TRANSFER AND COMPLEX ORIENTED COHOMOLOGY RINGS

MALKHAZ BAKURADZE AND STEWART PRIDDY

1. Introduction

For various examples of finite groups the complex oriented cohomology ring coincides with its subring generated by Chern classes [21],[22], [24]. Even more groups are good in the sense that their Morava Ktheory is generated by transferred Chern classes of complex representations of subgroups [10]. Special effort was needed to find an example of a group not good in this sense [14]. Thus the relations in the complex oriented cohomology ring of a finite group derived from formal properties of the transfer should play a major role. The purpose of this paper is to elucidate for finite coverings the interaction between transferred Chern classes and Chern classes of transferred bundles.

Let p be a prime and let G ≤ Σ_{p} be a subgroup of the symmetric group. In this paper we consider the complex oriented cohomology of homotopy orbit spaces X_{hG}^{p} = EG×_{G}X^{p}. Several authors have computed these cohomology groups, [15], [11], [12], [10], however we are particularly interested in the ring structure and thereby explicit formulas for the transfer. Thus we are led to consider Frobenius reciprocity, the relation between cup products and transfer:

T r^{∗}(x)y = T r^{∗}(xρ^{∗}(y))

(formula (i) of Section 2) where ρ : EG×X^{p} → X_{hG}^{p} is the covering projection and

T r^{∗} : E^{∗}(X^{p}) → E^{∗}(X_{hG}^{p})

is the associated transfer homomorphism.

Date: May 12, 2003.

1991 Mathematics Subject Classification. 55N22, 55R12.

Key words and phrases. transfer, Chern class, classifying space, complex cobordism, Morava K-theory.

The first author was supported by CRDF grant GM1 2083 and by the Max-Planck-Institut für Mathematik.

Let π ≤ Σ_{p} be the subgroup of cyclic permutations of order p. Our results for MU^{∗}(X_{hπ}^{p}), X = CP^{∞}, and ξ → CP^{∞} the canonical complex line bundle, provide a universal example which enables us to write explicitly the Chern classes c_{1}, ..., c_{p−1} of the transferred bundle ξ_{π} as certain formal power series in the Euler class c_{p}(ξ_{π}) with coefficients in E^{∗}(Bπ) plus certain transferred classes of the bundle ξ. In Section 3 we give an algorithm for computing these coefficients.

In paricular for E = BP, Brown-Peterson cohomology, the coefficients of this formal power series are invariant under the action of the normalizer of π in Σ_{p}. This enables us to give the similar results for Σ_{p} coverings. Moreover in Section 4 we compute the algebra BP^{∗}(X_{hΣ_{p}}^{p}) and show that its multiplicative structure is completely determined by Frobenius reciprocity.

In addition for E = K(s), Morava K-theory, the computations become easier: we show in Section 5 that the formal power series in the algorithm above descend to polynomials. We derive an alternative way for calculation and give some examples.

Section 6 is devoted to extending some results of [10] in Morava K-theory. In particular we show that if X is good then X_{hπ}^{p} is good.

In Section 7 we give some applications to classifying spaces of finite groups.

We would like to thank several people for their help during the course of this work: D. Ravenel for supplying us with the proof of Lemma 5.3, M. Jibladze for some Maple programs used in the examples of Section 5, and finally the referee who suggested many improvements.

A word about notation: in Sections 2, 3, 4 and 7 we denote CP^{∞} simply by X.

2. Preliminaries

We recall that a multiplicative cohomology theory E^{∗} is called complex oriented if there exists a Thom class, that is, a class u ∈ E^{2}(CP^{∞}) that restricts to a generator of the free one-dimensional E^{∗} module E^{2}(CP^{1}). The universal example is complex cobordism MU^{∗}. Then

E^{∗}(CP^{∞}) = E^{∗}[[x]],

where x is the Euler class of the canonical complex line bundle ξ over CP^{∞} = BU(1). Further

E^{∗}(BU(1)^{p}) = E^{∗}[[x_{1}, ..., x_{p}]],

where x_{i} = c_{1}(ξ_{i}) and ξ_{i} is the pullback bundle over BU(1)^{p} by the projection BU(1)^{p} → BU(1) on the i-th factor.

Much of our paper is written in terms of transfer maps [1, 13] and formal group laws. Let us give a brief review of formal properties of the transfer. For a finite covering

ρ : X → X/G

there is a stable transfer map

T r = T r(ρ) : X/G^{+} → X^{+}.

For any multiplicative cohomology theory E^{∗}, Frobenius reciprocity holds i.e., the induced map T r^{∗} is a map of E^{∗}(X/G) modules

(i) T r^{∗}(xρ^{∗}(y)) = T r^{∗}(x)y, x ∈ E^{∗}(X), y ∈ E^{∗}(X/G). For example

(ii) T r^{∗}(ρ^{∗}(y)) = T r^{∗}(1)y.

The element T r^{∗}(1) ∈ E^{0}(X/G) is called the index or stable Euler class of the covering ρ. The following additional properties of the transfer will be used:

(iii) The transfer is natural with respect to pullbacks;

(iv) T r(ρ_{1} × ρ_{2}) = T r(ρ_{1}) ∧ T r(ρ_{2});

(v) If ρ = ρ_{2}ρ_{1}, then T r(ρ) = T r(ρ_{2})T r(ρ_{1}).

More generally for a covering projection

ρ_{H,G} : X/H → X/G

with H ≤ G there is a stable transfer map

T r_{H,G} : X/G^{+} → X/H^{+}.

To ease notation if H = e, as above, we write projection and transfer in equivalent ways ρ = ρ_{G}, T r = T r(ρ) = T r_{G}.

The reverse composition to (ii) is given by

(vi) (Double coset formula) If K, H ≤ G then

ρ_{K,G}^{∗}T r_{H,G}^{∗} = ∑_{x} T r_{K∩H^{x},K}^{∗} ∘ x^{−1}^{∗} ∘ ρ_{K^{x^{−1}}∩H,H}^{∗}

where the sum is taken over a set of double coset representatives x ∈ K\G/H. Here H^{x} = xHx^{−1}.

For a regular covering ρ_{H,G}, i.e. H ⊴ G,

ρ_{H,G}^{∗}T r_{H,G}^{∗}(x) = N(x) = ∑_{g∈G/H} g^{∗}(x),

where N(x) is called the norm or trace of x.

In subsequent sections the reduced transfer T r_{H,G} : X/G → X/H is used.

We recall Quillen’s formula [16, 6]. First,

E^{∗}(BZ/p) = E^{∗}[[z]]/([p](z)),

where x is the Euler class of a faithful one-dimensional complex representation of Z/p and [p](z) is the p-series or p-fold iterated formal sum. Then

(1)  T r_{Z/p}^{∗}(1) = [p](z)/z,

where T r_{Z/p}^{∗} is the transfer homomorphism for the universal Z/p-covering EZ/p → BZ/p. The relation [p](z) = 0 is equivalent to the transfer relation

zT r_{Z/p}^{∗}(1) = T r_{Z/p}^{∗}(c_{1}(C)) = T r_{Z/p}^{∗}(0) = 0

obtained by applying (ii). Of course since the transfer is natural, Quillen’s formula enables us to compute the stable Euler class for any regular Z/p covering.

In this spirit, let

π = ⟨t⟩ ≤ Σ_{p}

be the subgroup of cyclic permutations of order p. For a given free action of π on a space Y with a given complex line bundle η → Y we have an equivariant map

η_{π} = (g_{1}, ..., g_{p}) : Y → BU(1)^{p},

where g_{i} classifies the line bundle t^{i−1}η.

So by naturality of the transfer, the computation of transferred Chern classes T r^{∗}(c_{1}^{i}(η)), i ≥ 1 for cyclic coverings can be reduced to the covering

ρ_{π} : Eπ × (BU(1))^{p} → Eπ ×_{π} (BU(1))^{p},

as the universal example.

Similarly for the symmetric group.

Let ξ be the canonical complex line bundle over CP^{∞} = BU(1) and ξ_{i} be the pullback bundle over BU(1)^{p} by the projection on the i-th factor as before. Then MU^{∗}(BU(1)^{p}) = MU^{∗}[[x_{1}, ..., x_{p}]], x_{i} = c_{1}(ξ_{i}) and x_{1} ⋯ x_{p} is the Euler class of the bundle ξ^{×p} = ⊕ξ_{i}.

