ALGEBRAIC CYCLES AND EQUIVARIANT COHOMOLOGY THEORIES

H. BLAINE LAWSON, JR., PAULO LIMA-FILHO, AND MARIE-LOUISE MICHELSOHN

Contents

1. Introduction 2 2. General Theory 6 2.1. Equivariant Cohomology Theories and G-Spectra 6 2.2. Equivariant infinite loop spaces 9 2.3. Coherence 10 2.4. GI_{∗}(U)-monoids 11 3. Constructions 13 3.1. The Chow monoid functor 13 3.2. The Grassmann functor and equivariant K-theory 16 3.3. Topological groups of cycles 19 3.4. Further constructions and KO-theory 20 4. Main Results 22 4.1. The equivariant cohomology theories 22 4.2. Total Chern classes and equivariant Segal questions 25 4.3. Transfers 26 4.4. Analogues in the real case 27 5. Coefficients 27 5.1. General results for abelian groups 28 5.2. Computations for finite groups 31 5.3. The general case 35 6. Embellishments and open problems 36 6.1. Applications to algebraic varieties 36 Appendix A. The Coherence Theorem 38 References 41

1991 Mathematics Subject Classification. Primary: 55P91 ; Secondary 14C05, 19L47, 55N91.

All authors were partially supported by the NSF..

1. Introduction

The main theme of this paper is that algebraic cycles provide interesting non-trivial invariants for finite groups, as well as new equivariant cohomology theories which answer natural questions in equivariant homotopy theory. Besides being quite computable, these theories carry Chern classes for representations and have deep relations with usual Borel cohomology theory. In fact their coefficients are simpler than standard group cohomology and have a geometric interpretation of independent interest. Among their main properties, we shall prove a full equivariant analogue of the Segal loop space conjecture proved in [3].

The genesis of the results is elementary and geometric. Nevertheless it directly yields a fully structured theory, graded over the representation ring and Mackey-functor-valued. Our construction is a natural extension of the standard construction of the classifying space for equivariant K-theory, and it fits beautifully into the theory of I_{∗}-functors, a machine which builds (equivariant) operad actions and infinite loop space structures. However, due to its elementary origins, the coefficients of the theory can be computed by geometric means. For example, using techniques of degeneration by (ℂ^{×})^{m}-actions we are able to compute π_{0}^{G} of our spectra when the group G is abelian. Its description relies on a remarkably simple and well-behaved graded ring ℍ^{∗}(G) characterized by the following properties:

(1) ℍ^{0}(G) = ℤ and ℍ^{1}(G) ≅ G.

(2) ℍ^{∗}(G_{1} ⊕ G_{2}) = ℍ^{∗}(G_{1}) ⊗ ℍ^{∗}(G_{2})

(3) If G is cyclic, then ℍ^{∗}(G) = H^{2∗}(G; ℤ) (group cohomology with coefficients in the trivial G-module ℤ).

For example, if G is abelian and ℨ_{G} denotes the first of our spectra, then the component ring π_{0}^{G}(ℨ_{G}) is exactly ℍ^{∗}(G) localized with respect to the multiplicative system engendered by the total Chern classes 1 + c_{1}(ρ) ∈ {1} × ℍ^{1}(G) of the irreducible complex representations ρ of G. Some of the higher degree coefficients π_{k}^{H}(ℨ_{G}), H < G, may be computed by combining a general equivariant algebraic suspension theorem from [12] and the equivariant Dold-Thom theorem in [15].

Our basic constructions are modeled on the following “classical” one. Let G be a finite group. Recall that the classifying space for G-equivariant K-theory is a limit BU_{G} of Grassmann manifolds Gr(W) ≡ Gr_{w}(W ⊕W) where w = dim(W) and where W ranges over a family of linear G-spaces for which the multiplicity of each irreducible representation tends to infinity. This limit carries an H-space structure induced by the direct sum ⊕ : Gr(W) × Gr(W′) → Gr(W ⊕W′). The internal coherence of this multiplication is succinctly captured in the fact that (Gr(•), ⊕) is an equivariant I_{∗}-functor. Hence, by general theory [4], the ⊕-multiplication enhances to an equivariant infinite loop space structure, and thereby BU_{G} becomes the zero’th space in a G-spectrum 𝔎𝔲_{G}, which gives connective equivariant Ktheory.

We mimic this construction by considering an analogous family Z^{w}(ℙ(W ⊕ W)) of algebraic cycles of codimension w on ℙ(W ⊕ W). Each Z(W) is a topological abelian group, and the construction comes equipped with a biadditive “external” pairing on these groups defined by the algebraic join ♯_{ℂ} of cycles. Taking the analogous limit over W defines the basic space of our theory Z(U_{G}), the group of “algebraic cycles on ℙ^{∞}”. This is a topological abelian group with a biadditive pairing induced by ♯_{ℂ}. Furthermore, at finite levels (Z(•), ♯_{ℂ}) is an I_{∗}-monoid. Our first main result, a combination of Propositions 2.4.2, 3.1.2 and 3.3.3 implies that

Z(U_{G}) is a G-group complete equivariant E_{∞}-ring space.

In particular, there is a G-equivalence ı : Z(U_{G}) → ℨ_{G}(0) with the degree-zero space of an equivariant E_{∞}-ring spectrum ℨ_{G}. These spectra ℨ_{G} form a coherent family over the category of finite groups (see §2.3), and each ℨ_{G} admits an augmentation δ : ℨ_{G} → ℤ to the “constant” ring spectrum ℤ.

The subset Z(U_{G})_{1} = ϵ^{−1}δ^{−1}(1) of cycles of degree one is closed under the (homotopy commutative) join multiplication, and the same machinery tells us that there is a second “multiplicative” G-spectrum 𝔐_{G} and a map

ϵ_{1} : Z(U_{G})_{1} ⟶ 𝔐_{G}(0)

which is a G-group completion with respect to join pairing. For the trivial group G = {1} this is actually an equivalence. However, the calculation of coefficients alluded to above shows that Z(U_{G})_{1} is not in general a group-like space over G, that is, π_{0}(Z(U_{G})_{1}^{H}) is not a group under ♯_{ℂ}_{∗} for non-trivial H < G. The completion 𝔐_{G}(0) is a connected space for which this is true. In a similar fashion to ℨ_{G}, the spectra 𝔐_{G}, for various G’s, form a coherent family; cf. Theorem 4.1.4. As a consequence, both ℨ_{G} and 𝔐_{G} are split spectra, in the sense of [14, I.8]; cf. Corollary 2.3.3.

We observe now that considering linear subspaces to be algebraic cycles of degree one gives an equivariant inclusion Gr(W) ⊂ Z(W) for any finite-dimensional representation W of G. Furthermore on linear subspaces the join pairing is simply the direct sum. Hence there is a G-equivariant map

c_{G} : BU_{G} = \underrightarrow{lim}_{W} Gr(W) ⟶ \underrightarrow{lim}_{W} Z(W) = Z(U_{G})  (1.1)

taking ⊕ to ♯_{ℂ} and having image in Z(U_{G})_{1}. In fact, the inclusions above constitute a natural transformation (Gr(•), ⊕) → (Z(•), ♯_{ℂ}) of I_{∗}-functors; cf. Proposition 3.2.1. From this one can prove that the map BU_{G} → Z(U_{G})_{1} enhances to a transformation of G-spectra

c_{G} : 𝔎𝔲_{G} ⟶ 𝔐_{G}.  (1.2)

Suppose we take G to be the trivial group. Then it was proved in [11] that there is a canonical homotopy equivalence

Z(W) ≅ K(ℤ, 0) × K(ℤ, 2) × K(ℤ, 4) × ⋯ × K(ℤ, 2w).  (1.3)

Furthermore, in [13] it was shown that under (1.3) the join pairing ♯_{ℂ} : Z(W) × Z(W′) → Z(W ⊕ W′) classifies the cup product, and the inclusion Gr(W) ⊂ Z(W) classifies the total Chern class of the tautological w-plane bundle over Gr(W). In particular, if one drops the group in (1.1) by forgetting the G-action, then one finds the transformation of classifying spaces

c : BU ⟶ {1} × ∏_{k>0} K(ℤ, 2k)  (1.4)

corresponding to the total Chern class. The compatibility of this map with ♯_{ℂ} corresponds to the standard formula c(E ⊕ E′) = c(E)c(E′) for vector bundles over a compact space. In 1975 G. Segal [23] conjectured that there exists an infinite loop space structure on {1} × ∏_{k>0} K(ℤ, 2k) such that (1.4) becomes an infinite loop map. The assertion (1.2) in the case where G = {1} constitutes a proof of this conjecture which was established in [3]. The general assertion (1.2) is the full equivariant analogue of Segal’s conjecture; see Definition 4.2.1.

It is natural to ask if there are analogues of the results above for standard Borel cohomology theory. In a second set of results we establish such analogues. Indeed given any G-spectrum k_{G} there are an associated Borel spectrum bk_{G} and a map of G-spectra

ϵ : k_{G} ⟶ bk_{G}

which is a non-equivariant homotopy equivalence; cf. [8]. The spectrum bk_{G} is simply defined by setting bk_{G} = F (EG_{+}, k_{G}) where EG denotes as usual a contractible CWcomplex on which G acts freely, and where F (X, Y) denotes the base-point preserving continuous maps from X to Y. The transformation ϵ is induced from the map EG → pt.

In §4 we compute the Borel spectra bℨ_{G} and b𝔐_{G} associated to our “algebraic cycle” spectra ℨ_{G} and 𝔐_{G} above. It turns out that for any compact G-space X there are natural isomorphisms

[X, bℨ_{G}(0)]_{G} ≅ H_{G}^{2∗}(X; ℤ)_{Borel} and [X, b𝔐_{G}(0)]_{G} ≅ {1} × ∏_{k>0} H_{G}^{2k}(X; ℤ)_{Borel};

see Theorems 4.1.5 and 4.1.6 and their corollaries.

A combination of Theorems 4.1.6 and 4.2.2 yields the following statement: Theorem A. Let G be a finite group. The map of G-spectra

c_{G}^{b} : 𝔎𝔲_{G} ⟶ b𝔐_{G}

obtained by composing ϵ with (1.2) above classifies the total Chern class

c : K_{G}(X) ⟶ {1} × ∏_{k>0} H_{G}^{2k}(X; ℤ)

in Borel cohomology. The multiplications on the zero’th level, which are respected by this map of spectra, correspond to the formula c(E ⊕ E′) = c(E)c(E′) for equivariant bundles over a compact G-space X.

Said in another way, the multiplication of units in Borel cohomology, considered as an H-space structure on the classifying space, enhances to to an equivariant infinite loop space structure so that the total Chern class from 𝔎𝔲_{G} becomes an infinite loop map in the category of G-spaces. This is precisely the Segal loop space conjecture for Borel cohomology.

We also obtain analogous results for equivariant KO-theory and Stiefel-Whitney classes, by using real algebraic cycles and the work of T. K. Lam [10]; cf. §4.4. We construct spectra ℨ_{ℝ,G} and 𝔐_{ℝ,G}, together with maps of spectra w_{G} : 𝔎𝔬_{G} ⟶ 𝔐_{ℝ,G}, where 𝔎𝔬_{G} is the spectrum of equivariant connective KO-theory. A similar combination of results yields the corresponding statement: Theorem B. Let G be a finite group. The map of G-spectra

w_{G}^{b} : 𝔎𝔬_{G} ⟶ b𝔐_{ℝ,G}

obtained by composing ϵ with (1.2) above classifies the total Stiefel-Whitney class

w : KO_{G}(X) ⟶ {1} × ∏_{k>0} H_{G}^{k}(X; ℤ/2ℤ)

in Borel cohomology with ℤ/2ℤ coefficients. The multiplications on the zero’th level, which are respected by this map of spectra, correspond to the formula w(E ⊕ E′) = w(E)w(E′) for equivariant bundles over a compact G-space X.

There are several consequences of the naturality of these spectra and the maps relating them, which remain to be fully explored. For example, these maps commute with the transfers associated to equivariant bundles, for the various theories. In particular, one has multiplicative transfers for even group cohomology which naturally compute Chern classes of induced representations; one computes equivariant Chern classes for the push-forward of bundles under finite equivariant covering spaces; naturality properties for the composition of coverings follow trivially from the nature of the constructions, etc. See §4.3 for a brief discussion.

With an aim at algebro-geometric applications, we end the paper with a generalization of these constructions which associates to every complex algebraic variety X with a finite group of automorphisms G, the morphic spectra ℨ_{X,G} and 𝔐_{X,G}. These spectra reflect both the algebraic and topological properties of X and provide an extension of the notion of morphic cohomology introduced and studied in [6]. In short, we obtain a natural transformation from the category of algebraic varieties over ℂ to the category of spectra, respecting actions of finite groups of automorphisms and endowed with a natural theory of Chern classes. The spectra studied in this paper are obtained from these more general objects by taking X = pt. See §6.1.

Acknowledgements: The authors would like to thank I.H.E.S., M.S.R.I., The University of Chicago, and S.U.N.Y. Stony Brook for their hospitality and support during the preparation of this work. The second author would particularly like to thank J. P. May for his initiation to equivariant homotopy theory, and to thank the Universität Osnabrück/Sondersforschungsbereich 343 for their hospitality during the elaboration of portions of this paper.

2. General Theory

In this chapter we recall the fundamental concepts of equivariant cohomology theories and their associated Borel counterparts. This will include the equivariant theory of infinite loop spaces, with emphasis on a “recognition principle” for detecting such spaces. The main sources for background material are [14], [4] and [8].

2.1. Equivariant Cohomology Theories and G-Spectra. A universe U for a compact Lie group G is a unitary (or orthogonal) representation containing countably many copies of each of its irreducible subrepresentations. We fix, for every compact Lie group G, a complete G-universe U_{G}, in other words, a universe containing all irreducible representations of G, including the trivial one. The universes U_{G} will parameterize all representations we deal with, and hence, whenever we refer to a G-module V or an indexing G-space V, we will be implicitly working with a finite dimensional unitary subrepresentation of U_{G}.

Definition 2.1.1. We assume that all spaces are based with G-fixed base point, and use the symbol X_{+} to denote the disjoint union X ⨿ {∗} of X with a G-fixed base point. The function space of based maps between G-spaces X and Y is denoted by F (X, Y), and is considered as a G-space under the standard action by “conjugation”. For a G-space X and G-module V, the V -th suspension of X is the G-space Σ^{V} X:=X ∧ S^{V}, where S^{V} denotes the one-point compactification of V with base point at ∞, and the V -th loop space of X is Ω^{V} X:=F (S^{V} , X), both taken with their natural G-actions.

In this equivariant context, given an appropriate representation ring R, e.g. R = RO(G) or R = R(G), we are going to construct natural R-graded cohomology theories. These are equivariant cohomology theories 𝔥_{G}^{∗} satisfying well-known axioms, including contravariance and invariance under equivariant maps and homotopies; exactness for equivariant cofiber sequences etc. The reader is referred to [24, Defn. 6.8] for a full description of these axioms, although we would like to emphasize that the main additional feature of these theories, as opposed to the non-equivariant generalized cohomology theories, is the existence of suspension isomorphisms associated to arbitrary G-modules. More precisely, the theories 𝔥_{G}^{∗} considered here come equipped with natural isomorphisms σ_{V} : 𝔥_{G}^{α}(X) ⟶ 𝔥_{G}^{α+V} (Σ^{V} X) for each α ∈ R and G-module V.

Equivariant theories have been the focus of intense investigation in recent years, and have proven to be very useful in dealing with both equivariant and non-equivariant questions in stable homotopy theory. The monographs [14] and [8] contain a myriad of examples as well as an extensive bibliography. For further material, the reader may also consult [24] and [9].

