CHROMATIC CHARACTERISTIC CLASSES IN ORDINARY GROUP COHOMOLOGY

DAVID J. GREEN, JOHN R. HUNTON, AND BJÖRN SCHUSTER

Abstract. We study a family of subrings, indexed by the natural numbers, of the mod p cohomology of a finite group G. These subrings are based on a family of v_{n}-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H^{∗}(BG)), the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var(Ch(G)). Our subrings give rise to a ‘chromatic’ (co)filtration, which has both topological and algebraic definitions, of var(H^{∗}(BG)) whose final quotient is the variety var(Ch(G)).

Introduction

This paper develops a structure within the mod p cohomology ring H^{∗}(BG) = H^{∗}(BG; 𝔽_{p}) of a finite group G which we show to arise both topologically and algebraically. In general, the determination and description of the ring H^{∗}(BG) is a hard problem and we build upon two separate approaches, due respectively to Quillen [18] and to Thomas [22], incorporating also ideas from the chromatic point of view in homotopy theory (in particular [10]) and the theory of coalgebraic rings (Hopf rings) and the homology of infinite loop spaces [8, 11, 12, 19, 23]. We utilise the work of Leary and the first author [9] on categories associated to subrings of H^{∗}(BG), which first hinted at the possibility of a theorem such as our result (0.1) below.

Recall that Quillen showed [18] that the ring H^{∗}(BG) could be described up to F-isomorphism in terms of its elementary abelian subgroup structure. For its most elegant formulation, suppose k is an algebraically closed field of characteristic p and for an 𝔽_{p} algebra R denote by var(R) the variety of algebra morphisms from R to k, topologised with the Zariski topology. Quillen’s result describes var(H^{∗}(BG)) in terms of a colimit colim_{A} var(H^{∗}(BV)) over a certain category A of elementary abelian subgroups V of G. In particular recall that, for such V, the variety var(H^{∗}(BV)) is isomorphic to V ⊗_{𝔽_{p}} k where V is viewed as an 𝔽_{p} vector space.

On the other hand, Thomas has shown [22] that a frequently significant as well as computable subring of H^{∗}(BG) is the Chern subring, Ch(G), essentially that part of the cohomology ring given most immediately by the complex representation theory of G (specifically, it is the subring generated by all Chern classes of irreducible representations). Leary and the first author [9] have completed this picture by describing var(Ch(G)) as colim_{A^{(1)}} var(H^{∗}(BV)), the colimit over a further category A^{(1)} of elementary abelians. The categories A and A^{(1)} have the same objects but in general they have different morphisms and the natural surjection var(H^{∗}(BG)) ↠ var(Ch(G)) is not usually a homeomorphism.

Date: 20 Aug 2001.

However, Atiyah’s theorem [2] linking the (completed) representation theory of G with K^{∗}(BG), the complex K-theory of BG, allows a reinterpretation of Ch(G) in terms of (unstable) maps from BG to the spaces in the Ω-spectrum for Ktheory: the homology images of such maps can be identified with Chern classes, the generators of the Chern subring. See section 1 for details. Replacing K-theory by any other representable generalised cohomology theory E^{∗}(−), this construction allows for the definition of ‘E-type characteristic classes’ in H^{∗}(BG) based on the spaces in the Ω-spectrum for E. These generate a subring, Ch_{E}(G) say, of H^{∗}(BG). In many instances (including, by the work of Bendersky and the second author [8], those when E is Landweber exact [16]) there are inclusions of rings

Ch(G) ⊂ Ch_{E}(G) ⊂ H^{∗}(BG).

Our main results concern a family of such subrings, defined by a family of cohomology theories E. Specifically, we are interested in the subrings given by a family of v_{n}-periodic cohomology theories such as the Johnson-Wilson [14] theories E(n), n ∈ ℕ. However, to obtain our best results we concentrate on the subrings given by their I_{n}-adic completions { \atop E(n)}, as given by Baker and Würgler [6], and these subrings we denote Ch{ \atop E(n)}(G). The use of complete theories has the advantage of enabling us to apply the results of Hopkins, Kuhn and Ravenel [10]. In section 3 we define certain categories A^{(n)} of elementary abelian subgroups of G allowing us to prove

Theorem 0.1. Let G be a finite group and k an algebraically closed field of characteristic p. Then there is a homeomorphism of varieties

colim_{A^{(n)}} var(H^{∗}(BV)) ⟶ var(Ch{ \atop E(n)}(G)) .

The categories A^{(n)} appeared in [9] where, as mentioned, the first case A^{(1)} was proved to represent the classical Chern subring. It was also suggested that there may be a possible link between the A^{(n)} and certain v_{n}-periodic spectra; this theorem provides such a link and supplies a topological construction for subrings of H^{∗}(BG) associated to the A^{(n)}.

The same result almost certainly holds on replacing the spectra { \atop E(n)} by Morava Etheory or in fact by any Landweber exact spectrum satisfying the main hypotheses of [10]; the recent work of Baker and Lazarev [5] supplies the technical constructions needed for adapting the proof given below for { \atop E(n)} (the essential element being the analogue of the Baker-Würgler tower [7]). It is less clear that the analagous result for E(n) will hold as these spectra fail the completion hypotheses of [10].

A number of basic results on the Ch{ \atop E(n)}(G) can be proved.

Theorem 0.2. 1. The varieties var(Ch{ \atop E(n)}(G)) form a (co)filtered space

var(H^{∗}(BG)) ↠ ⋯ ↠ var(Ch{ \atop E(n+1)}(G)) ↠ var(Ch{ \atop E(n)}(G)) ↠ ⋯ ↠ k

2. For each n ⩾ 1 there is a finite group G for which var(Ch{ \atop E(n)}(G)) is distinct from var(Ch{ \atop E(n+1)}(G)), and hence Ch{ \atop E(n)}(G) from Ch{ \atop E(n+1)}(G);

3. For n at least the p-rank of G, var(Ch{ \atop E(n)}(G)) = var(H^{∗}(BG)); 4. var(Ch{ \atop E(1)}(G)) = var(Ch(G)), the Chern subring of H^{∗}(BG);

5. For each n there is an inclusion of rings Ch(G) ⊂ Ch{ \atop E(n)}(G).

Given Theorem 0.1, the statements (1) and (3) follow immediately from the definition of the A^{(n)} while (2) and (4) follow from results of [9]. A more general version of statement (5) is proved in section 2.

For simplicity we shall assume, at least after section 1, that all our rings R, such as Ch_{E}(G), are defined as subrings of the even degree part of H^{∗}(BG), and hence are commutative. In fact, as noted in [9], this is not restrictive: if p ≠ 2 the odd degree elements of H^{∗}(BG) are nilpotent and any homomorphism H^{even}(BG) ⊃ R →^{f} k extends trivially for any odd degree element; if p = 2 such a map would extend uniquely on an odd degree element x to the unique square root of f(x^{2}).

The rest of the paper is organised as follows. We begin by defining our E-type characteristic classes in section 1, linking this construction to that of the classical Chern subring in the case E = K. We also introduce here some of the basic notation and language of coalgebraic algebra [12, 19] in order to discuss the homology of spaces in the Ω-spectrum for E.

In section 2 we recall and extend the main result of [9] and prove that, at least for E Landweber exact, var(Ch_{E}(G)) has a description as a colimit of var(H^{∗}(BV)) as V runs over some category, A_{E} say, of elementary abelian subgroups V of G. Here we also demonstrate that for such E there is an inclusion Ch(G) ⊂ Ch_{E}(G).

Most of the rest of the paper is devoted to the proof of the Theorem 0.1, i.e, to the identification of the category A_{E}. In section 3 we introduce the cateogry A^{(n)} together with an intermediate category C_{E}. We show that if E both is Landweber exact and satisfies the assumptions of the character isomorphism of [10] then it follows fairly easily that C_{E} = A^{(n)}. The harder part is to show that A_{E} = C_{E} and we prove that this will follow if certain properties of the ‘unstable Hurewicz homomorphism’

H: E^{∗}(X) = [X, E_{∗}] → Hom(H^{∗}(E_{∗}), H^{∗}(X))

can be established. Our most general result, which for E = { \atop E(n)} specialises to Theorem 0.1, is stated as Theorem 3.4 and the complete set of hypotheses needed on a spectrum E for its application are set out in (3.2).