Note that by transfer property (v), T r(ρ_{π})^{∗} has the same value on the Chern classes x_{1}, ..., x_{p}: the group π permutes the x_{i} and ρ_{π}t = ρ_{π}, t ∈ π. Thus in computations of the transfer we sometimes write these Chern classes in an equivalent way x, tx, ..., t^{p−1}x.

For the sphere bundle S(ξ^{×p}), one has

(2)  MU^{∗}(S(ξ^{×p})) = MU^{∗}[[x_{1}, ..., x_{p}]]/(x_{1} ⋯ x_{p}).

Then for the trace map

N = 1 + t + ... + t^{p−1}

we have kerN = Im(1 − t), t ∈ π in MU^{∗}BU(1)^{p} and after restricting N to MU^{∗}(S(ξ^{×p})) we have the exact sequence (3)

... ← MU^{∗}(S(ξ^{×p})) ←^{N} MU^{∗}(S(ξ^{×p})) ←^{1−t} MU^{∗}(S(ξ^{×p})) ←^{N} MU^{∗}(S(ξ^{×p})) ← ...

Then let ξ_{π} = Eπ ×_{π} ξ^{×p} be the Atiyah transfer bundle [2],

(4)  S(ξ_{π}) = Eπ ×_{π} S(ξ^{×p})

be its sphere bundle and

(5)  D(ξ_{π}) = Eπ ×_{π} D(ξ^{×p})

be its disk bundle. Let X = CP^{∞} then D(ξ_{π}) is homotopy equivalent to X_{hπ}^{p} = B(π ≀ U(1)).

The cofibration D(ξ_{π})/S(ξ_{π}) = (X_{hπ}^{p})^{ξ_{π}} gives a long exact sequence

(6) ... ← MU^{∗}(S(ξ_{π})) ← MU^{∗}(X_{hπ}^{p}) ←^{×c_{p}} MU^{∗}((X_{hπ}^{p})^{ξ_{π}}) ← ...

where (X_{hπ}^{p})^{ξ_{π}} is the Thom space of the bundle ξ_{π} and the right homomorphism is multiplication by the Euler class c_{p} = c_{p}(ξ_{π}).

Since the diagonal of BU(1)^{p} is fixed under the permutation action of π, the inclusion Eπ → Eπ × BU(1)^{p}; x → (x, fixpoint) defines the inclusions

i : Bπ → X_{hπ}^{p}
i_{0} : Bπ → S(ξ_{π}).

The projection φ : X_{hπ}^{p} → Bπ induced by π ≀ U(1) → π defines the projection

(7)  φ_{0} : S(ξ_{π}) → Bπ

and the compositions φ_{0}i_{0}, φi are the identity. We can consider S(ξ_{π}) as a bundle over Bπ with fiber S(ξ^{×p}).

Let η be the canonical line bundle over Bπ and

θ = φ^{∗}(η) → X_{hπ}^{p}

be the pullback bundle. Thus i^{∗}(θ) = η and i^{∗}(ξ_{π}) = C + η + ... + η^{p−1}.

Consider the pullback diagram

[figure: Eπ × S(ξ^{×p}); Eπ × BU(1)^{p}; S(ξ_{π}); ρ_{0}; X_{hπ}^{p}; ρ]

Let T r = T r(ρ) be the transfer of the covering ρ, and T r_{0} : S(ξ_{π}) → S(ξ^{×p}) the transfer map of ρ_{0}.

We will often refer to the following lemma which follows from (3) and Frobenius reciprocity.

Lemma 2.1. In MU^{∗}(X_{hπ}^{p}), ImT r^{∗} ⋂ Ker(ρ^{∗}) = 0.

Proof: ρ^{∗}(T r^{∗}(a)) = N(a) = 0 ⇒ a ∈ Im(1 − t) ⇒ T r^{∗}(a) = 0. □

Remark 2.2. Lemma 2.1 is valid only in complex oriented cohomology E^{∗} with torsion free coefficient ring. This lemma is used in the proof of Theorem 3.1 in complex cobordism and in the second statement of Theorem 4.6 in Brown-Peterson cohomology. By naturality, these results hold for all E^{∗} in the first case and all p-local E^{∗} in the second.

3. Transferred Chern classes for cyclic coverings

In this section we prove our main result for cyclic coverings, Theorem 3.2.

In the notation of the previous section the k-th Chern class of the bundle ξ^{×p} is the elementary symmetric function σ_{k}(x_{1}, ..., x_{p}) in Chern classes x_{i} and is the sum of \binom{p}{k} elementary monomials. The action of π on the set of these monomials gives us p^{−1}\binom{p}{k} orbits and the transfer homomorphism is constant on orbits by transfer property (v) or (iii).

Let E^{∗} be a complex oriented cohomology theory. For k = 1, ..., p−1, let

ω_{k} = ω_{k}(x_{1}, ..., x_{p}) ∈ E^{∗}(BU(1)^{p})

be the sum of representative monomials one from each of these orbits. The value of T r^{∗}(ω_{k}) does not depend on the choice of ω_{k} since ω_{k} is defined modulo Im(1−t) and on the elements of Im(1−t) the transfer homomorphism is zero again by (v). In other words we can take any ω_{k} for which Nω_{k} = σ_{k}(x_{1}, ..., x_{p}) holds. As we shall explain in Corollary 3.6 of Theorem 3.1, the following result enables us to calculate the transfer on all elements whose norm is symmetric.

For ease of notation let X = CP^{∞} and c_{j} = c_{j}(ξ_{π}), j = 1, ..., p.

Theorem 3.1. We can construct explicit elements

δ_{i}^{(k)} ∈ Ẽ^{∗}(Bπ), k = 1, ..., p − 1,

such that

T r^{∗}(ω_{k}) = c_{k} + ∑_{i≥0} φ^{∗}(δ_{i}^{(k)})c_{p}^{i}

for the transfer of the covering ρ : X^{p} → X_{hπ}^{p}.

Before constructing the elements δ_{i}^{(k)} in Section 3.2 we first prove their existence.

3.1. Complex cobordism of (CP^{∞})_{hπ}^{p}.

Theorem 3.2. In MU^{∗}(X_{hπ}^{p})

a) The annihilator of the Chern class c = c_{1}(θ) coincides with ImT r^{∗};

b) Multiplication by c_{p} = c_{p}(ξ_{π}) is a monomorphism;

c) Any element of Ker(ρ^{∗}) has the form ∑_{k≥0} φ^{∗}(δ_{k})c_{p}^{k}, for some elements δ_{k} ∈ M̃U^{∗}(Bπ).

d) For π = Z/2,

MU^{∗}B(π ≀ U(1)) = MU^{∗}[[c, c_{1}, c_{2}]]/(c_{1} − c_{1}^{∗}, c_{2} − c_{2}^{∗})

= MU^{∗}(Bπ)[[T r^{∗}(x), c_{2}]]/(cT r^{∗}(x)),

where c_{i} = c_{i}(ξ_{π}), c_{i}^{∗} = c_{i}(ξ_{π} ⊗_{C} θ), and x are Chern characteristic classes with x ∈ MU^{∗}(BU(1)^{2}) = MU^{∗}[[x, tx]].

We need the following

Lemma 3.3. The left homomorphism in the long exact sequence (6) is an epimorphism and thus gives a short exact sequence

0 ← MU^{∗}(S(ξ_{π})) ← MU^{∗}(X_{hπ}^{p}) ←^{×c_{p}} MU^{∗}(X_{hπ}^{p})^{ξ_{π}} ← 0.

Moreover there is a space X_{π} and a stable equivalence

φ_{0} ∨ f_{π} : S(ξ_{π}) → Bπ ∨ X_{π},

with f_{π} factoring through the following composite map

S(ξ_{π}) → X_{hπ}^{p} →^{T r} Eπ × BU(1)^{p}

and φ_{0} as in (7).

Proof: Consider the Serre spectral sequence for the fibration (7)

S(ξ^{×p}) → S(ξ_{π}) →^{φ_{0}} Bπ.

E_{2}^{i,j} = H^{i}(π, H^{j}(S(ξ^{×p}); F_{q})) with the action of π on H^{∗}(S(ξ^{×p}); F_{q}) by permutations of the cohomological Chern classes.

When q = p, E_{2}^{0,j} = H^{j}(S(ξ^{×p}); F_{p})^{π} and E_{2}^{i,0} = H^{i}(Bπ; F_{p}).

