RATIONAL O(2)-EQUIVARIANT COHOMOLOGY THEORIES.

J.P.C.GREENLEES

Abstract. We use Greenlees [1996, 1996a] to give a complete description of rational O(2)- equivariant cohomology theories.

1. Introduction.

This paper presents an algebraic model for O(2)-equivariant rational cohomology theories. The main mathematical input is the model for SO(2)-equivariant theories given in Part I of Greenlees [1996], together with special cases of results on rational Mackey functors from Greenlees [1996a]. Using these ingredients, and the usual formal framework of equivariant homotopy, it is not hard to assemble the model. Nonetheless, several features of the model were unexpected to the author, so it seems desirable to illustrate the issues which arise in this simple case. An exactly analagous model can be given with the continuous quaternion group as the group of equivariance, but it is most unlikely that such an explicit model can be given for groups of larger rank.

First, we should explain that the cohomology theories we refer to are those admitting suspension isomorphisms for arbitrary representations, often known as ‘RO(G)-graded’ or ‘genuine’ cohomology theories. These are represented by G-spectra as in Lewis et al [1986], and more specifically by those indexed on a complete G-universe. The advantages of considering representing objects rather than the cohomology theories are well understood, and we work with G-spectra without further comment, or more precisely in the homotopy category of G-spectra, in the sense of Quillen. We work exclusively with rational cohomology theories, and rational G-spectra, without always displaying this in the notation. Thus G-spectra denotes the homotopy category of rational G-spectra, and [X, Y]^{G} denotes the rational vector space of morphisms in this category. The G-spectrum representing the equivariant cohomology theory Y_{G}^{∗}(·) is denoted Y and the correspondence is given by Y_{G}^{∗}(X) = [X, Y]_{G}^{∗} in the usual way.

For brevity we let G = O(2), T = SO(2) and W = G/T. The other closed subgroups of G are the normal cyclic subgroups C_{n} for n ≥ 1, and the dihedral groups. For each n ≥ 1 there is a single conjugacy class of dihedral groups of order 2n, and we let D_{2n} denote the representative containing reflection in the x-axis. One calculates N_{G}(D_{2n}) = D_{4n}, so that the conjugacy class of D_{2n} is G-homeomorphic to the circle S(1/n) := G/D_{4n}. We may occasionally refer to T as the continuous cyclic group, and consider the set

𝒞 = {C_{n} | n ≥ 1} ∪ {T }

1The author is grateful to the Nuffield Foundation for support.

2This paper is in final form, and no substantially similar paper will be submitted elsewhere.

31991 Mathematics Subject Classification:

Primary 55N91 Secondary 55P42, 55P91.

of all cyclic groups. Similarly we may refer to G as the continuous dihedral group, and consider the set

𝒟 = {D | D is conjugate to D_{2n} for some n ≥ 1} ∪ {G}

of all dihedral groups. We consider 𝒞 as a discrete topological space, and 𝒟 as the union of concentric circles S(1/n) of radius 1/n with G at the centre. We consider the set of closed subgroups of G as the topological sum

𝒮_{f}G = 𝒞 ⨿ 𝒟.

The notation is chosen to correspond to that of Greenlees [1996a], and the subscript f refers to the fact that we have not used the Hausdorff metric topology. Note that G acts continuously on 𝒮_{f}G by conjugation.

Roughly speaking, the way to find the model of a rational G-spectrum X is to consider the family of geometric fixed point functors H ⟼ Φ^{H}X as giving an equivariant sheaf over 𝒮_{f}G. With this point of view it is obvious that we should separate behaviour over 𝒞 from that over 𝒟. It turns out that the model of X over 𝒞 is that of a T -spectrum with W -action, which is modelled by considering W -equivariant objects in either of the models of Greenlees [1996]. The model over 𝒟 splits as a product of Eilenberg-MacLane spectra, and is thus modelled by graded rational Mackey functors. By Greenlees [1996a], this is equivalent to using graded equivariant sheaves over 𝒟.

