since the algebraic and topological constructions in our general approach demand that we work with finitely generated ideals, we need to modify our strategy of proof. We begin work by describing our modified strategy.

Thom isomorphisms and Euler classes are essential to the strategy, and we begin with these. Consider real representations V of G. The inclusion

e_{V} : S^{0} ⟶ S^{V}

induces a map

e_{V}^{∗} : R_{G}^{∗}(S^{V}) ⟶ R_{G}^{∗}(S^{0}) = R_{G}^{∗}.

Let 1 ∈ R_{G}^{0} be the identity element; it is represented by the unit η : S_{G} ⟶ R_{G}. Its suspension Σ^{V} 1 is an element of R_{G}^{V}(S^{V}), and we define

e(V) = e_{V}^{∗} (Σ^{V} 1) ∈ R_{G}^{V}(S^{0}) = R_{G}^{V}.

We need to be able to shift these classes into integer degrees.

Definition 2.1. The theory R_{G}^{∗} has Thom isomorphisms if, for each complex representation V of G, there is a natural isomorphism

ϕ_{V} : R_{G}^{∗}(X ∧ S^{|V |}) ⟶ R_{G}^{∗}(X ∧ S^{V})

of R_{G}^{∗}-modules, where |V | is the real dimension of V. We may view ϕ_{V} as giving isomorphisms

ϕ_{V} : R_{G}^{W −|V |}(X) ⟶ R_{G}^{W −V} (X)

for W ∈ RO(G). Taking W = V + |V | and X = S^{0}, we define

χ(V) = ϕ_{V} (e(V)) ∈ R_{G}^{|V |}.

Remark 2.2. Let μ(V) = ϕ_{V} (1) ∈ R_{G}^{|V |−V}. Since ϕ_{V} is an isomorphism of R_{G}^{∗}- modules, μ(V) is a unit in R_{G}^{∗}, and we may as well insist that ϕ_{V} (x) = x·μ(V) for all x ∈ R^{∗}(X ∧ S^{|V |}) and all G-spectra X. That is, we take our Thom isomorphisms to be given by right multiplication by Thom classes. In particular, χ(V) = e(V)μ(V).

Remark 2.3. If V contains a trivial representation, so that V^{G} ≠ 0, then e_{V} is null homotopic and therefore e(V) = 0 and χ(V) = 0.

Now let I be a given finitely generated subideal of J_{G}. For H ⊆ G, let r_{H}^{G}(I) denote the resulting subideal res_{H}^{G}(I) · R_{H}^{∗} of J_{H}.

Definition 2.4. Assume that each R_{H}^{∗} has Thom isomorphisms. The ideal I is sufficiently large at H if there is a non-zero complex representation V of H such that V^{H} = 0 and the Euler class χ(V) ∈ R_{H}^{∗} is in the radical \sqrt{r_{H}^{G}(I)}. The ideal I is sufficiently large if it is sufficiently large at all H ⊆ G.

As explained in the introduction, we have a canonical map of R_{G}-modules

κ : EG_{+} ∧ R_{G} ⟶ K(I),

and our goal is to prove that it is an equivalence. The essential point of our strategy is the following result, which reduces the problem to the construction of a sufficiently large finitely generated subideal I of J_{G}.

Theorem 2.5. Assume that R_{H}^{∗} has Thom isomorphisms for all H ⊆ G. If I is a sufficiently large finitely generated subideal of J_{G}, then

κ : EG_{+} ∧ R_{G} ⟶ K(I)

is an equivalence. Therefore,

EG_{+} ∧ M_{G} → Γ_{I}(M_{G}) and (M_{G})_{I}^{∧} → F (EG_{+}, M_{G})

are equivalences for any R_{G}-module M_{G}.

Proof. Let ẼG be the cofiber of the projection EG_{+} ⟶ S^{0}. Then the cofiber of κ is equivalent to ẼG ∧ K(I), and we must prove that this is contractible. Using the transitivity of restriction maps to see that r_{H}^{G}(I) is a large enough subideal of R_{H}^{∗}, we see that the hypotheses of the theorem are inherited by any subgroup. Using the fact that there is no infinite descending chain of compact Lie groups, we see that we may assume that the theorem holds for H ∈ 𝒫, where 𝒫 is the family of proper subgroups of G. Thus ẼH ∧ K(r_{H}^{G}(I)) is contractible for H ∈ 𝒫, and clearly

(ẼG ∧ K(I))|_{H} = ẼH ∧ K(r_{H}^{G}(I)).

From here, the proof is just like that of [16, 7.5]. We have that

G/H_{+} ∧ ẼG ∧ K(I)

is contractible for H ∈ 𝒫. We take Ẽ𝒫 to be the colimit of spheres S^{V}, where V runs through the complex representations such that V^{G} = {0} in a complete complex universe U. Since Ẽ𝒫/S^{0} is triangulable as a G-CW complex whose cells have proper orbit type, the induction hypothesis implies that

(Ẽ𝒫/S^{0}) ∧ ẼG ∧ K(I)

is contractible. By the cofiber sequence S^{0} ⟶ Ẽ𝒫 ⟶ Ẽ𝒫/S^{0} and the fact that Ẽ𝒫 ∧ S^{0} ⟶ Ẽ𝒫 ∧ ẼG is an equivalence, it suffices to show that Ẽ𝒫 ∧ K(I) is contractible. For any R_{G}-module M_{G}, the construction of Ẽ𝒫 and our translation of Euler classes to integer gradings imply directly that π_{∗}^{G}(Ẽ𝒫 ∧M_{G}) is the localization π_{∗}^{G}(M_{G})[{χ(V)^{−1}}] obtained by inverting the Euler classes χ(V) (see [16, 3.20]). With M_{G} = K(I), we have a spectral sequence that converges from the local cohomology groups H_{I}^{∗}(R_{∗}^{G}) to π_{∗}^{G}(K(I)). Localizing by inverting the χ(V), we obtain a spectral sequence that converges from the localization H_{I}^{∗}(R_{∗}^{G})[{χ(V)^{−1}}] to π_{∗}^{G}(Ẽ𝒫 ∧K(I)). Since the local cohomology of a ring at an ideal vanishes when localized by inverting an element in that ideal, our assumption that I is sufficiently large at G ensures that the E^{2}-term of our spectral sequence is zero.

3. Constructing sufficiently large finitely generated ideals

The idea is to obtain enough elements of J_{G} to give a good approximation to it. For those groups G that act freely on a finite product of spheres, such as the finite pgroups, there are enough representations that we can simply use finitely many Euler classes χ(V). However, even for general finite groups, the ideal generated by the χ(V) is usually not sufficiently large. We need to add in other elements, and we shall do so by exploiting norm, or “multiplicative transfer”, maps that are analogous to Evens’ norm maps in the cohomology of groups [11, Ch. 5]. This is a new construction in equivariant stable homotopy theory and should have other applications. However, we explain how our strategy of proof works before stating what it means for a theory to have such norm maps.

We assume given a toral group G, namely an extension of the form

1 ⟶ T ⟶ G ⟶ F ⟶ 1,

where T is a torus and F is a finite group. Most of our work will be necessary even when T is trivial and we are dealing only with the finite group F.

Theorem 3.1. If G is toral and the R_{H}^{∗} for H ⊆ G admit norm maps and natural Thom isomorphisms, then J_{G} contains a sufficiently large finitely generated subideal.

The proof of the theorem depends on two lemmas, the first of which does not require norm maps. The second requires both norm maps and the as yet unstated naturality condition on Thom isomorphisms, which will be specified in its proof. We will explain the proofs of the lemmas after showing that they imply the theorem. For H ⊆ G, we write

res_{H}^{G} : R(G) ⟶ R(H)

for the restriction homomorphism. When H has finite index in G, we write

ind_{H}^{G} : R(H) ⟶ R(G)

for the induction homomorphism. Recall that ind_{H}^{G}V = ℂ[G] ⊗_{ℂ[H]} V.

Lemma 3.2. There are non-zero complex representations V_{1}, ⋯, V_{s} of T such that T acts freely on the product of the unit spheres of the representations res_{T}^{G}ind_{T}^{G}V_{i}.

Lemma 3.3. Let F′ be a subgroup of F with inverse image G′ in G. There is an element ξ(F′) of J_{G} such that

res_{G′}^{G}(ξ(F′)) = χ(V′)^{w′},

where V′ is the reduced regular complex representation of F′ regarded by pullback as a representation of G′ and w′ is the order of W G′ = NG′/G′.

Proof of Theorem 3.1. We claim that the ideal

I = (χ(ind_{T}^{G}V_{1}), ⋯, χ(ind_{T}^{G}V_{s})) + (ξ(F′)|F′ ⊆ F)

is sufficiently large.