Then in positive dimensions H^{∗}(S(ξ^{×p}); F_{q}) = F_{q}[x_{1}, ..., x_{p}]/(x_{1} ⋯ x_{p}) is a permutation representation of π acting on monomials which have degree zero in at least one indeterminate. This is a free F_{q}[π]-module since all the monomials that are fixed under this action have been factored out after quotienting by the ideal (x_{1} ⋯ x_{p}). Hence the cohomology of π with coefficients in this module is trivial in positive dimensions, i.e. E_{2}^{i,j} = 0 when i, j > 0. Thus the spectral sequence collapses and we have

H^{∗}(S(ξ_{π}); F_{p}) ≈ H^{∗}(Bπ; F_{p}) ⊕ H̃^{∗}(S(ξ^{×p}); F_{p})^{π}.

Also if q ≠ p we have H^{∗}(S(ξ_{π}); F_{q}) ≈ H^{∗}(S(ξ^{×p}); F_{q})^{π}.

Let X_{π} be a stable summand of BU(1)^{p} defined as follows. The action of π on BU(1)^{p} induces an action of π on the stable decomposition of BU(1)^{p} as a wedge of all smash products of length 1, ..., p − 1, say Y_{π}, and a smash product of length p. Then choose X_{π} such that NX_{π} = Y_{π}, where N = 1 + t + ... + t^{p−1}. By the stable equivalence

(8)  S(ξ^{×p}) → BU(1)^{p} → Y_{π}

we can consider X_{π} as a stable summand of S(ξ^{×p}). For any choice of X_{π}, consider the composition of stable maps

(9)  f_{π} : S(ξ_{π}) → X_{hπ}^{p} →^{T r} Eπ × BU(1)^{p} → BU(1)^{p} → X_{π}.

We have to show that the stable map φ_{0} ∨ f_{π} induces an isomorphism in cohomology for any group of coefficients F_{q}, q a prime, and hence gives a stable equivalence by the stable Whitehead lemma. It follows from the above arguments that

H̃^{∗}(S(ξ_{π}); F_{p}) = φ_{0}^{∗}H̃^{∗}(Bπ; F_{p}) ⊕ T r_{0}^{∗}H̃^{∗}(S(ξ^{×p}); F_{p}),

and H̃^{∗}(S(ξ_{π}); F_{q}) = T r_{0}^{∗}H̃^{∗}(S(ξ^{×p}); F_{q}), when q ≠ p. The restriction of T r_{0} on X_{π} induces a monomorphism on ImT r_{0}^{∗} since by the transfer property (iv), ρ_{0}^{∗}T r_{0}^{∗} = N and the restriction of N on H̃^{∗}(X_{π}; F_{q}) is a monomorphism. Hence (φ_{0} ∨ T r_{0}|X_{π})^{∗} is an isomorphism and so is (φ_{0} ∨ f_{π})^{∗} by the commutative diagram

[figure: S(ξ_{π}); X_{hπ}^{p}; S(ξ^{×p}); T r_{0}; Eπ × BU(1)^{p}; T r]

This proves Lemma 3.3.

Proof of Theorem 3.2:

a) We consider the restriction of any element y ∈ MU^{∗}(X_{hπ}^{p}) to MU^{∗}(S(ξ_{π})). By Lemma 3.3 we see this restriction has the form φ_{0}^{∗}(u) + f_{π}^{∗}(w) for some u ∈ M̃U^{∗}(Bπ), w ∈ MU^{∗}(X_{π}). Since the composition S(ξ_{π}) → X_{hπ}^{p} →^{φ} Bπ coincides with φ_{0}, φ^{∗}(u) also restricts to φ_{0}^{∗}(u). By diagram (10) there is an element v ∈ MU^{∗}(BU(1)^{p}) such that T r^{∗}(v) restricts to f_{π}^{∗}(w). By exactness

y = φ^{∗}(u) + T r^{∗}(v) + y_{1}c_{p},

for some y_{1} ∈ MU^{∗}(X_{hπ}^{p}). For use in the proof of c) we observe that (2),

(8) imply v can be chosen in the direct summand MU^{∗}[[x_{1}, ..., x_{p}]]/(x_{1} ⋯ x_{p}). Thus we can assume this expression for y is unique and if v ≠ 0 then T r^{∗}(v) restricts nontrivally in MU^{∗}(S(ξ_{π})).

Then suppose cy = 0. We know that ρ^{∗}(θ) = C, hence ρ^{∗}(c) = 0 and cT r^{∗}(v) = T r^{∗}(ρ^{∗}(c)v) = 0 by Frobenius reciprocity. So we have cφ^{∗}(u) + cy_{1}c_{p} = 0. We want to prove φ^{∗}(u) ∈ ImT r^{∗}. Applying i^{∗} we have 0 = i^{∗}(cφ^{∗}(u)) + i^{∗}(cy_{1}c_{p}) = zu since i^{∗}φ^{∗} = id, c = φ^{∗}(z), and i^{∗}(c_{p}) = 0. Hence u ∈ Ann(z) = ImT r_{Z/p}^{∗}. By naturality of the transfer φ^{∗}(u) ∈ ImT r^{∗}. Thus cφ^{∗}(u) = 0 and therefore cy_{1}c_{p} = 0. Multiplication by c_{p} is injective by Lemma 3.3, hence cy_{1} = 0. Since dim(y_{1}) = dim(y) − 2p iterating this argument gives us statement a).

b) This follows from the fact that the right homomorphism in the short exact sequence from Lemma 3.3 is multiplication by the Euler class c_{p}(ξ_{π}).

c) Let y ∈ Kerρ^{∗}. Since φρ = ∗ we have

0 = ρ^{∗}(T r^{∗}(v)) + ρ^{∗}(y_{1}c_{p}).

If the first summand is not zero it restricts nontrivially in MU^{∗}(S(ξ^{×p})) by definition of v. However the second summand restricts to zero since ρ^{∗}(c_{p}) = x_{1} ⋯ x_{p}, ρ^{∗}(y_{1}c_{p}) = ρ^{∗}(y_{1})x_{1} ⋯ x_{p} and x_{1} ⋯ x_{p} restricts to zero as the Euler class. Hence both summands are zero. Furthermore multiplication by x_{1} ⋯ x_{p} is a monomorphism hence ρ^{∗}(y_{1}) = 0. So

y = φ^{∗}(u) + y_{1}c_{p} = φ^{∗}(u) + (φ^{∗}(u_{1}) + y_{2}c_{p})c_{p} = φ^{∗}(u) + φ^{∗}(u_{1})c_{p} + y_{2}c_{p}^{2}.

Repetition of this process proves c).

d) The fact that c, c_{1}, c_{2} multiplicatively generate MU^{∗}B(π ≀ U(1)) follows from Lemma 3.3. The relations c_{1} = c_{1}^{∗}, c_{2} = c_{2}^{∗} follow from the bundle relation

ξ_{π} ⊗_{C} θ = (ξ ⊗_{C} ρ^{∗}(θ))_{π} = ξ_{π},

which in turn follows from transfer property (i).

So we have to prove that the Chern classes c, c_{1}, c_{2} with these relations are a complete system of generators and relations. Let us use the splitting principle to write formally

ξ_{π} = η_{1} + η_{2};
u_{1} = c_{1}(η_{1});
u_{2} = c_{1}(η_{2}).

Let F (x, y) = ∑ α_{ij}x^{i}y^{j} be the formal group law. Using the bundle relation above and applying the Whitney formula for the first and second Chern classes, we obtain two relations of the form:

F (u_{1}, c) + F (u_{2}, c) = c_{1}

and

(11)  F (u_{1}, c)F (u_{2}, c) = c_{2};

or in terms of c, c_{1} = u_{1} + u_{2}, c_{2} = u_{1}u_{2}

(12)  F (u_{1}, c) + F (u_{2}, c) − c_{1} = c(2 + ∑ β_{ijk}c^{i}c_{1}^{j}c_{2}^{k}) = 0

and

(13)  F (u_{1}, c)F (u_{2}, c) − c_{2} = c(c_{1} + ∑ γ_{ijk}c^{i}c_{1}^{j}c_{2}^{k}) = 0,

for some coefficients β_{ijk} , γ_{ijk} ∈ MU^{∗}(pt).

We claim that relations (12) and (13) are equivalent to the following two obvious transfer relations for T r^{∗} : MU^{∗}[[x, tx]] → MU^{∗}(B(π ≀ U(1)))

cT r^{∗}(1) = 0

and

cT r^{∗}(x) = 0.