Proof that these hypotheses are satisfied by { \atop E(n)} is given in section 4 which examines in detail the homology of the spaces in its Ω-spectrum, building on Baker and Würgler’s description [7] of { \atop E(n)} as a homotopy limit and Wilson’s calculation of the Hopf ring for Morava K-theory [23].

We conclude with some example computations in section 5.

The construction of the subrings Ch_{E}(G) is firmly based in the world of unstable homotopy through the use of the infinite loop spaces representing E-cohomology. We rely extensively on the techniques of coalgebraic algebra for our proofs and computations; see [12, 19] for introductory material on this field. We also rely on the work of Wilson [23] as the major computational input, but we note one more recent aspect of our work. The results of [10] needed in section 3 demand we work with completed spectra E and consequently with the spaces in their Ωspectra. These are truly huge spaces whose homologies have only recently become accessible; see [8, 11, 13] for the methods of handling such spaces and for the results we use below.

Acknowledgements We are grateful for the suggestions and advice of Andy Baker, Dave Benson, John Greenlees, Ian Leary, Erich Ossa and Steve Wilson. The second author thanks the Bergische Universität Wuppertal for financial support and the University of Leicester for sabbatical leave while much of this work was completed.

1. E-type characteristic classes

In this section we define the class of subrings considered and show that the classical Chern subring arises as a special case. Here and in the next section we can take G to be any compact Lie group; later we shall need to restrict to finite groups. We begin with a very general construction.

Definition 1.1. Suppose {X_{i}} is a family of spaces and FG is a set of maps of the form f: BG → X_{i}. Let Ch_{FG} be the subring of H^{∗}(BG) generated by all elements of the form f^{∗}(x) as f runs over the elements of FG and x over the homogeneous elements of all the H^{∗}(X_{i}).

Example 1.2. Take {X_{i}} to be the set containing just one space, BU. Let F be the set of maps BG → BU given by irreducible representations G → U(n) on embedding U(n) in the infinite unitary group U and taking classifying space. The subring in this instance is the classical Chern subring Ch(G) [22].

Remark 1.3. It is not hard to show that the Chern subring, as given in the last example, can equivalently be defined by taking FG as the maps given by all representations of G, or even as the maps given by all elements of the representation ring R(G). Clearly enlarging FG will only if anything enlarge the subring of H^{∗}(BG) it defines, so it suffices to show that elements of R(G) give no new elements than are already in Ch(G) as defined in (1.2). For example, if ρ_{1} : G → U(n) and ρ_{2} : G → U(m) are two representations, the H-space structure on BU induced by U(n) × U(m) → U(n + m) gives, in cohomology, the commutative diagram

H^{∗}(BU) ⊗ H^{∗}(BU) ⟶^{Bρ_{1}^{∗}⊗Bρ_{2}^{∗}} H^{∗}(BG) ⊗ H^{∗}(BG)
↑  ↓
H^{∗}(BU)  ⟶^{B(ρ_{1}+ρ_{2})^{∗}}  H^{∗}(BG)

sending an element x ∈ H^{∗}(BU) to ∑ Bρ_{1}^{∗}(x′)·Bρ_{2}^{∗}(x′′) in H^{∗}(BG) where ∑ x′⊗x′′ denotes the image of x in H^{∗}(BU)⊗H^{∗}(BU). Thus the representation ρ_{1} +ρ_{2} fails to give anything in H^{∗}(BG) that was not already in the subring defined by ρ_{1} and ρ_{2}. The rest of the proof is similar.

To set up notation that will be useful later, recall that H^{∗}(BU) is the polynomial ring on generators c_{1}, c_{2}, …, the universal Chern classes, where c_{i} ∈ H^{2i}(BU). The pull-back Bρ^{∗}(c_{i}) of a representation ρ: G → U is the called the i^{th} Chern class of ρ, also written c_{i}(ρ), and is an element of H^{2i}(BG).

For the remainder of this paper we assume E^{∗}(−) is a representable cohomology theory on the category of CW complexes with coefficients E_{∗} concentrated in even dimensions. In particular, there is a representing Ω-spectrum with spaces E_{r}, r ∈ ℤ, and, for all CW complexes X, natural equivalences between E^{r}(X) and the set of homotopy classes of maps from X to E_{r}. The following is the main definition of this paper.

Definition 1.4. Let E be as just described and let G be a compact Lie group. Let FG be the set of all homotopy classes of maps BG → E_{2r}, allowing all r ∈ ℤ. The E-Chern subring, Ch_{E}(G), is defined as the subring Ch_{FG} of H^{∗}(BG). Also, define an E-type characteristic class of G to be an element f^{∗}(x) ∈ H^{∗}(BG) for some f: BG → E_{2r} and some homogeneous x ∈ H^{∗}(E_{2r}).

Remark 1.5. Note that it does not matter in this definition whether we take the whole spaces E_{2r} or just the connected components of their basepoints, usually denoted E′_{2r}. This follows since all components of the E_{2r} are the same, with E_{2r} = π_{0}(E_{2r}) × E′_{2r} = E^{2r} × E′_{2r}.

The following result, building on Remark 1.3, shows that this definition includes that of the classical Chern subring.

Proposition 1.6. Let G be a compact Lie group. Then the K-Chern subring Ch_{K}(G) is equal to Ch(G), the classical Chern subring.

Proof. As any element of the representation ring R(G) gives rise to a homotopy class of maps BG → BU, it is immediate by (1.3) and (1.5) that every element of Ch(G) is also an element of Ch_{K}(G) since the only even graded spaces in the Ω-spectrum for K-theory are of the homotopy type of ℤ × BU. However, not every class of map BG → BU necessarily arises as the image under B(−) of something from the representation ring. Thus we must check further that Ch_{K}(G) ⊂ Ch(G).

Recall from the work of Atiyah and Segal [3] (or Atiyah [2] for the case of finite groups) that the homomorphism α: R(G) → K^{0}(BG) given by sending the virtual representation ρ to the corresponding map BG → ℤ × BU is continuous with respect to the I-adic topology on R(G), where I is the augmentation ideal, and the filtration topology on K^{0}(BG) given by

F_{m}K^{0}(BG) = ker(K^{0}(BG) → K^{0}(BG^{m−1})) ,

where BG^{m−1} is the m − 1 skeleton of BG. Moreover, K^{0}(BG) is a complete ring and α is an isomorphism after I-adic completion of R(G).

Note that the degree m cohomology of BG is determined by its (m + 1)-skeleton BG^{m+1}, and so in particular the r^{th} Chern class c_{r}(g) of a map g: BG → BU is determined by its restriction to BG^{2r+1} → BU. It follows that c_{r}(g) is zero for all 1 ⩽ r ⩽ n − 1 if g ∈ F_{2n}K^{0}(BG).

Now consider a map f: BG → BU in FG. As an element of K^{0}(BG) it is represented by a Cauchy sequence ρ_{1}, ρ_{2}, … in R(G). By the Whitney sum formula, the total Chern class of f satisfies c(f) = c(ρ_{i})c(f − Bρ_{i}). Hence if f − Bρ_{i} ∈ F_{2n}K^{0}(BG), then c_{r}(f) = c_{r}(ρ_{i}) for 1 ⩽ r ⩽ n − 1. Thus, for each r ⩾ 0, the sequence c_{r}(ρ_{1}), c_{r}(ρ_{2}), … in H^{2r}(BG) is eventually constant at c_{r}(f) = f^{∗}(c_{r}).

As any homogeneous x ∈ H^{∗}(BU) can involve only finitely many of the c_{r}, each element f^{∗}(x) ∈ H^{∗}(BG) can be realised in the form Bρ^{∗}(x) for some virtual representation ρ ∈ R(G). This completes the proof.

Remark 1.7. These definitions of E-type characteristic classes can also be made in the integral or p-local rings H^{∗}(BG; ℤ) and H^{∗}(BG; ℤ_{(p)}). If H^{∗}(E_{2∗}; ℤ) is torsion free (as is the case when E is Landweber exact, [8]), then the (mod-p)

E-type characteristic classes are just the mod-p reductions of the integral E-type characteristic classes. In general however this need not be the case. As H^{∗}(BU; ℤ) is torsion free, the integral analogue of (1.6) holds.