Definition 2.1.2. Let U be a universe for G, not necessarily complete.

a: A G-prespectrum k_{G} indexed on U, consists of a collection {k_{G}(V) | V ⊂ U} of based G-spaces, together with a transitive system of G-maps σ_{V}^{W} : k_{G}(V) ⟶ Ω^{W −V} k_{G}(W), where V is a G-submodule of W and W −V denotes the orthogonal complement of V in W; cf. [14, Defn. 2.1]. A morphism of G-prespectra f : k_{G} ⟶ k′_{G} is a collection of G-maps f_{V} : k_{G}(V) ⟶ k′_{G}(V) which is strictly compatible with the corresponding structural maps.

The collection of all G-prespectra indexed on U forms a category which we denote by GPU, in which the set of morphims from k_{G} to k′_{G} is denoted by GPU(k_{G}, k′_{G}).

b: A G-prespectrum k_{G}, indexed on U is called a G-spectrum whenever the structural maps σ_{V}^{W} are G-homeomorphisms. The category of G-spectra indexed on the universe U, denoted by GSU, forms a full subcategory of the category of G-prespectra.

c: Given a G-spectrum k_{G} and a G-space X, the collection F (X, k_{G}(V)) is again a G-spectrum under the evident natural maps. In particular, one can define for every Gspectrum k_{G} and G-module V, the V -th loop spectrum Ω^{V} k_{G}:=F (S^{V} , k_{G}) of k_{G}. Example 2.1.3. Let k_{G} be a G-spectrum and let EG denote a contractible CW-complex on which G acts freely. The associated Borel spectrum bk_{G} to k_{G} is the G-spectrum F (EG_{+}, k_{G}). We shall see afterwards that this notion represents the adequate notion of Borel cohomology associated to k_{G}. Notice that the projection EG ⟶ {pt} induces a natural map of G-spectra ϵ : k_{G} ⟶ bk_{G}.

A G-prespectrum is automatically a G-spectrum, and hence there is a “forgetful” functor ℓ : GSU ⟶ GPU. This functor admits a left adjoint L which solves the problem that the category of G-spectra is not closed under basic space-level constructions, such as taking smash products, wedges, cofibers, etc. This is the content of the following fundamental result; cf. [14, Thm I.2.2]. Theorem 2.1.4. There is a left adjoint L : GPU ⟶ GSU to the forgetful inclusion functor ℓ : GSU ⟶ GPU. That is, GPU(k_{G}, ℓh_{G}) ≅ GSU(Lk_{G}, h_{G}) for k_{G} ∈ GPU and h_{G} ∈ GSU. Let η : k_{G} ⟶ ℓLk_{G} and ϵ : Lℓh_{G} ⟶ h_{G} be the unit and counit of the adjunction. Then ϵ is an isomorphism for each G-spectrum h_{G}, hence η is an isomorphism if k_{G} = ℓh_{G}.

The functor L allows one to perform constructions with spectra whose properties are derived easily from the fact that L is a left adjoint. For example, given a G-space X and k_{G} ∈ GPU one defines a G-prespectrum X ∧ k_{G} by sending V to X ∧ k_{G}(V). If k_{G} is a spectrum, then the G-spectrum X ∧ k_{G} ∈ GSU is defined as L(X ∧ ℓk_{G}). Similar procedures allow one to define cones, suspensions and homotopies in the category of spectra. In particular, the suspension spectrum Σ^{∞}X of a G-space X is defined as L(S^{∞}X), where S^{∞}X is the prespectrum V ↦ X ∧ S^{V}. Definition 2.1.5. Given a G-spectrum h_{G} ∈ GSU_{G}, one defines an R(G)-graded cohomology theory h_{G}^{∗} on G-spectra by assigning to k_{G} ∈ GSU_{G} the groups

h_{G}^{α}(k_{G}):={Σ^{W} k_{G}, Σ^{V} h_{G}}_{G},

where one writes the virtual representation α ∈ R(G) as a formal difference of two Gmodules α = V − W. The symbol {Σ^{W} k_{G}, Σ^{V} h_{G}}_{G} denotes equivariant homotopy classes of maps of G-spectra, and Σ^{W} k_{G}, Σ^{V} h_{G} denote the appropriate suspension of the spectra k_{G} and k′_{G}, respectively. In particular, the V -th cohomology group h_{G}^{V}(X) of a G-space X of the homotopy type of a G-CW-complex is retrieved by h_{G}^{V}(Σ^{∞}X_{+}) = [X, h_{G}(V)]_{G}, where [X, h_{G}(V)]_{G} denotes equivariant homotopy classes of maps of G-spaces.

Changes of Universe

Given a G-linear isometric embedding ȷ : U ⟶ V between two G-universes U and V, one can define a pair of adjoint change of universe functors ȷ^{∗} : GSV ⟶ GSU and ȷ_{∗} : GSU ⟶ GSV as follows.

For k_{G} ∈ GSV, one defines ȷ^{∗}k_{G} ∈ GSU by ȷ^{∗}k_{G}(V) = k_{G}(ȷ(V)), where the adjoints of the structural maps are given by the compositions

Σ^{W −V} ȷ^{∗}k_{G}(V) = k_{G}(ȷ(V)) ∧ S^{W −V} ⟶^{1∧ȷ} k_{G}(ȷ(V)) ∧ S^{ȷ(W)−ȷ(V)} ⟶^{σ̃} k_{G}(ȷ(W)) = ȷ^{∗}k_{G}(W).

It is evident that this construction takes spectra to spectra.

We first define the covariant functor ȷ_{∗} in the level of prespectra. Given a G-prespectrum k_{G} indexed on U, the assignment V ↦ k_{G}(ȷ^{−1}V) ∧ S^{V −ȷ(ȷ^{−1}V)} naturally defines a Gprespectrum ȷ_{∗}k ∈ GPV. In the level of spectra, one defines ȷ_{∗}(E_{G}):=L(ȷ_{∗}(ℓE_{G})), and it is easy to see that ȷ_{∗} is a left adjoint of ȷ^{∗}; cf. [14, I.1.2].

Naïve G-spectra

An important instance of change of universes is given by the G-linear isometry ı : U^{G} ⟶ U, where U^{G} is the G-fixed point space of U, considered as a “trivial” G-universe identifiable with ℝ^{∞}. The G-spectra indexed on U^{G} are simply ordinary spectra with actions of G on their structural spaces. These spectra are called naïve G-spectra, and the usual, nonequivariant spectra can be considered as naïve G-spectra with trivial G-action. Following [8], we denote the category of naïve G spectra by GSU^{G}, and recall the following result from [14, II.1.8], which relates the passage from naïve to genuine G-spectra and vice-versa, given by the change of universe functors induced by ı : U^{G} ⟶ U.

Lemma 2.1.6. For k_{G} ∈ GSU^{G}, the unit map η : k_{G} ⟶ ı^{∗}ı_{∗}k_{G} of the (ı^{∗}, ı_{∗}) adjunction is a nonequivariant equivalence. For k_{G} ∈ GSU, the counit map ϵ : ı_{∗}ı^{∗}k_{G} ⟶ k_{G} is a nonequivariant equivalence.

Split Spectra

Since the indexing sets for naïve G-spectra are trivial G-spaces, one can naturally associate to k_{G} ∈ GSU^{G} its fixed point (nonequivariant) spectrum (k_{G})^{G} ∈ SU^{G}, which sends V ⊂ U^{G} to (k_{G}(V))^{G}. If k_{G} is an actual G-spectrum, one defines (k_{G})^{G} = (ı^{∗}k_{G})^{G}.

Definition 2.1.7. A naïve G-spectrum is said to be split if there is a map of spectra ξ : k ⟶ (k_{G})^{G}, from the underlying nonequivariant spectrum k of k_{G} to its fixed point spectrum (k_{G})^{G}, whose composition with the inclusion of (k_{G})^{G} into k is homotopic to the identity map. A G-spectrum k_{G} ∈ GSU is called split if ı^{∗}k_{G} ∈ GSU^{G} is split.

Split spectra play an important role in both equivariant and nonequivariant contexts, and among their pleasant characteristics is the following result; cf. [14, I.8.4] or [9].

Proposition 2.1.8. Let k_{G} be a split G-spectrum, and let X be a space on which G acts freely. Then k_{G}^{∗}(X) ≅ k^{∗}(X/G), where k^{∗} is the nonequivariant cohomology theory associated with the underlying nonequivariant spectrum k of k_{G}.

Corollary 2.1.9. If k_{G} is a split G-spectrum, and b(k_{G})^{∗} denotes the Borel cohomology associated to k_{G}, then for any G-space X one has

b(k_{G})^{∗}(X) ≅ k^{∗}(EG ×_{G} X).

Remark 2.1.10. This corollary justifies the assertion that b(k_{G})^{∗} is the adequate Borel cohomology associated to k_{G}^{∗}.

2.2. Equivariant infinite loop spaces. A good source of non-equivariant (connective) spectra is the classical theory of infinite loop spaces [1], [18]. Unfortunately, to our knowledge, in the equivariant context such machines are only well-understood in the case of finite groups. Therefore, despite the fact that our constructions hold for arbitrary compact Lie groups, we need to restrict ourselves to the case of finite groups in order to apply the existing equivariant infinite loop space machines. Throughout the rest of this section we work with a fixed finite group G and its universe U = U_{G}. Definition 2.2.1. An equivariant infinite loop space, indexed by U, is a G-space X such that for every indexing space V ⊂ U there is a G-space Y (V) together with a homeomorphism X ≅ Ω^{V} Y (V). Furthermore, if for a given submodule W ⊂ V one denotes by V −W the orthogonal complement of W in V, then there are compatible G-homeomorphisms Y (W) ≅ Ω^{V −W} Y (V). It is evident that an infinite loop space X is the zero-th space 𝕏(0) of an equivariant G-spectrum 𝕏.

Infinite loop space theories come equipped with a “recognition principle” which tells when a space is an infinite loop space or, more generally, with an “infinite loop machine” which produces infinite loop spaces out of spaces with a certain structure. The machine that we use requires the notion of a G-operad acting on a G-space X. We refer the reader to [14, ] or [4] for the precise definition of such object.

Presently we only need to introduce the linear isometries operad L(U) associated to a G-universe U, which consists of the collection of spaces {L(U)_{k} | k ∈ ℤ_{+}} and structural maps γ : L(U)_{k}×L(U)_{j_{1}} ⋯×⋯ L(U)_{j_{k}} ⟶ L(U)_{j_{1}+···+j_{k}}, where L(U)_{k} is the space I(U^{⊕k}, U) of linear isometries from U^{⊕k} into U and γ is the natural map given by composition γ : (f; g_{1}, …, g_{k}) ↦ f ∘ (g_{1} ⊕ ⋯ ⊕ g_{k}). The spaces L(U)_{k} admit a natural action of G × 𝔖_{k}, where 𝔖_{k} is the symmetric group on k letters, and the structural maps satisfy suitable equivariant compatibility conditions. Furthermore, L(U)_{k} is a G-CW-complex on which 𝔖_{k} acts freely and, if U is complete universe, the fixed point set L(U)_{k}^{Λ} is contractible, for each subgroup Λ ⊂ G × 𝔖_{k} such that Λ ∩ 𝔖_{k} is trivial. In other words, L(U)_{k} is a universal principal (G, 𝔖_{k})-bundle in the sense of [24].

One says that a G-space X is an algebra over an operad L, or that L acts on X if there is a collection of G×𝔖_{k}-maps λ : L_{k} ×(X ×…×X) ⟶ X satisfying appropriate compatibility relations with the structural maps of the operad.

The relevance of the notion of L(U)-spaces for our work lies in the following recognition principle, due to Hauschild, May and Waner, which generalizes earlier work of May [18], Theorem 2.2.2 ([4, Thm.1]). Let X be a group-like based G-space of the homotopy type of a G-CW-complex. If X is algebra over L(U), where U is a complete G-universe, then X is G-homotopy equivalent to an equivariant infinite loop space.

Now recall that a Hopf space^{1} map M ⟶ M̃ is a called a group completion (in the homotopy-theoretic sense) if for any coefficient ring R, the induced homomorphism of Pontrjagin rings H_{∗}(M, R) ⟶ H_{∗}(M̃, R) is the localization of H_{∗}(M, R) with respect to the multiplicative subset generated by π_{0}(M).

In the category of G-spaces, a G-map X ⟶ Y between two Hopf G-spaces is a G-group completion if each induced map X^{H} ⟶ Y^{H} is a (non-equivariant) group completion, for all H ≤ G.

In this framework, the proof of the previous theorem actually yields the following result, where we assume that X is any space of the homotopy type of a G-CW-complex. Theorem 2.2.3. Let X be a L(U)-space, where U is a complete G-universe. Then there is an equivariant infinite loop space 𝕏(0), and a G-map of L(U)-spaces ı : X ⟶ 𝕏(⊬) which is a G-group completion.

2.3. Coherence. In this section we shall analyze L(U)-spaces which have particularly nice properties with respect to the subgroups of G. Our main result asserts that such spaces yield spectra which are split, and whose Borel counterparts are therefore “classical” by (2.1.9).

In the following exposition we emphasize the dependence of U on G by writing U_{G} in place of U.

Let G denote the category whose objects are the subgroups of G, and whose morphisms are the inclusions α_{H′,H} : H ↪ H′, the projections β_{H} : H ⟶ {e}, and their compositions. For each H ∈ G one can take U_{H} = α_{G,H}^{∗}(U_{G}). To each inclusion α = α_{H′,H} and each projection β = β_{H} in G we associate the H-isometries

j_{α} : α^{∗}U_{H′} ⟶ U_{H} and j_{β} : β^{∗}U_{e} ⟶ U_{H}  (2.1)

by letting j_{α} be the identity map, and by defining j_{β} to be the composition U_{e} ⟶^{≅} (U_{H})^{H} ↪ U_{H}. These homomorphisms and isometries induce change of group functors among spectra, as follows; cf. [14].

A group homomorphism α : H ⟶ G naturally defines a functor α^{∗} : GSU_{G} ⟶ HS(α^{∗}U_{G}) which considers an indexing G-space V ⊂ U_{G} as an indexing H-space via α. For α in our category G we now define the first change of group functor α^{♯} : GSU_{G} ⟶ HSU_{H} to be the composition GSU_{G} ⟶^{α^{∗}} HS(α^{∗}U_{G}) ⟶^{j_{α∗}} HSU_{H}, where j_{α∗} is induced by the change of universe j_{α} : α^{∗}U_{G} ⟶ U_{H} defined above.

The functor α^{♯} has a natural “right homotopy adjoint” α_{♯} : HSU_{H} ⟶ GSU_{G} with {α^{♯}h_{G}, k_{H}}_{H} ≅ {h_{G}, α_{♯}k_{H}}_{G}, defined as follows. Given an H-prespectrum t_{H} ∈ HP(α^{∗}U_{G}), one can define a G-prespectrum F_{α}[G, t_{H}) which sends V ∈ U_{G} to the function space F_{α}(G_{+}, t_{H}(V)) of left H-maps G_{+} ⟶ t_{H}(V) with left action on F_{α}(G, t_{H}(V)) induced by the right action of G on itself. At the level of spectra, one then defines a functor α_{∗} : HS(α^{∗}U_{G}) ⟶ GSU_{G} as α_{∗}(k) = L(F_{α}[G, ℓk)). One finally introduces the desired change of

1An algebra over an operad (in the category of spaces) is automatically a Hopf space.

group functor α_{♯} : HSU_{H} ⟶ GSU_{G} as the composition HSU_{H} ⟶^{j_{α}^{∗}} HS(α^{∗}U_{G}) ⟶^{α_{∗}} GSU_{G},

where j_{α}^{∗} is the contravariant change of universe.

Now, let X_{G} be a given L(U_{G})-space. Then, by “neglect of structure” one obtains an L(U_{H})-space X_{H} = α_{G,H}^{∗}X_{G} for each H ∈ G. Notice that the (non-equivariant) operad L(U_{H})^{H} of H-equivariant isometries acts on X_{H}^{H} and can be thought of as a suboperad of L(U_{e}) via the non-equivariant identification of U_{H} with U_{e}. It follows that the inclusion X_{H}^{H} ↪ X_{e} is a map of L(U_{H})^{H}-spaces.