2. Burnside splitting.

Let ℱG denote the subspace of 𝒮_{f}G consisting of subgroups of finite index in their normalizers. (For conceptual purposes, it may be better to think of ℱG as the quotient of 𝒮_{f}G in which any pair of subgroups H ⊴ Ĥ with Ĥ/H a torus are identified.)

For any stable G-map f : S^{0} ⟶ S^{0}, the function function H ⟼ deg(Φ^{H}f) is a continuous equivariant function on 𝒮_{f}G, which passes to the quotient ℱG to give the isomorphism

ϕ : A(G) := [S^{0}, S^{0}]^{G} ⟶^{≅} C^{G}(ℱG, ℚ).

Thus the idempotents of A(G) correspond to open and closed G-invariant subspaces of 𝒮_{f}G.

In particular, if we let c and d denote the characteristic functions of 𝒞 and 𝒟, we have the decomposition 1 = c + d into orthogonal idempotents. Each idempotent acts on all G-spectra, and thus we have a corresponding decomposition

G-spectra ≃ (G, c)-spectra × (G, d)-spectra,

where the category (G, c)-spectra is the full subcategory of G-spectra X with X ≃ cX, and similarly for d.

It is easy to see by checking fixed points that this idempotent splitting is a splitting of the cofibre sequence EW_{+} ⟶ S^{0} ⟶ ẼW, so that cS^{0} = EW_{+} and dS^{0} = ẼW. Thus we have cX = EW_{+} ∧ X and dX = ẼW ∧ X for any G-spectrum X.

3. The cyclic part of the model.

In this section we describe the category of (G, c)-spectra.

RATIONAL O(2)-EQUIVARIANT COHOMOLOGY 3

Note that for any two G-spectra X and Y, we have an action of W on [X, Y]^{T} by conjugation. One way to package is this is via the action of W on G/T together with the change of groups isomorphism [X ∧ G/T_{+}, Y]^{G} = [X, Y]^{T}. Note that X ∧ EW_{+} is T -equivalent to X.

Proposition 3.1. If X and Y are G-spectra then the forgetful map induces a natural isomorphism

[EW_{+} ∧ X, Y]^{G} ⟶^{≅} {[X, Y]^{T} }^{W} .

Proof: Since any strictly equivariant map is homotopy equivariant, we have a natural map [EW_{+} ∧ X, Y]^{G} ⟶ {[X, Y]^{T} }^{W} . Since we are working over the rationals, both sides are cohomology theories of X, so that it suffices to deal with the case of a finite spectrum X. If X is finite, both sides preserve direct limits in the second variable, so it suffices to deal with the case when Y is finite. But in that case the spectral sequence

H^{∗}(W; [X, Y]^{T}) ⟹ [EW_{+} ∧ X, Y]^{G}

obtained from the skeletal filtration of EW_{+} is obviously convergent. Over the rationals, the spectral sequence collapses to give the required isomorphism.

Corollary 3.2. The category of (G, c)-spectra is equivalent to the category of T -spectra with an action of W:

(G, c)-spectra ≃ (T -spectra)[W].

Of course we can replace the category of T -spectra by either of the algebraic models from Greenlees [1996], to see that the category of (G, c)-spectra is the derived category of a suitable abelian category of objects with W -action. We are again using the fact that a W - action in a model category is equivalent to a W -action on the homotopy category, provided 2 is invertible.

4. 𝒟-Mackey functors.

Before describing the model of (G, d)-spectra, we must summarise the relevant properties of Mackey functors. This is the special case of the results of Greenlees [1996a] in which the group is O(2); alternatively, it is easy to deal with this special case directly.