Remark 1.8. The reader may wonder as to the restriction in Definition 1.4 to even graded spaces E_{2r} in the Ω-spectrum for E. In the case of complex K-theory and finite groups G this restriction is vacuous: there are no non-trivial maps from BG to the odd graded spaces [2]. For more general compact groups Proposition 1.6 would however fail without this hypothesis and so (1.4) would not have been the correct extension of the concept of Chern subring. Below we shall restrict to finite G and Landweber exact E; in light of Kriz’s examples such as presented in [15], it is conceivable that even here relaxing Definition 1.4 to include all spaces could enlarge the subring it defines, but the effect remains unclear. The theorems below are proved considering the even spaces alone.

Remark 1.9. There is of course no a priori reason why the subring Ch_{E}(G) defined by (1.4) should lie in H^{even}(BG), since there may well be odd dimensional elements of H^{∗}(E_{2r}) which pass to non-trivial elements of H^{∗}(BG). However, we shall need from the main result of the next section, Theorem 2.7, and onwards that the spectrum E considered is Landweber exact. We can thus appeal to the work of [13] from which it can easily be deduced that whenever E is Landweber exact H^{∗}(E_{2r}) has no odd dimensional elements.

2. Representations by categories of elementary abelians

Quillen proved in [18] that the variety of H^{∗}(BG) can be built from the elementary abelian p-subgroups using conjugacy information. This result was extended in [9] to cover the varieties of certain large subrings of H^{∗}(BG) as well. The object of this section is to show that for a wide range of theories E, the varieties of the E-Chern subrings Ch_{E}(G) have a similar description. We start by recalling the definitions and the main result of [9], in a slightly sharpened form.

Definition 2.1. For a compact Lie group G and homogeneously generated subring R of H^{∗}(BG), define the category C(R) as that with objects the elementary abelian p-subgroups V of G, and morphisms from W to V to be the set of injective group homomorphisms f: W → V satisfying

f^{∗} Res_{V} (x) = Res_{W} (x)  (2.1)

for every homogeneous x ∈ R.

Remark 2.2. This is a variation on the definition given in [9] where the condition on morphisms f: W → V was weakened to satisfying equation (2.1) modulo the nilradical. However, for subrings R that are closed under the action of the Steenrod algebra, the two definitions are equivalent. To see this, suppose that f is a morphism in the “modulo nilradical” version of C(R) with the property that, for some x ∈ R, the class w = f^{∗} Res_{V} (x) − Res_{W} (x) is a non-zero nilpotent in H^{∗}(BW). For any non-zero homogeneous element of H^{∗}(BW) there is an operation θ in the mod p Steenrod algebra such that θ(w) is non-nilpotent: this is immediate for p = 2; for p odd, see Lemma 2.6.5 of [21]. Naturality of θ implies that

f^{∗} Res_{V} (θx) − Res_{W} (θx) = θ(w) .

As R is closed under the Steenrod algebra, θ(x) lies in R, contradicting f being a morphism in C(R).

Definition 2.3. Let G be a compact Lie group. A virtual representation ρ of G is called p-regular if it has positive virtual dimension and restricts to every elementary abelian p-subgroup as a direct sum of copies of the regular representation. A subring R of H^{∗}(BG) is called large if it contains the Chern classes of some p-regular representation.

Theorem 2.4. [9, 6.1] Let G be a compact Lie group and R a homogenously generated subring of H^{∗}(BG) which is both large and closed under the Steenrod algebra. Then the natural map R → lim_{V ∈C(R)} H^{∗}(BV) induces a homeomorphism

colim_{V ∈C(R)} var(H^{∗}(BV)) → var(R) .

The problem, however, is to identify the category C(R) for a given R.

Example 2.5. The Quillen category A = A(G) is the category with objects the elementary abelian p-subgroups of G, and morphisms generated by inclusion and conjugation. It is shown in [9, 9.2] that A is the category C(H^{∗}(BG)), thus recovering one of the main results of [18]:

colim_{V ∈A} var(H^{∗}(BV)) ≅ var(H^{∗}(BG)) .

Example 2.6. It is shown in [9] that the category A^{(1)} of elementary abelian psubgroups of G and morphisms the injective homomorphisms f: W → V which take each element w ∈ W to a conjugate of itself is the category C(Ch(G)) associated to the Chern subring.

We come to the main result of this section. Recall that the class of Landweber exact spectra [16] includes examples such as BP -theory, complex cobordism, the Johnson-Wilson theories E(n) and their I_{n}-adic completions { \atop E(n)} as well as Morava E-theory, complex K-theory, various forms of elliptic cohomology [17] and their completions (in particular the completions (Ell) ˆ℘ of Baker [4]).

Theorem 2.7. Let E be a Landweber exact spectrum and let G be a compact Lie group. Then Ch_{E}(G) is closed under the Steenrod algebra and is large in the sense of (2.3). Hence there is a category C(Ch_{E}(G)), which we shall write as A_{E}(G) or just A_{E}, such that

var(Ch_{E}(G)) ≅ colim_{V ∈A_{E}} var(H^{∗}(BV)) .

Proof. Closure of Ch_{E}(G) under the Steenrod algebra is immediate from its definition and it suffices to show that it is also large. This follows from the next result and the fact that the Chern subring Ch(G) is large [9].

Proposition 2.8. For E Landweber exact the homomorphism (c_{1}^{E})^{∗}: H^{∗}(E′_{2}) → H^{∗}(BU) is surjective for all n ⩾ 1. Hence for such E there is an inclusion of subrings Ch(G) ⊂ Ch_{E}(G) for all compact Lie groups G and in particular Ch_{E}(G) contains all Chern classes of complex representations of G.

Here c_{1}^{E} is the first E-theory Chern class: recall that

E^{∗}(BU) = E^{∗}[[c_{1}^{E}, c_{2}^{E}, …]] with c_{i}^{E} ∈ E^{2i}(BU) ;

thus c_{1}^{E}: BU → E′_{2} is an element of Ẽ^{2}(BU) where, as before, E′_{2} means the base point component of E_{2}.

We need to introduce further notation. Recall that any Landweber exact spectrum E is certainly complex oriented. For any complex oriented spectrum, denote by x^{E}: ℂP^{∞} → E′_{2} in Ẽ^{2}(ℂP^{∞}) the orientation map. The definition of c_{1}^{E} is not independent of this and in fact c_{1}^{E}: BU → E′_{2} restricts to the complex orientation x^{E} on ℂP^{∞} [1, Part II, 4.3(ii)]. We assume that the orientation x^{K} has been chosen compatibly so that the diagram

BU  \overline{c_{1}^{E}} E′_{2}
[figure: x^{K}; x^{E}]
ℂP^{∞}

commutes.

Take basis elements β_{n}, n ∈ ℕ, of H_{∗}(ℂP^{∞}) with β_{n} ∈ H_{2n}(ℂP^{∞}), dual to the basis of H^{∗}(ℂP^{∞}) given by the powers of the orientation x^{H}. For a complex oriented theory E define the elements b_{n}^{E} ∈ H_{n}(E′_{2}) by b_{n}^{E} = (x^{E})_{∗}(β_{n}).

Finally, as E^{r}(X) is an abelian group for any space X, the space E_{r} has an H-space product E_{r} × E_{r} → E_{r} which leads, in mod p homology, to a product

∗: H_{∗}(E_{r}) ⊗ H_{∗}(E_{r}) → H_{∗}(E_{r}).

Lemma 2.9. Let E be any complex oriented spectrum and let K denote complex K-theory. Then (c_{1}^{E})_{∗} ((b_{1}^{K})^{∗r}) = (b_{1}^{E})^{∗r} for all r.

Proof. The diagram above, in homology, says (c_{1}^{E})_{∗}(b_{1}^{K}) = b_{1}^{E}. The lemma follows by noting that c_{1}^{E} ∈ E^{2}(BU) is a primitive element in the Hopf algebra E^{∗}(BU), and hence represents an (unstable) additive cohomology operation K^{0}(−) → E^{2}(−). Thus the map c_{1}^{E} commutes with ∗-products and

(c_{1}^{E})_{∗} ((b_{1}^{K})^{∗r}) = ((c_{1}^{E})_{∗}(b_{1}^{K}))^{∗r} = (b_{1}^{E})^{∗r}

for all r.