Let us denote by k_{H} the spectrum associated to X_{H} via Theorem 2.2.3. The following result applies to various situations in equivariant homotopy theory, as we shall see later on. Proposition 2.3.1. Let X be an L(U_{G})-space and suppose that for each H ∈ G one has a map f_{H} : X_{e} ⟶ X_{H}^{H} of L(U_{H})^{H}- spaces so that the composition X_{e} ⟶^{f_{H}} X_{H}^{H} ⟶^{i_{H}} X_{e} is a homotopy equivalence in the category of L(U_{H})^{H}-spaces. Then the family {k_{H} ∈ HS(U_{H}) | H ∈ G} of associated spectra to {X_{H} | H ∈ G} satisfies:

a. For each map θ : H ⟶ H′ in G there is a map of spectra η_{θ} : θ^{♯}k_{H′} ⟶ k_{H} which is the identity if θ is the identity and an equivalence if θ is an inclusion.

b. The diagram

β^{♯}α^{♯}k_{N} ⟶^{β^{♯}η_{α}} β^{♯}k_{H}
≅↓  ↓^{η_{β}}
(αβ)^{♯}k_{N} ⟶^{η_{αβ}} k_{K}.

homotopy commutes for every α : H ⟶ N and β : K ⟶ H.

Proof. See Appendix A. □

Remark 2.3.2. A collection of spectra satisfying the properties described in Proposition 2.3.1 is called a coherent family of spectra indexed on G or a G-spectrum in [14, II.8]. Corollary 2.3.3. Under the hypothesis of the proposition, all spectra k_{H} ∈ HS(U_{H}), with H ∈ G, are split.

Proof. This follows directly from the appropriate commutative diagram associated with the maps H ⟶^{β} {e} ⟶^{α} H; cf. [14, Ex. 8.7(i)]. □

2.4. GI_{∗}(U)-monoids. A simple, geometric way of constructing L(U)-spaces, for a fixed G-universe U, is given by GI_{∗}(U)-spaces. Start with the category I_{∗} of finite dimensional hermitian vector spaces and linear isometries between them. Denote by GI_{∗} the subcategory of I_{∗} consisting of G-modules and G-module morphisms and denote by GI_{∗}(U) the full subcategory of GI_{∗} consisting of finite dimensional G-modules isomorphic to submodules of U.

Definition 2.4.1. A GI_{∗}(U)-space (T, ω) is a continuous covariant functor^{2}

T : GI_{∗}(U) ⟶ G-T op

from GI_{∗}(U) to the category of non-degenerately based G- spaces together with a (coherently) commutative, associative, and continuous natural transformation

ω : T × T ⟶ T ∘ ⊕

satisfying:

(a) If x ∈ T V and if 1 ∈ T {0} is the basepoint, then ω(x, 1) = x ∈ T (V ⊕ {0}) = T V.

(b) If V = V′ ⊕ V′′, then the map T V′ ⟶ T V given by x ↦ ω(x, 1) is a homeomorphism onto a closed subset.

(c) Each sum map ω : T (V) × T (W) ⟶ T (V ⊕ W) is a G-map.

(d) Each evaluation map e : I_{∗}(V, W) × T (V) ⟶ T (W) is a G-map.

One can formulate the notion of morphisms of GI_{∗}(U)-spaces as a compatible collection of G-maps making the appropriate diagrams commute, and hence one obtains the category GI_{∗}-Top consisting of GI_{∗}-spaces and their maps.

The output of the formalism of GI_{∗}-spaces is the following simple, albeit very useful, result of J. P. May [18]. Proposition 2.4.2. For any GI_{∗}(U)-space T, the colimit T (U):= \underrightarrow{lim}_{V ⊂U}T (V), is an L(U)-

space. Furthermore, the assignment T ↦ T (U) is functorial, i.e., sends maps of GI_{∗}(U)- spaces to maps of L(U)-spaces. Remark 2.4.3. I_{∗}-functors T : I_{∗} ⟶ F may have values on any symmetric monoidal category F which admits colimits, in which case the colimit T (U) becomes an L(U)-object with values in F.

It is unnecessary to stress the relevance of ring spectra in homotopy theory and how rich a theory becomes when one has an E_{∞}-operad parameterizing the ring structure of a spectrum. A precise formulation of the notion of E_{∞}-ring spectra can be found in [19] and, for our purposes, it suffices only to exhibit particular instances where such spectra occur. We need only the concrete manifestation of multiplicative infinite loop space theory, which is explained in the next few paragraphs.

Let G-Atom be the subcategory of G-Top whose objects are abelian topological monoids on which G acts by monoid morphisms, and whose morphisms are equivariant continuous monoid homomorphisms. Definition 2.4.4. A GI_{∗}(U)-monoid is a GI_{∗}(U)-space (T, ω) with values in G-Atom, where ω is distributive over the monoid operation. The collection of GI_{∗}(U)-monoids, together with those GI_{∗}(U)-space morphisms which are spacewise monoid morphisms, form a subcategory GI_{∗}-Atom of the category of all GI_{∗}(U)-spaces.

2Recall that a topological category is one where both the objects and the set of morphisms are spaces, and whose structural operations such as composition and evaluation are continuous. A continuous functor between topological categories is one which induces continuous maps between morphism spaces. See [21] for details.

In the same fashion as the nonequivariant case [19], the resulting G-monoid T (U) = \underrightarrow{lim}_{V ⊂U}T (V) becomes an equivariant L(U)-ring space. We summarize, for future convenience

the main consequences of this fact, which follow from the application of the infinite loop space machines described above, in the following statements.

Proposition 2.4.5. Let G be a finite group, and let T be a GI_{∗}(U)-monoid, where U is a complete G-universe. Then there is an L(U)-ring spectrum k_{G}^{T}, together with a morphism η : T (U) ⟶ k_{G}^{T}(0) of L(U)-ring spaces which is a G-group completion of the additive structure of T (U). In particular, if T (U) is additively G-group-complete then η is a Ghomotopy equivalence.

Any commutative topological ring R yields a group-complete “constant” GI_{∗}(U)-monoid \underline{R}, on which G acts trivially. We then say that a GI_{∗}(U)- monoid T is augmented over R if there is a morphism T ⟶ \underline{R} of GI_{∗}(U)-monoids.

Proposition 2.4.6. Let ϕ : T ⟶ \underline{ℤ} be a GI_{∗}(U)-monoid augmented over ℤ, and assume that T (U) is additively G-group complete. Then ϕ induces a map of L(U)-ring spaces ϕ : T (U) ⟶ ℤ and T (U)_{1}:=ϕ^{−1}(1) is a L(U)-subspace of T. Therefore T (U)_{1} admits a multiplicative G-group completion η : T (U)_{1} ⟶ m_{G}^{T}(0) into the zero-th space of a G-spectrum m_{G}^{T}. In particular, if T (U)_{1} is a connected space, then η is a nonequivariant homotopy equivalence.

We have thus seen how the machinery of GI_{∗}-monoids automatically produces examples of highly structured equivariant ring spectra, and in the next chapter we will exhibit various examples of such objects.

3. Constructions

In this chapter we present the central constructions of the paper. We introduce certain concrete, geometric I_{∗}(U)-functors with values both in G-Top and in G-Atom, the category of abelian topological G-monoids and equivariant monoid morphisms. Our constructions involve the use of algebraic cycles on complex projective spaces. Recall that an algebraic p-cycle σ on ℙ^{n} is a finite formal sum σ = ∑ n_{i}V_{i}, n_{i} ∈ ℤ, where V_{i} is an irreducible p-dimensional subvariety of ℙ^{n}. An algebraic cycle is said to be effective if its coefficients are positive integers. The collection C_{p,d}(ℙ^{n}), of effective p-cycles on ℙ^{n} of degree d, forms a projective variety called a Chow variety. The degree of a p-cycle σ = ∑ n_{i}V_{i} is defined as deg σ = ∑ n_{i} deg V_{i}, where deg V_{i} ∈ ℤ is the topological degree of V_{i} determined as the fundamental class [V_{i}] ∈ H_{2p}(ℙ^{n}, ℤ) ≅ ℤ, which coincides with the number of points in the intersection of V_{i} with a generic linear subspace of ℙ^{n} of complementary dimension. We refer the reader to [11] and [17] for further details on topological properties of spaces of algebraic cycles.

3.1. The Chow monoid functor. Let us fix, momentarily, a compact Lie group G and its universe U = U_{G}.

Definition 3.1.1. A. Given an hermitian inner product space V of dimension v, let

C(V):= ∐_{d≥0} C_{d}^{v}(ℙ(V ⊕ V))  (3.1)

be the Chow monoid C^{v}(ℙ(V ⊕ V)) of effective algebraic cycles of codimension v in ℙ(V ⊕ V), with distinguished element 1 = ℙ(V ⊕ {0}). Here C_{d}^{v}(ℙ(V ⊕ V)) denotes the effective cycles of degree d and codimension v in ℙ(V ⊕ V). In dimension 0 we set C(0) = ℕ with distinguished element 1.

B. For a linear isometric embedding f : V ⟶ W, we define C(f) : C(V) ⟶ C(W) on a codimension v cycle c by

C(f)c = ℙ (W − f(V) ⊕ {0}) ♯_{ℂ} (f ⊕ f)_{∗}(c),  (3.2)

where W − f(V) denotes the orthogonal complement of f(V) in W and where ♯_{ℂ} denotes the complex join, which is defined on any pair of cycles which live in disjoint linear subspaces of ℙ(V ⊕ V). Recall that the complex join is given in homogeneous coordinates by taking the direct product of the homogeneous (conical) varieties. In particular, it is a strictly associative pairing. Note that C(f)(1) = 1.

C. Using the complex join we define a pairing ω : CV × CV′ ⟶ C(V ⊕ V′) by setting

ω(c, c′) = τ_{∗}(c ♯_{ℂ} c′)  (3.3)

where τ : V ⊕ V ⊕ V′ ⊕ V′ → V ⊕ V′ ⊕ V ⊕ V′ is the shuffle map which interchanges the two middle factors. The pairing ω is strictly associative, coherently commutative and continuous.

The “diagonal” action of G in V ⊕ V induces an action of G on the Chow monoid C^{v}(ℙ(V ⊕ V)) which makes it into a G-Atom. For an element g ∈ G and V ∈ I_{∗}(U) we let

g_{♯} : C(V) ⟶ C(V)  (3.4)

denote the action of g on the cycles of ℙ(V ⊕ V) via the given representation. We make C^{0}(ℙ(0 ⊕ 0)) into a trivial G-set. Proposition 3.1.2. For every compact Lie group G, the functor

(C_{G}, ω) : GI_{∗}(U_{G}) ⟶ G-Atom
V ↦ C(V)

is a GI_{∗}(U_{G})-monoid which is augmented over ℤ.

Proof. The fact that C is an I_{∗}-monoid in the nonequivariant situation is shown in [3], and the degree map is easily seen to give the desired augmentation. Therefore we just need to show the following:

Step 1: The points \underline{0} and \underline{1} in C(V) are fixed by the action of G.

Since V ⊕ {0} is a subrepresentation of V ⊕ V then ℙ(V ⊕ {0}) is an invariant subspace of ℙ(V ⊕ V) under the action of G, and hence 1:=ℙ(V ⊕ {0}) is fixed as a cycle in C(V). The fact that \underline{0} is fixed under the action follows from its definition.

Step 2: The evaluation maps

e : I_{∗}(V, W) × C(V) ⟶ C(W)

are G-maps.

Recall that G acts by isometries on V and W ∈ U and by conjugation on I_{∗}(V, W). In other words, given f ∈ I_{∗}(V, W) and g ∈ G, then gf : V ⟶ W is given by the formula (gf)(v):=g(f(g^{−1}v)) for v ∈ V. As a consequence one immediately gets, for all g ∈ G :

g (W − f(V)) = W − (gf)(V)  (3.5)

Now, for f ∈ I_{∗}(V, W) , g ∈ G and c ∈ C(V) one has:

g_{♯} (e(f, c)) = g_{♯} (C(f)(c)) = g_{♯} (ℙ ((W − f(V)) ⊕ {0}) ♯_{ℂ} (f ⊕ f)_{∗}(c))
= g_{♯}ℙ ((W − f(V)) ⊕ {0}) ♯_{ℂ} g_{♯} ((f ⊕ f)_{∗}(c))
= ℙ ((W − (gf)(V)) ⊕ {0}) ♯_{ℂ} ((gf) ⊕ (gf))_{∗} (g_{♯}c)
= e(gf , g_{♯}c),

where the fourth equality follows from 3.5 and from the action by conjugation of G on I_{∗}(V, W). Step 3: The Whitney sum

ω : C(V) × C(W) ⟶ C(V ⊕ W)

is a G-map, for all V , W ∈ I_{∗}(U) and V ⊥ W.

Given c ∈ C(V), c′ ∈ C(W) and g ∈ G one has:

g_{♯}ω(c, c′) = g_{♯} (τ(c ♯_{ℂ} c′))
= τ(g_{♯}c ♯_{ℂ} g_{♯}c′)
= ω(g_{♯}c, g_{♯}c′)
□

The construction of the GI_{∗}(U_{G})-monoid C_{G} presents an exceptional behavior with respect to subgroups. More precisely, if α : H ↪ G is the inclusion of a subgroup, then α^{∗}C_{G}(U_{G}) = C_{H}(U_{H}) under the identification of U_{H} with α^{∗}U_{G}. Furthermore, if V ⊂ U_{H}^{H} then C_{H}(V) = C_{H}(V)^{H} and the isometry j_{β} : β^{∗}U_{e} ⟶ U_{H}, associated to the map β : H ⟶ {e} in (2.1), induces a natural map τ_{H} : C_{e}(U_{e}) ⟶ C_{H}(U_{H})^{H} of ℤ-augmented L(U_{H})^{H}-monoids. Proposition 3.1.3. The composition C_{e}(U_{e}) ⟶^{τ_{H}} C_{H}(U_{H})^{H} ⟶ C_{H}(U_{H}) is a non-equivariant homotopy equivalence for each subgroup H of G.

Proof. First recall that one can assume that the H-linear isometric embedding j_{β} : β^{∗}U_{e} ⟶ U_{H} is induced by a (non-equivariant) identification of U_{e} with U_{H}^{H}, followed by an inclusion. We use the letter C to denote the Chow monoid functor when no mention to group actions is necessary, and identify C_{e}(U_{e}) with C(U_{H}^{H}). Therefore, one just needs to prove that the map C(U_{H}^{H}) ⟶ C(U_{H}) induced by the inclusion U_{H}^{H} ↪ U_{H} is a non-equivariant homotopy equivalence.

Let us denote U_{H} by U and U_{H}^{H} by U_{e}, and write U = U_{e}^{⊥} ⊕ U_{e}. Fix an isometric isomorphism l : U_{e} ⟶ U_{e} ⊕ U_{e} and observe that L:=1_{U_{e}^{⊥}} ⊕ l : U ⟶ U ⊕ U_{e} is an H-linear isometric isomorphism. Let i_{0}, i_{1} : U_{e} ↪ U_{e}⊕U_{e} denote the inclusions in the first and second factors, respectively, and let γ_{0}, γ_{1} : [0, 1] ⟶ I(U_{e}, U_{e}⊕U_{e}) be paths in the contractible space of linear isometries I(U_{e}, U_{e} ⊕ U_{e}) joining i_{0} and i_{1} to l, respectively.

Notice that 1_{U_{e}^{⊥}} ⊕γ_{0} induces an equivariant homotopy from the inclusion j_{0} : U ↪ U ⊕U_{e} to the H-linear isomorphism L, introduced above. Furthermore, under the identification of U with U ⊕ U_{e} via L, one sees that 1_{U_{e}^{⊥}} ⊕ γ_{1} induces a homotopy between the inclusion U_{e} ↪ U and the inclusion in the second factor j_{1} : U_{e} ↪ U ⊕ U_{e}.