If H is a subgroup of G that intersects T non-trivially, then, by Lemma 3.2, (res_{T}^{G}ind_{T}^{G}V_{i})^{H∩T} = {0} for some i and therefore (ind_{T}^{G}V_{i})^{H} = {0}. Since

χ(res_{H}^{G}ind_{T}^{G}V_{i}) = res_{H}^{G}(χ(ind_{T}^{G}V_{i})) ∈ r_{H}^{G}(I),

this shows that I is sufficiently large at H in this case.

If H is a subgroup of G that intersects T trivially, as is always the case when G is finite, then H maps isomorphically to its image F′ in F. If G′ is the inverse image of F′ in G and V′ is the reduced regular complex representation of F′ regarded as a representation of G′, then res_{H}^{G′}(V′) is the reduced regular complex representation of H and (res_{H}^{G′}(V′))^{H} = 0. By Lemma 3.3, we have res_{G′}^{G}(ξ(F′)) = χ(V′)^{w′} and therefore

χ(res_{H}^{G′}(V′))^{w′} = res_{H}^{G′}(χ(V′)^{w′}) = res_{H}^{G′}res_{G′}^{G}(ξ(F′)) = res_{H}^{G}(ξ(F′)) ∈ r_{H}^{G}(I).

This shows that I is sufficiently large at H in this case.

We must consider conjugation homomorphisms in the proofs of the lemmas. For g ∈ G and H ⊆ G, let ^{g}H = gHg^{−1} and c_{g} : ^{g}H ⟶ H be the conjugation isomorphism. For a representation V of H, let ^{g}V be the pullback of V along c_{g}.

Proof of Lemma 3.2. Let us say that a representation V of T detects a subgroup H if V ≠ 0 but V^{H} = 0. Then Lemma 3.2 states that we can find finitely many representations V_{i} such that every subgroup of T is detected by one of the induced representations ind_{T}^{G}V_{i}.

Consider the irreducible representations V of T. They may be viewed as elements of T^{∗} = Hom(T, S^{1}). Clearly V detects H unless H ⊆ ker(V). It is clear from the definition of induction that

res_{T}^{G}ind_{T}^{G}V = ⨁_{f∈F} ^{f}V .

Thus H is detected by ind_{T}^{G}V if and only if it is detected by all of the conjugate representations ^{f}V, that is, if and only if, for all f ∈ F, H is not in the kernel of ^{f}V. Note that ker(^{f}V) = ^{f}(ker V).

Thus, for any list of irreducible representations V_{1}, ⋯, V_{q}, if K_{i} is the kernel of V_{i}, then each subgroup not detected by the V_{i} is a subgroup of

f_{1}K_{1} ∩ ⋯ ∩^{f_{q}}K_{q}

for some list of elements f_{i} of F.

Let T have rank r. Proceeding inductively, we choose irreducible representations V_{1}, ⋯, V_{r} such that, for 1 ≤ q ≤ r, each displayed intersection of conjugated kernels has rank r − q. We begin the induction by choosing any V_{1}. Certainly each ^{f_{1}}K_{1} has rank r − 1. Assume that V_{1}, ⋯, V_{q} have been chosen, where q < r. Thinking on the Lie algebra level, and noting that conjugations induce translations of Lie algebras, we see that it suffices to choose the kernel K_{q+1} of V_{q+1} so that none of the (r − q)-dimensional subspaces

f_{1}(LK_{1}) ∩ ⋯ ∩ f_{q}(LK_{q})

of LT is contained in any f_{q+1}(LK_{q+1}). Translating by f_{q+1}^{−1}, we see that each such condition excludes an (r−q)-dimensional subspace of LT from lying in LK_{q+1}. Dually, consider the finitely many q-dimensional subspaces

{α | α(f_{q+1}^{−1}(f_{1}(LK_{1}) ∩ ⋯ ∩ f_{q}(LK_{q}))) = 0} ⊂ (LT)^{∗}.

Since q < r, the set theoretic union of these subspaces cannot be dense in (LT)^{∗}. Therefore we can choose β ∈ (LT)^{∗} such that β is in none of these subspaces and such that the kernel of β has rational basis with respect to the lattice on LT given by the kernel of the exponential. The rationality condition ensures that this kernel is the Lie algebra of the kernel K_{q+1} of a representation V_{q+1} : T ⟶ S^{1} whose induced map LT ⟶ ℝ of Lie algebras is β. At the rth stage, all of the intersections

f_{1}K_{1} ∩ ⋯ ∩^{f_{r}}K_{r}

have dimension zero and are therefore finite. To detect the finitely many subgroups in these finitely many intersections, we need only detect their nonidentity elements g. However, if V_{g} is a representation whose kernel K_{g} does not contain all of the conjugates ^{f}g, then ind_{T}^{G}V_{g} detects g, so this is easily done.

For H ⊆ G, we have a natural restriction homomorphism

res_{H}^{G} : R_{∗}^{G}(X) ⟶ R_{∗}^{H}(X)

on based G-spaces X. For g ∈ G, we also have a natural isomorphism

c_{g} : R_{∗}^{H}(X) ⟶ R_{∗}^{^{g}H}(^{g}X),

where X is a based H-space and ^{g}X denotes X regarded as a ^{g}H-space by pullback along c_{g}. To give content to the proof of Lemma 3.3, we must explain our hypothesis that R_{∗}^{G} also has norm maps. We give a crude and perhaps unilluminating definition that prescribes exactly what we shall use in the proof.

Definition 3.4. We say that R_{∗}^{G} has norm maps if, for a subgroup H of finite index n in G and an element y ∈ R_{−r}^{H}, where r ≥ 0 is even, there is an element

\overline{norm}_{H}^{G}(1 + y) ∈ ∑_{i=0}^{n} R_{−ri}^{G}

that satisfies the following properties; here 1 = 1_{H} ∈ R_{0}^{H} denotes the identity element. (i) \overline{norm}_{G}^{G}(1 + y) = 1 + y. (ii) \overline{norm}_{H}^{G}(1) = 1. (iii) [The double coset formula]

res_{K}^{G} \overline{norm}_{H}^{G} (1 + y) = ∏_{g} \overline{norm}_{^{g}H∩K}^{K} res_{^{g}H∩K}^{^{g}H} c_{g}(1 + y),

where K is any subgroup of G and {g} runs through a set of double coset representatives for K\G/H.

Proof of Lemma 3.3. Since the restriction of the reduced regular representation of F′ to any proper subgroup contains a trivial representation, the restriction of χ(V′) ∈ R_{G′}^{∗} to a subgroup that maps to a proper subgroup of F′ is zero. In R_{G′}^{∗}, the double coset formula gives

res_{G′}^{G} \overline{norm}_{G′}^{G} (1 + χ(V′)) = ∏_{g} \overline{norm}_{^{g}G′∩G′}^{G′} res_{^{g}G′∩G′}^{^{g}G′} c_{g}(1 + χ(V′)), (3.5)

where g runs through a set of double coset representatives for G′\G/G′. It is clear that c_{g}(e(V)) = e(^{g}V) for any subgroup H of G and representation V of H. We require that our Thom classes satisfy the naturality condition

μ(^{g}V) = c_{g}(μ(V)) ∈ R_{H}^{|V |−^{g}V} . (3.6)

Taking V′ as in the statement of the lemma, this ensures that

c_{g}(1 + χ(V′)) = 1 + χ(^{g}V′).

Here ^{g}V′ is the reduced regular representation of ^{g}G′. Clearly ^{g}G′ ∩ G′ is the inverse image in G of ^{g}F′ ∩ F′. If ^{g}F′ ∩ F′ is a proper subgroup of F′, then the restriction of χ(V′) to ^{g}G′ ∩ G′ is zero. Therefore all terms in the product on the right side of (3.5) are 1 except for those that are indexed on elements g ∈ NG′. There is one such g for each element of W G′ = NG′/G′, and the term in the product that is indexed by each such g is just 1 + χ(V′). Therefore (3.5) reduces to

res_{G′}^{G} \overline{norm}_{G′}^{G}(1 + χ(V′)) = (1 + χ(V′))^{w′}. (3.7)

If V′ has real dimension r, then the summand of (1+χ(V′))^{w′} in degree rw′ is χ(V′)^{w′}. Since res_{G′}^{G} preserves the grading, we may take ξ(F′) to be the summand of degree rw′ in \overline{norm}_{G′}^{G}(1 + χ(V′)).

4. The idea and properties of norm maps

We give an intuitive idea of the construction, leaving details and rigor to later sections. Let H be a subgroup of finite index n in a compact Lie group G. For based H-spaces X, we can give the smash power X^{n} an action of G. Intuitively, this is done in exactly the same way that one induces up a representation of H to a representation of V, and the analogy will guide much of our work.

To begin with, the norm map will be a natural function

norm_{H}^{G} : R_{0}^{H}(X) ⟶ R_{0}^{G}(X^{n}). (4.1)

Norm maps \overline{norm}_{H}^{G} in the sense of Definition 3.4 will be obtained by taking X to be the wedge S^{0} ∨ S^{r}, studying the decomposition of X^{n} into wedge summands of G-spaces described in terms of representations, and using Thom isomorphisms to translate the result to integer gradings. The norm map norm_{H}^{G} will satisfy the following properties.