Rewrite relations (12) and (13) as follows:

ca = 0, where a = 2 + α_{11}c_{1} + ∑_{k≥2} α_{k1}(u_{1}^{k} + u_{2}^{k}) + o(c),

cb = 0, where b = c_{1} + 2α_{11}c_{2} + ∑_{k≥2} α_{k1}(u_{1}^{k−1} + u_{2}^{k−1})c_{2} + o(c),

and the α_{ij} are the coefficients of the formal group law.

By the first part of Theorem 3.2, a ∈ ImT r^{∗}. Also by transfer property (vi)

ρ^{∗}(a) = ρ^{∗}(T r^{∗}(1) + α_{11}T r^{∗}(x) + ∑_{k≥2} α_{k1}T r^{∗}(x^{k})).

Thus by Lemma 2.1

T r^{∗}(1) + α_{11}T r^{∗}(x) + ∑_{k≥2} α_{k1}T r^{∗}(x^{k}) = (F (u_{1}, c) + F (u_{2}, c) − c_{1})/c;

similarly b ∈ ImT r^{∗} and

T r^{∗}(x) + α_{11}T r^{∗}(1)c_{2} + ∑_{k≥2} α_{k1}T r^{∗}(x^{k−1})c_{2} = (F (u_{1}, c)F (u_{2}, c) − c_{2})/c.

Now since

x^{k} = x^{k−1}(x + tx) − x^{k−2}(xtx),

transfer property (i) and the computation of T r^{∗}(x) is sufficient for the computation of T r^{∗}(x^{k}), k ≥ 2 (see also Corollary 3.6, Remark 3.7). So we have

(14) T r^{∗}(1)(1 + g_{0}) + T r^{∗}(x)h_{0} = (F (u_{1}, c) + F (u_{2}, c) − c_{1})/c,

and

(15)  T r^{∗}(1)g_{1} + T r^{∗}(x)(1 + h_{1}) = (F (u_{1}, c)F (u_{2}, c) − c_{2})/c,

where g_{0}, h_{0}, g_{1}, h_{1} ∈ MU^{∗}(B(π ≀ U(1))). This proves d).

This completes the proof of Theorem 3.2.

Formula (15) for computing T r^{∗}(x) is complicated; let us give a simpler form. Consider again (13). Note that the coefficient γ_{00k} ∈ MU^{∗}(pt) contains a factor 2: the element

c_{1} + ∑ γ_{ijk}c^{i}c_{1}^{j}c_{2}^{k}

annihilates c and hence belongs to ImT r^{∗}. On the other hand

ρ^{∗}T r^{∗} = 1 + t; ρ^{∗}(c) = 0; ρ^{∗}(c_{1}) = x + tx; ρ^{∗}(c_{2}) = xtx,

hence applying ρ^{∗} we have that

x + tx + ∑ γ_{0jk}(x + tx)^{j}(xtx)^{k}

belongs to Im(1 + t). So γ_{00k}(xtx)^{k} = 2γ_{k}(xtx)^{k}, that is, γ_{00k} = 2γ_{k} for some coefficient γ_{k}.

Recall that on the other hand F (c, c) = 0 that is 2c = o(c^{2}). So γ_{00k}c = o(c^{2}), hence taking into account the relation F (c, c) = 0 we can rewrite (13) after division by

1 + ∑_{i,k≥0} γ_{i1k}c^{i}c_{2}^{k}

(the coefficient at cc_{1}) as follows

(16)  cc_{1} = d_{0}c + d_{2}cc_{1}^{2} + ... + d_{n}cc_{1}^{n} + ...,

where d_{k} = d_{k}(c, c_{2}) ∈ MU^{∗}[[c, c_{2}]] and d_{0}(0, c_{2}) = 0; the lower index n indicates the coefficient at cc_{1}^{n}. Since

ρ^{∗}(T r^{∗}(x) − c_{1}) = 0,

it follows from Theorem 3.2c) that there exist elements

δ_{j} ∈ M̃U^{∗}(Bπ)

such that

T r^{∗}(x) = c_{1} + ∑_{j≥0} φ^{∗}(δ_{j})c_{2}^{j}.

Using the inclusion i : Bπ → B(π ≀ U(1)) we have

i^{∗}(c_{1}) = i_{0}^{∗}(c); i^{∗}T r^{∗}(x) = 0; i^{∗}(c_{2}) = 0,

thus

φ^{∗}(δ_{0}) = −c.

For the calculation of the other elements δ_{j} recall that cT r^{∗}(x) = 0, hence

(17)  cc_{1}^{n} = −cφ^{∗}(δ^{n}); n ≥ 1,

where

δ = −c + ∑_{j≥1} δ_{j}c_{2}^{j}.

Combining (16) and (17), we have

Proposition 3.4. The elements δ_{j}, j > 0 can be determined from the recurrence relations which arise from the following formula in MU^{∗}(Bπ)[[c_{2}]]

δ = d_{0} + ∑_{i≥2} d_{i}δ^{i}.

Proof: By definition the element δ − d_{0} − ∑_{i≥2} d_{i}δ^{i} belongs to Kerρ^{∗}. On the other hand this element is annihilated by c hence

δ − d_{0} − ∑_{i≥2} d_{i}δ^{i} ∈ ImT r^{∗} ⋂ Ker(ρ^{∗}) = 0

by Lemma 2.1. □

Corollary 3.5. For the elements δ_{j} ∈ M̃U^{∗}(BZ/2), constructed in 3.4, the following formula holds in MU^{∗}B(Z/2 ≀ U(1))

T r^{∗}(x) = c_{1} − c + ∑_{j≥1} φ^{∗}(δ_{j})c_{2}^{j}.

In fact we have proved Theorem 3.1 for p = 2. The general case, analogous but more technical, is given next.

3.2. Proof of Theorem 3.1. Note that by the definition of ω_{k} the difference T r^{∗}(ω_{k}) − c_{k} is an element of Ker(ρ^{∗}). Thus Theorem 3.2 c) implies existence of the elements δ_{i}^{(k)} in Theorem 3.1.

First let us elucidate the meaning of the relations

ξ_{π} ⊗_{C} θ = ξ_{π}

in the general case of B(π ≀ U(1)).

Again, we can use the splitting principle and write formally

ξ_{π} = η_{1} + η_{2} + ... + η_{p}; u_{m} = c_{1}(η_{m}), m = 1, ..., p.

Applying the Whitney formula for the relation

η_{1} ⊗_{C} θ + ... + η_{p} ⊗_{C} θ = η_{1} + ... + η_{p},

and taking into account that c_{m} = c_{m}(ξ_{π}) is the elementary symmetric function σ_{m}(u_{1}, ..., u_{p}) we have

(18)  σ_{m}(F (u_{1}, c), ..., F (u_{p}, c)) = c_{m},

m = 1, ..., p, or in terms of c, c_{1}, ..., c_{p} we have

c(p + ∑ β_{i_{0},i_{1},...,i_{p}}^{0}c^{i_{0}}c_{1}^{i_{1}}...c_{p}^{i_{p}}) = 0;

and

(19)  c((p − k)c_{k} + ∑ β_{i_{0},i_{1},...,i_{p}}^{k}c^{i_{0}}c_{1}^{i_{1}}...c_{p}^{i_{p}}) = 0;

for k = 1, ..., p − 1 and some β_{i_{0},i_{1},...,i_{p}}^{0}, β_{i_{0},i_{1},...,i_{p}}^{k} ∈ MU^{∗}(pt).

We claim that these relations are equivalent to the following obvious relations

cT r^{∗}(1) = 0,

and

cT r^{∗}(ω_{k}) = 0,

for the elements ω_{k} ∈ MU^{∗}(BU(1))^{p}, k = 1, ..., p − 1 defined above.

For the proof of our claim multiply the k-th relation from (19) by p_{k} = (p − k)^{−1} in F_{p}. Then by Theorem 3.2, Ann(c) coincides with ImT r^{∗} hence (18) implies that

p_{k}(σ_{k+1}(F (u_{1}, c), ..., F (u_{p}, c)) − c_{k+1})/c = T r^{∗}(a_{k}),

for some a_{k} which we have to find. Let us write

ρ^{∗}(p_{k}(σ_{k+1}(F (u_{1}, c), ..., F (u_{p}, c)) − c_{k+1})/c) = g^{(k)}(σ_{1}, ..., σ_{p})

= σ_{k}(1 + g_{k}^{(k)}(σ_{1}, ..., σ_{p})) + ∑_{j≠k,1≤j≤p−1} σ_{j}g_{j}^{(k)}(σ_{j}, σ_{j+1}, ..., σ̌_{k}, ..., σ_{p})

= N(ω_{k})(1+g_{k}^{(k)}(σ_{1}, ..., σ_{p}))+ ∑_{j≠k,1≤j≤p−1} N(ω_{j})g_{j}^{(k)}(σ_{j}, σ_{j+1}, ..., σ̌_{k}, ..., σ_{p}).