Given a linear isometry f ∈ I(U, U ⊕ U_{e}), it naturally defines a monoid morphism f_{∗} : C(U) ⟶ C(U ⊕ U_{e}), as follows. For each V ⊂ U, f induces a natural map f_{∗,V} : C(V) ⟶ C(f(V)) which is seen to be a map of directed systems, once one observes that for V ⊂ W, one has f(W − V) = f(W) − f(V), since f is an isometry. It is not hard to see that the assignment f ↦ f_{∗} induces a continuous map Γ : I(U, U ⊕ U_{e}) × C(U) ⟶ C(U ⊕ U_{e}).

The composition

[0, 1] × C(U) ⟶^{1_{U^{⊥}_{e}}⊕γ_{1}} I(U, U ⊕ U_{e}) × C(U) ⟶^{Γ} C(U ⊕ U_{e}) ⟶^{l_{∗}^{−1}} C(U)

gives an equivariant homotopy between C(U) ↪ C(U ⊕ U_{e}) and C(U) ⟶^{l_{∗}} C(U ⊕ U_{e}) .

At this point we can identify, non-equivariantly, U with U_{e} and U_{e} with U_{e}^{⊥} in the formula above, to avoid repetition of arguments, and reinterpret the same homotopy as providing the desired homotopy equivalence. □

Remark 3.1.4. Notice that C_{H}(U_{H}) is non-equivariantly identified with C_{e}(U_{e}) for every H ∈ G, and any group G. In particular, the Proposition above shows that C_{G}(U_{G}) is an L(U_{G})-space satisfying the conditions of Proposition 2.3.1.

3.2. The Grassmann functor and equivariant K-theory. We start by observing that the Chow variety C_{1}^{v}(ℙ(V ⊕ V)) of cycles of degree 1 in ℙ(V ⊕ V) is precisely the Grassmannian variety Gr^{v}(V ⊕ V), of v-planes in V ⊕ V, where v = dim_{ℂ} V.

In the setting of the previous constructions, the complex join, when restricted to linear subspaces, becomes the direct sum of planes. As a consequence, the following result is just an easy corollary of the previous Proposition. Proposition 3.2.1. For each compact Lie group G, the assignment

(Gr, ⊕) : GI_{∗}(U_{G}) ⟶ G-Top

given by

V ↦ C_{1}^{v}(ℙ(V ⊕ V)) ≅ Gr^{v}(V ⊕ V),

defines a GI_{∗}(U_{G})-subspace of C_{G}. Furthermore, the map τ_{G} of Proposition 3.1.3 restricts to a map τ_{G} : Gr(U_{e}) ⟶ Gr(U_{G})^{G} whose composition with the inclusion Gr(U_{G})^{G} ⟶ Gr(U_{G}) is also a non-equivariant homotopy equivalence.

Proof. The first assertion follows the nonequivariant analogue, cf. [3] once we observe that Gr(V) is a G-subspace of C(V), for every G-module V and that the complex join operation is equivariant. Furthermore, the homotopy equivalence obtained in Proposition 3.1.3 restricts to a homotopy equivalence in the level of the Grassmann subspaces, since it preserves degrees. □

Remark 3.2.2. Notice that Gr is not a GI_{∗}-submonoid of C, but only a subspace. However, the assignment of the infinite symmetric product SP (Gr(V)) to V, is naturally identified with a GI_{∗}-submonoid of C. It is important to note that SP (Gr(V)^{G}) is not necessarily equal to SP (Gr(V))^{G}. Actually, after group completion this difference is explained in terms of Bredon homology in [15]. Remark 3.2.3. It follows immediately from 3.2.1 and Theorem 2.2.3 that the direct limit

BU_{G} := Gr(U_{G}) = \underrightarrow{lim}_{V} Gr(V)

is an L(U_{G})-space and therefore inherits the sturcture of an equivariant infinite loop space. In particular, there is a G- spectrum 𝔎𝔲_{G}(∗) and a G-map

BU_{G} ⟶ 𝔎𝔲_{G}(0)

into the zero’th space of the spectrum, which is a G-group completion.

The spectrum 𝔎𝔲_{G} corresponds to connective equivariant K-theory (cf. [14]). For completeness we show here directly that the zero’th space BU_{G} classifies the functor K_{G}(X). We begin by recalling the basic definitions and results (cf. [22]). Let X be a compact G-space, and denote by V_{G}(X) the set of isomorphism classes of complex G-vector bundles over X. Under the Whitney sum ⊕ this is an abelian monoid whose naïve group completion K_{G}(X) is called the equivariant K-theory of X. Given a G-module V, we denote by 𝕍 the “trivial” G- vector bundle X × V ⟶ X. Elements in K_{G}(X) can be canonically identified with stable equivalence classes in V_{G}(X), where E, E′ are called stably equivalent if there exist G-modules V, V′ so that E ⊕ 𝕍 ≅ E′ ⊕ 𝕍′. This is a consequence of the following. Proposition 3.2.4. ([22], Prop. 2.4) If E is a G-vector bundle over a compact space X, there is a G-module V and a G-vector bundle E^{⊥} such that E ⊕ E^{⊥} ≅ 𝕍.

There is an equivariant augmentation ϕ : K_{G}(X) ⟶ ℤ given by the virtual rank, and we set K_{G}(X)_{o} = ker(ϕ).

Given a G-module W, let ξ_{W} denote the “universal” G- vector bundle over Gr(W) whose fibre at V is V^{⊥}. Given a GI_{∗}- map j : U ⟶ W, one has

j^{∗}ξ_{W} ≅ ξ_{U} ⊕ (𝕎 − j(𝕌)),

where 𝕎−f(𝕌) is the “trivial” bundle associated to the orthogonal complement of {0}×j(U) in {0} × W.

Consider now a G-map f : X ⟶ BU_{G}. Since X is compact and BU_{G} carries the compactly generated topology, there is a G-module W ⊂ U_{G} and a G-map f_{W} : X ⟶ Gr(W) which represents f in the limit. We associate to f the element

f_{W}^{∗} − 𝕎 ∈ K_{G}(X).  (3.6)

This element depends only on the G-homotopy class of f ([22], Prop. 1.3). Furthermore, this element restricts to zero at the base point. Thus, the assignment (3.6) determines a well-defined homomorphism

i : [X^{+}, BU_{G}]_{G} ⟶ K_{G}(X),  (3.7)

where X_{+} denotes the union of X with a disjoint base point. Proposition 3.2.5. The following sequence is exact:

0 ⟶ [X_{+}, BU_{G}]_{G} ⟶^{i} K_{G}(X) ⟶^{ϕ} ℤ ⟶ 0,

and hence [X_{+}, BU_{G}]_{G} ≅ K_{G}(X)_{o}.

Proof. It is clear from its construction that image(i) ⊆ ker(ϕ). To show surjectivity onto ker(ϕ), consider E − E′ ∈ ker(ϕ). Then rank(E) = rank(E′) = m and we extend E and E′ to X_{+} by defining them to be the trivial representation of rank m at the disjoint base point. By Proposition 3.2.4 there is a G-vector bundle E′^{⊥} over X_{+} and a G-module U so that E′ ⊕ E′^{⊥} ≅ 𝕌. By adding and subtracting this element in E − E′ we may assume that E′ = 𝕌.

Similarly we may choose E^{⊥} over X_{+} with E ⊕ E^{⊥} ≅ 𝕎 for some G-module W. The inclusion E^{⊥} ⊂ 𝕎 induces a G-map f : X ⟶ Gr_{n}(W) (n = rank(E^{⊥})) which classifies E in the sense that f^{∗}ξ_{W} = E, where ξ_{W} is the universal bundle as above. Let ϕ : Gr_{n}(W) ⟶ Gr(W ⊕ U) be defined by ϕ(L) = L ⊕ {0} ⊕ {0} ⊕ U ⊂ (W ⊕ U) ⊕ (W ⊕ U). Then one has (ϕ ∘ f)^{∗}ξ_{W ⊕U} ≅ E ⊕ 𝕎, and hence i([ϕ ∘ f]) = (E ⊕ 𝕎) − (𝕎 ⊕ 𝕌) = E − 𝕌 (restricted to X).

Proof of the injectivity of i is analogous to the non-equivariant case. □

Remark 3.2.6. One knows that R(G) is generated as an abelian group by the classes ξ_{1}, …, ξ_{m} of the irreducible representations of G. Later on, especially in Section 5, the group ring of this abelian group will play an important role in our computations. For this reason, we shall write R(G) multiplicatively, i.e., its elements will be expressed as Laurent monomials ξ_{1}^{n_{1}}ξ_{2}^{n_{2}} ⋯ ξ_{m}^{n_{m}} for n_{i} ∈ ℤ. Example 3.2.7. Let X = pt so that X^{+} = S^{0} and K_{G}(X) = R(G) and K_{G}(X)_{o} = R(G)_{o} = the kernel of the augmentation map R(G) ⟶ ℤ. Then the previous proposition shows that

π_{0}((BU_{G})^{G}) = [S^{0}, (BU_{G})^{G}] = [S^{0}, BU_{G}]_{G} ≅ R(G)_{o}.  (3.8)

Note that in the multiplicative notation above we can write R(G)_{o} as

R(G)_{o} = {ξ_{1}^{n_{1}} ⋯ ξ_{m}^{n_{m}} | ∑ n_{i} dim ξ_{i} = 0}  (3.9)

i.e., as the multiplicative group of “Laurent monomials of weighted degree zero” in the irreducible representations of G.

3.3. Topological groups of cycles. For any vector space V the monoid C(V) completes, in the naïve Grothendieck sense, to become the free abelian group Z(V) of all codimension-v cycles on ℙ(V ⊕ V) with topology induced from C(V). In [16] it is proved that this naïve topological group completion C(V) ⟶ Z(V) is equivalent to the homotopy-theoretic one C(V) ⟶ ΩBC(V).

In this section we carry this result over to the current context, proving that the induced map of L(U_{G})-spaces C(U_{G}) ⟶ Z(U_{G}) is a G-group completion equivalent to the one constructed above. This result is useful for computations and localization.

To begin we note that the assignment V ↦ Z(V), together with the join pairing as above, defines a GI_{∗}(U_{G})-monoid which is additively group complete. Furthermore, from [16] we have the following.

Proposition 3.3.1. The induced map C(U) ⟶ Z(U) of direct limit spaces is a nonequivariant additive group completion.

Proof. Homotopy and homology group functors commute with direct limits. □

Let us consider the equivariant case. The following result follows from the freeness of C(V):

Lemma 3.3.2. For every G-module V and subgroup H ≤ G, the naïve topological group completion of the monoid C(V)^{H} of effective cycles fixed by H is the group of fixed cycles Z(V)^{H} of the naïve group completion Z(V) of C(V). In particular, the naïve group completion of C(V)^{H} is a Hausdorff topological group with the natural quotient topology.

Proof. Obvious. □

For each h ∈ H, the action h : C(V) ⟶ C(V) preserves degree and is an algebraic map. Therefore, the intersection C_{d}^{v}(ℙ(V ⊕ V)) ∩ C(V)^{H} is an algebraic variety, and the restriction of the monoid operation to the pieces of bounded degree is an algebraic map. This implies that C(V)^{H} is a properly c-graded monoid in the sense of Definition 4.1 of [16]. Algebraically C(V)^{H} is a free monoid generated by H-orbits of irreducible subvarieties of C(V). Therefore, by [16, Thm. 4.4] the naïve group completion C(V)^{H} ⟶ Z(V)^{H} is equivalent to the homotopy-theoretic one. Thus the assignment V ⟶ Z(V) defines a ℤaugmented GI_{∗}(U_{G})-space taking values in the category of abelian topological groups, and the natural inclusions C(V) ↪ Z(V) induce a map of augmented GI_{∗}(U_{G})-monoids so that for every H ∈ G and V ⊂ U_{H}, the map C(V)^{H} ↪ Z(V)^{H} is an additive group completion. This gives the following.

Proposition 3.3.3. The map of L(U_{G})-spaces C(U_{G}) ⟶ Z(U_{G}) is a G-group completion.

Remark 3.3.4.

a. The GI_{∗}-functors (C, ♯) and (Z, ♯) determine G-spectra k_{G}^{C}(∗) and k_{G}^{Z}(∗) respectively. Furthermore, the natural transformation (C, ♯) ⟶ (Z, ♯) induces a morphism j : k_{G}^{C}(∗) ⟶ k_{G}^{Z}(∗) of G-spectra and a commutative diagram

C(U) ⟶^{α_{0}} k_{G}^{C}(0)
j_{0}↓  ↓^{j}
Z(U) ⟶^{α} k_{G}^{Z}(0)

where by Proposition 2.4.5, α_{0} is an additive G-group completion and α is a G-homotopy equivalence. By 3.3.3 j_{0} is a G-group completion. It follows that j : k_{G}^{C}(∗) ⟶ k_{G}^{Z}(∗) is an equivalence of G-spectra.

b. The functors Z_{G} suffice for the construction of all spectra we shall deal with. Nevertheless, we need the properties of the Chow monoid functor as an algebraic geometric object, together with the Proposition 3.3.3, in order to perform the computations in Chapter 5.

c. It is clear from its very definition that the homotopies introduced in the proof of Proposition 3.1.3 extend to degree preserving homotopies in the level of Z_{G}. In particular, Proposition 3.1.3 remains true if one replaces C_{G} by Z_{G} in its statement.

Our next step is the analysis of the inclusion C(V)^{G} ⊂ C(V)^{H} for a subgroup H ≤ G. In this case C(V)^{G} is a closed submonoid of C(V)^{H}, and C(V)^{G} ∩ C_{d}^{v}(ℙ(V ⊕ V)) is a subvariety of C(V)^{H} ∩ C_{d}^{v}(ℙ(V ⊕ V)).

We prove the following Proposition 3.3.5. The sequence

Z(V)^{G} ⟶ Z(V)^{H} ⟶ Z(V)^{H}/Z(V)^{G}

is a principal fibration. In particular, after taking limits, one gets a principal fibration

Z(U)^{G} ⟶ Z(U)^{H} ⟶ Z(U)^{H}/Z(U)^{G}

Proof. The proposition would be a straightforward consequence of Theorem 5.2 of [16] if the following condition were satisfied: C(V)^{G} is freely generated by a subset of the generators of C(V)^{H}. However, this assertion is not true.

On the other hand, the role played by such condition in the proof of the aforementioned theorem was to show that the group completion Z(V)^{G} is a closed subgroup of Z(V)^{H} and that the quotient Z(V)^{H}/Z(V)^{G} is filtered by cofibrations under the quotient topology. This is clearly satisfied in the present situation.

Now, the proof of the existence of a local cross section for the quotient map Z(V)^{H} ⟶ Z(V)^{H}/Z(V)^{G} follows mutatis mutandis the proof of Theorem 5.2 of [16]. □

3.4. Further constructions and KO-theory. Let T be a GI_{∗}(U_{G})-monoid and let K be a finite group which acts on T in the sense that it acts on T (V) by monoid morphisms, for all V ∈ I_{∗}(U_{G}), and its action commutes with the action of G and is compatible with the structure maps of T. Example 3.4.1. Let K be the center of G.

One can naturally define two I_{∗}(U_{G})-submonoids of T associated to the action of K as follows.

Definition 3.4.2.

T^{K} T^{K} T^{K} : (Fixed point functor) For every V ∈ I_{∗}(U_{G}) let T^{K}(V) denote the submonoid of T (V) consisting of those elements in T (V) which are fixed by the group K.

KT KT KT : (Averaged elements functor) For every V ∈ I_{∗}(U_{G}) let KT (V) denote the submonoid of T^{K} consisting of the “averaged” elements over K. More precisely,

KT (V):={ ∑_{z∈K} z_{♯}σ | σ ∈ T (V)}.