(4.2) norm_{G}^{G} is the identity function.

(4.3) norm_{H}^{G}(1_{H}) = 1_{G}, where 1_{H} ∈ R_{0}^{H}(S^{0}) is the identity element.

norm_{H}^{G}(xy) = norm_{H}^{G}(x)norm_{H}^{G}(y) if x ∈ R_{0}^{H}(X) and y ∈ R_{0}^{H}(Y). (4.4)

Here the product xy on the left is defined by use of the evident map

R_{0}^{H}(X) ⊗ R_{0}^{H}(Y) ⟶ R_{0}^{H}(X ∧ Y) (4.5)

and similarly on the right, where we must also use the isomorphism

R_{0}^{G}(X^{n} ∧ Y^{n}) ≅ R_{0}^{G}((X ∧ Y)^{n}).

Its most important property will be the double coset formula

res_{K}^{G} norm_{H}^{G}(x) = ∏_{g} norm_{^{g}H∩K}^{K} res_{^{g}H∩K}^{^{g}H} c_{g}(x), (4.6)

where K is any subgroup of G and {g} runs through a set of double coset representatives for K\G/H. Here, if ^{g}H ∩ K has index n(g) in ^{g}H, then n = ∑ n(g) and the product on the right is defined by use of the evident map

⨂_{g} R_{0}^{K}(X^{n(g)}) ⟶ R_{0}^{K}(X^{n}). (4.7)

An element of R_{0}^{H}(X) is represented by an H-map x : S_{G} ⟶ R_{G} ∧ X. There is no difficulty in using the product on R_{G} to produce an H-map

S_{G} ≅ (S_{G})^{n}⟶^{x^{n}}(R_{G} ∧ X)^{n} ≅ (R_{G})^{n} ∧ X^{n} ⟶ R_{G} ∧ X^{n}.

The essential point of our construction is that this may be done in such a way as to produce a G-map: this will be norm_{H}^{G}(x). Here and later, all powers are understood to be smash powers.

This is the basic idea, but carrying it out entails several difficulties. Since our group actions involve permutations of smash powers, we cannot hope to control equivariance unless we are using a smash product that is strictly associative and commutative and a multiplication on R_{G} that is strictly associative and commutative. Moreover, our G-actions come by restriction of actions of wreath products Σ_{n} ∫ H, and it turns out to be essential to work with (Σ_{n} ∫ H)-spectra. If we start just with a G-spectrum R_{G}, then it is not clear how to proceed. Similarly, to make our spectra precise, we must specify appropriate universes on which to index them, and we find that the norm map acts nontrivially on universes.

We explain why the last phenomenon occurs and at the same time explain the perhaps surprising restriction of our initial description of the norm map to degree zero. The point is that, for q ≥ 0 at least, the norm gives a map

R_{−q}^{H}(X) = R_{0}^{H}(X ∧ S^{q}) ⟶ R_{0}^{G}((X ∧ S^{q})^{n}) ≅ R_{0}^{G}(X^{n} ∧ (S^{q})^{n}).

However, the sphere (S^{q})^{n} is twisted: in fact, it is the sphere S^{V} associated to the representation V = ind_{H}^{G}(q), where q denotes the q-dimensional trivial representation of H. Thus the target is R_{−V}^{G} (X^{n}), which is not in the integer graded part of the theory. Of course, if q is even and R_{G}^{∗} has Thom isomorphisms, then we can use them to translate to integer degrees and thus to obtain the translated norm map

R_{−q}^{H}(X) ⟶ R_{−nq}^{G}(X^{n}).

An elaboration of this idea to sums of elements will give the modified norm maps \overline{norm}_{H}^{G}. We shall explain this elaboration of the definition after making sense of the geometric construction of the norm map norm_{H}^{G} and proving its double coset formula.

5. Global I_{∗}-functors with smash product

To deal with the difficulties that we have indicated, we shall assume that R_{G} arises from a 𝒢ℐ_{∗}-FSP, where FSP stands for functor with smash product. This is a global version of the notion of an ℐ_{∗}-prefunctor that was introduced in [24, IV.2.1]. The earlier notion was defined nonequivariantly but transcribes directly to a definition in which we restrict attention to a given compact Lie group G acting on everything in sight. The adjective “global” means that we allow G to range through all compact Lie groups G (or through all G in some suitably restricted class). Let 𝒢 denote the category of compact Lie groups and their homomorphisms; for the purposes of the present theory, it would suffice to restrict attention to injective homomorphisms, but our examples are defined on the larger category.

Definition 5.1. Define the global category 𝒢𝒯 of equivariant based spaces to have objects (G, X), where G is a compact Lie group and X is a based G-space. The morphisms are the pairs

(α, f) : (G, X) ⟶ (G′, X′)

where α : G ⟶ G′ is a homomorphism of Lie groups and f : X ⟶ X′ is an α-equivariant map, in the sense that f(gx) = α(g)f(x) for all x ∈ X and g ∈ G. Topologize the set of maps (G, X) ⟶ (G′, X′) as a subspace of the evident product of mapping spaces and observe that composition is continuous.

Definition 5.2. Define the global category 𝒢ℐ_{∗} of finite dimensional equivariant inner product spaces to have objects (G, V), where G is a compact Lie group and V is a finite dimensional inner product space with an action of G through linear isometries. The morphisms are the pairs

(α, f) : (G, V) ⟶ (G′, V′)

where α : G ⟶ G′ is a homomorphism and f : V ⟶ V′ is an α-equivariant linear isomorphism.

We often find it convenient to work with complex rather than real inner product spaces. Our definitions apply equally well under either interpretation.

Observe that each morphism (α, f) in 𝒢ℐ_{∗} factors as a composite

(G, V)⟶^{(id,f)}(G, W)⟶^{(α,id)}(H, W),

where G acts through α on W. We have a similar factorization of morphisms in 𝒢𝒯 . Observe too that we have forgetful functors 𝒢ℐ_{∗} ⟶ 𝒢 and 𝒢𝒯 ⟶ 𝒢. We shall be interested in functors 𝒢ℐ_{∗} ⟶ 𝒢𝒯 over 𝒢, that is, functors that preserve the group coordinate. For example, one-point compactification of inner product spaces gives such a functor, which we shall denote by S^{•}. As in this example, the space coordinate of our functors will be the identity on morphisms of the form (α, id). For these reasons, we shall usually omit the group coordinate from the notation for functors.

Definition 5.3. A 𝒢ℐ_{∗}-functor is a continuous functor T : 𝒢ℐ_{∗} ⟶ 𝒢𝒯 over 𝒢, written (G, T V) on objects (G, V), such that

T (α, id) = (α, id) : (G, T W) ⟶ (H, T W)

for a representation W of H and a homomorphism α : G ⟶ H.

The following observation will be the germ of the definition of the norm map.

Lemma 5.4. Let A = Aut(G, V) be the group of automorphisms of (G, V) in the category 𝒢ℐ_{∗}. For any 𝒢ℐ_{∗}-functor T, the group A ⋉ G acts on the space T V.

Proof. An element of A is a map (α, f) : (G, V) ⟶ (G, V) such that f(gv) = α(g)v for all v ∈ V and g ∈ G and α is an isomorphism. The semi-direct product A ⋉ G is the set A × G with the multiplication

((α, f), g))((β, ℓ), h) = ((αβ, fℓ), β^{−1}(g)h),

and G is contained in A ⋉ G as the subgroup of elements (id, g). The action of A ⋉ G on T V is specified by

((α, f), g)x = T (α, f)(gx).

This does define an action since functoriality and equivariance imply that

T (α, f)(gT (β, ℓ)(hx)) = α(g)αβ(h)T (αβ, fℓ)(x) = T (αβ, fℓ)(β^{−1}(g)hx).

Define the direct sum functor ⊕ : 𝒢ℐ_{∗} × 𝒢ℐ_{∗} ⟶ 𝒢ℐ_{∗} by

(G, V) ⊕ (H, W) = (G × H, V ⊕ W).

Define the smash product functor ∧ : 𝒢𝒯 × 𝒢𝒯 ⟶ 𝒢𝒯 by

(G, X) ∧ (H, Y) = (G × H, X ∧ Y).

These functors lie over the functor × : 𝒢 × 𝒢 ⟶ 𝒢.

