RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES.

J.P.C.GREENLEES

Abstract. We apply the results of [4, 2] to give an explicit algebraic model of the category of rational SO(3)-spectra. This gives a complete classification of rational SO(3)-equivariant cohomology theories.

For J.P.May on his 60th birthday.

Contents

1. Introduction. 1 2. The closed subgroups of SO(3). 2 3. Burnside splitting. 5 4. Stable isotropy groups. 6 5. Free G-spectra. 7 6. Reduction to normalizers. 8 7. The algebraic model of the category of (G, c)-spectra. 11 8. The fibres. 13 9. Mackey functors. 15 10. The ordinary part of the model. 17 11. Reduction to the Weyl group. 19 References 20

1. Introduction.

This paper presents an algebraic model for SO(3)-equivariant rational cohomology theories. The main mathematical input is the model for SO(2)-equivariant theories given in Part I of [4], and the analysis of O(2)-equivariant cohomology theories in [3], together with special cases of results on rational Mackey functors from [2]. Using these ingredients, and the usual formal framework of equivariant homotopy, it is not hard to assemble the model. Nonetheless, new phenomena occur, and it is worth illustrating them in a case where we can be completely explicit.

The new feature of SO(3) is that it is semisimple. Thus the maximal torus is no longer normal and there is significant extra complication due to conjugacy. An exactly analagous model can be given with SU(2) as the group of equivariance. For groups of larger rank one expects a similar reduction to the case of a torus and equivariant sheaves over spaces of subgroups, however one cannot expect such an explicit analysis.

1This paper is in final form, and no substantially similar paper will be submitted elsewhere.

21991 Mathematics Subject Classification:

Primary 55N91

Secondary 55P42, 55P91.

Before proceeding, 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 [6], 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 say that a cohomology theory is rational if its values are graded rational vector spaces. A spectrum is rational if the cohomology theory it represents is rational. It suffices to check the values on the homogeneous spaces G/H for closed subgroups H, since all spaces are built from these. 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. Convention 1.1. Henceforth all spaces, groups and spectra are rationalized whether or not this is indicated in the notation.

We will outline the nature of the model at the end of Section 2 when we have summarized the subgroup structure of SO(3) and introduced appropriate terminology.

2. The closed subgroups of SO(3).

We take G = SO(3) to be the group of rotations of ℝ^{3}. We need to describe the space of closed subgroups H of G. For SO(3), the conjugacy class of a subgroup H is determined by the isomorphism type of H; quite generally the G-space of conjugacy classes of H is (H)_{G} ≅ G/N_{G}(H).

We may choose a maximal torus T = SO(2) consisting of rotations around the z-axis, and hence its normalizer N = O(2), which includes all half turns with axis in the xy-plane. Any positive dimensional proper subgroup of G is conjugate to N = O(2) or T = SO(2). Notation 2.1. For the rest of the paper we let G = SO(3), N = O(2), T = SO(2) and W = O(2)/SO(2).

There are three exceptional conjugacy classes of finite subgroups: the symmetry group A_{5} of the dodecahedron, the symmetry group Σ_{4} of the cube and the symmetry group A_{4} of the tetrahedron. We choose the cube to have vertices (±1, ±1, ±1), the dodecahedron to have these amongst its vertices and the tetrahedron to have its vertices at vertices of the cube. The group A_{5} is maximal and equal to its own normalizer. Hence its space of conjugates is the rational homology 3-sphere G/A_{5}. The cubical group Σ_{4} is equal to its own normalizer, and its space of conjugates is the rational homology 3-sphere G/Σ_{4}. The normalizer of the tetrahedral group A_{4} is the cubical group Σ_{4}, and its space of conjugates is the rational homology 3-sphere G/Σ_{4}.

All other finite subgroups of G are conjugate to subgroups of N = O(2). The finite subgroups of O(2) are the finite cyclic groups C_{n} for n ≥ 1, which are normal in O(2) and the finite dihedral groups. All the dihedral groups of order 2n are conjugate to the standard one D_{2n}, which we take to be generated by C_{n} and reflection in the x-axis. The normalizer of D_{2n} in O(2) is D_{4n}. Since O(2) is maximal in G, and C_{n} is not normal in G for n ≥ 2 the normalizer of C_{n} in G is O(2) for n ≥ 2, and the space of conjugacy classes is G/N (a copy of ℝP^{2}). Since the normalizer in G of D_{2n} for n ≥ 3 must preserve the axis of a rotation of order n, it must lie in O(2) and hence it remains D_{4n}.

The remaining two dihedral subgroups are exceptional. The normalizer of D_{4} in G is the cubical group Σ_{4}. The group D_{2} is conjugate in G to C_{2}, and the normalizer of D_{2} in G is a conjugate of O(2).

We divide these subgroups up into the exceptional conjugacy classes

ℰ = (G) ∪ (A_{5}) ∪ (Σ_{4}) ∪ (A_{4}) ∪ (D_{4}),

the cyclic conjugacy classes

𝒞 = ⋃_{n≥1}(C_{n}) ∪ (T),

and the typical dihedral conjugacy classes

𝒟′ = ⋃_{n≥3}(D_{2n}) ∪ (N).

Note that the dihedral groups D_{2} and D_{4} are excluded from 𝒟′, and that the cyclic group C_{1} is somewhat exceptional in that its normalizer is G rather than N. These sets of conjugacy classes can all be given various natural topologies: the relevant one for us is the f topology of [2]. This makes each of ℰ and 𝒞 the topological sum of the conjugacy classes G/N_{G}(H) it contains. However

𝒟′ = G ×_{N} 𝒟′_{N}

where 𝒟′_{N} is the set of dihedral subgroups of N of order ≥ 6. The N-space 𝒟′_{N} is topologized so that the conjugacy class N/D_{4n} of D_{2n} in N is the circle S(1/n) of radius 1/n centred at the origin and the conjugacy class of N is the origin itself. The space of all closed subgroups with the f topology of [2] is the topological sum

S_{f}G = 𝒞 ⨿ 𝒟′ ⨿ ℰ.

Note that G acts continuously on 𝒮_{f}G by conjugation.

We say a subgroup is topologically isolated if its conjugacy class is isolated in the ftopology. All subgroups of SO(3) except those conjugate to N are topologically isolated. We say a subgroup H is torally isolated if W_{G}(H) is finite and there is no subgroup K ⊴ H with H/K a torus. The set of torally isolated subgroups of SO(3) is 𝒟′ ⨿ ℰ. Finally, a subgroup is totally isolated if it is both torally and topologically isolated.