Proposition 3.4.3.

a: T^{K} is a GI_{∗}(U_{G})-submonoid of T.

b: KT is a GI_{∗}(U_{G})-submonoid of T^{K}. Furthermore, KT is an ideal of T^{K}, in the

sense that

ω(T^{K}(V) , KT (W)) ⊆ KT (V ⊕ W),

for all V, W ∈ I_{∗}(U_{G}).

Proof. a: Given g ∈ G and c , c′ ∈ T (V) one has

g_{♯}(c + c′) = g_{♯}c + g_{♯}c′  (3.10)

and

\begin{aligned}g_{♯} ∘ ω(c, c′) & = g_{♯} (τ(c ♯_{ℂ} c′)) \\ & = τ ∘ g_{♯} (c ♯_{ℂ} c′) = τ ((g_{♯}c) ♯_{ℂ} (g_{♯}c′)) \\ & = ω(g_{♯}c , g_{♯}c′). \end{aligned}  (3.11)

The combination of (3.10) and (3.11) proves the assertion.

b: For c ∈ T^{K}(V) and c′ ∈ KT (W) one has, by definition: h_{♯}c = c for all h ∈ K and c′ = ∑_{h∈K} h_{♯}c′′, for some c′′ ∈ T (W).

Therefore, using 3.10 and 3.11 one obtains

ω(c, c′) = ω(c , ∑_{h∈K} h_{♯}c′′)

= ∑_{h∈K} ω(c , h_{♯}c′′) = ∑_{h∈K} ω(h_{♯}c , h_{♯}c′′)

= ∑_{h∈K} h_{♯}ω(c, c′′) ∈ KT (V ⊕ W).

□

Using the results above and the topological bar construction of [19, Defn. 2.2], one can define the following I_{∗}(U_{G})-monoid. Definition 3.4.4. Given a group K acting on the GI_{∗}(U_{G})-monoid T as above, and V ∈ I_{∗}(U_{G}), let (T/K)(V) be the monoid defined as

(T/K)(V):=B(T^{K}(V), KT (V), ∗),

where B(−, −, −) denotes the toplogical triple bar construction, and where KT (V) is acting on T^{K}(V) by right translations.

Proposition 3.4.5. The assignment V ↦ (T/K)(V) naturally inherits the structure of a I_{∗}(U_{G})-monoid from T.

Proof. It follows straightforwardly from Proposition 3.4.3 and [19, §2]. □

A simple example of this set-up arises, in the same spirit of [3], when one uses the real linear isometries operad and Chow monoids to construct RO(G)-graded theories.

We denote by I_{∗}(U_{G}) the category of real finite-dimensional inner product spaces of a fixed real G-universe U and their linear isometries. Complexification (plus sesquilinear extension of inner products) identifies I_{∗}(U_{G}) as a subcategory of I_{∗}(U_{G} ⊗ℂ). In particular, every GI_{∗}(U_{G} ⊗ ℂ)-functor becomes a GI_{∗}(U_{G})-functor.

Consider now the Chow GI_{∗}-monoid C described in (3.1). Note that for any real orthogonal representation V ⊂ U_{G} the G-monoid C_{G}^{v}(ℙ(V_{ℂ} ⊕ V_{ℂ})) (where V_{ℂ} = V ⊗ ℂ) admits a ℤ/2ℤ action induced by complex conjugation which commutes with the action of G. We now invoke Definition 3.4.4 with K = ℤ/2ℤ. Definition 3.4.6. By the real Chow functor we mean C_{ℝ}:=C/(ℤ/2ℤ) with its natural structure of a I_{∗}(U_{G})- monoid provided by 3.4.5.

In parallel fashion one can construct a real Grassmann functor Gr_{ℝ} together with a natural transformation (Gr_{ℝ}, ⊕) ⟶ C_{ℝ}, ♯) of GI_{∗}(U_{G})- functors. This induces a map

Gr_{ℝ}(U_{G}) ⟶ C_{ℝ}(U_{G})  (3.12)

of L(U_{G})-spaces. One checks directly that Gr_{ℝ}^{v}(V) is exactly the Grassmannian of real codimension-v planes in V ⊕ V where v = dim_{ℝ}(V). Therefore, just as in §3.2 we may identify

BO(U_{G}):=Gr_{ℝ}(U_{G}) = \underrightarrow{lim}_{V} Gr_{ℝ}(V)

with the classifying space for reduced KO_{G}-theory. In fact we have the following analogue of 3.2.5. Proposition 3.4.7. For any compact G-space X, the following sequence is exact:

0 ⟶ [X_{+}, BO(U_{G})]_{G} ⟶^{i} KO_{G}(X) ⟶^{ϕ} ℤ ⟶ 0,

and hence [X_{+}, BO(U_{G})]_{G} ≅ KO_{G}(X)_{o} = the reduced KO_{G}-group of X.

4. Main Results

In this chapter we apply the machinery and constructions developed above in order to obtain the cohomology theories of primary interest in this paper, and to study their properties.

4.1. The equivariant cohomology theories. Definition 4.1.1. For each finite group G, let ℨ_{G} be the L(U_{G})-ring-spectrum associated to the GI_{∗}(U_{G})- functor Z introduced in §3.3, and let 𝔐_{G} denote the multiplicative spectrum associated to Z(U_{G})_{1}, as in Proposition 2.4.6. Denote by δ_{G} : ℨ_{G} ⟶ bℨ_{G} and ϵ_{G} : 𝔐_{G} ⟶ b𝔐_{G} the natural maps into their associated Borel theories, respectively; cf. Example 2.1.3.

Remark 4.1.2. It follows from Proposition 3.3.3 and universal properties of group completions that ℨ_{G} is the same spectrum as the one associated with the Chow monoid functor of Proposition 3.1.2.

We first describe the non-equivariant properties of our spectra.

Theorem 4.1.3.

a. The underlying non-equivariant spectra ℨ and 𝔐 associated to ℨ_{G} and 𝔐_{G}, respectively, are independent of the group G.

b. The zero-th space ℨ(0) of ℨ is a weak product ∏_{j≥0} K(ℤ, 2j) of Eilenberg-MacLane spaces whose additive structure is the usual one, and whose multiplicative structure is induced by the cup product.

c. The map Z(U)_{1} ⟶ 𝔐(0) is a homotopy equivalence and the underlying non-equivariant spectrum 𝔐 is the BLLMM spectrum of [3]. In particular, 𝔐(0) ≅ 1 × ∏_{j≥1} K(ℤ, 2j) and its infinite loop space structure enhances the cup product operation.

Proof. a. Let α : {e} ↪ G be the inclusion. By definition, ℨ = α^{♯}ℨ_{G}. On the other hand, it follows from Propositions 3.1.3 and 2.3.1, and Remark 3.1.4 that α^{♯}ℨ_{G} = ℨ_{e} is the spectrum associated to the non-equivariant Chow monoid functor C of Definition 3.1.1, independently of the group G. Similar argument applies to 𝔐.

b. This follows from the previous item, together with [11] and [13].

c. This follows from item (a) and [3]. □

The coherence properties with respect of changes of groups are expressed in the following result:

Theorem 4.1.4.

a. For every finite group G, the natural map Z_{G}(U_{G}) ⟶ ℨ_{G}(0) is an equivariant homotopy equivalence.

b. For a given finite group G, the collections of spectra {ℨ_{H} | H ∈ G} and {𝔐_{H} | H ∈ G} form coherent families of spectra, in the sense of Remark 2.3.2. In particular, if H ∈ G and α : H ⟶ G is the inclusion, then α^{♯}ℨ_{G} = ℨ_{H} and α^{♯}𝔐_{G} = 𝔐_{H}, and both ℨ_{G} and 𝔐_{G} are split spectra.

Proof. a. This follows from Propositions 2.4.5 and 3.3.3.

b. This follows from Propositions 3.1.3, 2.3.1 and Remark 3.3.4(c). □

The cohomology theories ℨ_{G}^{∗} and 𝔐_{G}^{∗} (see Definition 2.1.5) are quite rich and intriguing, and even the computation of their coefficients present several technical difficulties. On the other hand, their associated Borel theories are much more approachable and amenable to calculations, as the following results indicate.

Theorem 4.1.5. Let X be a compact G-space.

a. The zero-th cohomology ring bℨ_{G}^{0}(X) is isomorphic to the completion at the augmentation ideal of the ordinary even Borel cohomology ring of X, with coefficients in ℤ. In other words,

bℨ_{G}^{0}(X) ≅ ∏_{j≥0} H_{G}^{2j}(X, ℤ)  (4.1)

with the multiplicative structure induced by the cup product.

b. If k > 0 is a positive integer, then

bℨ_{G}^{−k}(X) ≅ ∏_{j≥0} H_{G}^{2j−k}(X, ℤ)  (4.2)

as a module over bℨ_{G}^{0}(X).

Proof. a. The usual adjunction gives [X_{+}, bℨ_{G}(0)]_{G} ≅ [X_{+} × EG_{+}, ℨ_{G}(0)]_{G} and the fact that the spectrum is split shows that the latter group is [X_{+} ∧_{G} EG_{+}, ℨ(0)]. It follows from Theorem 4.1.3(b) that bℨ_{G}^{0}(X) ≅ ∏ H^{2j}(X ×_{G} EG, ℤ), proving the assertion.

b. For k > 0, the same argument shows that bℨ_{G}^{−k}(X) ≅ ∏ H^{2j}((S^{k}X) ×_{G} EG, ℤ), and since (S^{k}X_{+}) ∧_{G} EG_{+} ≅ S^{k}(X_{+} ∧_{G} EG_{+}), the usual suspension isomorphism implies that bℨ_{G}^{−k}(X) ≅ ∏ H_{G}^{2j−k}(X, ℤ). The module structure is the evident one. □

Similarly, one has the following theorem. Theorem 4.1.6. Let X be a compact G-space.

a. The zero-th cohomology group b𝔐_{G}^{0}(X) is isomorphic to the “units” of bℨ_{G}^{0}(X). In other words,

b𝔐_{G}^{0}(X) ≅ 1 × ∏_{j≥1} H_{G}^{2j}(X, ℤ)  (4.3)

under the cup product structure.

b. If k > 0 is positive integer, then

b𝔐_{G}^{−k}(X) ≅ ∏_{j≥1} H_{G}^{2j−k}(X, ℤ)  (4.4)

with the usual additive structure.

Proof. a. The proof of this assertion follows the same steps of the proof of the corresponding statement in the previous theorem.

b. Write an element x ∈ 1×∏_{j≥1} H^{2j}(S^{k}(X ×_{G}EG), ℤ) as a formal sum 1+x_{1}+x_{2}+⋯, with x_{j} ∈ H^{2j}(S^{k}(X×_{G}EG), ℤ), so that the group operation follows the usual mutiplication law for formal power series. Notice that given x, y ∈ 1 × ∏_{j≥1} H^{2j}(S^{k}(X ×_{G} EG), ℤ) one has x_{i} ∪y_{j} = 0, for all i, j ≥ 1, since these are singular cohomology classes on the suspension of some space. It follows that x ∪ y = 1 + (x_{1} + y_{1}) + (x_{2} + y_{2}) + ⋯, and the theorem follows. □

As a corollary one computes the coefficients of the associated Borel theories bℨ_{G}^{∗} and b𝔐_{G}^{∗}, namely, the cohomology groups bℨ_{G}^{k}(G/K) and b𝔐_{G}^{k}(G/K), where k ∈ ℤ and G/K denotes an orbit space for a subgroup K ⊆ G. Recall that for a G-space X, one denotes by π_{n}^{K}(X) the homotopy group [S_{K}^{n}, X]_{G}, where S_{K}^{n} is the “generalized sphere” G/K_{+} ∧ S^{n}; cf. [14, I.1]. One can similarly define π_{n}^{K}(k_{G}) for any G-spectrum k_{G}.

Corollary 4.1.7. a. Given a subgroup K of the finite group G, there is a ring isomorphism

bℨ_{G}^{0}(G/K) = π_{0}^{K}(bℨ_{G}) ≅ ∏_{j≥0} H^{2j}(BK, ℤ),

where the multiplicative structure is induced by the cup product on singular cohomology. b. For k > 0, one has bℨ_{G}^{k}(G/K) = 0 and

bℨ_{G}^{−k}(G/K) = π_{k}^{K}(bℨ_{G}) ≅ ∏_{j≥0} H^{2j−k}(BK, ℤ)

as a graded module over π_{0}^{K}(bℨ_{G}).

c. The group b𝔐_{G}^{0}(G/K) = π_{0}^{K}(b𝔐_{G}) is isomorphic to 1 × ∏_{j≥1} H^{2j−k}(BK, ℤ) under the cup product operation. Furthermore, for k > 0, one has b𝔐_{G}^{k}(G/K) = 0 and b𝔐_{G}^{−k}(G/K) = π_{k}^{M}(bℨ_{G}) ≅ ∏_{j≥1} H^{2j−k}(BK, ℤ) with its usual additive structure.

The results above show, in particular, that the “units” 1 × ∏_{j≥1} H_{G}^{2j}(X, ℤ) of the even Borel cohomology of X are the zero-th group of the R(G)-graded equivariant cohomology theory b𝔐_{G}^{∗}.

4.2. Total Chern classes and equivariant Segal questions. Combining Propositions 3.2.1 and 3.3.3 one obtains a map of L(U_{G})-spaces BU_{G} ⟶ Z(U_{G})_{1} which in turn induces a map c_{G} : 𝔎𝔲_{G} ⟶ 𝔐_{G} of associated spectra. Passing to Borel counterparts then induces maps of the associated Borel spectra. We collect the various transformations obtained by this construction in the following definition. Definition 4.2.1. For each finite group G, the map of spectra c_{G} : 𝔎𝔲_{G} ⟶ 𝔐_{G} described above is called the equivariant total Chern class. The composition 𝔎𝔲_{G} ⟶^{c_{G}} 𝔐_{G} ⟶^{ϵ_{G}} b𝔐_{G}, denoted by c_{G}^{b} : 𝔎𝔲_{G} ⟶ b𝔐_{G}, is called the total equivariant Borel-Chern class. We denote by ĉ_{G} : b𝔎𝔲_{G} ⟶ b𝔐_{G} the map induced by c_{G}. These maps fit into a commutative diagram

[figure: 𝔐_{G}; b𝔐_{G}; 𝔎𝔲_{G}; b𝔎𝔲_{G}; ϵ′_{G}; ĉ_{G}; c_{G}; c_{G}^{b}; ϵ_{G}]

The next result represents an extension to the equivariant situation of a conjecture of G. Segal [23], which was positively answered in the non-equivariant case in [3]. This result justifies the terminology in Definition 4.2.1. Theorem 4.2.2. The zero-th level map c_{G}^{b} : 𝔎𝔲_{G}^{0}(X) = K_{G}(X)_{o} ⟶ b𝔐_{G}^{0}(X) coincides with the total Chern class map from the equivariant K- theory of X to its Borel cohomology.

Proof. Let E ⟶ X be an equivariant virtual vector bundle of degree zero over X. The pull back bundle pr_{1}^{∗}E ⟶ X × EG, under the projection onto first term, descends to a virtual bundle (pr_{1}^{∗}E)/G over X ×_{G} EG. The total Chern class c((pr_{1}^{∗}E)/G) ∈ 1× ∏_{j≥1} H^{2j}(X ×_{G} EG; ℤ) is, by definition, the total Chern class of E in Borel cohomology.

In order to compare c((pr_{1}^{∗}E)/G) with c_{G}^{b} one has only to rephrase the previous paragraph in terms of classifying maps into the appropriate spaces, and use the corresponding definitions together with [13]. □

Corollary 4.2.3. The group of units in even Borel cohomology extends to a generalized equivariant R(G)-graded cohomology theory b𝔐_{G}^{∗}, which has the property that the total Chern class map extends to a map of equivariant cohomology theories.

4.3. Transfers. It is a general fact that whenever one has a coherent family of spectra, then the ‘transfers’ associated to equivariant coverings for the various cohomology theories in the family, are functorialy related; cf. [14, IV.4]. In general, given an equivariant spectrum k_{G} and an equivariant covering E ⟶^{ξ} X, we denote by τ_{ξ} : k_{G}^{∗}(E) ⟶ k_{G}^{∗}(X) the associated transfer homomorphism.