Definition 5.5. A 𝒢ℐ_{∗}-FSP is a 𝒢ℐ_{∗}-functor together with a continuous natural unit transformation η : S^{•} ⟶ T of functors 𝒢ℐ_{∗} ⟶ 𝒢𝒯 and a continuous natural product transformation ω : T ∧ T ⟶ T ∘ ⊕ of functors 𝒢ℐ_{∗} × 𝒢ℐ_{∗} ⟶ 𝒢𝒯 such that the composite

T V ≅ T V ∧ S^{0}⟶^{id ∧η}T V ∧ T (0)⟶^{ω}T (V ⊕ 0) ≅ T V

is the identity map and the following unity, associativity, and commutativity diagrams commute:

[figure: S^{V} ∧ S^{W}; ≅; η∧η; T V ∧ T W; ω; S^{V ⊕W}; η; T (V ⊕ W),]

[figure: T V ∧ T W ∧ T Z; id ∧ω; ω∧id; T (V ⊕ W) ∧ T Z; ω; T V ∧ T (W ⊕ Z); ω; T (V ⊕ W ⊕ Z),]

and

[figure: T V ∧ T W; ω; τ; T (V ⊕ W); T (τ); T W ∧ T V; ω; T (W ⊕ V).]

Actually, the notion that we have just defined is that of a commutative 𝒢ℐ_{∗}-FSP; For the more general non-commutative notion, the commutativity diagram must be replaced by a weaker centrality of unit diagram. Observe that the space coordinate of each map T (α, f) is necessarily a homeomorphism since (α, f) = (α, id) ∘ (id, f) and f is an isomorphism. We record the following observation for later use.

Remark 5.6. For objects (H_{i}, V_{i}) of 𝒢ℐ_{∗}, 1 ≤ i ≤ n, and a permutation σ ∈ Σ_{n}, the associativity and commutativity diagrams imply the following commutative diagram:

[figure: T V_{1} ∧ ⋯ ∧ T V_{n}; σ; ω; T (V_{1} ⊕ ⋯ ⊕ V_{n}); T (σ); T V_{σ^{−1}(1)} ∧ ⋯ ∧ T V_{σ^{−1}(n)}; ω; T (V_{σ^{−1}(1)} ⊕ ⋯ ⊕ V_{σ^{−1}(n)}).]

Here and in the commutativity axiom, we have suppressed the group coordinate from the notation and T (σ) means T (σ, σ): we must permute the groups in the same way that we permute the inner product spaces.

Example 5.7. The sphere functor S^{•} is a 𝒢ℐ_{∗}-FSP with unit given by the identity maps of the S^{V} and product given by the isomorphisms S^{V} ∧ S^{W} ≅ S^{V ⊕W}. For any 𝒢ℐ_{∗}-FSP T, the unit η : S^{•} ⟶ T is a map of 𝒢ℐ_{∗}-FSP’s.

Example 5.8. Let dim V = n and define T V to be the one-point compactification of the canonical n-plane bundle EV over the Grassmann manifold Gr_{n}(V ⊕ V). An action of G on V induces an action of G that makes EV a G-bundle and TV a based G-space. Take V = V ⊕ {0} as a canonical basepoint in Gr_{n}(V ⊕ V). The inclusion of the fiber over the basepoint induces a map η : S^{V} ⟶ T V. The canonical bundle map EV ⊕ EW ⟶ E(V ⊕ W) induces a map ω : T V ∧ T W ⟶ T (V ⊕ W). With the evident definition of T on morphisms, T is a 𝒢ℐ_{∗}-functor. Actually, there are two variants: we write T O when dealing with real inner product spaces and T U when dealing with complex inner product spaces.

6. The passage to spectra

It is useful to regard a 𝒢ℐ_{∗}-FSP as a 𝒢ℐ_{∗}-prespectrum with additional structure.

Definition 6.1. A 𝒢ℐ_{∗}-prespectrum is a 𝒢ℐ_{∗}-functor T : 𝒢ℐ_{∗} ⟶ 𝒢𝒯 together with a continuous natural transformation σ : T ∧ S^{•} ⟶ T ∘ ⊕ of functors 𝒢ℐ_{∗} × 𝒢ℐ_{∗} ⟶ 𝒯 such that the composites

T V ≅ T V ∧ S^{0}⟶^{σ}T (V ⊕ 0) ≅ T V

are identity maps and each of the following diagrams commutes:

[figure: T V ∧ S^{W} ∧ S^{Z}; ≅; σ∧id; T (V ⊕ W) ∧ S^{Z}; σ; T V ∧ S^{W ⊕Z}; σ; T (V ⊕ W ⊕ Z).]

We say that a 𝒢ℐ_{∗}-prespectrum is an inclusion 𝒢ℐ_{∗}-prespectrum if each adjoint map

σ̃ : T V ⟶ F (S^{W} , T (V ⊕ W))

is an inclusion.

Lemma 6.2. If T is a 𝒢ℐ_{∗}-FSP, then T is a 𝒢ℐ_{∗}-prespectrum with respect to the composite maps

σ : T V ∧ S^{W} ⟶^{id ∧η}T V ∧ T W ⟶^{ω}T (V ⊕ W).

Now fix a group G and a G-universe U, namely a countably infinite dimensional inner product space that contains a trivial representation and contains each of its finite dimensional representations infinitely often. A G-prespectrum indexed on U consists of based G-spaces T V for finite dimensional inner product spaces V ⊂ U and a transitive system of structure G-maps σ : Σ^{W −V} T V ⟶ T W for V ⊂ W, where W − V is the orthogonal complement of V in W. A spectrum E is a prespectrum whose adjoint structure maps EV ⟶ Ω^{W −V} EW are homeomorphisms. There is a spectrification functor L from prespectra to spectra that is left adjoint to the evident forgetful functor. See [21, 17, 26] for the development of equivariant stable homotopy theory from this starting point. It is evident that a 𝒢ℐ_{∗}-prespectrum restricts to a G-prespectrum indexed on U for every G and U.

Notations 6.3. Let T_{(G,U)} denote the G-prespectrum indexed on U associated to a 𝒢ℐ_{∗}-FSP T. Write R_{(G,U)} for the spectrum LT_{(G,U)} associated to T_{(G,U)}.

Let ℒ (j) be the G-space of linear isometries U^{j} ⟶ U, with G acting by conjugation. As discussed in the cited sources and [8], ℒ is a G-operad and is an E_{∞} G-operad when U contains all irreducible representations of G; we then say that U is complete. There is a notion of an ℒ -prespectrum [24, IV.1.1] (amended slightly in [21, VII.2.4-2.6]). Exactly as in [24, IV.2.2], T_{(G,U)} is an ℒ -prespectrum. The essential point is that if f : U^{j} ⟶ U is a linear isometry and V_{i} are indexing spaces in U, then we have maps

ξ_{j}(f) : T V_{1} ∧ ⋯ T V_{j}⟶^{ω}T (V_{1} ⊕ ⋯ ⊕ V_{j})⟶^{T f}T f(V_{1} ⊕ ⋯ ⊕ V_{j}).

The notion of an ℒ -prespectrum is defined in terms of just such maps.

The notion of an ℒ -spectrum E is defined more conceptually in terms of maps

ℒ (j) ⋉ E^{j} ⟶ E.

(In fact, the twisted half-smash product ⋉ was not known when [24] was written.) However, by [24, IV.1.6] and, in current terms, [21, VII§2], the functor L converts ℒ -prespectra to ℒ -spectra. We conclude that, for every G and every complete Guniverse U, R_{(G,U)} is an ℒ -spectrum and thus an E_{∞} ring G-spectrum. Moreover, as explained in [9, 8], ℒ -spectra functorially determine weakly equivalent commutative S_{G}-algebras.

While the constructions of [16, 17] on which our work is based depend on the fact that the G-spectra we are working with admit S_{G}-algebra structures, we will not need to make explicit use of these structures in our study of the norm map.

Remark 6.4. Let T be a global 𝒢 ℐ_{∗}-functor, choose a complete G-universe U_{G} for each G, and consider the G-spectra R_{(G,U_{G})} as G varies. It is a routine exercise to verify that the collection {R_{(G,U_{G})}} defines a 𝒢 -spectrum, in the sense defined in [21, II.8.5]. In particular, it follows immediately from [21, II.8.6 and II.8.7] that R_{(G,U_{G})} is a split G-spectrum for each G. This shows that our Thom G-spectra are split, as stated in the introduction.

Remark 6.5. If U is a complete complex G-universe, we may regard it as a complete real G-universe by neglect of structure. This gives two variants of all definitions in sight, one in which we restrict attention to complex finite dimensional inner product spaces V in U and the other in which we allow all real finite dimensional inner product spaces. The resulting categories of G-spectra are canonically equivalent [21, I.2.4] because the adjoint structure maps of G-spectra are homeomorphisms and complex inner product spaces are cofinal among real ones. We always understand G-spectra to be indexed on the real inner product spaces, but G-prespectra can be taken in whichever sense is appropriate to the context.

7. Wreath products and the definition of the norm map

Recall that the wreath product Σ_{n} ∫ H is the set Σ_{n} × H^{n} with the product

(σ, h_{1}, …, h_{n})(τ, h′_{1}, …, h′_{n}) = (στ, h_{τ1}h′_{1}, ⋯, h_{τn}h′_{n}). (7.1)

We have the following evident actions of this group. We display them explicitly because of their centrality in our work.