Roughly speaking, our description of G-spectra will be in terms of its geometric fixed points Φ^{H}X, viewed as the stalks of a sheaf on S_{f}G. We say that X is lies over K if these stalks are only non-trivial when K ⊆_{G} H. From this point of view, it is natural to separate behaviour over the terms 𝒞, 𝒟′ and ℰ. To describe the behaviour, we first recall the analysis for N given in [3]. The space of all closed subgroups of N is the topological sum

S_{f}N = 𝒞_{N} ⨿ 𝒟_{N},

where

𝒞_{N} = {C_{n} | n ≥ 1} ∪ {T }

is the discrete space of cyclic subgroups of N, and

𝒟_{N} = ⋃_{n≥1}(D_{2n})_{N} ∪ {N}

is the space of all dihedral subgroups of N. Thus

𝒟_{N} = 𝒟′_{N} ∪ (D_{2})_{N} ∪ (D_{4})_{N}.

The part of an N-spectrum over 𝒟_{N} is described by an N-equivariant sheaf of graded rational vector spaces on 𝒟_{N}. The part over 𝒞_{N} is given by a T -spectrum with homotopy W -action, which can in turn be described in algebraic terms as in [4].

Returning to G, the part of the model over 𝒟′⨿ℰ splits as a product of Eilenberg-MacLane spectra. Over one of these subgroups, the stalk is simply a graded rational W_{G}(H)-module (Section 8). Since N is the only accumulation point, it is easy to be explicit about the algebraic nature of such a sheaf as in [3]. This analysis is carried out in Sections 9 and 10.

The other conjugacy classes are the finite cyclic groups and T: this is the interesting part of the model. Its form is exactly similar to the form of the model for O(2), but the forgetful map from G-equivariance to N-equivariance is not an isomorphism. This is because the finite subgroups are normal in N but not in G, and because D_{2} and C_{2} are conjugate in G. Instead, passage to geometric fixed points gives an isomorphism

[X, Y]^{G} ⟶ [Φ^{H}X, Φ^{H}Y]^{N/H}

when X and Y lie over H. For H = T this identifies spectra over T with graded W -modules. If H = C is a finite cyclic group then N/C ≅ O(2), so the right hand side is known. If C is a non-trivial finite cyclic group the group N/C is the Weyl group W_{G}(C), but if C = 1 reduction to the Weyl group W_{G}(1) = G would give no information, and a quite separate argument shows that restriction from G-equivariance to N-equivariance is an isomorphism for free G-spectra.

The decomposition of S_{f}G gives rise to a corresponding equivalence

G-spectra ≃ (G, 𝒞)-spectra × (G, 𝒟′)-spectra × (G, ℰ)-spectra,

and the main theorem gives algebraic models for each of the factors. Theorem 2.2. The homotopy category of rational G-spectra is described by the equivalence

G-spectra ≃ D(𝒜_{s}(G, c)) × D(Sheaves_{G}(𝒟′)) × D(Sheaves_{G}(ℰ))

of triangulated categories. Here 𝒜_{s}(G, c) is a certain abelian category of injective dimension 1 to be described in Section 7, for a G-space X, Sheaves_{G}(X) denotes the category of Gequivariant sheaves of vector spaces on X, and for an abelian category 𝔸, D(𝔸) denotes its derived category. Since ℰ is a topological sum of orbits, there is a further equivalence

Sheaves_{G}(ℰ) = ∏_{(E)} ℚ[W_{G}(E)]-modules

where the product is over conjugacy classes of exceptional subgroups.

The paper is layed out as follows. First, Sections 3 and 4 recall basic phenomena: idempotents in the Burnside ring enable us to separate the isotropy groups. We then give the analysis of free G-spectra in Section 5, since this is the only completely new ingredient. The next general tool is reduction to the normalizer in Section 6. This provides all the ingredients to finish the analysis of G-spectra concentrated over subgroups of SO(2): we could simply transpose the method of [4], but instead we identify this category as a subcategory of the algebraic model of O(2)-spectra identified in [3]. This analysis occupies Sections 7 and 8. We then turn to the complementary set of spectra in Sections 9 and 10, showing that they split as a product of Eilenberg-MacLane spectra. Finally, in Section 11 we include a section proving that geometric fixed points give a reduction to the Weyl group: this is not necessary to the analysis, but nonetheless is illuminating.

3. Burnside splitting.

A characteristic feature of the stable setting is the (rationalized) Burnside ring

A(G) = [S^{0}, S^{0}]^{G}.

Since S^{0} is a unit object, this acts on all rational G-spectra. For each non-trivial idempotent e ∈ A(G) we may consider the full subcategory (G, e)-spectra of rational G-spectra of the form eX, and any orthogonal decomposition

1 = ∑_{i=1}^{n} e_{i}

gives a corresponding decomposition

G-spectra ≃ ∏_{i=1}^{n}(G, e_{i})-spectra.

Let ℱG denote the subspace of 𝒮_{f}G consisting of subgroups of finite index in their normalizers. Consider the equivalence relation ∼ on 𝒮_{f}G in which any pair of subgroups H ⊴ Ĥ with Ĥ/H a torus are ∼-related; it turns out that each conjugacy class in this quotient is represented by a unique conjugacy class of elements of ℱG so that ℱG/G ⟶^{≅} (𝒮_{f}G/ ∼)/G. This is easily checked directly in the present case. We have

ℱG/G = {(G), (A_{5}), (Σ_{4}), (A_{4}), (D_{4})} ∪ {(D_{2n}) | n ≥ 3} ∪ {(N), (T)},

which is compact and totally disconnected with (N) as the only limit point.

For any stable G-map f : S^{0} ⟶ S^{0}, the function

ϕ(f) : 𝒮_{f}G ⟶ ℚ

defined by ϕ(f)(H) = deg(Φ^{H}f) is, continuous and equivariant. Furthermore if H ⊴ Ĥ with Ĥ/H a torus we have ϕ(f)(H) = ϕ(f)(Ĥ) so ϕ passes to the quotient to give tom Dieck’s isomorphism

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

where C^{G}(X, ℚ) denotes continuous equivariant functions f : X ⟶ ℚ, and ℚ has the discrete topology. Thus an idempotent e of A(G) corresponds to the open and closed Ginvariant subspaces

Supp(e) = {H | e(H) = 1}

of 𝒮_{f}G which are unions of ∼-equivalence classes.

It is appropriate to let c be the idempotent supported on 𝒞, and we let m = 1 − c denote the complementary idempotent. This gives a correseponding decomposition

G-spectra ≃ (G, c)-spectra × (G, m)-spectra.

The letter m stands for Mackey, since it turns out that (G, m)-spectra split as products of Eilenberg-MacLane spectra and are therefore given by graded Mackey functors on ℰ ⨿ 𝒟′.

Furthermore, each topologically isolated conjugacy class (H)_{G} with N_{G}(H)/H finite gives rise to an idempotent e_{H} ∈ A(G). If ℰ^{+} is any finite collection of conjugacy classes from 𝒟′ ∪ ℰ not containing (N), we have an orthogonal decomposition

1 = c + m^{−} + ∑_{(H)∈ℰ^{+}} e_{H},

where Supp(m^{−}) = 𝒟′ ⨿ ℰ \ ℰ^{+}. This gives a corresponding decomposition

G-spectra ≃ (G, c)-spectra × (G, m^{−})-spectra × ∏_{(H)∈ℰ^{+}}(G, e_{H})-spectra.