Here the symbol σ̌_{k} indicates absence of the corresponding term. So we have

p_{k}(σ_{k+1}(F (u_{1}, c), ..., F (u_{p}, c)) − c_{k+1})/c

= T r^{∗}(ω_{k})(1+g_{k}^{(k)}(c_{1}, ..., c_{p}))+ ∑_{j≠k,1≤j≤p−1} T r^{∗}(ω_{j})g_{j}^{(k)}(c_{j}, c_{j+1}, ..., č_{k}, ..., c_{p}),

and

[σ_{1}(F (u_{1}, c), ..., F (u_{p}, c)) − c_{1}]/c

= T r^{∗}(1)(1 + g_{0}^{(0)}(c_{1}, ..., c_{p})) + ∑_{1≤j≤p−1} T r^{∗}(ω_{j})g_{j}^{(0)}(c_{j}, c_{j+1}, ..., c_{p}).

This proves our claim.

For computing δ_{i}^{(k)} we start with the equations (19) and rewrite them as

(20)  cf_{k}(c, c_{1}, ..., c_{p}) = 0, k = 1, ..., p − 1.

These are equations in a power series algebra MU^{∗}(Bπ)[[c_{p}]], since we know cc_{k} ∈ cMU^{∗}(Bπ)[[c_{p}]].

We now want to find explicitly formal series

(21)  δ^{(k)}(c_{p}) = ∑_{i≥0} δ_{i}^{(k)}(c)c_{p}^{i}

such that

T r^{∗}(ω_{k}) = c_{k} + δ^{(k)}(c_{p})

and hence

(22)  cc_{k}^{j} = −c(δ^{(k)}(c_{p}))^{j}, j ≥ 1.

For this we want to replace the equations (20) by the equations

(23)  cf̃_{k}(c, δ^{(1)}(c_{p}), ..., δ^{(p−1)}(c_{p}), c_{p}) = 0,

where f̃_{k} ∈ Kerρ_{π}^{∗} is a series whose coefficient at δ^{(k)} is invertible. In fact f̃_{k} = 0 since we know that Ann(c) = ImT r^{∗} and Ker(ρ^{∗}) ∩ ImT r^{∗} = 0 by Lemma 2.1.

Then equating each coefficient of the resulting series

(24)  g_{k}(c_{p}) = f̃_{k}(c, δ^{(1)}(c_{p}), ..., δ^{(p−1)}(c_{p}), c_{p}) = 0

in the ring MU^{∗}(Bπ)[[c_{p}]] to zero we will obtain p − 1 infinite strings of equations in MU^{∗}(Bπ). Assuming δ_{i}^{(l)} are already found for i < n we get

(25)  δ_{n}^{(k)} = ψ_{n,k}((δ_{i}^{(1)})_{i≤n}, ..., (δ_{i}^{(k)})_{i<n}, ..., (δ_{i}^{(p−1)})_{i≤n}),

a system of linear equations in δ_{n}^{(l)}, l = 1, ..., p − 1 with invertible determinant and coefficients in MU^{∗}[[c]]. Since the δ_{0}^{(l)} are already known as l-th Chern classes of the bundle 1+θ+...+θ^{p−1}, by induction on n we can solve formally (25) to get

(26)  δ_{n}^{(k)}(c) = ψ̃_{n,k}((δ_{i}^{(l)})_{i<n}).

This gives δ_{n}^{(k)} = δ_{n}^{(k)}(z) ∈ MU^{∗}(Bπ) obviously satisfying our equations.

Now for the remaining equation (23) we proceed as follows: let us look at the term cf_{k}(0, 0, ..., 0, c_{p}) in equations (20). Note that f_{k}(0, 0, ..., 0, c_{p}) is divisible by p:

f_{k} ∈ Ann(c) = ImT r^{∗} ⇒ ρ_{π}^{∗}f_{k} ∈ ImN ⇒ f_{k}(0, ..., 0, σ_{p}) ∈ ImN ⇒ f_{k}(0, ..., 0, σ_{p}) is divisible by p.

Next using the relation [p]_{F} (c) = 0 we know that pc is divisible by c^{2}; hence each occurrence of pc in these equations can be replaced by terms with higher powers of c. So cf_{k}(0, 0, ..., 0, c_{p}) can be replaced by a term divisible by c^{2}.

Also the k-th relation from (20) contains the term c(p−k)c_{k}, and for the condition (24) we have to multiply the k-th equation from (20) by (p − k)^{−1}, the inverse of p − k in F_{p}, and as above we can replace c(p − k)c_{k} by cc_{k}+(terms divisible by c^{2}). Then we use (22) and substitute the series δ^{(k)} in the resulting equations, thus obtaining (23).

This completes the proof for E = MU, which is the universal example of complex oriented cohomology theories. From this result we can descend to all E. □

We now turn to computation of T r^{∗} in general.

Corollary 3.6. For all primes p, Theorem 3.1 enables us to explicitly compute the transfer homomorphism for those polynomials a ∈ M̃U^{∗}[[x_{1}, ..., x_{p}]] for which Na = a + ta + ... + t^{p−1}a is symmetric in x_{1}, ..., x_{p}.

Proof: If Na = σ_{1}a_{1}(σ_{1}, ..., σ_{p}) + ... + σ_{p−1}a_{p−1}(σ_{1}, ..., σ_{p}), then

T r^{∗}(a) = T r^{∗}(ω_{1})a_{1}(c_{1}, ..., c_{p}) + ... + T r^{∗}(ω_{p−1})a_{p−1}(c_{1}, ..., c_{p})).

To see this let â = ω_{1}a_{1}(σ_{1}, ..., σ_{p}) + ... + ω_{p−1}a_{p−1}(σ_{1}, ..., σ_{p}). Then N(a − â) = 0, that is, a − â ∈ Im(1 − t), hence T r^{∗}(a) = T r^{∗}(â). □

Remark 3.7: For p = 2 one has recurrence formulas for T r^{∗}(x^{k}), k ≥ 1.

T r^{∗}(x) = T r^{∗}(ω_{1})
T r^{∗}(x^{k}) = T r^{∗}(x^{k−1})c_{1} − T r^{∗}(x^{k−2})c_{2}

This follows using the formula x^{k} = x^{k−1}(x + tx) − x^{k−2}(xtx).

4. Transferred Chern classes for Σ_{p}-coverings

If we consider a p-local complex oriented cohomology E^{∗} then by standard transfer arguments (see Lemma 4.3 below) E^{∗}(BΣ_{p}) is isomorphic to the subring of E^{∗}(Bπ) invariant under the action of the normalizer of π in Σ_{p}. The results of this section imply the elements δ_{i}^{(k)} ∈ Ẽ^{∗}(Bπ) from Theorem 3.1 are invariant under this action. This defines elements δ̃_{i}^{(k)} ∈ Ẽ^{∗}(BΣ_{p}) which we use for computing the transfer.

In this section we consider BP^{∗}(X_{hΣ_{p}}^{p}) for X = CP^{∞} and for the covering projection

ρ_{Σ_{p}} : EΣ_{p} × X^{p} → X_{hΣ_{p}}^{p}

we give a formula for the transfer homomorphism

(27)  T r_{Σ_{p}}^{∗} : BP^{∗}(X^{p}) → BP^{∗}(X_{hΣ_{p}}^{p})

using the elements δ̃_{i}^{(k)}.

4.1. Brown-Peterson cohomology of (CP^{∞})_{hΣ_{p}}^{p}. We need definitions analogous to those of Section 2, with the cyclic group replaced by the symmetric group. The p-fold product, ξ^{×p}, of the canonical line bundle over X^{p} extends to an p-dimensional bundle

(28)  ξ_{Σ_{p}} = EΣ_{p} ×_{Σ_{p}} ξ^{×p}

over X_{hΣ_{p}}^{p} classified by the inclusion X_{hΣ_{p}}^{p} = B(Σ_{p} ≀ U(1)) ↪ BU(p). Let c_{i} = c_{i}(ξ_{Σ_{p}}). Then ρ_{Σ_{p}}^{∗}(c_{i}) = c_{i}(ξ^{×p}) = σ_{i}, the i-th symmetric polynomial in the x_{j}, where BP^{∗}(X^{p}) = BP^{∗}[[x_{1}, ..., x_{p}]].