Proof of Proposition 2.8. By the work of Bendersky and the second author [8], H_{∗}(E_{2r}) is polynomial (under the ∗-product) for any Landweber exact E and b_{1}^{E} ≠ 0. It follows that (c_{1}^{E})_{∗}(b_{1}^{K}^{∗r}) = (b_{1}^{E})^{∗r} is never zero. But c_{r}^{K} ∈ H^{2r}(BU) is dual to (b_{1}^{K})^{∗r} ∈ H_{2r}(BU), with respect to the basis (b_{1}^{K})^{∗r_{1}} ∗ (b_{2}^{K})^{∗r_{2}} ∗ ⋯ ∗ (b_{m}^{K})^{∗r_{m}}. Hence there is an η_{r} ∈ H^{2r}(E′_{2}) such that (c_{1}^{E})^{∗}(η_{r}) ≡ c_{r}^{K} mod (c_{1}^{K}, …, c_{r−1}^{K}). This proves (c_{1}^{E})^{∗} is surjective.

The Chern subring Ch(G) can be generated by elements of the form Bρ^{∗}(x) for some representations ρ: G → U and homogeneous elements x ∈ H^{∗}(BU). By the first part of the proposition for any such x there is a homogeneous y ∈ H^{∗}(E′_{2}) with (c_{1}^{E})^{∗}(y) = x. Hence Bρ^{∗}(x) = (c_{1}^{E}Bρ)^{∗}(y) lies in Ch_{E}(G).

Remark 2.10. For the theory { \atop E(n)}, in terms of which our main theorem is stated, we could have appealed to [11, 3.11] instead of [8].

3. The category A_{E}

In this section, under the restriction to finite groups G, we identify the category A_{E} for theories E satisfying certain conditions. The identification of A_{E} will depend on a positive natural number n associated to E, and thus we begin by defining a family of categories of elementary abelian subgroups of G, indexed by such n.

Definition 3.1. For 0 ⩽ n ⩽ ∞, define A^{(n)} = A^{(n)}(G) to be the category with objects the elementary abelian p-subgroups of G and morphisms the injective group homomorphisms f: W → V such that

∀w_{1}, …, w_{n} ∈ W ∃g ∈ G ∀1 ⩽ i ⩽ n f(w_{i}) = gw_{i}g^{−1} .

In particular, if t is the p-rank of G, there are equivalences of categories

A^{(t)} = A^{(t+1)} = ⋯ = A^{(∞)}

and this common category is the Quillen category A. As noted in Example 2.6, it is proved in [9, 7.1] that A^{(1)} is the category C(Ch(G)) of elementary abelians associated to the Chern subring. Moreover, by [9, 9.2], for each 2 ⩽ n < ∞ there is a subring R which satisfies the conditions of Theorem 2.4 and has category C(R) = A^{(n)}. The current paper arose from the desire to find a topological construction of such a ring R.

One way to explain why A^{(1)} is the right category for Ch_{K}(G) is via group characters. On the one hand, K^{0}(BG) is a completed ring of (virtual) characters. On the other hand, the morphisms in A^{(1)} are the group homomorphisms which preserve the values of characters for G. That is, f: W → V lies in A^{(1)} if and only if it satisfies the equation

f^{∗} Res_{V} (χ) = Res_{W} (χ)

for every character χ of G.

Switching attention to A^{(n)} we recall the work of Hopkins, Kuhn and Ravenel [10]. The morphism f: W → V lies in A^{(n)} if and only if it satisfies the analagous equation for every generalised character (class function) χ of G, i.e., for every function on the set G_{n,p} of commuting n-tuples of elements of G having p-power order which is constant on conjugacy classes and takes values in a certain E^{∗}-algebra L(E^{∗}), which is, roughly speaking, the smallest E^{∗}-algebra which contains all roots of each equation of the form [p^{k}](x) = 0: let E_{cont}^{∗}(Bℤ_{p}^{n}) = colim_{r} E^{∗}(B(ℤ/p^{r})^{n}) and S the multiplicatively closed subset generated by Euler classes of continuous homomorphisms ℤ_{p}^{n} → S^{1}, then L(E^{∗}) = S^{−1}E_{cont}^{∗}(Bℤ_{p}^{n}).

For suitable complex oriented E, an element x ∈ E^{∗}(BG) gives rise to such a class function in essentially the following way: a commuting n-tuple (g_{1}, …, g_{n}) in G as above can be thought of as a homomorphism α: ℤ_{p}^{n} → G. The value of a generalised character (class function) afforded by x on α is then given by the

composite E^{∗}(BG) ⟶^{(Bα)^{∗}} E_{cont}^{∗}(Bℤ_{p}^{n}) ⟶ L(E^{∗}). Theorem C of [10] asserts that the character map χ_{G} associating to each x ∈ E^{∗}(BG) the character it affords induces an isomorphism

L(E^{∗}) ⊗_{E^{∗}} E^{∗}(BG) ⟶^{χ_{G}} Cl_{n,p}(G; L(E^{∗})) .

Thus E^{∗}(BG) is related to the ring of generalised class functions in the same way as ordinary characters to the K-theory of BG.

Here a suitable theory means a complex oriented ring spectrum E, whose coefficients E^{∗} are a complete, graded, local ring with maximal ideal 𝔪, and residue characteristic p > 0, such that the mod 𝔪 reduction of its formal group law has height n and p^{−1}E^{∗} is non-zero. These, together with Landweber exactness condition needed to apply Theorem 2.7, constitute some of our requirements on E.

We need however E to satisfy one further property. Given a space X and any cohomology theory E^{∗}(−) represented by an Ω-spectrum E_{∗}, we have a ‘Hurewicz’ construction

H_{E}(X): E^{∗}(X) ⟶ Hom(H^{∗}(E_{∗}), H^{∗}(X))

where Hom denotes, say, morphisms of graded 𝔽_{p} algebras. The map H_{E}(X) is given by sending an element α ∈ E^{r}(X), thought of as a homotopy class of maps α: X → E_{r}, to its corresponding cohomology homomorphism α^{∗}: H^{∗}(E_{r}) → H^{∗}(X).

Property 3.2. Say the theory E satisfies Property 3.2(n) if

(a) E is Landweber exact and satisfies the hypotheses of [10], namely: the coefficients E^{∗} are a complete, graded, local ring with maximal ideal 𝔪 and residue characteristic p > 0, such that p^{−1}E^{∗} is nonzero and the mod 𝔪 reduction of the formal group law has height n;

(b) the Hurewicz map H_{E}(X) is injective when X = BV, the classifying space of an elementary abelian group of finite rank.

Remark 3.3. In the next section we prove that the Baker-Würgler theory { \atop E(n)} satisfies Property 3.2(n); it appears that a similar proof works for Morava E theory.

It may well be that the property (b) follows from (a); we know of no examples satisfying (a) which do not also satisfy (b). On the other hand, the Johnson-Wilson (incomplete) theories E(n) are examples of Landweber exact spectra for which property (b) holds, but not (a).

Property (b) is similar to a result of Lannes and Zarati (see, for example, [21, 8.1]) who prove such an injectivity property under the additional assumption that the infinite loop space E_{r} concerned has finite type cohomology. However, any theory E satisfying part (a) will be far from having finite type cohomology and we use very different methods in the next section to establish the result for the { \atop E(n)}.

Theorem 3.4. Let G be a finite group and suppose E satisfies Property 3.2(n). Then there is a homeomorphism of varieties

var(Ch_{E}(G)) → colim_{A^{(n)}} var(H^{∗}(BV)) .

Theorem 0.1 will of course follow from this and Theorem 4.1.

To prove Theorem 3.4 we introduce a further category C_{E} = C_{E}(G) of elementary abelians, defined in terms of the theory E.

Definition 3.5. For a finite group G, let C_{E} = C_{E}(G) be the category with objects the elementary abelian p-subgroups of G and morphisms the injective group homomorphisms f: W → V such that f^{∗} Res_{V} = Res_{W} holds in E-cohomology:

E^{∗}(BW) ^{f ∗} E^{∗}(BV)
[figure: Res_{W}; Res_{V}]
E^{∗}(BG)

We prove Theorem 3.4 in two stages, identifying respectively A^{(n)} with C_{E}, and C_{E} with A_{E}. The former uses part (a) of (3.2); the latter needs part (b).

Proposition 3.6. Suppose the theory E satisfies part (a) of Property 3.2(n). Then C_{E} = A^{(n)} for all finite groups.