We illustrate the general picture with the following “classical” examples. Consider the trivial bundles ξ : EG × G/H ⟶ EG and ζ : G/H ⟶ pt, together with the evident equivariant bundle map ϵ : ξ ⟶ ζ, and let θ : BH ⟶ BG denote the non-equivariant bundle, with fiber G/H obtained as the quotient ξ/G : BH = EG ×_{G} G/H ⟶ EG/G = BG.

The following result summarizes the relations among the various transfer homomorphisms associated with these bundles. Theorem 4.3.1. Given a subgroup H ⊂ G, one has a commutative diagram

[figure: R(G)_{o} = 𝔎𝔲_{G}^{0}(pt); 𝔎𝔲^{0}(BG); R(H)_{o} = 𝔎𝔲_{H}^{0}(pt); 𝔎𝔲^{0}(BH); 𝔐_{G}^{0}(pt); b𝔐_{G}^{0}(pt); 𝔐_{H}^{0}(pt); b𝔐_{H}^{0}(pt); ϵ′_{G}^{∗}; ϵ_{G}^{∗}; ĉ_{G}; c_{G}; Ind_{H}^{G}; τ_{θ}; τ_{ξ}; τ_{ξ}; ϵ′_{H}^{∗}; ĉ_{H}; ϵ_{H}^{∗}; c_{H}; c_{G}^{b}; c_{H}^{b}]

where the horizontal maps in the front and back faces of the cube are natural maps into respective Borel theories, the vertical maps are transfer homomorphisms and all the other maps are Chern classes. In particular, Ind_{H}^{G} is the induction homomorphim in representation theory and the total Chern class c_{H}^{b}(Ind_{H}^{G}(V)) of an induced representation is given by applying the transfer τ_{ξ} to the Chern class c_{g}^{b}(V) of V.

Proof. The groups appearing in the diagram and their relation with the bundles above are explained by observing that if k_{G}^{∗} denotes either ℨ_{G}^{∗} or 𝔐_{G}^{∗}, and H ⊂ G, then one can identify k_{G}^{0}(G/H) with k_{H}^{0}(pt). This follows from the coherence properties of our spectra, which give the isomorphisms k_{G}^{0}(G/H) = [G/H_{+}, k_{G}(0)]_{G} ≅ [S^{0}, α^{♯}k_{G}(0)]_{H} ≅ [S^{0}, k_{H}(0)]_{H} = k_{H}^{0}(pt), where α : H ↪ G is the inclusion.

The commutativity follows directly from the coherence properties of our spectra. See [14, IV.4.8]. The identification of the transfer in K-theory with Ind_{H}^{G} is given in [20]. □

Note that the transfers above, when restricted to zero-th cohomology groups are multiplicative, in the sense that they commute with the ordinary cup product. In general, there are several other non-trivial properties which can be inferred from the fact that the transfers come from a generalized cohomology theory. See [20].

4.4. Analogues in the real case. All the results established in this chapter can be carried over to the real case by using the constructions and results of section §3.4 together with the results of [10].

The real Chow functor C_{ℝ} defined in (3.4.6) has a naïve group completion Z_{ℝ}. This functor determines an equivariant ring spectrum ℨ_{ℝ} and a multiplicative spectrum 𝔐_{ℝ} in analogy with the complex case above. The underlying non-equivariant spectra are independent of the group G. The zero’th space ℨ_{ℝ}(0) is non-equivariantly the weak product

∏_{j≥0} K(ℤ/2ℤ, j)

with multiplicative structure induced by the cup product (cf. [10]). Furthermore, we have the non-equivariant equivalence 𝔐_{ℝ} ≅ {1} × ∏_{j≥1} K(ℤ/2ℤ, j) with infinite loop space structure which enhances the cup product. The analogues of Theorems 4.1.4, 4.1.5, 4.1.6 and Corollary 4.1.7 hold for these theories.

The transformation

Gr_{ℝ}(U_{G}) ⟶ Z_{ℝ}(U_{G})

of L(U_{G})-spaces induces a map of equivariant spectra

w : 𝔎𝔬_{G} ⟶ ℨ_{ℝ,G}

called the equivariant total Stiefel-Whitney class. Composition with ℨ_{ℝ,G} ⟶ bℨ_{ℝ,G} gives an equivariant total Stiefel-Whitney class in the associated Borel theory. In the zero-th level one has an equivariant total Stiefel-Whitney class with values in the Borel cohomology with ℤ/2ℤ-coefficients, which coincides with the classical one. Thus the equivariant version of the Segal question holds for this Borel Stiefel-Whitney class.

The results on transfers also hold in this case.

5. Coefficients

In this chapter we shall compute the coefficients ℨ_{G}^{0}(pt) = [S^{0}, Z(U)]_{G} = π_{0}(Z(U)^{G}) of our equivariant theory ℨ_{G} in the case where G is abelian. Our main result, Theorem 5.2.5, states that in this case the coefficients come from a simple graded ring functor ℍ^{∗}(G) with particularly nice properties, and they always embed into ∏_{k} H^{2k}(G; ℤ). To carry out these computations we use techniques of degeneration by torus actions on Chow varieties.

5.1. General results for abelian groups. In this section we derive general results relating π_{0}(Z(U)^{G}) of the theory to the representation ring of G, where U = U_{G} stands for our fixed complete G-universe. A key factor in this computation is Theorem 3.3.3 which allows one to work with the Chow monoids C(W) themselves in order to approximate Z(U). Our main trick is to use the equivariant degeneration of varieties under action of a maximal complex torus commuting with the action of G.

The first result is a basic reduction formula. Start with a compact abelian (Lie) group G and a finite dimensional G-module W . Fix p ≥ 0 and denote by Gr_{p}(W) the Grassmannian of projective p-planes in ℙ(W). Let C_{p}(W) denote the Chow monoid of effective p-cycles in ℙ(W). The inclusion i_{W} : Gr_{p}(W) ⟶ C_{p}(W) is a G-equivariant map and induces a continuous monoid homomorphism j_{W} : ℤ^{+} {Gr_{p}(W)^{G}} ⟶ C_{p}(W)^{G} (where ℤ^{+}Y = ∐_{d} SP^{d}Y denotes the free abelian topological monoid on the space Y). Elements of Gr_{p}(W)^{G} are invariant p-planes, and elements of ℤ^{+} {Gr_{p}(W)^{G}} are called invariant linear p-cycles.

For each (p+1)-dimensional invariant linear subspace U ⊂ W we consider the submonoid

D_{0}(U):=ker (j_{U})_{∗} ⊂ π_{0}ℤ^{+} {Gr_{p}(U)^{G}}

where (j_{U})_{∗} denotes the induced map on π_{0}. These are the relations on homotopy classes of sums of G-invariant hyperplanes induced by homotopy through invariant divisors. Let

D(U) ⊂ π_{0}ℤ^{+} {Gr_{p}(W)^{G}}  (5.1)

be the image of D_{0}(U) induced by the inclusion U ⊂ W. Theorem 5.1.1. For any compact abelian group, the map

(j_{W})_{∗} : π_{0}ℤ^{+} {Gr_{p}(W)^{G}} ⟶ π_{0} {C_{p}(W)^{G}}

induced by j_{W} is a monoid surjection whose kernel is generated by the “divisor” relations D(U) for U ∈ Gr_{p+1}(W)^{G}.

To prove this result we need a preliminary lemma. Consider ℙ^{N} = ℙ(ℂ^{N+1}) with homogeneous coordinates [w] = [w_{0} : w_{1} : ⋯: w_{N}] and the canonical action of the complex N-torus 𝕋 = (ℂ^{×})^{N+1}/Δ^{×} given by

ϕ_{t}([w]) = [t_{0}w_{0} : t_{1}w_{1} : ⋯: t_{N}w_{N}],  (5.2)

where Δ^{×} denotes the diagonal subgroup (t, t, …, t), for t ∈ ℂ^{×}. This action on ℙ^{N} induces an action on each component of the Chow monoid C_{p}(ℙ^{N}) for any p ≥ 0.

Fix each integer k ≥ 0 and for each multi-index I = {i_{0} < i_{1} < ⋯ < i_{p}} of length k + 1 consider the linear subspace

ℂ^{I} = {w ∈ ℂ^{N+1} : w_{i} = 0 if i ≠ i_{k} for some k}.

The subspace ℙ_{I}^{p} = ℙ(ℂ^{I}) is called the I^{th} coordinate k-plane. It is clearly 𝕋-invariant. Proposition 5.1.2. Fix p ≥ 0 and d > 0. Then (1) The fixed-point set of 𝕋 acting on C_{p,d}(ℙ^{N}) is finite and consists of coordinate linear

cycles ∑_{I} n_{I}ℙ_{I}^{p} where each n_{I} ∈ ℤ^{+} and ∑ n_{I} = d.

(2) Let σ be an irreducible, 𝕋-fixed algebraic curve on C_{p,d}(ℙ^{N}). Then an open, dense subset of σ can be written as a parameterized curve of the form

σ(t) = ∑_{i} n_{i}σ_{i}(t) t ∈ S

where S is an irreducible algebraic curve, and where for each i, n_{i} ∈ ℤ and

⋃_{t} |σ_{i}(t)| ⊆ ℙ_{I}^{p+1}

for some coordinate (p + 1)-plane ℙ_{I}^{p+1}. In particular, each cycle σ_{i}(t) is a divisor in ℙ_{I}^{p+1}.

Proof. Let 𝕃 ≅ ℂ^{N} denote the Lie algebra of 𝕋 and for each x ∈ ℙ^{N} let 𝕃(x) ⊂ T_{x}(ℙ^{N}) denote the tangent space to the 𝕋-orbit 𝕋(x) at x. Set

Σ_{p} = {x ∈ ℙ^{N} : dim_{ℂ} 𝕃(x) ≤ p}.

then it is straightforward to see that

Σ_{p} = ⋃_{I} ℙ_{I}^{p}.  (5.3)

Now if c ∈ C_{p,d}(ℙ^{N}) is 𝕋-fixed, then its support |c| is a 𝕋-invariant algebraic subset of dimension p. Let V be an irreducible component of |c| and fix any x ∈ Reg(V). Then V is 𝕋-invariant and so

𝕃(x) ⊂ T_{x}V.

It follows (since dim(T_{x}V) = p) that x ∈ Σ_{p}. We conclude that |c| ⊂ Σ_{p} and (1) now follows from (5.3).

For (2) we begin by observing that the subvariety

W := ⋃_{c∈σ} |c| ⊂ ℙ^{N}

is 𝕋-invariant and of dimension p + 1. It therefore follows as above that

W ⊂ Σ_{p+1} = ⋃_{I} ℙ_{I}^{p+1}.  (5.4)

Now the irreducibility of σ together with the finiteness of the fixed-point set of 𝕋 on C_{p,d}(ℙ^{N}) imply that σ is the closure of the 𝕋-orbit of a generic point c in its support. Thus W is the closure of the set

W^{0}:= ⋃_{t∈𝕋/𝕋_{c}} |t_{∗}c| ⊂ ⋃_{I} ℙ_{I}^{p+1},

where 𝕋_{c} is the isotropy subgroup of c. We set S:=𝕋/𝕋_{c} and note that S embeds as an open dense subset of σ (as the 𝕋-orbit of c). Consequently σ can be written generically as the parameterized curve σ(t) = t_{∗}c for t ∈ S. Write c = ∑_{i} n_{i}V_{i} where the n_{i} are positive integers and the V_{i} are irreducible p-dimensional subvarieties of ℙ^{N}. Then

σ(t) = t_{∗}c = ∑ n_{i}t_{∗}V_{i} ≡ ∑_{i} n_{i}σ_{i}(t)  for  t ∈ S.

Each V_{i} is contained in some ℙ_{I}^{p+1} by (5.4), and so the curve σ_{i} is also contained in ℙ_{I}^{p+1}. □

Proof. [of Theorem 5.1.1] Choose homogeneous coordinates [w] = [w_{0} : w_{1} : ⋯: w_{N}] so that each coordinate line is G-invariant. (This is possible since the irreducible representations of G are all 1-dimensional.) Let 𝕋 be the complex N-torus acting on ℙ(W) via this coordinate decomposition as in (5.2). Note that the actions of G and 𝕋 commute. It follows for example that 𝕋 preserves the components of ℤ^{+}(Gr_{p}(W))^{G} and C_{p}(W)^{G}.

We first prove surjectivity. Fix c ∈ C_{p}(W)^{G}. It is a basic result (cf. [2]) that the closure of this orbit \overline{𝕋(c)} contains 𝕋-fixed points. (This result is usually stated for 𝕋-actions on smooth varieties. However each component of C_{p}(W) admits a 𝕋-equivariant embedding into some projective space ℙ^{M}. In fact the standard Chow embedding is 𝕋-equivariant. One can then ignore C_{p}(W) and apply the theorem to 𝕋-orbits in ℙ^{M}.)

Now by Proposition 5.1.2(1) every 𝕋-fixed cycle is a sum of G-invariant coordinate planes. This proves the surjectivity in Theorem 5.1.1.

We now prove the second assertion. Suppose a, b ∈ ℤ^{+}(Gr(W)^{G}) lie in the same connected component of C_{p,d}(W)^{G} under the inclusion j_{W}. Then they can be joined in C_{p,d}(W)^{G} by a connected algebraic curve σ. Consider σ as a 1-cycle in C_{p,d}(W)^{G}, and hence as a point in some Chow variety C_{1}(C_{p,d}(W)^{G}) ⊂ C_{1}(ℙ^{M}). Choose a fixed-point σ_{0} in the orbit closure \overline{𝕋(σ)} of σ in C_{1}(ℙ^{M}). The point σ_{0} corresponds to a 𝕋-fixed 1-cycle in ℙ^{M}. Now, since 𝕋 preserves the subvariety C_{p,d}(W)^{G} ⊂ ℙ^{M}, we have |t_{∗}σ| ⊂ C_{p,d}(W)^{G} for all t ∈ 𝕋, and therefore |σ_{0}| ⊂ C_{p,d}(W)^{G}. Since each curve t_{∗}σ is connected, so is σ_{0}, and since a and b are 𝕋-fixed, we have a, b ∈ |σ_{0}|. Thus |σ_{0}| is a connected, 𝕋-invariant algebraic curve in C_{p,d}(W)^{G} joining a to b. Each irreducible component of σ_{0} is 𝕋-invariant. Furthermore, any point in the intersection of two distinct components must be a 𝕋-fixed point, i.e., an element of ℤ^{+}(Gr_{p}(W)^{G}) by Proposition 5.1.2(1). The result now follows from Proposition 5.1.2(2) □

Suppose now that W is a complex linear G-space of dimension w. Then we can apply Theorem 5.1.1 to the map

Gr(W):=Gr_{w−1}(W ⊕ W) ⟶ C_{w−1}(W ⊕ W):=C(W)  (5.5)

Let Z(W) = Z_{w−1}(W ⊕ W) denote the naïve group completion of C(W). Note that for any space Y there is an isomorphism ℤ^{+}π_{0}(Y^{G}) ≅ π_{0}ℤ^{+}(Y^{G}). Therefore taking group completions in (5.1) gives us the following.

Corollary 5.1.3. For any invariant subspace W ⊂ U the natural inclusion induces a surjective homomorphism of abelian groups

ℤπ_{0} {Gr(W)^{G}} ⟶ π_{0} {Z(W)^{G}}

whose kernel is generated by the “divisor” relations D(U) for U ∈ Gr_{w}(W ⊕ W)^{G}. We now consider the direct limits

BU:= \underrightarrow{lim}_{W ⊂U}Gr(W)  and  ZU:= \underrightarrow{lim}_{W ⊂U}Z(W)  (5.6)

and recall that under the join pairing, ZU and ZU^{G} are homotopy-commutative rings. Corollary 5.1.3 immediately gives the following result. Theorem 5.1.4. For any compact abelian group G, the inclusion (5.5) induces a surjective ring homomorphism

ℤπ_{0} {BU^{G}} ⟶ π_{0} {ZU^{G}}

whose kernel is generated by the images of the “divisor” relations D(U) in the direct limit.