Lemma 7.2. If V is a representation of H, then the sum V^{n} of n copies of V is a representation of Σ_{n} ∫ H with action given by

(σ, h_{1}, …, h_{n})(v_{1}, …, v_{n}) = (h_{σ^{−1}(1)}v_{σ^{−1}(1)}, …, h_{σ^{−1}(n)}v_{σ^{−1}(n)}).

Lemma 7.3. If X is a based H-space, then the smash power X^{n} is a (Σ_{n} ∫ H)-space with action given by

(σ, h_{1}, …, h_{n})(x_{1} ∧ … ∧ x_{n}) = h_{σ^{−1}(1)}x_{σ^{−1}(1)} ∧ … ∧ h_{σ^{−1}(n)}x_{σ^{−1}(n)}.

This leads to the following crucial observation.

Proposition 7.4. Let T be a 𝒢ℐ_{∗}-FSP. For an H-representation V, (T V)^{n} and T (V^{n}) are Σ_{n} ∫ H-spaces and the map

ω : (T V)^{n} ⟶ T (V^{n})

is (Σ_{n} ∫ H)-equivariant. If U is an H-universe, then U^{n} is a (Σ_{n} ∫ H)-universe and the maps ω define a map of (Σ_{n} ∫ H)-prespectra indexed on U^{n}

ω : (T_{(H,U)})^{n} ⟶ T_{(Σ_{n} ∫ H,U^{n})},

where (T_{(H,U)})^{n} is the nth external smash power of T_{(H,U)}. If T = S^{•}, then ω is an isomorphism of prespectra. If n = ∑ n(i), where n(i) ≥ 1 and 1 ≤ i ≤ m, then the following diagram of prespectra commutes:

[figure: ⋀_{i=1}^{m}(T_{(H,U)})^{n(i)}; ∧_{i}ω; ⋀_{i=1}^{m} T_{(Σ_{n(i)} ∫ H,U^{n(i)})}; ω; (T_{(H,U)})^{n}; ω; T_{(Σ_{n} ∫ H,U^{n})}.]

Proof. For σ ∈ Σ_{n}, (σ, σ) is an automorphism of (H^{n}, V^{n}) and thus Σ_{n} maps to A = Aut(H^{n}, V^{n}). This induces a map from Σ_{n} ∫ H to A ⋉ H^{n} (which is an injection unless V = {0}). Now Lemma 5.4 restricts to give the action of Σ_{n} ∫ H on T (V^{n}). We see that ω is (Σ_{n} ∫ H)-equivariant by taking each (H_{i}, V_{i}) to be (H, V) in the diagram of Remark 5.6. We may index our prespectra on the cofinal family of indexing spaces of the form V^{n} in U^{n} , and the external smash product has V^{n}th space (T V)^{n}. The prespectrum level statements are now easily verified from the definition of a 𝒢ℐ_{∗}-FSP.

We shall use the proposition to define the norm map, but we first need a bit of algebra. For the rest of the section, assume given a subgroup H of finite index n in a compact Lie group G. Choose coset representatives t_{1}, t_{2}, ⋯, t_{n} for H in G, taking t_{1} = e, and define the monomial representation

α : G ⟶ Σ_{n}∫ H (7.5)

by the formula

α(γ) = (σ(γ), h_{1}(γ), …, h_{n}(γ)), (7.6)

where σ(γ) and h_{i}(γ) are defined implicitly by the formula

γt_{i} = t_{σ(γ)(i)}h_{i}(γ). (7.7)

Lemma 7.8. The map α is a homomorphism of groups. If α′ is defined with respect to a second choice of coset representatives {t′_{i}}, then α and α′ differ by a conjugation in Σ_{n} ∫ H.

Proof. The first statement holds by the definition (7.1) of the product in Σ_{n} ∫ H and the observation that

(γγ′)t_{i} = γt_{σ(γ′)(i)}h_{i}(γ′) = t_{σ(γ)(σ(γ′)(i))}h_{σ(γ′)(i)}(γ)h_{i}(γ′).

For the second statement, if t′_{i} = t_{i}k_{i}, then

γt′_{i} = t_{σ(γ)(i)}h_{i}(γ)k_{i} = t′_{σ(γ)(i)}k_{σ(γ)(i)}^{−1}h_{i}(γ)k_{i}

and therefore

α′(γ) = (1, k_{1}, …, k_{n})^{−1}α(γ)(1, k_{1}, …, k_{n}).

The homomorphism α is implicitly central to induction in representation theory, as the following lemma explains. Write α^{∗}W for a representation W of Σ_{n} ∫ H regarded as a representation of G by pullback along α.

Lemma 7.9. If V is a representation of H, then α^{∗}V^{n} ≅ ind_{H}^{G}V.

Proof. Recall that ind_{H}^{G}V = ℂ[G] ⊗_{ℂ[H]} V. The isomorphism is given by mapping t_{i} ⊗ v on the left to v in the ith summand on the right, as is dictated by the case i = 1 and equivariance.

Analogously, for a based Σ_{n} ∫ H-space Y, write α^{∗}Y for Y regarded as a G-space by pullback along α. In particular, α^{∗}((S^{V})^{n}) ≅ S^{ind_{H}^{G}V}.

Here, finally, is the definition of the norm map. Recall Notations 6.3.

Definition 7.10. Let T be a 𝒢ℐ_{∗}-FSP, let X be a based H-space, and let U be a complete H-universe. An element x ∈ R_{0}^{H}(X) is given by a map of H-spectra x : S_{(H,U)} ⟶ R_{(H,U)} ∧ X. Define the norm of x to be the element of R_{0}^{G}(α^{∗}X^{n}) given by the pullback along α of the composite map of Σ_{n} ∫ H-spectra indexed on U^{n} displayed in the commutative diagram:

[figure: S_{(Σ_{n}∫ H,U^{n})}; ω^{−1}; (S_{(H,U)})^{n}; x^{n}; (R_{(H,U)} ∧ X)^{n}; ≅; R_{(Σ_{n}∫ H,U^{n})} ∧ X^{n}; (R_{(H,U)})^{n} ∧ X^{n}.; ω∧id]

If we take it as understood that G acts on U^{n} through α, then the composite defining norm_{H}^{G}(x) may be rewritten more simply as

[figure: S_{(G,U^{n})}; ω^{−1}; norm_{H}^{G}(x); (S_{(H,U)})^{n}; x^{n}; (R_{(H,U)} ∧ X)^{n}; ≅; R_{(G,U^{n})} ∧ X^{n}; (R_{(H,U)})^{n} ∧ X^{n}.; ω∧id] (7.11)

Observe that the G-universe U^{n} is complete; for example, if G is finite, this holds because the regular representation of H induces up to the regular representation of G. Strictly speaking, if we start with H-spectra defined in fixed complete H-universes U_{H} for all H, then we must choose an isomorphism U_{G} ≅ U_{H}^{n} to transfer the norm to a map of spectra indexed on U_{G}. This is a standard procedure that must be applied to various of the maps that we shall construct; compare Remark 6.4.

Property (4.2) of the norm is obvious. Property (4.3) is an easy consequence of the unity and associativity diagrams in the definition of a 𝒢ℐ_{∗}-FSP. Property (4.4) also follows easily from the definition of a 𝒢ℐ_{∗}-FSP, once we make precise how to interpret the product (4.5). Thus suppose given H-universes U and U′ and maps x : S_{(H,U)} ⟶ R_{(H,U)} and y : S_{(H,U′)} ⟶ R_{(H,U′)}. For the present purpose, U and U′ could be the same, but we will want to allow them to be different in the next section. We then define xy to be the composite displayed in the diagram

[figure: S_{(G,U⊕U′)}; ω^{−1}; xy; S_{(G,U)} ∧ S_{(G,U′)}; x∧y; R_{(G,U)} ∧ X ∧ R_{(G,U′)} ∧ Y; ≅; R_{(G,U⊕U′)} ∧ X ∧ Y; R_{(G,U)} ∧ R_{(G,U′)} ∧ X ∧ Y; ω∧id] (7.12)

Of course, when U = U′, we can internalize this external multiplication by use of a linear isometry f : U ⊕ U ⟶ U; it is then obvious that it agrees with the standard homotopical definition of such a product. The external form makes the verification of (4.4) transparent.

8. The proof of the double coset formula

The proof of the double coset formula requires a more precise combinatorial discussion of double cosets than we have found in the literature. We again suppose given a subgroup H of finite index n in a compact Lie group G. We fix coset representatives t_{j} and use them to define the monomial representation α, as in the previous section. We suppose given a second subgroup K of G and we choose representatives g_{1}, ⋯, g_{m} for the double cosets K\G/H. We shall choose the g_{i}’s from among the t_{j}’s, in a manner to be specified.