Then we have the projection

(29)  φ : X_{hΣ_{p}}^{p} → BΣ_{p},

induced by the factorization Σ_{p} ≀ U(1)/U(1)^{p} = Σ_{p} and the inclusion

(30)  i : BΣ_{p} → X_{hΣ_{p}}^{p},

induced by the inclusion of Σ_{p} in Σ_{p} ≀ U(1).

Definition 4.1. Let c̃_{i} = T r_{Σ_{p}}^{∗}(x_{1}x_{2} ⋯ x_{i}) for i = 1, ..., p − 1.

Lemma 4.2. ρ_{Σ_{p}}^{∗}(c̃_{i}) = i!(p − i)!σ_{i}.

Proof: ρ_{Σ_{p}}^{∗}(c̃_{i}) = ρ_{Σ_{p}}^{∗}T r_{Σ_{p}}^{∗}(x_{1}x_{2} ⋯ x_{i}) = N_{Σ_{p}}(x_{1}x_{2} ⋯ x_{i}). For each subset of i integers {j_{1}, j_{2}, ..., j_{i}} with 1 ≤ j_{k} ≤ p, there are i! bijections {1, 2, ..., i} → {j_{1}, j_{2}, ..., j_{i}} and (p − i)! bijections {i + 1, i + 2, ..., p} → {1, 2, ..., p}\{j_{1}, j_{2}, ..., j_{i}}. Thus there are i!(p − i)! summands of x_{j_{1}}x_{j_{2}}...x_{j_{i}} in N_{Σ_{p}}(x_{1}x_{2}...x_{i}). □

We recall

BP^{∗}(Bπ) = BP^{∗}[[z]]/([p]z)

with |z| = 2. The corresponding computation for BP^{∗}(BΣ_{p}) is also known [19]. For the reader’s convenience we derive the result in a form useful for our purposes.

Lemma 4.3. As a BP^{∗} algebra

(i) BP^{∗}(BΣ_{p}) = BP^{∗}[[y]]/(yT r_{Σ_{p}}^{∗}(1)),

where y and T r_{Σ_{p}}^{∗}(1) are uniquely determined by ρ_{π,Σ_{p}}^{∗}(y) = z^{p−1} and

(ii) ρ_{π,Σ_{p}}^{∗}(T r_{Σ_{p}}^{∗}(1)) = (p − 1)!T r_{π}^{∗}(1) = (p − 1)![p](z)/z.

In particular |y| = 2(p − 1).

Proof: (ii) Applying the double coset formula (transfer property (vi)) to

BP^{∗}(Be) ⟶^{T r_{e,Σp}^{∗}} BP^{∗}(BΣ_{p}) ⟶^{ρ_{π,Σp}^{∗}} BP^{∗}(Bπ),

the statement follows from Quillen’s formula (1).

(i) The relation yT r_{Σ_{p}}^{∗}(1) = 0 is a consequence of Frobenius reciprocity. To see that it is the defining relation we recall that the cohomology of BΣ_{p} with simple coefficients in Z_{(p)} is

H^{∗}(BΣ_{p}; Z_{(p)}) = Z_{(p)}[y]/(py)

where |y| = 2(p − 1). This follows easily from the mod-p cohomology and the Bockstein spectral sequence.

Also H^{∗}(Bπ; Z_{(p)}) = Z_{(p)}[z]/(pz) where |z| = 2. The map ρ_{π,Σ_{p}} : Bπ → BΣ_{p} yields ρ_{π,Σ_{p}}^{∗}(y) = x^{p−1}.

Now the Atiyah-Hirzebruch-Serre spectral sequence for BP^{∗}(BΣ_{p}) is

E_{2} = H^{∗}(BΣ_{p}; BP^{∗}) = BP^{∗}[y]/(py) ⟹ BP^{∗}(BΣ_{p}).

Since y is even dimensional, the sequence collapses at E_{2} = E_{∞}. Thus BP^{∗}(BΣ_{p}) is generated by y as a BP^{∗} algebra.

For the group W = N_{Σ_{p}}(π)/π ≈ Z/(p − 1), |W | is prime to p, hence by the standard transfer argument ρ_{π,Σ_{p}}^{∗} : BP^{∗}(BΣ_{p}) → BP^{∗}(Bπ) is an injective map of BP^{∗} algebras. Since ρ_{π,Σ_{p}}^{∗}(yT r_{Σ_{p}}^{∗}(1)) = p!z^{p−1} plus terms of higher filtration, yT r_{Σ_{p}}^{∗}(1) = 0 is the only relation. □

Relating π and Σ_{p} we have a lift of ρ_{π,Σ_{p}}

[figure: X_{hπ}^{p}; ρ̃_{π,Σ_{p}}; X_{Σ_{p}}^{p}; Bπ; φ; ρ_{π,Σ_{p}}; BΣ_{p}; φ]

Lemma 4.4. ρ̃_{π,Σ_{p}}^{∗}(c̃_{k}) = k!(p − k)!T r_{π}^{∗}(ω_{k}).

Proof: Note that modulo Im(1 − t) we have Σg^{∗}(x_{1}x_{2} ⋯ x_{k}) = k!(p − k)!ω_{k} summed over Σ_{p}/π. Applying the double coset formula

ρ_{π,Σ_{p}}^{∗}(c̃_{k}) = ρ_{π,Σ_{p}}^{∗}T r_{Σ_{p}}^{∗}(x_{1}x_{2} ⋯ x_{k}) = T r_{π}^{∗} ∑_{g∈Σ_{p}/π} g^{∗}(x_{1}x_{2} ⋯ x_{k}) = k!(p−k)!T r_{π}^{∗}(ω_{k}).

□

Let c = φ^{∗}(y) ∈ BP^{2(p−1)}(X_{hΣ_{p}}^{p}).

Lemma 4.5. ImT r_{Σ_{p}}^{∗} is contained in the BP^{∗} algebra generated by c, c̃_{1}, ..., c̃_{p−1}, c_{p}.

Proof: By the Künneth isomorphism,

(31)  BP^{∗}(X^{p}) = BP^{∗}(X)^{⊗p} = F ⊕ T

as a π-module, where F is free and T is trivial. Explicitly a BP^{∗} basis for T is {x_{1}^{i} ⋯ x_{p}^{i}, i ≥ 0}, while a BP^{∗} basis for F is {x_{1}^{i_{1}} ⋯ x_{p}^{i_{p}}, i_{j} ≥ 0} where not all the exponents are equal.

By Lemma 4.3 T r_{Σ_{p}}^{∗}(1) is a power series in c. Now recall from [9], p. 44, that we can consider BP^{∗}(X^{p}), X = CP^{∞} as a free BP^{∗}[[σ_{1}, ..., σ_{p}]] module generated by 1 and the elements x_{1}^{i_{1}} ⋯ x_{p}^{i_{p}} ∈ F, with 0 ≤ i_{j} ≤ p−j. So by Frobenius reciprocity it suffices to compute the transfer on these monomials. Summed over the symmetric group ∑ g^{∗}(x_{1}^{i_{1}} ⋯ x_{p}^{i_{p}}) is a symmetric function and hence has the form

∑_{π} ∑_{Σ_{p}/π} g^{∗}(x_{1}^{i_{1}} ⋯ x_{p}^{i_{p}}) = σ_{1}s_{1}+...+σ_{p−1}s_{p−1} = ∑_{π}(ω_{1}s_{1}+...+ω_{p−1}s_{p−1}),

for the elements ω_{k} from Theorem 3.1 and some symmetric functions s_{1}, ..., s_{p−1}. Hence modulo kerN_{π} = Im(1 − t), t ∈ π, we have the following equation in F

∑_{Σ_{p}/π} g^{∗}(x_{1}^{i_{1}} ⋯ x_{p}^{i_{p}}) = ω_{1}s_{1} + ... + ω_{p−1}s_{p−1}.

The left sum consists of (p−1)! elements each having the same transfer value. Also ω_{k} is the sum of p^{−1}\binom{p}{k} elements x_{i_{1}} ⋯ x_{i_{k}}; on each of these elements the transfer evaluates to T r_{Σ_{p}}^{∗}(x_{1} ⋯ x_{k}) = c̃_{k}. Thus Frobenius reciprocity and Lemma 4.2 is all that is needed for computing T r_{Σ_{p}}^{∗}.