Proof. Firstly recall from Theorem C of [10] that the kernel of the character map consists entirely of p-torsion, as L(E^{∗}) is faithfully flat over p^{−1}E^{∗}. So the character map is injective for elementary abelian groups.

Suppose f: W → V is in A^{(n)} but not in C_{E}. Then there is an x ∈ E^{∗}(BG) such that y := f^{∗} Res_{V} (x) − Res_{W} (x) ≠ 0. The injectivity of the character map implies that W has a rank (at most) n subgroup S such that Res_{S}(y) ≠ 0: just take S to be the subgroup generated by any commuting n-tuple on which the generalised class function associated to y does not vanish. Now let T = f(S) and h = f|_{S}. Then h: S → T is an isomorphism, and is induced by conjugation by some g ∈ G. Since conjugation by an element of G leaves x fixed, we arrive at

Res_{S}(y) = h^{∗} Res_{T} (x) − Res_{S}(x) = Res_{S}(g^{∗}x − x) = 0 ,

a contradiction.

Now suppose that f: W → V is in C_{E} but not in A^{(n)}. Then there is an elementary abelian S ⊂ W of rank at most n, such that h = f|_{S}: S → T = f(S) is an isomorphism not induced by conjugation in G. From the definition of C_{E} it is clear that h lies in C_{E}. Let g_{1}, …, g_{m} be a (minimal) generating set for S, and set g_{m+1} = … = g_{n} = 1 if necessary. Then (g_{1}, …, g_{n}) and (h(g_{1}), …, h(g_{n})) are two non-conjugate n-tuples in G, and hence are separated by a generalised class function. Surjectivity of the character map gives us a class x ∈ E^{∗}(BG) with Res_{S}(x) − h^{∗} Res_{T} (x) ≠ 0, i.e. Res_{V} (x) and f^{∗} Res_{W} (x) are distinct.

Remark 3.7. (1) In the definition of C_{E}, the condition that morphisms be monomorphisms is redundant: suppose f: W → V has kernel K. Then Res_{K} is trivial on E^{∗}(BG), but this cannot happen unless K = 1, as the character isomorphism gives a nontrivial class in the image of restriction.

(2) Instead of C_{E} it might seem more appropriate to consider a category C′_{E} consisting of all abelian subgroups and group homomorphisms inducing commutative triangles as in (3.5). The respective variant of Proposition 3.6 would still hold, by essentially the same arguments. We have refrained from doing so since our construction ultimately ends up in mod p cohomology, where the difference cannot be seen. This other approach would be relevant were we using p-local or integral cohomology; compare the final remarks in section 17 of Quillen’s paper [18].

(3) By construction, every morphism in C_{E} is also in A_{E}. Thus combining Proposition 3.6 with Theorem 2.7 yields the chain of inclusions A ⊆ A^{(n)} ⊆ A_{E} ⊆ A^{(1)} .

Proposition 3.8. Suppose the theory E is Landweber exact and satisfies part (b) of Property 3.2(n). Then C_{E} = A_{E} for all finite groups.

Proof. As just noted, it is immediate that C_{E} ⊆ A_{E} and it is the reverse inclusion we must show. Equivalently, we need to show that the category C_{E} does not change upon passing to the subrings of mod p cohomology generated by E-type characteristic classes.

So, assume f: W → V is in A_{E} but not in C_{E}. Consider the diagram

E^{∗}(BW) ^{f ∗}  E^{∗}(BV)
[figure]
E^{∗}(BG)

[figure: Hom(H^{∗}(E_{∗}), H^{∗}(BW)); H; ∩; \overline{f^{∗}}; Hom(H^{∗}(E_{∗}), H^{∗}(BV)); H; ∩]
[figure]
Hom(H^{∗}(E_{∗}), H^{∗}(BG)) \begin{aligned}H \\ &  \end{aligned}

Injectivity of the two outside vertical maps is the assumption that E satisfies part (b) of Property 3.2(n). That f lies in A_{E} implies the commutativity of the image of the top triangle in the bottom one; that f does not lie in C_{E} means that the top triangle does not commute. Commutativity of the ‘sides’ of the prism follows from the naturality of the H_{E}(X) construction with respect to maps of the space X. A contradiction now follows by chasing round an element x ∈ E^{∗}(BG) for which Res_{W} (x) − f^{∗} Res_{V} (x) ≠ 0.

4. Injectivity of HÊ(n)(BV)

The goal of this section is to prove that the I_{n}-adically complete theory { \atop E(n)} [6] satisfies Property 3.2(n). Satisfaction of part (a) is well established (we see that it is Landweber exact in [6] and it is noted in [10] that the other properties listed in part (a) also hold). Thus we must demonstrate that the maps

H{ \atop E(n)}(BV): { \atop E(n)}^{∗}(BV) → Hom(H^{∗}({ \atop E(n)}_{∗}), H^{∗}(BV))

are injective for all finite rank elementary abelian p-groups V.

In fact, we shall prove the following, equivalent result in homology.

Theorem 4.1. Let V be a finite rank elementary abelian p-group. Then

H{ \atop E(n)}(BV): { \atop E(n)}^{∗}(BV) → Hom(H_{∗}(BV), H_{∗}({ \atop E(n)}_{∗}))

defined by (α: BV → { \atop E(n)}_{r}) ↦ (α_{∗}: H_{∗}(BV) → H_{∗}({ \atop E(n)}_{r})) is injective. Here Hom denotes the morphisms in the category of graded cocommutative 𝔽_{p} coalgebras.

We start by noting that the map H_{E}(X) (in either homology or cohomology, but from now we shall work with the homology variant) satisfies good algebraic properties.

Proposition 4.2. Suppose that E is a ring spectrum and X is any space. Then Hom(H_{∗}(X), H_{∗}(E_{∗})) has a natural E_{∗}-algebra structure and

H_{E}(X): E^{∗}(X) ⟶ Hom(H_{∗}(X), H_{∗}(E_{∗}))

is a graded E_{∗}-algebra homomorphism.

Proof. This is essentially formal, but it is a good opportunity to introduce the notation and operations from coalgebraic algebra that are needed in the main proof of Theorem 4.1 together with explicit formulæ. For basic references to the algebraic properties and rules of manipulation in coalgebraic algebras (Hopf rings with further structure), see [12, 19].

As each E_{r} is an H-space, each H_{∗}(E_{r}) is a Hopf algebra with product ∗ as introduced in section 2 and coproduct

ψ: H_{∗}(E_{r}) → H_{∗}(E_{r}) ⊗ H_{∗}(E_{r}) .

However, as E is a ring spectrum the graded product in E^{∗}(X) is represented by maps E_{r} × E_{s} → E_{r+s} giving a further product

∘: H_{m}(E_{r}) ⊗ H_{n}(E_{s}) → H_{m+n}(E_{r+s}) .

An element e ∈ E_{−r} = E^{r} = π_{0}(E_{r}), thought of as a map from a point into E_{r}, gives rise to an element [e] ∈ H_{0}(E_{r}) as the image of 1 ∈ H_{0}(point). Such an element is grouplike and satisfies [d]∗[e] = [d+e] (where defined) and [d]∘[e] = [de]; the subobject of all such elements, written 𝔽_{p}[E^{∗}], forms a sub-coalgebraic ring and H_{∗}(E_{∗}) is a coalgebraic algebra over this coalgebraic ring. In fact, H_{0}(E_{∗}) = 𝔽_{p}[E^{∗}] and this should be thought of as the classical group-ring construction, endowed with extra structure. Note that [0] = b_{0} and is the ∗ unit (and is distinct from 0) and [1] is the ∘ unit (and is distinct from 1, which, however, is identical to [0]). Note also that a ∘ [0] is [0] if a ∈ H_{0}(E_{∗}) but is 0 otherwise.

It is of course entirely formal that the set of coalgebra maps from H_{∗}(X), an 𝔽_{p} coalgebra, to H_{∗}(E_{∗}), an algebra in the category of 𝔽_{p} coalgebras, carries itself an algebra structure. However, it will be useful to identify the operations explicitly. Addition of say f, g ∈ Hom(H_{∗}(X), H_{∗}(E_{∗})) is given by the composite

H_{∗}(X) ⟶^{Δ_{∗}} H_{∗}(X) ⊗ H_{∗}(X) ⟶^{f⊗g} H_{∗}(E_{∗}) ⊗ H_{∗}(E_{∗}) ⟶^{∗} H_{∗}(E_{∗})

and the product is described similarly using the ∘ product. The zero element is given by the composite

H_{∗}(X) → H_{∗}(point) → H_{∗}(E_{∗})

where the second map is that which in H_{0}(−) sends 1 to [0]; the unit is similar, using the map representing [1]. Finally, the E_{∗} action is given as follows: if e ∈ E^{∗} = E_{−∗} and f: H_{∗}(X) → H_{∗}(E_{∗}), then ef is the map H_{∗}(X) → H_{∗}(E_{∗}) sending x ∈ H_{∗}(X) to [e] ∘ f(x). It is left to the reader to check that, with these operations, H_{E}(X) is an E_{∗}-algebra homomorphism.