In the special case ℰ^{+} = ℰ we let d′ denote the idempotent with support 𝒟′ and e denote the idempotent with support ℰ. We give an analysis of (G, e_{H})-spectra for a totally isolated conjugacy class (H) in Section 8, and an analysis of all (G, m)-spectra in Sections 9 and 10. The analysis of (G, c)-spectra occupies Section 7 and 8.

4. Stable isotropy groups.

Since our analysis is based on geometric fixed points we introduce some convenient terminology. For a G-spectrum X we say that H is a stable isotropy group if Φ^{H}X is nonequivariantly essential, and we let 𝒮ℐ(X) denote the set of stable isotropy groups. This set is closed under conjugation, but not under passage to subgroups or supergroups. Example 4.1. If ℱ is a family of subgroups we let Eℱ denote the universal ℱ-space, and Ẽℱ denote its unreduced suspension. The defining properties of the universal space give

• 𝒮ℐ(Eℱ_{+}) = ℱ and

• 𝒮ℐ(Ẽℱ) = All \ ℱ.

If e ∈ A(G) is idempotent then

• 𝒮ℐ(eS^{0}) = Supp(e). We record some basic properties.

Lemma 4.2. (i) The set 𝒮ℐ(X) is empty if and only if X ≃ ∗. (ii) If X is built from cells G/K_{+} with K ∈ ℱ then 𝒮ℐ(X) consists of subconjugates of ℱ. (iii) If X is rational and e ∈ A(G) is an idempotent with X ≃ eX then 𝒮ℐ(X) ⊆ Supp(e). (iv) For any spectra X and Y

𝒮ℐ(X ∨ Y) = 𝒮ℐ(X) ∪ 𝒮ℐ(Y)

and

𝒮ℐ(X ∧ Y) ⊆ 𝒮ℐ(X) ∩ 𝒮ℐ(Y).

(v) If X ⟶ Y ⟶ Z is a cofibre sequence then

𝒮ℐ(Z) ⊆ 𝒮ℐ(X) ∪ 𝒮ℐ(Y).
□

It is useful to connect stable isotropy groups to the more familiar notion of an ℱ-spectrum when ℱ is closed under passage to subgroups. An ℱ-spectrum is one constructed from cells G/K_{+} with K in ℱ. Lemma 4.3. If ℱ is closed under passage to subgroups and 𝒮ℐ(X) ⊆ ℱ then X is equivalent to an ℱ-spectrum. Proof: Use the usual procedure for killing homotopy groups. □

For any set ℋ of subgroups of G we let (G, ℋ)-spectra denote the full subcategory of G-spectra with stable isotropy in ℋ. Thus for example (G, 1)-spectra are free G-spectra and (G, G)-spectra X have the property that

X ≃ X ∧ Ẽ𝒫 ≃ Φ^{G}X ∧ Ẽ𝒫,

where 𝒫 is the family of proper subgroups of G. It follows that the geometric fixed point functor gives an equivalence (G, G)-spectra ≃ 1-spectra.

Note that if e is a non-trivial idempotent (i.e. e ≠ 0 or 1), (G, Supp(e))-spectra is another name for the category of (G, e)-spectra.

5. Free G-spectra.

In this section and the next we consider conditions under which the restriction map

[X, Y]^{G} ⟶ [X, Y]^{N}

is an isomorphism. The type of conditions we have in mind are restrictions on the isotropy groups of the spectra X and Y. In this section we deal with the case of free spectra, which is rather different from the case when X has isotropy groups with normalizer equal to N. We concentrate on the case in hand, when G = SO(3) and N = O(2), but the method is quite general. Theorem 5.1. If X is a free G-spectrum then the forgetful map

U : [X, Y]_{∗}^{G} ⟶ [X, Y]_{∗}^{N}

is an isomorphism. Proof: Since both sides are cohomology theories in X, it suffices to consider the case X = G/1_{+}. In this case we have a diagram

\xymatrix{ & [G/1_{+}, Y]^{G} \ar[r]^{≅} \ar@{-->}[dl]_{U} & [S^{0}, Y]^{1} \\ [G/1_{+}, Y]^{N} \ar[r]^{i^{∗}} & [N/1_{+}, Y]^{N} \ar[u]^{≅}_{ind_{1}^{G}} \ar[r]^{≅} & [S^{0}, Y]^{1}. \ar[u]_{=}}

Since i^{∗} ∘ U ∘ ind_{1}^{G}(f) = f, it follows that the forgetful map U is an isomorphism if and only if i^{∗} is an isomorphism. Thus it suffices to show that for the N-space Q = (G_{+})/(N_{+})

[Q, Y]_{∗}^{N} = 0

for all G-spaces Y.

A general approach might use a transfer argument here, but we give an explicit analysis. In fact it is simpler to begin with the periodic group G̃ = SU(2) and pass to the quotient by its centre.

We extend the use of a tilde, so that Ñ is the normalizer of the maximal torus in G̃, and we consider the quotient space Q̃ = (G̃_{+})/(Ñ_{+}) as a left and right Ñ-space. Lemma 5.2. (i) As a G̃-space, G̃_{+} = S(V)_{+} where V is the natural 2-dimensional complex representation of G̃. (ii) As an Ñ-space

G̃_{+} = Ñ_{+} ∪ Ñ_{+} ∧ e^{1} ∪ Ñ_{+} ∧ e^{2}.

Proof: The space S(V) is free, so the orbit of each point is G/1. Since S(V) is a connected 3-manifold, the orbit is S(V). This proves Part (i), and Part (ii) follows easily. □

To describe the attaching map we observe that passage to homology gives an isomorphism

[Ñ_{+}, Ñ_{+}]Ñ ⟶^{≅} Hom_{ℤW}(H_{0}(Ñ_{+}), H_{0}(Ñ_{+})).

The stable maps on the left are realized unstably after one suspension. A basis for this set of maps is given by the identity and right multiplicaton R_{t} where t is any element of Ñ not in the identity component.

The attaching maps in G̃_{+} may be taken to be 1 − R_{t} and 1 + R_{t}. Corollary 5.3. There is a cofibre sequence

ΣÑ_{+} ⟶^{1+R_{t}} ΣÑ_{+} ⟶ Q̃

of left Ñ-spaces. Factoring out the centre, there is a cofibre sequence

ΣN_{+} ⟶^{1+R_{t}} ΣN_{+} ⟶ Q

of left N-spaces. □

Remark 5.4. It is tempting to try to analyze Q̃_{K} = (G̃/K_{+})/(Ñ/K_{+}) by factoring out the right action on K, and hence to obtain results when X has isotropy K. However if K is not central, the cell decomposition of Q̃_{K} is a little more complicated.