□

Recall the elements δ_{i}^{(k)} ∈ B̃P^{∗}(Bπ) derived from Theorem 3.1 by naturality. By the standard transfer argument again the map induced by ρ̃_{π,Σ_{p}} : X_{hπ}^{p} → X_{hΣ_{p}}^{p}, the lift of ρ_{π,Σ_{p}} : Bπ → BΣ_{p}, is also injective. Moreover for BP^{∗}(X_{hΣ_{p}}^{p}) the ring structure is completely determined by the following

Theorem 4.6. As a BP^{∗} algebra

BP^{∗}(X_{hΣ_{p}}^{p}) = BP^{∗}[[c, c̃_{1}, ..., c̃_{p−1}, c_{p}]]/(cT r_{Σ_{p}}^{∗}(1), cc̃_{i})

and one has the formula

c̃_{k} − k!(p − k)!c_{k} = Σ_{i≥0}φ^{∗}(δ̃_{i}^{(k)})c_{p}^{i}, k = 1, ..., p − 1,

where the elements δ̃_{i}^{(k)} ∈ B̃P^{∗}(BΣ_{p}) are determined by

ρ_{π,Σ_{p}}^{∗}(δ̃_{j}^{(k)}) = k!(p − k)!δ_{j}^{(k)}, j ≥ 0.

For the proof we follow that of Theorem 3.2. Let

S(ξ_{Σ_{p}}) = EΣ_{p} ×_{Σ_{p}} S(ξ^{×p})

be the sphere bundle of the bundle ξ_{Σ_{p}} of (28). X_{hΣ_{p}}^{p} is homotopy equivalent to the disk bundle D(ξ_{Σ_{p}}) = EΣ_{p} ×_{Σ_{p}} D(ξ^{×p}). Then we have the obvious inclusion i_{0} : BΣ_{p} → S(ξ_{Σ_{p}}) and projection φ_{0} : S(ξ_{Σ_{p}}) → BΣ_{p} with fiber S(ξ^{×p}). φ_{0}i_{0} is the identity. Thus stably BΣ_{p} is a wedge summand of S(ξ_{Σ_{p}}). As for the other summand let

X_{Σ_{p}} = ∨_{i=1}^{p−1}EΣ_{i}×_{Σ_{i}}BU(1)^{∧i}.

By the standard transfer argument, localized at p, X_{Σ_{p}} is a stable summand of ∨_{i=1}^{p−1}EΣ_{i} × BU(1)^{∧i} and hence of EΣ_{p} × BU(1)^{×p}. From this we derive the following result.

Lemma 4.7. One has a stable equivalence localized at p

φ_{0} ∨ f_{Σ_{p}} : S(ξ_{Σ_{p}}) → BΣ_{p} ∨ X_{Σ_{p}},

with f_{Σ_{p}}, the composition of stable maps

f_{Σ_{p}} : S(ξ_{Σ_{p}}) → X_{hΣ_{p}}^{p} →^{T r_{Σp}} EΣ_{p} × BU(1)^{p} → X_{Σ_{p}}.

Proof: The inclusion i_{0} splits off φ_{0}^{∗}H^{∗}(BΣ_{p}) in H^{∗}(S(ξ_{Σ_{p}})). Furthermore in mod-p cohomology

H^{∗}(S(ξ^{×p})) = F_{p}[x_{1}, ..., x_{p}]/(σ_{p}),

hence

H^{∗}(S(ξ^{×p}))^{Σ_{p}} ≈ F_{p}[c̃_{1}, ..., c̃_{p−1}],

by Lemma 4.2.

Then H := H̃^{∗}(S(ξ^{×p})) is a free π module and H^{∗}(Σ_{p}; H) ⊆ H^{∗}(π; H). Thus

H^{∗}(Σ_{p}; H) = H^{Σ_{p}} if ∗ = 0
= 0 if ∗ > 0.

Therefore there is an isomorphism

H^{∗}(S(ξ_{Σ_{p}})) ⟶^{ρ^{∗}⊕i_{0}^{∗}} H̃^{∗}(S(ξ^{×p}))^{Σ_{p}} ⊕ H^{∗}(BΣ_{p})

where ρ : S(ξ^{×p}) → S(ξ_{Σ_{p}}) is the projection. We have to prove that the first summand is f_{Σ_{p}}^{∗}H̃^{∗}(X_{Σ_{p}}).

By naturality of the transfer we have the commutative diagram

\begin{matrix}S(ξ^{×p}) & \overline{} & BU(1)^{p} & ^{}X_{Σ_{p}} \\ S(ξ_{Σ_{p}}) & ^{T r_{0} } & \overline{} & X_{hΣ_{p}}^{p} & ^{T r_{Σ_{p}} } \end{matrix}

Thus f_{Σ_{p}} coincides with f̃_{Σ_{p}}, the map T r_{0} followed by the horizontal maps in the above diagram. We wish to show the restriction of T r_{0}^{∗} to the image of H^{∗}(X_{Σ_{p}}) is an isomorphism onto H^{∗}(S(ξ^{×p}))^{Σ_{p}}.

Now considering the transfer for the Σ_{i} coverings

EΣ_{i} × BU(1)^{∧i} → EΣ_{i} ×_{Σ_{i}} BU(1)^{∧i},

it follows from transfer properties (ii) and (vi) that H^{∗}(EΣ_{i}×_{Σ_{i}}BU(1)^{∧i}) is a submodule of H^{∗}(EΣ_{i} ×BU(1)^{∧i}) generated by Σ_{i} norms of monomials in x_{1}, x_{2}, ..., x_{i}, with non-increasing degrees. From this it is straightforward that H^{∗}(X_{Σ_{p}}) and H^{∗}(S(ξ^{×p}))^{Σ_{p}} have the same ranks in each dimension. Thus we are reduced to showing the desired map is injective.

However, for any monomial x in x_{1}, x_{2}, ..., x_{i}, we have

T r_{0}^{∗}(N_{Σ_{i}}(x)) = i!T r_{0}^{∗}(x)

by naturality of the transfer. Thus the restriction of T r_{0}^{∗} to the image of H^{∗}(X_{Σ_{p}}) will be a monomorphism if T r_{0}^{∗} is non-zero on polynomials consisting of monomials with non-increasing degrees. This in turn will follow if the norm N_{Σ_{p}} is non-zero on such polynomials. In fact we claim: 1) N_{Σ_{p}} is non-zero on any monomial x^{I} = x_{1}^{i_{1}} ⋯ x_{p−1}^{i_{p−1}}, and 2) different monomials with non-increasing degrees in x_{1}, ..., x_{i}, i < p are in different Σ_{p} orbits.

Claim 2) is clear. To see 1) let J = (j_{1}, ..., j_{p}) and x^{J} = x_{1}^{j_{1}} ⋯ x_{p}^{j_{p}}, all of whose exponents are not equal. Then we will show the coefficient of x^{J} in N_{Σ_{p}}(x^{J}) is prime to p. The isotropy subgroup of x^{J} is the finite product Σ_{n_{1}} × Σ_{n_{2}} × ... < Σ_{p} where n_{j} is the number of terms of J equaling j. This group has order n_{1}!n_{2}!... which is prime to p. Hence N_{Σ_{p}}(x) = (n_{1}!n_{2}!...)x^{J} + other monomials proving the claim.

Thus φ_{0} ∨ f_{Σ_{p}} induces an isomorphism and hence is a p-local stable equivalence. □

This implies Lemma 4.8. The long exact sequence for the pair (D(ξ_{Σ_{p}}), S(ξ_{Σ_{p}})) gives the following short exact sequence

0 ← BP^{∗}(S(ξ_{Σ_{p}})) ← BP^{∗}(X_{hΣ_{p}}^{p}) ← BP^{∗}((X_{hΣ_{p}}^{p})^{ξ_{Σp}}) ← 0.

Indeed the left arrow is an epimorphism by Lemma 4.7 and hence the right arrow is a monomorphism. □

Now the proof of Theorem 4.6 is completely analogous to that of Theorem 3.2 taking into account additionally that any element y ∈ BP^{∗}(X_{hπ}^{p}) has the form

y = φ^{∗}(u) + g(c̃_{1}, ..., c̃_{p−1}) + y_{1}c_{p}

for some u ∈ BP^{∗}(X_{hπ}^{p}), where g denotes some formal power series and y_{1} ∈ BP^{∗}(X_{hπ}^{p}). This follows by Lemma 4.5 and Lemma 4.8. □

5. Calculation of the elements δ_{i}^{(k)} and δ̃_{i}^{(k)} in Morava K-theory

In this section we work in Morava K-theory K(s)^{∗} and give an alternative, better algorithm for explicit computations.