The main result states that the category of rational Mackey functors M is equivalent to the category of equivariant sheaves V on 𝒮_{f}G equipped with restriction maps V (T) ⟶ V (C_{n}) for all n. More precisely, we should make 𝒮_{f}G into a topological category with one nonidentity morphism C_{n} ⟶ T for each n ≥ 1, and ask that V be a sheaf on the category. However, since we are only interested in sheaves supported on 𝒟, where there are no nonidentity morphisms, we do not need this extra precision. To be explicit, the stalks of V may be constructed from M as follows, if H is a finite subgroup V (H) = e_{H}M(H), where e_{H} ∈ A(H) is the idempotent of H supported at H, V (T) = M(T) and V (G) = lim_{→}_{e(G)=1} eM(G),

where the direct limit is over all idempotents e with e(G) = 1.

Notice that for any Mackey functor M, an idemptent e ∈ A(G) gives a new Mackey functor eM, defined by (eM)(H) = (eM)(G/H_{+}) := M(eG/H_{+}). In particular, the category of rational Mackey functors splits:

G-Mackey ≃ (G, c)-Mackey × (G, d)-Mackey.

In terms of sheaves, we find (G, c)-Mackey functors correspond to equivariant sheaves supported on the subcategory 𝒞: they are thus specified by W -modules V (T), V (C_{n}) for all n ≥ 1, and W -maps V (T) ⟶ V (C_{n}). We have no further need to consider these.

Similarly, we find (G, d)-Mackey functors correspond to equivariant sheaves supported on the space 𝒟. Since we need to perform calculations, it is useful to be more explicit. The stalks are V (G), and V (D) for finite dihedral groups D. By G-equivariance, the stalks V (D) are specified by W_{G}(D_{2n})-modules V (D_{2n}) for each n. Since W_{G}(D_{2n}) is of order 2, we abbreviate it too by W, relying on context to determine which group of order 2 is intended. Now the orbit of D_{2n} is isolated for each n, so the only piece of data required to recover the sheaf from its stalks is the way a germ from V (G) may be represented in some neighbourhood of G. Each germ is defined on D_{2n} for all sufficiently large n, so we consider the end space

V (D_{∞}) := lim_{→}_{i} ∏_{n≥i} V (D_{2n}),

of sections of 𝒟 \ {D} near infinity, and the behaviour of sections is described by a W -map σ : V (G) ⟶ V (D_{∞}).

Summary 4.1. A (G, d)-Mackey functor is given by a vector space V (G) together with W - modules V (D_{2n}) for n ≥ 1 and a W -map σ : V (G) ⟶ V (D_{∞}). This correspondence is natural and gives an equivalence of categories.

It is then not hard to give a sufficient condition for a Mackey functor to be injective, and to conclude that all (G, d)-Mackey functors have an injective resolution of length 1.

Lemma 4.2. A (G, d)-Mackey functor is injective if the map σ is an epimorphism, and hence the category of (G, d)-Mackey functors is of injective dimension 1.

Proof: Consider the category ℂ of diagrams 𝕍 = (V_{0} ⟶^{s} V_{1}) of ℚW -modules with V_{0} fixed by W. It is easy to see that an object is injective if and only if the structure map s is surjective. Indeed, given a monomorphism 𝔸 ⟶ 𝔹 and a map f : 𝔸 ⟶ 𝕍 we first choose an extension f̂_{1} of f_{1} over B_{1}, then choose a splitting V_{0} = V_{1} ⊕ V′_{0}. This determines the component of f̂_{0} mapping into V_{1}, and the other component may be any extension of the V′_{0} component of f_{0}.

In the case of a Mackey functor, we first choose the extensions f̂_{D} for the finite dihedral groups D. This gives the extension f̂_{1} for the object V (G) ⟶ V (D_{∞}), and we proceed as before.

It is now clear that an arbitrary sheaf V can be embedded in an associated injective sheaf I_{V} with I_{V} (D) = V (D) and I_{V} (G) = V (G) ⊕ V (D_{∞}). The cokernel is concentrated at G and thus injective.

For completeness we record the situation for projective functors, although we do not use it elsewhere.