It follows that the forgetful map

U : [X, Y]_{∗}^{G} ⟶ [X, Y]_{∗}^{N}

is an isomorphism for X = G_{+} provided

1 + R_{t}^{∗} : π_{∗}(Y) ⟶ π_{∗}(Y)

is an isomorphism. However, R_{t}^{∗} is multiplication by t on non-equivariant homotopy; since G is connected R_{t}^{∗} = 1. □

6. Reduction to normalizers.

It is often more convenient to work with normal subgroups. In this section we prove a result allowing us to reduce to this case in favourable circumstances. Combining this with the analysis of free spectra in Section 5 we obtain the key reduction. Recall that c is the idempotent supported on conjugates of T and its subgroups.

Theorem 6.1. If X is a (G, c)-spectrum then restriction

[X, Y]^{G} ⟶^{≅} (1 − e_{D_{2}})[X, Y]^{N}

is an isomorphism. Remark 6.2. (i) Of course the idempotent (1 − e_{D_{2}}) can be replaced by c since X is a (G, c)-spectrum. However the more precise statement shows that it is only the fusion of C_{2} and D_{2} that causes difficulty. (ii) The special case X = (G/C_{2})_{+} and Y = Ẽ[_{G} ⊉ C_{2}] shows that the idempotent (1 − e_{D_{2}}) is necessary; here [_{G} ⊉ C_{2}] is the family of subgroups containing no conjugate of C_{2}. Indeed (G/C_{2})^{D_{2}} ≅ (G/C_{2})^{C_{2}} = N/C_{2}, so as an N-space G/C_{2} is obtained from N/C_{2} by attaching cells with isotropy D_{2}. To see that the cells with isotropy D_{2} make a non-zero change we need only show the obstruction group

R = [e_{D_{2}}π_{∗}^{D_{2}}(G/C_{2})/(N/C_{2}), Ẽ[_{G} ⊉ C_{2}]]_{∗}^{N}

is non-zero. One easy way to do this is to use the theory of [3], which shows that since the domain lies over D_{2}, we need only calculate with homotopy Mackey functors:

R = Hom_{Mackey}(\underline{π}_{∗}^{N}((G/C_{2})/(N/C_{2})), \underline{π}_{∗}^{N}(Ẽ[_{G} ⊉ C_{2}]).

This calculation is in turn reduced to calculating W -maps of the W -modules obtained by applying

e_{D_{2}}π_{∗}^{D_{2}}(·) = π_{∗}(Φ^{D_{2}}(·)).

Thus

R = Hom_{W} (e_{D_{2}}π_{∗}^{D_{2}}((G/C_{2})/(N/C_{2})), e_{D_{2}}π_{∗}^{D_{2}}(Ẽ[_{G} ⊉ C_{2}]).

Now evidently

e_{D_{2}}π_{∗}^{D_{2}}(Ẽ[_{G} ⊉ C_{2}]) = ℚ,

so the obstruction group R is non-zero provided the trivial representation occurs in the domain. However, since (N/C_{2})^{D_{2}} = ∅,

e_{D_{2}}π_{∗}^{D_{2}}((G/C_{2})/(N/C_{2})) = e_{D_{2}}π_{∗}^{D_{2}}((G/C_{2})_{+}) = π_{∗}((G/C_{2})_{+}^{D_{2}}),

which has non-zero W -fixed points in degree 0. Proof of 6.1: Since both sides are cohomology theories of X, it suffices to consider the case X = G/H_{+} with H a cyclic subgroup of T. By the 5-lemma and the cofibre sequence EG_{+} ⟶ S^{0} ⟶ ẼG, it suffices to show the result when Y is 1-contractible and when Y is free.

In the case Y is G-free, it suffices to suppose Y = G_{+}, since X is finite. Now EG_{+} ∧X ⟶ X induces an isomorphism in domain and range so we may suppose X is also free. This case was dealt with in 5.1.

The case where Y is 1-contractible will be covered in 6.8 (ii) below. □

We are considering the case X = G/H_{+}, especially if H is a non-trivial cyclic group. Definition 6.3. Suppose H is a closed subgroup of G. We say that K ⊆ N_{G}(H) is a bad subgroup for H (or simply that K is H-bad) if there is a g ∉ N_{G}(H) so that g^{−1}Kg ⊆ H.

Remark 6.4. (i) A subgroup K is H-bad if and only if (G/H)^{K} ≠ N_{G}(H)/H. (ii) Only proper subgroups of H can be H-bad and the trivial subgroup 1 is always H-bad (unless H is normal in G). Lemma 6.5. If Y is K-contractible whenever K is H-bad, then the restriction

[G/H_{+}, Y]^{G} ⟶^{≅} [G/H_{+}, Y]^{N_{G}(H)}

is an isomorphism Proof: Let M = N_{G}(H) and consider the diagram

\xymatrix{ & [G/H_{+}, Y]^{G} \ar[r]^{≅} \ar[dl]_{U} & [S^{0}, Y]^{H} \\ [G/H_{+}, Y]^{M} \ar[r]^{i^{∗}} & [M/H_{+}, Y]^{M} \ar[u]^{≅}_{ind_{M}^{G}} \ar[r]^{≅} & [S^{0}, Y]^{H} \ar[u]_{=}}

where

i : M/H_{+} ⟶ G/H_{+}

is the inclusion of M-spaces. Since i^{∗} ∘ U ∘ ind_{M}^{G}(f) = f, it suffices to show that i induces an isomorphism. However, by Remark 6.4 (i), we need only attach cells M/K_{+} when K is H-bad, and by hypothesis these make no contribution. □

Remark 6.6. There are analogous results if we replace N_{G}(H) by an arbitrary subgroup M normalizing H. Lemma 6.7. If H is a non-trivial cyclic subgroup, the only H-bad subgroups are 1 and D_{2}, and D_{2} only occurs if H contains C_{2}. Proof: Any subgroup K ⊆_{G} T is cyclic. If K ≠ 1 and K^{g} ⊆ H then g takes the axis of a generator of K to that for H. Thus if K = T or K = C_{n} for some n we deduce g ∈ N = N_{G}(H). The only other cyclic subgroups K are conjugate to D_{2}. □

This gives a useful conclusion. Corollary 6.8. (i) If X is a (G, c)-spectrum and Y is C_{2}-contractible then restriction

[X, Y]^{G} ⟶^{≅} [X, Y]^{N}

is an isomorphism. (ii) If X is a (G, c)-spectrum and Y is 1-contractible then restriction

[X, Y]^{G} ⟶^{≅} (1 − e_{D_{2}})[X, Y]^{N}

is an isomorphism. Proof: Part (i) is a special case of Part (ii), since if Y is C_{2}-contractible, e_{D_{2}} acts as 0 on the N-spectrum Y. We therefore suppose Y is 1-contractible and prove Part (ii).