Remark 4.3. It will be useful to note that the construction H_{E}(X) is not only natural in the space X (as used in the previous section), but is also natural in the spectrum E. The strategy of the proof of Theorem 4.1 will be to prove the analagous result for H_{K(n)}(Bℤ/p), i.e. in Morava K-theory for the rank 1 case, and then deduce (4.1) from naturality in the spectrum via the Baker-Würgler tower [7] linking K(n) and { \atop E(n)}, and the application of an appropriate Künneth theorem.

Recall that the group monomorphism ℤ/p → S^{1} induces a homomorphism E^{∗}(ℂP^{∞}) → E^{∗}(Bℤ/p). For a complex oriented theory E we shall just write x ∈ E^{2}(Bℤ/p) for the image of the complex orientation x^{E}.

Proposition 4.4. [7, 10, 20] For E = K(n) or { \atop E(n)}, E^{∗}(Bℤ/p) is the free E^{∗} module on basis {1, x, ⋯, x^{p^{n}−1}}. Moreover, for both these theories there is a Künneth isomorphism

E^{∗}(B(ℤ/p)^{r}) = E^{∗}(Bℤ/p) ⊗_{E^{∗}} ⋯^{(r)} ⊗_{E^{∗}} E^{∗}(Bℤ/p).

As the homomorphism H_{∗}(Bℤ/p) → H_{∗}(ℂP^{∞}) is an isomorphism in even degrees, we shall extend the notation of section 2 and write β_{r} ∈ H_{2r}(Bℤ/p) for the corresponding elements. Note that the coproduct ψ: H_{∗}(Bℤ/p) → H_{∗}(Bℤ/p) ⊗ H_{∗}(Bℤ/p) acts by ψ(β_{r}) = ∑_{i+j=r} β_{i} ⊗ β_{j}. Note also that β_{0} = 1.

Proposition 4.5. (a) Suppose r > 0. The following formulæ describe some of the action of the map H_{E}(Bℤ/p) on the class x^{r} ∈ E^{2r}(Bℤ/p) for a complex oriented theory E.

H_{E}(Bℤ/p)(x^{r}):  β_{r} ↦ b_{1}^{∘r},
β_{t} ↦ 0,  if 0 < t < r,
β_{0} = 1 ↦ [0] = 1.

(b) (Action on x^{0}.) Suppose e ∈ E_{∗}. Then

\begin{aligned}H_{E}(Bℤ/p)(e): & β_{t} & ↦ & 0, & if 0 < t, \\ & β_{0} = 1 & ↦ & [e]. \end{aligned}

Proof. Given the description of the map H_{E}(X) in the proof of Proposition 4.2, these are essentially straightforward calculations in the ‘Hopf ring calculus’ of [19]. For example, H_{E}(Bℤ/p)(x)(β_{t}) = b_{t}, by definition of b_{t}. Then H_{E}(Bℤ/p)(x^{r})(β_{t}) is computed by the composite

β_{t} ↦ ∑_{j_{1}+···+j_{r}=t} β_{j_{1}} ⊗⋯⊗β_{j_{r}} ↦ ∑_{j_{1}+···+j_{r}=t} b_{j_{1}} ⊗⋯⊗b_{j_{r}} ↦ ∑_{j_{1}+···+j_{r}=t} b_{j_{1}} ∘⋯∘b_{j_{r}} .

If t = 0 there is just one term in the sum – all j_{i} = 0 and [0] ∘ ⋯ ∘ [0] = [0]. If 0 < t < r then, in every term in the sum, at least one j_{i} = 0 and at least one j_{k} > 0; thus each summand contains the element b_{j_{i}} ∘ b_{j_{k}} = [0] ∘ b_{j_{k}} = 0 and so the whole sum is zero. If t = r then the sum has one term in which all the j_{i} = 1, giving the b_{1}^{∘r} of the proposition, and all other summands are zero, as in the previous case. Part (b) follows immediately from the definitions.

Proposition 4.6. The morphism

H_{K(n)}(Bℤ/p): K(n)^{∗}(Bℤ/p) ⟶ Hom(H_{∗}(Bℤ/p), H_{∗}(K(n)_{∗}))

is injective.

Proof. By (4.4) a typical element of K(n)^{∗}(Bℤ/p) is of the form ∑_{i=0}^{p^{n}−1} c_{i}x^{i} where c_{i} ∈ K(n)^{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}]. In fact, it suffices to consider homogeneous elements, given the construction of H. We shall suppress the unit v_{n} (and corresponding [v_{n}] ∈ H_{∗}(K(n)_{∗}), as in [23]) for simplicity of notation and so shall assume c_{i} ∈ 𝔽_{p}. Then generally homogeneous elements of degree 2i are of the simple form c_{i}x^{i}, c_{i} ∈ 𝔽_{p} and 0 ⩽ i < p^{n}; this is not quite true if i ≡ 0 mod 2(p^{n} − 1) in which case the general element is c_{0} + c_{p^{n}−1}x^{p^{n}−1}.

So, for i ≢ 0 mod 2(p^{n} − 1), it suffices to show that, for c_{i} ≠ 0,

H_{K(n)}(Bℤ/p) (c_{i}x^{i})

acts non-trivially on some β_{t}. By Proposition 4.5 it sends β_{i} to [c_{i}] ∘ b_{1}^{∘i}; as b_{1} is primitive and i > 0, this is just c_{i}b_{1}^{∘i}. From [23] we know that all the b_{1}^{∘r} ∈ H_{2r}(K(n)_{2r}) are non-trivial if r < p^{n}.

For i ≡ 0 mod 2(p^{n} − 1) we must consider the general element c_{0} + c_{p^{n}−1}x^{p^{n}−1}. If c_{p^{n}−1} = 0 then (4.5)(b) shows H_{K(n)}(Bℤ/p)(c_{0}) is non-zero on β_{0}. Otherwise, assuming c_{p^{n}−1} ≠ 0,

\begin{aligned}H_{K(n)}(Bℤ/p)(c_{0} + c_{p^{n}−1}x^{p^{n}−1}): β_{p^{n}−1} & ↦ & [c_{0}] ∗ ([c_{p^{n}−1}] ∘ b_{1}^{∘p^{n}−1}) \\ & = & [c_{0}] ∗ (c_{p^{n}−1}b_{1}^{∘p^{n}−1}) . \end{aligned}

As [c_{0}] is a ∗ unit (with ∗ inverse [−c_{0}]), this last expression is non-zero and the proof is complete.

It is interesting to note that in H_{∗}(K(n)_{∗}) although b_{1}^{∘p^{n}−1} ≠ 0, one more ∘ power of b_{1} (or even one more suspension) kills this element. In this sense the above proof only ‘just’ works.

We now recall the tower of spectra defined in [7] (and implicitly in [6]). This is a tower

⋯ ⟶ E(n)/I_{n}^{k+1} ⟶ E(n)/I_{n}^{k} ⟶ ⋯ ⟶ K(n) = E(n)/I_{n}^{1}

where the spectrum E(n)/I_{n}^{k} has homotopy E(n)_{∗}/I_{n}^{k}. The homotopy limit of this tower is the Baker-Würgler spectrum { \atop E(n)}. Unstably, (i.e., passing to Ω spectra), this corresponds to a tower of fibrations of the relevant spaces; the fibre of the map

(E(n)/I_{n}^{k+1})_{∗} ⟶ (E(n)/I_{n}^{k})_{∗}

is a product of copies of K(n)_{∗} indexed by a basis of I_{n}^{k+1}/I_{n}^{k}, i.e., by monomials in the v_{l}, 0 ⩽ l < n (using the convention of putting v_{0} = p) of degree k + 1.