The g_{i} give a decomposition of the finite K-set G/H as

∐_{i=1}^{m} K/^{g_{i}}H ∩ K⟶^{≅} ∐_{i=1}^{m} Kg_{i}H = K\G/H.

Explicitly, the ith component is the isomorphism

K/^{g_{i}}H ∩ K⟶^{≅}Kg_{i}H

that sends k(^{g_{i}}H ∩ K) to kg_{i}H. Let the ith double coset have n(i) elements, define q(i) = n(1) + ⋯ + n(i − 1), and label the g_{i} and t_{j} so that g_{i} = t_{q(i)+1} and the t_{q(i)+r}, 1 ≤ r ≤ n(i), run through the coset representatives of G/H that are in the ith double coset Kg_{i}H. Thus

Kg_{i}H = ∐_{r=1}^{n(i)} t_{q(i)+r}H.

Define

s_{i,r} = t_{q(i)+r}g_{i}^{−1} ∈ K. (8.1)

Thus the s_{i,r} are coset representatives for K/^{g_{i}}H ∩ K that map to our chosen coset representatives t_{q(i)+r} in Kg_{i}H. With this choice, we define homomorphisms

β_{i} : K ⟶ Σ_{n(i)}∫^{g_{i}}H ∩ K (8.2)

by the formula

β_{i}(κ) = (τ_{i}(κ), ℓ_{1}(κ), …, ℓ_{n(i)}(κ)), (8.3)

where τ_{i}(κ) and ℓ_{r}(κ) are defined implicitly by the formula

κs_{i,r} = s_{i,τ_{i}(κ)(r)}ℓ_{r}(κ). (8.4)

Lemma 8.5. With these choices of α and the β_{i}, the permutations σ(κ) for κ ∈ K decompose as block sums σ_{1}(κ) ⊕ ⋯ ⊕ σ_{m}(κ) with σ_{i}(κ) ∈ Σ_{n(i)}, and

σ_{i}(κ) = τ_{i}(κ) and h_{q(i)+r}(κ) = g_{i}^{−1}ℓ_{r}(κ)g_{i}.

Proof. The first statement holds since κ must permute the t_{q(i)+1}, ⋯, t_{q(i)+n(i)} among themselves. For the second statement, we obtain

κt_{q(i)+r}g_{i}^{−1} = t_{q(i)+τ_{i}(κ)(r)}g_{i}^{−1}ℓ_{r}(κ)

by inserting the formula (8.1) into (8.4), while (7.7) gives

κt_{q(i)+r} = t_{q(i)+σ_{i}(κ)(r)}h_{q(i)+r}(κ).

The lemma can be rewritten in terms of homomorphisms, giving a description of the restriction of α to K in terms of the β_{i}.

Lemma 8.6. The restriction of α to K maps into ∏_{i} Σ_{n(i)} ∫ H, and the following diagram commutes:

[figure: G; α; Σ_{n} ∫ H; K; (α_{1},... ,α_{m}); (β_{1},... ,β_{m}); ∏_{i} Σ_{n(i)} ∫ H; ∏_{i} Σ_{n(i)} ∫ ^{g_{i}}H ∩ K,; _{∏_{i} id ∫ c_{g_{i}}}]

where α_{i}(κ) = (σ_{i}(κ), h_{q(i)+1}(κ), …, h_{q(i)+n(i)}(κ)) and c_{g_{i}}(g_{i}hg_{i}^{−1}) = h.

We can now prove (4.6). For a map x : S_{(H,U)} ⟶ R_{(H,U)} ∧ X, res_{K}^{G}norm_{H}^{G}(x) is the composite map of K-spectra indexed on U^{n} that is displayed in the following diagram, where K acts on U^{n} and X^{n} through the restriction of α to K:

[figure: S_{(K,U^{n})}; ω^{−1}; S_{(H,U)}^{n}; x^{n}; (R_{(H,U)} ∧ X)^{n}; ≅; R_{(K,U^{n})} ∧ X^{n}; (R_{(H,U)})^{n} ∧ X^{n}.; ω∧id]

Define ν_{i}(x) to be the composite displayed in the following diagram, in which K acts through α_{i} on U^{n(i)} and X^{n(i)}:

[figure: S_{(K,U^{n(i)})}; ω^{−1}; ν_{i}(x); (S_{(H,U)})^{n(i)}; x^{n(i)}; (R_{(H,U)} ∧ X)^{n(i)}; ≅; R_{(K,U^{n(i)})} ∧ X^{n(i)}; (R_{(H,U)})^{n(i)} ∧ X^{n(i)}.; ω∧id] (8.7)

In view of the transitivity diagram for ω given in Proposition 7.4, applied to both R = LT and S, we see immediately that the following diagram commutes:

[figure: S_{(K,U^{n})}; ω^{−1}; res_{K}^{G}norm_{H}^{G}(x); ⋀_{i=1}^{m} S_{(K,U^{n(i)})}; ∧_{i}ν_{i}(x); ⋀_{i=1}^{m}(R_{(K,U^{n(i)})} ∧ X^{n(i)}); ≅; R_{(K,U^{n})} ∧ X^{n}; (⋀_{i=1}^{m} R_{(K,U^{n(i)})}) ∧ (⋀_{i=1}^{m} X^{n(i)}).; ω∧id]

Comparing with (7.12), we see that precisely such a diagram makes sense of the iterated product (4.7), and we conclude that res_{K}^{G}norm_{H}^{G}(x) is the product of the ν_{i}(x). Abbreviating notation (as in (4.6)), let g = g_{i}. To complete the proof of (4.6), we need only show that

ν_{i}(x) = norm_{^{g}H∩K}^{K} res_{^{g}H∩K}^{^{g}H} c_{g}(x).

This means that ν_{i}(x) coincides with the composite displayed in the following diagram, in which ^{g}X denotes the H-space X regarded as a ^{g}H-space by pullback along c_{g} : ^{g}H ⟶ H, ^{g}U denotes the H-universe U regarded as a ^{g}H-universe by pullback along c_{g}, and K acts through β_{i} on ^{g}U^{n(i)} and ^{g}X^{n(i)}:

[figure: S_{(K,^{g}U^{n(i)})}; ω^{−1}; (S_{(^{g}H∩K,^{g}U)})^{n(i)}; c_{g}(x)^{n(i)}; (R_{(^{g}H∩K,^{g}U)} ∧^{g}X)^{n(i)}; ≅; R_{(K,^{g}U^{n(i)})} ∧^{g}X^{n(i)}; R_{(^{g}H∩K,^{g}U)}^{n(i)} ∧^{g}X^{n(i)}.; ω∧id] (8.8)

It is immediate from the definition of a 𝒢ℐ_{∗}-functor that

R_{(^{g}H,^{g}U)} = ^{g}R_{(H,U)},

where ^{g}R_{(H,U)} denotes R_{(H,U)} regarded as a ^{g}H-spectrum by pullback along c_{g}. The same is true for S, and c_{g}(x) = ^{g}x is just the map x regarded as a^{g}H-map by pullback along c_{g}. By Lemma 8.6,

α_{i} = c_{g} ∘ β_{i} : K ⟶ Σ_{n(i)}∫ H.

Therefore X^{n(i)} regarded as a K-space via α_{i} is identical to ^{g}X^{n(i)} regarded as a Kspace via β_{i} and U^{n(i)} regarded as a K-universe via α_{i} is identical to ^{g}U^{n(i)} regarded as a K-universe via β_{i}. Except that we have used that β_{i} takes values in Σ_{n(i)} ∫ ^{g}H ∩K to restrict the group action in some of the terms of (8.8), we see that the diagrams (8.7) and (8.8) display one and the same map.

9. The norm map on sums and its double coset formula

Consider norm_{H}^{G}(x + y), where x ∈ R_{−q}^{H} and y ∈ R_{−r}^{H} for even integers q ≥ 0 and r ≥ 0. Here we are considering the case X = S^{q} ∨ S^{r} of the norm map that we defined in Definition 7.10. For based H-spaces X and Y, we have

(X ∨ Y)^{n} ≅ ⋁_{i=0}^{n}(Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} X^{i} ∧ Y^{n−i} (9.1)

as Σ_{n} ∫ H-spaces. For any subgroup K ⊆ G, we therefore have

(X ∨ Y)^{n} ≅ ⋁_{i=0}^{n} ⋁_{γ} K ⋉_{^{γ}((Σ_{i}×Σ_{n−i}) ∫ H)∩K} ^{γ}(X^{i} ∧ Y^{n−i}) (9.2)

as K-spaces, where γ runs through a set of double coset representatives for

K \ (Σ_{n} ∫ H) / ((Σ_{i} × Σ_{n−i}) ∫ H).