It suffices to consider the case X = G/H_{+} with H a cyclic subgroup of T. If H = 1 the result is obvious since both domain and codomain are 0, so we suppose H is non-trivial. By 6.7 the only possible H-bad subgroups are 1 and D_{2}. Now the argument of 6.5 applies to show that

[G/H_{+}, Y]^{G} ⟶^{≅} (1 − e_{D_{2}})[G/H_{+}, Y]^{N}

is an isomorphism: simply apply the idempotent (1 − e_{D_{2}}) to all the N-equivariant groups in the proof. □

7. The algebraic model of the category of (G, c)-spectra.

We proved in 6.1 that the algebraic model of (G, c)-spectra is a full subcategory of the algebraic model of (N, c)-spectra, given in [3]. We have an algebraic model of (N, c)-spectra, and we construct a subcategory modelling (G, c)-spectra. In this section we recall the relevant algebra, and in the next we complete the proof that we do have a model.

There are two models for rational (G, c)-spectra, as derived categories of abelian categories:

(G, c)-spectra ≃ D(𝒜_{s}(G, c)) ≃ D(𝒜_{t}(G, c)).

The standard abelian category 𝒜_{s}(G, c) has injective dimension 1, and the torsion abelian category 𝒜_{t}(G, c) is of injective dimension 2. It is usually easiest to identify the model for a (G, c)-spectrum in D(𝒜_{t}(G, c)), at least providing its model has homology of injective dimension 1. This is then transported to the standard category, where calculations are sometimes easier.

To describe the categories, we should begin by describing the corresponding categories 𝒜_{s}(T) and 𝒜_{t}(T) for the subgroup T. This is constructed from simpler categories, one for each closed subgroup of T: for finite cyclic groups C the fibre category is the category of ℚ[c]-modules with c of degree −2, and for T itself the fibre category is the category of graded ℚ-modules. There is some subtlety in how these are assembled, and it can be viewed as considering a category of sheaves over the topological category S_{f}T of subgroups of T, although we will not emphasize this view here.

We need to use the discrete set ℱ_{T} of finite cyclic subgroups of T. On this we consider the sheaf ℛ of rings with stalks ℚ[c] where c has degree −2. We need to consider the ring R = map(ℱ_{T} , ℚ[c]) of global sections. For each subgroup H, we let e_{H} ∈ R denote the idempotent with support H. If w : ℱ_{T} ⟶ ℤ is a function, we write c^{w} for the element of R with c^{w}(H) = c^{w(H)}. Now consider the multiplicative set ℰ generated by the universal Euler classes e(V) for the representations V of T with V^{T} = 0. These are defined by e(V) = c^{v}, where v(H) = dim_{ℂ}(V^{H}). In particular if V is a 1-dimensional representation with kernel of order n, we have e(V) = c^{δ(n)} where δ(n)(H) = 1 if |H| divides n and 0 otherwise. Equivalently,

ℰ = {c^{w} | w : ℱ_{T} ⟶ ℤ_{≥0} of finite support}.

We let t_{ℱ}^{∗} = ℰ^{−1}R: as a graded vector space this is ⨁_{H} ℚ in positive degrees and ∏_{H} ℚ in degrees zero and below.

The objects of the standard model 𝒜_{s}(T) are triples (N, β, V) where N is an R-module (called the nub), V is a graded rational vector space (called the vertex) and β : N ⟶ t_{ℱ}^{∗} ⊗V is a morphism of R-modules (called the basing map) which becomes an isomorphism when ℰ is inverted. When no confusion is likely, we simply say that N ⟶ t_{ℱ}^{∗} ⊗ V is an object of the standard abelian category. An object of 𝒜_{s}(T) should be viewed as the module N with the additional structure of a trivialization of ℰ^{−1}N. A morphism (N, β, V) ⟶ (N′, β′, V′) of objects is given by an R-map θ : N ⟶ N′ and a ℚ-map ϕ : V ⟶ V′ compatible under the basing maps.

We then form the derived category D(𝒜_{s}(T)) by taking differential graded objects of 𝒜_{s}(T) and inverting homology isomorphisms. Thus homology gives a functor H_{∗} : D(𝒜_{s}(T)) ⟶ 𝒜_{s}(T). Since the category 𝒜_{s}(T) has injective dimension 1, homotopy types of objects of the derived category D(𝒜_{s}(T)) are classified by their homology in 𝒜_{s}(T), so that homotopy types correspond to isomorphism classes of objects of the abelian category 𝒜_{s}(T). In the sheaf theoretic approach, N is the space of global sections of a sheaf on the space of closed subgroups T, the vertex V is the value of the sheaf at the subgroup T and the fact that the basing map β : N ⟶ t_{ℱ}^{∗} ⊗ V is an isomorphism away from ℰ is the manifestation of the patching condition for sheaves together with the localization theorem for each individual finite subgroup.

The objects of the torsion abelian category 𝒜_{t}(T) are triples (V, q, Q) where V is a graded rational vector space, Q is an ℰ-torsion R-module and q : t_{ℱ}^{∗} ⊗ V ⟶ Q is a morphism of R-modules. The condition on Q is equivalent to requiring (i) that Q is the sum of its idempotent factors Q(H) = e_{H}Q in the sense that Q = ⨁_{H} Q(H) and (ii) that each Q(H) is a torsion ℚ[c]-module. When no confusion is likely, we simply say that t_{ℱ}^{∗} ⊗ V ⟶ Q is an object of the torsion abelian category. In the sheaf theoretic approach, the module Q(H) is the cohomology of the structure sheaf with support at H. By contrast with the standard abelian category, the torsion abelian category has injective dimension 2. Thus not every object X of the derived category D(𝒜_{t}) is determined up to equivalence by its homology H_{∗}(X) in the abelian category 𝒜_{t}(T). We say that X is formal if it is determined up to isomorphism by its homology. It is not hard to check that X is formal if its homology has injective dimension 0 or 1 in 𝒜_{t}(T). In general, if H_{∗}(X) = (t_{ℱ}^{∗} ⊗ V ⟶ Q), the object X is equivalent to the fibre of a map (t_{ℱ}^{∗} ⊗ V ⟶ 0) ⟶ (t_{ℱ}^{∗} ⊗ 0 ⟶ ΣQ) (in the derived category) between objects in 𝒜_{t} of injective dimension 1. This map is classified by an element of Ext(t_{ℱ}^{∗} ⊗ V, ΣQ), so that X is formal if the Ext group is zero in even degrees. Thus X is formal if both V and Q are in even degrees or if Q is injective in the sense that each Q(H) is an injective ℚ[c]-module.

Notice that the action of N on T by conjugation gives an action of W on T -maps between N-spectra. It is shown in [3, 3.2] that the abelian categories for (N, c) are described by

𝒜_{s}(N, c) = 𝒜_{s}(T)[W] and 𝒜_{t}(N, c) = 𝒜_{t}(T)[W].

Thus the fibre category over a finite cyclic group C is the category of ℚ[c][W]-modules, and the fibre category over T itself is the category of graded ℚ[W]-modules.