Remark 4.3. By contrast, the surjectivity of σ is not a sufficient condition for the corresponding (G, d)-Mackey functor to be projective. However all (G, d)-Mackey functors admit projective resolution of length 1.

Proof: In the category ℂ, an object is projective if and only if s is a monomorphism. The problem is that the proof starts by lifting the component on V_{0}. If the constant sheaf is denoted \underline{ℚ}, one may check that the sheaf V with V (D) = ℚ, V (G) = \underline{ℚ}(D_{∞}) and whose structural map σ is the identity, is not projective.

However, the constant sheaf is projective, as are the sheaves supported away from G, and hence all (G, d)-Mackey functors admit projective resolution of length 1.

5. The dihedral part of the model.

We may now show that any (G, d)-spectrum splits as a product of Eilenberg-MacLane spectra. Indeed, we may use the same proof as given in Appendix A of Greenlees-May [1995] for finite groups.

Thus, for any injective rational Mackey functor I we may define a cohomology theory hI_{G}^{∗}(·) by

hI_{G}^{n}(X) = Hom(\underline{π}_{n}^{G}(X), I),

where the homotopy Mackey functor \underline{π}_{n}^{G}(X) is defined by H ⟼ π_{n}^{H}(X). Evidently, if I is supported over 𝒟, the cohomology theory vanishes on (G, c)-spectra.

Lemma 5.1. If I is an injective (G, d)-Mackey functor then hI is an Eilenberg-MacLane spectrum: hI = HI.

Proof: Since \underline{π}_{0}^{G}(G/H_{+}) = [·, G/H_{+}]^{G} is the free functor, it is clear that hI has the correct homotopy groups in degree 0. We must calculate hI_{G}^{n}(G/H_{+}) for each subgroup H, and show that it is zero if n ≠ 0. We have already observed that hI_{G}^{∗}(·) is zero on (G, c)-spectra, so this deals with the case in which H lies in 𝒞. This leaves us to deal with the cases H = G and H = D_{2n} for some n ≥ 1.

For this we need to examine the functor \underline{π}_{n}^{G}(G/H_{+}), which is made up from the groups π_{n}^{K}(G/H_{+}). Since I is a (G, d)-Mackey functor, the groups with K in 𝒞 have no effect, and we need only consider K = G and K = D_{2m} for some m ≥ 1.

The tom Dieck splitting theorem for the G-space X states

π_{n}^{G}(X) = π_{n}(X^{G}) ⊕ ⨁_{(D)} π_{n}(EW_{+} ∧_{W} X^{D})

⊕ π_{n}(EW_{+} ∧_{W} X^{T}) ⊕ ⨁_{C} π_{n}(E(G/C)_{+} ∧_{G/C} Σ^{ξ}X^{C}),

where ξ is the non-trivial representation of W on ℝ. The summands involving X^{L} with L in 𝒞 do not contribute, since I is concentrated over 𝒟. Now if X = G/G_{+} then X^{G} = X^{D} = S^{0}, and the relevant part is zero if n ≠ 0, since W has no positive dimensional homology over ℚ. If X = (G/D_{2n})_{+} then X^{G} = ∗, and X^{D} is either ∗ if D_{2n} is not subconjugate to D, or W_{+} if it is subconjugate. In either case the relevant part is zero if n ≠ 0.

Similarly, tom Dieck splitting for the G-space X states

π_{n}^{D_{2m}}(X)  =  ⨁_{(D)} π_{n}(EW_{+} ∧_{W} X^{D}) ⊕ ⨁_{C} π_{n}(E(D_{2m}/C)_{+} ∧_{D_{2m}/C} X^{C}),

where the sums are now over conjugacy classes of subgroups of D_{2m}. The argument that the relevant part is zero if n ≠ 0 is a little easier than that above.