Taking K = G, we see that, in general, the norm of the sum of elements of R_{0}^{H}(X) and R_{0}^{H}(Y) is an element of

R_{0}^{G}((X ∨ Y)^{n}) = ∑_{i=0}^{n} ∑_{γ} R_{0}^{^{γ}((Σ_{i}×Σ_{n−i}) ∫ H)∩G}(^{γ}(X^{i} ∧ Y^{n−i})). (9.3)

Now return to our elements x ∈ R_{−q}^{H} and y ∈ R_{−r}^{H}. We are thinking of x = 1. In order to obtain the norm of Definition 3.4, we must transform the element norm_{H}^{G}(x + y) to an element of R_{∗}^{G}, and we must do so in a fashion that makes sense of and validates the double coset formula of Definition 3.4. This requires the use of Thom isomorphisms, and by working on the level of Σ_{n} ∫ H as long as possible, it allows us to circumvent any use of the complexities displayed in formulas (9.2) and (9.3). We shall tacitly assume given Thom isomorphisms that satisfy the properties that we use in our proof of (3.4), and we shall justify the assumption in the case of Thom spectra in the next section. That will complete the proof of Theorem 1.3.

Let V_{n} denote ℝ^{n} with its permutation action by Σ_{n} and regard V_{n} as a Σ_{n} ∫ Hrepresentation with trivial action by H. Thus α^{∗}V_{n} ≅ ind_{H}^{G}ℝ. Writing V^{q} for the sum of q copies of V, we have (S^{q})^{n} = S^{V_{n}^{q}} as a Σ_{n} ∫ H-space. Therefore (9.1) gives

(S^{q} ∨ S^{r})^{n} ≅ ⋁_{i=0}^{n}(Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} S^{V_{i}^{q}⊕V_{n−i}^{r}} (9.4)

as Σ_{n} ∫ H-spaces. Define a translated induction map

ψ_{i} : R_{∗}^{Σ_{n} ∫ H}((Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} S^{V_{i}^{q}⊕V_{n−i}^{r}}) ⟶ R_{∗}^{Σ_{n} ∫ H}(S^{qi+r(n−i)})

by commutativity of the following diagram:

[figure: R_{∗}^{Σ_{n} ∫ H}((Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} S^{V_{i}^{q}⊕V_{n−i}^{r}}); ≅; ψ_{i}; R_{∗}^{(Σ_{i}×Σ_{n−i}) ∫ H}(S^{V_{i}^{q}⊕V_{n−i}^{r}}); ϕ_{V{q \atop i}⊕V{r \atop n−i}}; R_{∗}^{Σ_{n} ∫ H}(S^{qi+r(n−i)}); R_{∗}^{(Σ_{i}×Σ_{n−i}) ∫ H}(S^{qi+r(n−i)}),; ind_{(Σ_{i}×Σ_{n−i})∫ H}^{Σn ∫ H}]

(9.5)

where we have written ind_{H}^{G} for the ordinary transfer homomorphism associated to H ⊆ G. In particular, restricting to degree zero, this gives

ψ_{i} : R_{0}^{Σ_{n} ∫ H}((Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} S^{V_{i}^{q}⊕V_{n−i}^{r}}) ⟶ R_{−qi−r(n−i)}^{Σ_{n} ∫ H}.

We continue to write ψ_{i} for this map regarded as a G-map by pullback along α. Observe that V_{i}^{q} and V_{n−i}^{r} are not representations of G, so that we must start with (9.1) and not (9.2) in order for the Thom isomorphisms that we use here to make sense. Note too that we require Thom isomorphisms for Σ_{n} ∫ H and its subgroups, not just for G and its subgroups.

Use of the ψ_{i} allows us to redefine the norm of sums in a ℤ-graded form. We have

norm_{H}^{G}(x + y) ∈ R_{0}^{G}(α^{∗}(S^{q} ∨ S^{r})^{n}).

It is the restriction to G of an element of

R_{0}^{Σ_{n} ∫ H}((S^{q} ∨ S^{r})^{n}) ≅ ∑_{i=0}^{n} R_{0}^{Σ_{n} ∫ H}((Σ_{n} ∫ H) ⋉_{(Σ_{i}×Σ_{n−i}) ∫ H} S^{V_{i}^{q}⊕V_{n−i}^{r}}).

We write norm_{H}^{G}(x + y)_{i} for the component in the ith summand and define

\overline{norm}_{H}^{G}(x + y) = ∑_{i=0}^{n} ψ_{i}(norm_{H}^{G}(x + y)_{i}). (9.6)

Modulo a hypothesis about Thom isomorphisms, the precise meaning of which will be apparent from the proof, the double coset formula is still valid for these modified norm maps.

Proposition 9.7. Assume that, for subgroups J of Σ_{n} ∫ H, the theories R_{J}^{∗} have Thom isomorphisms which commute with restrictions and products. For elements x ∈ R_{−q}^{H} and y ∈ R_{−r}^{H},

res_{K}^{G} \overline{norm}_{H}^{G}(x + y) = ∏_{g} \overline{norm}_{^{g}H∩K}^{K} res_{^{g}H∩K}^{^{g}H} c_{g}(x + y),

where K is any subgroup of G and {g} runs through a set of double coset representatives for K\G/H.

Proof. To simplify the notation, we restrict attention to the case q = 0, which is the case of interest. Since V_{i}^{0} = {0}, Σ_{i} acts trivially on S^{0} = S^{V_{i}^{0}}. Reverse the roles of i and n − i in the notations above and let Σ′_{i} ⊂ Σ_{n} be the subgroup of permutations that fix the first n − i letters. Recall that if ^{g}H ∩ K has index n(g) in ^{g}H, then n = ∑ n(g). Fix an ordering of the g’s and write Σ_{{n(g)}} for ∏_{g} Σ_{n(g)} regarded as a subgroup of Σ_{n}. The left side of the equation is the restriction to K of an element of

∑_{i=0}^{n} R_{−ri}^{Σ_{n} ∫ H}.

The right side is the product over {g} of the restrictions to K of elements of

∑_{a(g)=0}^{n(g)} R_{−ra(g)}^{Σ_{n(g)} ∫ H}.

In view of Lemma 8.6, we see that the relevant products are obtained by adding up restrictions to K of products

\xymatrix{⨂_{g} R_{0}^{Σ_{n(g)} ∫ H}(S^{ra(g)}) \ar[r] & R_{0}^{Σ_{\{n(g)\}} ∫ H}(S^{ri}),}

where 0 ≤ a(g) ≤ n(g) and ∑ a(g) = i. For such a sequence {a(g)}, let Σ′_{{a(g)}} = ∏_{g} Σ′_{a(g)} regarded as a subgroup of Σ′_{i}. The original double coset formula (4.6) made use of the restriction to K of the product

\xymatrix{⨂_{g} R_{0}^{Σ_{n(g)} ∫ H}((S^{0} ∨ S^{r})^{n(g)}) \ar[r] & R_{0}^{Σ_{\{n(g)\}} ∫ H}((S^{0} ∨ S^{r})^{n}).}

Under the wedge decomposition (9.1) and change of groups isomorphisms like those in the top line of (9.5), this product agrees with the sum over sequences {a(G)} of the product maps

\xymatrix{⨂_{g} R_{0}^{Σ′_{a(g)} ∫ H}(S^{V_{a(g)}^{r}}) \ar[r] & R_{0}^{Σ′_{\{a(g)\}} ∫ H}(S^{V_{i}^{r}}).}

We claim that the following diagram commutes for each such sequence {a(g)}, where the left horizontal arrows are products and the right horizontal arrows are restrictions:

[figure: ⨂_{g} R_{0}^{Σ′_{a(g)} ∫ H}(S^{V_{a(g)}^{r}}); ⨂ ϕ_{V{r \atop a(g)}}; R_{0}^{Σ′_{\{a(g)\}} ∫ H}(S^{V_{i}^{r}}); ϕ_{V^{r}_{i}}; R_{0}^{Σ′_{i} ∫ H}(S^{V_{i}^{r}}); ϕ_{V^{r}_{i}}; ⨂_{g} R_{0}^{Σ′_{a(g)} ∫ H}(S^{ra(g)}); ⨂ ind_{Σ{′ \atop a(g)} ∫ H}^{Σ_{n(g)} ∫ H}; R_{0}^{Σ′_{\{a(g)\}} ∫ H}(S^{ri}); ind_{Σ{′ \atop \{a(g)\}} ∫ H}^{Σ_{\{n(g)\}} ∫ H}; R_{0}^{Σ′_{i} ∫ H}(S^{ri}); ind_{Σ′_{i} ∫ H}^{Σn ∫ H}; ⨂_{g} R_{0}^{Σ_{n(g)} ∫ H}(S^{ra(g)}); R_{0}^{Σ_{\{n(g)\}} ∫ H}(S^{ri}); R_{0}^{Σ_{n} ∫ H}(S^{ri})]

The top two squares commute since our Thom isomorphisms commute with products and restriction, and the bottom two squares commute since transfer commutes with products and restriction by [21, IV.4.4 and IV.5.2]. It follows directly that the present version of the double coset formula follows from the original version.