For (G, c)-spectra, 𝒜_{s}(G, c) is a full subcategory of 𝒜_{s}(N, c) and 𝒜_{t}(G, c) is a full subcategory of 𝒜_{t}(N, c). The additional condition only involves the isotropy group 1. Theorem 5.1 states that the forgetful map

(G, 1)-spectra ⟶ (N, 1)-spectra

is full and faithful. A very special case of [3, 3.2] (together with [4, 4.4.1]) shows that

(N, 1)-spectra ≃ D(tors-ℚ[c][W]-mod).

However the composite is not essentially surjective.

To motivate the algebraic model of the image, we return to topology. Note that H^{∗}(BG; ℚ) = ℚ[c_{2}] where c_{2} has degree −4. and that the restriction H^{∗}(BG; ℚ) ⟶ H^{∗}(BT; ℚ) takes c_{2} to c^{2}. Furthermore the conjugation action of W on T takes c to −c.

Lemma 7.1. The functor r_{N}^{G} : ℚ[c_{2}]-modules ⟶ ℚ[c] [W]-modules defined by

r_{N}^{G}(M) = ℚ[c] ⊗_{ℚ[c_{2}]} M

is an exact functor of abelian categories. It is also full and faithful, and takes c_{2}-power torsion modules to c-power torsion modules. Proof: The functor is faithful because ℚ[c] is a free ℚ[c_{2}]-module. It is full because any W -map f : r_{N}^{G}(M) ⟶ r_{N}^{G}(M′) is r_{N}^{G}(f^{W}). The rest of the lemma follows. □

Now 𝒜_{t}(G, c) is the full subcategory of objects in which e_{1}Q lies in r_{N}^{G}(ℚ[c_{2}]-modules), and 𝒜_{s}(G, c) is the full subcategory of objects in which e_{1} ker(β : N ⟶ t_{ℱ}^{∗} ⊗ V) lies in r_{N}^{G}(ℚ[c_{2}]-modules). Lemma 7.2. The categories 𝒜_{s}(G, c) and 𝒜_{t}(G, c) are abelian. □

Theorem 7.3. There are equivalences of triangulated categories

(G, c)-spectra ≃ D(𝒜_{t}(G, c)) ≃ D(𝒜_{s}(G, c)).

Proof: It suffices to prove the first equivalence, since the second equivalence is immediate from the definitions of the algebraic categories.

We have shown that restriction displays (G, c)-spectra as a full triangulated subcategory of (N, c)-spectra. It remains to show that (N, c)-spectra which are restrictions of G-spectra have models in the subcategory D(𝒜_{t}(G, c)) and that all objects in this subcategory occur.

The model for (N, c)-spectra is built up one subgroup at a time, and the subcategory is defined by imposing a condition on the isotropy group 1. Accordingly, by working through the proof that D(𝒜_{t}(N, c)) is equivalent to the category of (N, c)-spectra, we may verify the above two statements by looking at one subgroup at a time. Since this fibrewise analysis is interesting in itself, we devote Section 8 to it. □

8. The fibres.

In this section we summarize the analysis of (G, H)-spectra for each subgroup H. This is not a necessary step if H is exceptional or dihedral (since it follows from Section 10 and the theory of Mackey functors), but it is clearer to treat all groups. Theorem 8.1. The categories of G-spectra with a single stable isotropy group are described as follows.

(1) If H is of finite index in its normalizer then

(G, H)-spectra ≃ D(ℚ[W_{G}(H)]-modules).

The graded ℚ[W_{G}(H)]-module classifying a (G, H)-spectrum X is π_{∗}(Φ^{H}X).

(2) If C is a non-trivial finite cyclic group

(G, C)-spectra ≃ D(tors-ℚ[c][W]-mod),

where c has degree −2. The torsion ℚ[c][W]-module classifying a (G, C)-spectrum X is π_{∗}(E(N/C)_{+} ∧_{T/C} Φ^{C}X).

(3) Free spectra are described by

(G, 1)-spectra ≃ D(tors-ℚ[c_{2}]-mod)

where c_{2} has degree −4. The torsion ℚ[c_{2}]-module classifying a (G, 1)-spectrum X is π_{∗}(EG_{+} ∧_{G} X). Restriction is described by the commutative diagram

(G, 1)-spectra ⟶^{res_{N}^{G}}  (N, 1)-spectra
≃↓  ↓≃
D(tors-ℚ[c_{2}]-mod) ⟶^{r_{N}^{G}} D(tors-ℚ[c][W]-mod).

The algebraic restriction functor is described by r_{N}^{G}(M) = ℚ[c] ⊗_{ℚ[c_{2}]} M, using the ring map ℚ[c_{2}] ⟶ ℚ[c] defined by taking c_{2} to c^{2}, and letting W act as the sign representation on c.

Remark 8.2. The equivalences of categories in the theorem each state a triangulated category Ho(ℂ) is equivalent to the derived category D(𝔸) of an abelian category 𝔸 of injective dimension 0 or 1. The equivalence arises from an Adams spectral sequence based on 𝔸, for calculating maps in Ho(ℂ): because of the injective dimension of 𝔸 is ≤ 1, the spectral sequence collapses, and because of parity one may choose splittings giving an equivalence. The proof is given in [4, 4.3.2]. Because of the low injective dimension, all objects are classified by their 𝔸-valued homology, and this is the functor mentioned in the statement. Ideally the equivalence would be constructed as a Quillen equivalence of model categories, and joint work with B.E. Shipley now suggests this will be possible. Proof: First consider the subgroups of finite index in their normalizer. We will show in 11.4 and 11.5 that if X and Y are (G, H)-spectra then

[X, Y]^{G} ⟶^{≅} [Φ^{H}X, Φ^{H}Y]^{W_{G}(H)}.

Also, since X and Y are (G, H)-spectra, Φ^{H}X and Φ^{H}Y are free W_{G}(H)-spectra. Now if F is finite we may consider the descent spectral sequence

H^{∗}(F; [A, B]_{1}^{∗}) ⇒ [EF_{+} ∧ A, B]_{∗}^{F} .

If we work over the rationals this collapses and converges to show

H^{0}(F; [A, B]_{∗}^{1}) = [EF_{+} ∧ A, B]_{∗}^{F} .

This shows that if W_{G}(H) is finite,

X ⟼ π_{∗}(Φ^{H}X)

defines a full and faithful functor

(G, H)-spectra ⟶ D(ℚ[W_{G}(H)]-modules).

To see the functor is essentially surjective, note all objects of ℚ[W_{G}(H)] are retracts of sums of copies of ℚ[W_{G}(H)], so it suffices to show ℚ[W_{G}(H)] is in the image. For this we use the fact that (G/H)^{H} = W_{G}(H). To obtain a spectrum with stable isotropy (H) we use the space G/H_{+} ∧ Ẽ[⊂_{G} H] where [⊂_{G} H] is the family of conjugates of proper subgroups of H.