Now if M is any (G, d)-Mackey functor, we may choose an injective resolution 0 ⟶ M ⟶ I ⟶ J ⟶ 0, and by definition of hI there is a map HM ⟶ hI lifting the embedding M ⟶ I. Thus by the lemma we have a cofibre sequence HM ⟶ hI ⟶ hJ, and this gives a Universal Coefficient Theorem.

Corollary 5.2. For any (G, d)-Mackey functor M there is a short exact sequence

0 ⟶ Ext(\underline{π}_{n−1}^{G}(X), M) ⟶ HM_{G}^{n}(X) ⟶ Hom(\underline{π}_{n}^{G}(X), M) ⟶ 0.

Corollary 5.3. Any (G, d)-spectrum X splits as a product of Eilenberg-MacLane spectra

X ⟶^{≃} ∏_{n} Σ^{n}H\underline{π}_{n}^{G}(X).

Proof: By 5.2 we may lift the identity of \underline{π}_{n}^{G}(X) to give a map X ⟶ Σ^{n}H\underline{π}_{n}^{G}(X) inducing the identity of \underline{π}_{n}^{G}. Using these as components we obtain a map into the product. The resulting map is an isomorphism in homotopy groups π_{n}^{H} for all n and H by construction, and thus a weak equivalence.

Note that this gives a short exact sequence for calculating [X, Y]^{G} for any two (G, d)- spectra X and Y.

Corollary 5.4. For any (G, d)-spectra, X and Y there is a short exact sequence

0 ⟶ Ext(\underline{π}_{∗}^{G}(ΣX), \underline{π}_{∗}^{G}(Y)) ⟶ [X, Y]^{G} ⟶ Hom(\underline{π}_{∗}^{G}(X), \underline{π}_{∗}^{G}(Y)) ⟶ 0.

We may go a little further.

Corollary 5.5. The category of (G, d)-spectra is equivalent to the derived category of graded (G, d)-Mackey functors:

(G, d)-spectra ≃ D((G, d)-Mackey).

Proof: It remains to comment on the naturality of the splitting. First choose a splitting of each (G, d)-spectrum X as a product of Eilenberg-MacLane spectra. This determines a splitting of the short exact sequence of 5.4, and this allows us to define a functor

(G, d)-spectra ⟶ D((G, d)-Mackey)

to the derived category. The functoriality follows from the fact that the composite of two elements of Ext is zero. It is easy to see the map is an equivalence of categories.

6. Sheaf theoretic formulation.

We may package 5.5 by saying that a rational (G, d)-spectrum is modelled by an equivariant sheaf of graded vector spaces on 𝒟, and morphisms are given by morphisms in the derived category of such sheaves. The functor is simply given by X ⟼ \underline{π}_{∗}^{G}(X) on objects.

To give the best possible picture, we want to package 3.2 as showing (G, c)-spectra may be viewed as sheaves on the category 𝒞. Indeed, since 𝒞 is discrete as a topological space, we simply need to specify the stalks over the finite cyclic subgroups C, the stalk over T, and the restriction maps relating them. For this we can adapt the torsion model of Greenlees [1996].