5.2. Computations for finite groups. In this section we shall explicitly compute π_{0} {ZU^{G}} for finite G. The main point is to compute the relations D(U) given by invariant divisors; cf. (5.1). We set the stage by recalculating π_{0}(BU^{G}) in a specific manner which is subsequently generalized to cycles. This approach is similar to calculations in [22], and its relevance here lies on its extension to arbitrary cycles.

Let G be a finite (not necessarily abelian) group of order γ and let BU be defined as in (5.6). We want to compute

π_{0}(BU^{G}) = π_{0} \{\underrightarrow{lim}_{W ⊂U}Gr(W)^{G}\} .

Let ξ_{1}, …, ξ_{m} denote the distinct irreducible complex representations of G and set |ξ_{i}| = dim ξ_{i}. For each invariant w-plane P ⊂ W ⊕W there is a unique G-equivalence P ≅ ⨁ ξ_{i}^{⊕a_{i}} where ∑ a_{i}|ξ_{i}| = w. Furthermore, P is homotopic to P′ ≅ ⨁ ξ_{i}^{⊕a′_{i}} through G-invariant planes if and only if a_{i} = a′_{i} for all i. (this follows from the continuity of characters.) Thus if W ≅ ⨁ ξ_{i}^{⊕n_{i}} we see that

π_{0} {Gr(W)^{G}} ≅ {(a_{1}, …, a_{m}) ∈ ℤ^{m} : 0 ≤ a_{i} ≤ 2n_{i} and ∑ a_{i}|ξ_{i}| = w} .

We now set

V = ξ_{1} ⊕ ⋯ ⊕ ξ_{m}

and consider the cofinal family V^{⊕n} = V ⊕ ⋯ ⊕ V in U. Then

\begin{matrix}π_{0} \{Gr(V^{⊕n})^{G}\} & ≅ F_{n} \\ := & \{(a_{1}, …, a_{m}) ∈ ℤ^{m} : 0 ≤ a_{i} ≤ 2n and ∑ a_{i}|ξ_{i}| = nv\} \end{matrix}

where v = dim V = ∑ |ξ_{i}|. Since U is a limit of V^{⊕n} ⊕ V^{⊕n}, we see that π_{0}(BU^{G}) = \underrightarrow{lim} F_{n} under the family of inclusion maps ⋯ ⊂ F_{n} ⊂ F_{n+1} ⊂ … given by

(a_{1}, …, a_{m}) ↦ (a_{i} + 1, …, a_{m} + 1),  (5.7)

since at each stage we sent P ∈ Gr(V^{⊕n}) to P ⊕ (V ⊕ {0}) ≅ P + (1, 1, ⋯, 1) ∈ Gr(V^{⊕n}). We now write R(G) as the free abelian group with generators ξ_{1}, ⋯, ξ_{m}, and we write this multiplicatively as in 3.2.7. Recall the subgroup

R(G)_{0} = {ξ_{1}^{a_{1}} … ξ_{m}^{a_{m}} ∈ R(G) : ∑ a_{i} dim ξ_{i} = 0} .

of representations of virtual dimension zero. Then we have the following; cf. (3.9). Proposition 5.2.1. R(G)_{0} ≅ \underrightarrow{lim} F_{n} = π_{0} (BU^{G}).

Proof. The map F_{n} ↪ R(G)_{0} given by

(a_{1}, ⋯, a_{m}) ↦ ∏_{i=1}^{m} ξ_{i}^{(a_{i}−n)}

is compatible with the inclusions (5.7) and therefore induce a mapping \underrightarrow{lim} F_{n} → R(G)_{0} which is easily seen to be an isomorphism. □

By Proposition 5.2.1 we have

\begin{matrix}ℤπ_{0} \{(BU)^{G}\} & = ℤ [R(G)_{0}] \\ = ℤ [ξ_{1}, ⋯, ξ_{m}, ξ_{1}^{−1}, ⋯, ξ_{m}^{−1}]_{0} \\ :=ℤ [ξ, ξ^{−1}]_{0} \end{matrix}  (5.8)

where ℤ [ξ, ξ^{−1}]_{0} denotes the ring of all Laurent polynomials in ξ_{1}, ⋯, ξ_{m} of degree zero, i.e., integral linear combinations on monomials ξ_{1}^{a_{1}} … ξ_{m}^{a_{m}} with ∑ a_{i} = 0.

We now assume that G is abelian so that m = γ. For convenience we shall reindex the irreducible representations by ξ_{ρ} for ρ ∈ Ĝ where Ĝ ≅ G denotes the character group of G. Thus we have

V = ⨁_{ρ∈_{G}} ξ_{ρ}.

Fix now an element a^{→} = {a_{ρ}}_{ρ∈_{G}} ∈ ℤ^{+}Ĝ, and consider the corresponding invariant subspace

U = U(a^{→}) = ⨁_{ρ} ξ_{ρ}^{⊕a_{ρ}} ⊂ V^{⊕n} ⊕ V^{⊕n}  where
dim(U) = nγ + 1 = ∑ a_{ρ}

For fixed d let Div_{d}(U) denote the set of divisors of degree d on ℙ(U). Then Div_{d}(U) = ℙ(S^{d}(U)) where S^{d}(U) is the d^{th} symmetric tensor power of U. Note that Div_{d}(U) is a G-space with fixed-point set

Div_{d}(U)^{G} = ∐_{ρ∈_{G}} ℙ (S^{d}(U)_{ρ})  (5.9)

where S^{d}(U)_{ρ} ⊂ S^{d}(U) denotes the G-invariant subspace with character ρ. Observe also that for σ ∈ Ĝ,

S^{d}(U)_{σ} = _{⨁ ; ∑ d_{ρ}=d ; ∑ d_{ρ}ρ=σ} {⨂_{ρ∈_{G}} S^{d_{ρ}} (ξ_{ρ}^{⊕a_{ρ}})} .  (5.10)

Now the coordinate linear divisors lying in ⨂_{ρ} S^{d_{ρ}} (ξ_{ρ}^{⊕a_{ρ}}) correspond to monomials ∏_{ρ} t_{ρ}^{d_{ρ}} where t_{ρ} is the coordinate on ξ_{ρ}. This divisor on ℙ(U(a^{→})) can be written as the linear sum of hyperplanes ∑ d_{ρ} Div(t_{ρ}) = ∑ d_{ρ}ℙ(U(a^{→} − e_{ρ}^{→})) where e_{ρ}^{→} = {δ_{ρρ′}}_{ρ′} and where δ_{ρρ′} = 1 if ρ = ρ′ and 0 otherwise. In the notation of (5.8) this is the coordinate linear cycle

∑_{ρ} d_{ρ} ∏_{σ} ξ_{σ}^{(a_{σ}−δ_{σρ})} = (∏_{σ} ξ_{σ}^{a_{σ}})(∑_{ρ} d_{ρ}ξ_{ρ}^{−1}).

By (5.9) we see that two such cycles are equivalent in π_{0}{CU^{G}} if they correspond to the same invariant subspace S^{d}(U)_{σ}. By (5.10) this means that

ξ a^{→} ∑_{ρ} d_{ρ}ξ_{ρ}^{−1} ≡ ξ a^{→} ∑_{ρ} d′_{ρ}ξ_{ρ}^{−1} in π_{0}\{CU\}  if  \{ \begin{matrix}∑_{ρ} d_{ρ} = ∑_{ρ} d′_{ρ} & and \\ ∑_{ρ} d_{ρ}ρ = ∑_{ρ} d′_{ρ}ρ \end{matrix}(5.11)

where ξ a^{→} ≡ ∏_{σ} ξ_{σ}^{a_{σ}}. We now pass to the group completion ZU of CU. From (5.11) and Theorem 5.1.4 it is straightforward to calculate the following.

Lemma 5.2.2. The homomorphism ℤπ_{0}{BU^{G}} ⟶ π_{0}{ZU^{G}} descends to an isomorphism

ℤ[ξ, ξ^{−1}]_{0}/D ≅ π_{0}{ZU^{G}}

where D is the ideal generated by the elements

ξ_{1} ∑_{ρ} d_{ρ}ξ_{ρ}^{−1} for \{ \begin{matrix}∑_{ρ} d_{ρ} = 0 & and \\ ∑_{ρ} d_{ρ}ρ = 0 \end{matrix}.

We now make a change of variables by setting ζ_{ρ}:=ξ_{r}/ξ_{1} for ρ ∈ Ĝ. Note that ζ_{1} = 1. This change of variables induces an isomorphism

ℤ[ξ, ξ^{−1}]_{0} ≅ ℤ[ζ, ζ^{−1}].

The isomorphism of Lemma 5.2.2 now becomes the isomorphism

π_{0}{ZU^{G}} ≅ ℤ[ζ, ζ^{−1}]/D  (5.12)

where D is the ideal generated by the elements

∑_{ρ} d_{ρ}ζ_{ρ}^{−1}  for \{ \begin{matrix}∑_{ρ} d_{ρ} = 0 & and \\ ∑_{ρ} d_{ρ}ρ = 0 \end{matrix}.  (5.13)

Note that the ring π_{0}{ZU^{G}} is graded by the augmentation deg : π_{0}{ZU^{G}} → ℤ, corresponding to projective degree, which sends ∑ d_{α}ζ^{α} to ∑ d_{α}.

We now make a further change of variables by setting

ζ_{ρ}^{−1} := 1 + x_{ρ}  and  ζ_{ρ}:= 1 + x̄_{ρ}

with the convention that x_{1} = x̄_{1} = 0. The element x_{ρ} is called the first Chern class of the representation ρ ∈ Ĝ in π_{0}{ZU^{G}} .

Proposition 5.2.3. In the variables x_{ρ}, x̄_{ρ} the isomorphism of Lemma 5.2.2 becomes

π_{0}{ZU^{G}} ≅ ℤ[x, x̄]/D

where D is the ideal generated by the relations (1 + x_{ρ})(1 + x̄_{ρ}) = 1 for ρ ∈ Ĝ and

∑_{ρ} d_{ρ}x_{ρ} = 0 whenever ∑_{ρ} d_{ρ}ρ = 0 in Ĝ.

Proof. The first set of relations reflect the fact that ζ_{ρ}ζ_{ρ}^{−1} = 1. The second is deduced easily from (5.13) □

Definition 5.2.4. Let ℍ^{∗}(G) ⊂ ℤ[x, x̄]/D ≅ π_{0}{ZU^{G}} denote the subring with unit generated by the classes x_{ρ}. Note that ℍ^{∗}(G) = ℤ[x]/D_{0} where D_{0} is the homogeneous ideal

D_{0} := ⟨∑ d_{ρ}x_{ρ} | ∑ d_{ρ}ρ = 0 in Ĝ⟩ .  (5.14)

Note that ℍ^{∗}(G) inherits a natural grading from the usual one on ℤ[x]. Theorem 5.2.5. ℍ^{∗}(G) is a graded ring functor on the category of finite abelian groups with the following properties.

(1) ℍ^{0}(G) ≅ ℤ and ℍ^{1}(G) ≅ Ĝ.

(2) ℍ^{∗}(G ⊕ G′) = ℍ^{∗}(G) ⊗ ℍ^{∗}(G′).

(3) If G is cyclic, then there exists a graded ring isomorphism

ℍ^{∗}(G) ≅ H^{2∗}(G; ℤ)

with the cohomology of G with coefficients in the trivial G- module ℤ. For any finite abelian group G there is a natural isomorphism

π_{0}{ZU^{G}} ≅ ℍ^{∗}(G)_{M}  (5.15)

where M ⊂ ℍ^{∗}(G) is the multiplicative system generated by the total Chern classes {1 + x_{ρ}}_{ρ∈_{G}} of the irreducible representations of G.

Proof. For (1) consider the homomorphism ϕ : ℤ ⟨x_{ρ}⟩_{ρ≠1} → Ĝ given by ϕ(∑ d_{ρ}x_{ρ}) = ∑ d_{ρ}ρ. By (5.14) this evidently descends to the desired isomorphism. To prove (3), suppose G = ℤ/mℤ and let x_{1}, …, x_{m−1} be the generators of ℤ[x]. Then D_{0} is generated by the relations x_{d} = dx_{1} for d = 2, …, m − 1. Thus setting x = x_{1}, one sees directly that ℍ^{∗}(ℤ/mℤ) = ℤ[x] ⟨mx⟩ ≅ H^{2∗}(ℤ/mℤ; ℤ).

For (2) we write ℍ^{∗}(G) = ℤ[x]/D_{0} and ℍ^{∗}(G′) = ℤ[y]/D′_{0} as in (5.2.4). Then ℍ^{∗}(G ⊕ G′) = ℤ[x, y]/E_{′} where

\begin{aligned}E_{′} & = ⟨∑ d_{ρ}x_{ρ} + ∑ d′_{ρ′}y_{ρ′} | ∑ d_{ρ}ρ + ∑ d′_{ρ′}ρ′ = 0 in Ĝ ⊕ Ĝ′⟩ \\ & = ⟨∑ d_{ρ}x_{ρ} | ∑ d_{ρ}ρ = 0 in Ĝ⟩ + ⟨∑ d′_{ρ′}y_{ρ′} | ∑ d′_{ρ′}ρ′ = 0 in Ĝ′⟩ = D_{0} + D′_{0}. \end{aligned}

This proves the first part of the theorem. The second part follows from Proposition 5.2.3. □

Note 5.2.6. For cyclic groups (in fact for products of cyclic groups whose orders are pairwise coprime) the functors ℍ(G) and H(G; ℤ) coincide, but in general ℍ(G) is simpler. There is however a canonical embedding of graded rings ℍ^{∗}(G) ⊂ H^{2∗}(G; ℤ). Note 5.2.7. Let ℍ̂^{∗}(G) = ∏_{k} ℍ^{k}(G) and Ĥ^{2∗}(G; ℤ) = ∏_{k} H^{2k}(G; ℤ) denote the completions of ℍ^{∗}(G) and H^{2∗}(G; ℤ) with respect to the augmentation ideals. Then there are natural embeddings

ℍ^{∗}(G)_{M} ⊂ ℍ̂^{∗}(G) ⊂ Ĥ^{2∗}(G; ℤ)  (5.16)

Combining (5.14) and (5.15) gives an embedding π_{0} {ZU^{G}} ⊂ Ĥ^{2∗}(G; ℤ) which is parallel to the fact that non-equivariantly we have ZU ≅ ∏_{j} K(ℤ, 2j) and so ZU classifies the functor Ĥ^{2∗}(•; ℤ)

5.3. The general case. The calculations above indicate that π_{0} {ZU} might be a useful functor for finite groups, and it would be interesting to compute it in some non-abelian cases.

The higher coefficients π_{k} {ZU} (and more generally π^{V} {ZU} for representations V of G) are also interesting and there are some tools in place for trying to compute them. In particular, there is an Equivariant Algebraic Suspension Theorem proved in [12] which asserts the following. Let G be a finite group of order γ, and suppose V is a complex G– module. Let V_{0} denote the regular representation of G. Then for any p < dim V, there is a sequence of continuous G-homomorphisms

\begin{matrix}Z_{p}(ℙ(V)) ⟶^{Σ /^{V_{0}}} & Z_{p+γ}(ℙ(V ⊕ V_{0})) ⟶^{Σ /^{V_{0}}} … ⟶^{Σ /^{V_{0}}} \\ Z_{p+nγ}(ℙ(V ⊕ V_{0}^{n})) ⟶^{Σ /^{V_{0}}} Z_{p+(n+1)γ}(ℙ(V ⊕ V_{0}^{n+1})) ⟶^{Σ /^{V_{0}}} … \end{matrix} (5.17)

which associate to a cycle c the join Σ /^{V_{0}}(c) = ℙ(V_{0})♯c. The theorem asserts that there is an integer n, depending explicitly on V and p, so that after the n^{th} iteration, all maps in (5.17) are G-homotopy equivalences.