Taking x = 1, we obtain norm maps as specified in Definition 3.4.

10. The Thom classes of Thom spectra

In this section, we write T for the Thom 𝒢 ℐ_{∗}-functor T U of Example 5.8. It is an inclusion 𝒢 -prespectrum by neglect of structure, as in Lemma 6.2. We restrict attention to complex G-universes U and complex inner product spaces, and we think of ℝ^{∞} as the underlying real inner product space of U^{G} ≅ ℂ^{∞}; compare Remark 6.5. For each compact Lie group G and G-universe U, we have the inclusion prespectrum T_{(G,U)}, and we write MU_{(G,U)} for its associated G-spectrum. We have seen in Section 6 that these are E_{∞} ring G-spectra and so determine weakly equivalent S_{(G,U)}-algebras. All that remains to complete the proof of Theorem 1.3 is to discuss the Thom classes

μ(V) ∈ MU_{(G,U)}^{2n−V} ≅ MU_{(G,U)}^{2n}(S^{V}), (10.1)

where V is a complex representation of complex dimension n. We take U to be a complete G-universe, and it suffices to assume that V is a finite dimensional subspace of U.

Let T′_{(G,U)}(V) be the Thom complex associated to the canonical complex n-plane G-bundle over the Grassmannian G_{n}(V ⊕ U) of n-planes in V ⊕ U. For V ⊆ W, let

σ′ : T′_{(G,U)}V ∧ S^{W −V} ⟶ T′_{(G,U)}W

be the map of Thom complexes induced by the evident map from the sum of the canonical n-plane bundle over G_{n}(V ⊕U) and the trivial bundle W −V ⟶ {W − V } to the canonical q-plane bundle over G_{n}(W ⊕ U), where dim(W) = q. The inclusion V ⊕ V ⟶ V ⊕ U induces a G-map T V ⟶ T′_{(G,U)}V, and these maps together define a map of inclusion G-prespectra

i : T_{(G,U)} ⟶ T′_{(G,U)}.

A standard and easy comparison of colimits of homotopy groups shows that the associated map

MU_{(G,U)} ⟶ MU′_{(G,U)}

of G-spectra is a spacewise G-equivalence and thus a weak equivalence. We also have evident unit and product maps

η′ : S^{V} ⟶ T′_{(G,U)}V

and

ω′ : T′_{(G,U)}V ∧ T′_{(G,U)}W ⟶ T′_{(G,U⊕U)}(V ⊕ W)

that are compatible with the unit and product maps of T. Of course, the advantage of T′_{(G,U)} and its associated G-spectrum MU′_{(G,U)} is that the Grassmannians G_{n}(V ⊕ U) are classifying spaces for complex n-plane G-bundles.

The inclusion of V in U may be viewed as a map of G-bundles from the trivial bundle over a point to the universal n-plane bundle over Gr_{n}(ℂ^{n} ⊕ U). On passage to Thom complexes, it gives a map

t(V) : S^{V} ⟶ T′_{(G,U)}ℂ^{n}.

Composing with the natural map to MU′_{(G,U)}ℂ^{n}, we see that t(V) represents an element μ(V) as in (10.1). It is a Thom class, as is standard (e.g. [6] or [5, 2.1]) and can be verified in various ways. Perhaps the simplest is to define

t^{−1}(V) : S^{2n} ⟶ T′_{(G,U)}V

by reversing the roles of V and ℂ^{n}. Composing with the natural map to MU′_{(G,U)}V, we see that t^{−1}(V) represents an element μ^{−1}(V) ∈ MU_{(G,U)}^{V −2n}. Via ω′, the smash product of t(V) and t^{−1}(V) induces a G-map

S^{V +2n} ⟶ T′_{(G,U⊕U)}(V ⊕ ℂ^{n})

that is homotopic to the map obtained by including V ⊕ ℂ^{n} as the base plane. The latter map is part of the unit map η′, and a standard unravelling of definitions shows that μ(V) and μ^{−1}(V) are inverse units of the RO(G)-graded ring MU′_{G}^{∗} ≅ MU_{G}^{∗}.

This completes the verification that each MU_{G}^{∗} has Thom classes. To verify the naturality condition μ(^{g}V) = c_{g}(μ(V)) of (3.6), where V is an H-space in an Huniverse U and g ∈ G, write ^{g}U for the universe U regarded as a ^{g}H universe by pullback along c_{g} :^{g} H ⟶ H, and observe that

t(^{g}V) = c_{g}^{∗}(t(V)) : S^{^{g}V} ⟶ T′_{(^{g}H,^{g}U)}ℂ^{n}. (10.2)

To complete the verification that each MU_{G}^{∗} has norm maps, as in the previous section, we must verify that our Thom classes commute with products and restrictions. For restrictions, there is nothing to prove: we consider a G-space V in a G-universe U and see that

t(V)|_{H} = t(V |_{H}) : S^{V} ⟶ T′_{(H,U)}ℂ^{n}, hence μ(V)|_{H} = μ(V |_{H}). (10.3)

For products, looking at the diagram in the proof of Proposition 9.8 and recalling the external form of our basic products displayed in (7.12), we see that the following immediate observation suffices.

Lemma 10.4. Let U be a G-universe and U′ be a G′-universe, and let V ⊂ U and V′ ⊂ U′. Then the following diagram commutes:

[figure: S^{V} ∧ S^{V′}; t(V)∧t(V′); ≅; T′_{(G,U)}V ∧ T′_{(G′,U′)}V′; ω′; S^{V ⊕V′}; t(V ⊕V′); T′_{(G×G′,U⊕U′)}(V ⊕ V′).]

Therefore μ(V)μ(V′) = μ(V ⊕ V′).

References

1. M.F. Atiyah and G.B. Segal. Equivariant K-theory and completion. J.Diff. Geom. 3 (1969), 1-18.

2. G. Carlsson. Equivariant stable homotopy and Segal’s Burnside ring conjecture. Ann. Math. 120 (1984), 189-224.

3. G. Carlsson. A survey of equivariant stable homotopy theory. Topology 31 (1992), 1-27.

4. G. Comezaña. Old and new results in complex equivariant bordism. Preprint, 1994. (To appear in [25].)

5. S.R. Costenoble. The equivariant Conner-Floyd isomorphism. Trans. Amer. Math. Soc. 304, 801-818.

6. T. tom Dieck. Bordism of G-manifolds and integrality theorems. Topology 9 (1970), 345-358.

7. A.D. Elmendorf, J.P.C. Greenlees, I. Kriz, and J.P. May. Commutative algebra in stable homotopy theory and a completion theorem. Math. Res. Letters 1 (1994), 225-239.

8. A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995.

9. A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Rings, modules, and algebras in stable homotopy theory. Preprint, 1995.

10. A.D. Elmendorf and J.P. May. Algebras over equivariant sphere spectra. Preprint, 1995.

11. L. Evens. The cohomology of groups. Oxford Mathematical Monographs. 1991.

12. J.P.C. Greenlees. K homology of universal spaces and local cohomology of the representation ring. Topology 32 (1993), 295-308.

13. J.P.C. Greenlees and J.P. May. Completions of G-spectra at ideals of the Burnside ring. Proc. Adams Memorial Symposium, Volume II, CUP (1992), 145-178.

14. J.P.C. Greenlees and J.P. May. Derived functors of I-adic completion and local homology. J. Algebra 149 (1992), 438-453.

15. J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Memoir American Math. Soc. No. 543. 1995.

16. J.P.C. Greenlees and J.P. May. Completions in algebra and topology. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995.

17. J.P.C. Greenlees and J.P. May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995.

18. A. Grothendieck (notes by R.Hartshorne). Local cohomology. Lecture notes in maths. Springer Lecture Notes Volume 42. 1967.

19. P.S. Landweber. Cobordism and classifying spaces. Proc. Symp. Pure Math. 22 (1971), 125-129.

20. P. S. Landweber. Coherence, flatness and cobordism of classifying spaces. Proc. Adv. Study Inst. Alg. Top. Aarhus, Denmark, 1970, 257-269.

21. L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Volume 1213. 1986.

22. P. Löffler. Equivariant unitary bordism and classifying spaces. Proc. Int. Symp. Topology and its applications, Budva, Yugoslavia, (1973), 158-160.

23. P. Löffler. Bordismengruppen unitärer Torusmannigfaltigkeiten. Manuscripta Math. 12 (1974), 307-327.

24. J. P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). E_{∞} ring spaces and E_{∞} ring spectra. Springer Lecture Notes in Mathematics Volume 577. 1977.

25. J. P. May. Equivariant and nonequivariant module spectra. In preparation.

26. J.P. May. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Proceedings. To appear.

School of Mathematics and Statistics, Hicks Building,, Sheffield S3 7RH, UK E-mail address: j.greenlees@sheffield.ac.uk

Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA E-mail address: may@math.uchicago.edu