This tower of spectra gives rise to Baker and Würgler’s K(n) Bockstein spectral sequence [7]. An example of this sequence is that for the space Bℤ/p in which the E_{2}-page is just

{ \atop E(n)}_{∗} ⊗_{K(n)_{∗}} K(n)^{∗}(Bℤ/p).

This is entirely in even dimensions and the sequence, converging to ({ \atop E(n)})^{∗}(Bℤ/p), collapses; cf. Proposition 4.4. We shall prove the rank 1 case of Theorem 4.1 by examining the Hurewicz image of this spectral sequence.

Theorem 4.7. The morphism

H{ \atop E(n)}(Bℤ/p): { \atop E(n)}^{∗}(Bℤ/p) ⟶ Hom(H_{∗}(Bℤ/p), H_{∗}({ \atop E(n)}_{∗}))

is injective.

Proof. Let 0 ≠ α ∈ ({ \atop E(n)})^{s}(Bℤ/p) and consider it as a map Bℤ/p → { \atop E(n)}_{s}. Either there is some integer k ⩾ 1 for which composition of α with the maps in the Baker-Würgler tower gives an essential map α̃: Bℤ/p → (E(n)/I_{n}^{k+1})_{s} but a null map to (E(n)/I_{n}^{k})_{s}, or else α maps to a non-zero element of K(n)^{s}(Bℤ/p) (at the bottom of the tower).

In the former case, the map α̃ lifts to an essential map to the fibre of the map (E(n)/I_{n}^{k+1})_{s} ⟶ (E(n)/I_{n}^{k})_{s}, a product of spaces from the Ω spectrum for K(n). Thus in both this or the second case, α gives rise to a non-trivial map

a: Bℤ/p ⟶ ∏_{i} K(n)_{r_{i}}

where the product is finite, indexed by the monomials, w_{i} say, in the v_{l} of degree k. (Note that the r_{i} will generally differ from the original s, their value depending on the dimension of the w_{i} concerned.)

Consider the commutative diagram

H_{∗}(Bℤ/p)  \overline{}H_{∗}({ \atop E(n)}_{s})

[figure: H_{∗}(∏_{i} K(n)_{r_{i}}); a_{∗}; H_{∗}((E(n)/I_{n}^{k+1})_{s}) .]

The top map is H{ \atop E(n)}(Bℤ/p)(α), the map we wish to show to be non-trivial. This will follow by finding an element β_{r} ∈ H_{∗}(Bℤ/p) which passes to something non-zero in H_{∗}((E(n)/I_{n}^{k+1})_{s}) and we do this by examining the action of a_{∗} (via 4.6) and the bottom horizontal map which is given by the inclusion of the fibre ι: ∏_{i} K(n)_{r_{i}} → (E(n)/I_{n}^{k+1})_{s}.

First examine the map a: Bℤ/p → ∏_{i} K(n)_{r_{i}}. Keeping track of the components and the monomials w_{i} they correspond to, we can write this map as a tuple (…, w_{i}c_{i}x^{t_{i}}, …) where, as before, we assume c_{i} ∈ 𝔽_{p} and 0 ⩽ t_{i} < p^{n} (and, strictly speaking, in the dimension congruent to 0 mod 2(p^{n} − 1), components may be of the form w_{i}(c_{0,i} + c_{p^{n}−1,i}x^{p^{n}−1})). The composite with the inclusion, in homology,

H_{∗}(Bℤ/p) ⟶^{a_{∗}} H_{∗}(∏ K(n)_{r_{i}}) ⟶ H_{∗}((E(n)/I_{n}^{k+1})_{s})

sends an element β_{r} to

∑ ∗_{i}[w_{i}] ∘ ι_{∗}H_{K(n)}(Bℤ/p) (c_{i}x^{t_{i}}) (β′_{i})

where the sum is over all the terms in the iterated coproduct of β_{r}, and where we write ψ(β_{r}) = ∑ ⋯ ⊗ β′_{i} ⊗ · · ·. (Again, reading the longer expression c_{0,i} + c_{p^{n}−1,i}x^{p^{n}−1} in the displayed formula where necessary.)

If all the powers of x in this expression are zero, so that we are just dealing with a ‘constant’ term, then the result is easy – taking β_{0} in the top left hand of the commutative square we map to ∑_{i}[c_{i}w_{i}] ≠ 0 ∈ H_{∗}((E(n)/I_{n}^{k+1})_{∗}), where the c_{i} ∈ 𝔽_{p} (not all zero) are the coefficients of x^{0} in each factor in the above expression for a.

So suppose r is the smallest positive power of x which appears in the expression for a above. By (4.6), the image of β_{r} in the bottom right of the commutative square is then (up to a ∗ multiple of ∗ invertible elements [c_{0,j}])

∑_{i}[w_{i}] ∘ c_{i}b_{1}^{∘r} = (∑_{i} c_{i}[w_{i}]) ∘ b_{1}^{∘r}

where the sum is now over only some of the indexing elements i (namely those for which t_{i} = r).

It suffices now to show that expressions of the form (∑_{i} c_{i}[w_{i}])∘b_{1}^{∘r} are non-zero in H_{∗}((E(n)/I_{n}^{k+1})_{∗}). This follows from the following lemma, thus completing the present proof.

Lemma 4.8. Let w_{j} be a set of monomials in the v_{l}, 0 < l < n, of degree k and of some fixed homotopy dimension and suppose c_{j} ∈ 𝔽_{p}, not all zero. Then (∑_{i} c_{j}[w_{j}]) ∘ b_{1}^{∘q} are non-zero in H_{∗}((E(n)/I_{n}^{k+1})_{∗}) for all 0 ⩽ q < p^{n}.

Proof. As ∘ product with b_{1} represents (double) suspension, one way to prove this would be to observe that (∑_{i} c_{j}[w_{j}]) was a non-zero element of H_{0}((E(n)/I_{n}^{k+1})_{∗}) and that this suspends to a non-zero element (∑_{i} c_{j}w_{j}) of H_{∗}(E(n)/I_{n}^{k+1}). Then every intermediate (∑_{i} c_{j}[w_{j}]) ∘ b_{1}^{∘q} must be non-zero as well. Although the statement about (∑_{i} c_{j}[w_{j}]) ∈ H_{0}((E(n)/I_{n}^{k+1})_{∗}) is true, it is not true that this suspends to a non-zero element of the stable gadget as, indeed, H_{∗}(E(n)/I_{n}^{k+1}) = 0. However, this proof would work if H was replaced with K(n) and the strategy of proof will be to deduce the H result from that for K(n) by arguing with the Atiyah-Hirzebruch spectral sequence.

First check the statements for K(n). It is a basic fact on the homology of Ω spectra that (∑_{i} c_{j}[w_{j}]) ≠ 0 ∈ K(n)_{0}((E(n)/I_{n}^{k+1})_{∗}) as this is just the group-ring [19]. To see the stable result it suffices to check that the right unit

E(n)_{∗}/I_{n}^{k+1} ⟶ K(n)_{∗}(E(n)/I_{n}^{k+1})

is an inclusion on sums of monomials of degree k. However, the cofibration of spectra

⋁ Σ^{•}K(n) ⟶ E(n)/I_{n}^{k+1} ⟶ E(n)/I_{n}^{k}

gives rise to a long exact sequence in Morava K-theory. The elements in question map to 0 in K(n)_{∗}(E(n)/I_{n}^{k}) but lift non-trivially in K(n)_{∗}(⋁ Σ^{•}K(n)).

Recall that in H_{∗}(K(n)_{∗}) the element b_{1} suspends by iterated ∘ product with itself to the elements b_{1}^{∘r} ≠ 0 ∈ H_{2r}(K(n)_{2r}). These are non-zero until we get to b_{1}^{∘p^{n}−1} (non-zero) suspending to b_{1}^{∘p^{n}−1} ∘ e_{1} = 0 ∈ H_{2p^{n}−1}(K(n)_{2p^{n}−1}). Here we write e_{1} ∈ H_{1}(K(n)_{1}) for the single suspension element following the notation of [19]. However, in K(n)_{∗}(K(n)_{∗}) we have b_{1}^{∘p^{n}−1} ∘ e_{1} = v_{n}e_{1} and suspension continues indefinitely, ultimately reaching the stable element given by the image of 1 under the right unit in K(n)_{∗}(K(n)).