If these maps were G-homotopy equivalences from the first iteration, then we would be able to compute all higher coefficients in the theory, since this computation would be reduced to the equivariant Dold-thom Theorem established in [15]. However, this is not true. The integers n in the Suspension Theorem are nearly sharp as shown by many examples.

6. Embellishments and open problems

We now examine some extensions of the results developed above. We begin by showing that our constructions can be carried out meaningfully on any algebraic variety. Thereby, one enhances the total Chern class in morphic cohomology developed in [6]

6.1. Applications to algebraic varieties. The basic constructions of Chapter 3 can be carried into algebraic geometry as follows. Let X be a projective algebraic variety of dimension n with a finite group of automorphisms G. Fix a complete G-universe U. Then to each G-module V ⊂ U we associate the abelian topological monoid

C_{X}(V) := Mor(X, C(V))

where C(V) is Chow monoid defined in (3.1); cf. [6] and [7]. The mappings C(f) defined in (3.2), and the biadditive join pairing defined in (3.3) extend in an obvious way to these monoids, and the arguments of §3.1 carry over directly to prove that (C_{X}, ♯) is a GI_{∗}-functor.

Let Z_{X}(V) denote the naïve topological group completion of the monoid C_{X}(V). It is proved in [7] that Z_{X}(V) is (non-equivariantly) homotopy equivalent to ΩBC_{X}(V). Just as above we see that (Z_{X}, ♯) is a GI_{∗}-functor, and the obvious inclusion (C_{X}, ♯) ⊂ (Z_{X}, ♯) yields a transformation of GI_{∗}- functors. Therefore, passing to the direct limits gives a map

C_{X}(U) ⟶ Z_{X}(U)

of L(U)-spaces which can be shown to be an additive G-group completion as in 3.3.3. Definition 6.1.1. The G-ring spectrum ℨ_{X} associated to the L(U)-space Z_{X}(U) is called the morphic spectrum of X.

We denote by 𝔐_{X} the associated multiplicative morphic spectrum with the property that the map Z_{X}(U)_{1} ⟶ 𝔐_{X}(0) is a G-group completion with respect to the cup product.

When G = e, the spectrum ℨ_{X} corresponds to the morphic cohomology of X introduced in [6]. In the notation of [6] we have

Z_{X}(U) = lim_{q→∞} Z^{q}(X; ℙ^{q}) = Z^{∞}(X; ℙ^{∞}) ≅ ∏_{s≥0} Z^{s}(X)

and therefore

π_{i}Z_{X}(U) = ⨁_{s≥0} L^{s}H^{2s−i}(X)

(cf. [6, Thm 2.10]). This direct sum is finite since L^{s}H^{k}(X) = 0 whenever k > 2n (see [6, 9.8]). Furthermore, if X is smooth, then L^{s}H^{2s−i}(X) ≅ H^{2s−i}(X; ℤ) whenever s ≥ n by [7]. When X is smooth we also have that

π_{0}Z_{X}(U) = ⨁_{s≥0} L^{s}H^{2s}(X) ≅ A_{0} × A_{1} × A_{2} × ⋯ × A_{n}

where A_{p} denotes the group of algebraic p-cycles on X modulo algebraic equivalence (see [7, Thm 5.1]). This shows that in general Z_{X}(U) depends strongly on the algebraic structure of X.

Note that by the above π_{0}Z_{X}(U) is a finitely graded, commutative ring with unit. It follows that when G = e, the space Z_{X}(U)_{1} is group-like with respect to the cup-product. Thus, in this case, there is a homotopy equivalence 𝔐_{X}(0) ≅ Z_{X}(U)_{1}. If X is smooth, then by duality [7] we have π_{0}𝔐_{X}(0) ≅ {1} × A_{n−1} × ⋯ × A_{0}.

From the discussion above we see that the coefficients in the theory ℨ_{X} could be justifiably called the equivariant morphic cohomology of X.

We now replace C_{X}(V) with the GI_{∗}-subspace

Gr_{X}(V) := Mor(X, Gr(V)),

and note that the direct limit Gr_{X}(U) is a L(U)-space. Definition 6.1.2. The G-spectrum 𝔎𝔲_{X} associated to the L(U)-space Gr_{X}(U) is called the Grassmann spectrum of X.

In the language of [6] we have that Gr_{X}(V) = Vect^{v}(X), and it is shown there that π_{0}Gr_{X}(V) coincides with algebraic equivalence classes of algebraic vector bundles of rank v on X which are generated by their global sections. Thus Gr_{X}(U) is the additive monoid of stable equivalence classes of such bundles under Whitney sum. The map Gr_{X}(U) → 𝔎𝔲_{X}(0) into the zero’th space of the spectrum is a G-group completion. In particular we see that (non-equivariantly) π_{0}𝔎𝔲_{X}(0) represents the group completion K_{alg}(X)_{o} of the monoid of equivalence classes of algebraic vector bundles generated by their global sections.

Observe now that the inclusion

(Gr_{X}, ⊕) ⟶ (C_{X}, ♯)

is a transformation of GI_{∗}-functors, which induces a map

Gr_{X}(U) → Z_{X}(U)_{1}

of L(U)-spaces. This in turn induces a map

c : 𝔎𝔲_{X} ⟶ 𝔐_{X}

of G-spectra, called the total Chern class in equivariant morphic cohomology

It is interesting to examine the “continuous analogue” of these constructions given by the transformation Map(X, Gr(V)) ⟶ Map(X, Z(V)) of GI_{∗}-functors. Here we use the notation Map(X, Y) instead of F (X, Y), for compatibility with [6]. Note that there is a commutative diagram of GI_{∗}-transformations

Mor(X, Gr(V)) ⟶ Mor(X, C(V))
↓  ↓
Map(X, Gr(V)) ⟶ Map(X, Z(V))

leading to a commutative diagram of L(U)-spaces

Gr_{X}(U)  ⟶  Z_{X}(U)_{1}
↓  ↓
Map(X, Gr(U)) ⟶ Map(X, Z(U)_{1})

and therefore a diagram of G-spectra

𝔎𝔲_{X}  ⟶  𝔐_{X}
↓  ↓
Map(X, 𝔎𝔲) ⟶ Map(X, 𝔐).

Suppose X is smooth and G = e. Then applying π_{0} at level zero in this diagram gives

K_{alg}(X)_{o} ⟶ {1} × ∏_{k=n−1}^{0} A_{k}(X)
↓  ↓
K(X)_{o} ⟶  H^{2∗}(X; ℤ).

where the top horizontal map is the total Chern class map into algebraic cycles modulo algebraic equivalence under intersection product [5], and the bottom horizontal arrow is the standard total Chern class [6]. The preceding diagram of G-spectra is a full extension of this classical diagram to equivariant theories.

Appendix A. The Coherence Theorem

The point of this appendix is to provide the following.

Proof. [of Proposition 2.3.1]

We first observe that any morphism θ : H ⟶ H′ in G has one of the following three forms:

(1) θ = α_{H′,H} : H ↪ H′ is an inclusion of subgroups;

(2) θ = β_{H} : H ⟶ {e} is a projection onto the trivial subgroup;

(3) θ is a composition of the type H ⟶^{β_{H}} {e} ⟶^{α_{H′,e}} H′. Therefore, we only need to define η_{θ} in the cases (1) and (2), satisfying suitable conditions, and use the natural equivalence of functors β^{♯}α′^{♯} ≅ (α′β)^{♯} = θ^{♯} to define η_{θ} in the third case by the commutativity of the diagram

β^{♯}α′^{♯}k_{H′}  ⟶^{β^{♯}η_{α′}} β^{♯}k_{e}
≅↓  ↓^{η_{β}}
θ^{♯} = (α′β)^{♯}k_{H′} ⟶_{η_{α′β}=η_{θ}} k_{H},

where α′ = α_{H′,e} and β = β_{H}.

Let us first recall how k_{H} is defined. Given an H-module V ⊂ U_{H}, let D_{V}^{H} denote the monad associated to the operad D_{V}^{H} = C_{V} × L(U_{H}), where C_{V} is the little discs operad (cf. [4, 2.2(b)]). By definition

k_{H}(V) = lim_{\begin{matrix}W ⊂U_{H} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{V} , D_{V}^{H}, X_{H}).  (A.1)

The first case (θ = α_{H′,H} : H ↪ H′) is quite simple, since j_{θ} : θ^{∗}U_{H′} ⟶ U_{H} is the identity map regarded as an H-module map. We will define a map τ_{θ} : θ^{∗}k_{H′} ⟶ j_{θ}^{∗}k_{H} in HS(θ^{∗}U_{H′}) and then apply the functor j_{θ∗} to τ_{θ} to obtain η_{θ} : θ^{♯}k_{H′} = j_{θ∗}θ^{∗}k_{H′} ⟶^{j_{θ∗}(τ_{θ})} (j_{θ∗}j_{θ}^{∗}k_{H}) = k_{H}. The last identification comes from the fact that j_{θ∗} = (j_{θ}^{−1})^{∗} in this particular situation.

The map τ_{θ} : θ^{∗}k_{H′} ⟶ j_{θ}^{∗}k_{H} is defined as follows. Given V ⊂ θ^{∗}U_{H′} then (θ^{∗}k_{H′})(V) = k_{H′}(V) as an H-space. On the other hand, there is a natural identification of monads D_{V}^{H} = D_{V}^{H′} in the category of H-spaces, when V is seen as an H-module and L(U_{H′}) is identified to L(U_{H}) as an H-operad. Therefore the identification θ^{∗}X_{H′} ⟶ X_{H} induces compatible homeomorphisms of H-spaces Ω^{W −V} B(Σ^{W} , D_{W}^{H′}, X_{H′}) ⟶ Ω^{W −V} B(Σ^{W} , D_{W}^{H} , X_{H}) for any W ⊂ U_{H} such that V ⊂ W. Furthermore, once one observes that the family of H′- submodules of U_{H′} is cofinal in the family of H-submodules of U_{H′}, then the above maps induce an isomorphism of spectra τ_{θ} : θ^{∗}k_{H′} ⟶ j_{θ∗}k_{H} when one passes to the appropriate limit; cf. (A.1). Therefore, η_{θ} = j_{θ∗}(τ_{θ}) is also an isomorphism, in this case. It is a tautology to check that when θ : H ⟶ H′ and θ′ : H′ ⟶ H′′ are inclusions, then the desired diagram commutes.

The definition of η_{θ}, when θ = β_{H} : H ⟶ {e}, is a bit more involved. Given W ⊂ U_{e}, denote by D̃_{W}^{H} the operad C_{W} ×L(U_{H})^{H}, and let D̃_{W}^{H} be its associated monad. The inclusion D̃_{W}^{H} ⟶ D_{W}^{e} is a local equivalence (see [18, §3]) of E_{∞}-operads which induces equivalences of spaces:

B(Σ^{W} , D̃_{W}^{H} , X_{e}) ⟶^{≅} B(Σ^{W} , D_{W}^{e} , X_{e}),

for all W ⊂ U_{e}. These equivalences assemble to define

Φ_{V} : lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{e}) ⟶^{≅} lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D_{W}^{e} , X_{e}):=k_{e}(V). (A.2)

We now introduce a series of intermediate natural maps. The given map f_{H} : X_{e} ⟶ X_{H}^{H} of L(U_{H})^{H}-spaces induces maps

B(Σ^{W} , D̃_{W}^{H} , X_{e}) ⟶ B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H})

for each W ⊂ U_{e}, which in turn define

f_{H∗} : lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{e}) ⟶ lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H}).  (A.3)

On the other hand, the inclusions L(U_{H})^{H} ↪ L(U_{H}) and j_{β} : β^{∗}U_{e} ⟶ U_{H} induce maps

B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H}) ⟶ B(Σ^{j_{β}W} , D_{j_{β}W}^{H} , X_{H}^{H})^{H},

for each W ⊂ U_{e}, with compatible associated maps

Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H}) ⟶ {Ω^{j_{β}W −j_{β}V} B(Σ^{j_{β}W} , D_{j_{β}W}^{H} , X_{H}^{H})}^{H} .

One then obtains, by passing to limits on both sides, a map:

Ψ_{V} : lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H}) ⟶ lim_{\begin{matrix}U⊂U_{H} \\ j_{β}(V)⊂U \end{matrix}} \{Ω^{U−j_{β}V} B(Σ^{U}, D_{U}^{H}, X_{H}^{H})\}^{H} . (A.4)

Finally, one can use j_{β} to define a non-equivariant map

Ω^{W −V} B(Σ^{W} , D_{W}^{e} , X_{e}) ⟶ Ω^{j_{β}W −j_{β}V} B(Σ^{j_{β}W} , D_{j_{β}W}^{H} , X_{H}),

where X_{e} is identified with X_{H} and D_{W}^{e} with D_{j_{β}W}^{H} when the action of H is ignored. This induces the equivalence

Ξ_{V} : k_{e}(V) = (βα)_{♯}k_{e}(V) ⟶ β_{♯}α_{♯}k_{e}(V) = lim_{\begin{matrix}U⊂U_{H} \\ j_{β}(V)⊂U \end{matrix}} Ω^{U−j_{β}V} B(Σ^{U}, D_{U}^{H}, X_{H}), (A.5)

where we use the particular description of j_{α} = j_{α_{H,e}} to obtain the equality on the right hand side of the equation. The following diagram commutes

lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{e}) ⟶_{(A.2)}^{Φ_{V}} lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D_{W}^{e} , X_{e})
f_{H∗}↓^{(A.3)}  ∥
lim_{\begin{matrix}W ⊂U_{e} \\ V ⊂W \end{matrix}} Ω^{W −V} B(Σ^{W} , D̃_{W}^{H} , X_{H}^{H})  k_{e}(V)
Ψ_{V} ↓^{(A.4)}  ^{(A.5)}↓^{Ξ_{V}}
lim_{\begin{matrix}U⊂U_{H} \\ V ⊂U \end{matrix}} \{Ω^{U−j_{β}V} B(Σ^{U}, D_{U}^{H}, X_{H})\}^{H} ⟶_{⊂} lim_{\begin{matrix}U⊂U_{H} \\ j_{β}V ⊂U \end{matrix}} Ω^{U−j_{β}V} B(Σ^{U}, D_{U}^{H}, X_{H})
∥  ∥
β_{♯}k_{H}(V)  ⟶^{β_{♯}ξ_{α}}  β_{♯}α_{♯}k_{e}(V),

where ξ_{α} : k_{e} ⟶ α_{♯}k_{H} is the adjoint of η_{α}, with α = α_{H,e}.

Define ξ_{β,V}:=Ψ_{V} ∘ f_{H∗} ∘ Φ_{V}^{−1} and notice that we have proven that the diagram

k_{e}(V) ⟶^{id} (βα)_{♯}k_{e}(V)
ξ_{β,V} ↓  ≅↓^{Ξ_{V}}
β_{♯}k_{H}(V) ⟶_{β_{♯}ξ_{α}} β_{♯}α_{♯}k_{e}(V)

commutes. If η_{β} : β^{♯}k_{e} ⟶ k_{H} denotes the adjoint of the map of spectra defined by ξ_{β}, we conclude that the adjoint diagram

β^{♯}(α^{♯}k_{e}) ⟶^{β^{♯}η_{α}} β^{♯}k_{H}
≅↓  ↓^{η_{β}}
(αβ)^{♯}k_{e} ⟶^{η_{αβ}} k_{e}

also commutes. A simple verification now shows the commutativity of all required diagrams.

□

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(Lawson) SUNY @ Stony Brook

E-mail address: blawson@sbccmail.bitnet

(Lima-Filho) Texas A&M University

E-mail address: plfilho@math.tamu.edu

(Michelsohn) SUNY @ Stony Brook

E-mail address: mmichelsohn@sbccmail.bitnet