It follows from 6.1 that if C is a cyclic group there is a full and faithful functor

(G, C)-spectra ⟶ (N, C)-spectra.

A special case of [3, 3.2] (together with [4, 4.4.1]) shows that

(N, C)-spectra ≃ D(tors-ℚ[c][W]-mod).

If C ≠ 1 the composite functor is essentially surjective. Indeed, N/T_{+} ∧ EN/C_{+} generates (N/C, 1)-spectra, and this space is the C-fixed point space of G/T_{+} ∧ E[⊆_{G} C]_{+} ∧ Ẽ[⊂_{G} C], where [⊆_{G} C] is the family of subgroups of G conjugate to a subgroup of C.

Finally, we come to the isotropy group 1, where the composite is not essentially surjective. We need only show that the image category is the image of the algebraic functor r_{N}^{G}. The G-spectrum G_{+} generates free G-spectra, so it suffices to observe that its image lies in the image of r_{N}^{G} and generates it. This is immediate since G_{+} corresponds to π_{∗}^{T} (G_{+}) = π_{∗}(G_{+}/T) = π_{∗}(ℂP_{+}^{1}). □

Remark 8.3. For the proof of Theorem 7.3 we only need the fibres over H with H a subgroup of T. For these cases we do not need to appeal to Section 11. The appeal for H = T can be replaced by a reference to Theorem 6.1 and the N-equivariant result.

9. Mackey functors.

We now begin the analysis relevant to spectra with no stable isotropy groups contained in a maximal torus. It will turn out these are all products of Eilenberg-MacLane spectra, so we start by analyzing the possible coefficients.

First recall that a Mackey functor is a contravariant additive functor

M : G-orbits ⟶ ℚ-modules,

where G-orbits is the full subcategory of G-spectra on the orbits G/H_{+}. Before describing the model of (G, m)-spectra, we need to summarize the relevant properties of Mackey functors. This is the special case of the results of [2] in which the group is SO(3). I am grateful to L.G.Lewis for pointing out that in [2] the category of h𝒪-sheaves must be replaced by a full subcategory in Theorems A and B to obtain correct statements. However in the present paper, we use only the composite equivalence

Mackey functors ≃ continuous Weyl-toral modules,

which is true as stated with the same proof. Alternatively it is easy to deal with the special case of SO(3) 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 ℰ⨿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 the Mackey functor M as follows. If H is totally isolated as a subgroup of itself 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 (N) = lim_{→}_{e(N)=1} eM(N), where the direct limit is over all idempotents

e with e(N) = 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 × (G, e)-Mackey,

where d′ is the idempotent with support 𝒟′ and e is the idempotent with support ℰ. 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, where the W -action on V (C_{1}) is trivial, and W -maps V (T) ⟶ V (C_{n}).

The (G, e)-Mackey functors are easy to describe, since they are equivariant sheaves over a topological union of orbits. They are thus specified by giving a W_{G}(E)-module V (E) for each E ∈ ℰ, and the category of (G, e)-Mackey functors is of projective and injective dimension 0 by Mashke’s theorem.

Finally, (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 following results are obtained by adapting Section 4 of [3]: we need only omit D_{2} and D_{4} from 𝒟 to obtain 𝒟′. The stalks are V (N), 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 ≥ 3.

Since W_{G}(D_{2n}) is of order 2 for n ≥ 3, 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 (N) 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 \ {N} near infinity, and the behaviour of sections is described by a W -map σ : V (N) ⟶ V (D_{∞}). Summary 9.1. A (G, d′)-Mackey functor is given by a vector space V (N) together with W -modules V (D_{2n}) for n ≥ 3 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 9.2. 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. □

For completeness we record the situation for projective functors, although we do not use it elsewhere. Remark 9.3. 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. □

10. The ordinary part of the model.

In this section we discuss the torally isolated subgroups, which are associated to the idempotent m = 1 − c supported on subgroups not conjugate to a subgroup of T. We show that any (G, m)-spectrum splits as a product of Eilenberg-MacLane spectra. Indeed, we may use the same proof as given in [5, Appendix A] for finite groups.

In the usual way, the nth H-equivariant stable homotopy group of a G-spectrum X is defined by π_{n}^{H}(X) = [G/H_{+}∧S^{n}, X]^{G} = [S^{n}, X]^{H}, and these are assembled into the homotopy Mackey functor \underline{π}_{n}^{G}(X) defined by by H ⟼ π_{n}^{H}(X). The Eilenberg-MacLane spectrum HM represents ordinary cohomology with coefficients in the Mackey functor M: it is characterized by the dimension axiom, which states that \underline{π}_{0}^{G}(HM) = M and \underline{π}_{n}^{G}(HM) = 0 for n ≠ 0.

For the proof, we define a related cohomology theory. Given 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),

Evidently, if I is supported on the set of torally isolated subgroups, the cohomology theory vanishes on (G, c)-spectra. Lemma 10.1. If I is an injective (G, m)-Mackey functor then hI is an Eilenberg-MacLane spectrum: hI = HI. Proof: Since \underline{π}_{0}^{G}(G/H_{+}) = [·, G/H_{+}]^{G} is a free (or representable) functor on the orbit category, 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 torally isolated groups

H ∈ {G, N, A_{5}, Σ_{4}, A_{4}} ∪ {D_{2n} | n ≥ 2}.

We use the generic letter B for these groups.

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, m)-Mackey functor, the groups with K in 𝒞 have no effect, and we need only consider torally isolated groups K. The case K = G gives the general pattern.

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

π_{n}^{G}(X) = ⨁_{(B)} π_{n}(EW_{G}(B)_{+} ∧_{W_{G}(B)} X^{B})

⊕ ⨁_{(C)} π_{n}(EW_{G}(C)_{+} ∧_{W_{G}(C)} Σ^{L(C)}X^{C}) ⊕ π_{n}(EG_{+} ∧_{W} Σ^{ad}X^{T}),

where B runs through torally isolated groups and C through the finite cyclic groups and L(C) is the representation of W_{G}(C) = N/C on the tangent space to G/C at the identity. The summands for T and the finite cyclic groups do not contribute, since I is concentrated on torally isolated groups. The other Weyl groups W_{G}(B) are all finite and hence over the rationals we have

mπ_{n}^{G}(X) = m ⨁_{(B)} π_{n}(X^{B})^{W_{G}(B)}.

We must now consider X = G/H_{+} for torally isolated groups H, and observe that (G/H)^{B} is 0-dimensional in all cases. This is a simple verification, and the same verification deals with the other subgroups K.