We begin by describing a target category Sh_{𝒞} of sheaves on 𝒞. First, we have a sheaf 𝒪 of rings on 𝒞: its stalk over the finite cyclic subgroup C is 𝒪_{C} = ℚ[c_{C}], where c_{C} is of degree −2. Thus its space of sections over the space ℱ of finite subgroups of T is 𝒪_{ℱ} = ∏_{C} ℚ[c_{C}], and we may consider the set ℰ of elements x = (x_{C})_{C} ∈ 𝒪_{ℱ} with the components x_{C} homogeneous, and almost all 1; we refer to these as (generalized) Euler classes. Alternatively, we may think of ℰ as the space of sections with compact multiplicative support. Now we may define the stalk over T to be 𝒪_{T} = ℰ^{−1}𝒪_{ℱ}. The structure maps for this sheaf are covariant and consist of the obvious maps 𝒪_{C} ⟶ ℰ^{−1}𝒪_{ℱ} = 𝒪_{T} making 𝒪_{T} into a module over 𝒪_{C} = ℚ[c_{H}]. The category Sh_{𝒞} is the full subcategory of the category of sheaves of modules over 𝒪, consisiting of sheaves 𝕏 so that the stalk 𝕏(C) over C is a torsion module over 𝒪_{C} = ℚ[c_{C}], the stalk 𝕏(T) is a free module over 𝒪_{T}, and so that the map 𝕏(T) ⟶ 𝕏(ℱ) = ∏_{C} 𝕏(C) of 𝒪_{ℱ}- modules, maps into the space ⨁_{C} 𝕏(C) of sections with compact support. The category Sh_{𝒞} is clearly equivalent to the torsion model category described in Greenlees [1996], and thus is an abelian category of injective dimension 2. Accordingly, the objects 𝕏̃ in its derived category are not necessarily determined by their homology in H_{∗}(𝕏̃) in Sh_{𝒞}. The main result of Greenlees [1996] Part I gives an equivalence T -spectra ≃ D(Sh_{𝒞}), and so we conclude that (G, c)-spectra ≃ D(Sh_{𝒞})[W] by 3.2. The homology of the object corresponding to the (G, c)-spectrum X is the sheaf 𝕏 with stalk π_{∗}^{T} (E⟨C⟩ ∧ ΣX) over C, where E⟨C⟩ is the cofibre of the natural map E[⊂ C]_{+} ⟶ E[⊆ C]_{+}, having nontrivial isotropy only at C. The presence of the suspension may be explained from one point of view by the fact that W_{T} (C) is one dimensional. The stalk at T is π_{∗}^{T} (Ẽℱ ∧ DEℱ_{+} ∧ X) ≅ 𝒪_{T} ⊗ π_{∗}(Φ^{T} X), and the structure map is induced by

Ẽℱ ∧ DEℱ_{+} ∧ X ⟶ ΣEℱ_{+} ∧ DEℱ_{+} ∧ X ≃ ΣEℱ_{+} ∧ X ≃ ⋁_{C} E⟨C⟩ ∧ ΣX.

To see the W -action is the obvious one, note that we may take Eℱ_{+} and Ẽℱ to be G-spaces. The other important point is that all the subgroups of T are normal in G, so that W preserves the splitting Eℱ_{+} ≃ ⋁_{C} E⟨C⟩.

Extending the sheaf 𝒪 to a sheaf on 𝒮_{f}G using the constant sheaf \underline{ℚ} on 𝒟, and combining the two parts of the model, we obtain a sheaf theoretic model for rational G-spectra.

Summary 6.1. The category of rational G-spectra is equivalent to the derived category of the category of equivariant sheaves of 𝒪-modules on the category 𝒮_{f}G whose stalks at finite cyclic groups are torsion, whose stalk at T is free, and whose restriction map from 𝒞 maps into sections with compact support.

The standard model for T -spectra consists of maps N ⟶ ℰ^{−1}𝒪_{ℱ} ⊗ V of 𝒪_{ℱ}-modules, which become an isomorphism when ℰ is inverted, but I do not see at present how to express this in sheaf theoretic terms. It would be more satisfactory to describe a sheaf theoretic model corresponding to this standard model, both because of the fact it is of injective dimension 1, and because of the intuition coming from the localization theorem.

References

L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) [1986] Equivariant stable homotopy theory” Lecture notes in mathematics, 1213, Springer-Verlag, Berlin, ix + 538pp J.P.C.Greenlees [1996] “Rational S^{1}-equivariant cohomology theories” Preprint, vi + 225 pp (Submitted for publication) J.P.C.Greenlees [1996a] “Rational Mackey functors for compact Lie groups I.” Preprint, 32 pp (Submitted for publication) J.P.C.Greenlees and J.P.May [1985] “Generalized Tate cohomology” Mem. American Math. Soc. 543 vi + 178 pp.

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.

E-mail address: j.greenlees@sheffield.ac.uk