Now consider the map of Atiyah-Hirzebruch spectral sequences

K(n)_{∗} ⊗_{𝔽_{p}} H_{∗}(∏ K(n)_{r_{i}}) ⟹  K(n)_{∗}(∏ K(n)_{r_{i}})
↓  ↓
K(n)_{∗} ⊗_{𝔽_{p}} H_{∗}((E(n)/I_{n}^{k+1})_{s}) ⟹ K(n)_{∗}((E(n)/I_{n}^{k+1})_{s}).

Take an element (∑_{j} c_{j}[w_{j}]) ∘ b_{1}^{∘q} ∈ H_{∗}(∏ K(n)_{r_{i}}) with 0 ⩽ q < p^{n}. We know it is a permanent (and non-trivial) cycle in the top spectral sequence [23] and that regarded as an element of K(n)_{∗}(∏ K(n)_{r_{i}}) it maps non-trivially to K(n)_{∗}((E(n)/I_{n}^{k+1})_{∗}). The only way the version of this element in H_{∗}(−) could map to zero in H_{∗}((E(n)/I_{n}^{k+1})_{∗}) would be if the K(n)_{∗}(−) version dropped Atiyah-Hirzebruch filtration in mapping to K(n)_{∗}((E(n)/I_{n}^{k+1})_{∗}). As dim([v] ∘ b_{1}^{∘q}) ⩽ 2p^{n} − 2 this cannot happen, for dimensional reasons.

Theorem 4.1 now follows from (4.4) and (4.7) together with the observation that H_{∗}(B(ℤ/p)^{r}) also satisfies a Künneth isomorphism.

Remark 4.9. The method of proof used for Theorem 4.1 can be adapted to establish analagous results for other theories. One of the key elements of our proof is the use of the Baker-Würgler tower, and the recent work of Baker and Lazarev [5] now allows such towers to be built in quite general circumstances. In particular, it would seem that height n Morava E-theory also satisfies Property 3.2(n).

Baker’s study [4] of the homotopy type of elliptic spectra shows that if ℘ is any suitable prime ideal of Ell_{∗}, the complete spectrum (Ell) ˆ℘ splits as a wedge

of suspensions of { \atop E(2)} (see [4] for precise details of the spectra Ell and ideals ℘ considered). Theorem 4.1 thus shows that all these complete spectra also satisfy Property 3.2(2) and thus, in some sense, identifies the subring of ‘elliptic characteristic classes’ in the mod p cohomology of a finite group (even though we are still not sure what an ‘elliptic object’ actually is).

5. Examples

We finish by sketching some of the calculations for the chromatic subrings Ch{ \atop E(n)}(G) for the simplest non-trivial example, that of G the alternating group A_{4}. We take the prime p to be 2.

As the 2-rank of A_{4} is two, the category A^{(2)} is the Quillen category A. The skeleton of A may be represented as

[figure: 1; 2; 1; C_{3}; 1; Rank of elem. abelian; Automorphism group]

Here, the nodes represent isomorphism classes of elementary abelians, labelled by 2-rank and automorphism group. The edges represent equivalence classes of morphisms under conjugacy, the label denoting the stabilizer in the automorphism group of the target group. Thus here there are 3 = |C_{3} : 1| morphisms from a rank 1 to a rank 2 elementary abelian.

The skeleton of the category A^{(1)} is

[figure: 1; 2; 1; S_{3}; C_{2}]

Thus A is strictly contained in A^{(1)} and so, by Theorem 0.1, the subring Ch{ \atop E(1)}(A_{4}) is strictly contained in Ch{ \atop E(2)}(A_{4}). The following calculations demonstrate an element in the latter not in the former as well as showing explicitly the equivalence var(Ch{ \atop E(2)}(A_{4})) = var(H^{∗}(BA_{4})).

Let V be the Sylow 2-subgroup of A_{4}. The Weyl group of V in A_{4} is the cyclic group C_{3}, permuting the non-trivial elements transitively. As V is abelian, H^{∗}(BA_{4}) is the ring of N_{A_{4}}(V)-invariants in H^{∗}(BV). Let x, y be the basis for V^{∗} dual to the basis (1 2)(3 4), (1 3)(2 4) for V. Then we have

H^{∗}(BA_{4}) ≅ 𝔽_{2}[x, y]^{C_{3}} |x| = |y| = 1

where the C_{3}-action is x ↦ y ↦ x + y. Over 𝔽_{4} we can diagonalize this action and so calculate the invariants: H^{∗}(BA_{4}) is generated by D_{1}, D_{0} and η, where

D_{1} = x^{2} + xy + y^{2} D_{0} = x^{2}y + xy^{2} η = x^{3} + x^{2}y + y^{3} .

Observe that D_{1} and D_{0} are Dickson invariants, and that η is the orbit sum of x^{2}y. The natural permutation representation π of A_{4} has Chern classes c_{2}(π) = D_{1}^{2} and c_{3}(π) = D_{0}^{2}. These generate the Chern subring Ch(A_{4}) = Ch{ \atop E(1)}(A_{4}), and are

{ \atop E(2)}-type classes as well by Proposition 2.8.

Now { \atop E(2)}^{∗}(BV) ≅ { \atop E(2)}^{∗}[[w, z]]/([2]_{F} (w), [2]_{F} (z)) where [2]_{F} (−) denotes the 2-series of the formal group law for { \atop E(2)}. Set θ = Tr_{BV}^{BA_{4}}(w^{2}z). By the Mackey formula, we have

Res_{BV} θ = w^{2}z + z^{2}(w +_{F} z) + (w +_{F} z)^{2}w: BV → { \atop E(2)}_{6} .

Write β(s) for the generating function of the β_{i} ∈ H_{2i}(Bℤ/2), that is β(s) is the formal power series ∑_{i⩾0} β_{i}s^{i} ∈ H_{∗}(Bℤ/2)[[s]], and similarly regard β(s) ⊗ β(t) as the corresponding generating function in H_{∗}(BV) = H_{∗}(Bℤ/2) ⊗_{𝔽_{2}} H_{∗}(Bℤ/2).

Likewise, write b(s) for the generating function of the b_{i} in H_{∗}({ \atop E(2)}_{2}). The standard Ravenel-Wilson relation [19, 3.6] in H_{∗}(E_{∗}) for any complex oriented theory E now allows us to compute

(Res_{BV} θ)_{∗}(β(s) ⊗ β(t))

= (b(s)^{∘2} ∘ b(t)) ∗ (b(t)^{∘2} ∘ b(s + t)) ∗ (b(s + t)^{∘2} ∘ b(s)) .

This reduces to

b_{0}^{∘3} + b̄(s)^{∘2} ∘ b̄(t) + b̄(t)^{∘2} ∘ b̄(s + t) + b̄(s + t)^{∘2} ∘ b̄(s) mod indecomposables

where b̄(t) is defined as b(t) − b_{0} = ∑_{r⩾1} b_{r}t^{r} (recall that b_{r} ∘ b_{0} = 0 for r > 0 and that b_{0} is the ∗ unit).

The (s, t)-degree 3 part of this expression is

b_{1}^{∘3}(s^{2}t + t^{2}(s + t) + (s + t)^{2}s) = b_{1}^{∘3}(s^{3} + s^{2}t + t^{3}) .

Note that b_{1}^{∘3} ∈ H_{6}({ \atop E(2)}_{6}) represents a non-zero indecomposable: for example, it maps to the corresponding element in H_{6}(K(2)_{6}) through the Baker-Würgler tower, and this represents a non-zero indecomposable here [23]. By duality, there is a class γ ∈ H^{6}({ \atop E(2)}_{6}) which kills all decomposables in H_{6} and sends b_{1}^{∘3} to 1. Then the { \atop E(2)}-type characteristic class θ^{∗}(γ) restricts to H^{∗}(BV) as η^{2}. Hence Ch{ \atop E(2)}(A_{4}) contains Ch{ \atop E(1)}(A_{4}) as a subring, and the inclusion of Ch{ \atop E(2)}(A_{4}) in H^{∗}(BA_{4}) is an inseparable isogeny.

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Department of Mathematics, University of Wuppertal, Gaußstr. 20, D-42097 Wuppertal, Germany

E-mail address: green@math.uni-wuppertal.de, schuster@math.uni-wuppertal.de

Department of Mathematics and Computer Science, University Road, Leicester, LE1 7RH, England

E-mail address: J.Hunton@mcs.le.ac.uk