The result is obvious for H = G, so suppose H is a proper subgroup. Now, if (G/H)^{B} were not finite there would be a one parameter family B^{g_{t}} ⊆ H, and hence H would be of higher dimension than B. This means we need only look at the special case H = N and B = D_{2n}. If n ≥ 3 then the only elements conjugating B into H normalize C_{n} and therefore lie in N. This leaves the case (G/N)^{D_{4}}. This has three points. Indeed, if D_{4}^{g} ⊆ N then there is exactly one of the three non-identity elements s of D_{4} so that s^{g} is the unique element ω of order 2 in T. If s = ω, g normalizes ω and hence lies in N. If s is rotation about the x-axis g is a rotation moving the x-axis to the z-axis, and its axis thus lies on one of the two planes bisecting the angle between the planes x = 0 and z = 0. If s is rotation about the y-axis the analysis is similar. □

Now if M is any (G, m)-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 10.2. For any (G, m)-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 10.3. Any (G, m)-spectrum X splits as a product of Eilenberg-MacLane spectra

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

Proof: By 10.2 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 since the product in the corollary is equivalent to a sum, this gives a short exact sequence for calculating [X, Y]^{G} for any two (G, m)-spectra X and Y. Corollary 10.4. For any (G, m)-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 10.5. The category of (G, m)-spectra is equivalent to the derived category of graded (G, m)-Mackey functors:

(G, m)-spectra ≃ D((G, m)-Mackey).

Proof: It remains to comment on the naturality of the splitting. First choose a splitting of each (G, m)-spectrum X as a product of Eilenberg-MacLane spectra. This determines a splitting of the short exact sequence of 10.4, and this allows us to define a functor

(G, m)-spectra ⟶ D((G, m)-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. □

11. Reduction to the Weyl group.

The purpose of this section is to consider the geometric fixed point map

Φ^{K} : [X, Y]^{G} ⟶ [Φ^{K}X, Φ^{K}Y]^{W_{G}(K)},

and to show that under various hypotheses on the isotropy of X and Y it is an isomorphism. This could be the basic ingredient in our analysis, since for our group G = SO(3), either K = 1 and we may use the result of Section 5 or we may apply known results since W_{G}(K) is finite, a circle or O(2). Because restriction to N is so close to being an isomorphism we have chosen to avoid using these results in the proof of the main equivalence, but they remain conceptually helpful.

The basic result is straightforward obstruction theory. Lemma 11.1. If K is normal in K̂, and Φ^{H}Y ≃ ∗ unless H ⊇ K then

[X, Y]K̂ ≅ [Φ^{K}X, Φ^{K}Y]^{W_{G}(K)}.

Proof: By the isotropy hypothesis on Y, the inclusion Y ⟶ Y ∧ Ẽ[⊉ K] is an equivalence, where [⊉ K] is the family of subgroups not containing K. Now we quote [6, II.9]. □

Corollary 11.2. If Φ^{H}Y ≃ ∗ unless H ⊇ K, and X is built from cells G/H_{+} with H ⊆ N_{G}(K) and so that if K ⊆ H we have

(G/H)^{K} = N_{G}(K)/H,

then

[X, Y]^{G} ≅ [Φ^{K}X, Φ^{K}Y]^{W_{G}(K)}.

Remark 11.3. Note that the condition on (G/H)^{K} is slightly different to the requirement that K is H-good, when we need (G/H)^{K} = N_{G}(H)/H. Proof: It suffices to consider the case X = G/H_{+} with H ⊆ N_{G}(K). If K ⊈ H then both sides are zero, so we may suppose K ⊆ H. Note that W_{H}(K) ⊆ W_{G}(K) and

W_{G}(K)/W_{H}(K) = (N_{G}(K)/K)/(N_{H}(K)/K) = N_{G}(K)/N_{H}(K) = N_{G}(K)/H

and since K ⊆ H it follows that

N_{G}(K)/H ⊆ (G/H)^{K}.

Now consider the diagram

\xymatrix{[G/H_{+}, Y]^{G} \ar[r] \ar[d]^{≅} & [Φ^{K}G/H_{+}, Φ^{K}Y]^{W_{G}(K)} \ar[r] & [N_{G}(K)/H_{+}, Φ^{K}Y]^{W_{G}(K)} \ar[d]^{≅} \\ [S^{0}, Y]^{H} \ar[r]^{≅} & [Φ^{K}S^{0}, Φ^{K}Y]^{W_{H}(K)} \ar[r]^{=} & [W_{G}(K)/W_{H}(K)_{+}, Φ^{K}Y]^{W_{G}(K)}}

The result follows since the inclusion

N_{G}(K)/H ⟶ (G/H)^{K}

is an equivalence. □

We restate the condition Φ^{H}Y ≃ ∗ unless H ⊇ K as saying that the stable isotropy groups of Y all contain K.

The crudest application of the result is exactly analogous to the Hopf classification theorem for maps of an n-dimensional complex into an (n − 1)-connected space. This is adequate to deal with subgroups K with W_{G}(K) finite. Corollary 11.4. If the stable isotropy groups of X are all contained in conjugates of K and the stable isotropy groups of Y all contain K then

[X, Y]^{G} ≅ [Φ^{K}X, Φ^{K}Y]^{W_{G}(K)}.

Proof: It suffices to observe that such a spectrum X may be constructed using cells G/K_{+}, and cells G/H_{+} with H not containing K. □

We now turn to the case of cyclic groups K = C; the case C = 1 is different and was dealt with in Section 5. Corollary 11.5. If Φ^{H}Y ≃ ∗ unless C ⊆ H and X is built from cells G/H_{+} with H ⊆ N then if either

• |C| ≥ 3 or

• |C| = 2 and all cells of X are G/H_{+} with H topologically cyclic then

[X, Y]^{G} ≅ [Φ^{C}X, Φ^{C}Y]^{N/C}.

Proof: We just need to verify that if C ⊆ H ⊆ N then (G/H)^{C} = N/H. For gH to be fixed by C we require C^{g} ⊆ H ⊆ N. Since N is the stabilizer of the z-axis, it suffices to show that either of the conditions imply g preserves the z-axis. However all elements of N with order ≥ 3 are rotations about the z-axis. If |C| ≥ 3 then g must preserve the z axis and hence lie in N; since C is normal in N, the condition is satisfied. However N contains elements of order 2 with many other axes, so if |C| = 2 we must require that all non-identity elements of H preserve the z axis. □

References

[1] J.P.C.Greenlees “A rational splitting theorem for the universal space for almost free actions.” Bull. London Math. Soc. 28 (1996) 183-189.

[2] J.P.C.Greenlees “Rational Mackey functors for compact Lie groups I” Proc. London Math. Soc 76 (1998) 549-578

[3] J.P.C.Greenlees “Rational O(2)-equivariant cohomology theories.” Fields Institute Communications 19 (1998) 103-110

[4] J.P.C.Greenlees “Rational S^{1}-equivariant cohomology theories” Mem. American Math. Soc. 661 (1999), xii + 289 pp

[5] J.P.C.Greenlees and J.P.May “Generalized Tate cohomology” Mem. American Math. Soc. 543 vi + 178 pp.

[6] L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) Equivariant stable homotopy theory” Lecture notes in mathematics, 1213, Springer-Verlag, Berlin, ix + 538pp

Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.

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