Fix a prime p and an integer s ≥ 0, then K(s)^{∗} = F_{p}[v_{s}, v_{s}^{−1}] with |v_{s}| = −2(p^{s} − 1). By a result of Würgler [23] there is no restriction on p: although K(s) is not a commutative ring spectrum for p = 2, we shall consider only those spaces whose Morava K-theory is even dimensional. This implies the deviation from commutativity is zero.

We recall

K(s)^{∗}(Bπ) = K(s)^{∗}[z]/(z^{p^{s}})

where |z| = 2.

As in Lemma 4.3. we have

Lemma 5.1. (i) ρ_{π,Σ_{p}} : Bπ → BΣ_{p} induces an isomorphism of K(s)^{∗} algebras

ρ_{π,Σ_{p}}^{∗} : K(s)^{∗}(BΣ_{p}) ⟶^{≈} {K(s)^{∗}(Bπ)}^{W} ,

where W = N_{Σ_{p}}(π)/π ≈ Z/(p − 1). Computing invariants yields

K(s)^{∗}(BΣ_{p}) = K(s)^{∗}[y]/(y^{m_{s}}),

where ρ_{π,Σ_{p}}^{∗}(y) = z^{p−1} and m_{s} = [(p^{s} − 1)/(p − 1)] + 1.

(ii) T r_{Σ_{p}}^{∗}(1) = −v_{s}y^{m_{s}−1}.

Then combining Theorem 4.6 and Remark 2.2 we have

K(s)^{∗}(X_{hΣ_{p}}^{p}) = K(s)^{∗}[[c, c̃_{1}, ..., c̃_{p−1}, c_{p}]]/(c^{m_{s}}, cc̃_{i}).

Our main result in this section is the following

Proposition 5.2. We can construct explicit elements δ_{i}^{(k)} ∈ K(s)^{∗}(Bπ) such that

1) In K(s)^{∗}(X_{hπ}^{p}) the following formula holds

c_{k}(ξ_{π}) = T r_{π}^{∗}(ω_{k}) − ∑_{0≤i≤p^{s}} φ_{π}^{∗}(δ_{i}^{(k)})c_{p}^{i}(ξ_{π}).

2) In K(s)^{∗}(X_{hΣ_{p}}^{p}) one has

c_{k}(ξ_{Σ_{p}}) = T r_{Σ_{p}}^{∗}(x_{1}...x_{k}) − ∑_{0≤i≤p^{s}} φ_{Σ_{p}}^{∗}(δ̃_{i}^{(k)})c_{p}^{i}(ξ_{Σ_{p}}),

with ρ̃_{π,Σ_{p}}^{∗}(δ̃_{i}^{(k)}) = k!(p − k)!δ_{i}^{(k)}k.

3) The value of T r_{π}^{∗}(c_{1}(ξ_{i})) is determined by

c_{1}(ξ_{π}) = T r_{π}^{∗}(c_{1}(ξ_{i})) + v_{s} ∑_{1≤j≤s−1} c^{p^{s}−p^{j}}c_{p}^{p^{j−1}}(ξ_{π})

where ξ_{i} is the pullback of the canonical line bundle ξ by projection BU(1)^{p} → BU(1) on the i-th factor.

We are grateful to D. Ravenel for supplying us with the proof of the following result.

Lemma 5.3. For the formal group law in Morava K-theory K(s), s > 1, we have

F (x, y) ≡ x + y − v_{s} ∑_{0<j<p} p^{−1}\binom{p}{j}(x^{p^{s−1}})^{j}(y^{p^{s−1}})^{p−j}

modulo x^{p^{2(s−1)}} (or modulo y^{p^{2(s−1)}}).

Proof: This result can be derived from the recursive formula for the FGL given in 4.3.9 [17]. For the FGL in Morava K-theory it reads

F (x, y) = ∑_{i≥0}^{F} v_{s}^{e_{i}}w_{i}(x, y)^{p^{i(s−1)}}

where w_{i} is a certain homogeneous polynomial of degree p^{i} defined by 4.3.5 [17] and e_{i} = (p^{is} − 1)/(p^{s} − 1). In particular ω_{0} = x + y,

w_{1} = − ∑_{0<j<p} p^{−1}\binom{p}{j}x^{j}y^{p−j},

and w_{i} ∉ (x^{p}, y^{p}).

We find it more convenient to express F (x, y) as

F (x, y) = F (x + y, v_{s}w_{1}(x, y)^{p^{s−1}}, v_{s}^{e_{2}}w_{2}(x, y)^{p^{2(s−1)}}, ...).

Then for s > 1 we can reduce modulo the ideal v_{s}^{e_{2}}(x^{p^{2(s−1)}}, y^{p^{2(s−1)}}) and get

F (x, y) ≡ F (x + y, v_{s}w_{1}(x, y)^{p^{s−1}})
= F (x + y + v_{s}w_{1}(x, y)^{p^{s−1}}, v_{s}w_{1}(x + y, v_{s}w_{1}(x, y)^{p^{s−1}})^{p^{s−1}}, …)
≡ F (x + y + v_{s}w_{1}(x, y)^{p^{s−1}}, v_{s}w_{1}(x^{p^{s−1}} + y^{p^{s−1}}, v_{s}^{p^{s−1}}w_{1}(x, y)^{p^{2(s−1)}})),

and modulo v_{s}^{1+p^{s−1}}(x^{p^{2(s−1)}}, y^{p^{2(s−1)}}) we have

F (x, y) ≡ x + y + v_{s}w_{1}(x, y)^{p^{s−1}}. □

Let us write for short σ_{k} = σ_{k}(x, F (x, z), ..., F (x, (p − 1)z)).

Corollary 5.4. The following formula holds in K(s)^{∗}(BU(1) × Bπ)

σ_{k} = − ∑_{0≤i≤p^{s}} λ_{i}^{(k)}σ_{p}^{i} + p^{−1}\binom{p}{k}x^{k}v_{s}z^{p^{s}−1},

where λ_{i}^{(k)} = λ_{i}^{(k)}(z^{p−1}) are polynomials in z^{p−1} and λ_{0}^{(j)} = 0, j = 1, ..., p − 2, λ_{0}^{(p−1)} = −z^{p−1}.

Proof. For 1 ≤ k ≤ p − 1, equating the coefficients of x^{ip}, 1 ≤ i ≤ p^{s} gives a system of linear equations with invertible matrix of the form Id + nilpotent. Thus the elements λ_{1}^{(k)}, ..., λ_{p^{s}}^{(k)} can be defined as the solution of this system. Of course equating the coefficients at x^{i} for i ≠ p, 2p, ..., p^{s+1} will produce other equations in λ_{j}^{(k)}, j = 1, ..., p^{s}. But these equations are derived from the old equations above. These additional equations make the matrix upper triangular. □

Now, let us prove Proposition 5.2 and show that one necessarily has δ_{i}^{(k)} = λ_{i}^{(k)}, i = 0, ..., p^{s} for λ_{i}^{(k)} encountered in Corollary 5.4. Thus by Lemma 5.1, δ_{i}^{(k)} is invariant under the action of W and we can define δ̃_{i}^{(k)} by ρ̃_{π,Σ_{p}}^{∗}(δ̃_{i}^{(k)}) = k!(p − k)!δ_{i}^{(k)}.

The diagonal map Δ : BU(1) → BU(1)^{p} induces an inclusion Bπ × BU(1) → X_{hπ}^{p} and the commutative diagram

[figure: Eπ × BU(1); 1 × Δ; Eπ × BU(1)^{p}; Bπ × BU(1); π × 1; X_{hπ}^{p}; ρ_{π}]

Then (1 × Δ)^{∗}(ω_{k}) = p^{−1}\binom{p}{k}x^{k}, x = c_{1}(ξ). Hence by transfer properties (i) and (iv) we have for the transfer T r = T r(π × 1):

T r^{∗}((1 × Δ)^{∗}(ω_{k})) = p^{−1}\binom{p}{k}x^{k}T r^{∗}(1) = p^{−1}\binom{p}{k}x^{k}v_{s}z^{p^{s}−1}.

On the other hand by the existence of the elements δ_{i}^{(k)} (Theorem 3.1) we have

T r^{∗}((1 × Δ)^{∗}(ω_{k})) =

σ_{k}(x, F (x, z), ..., F (x, (p−1)z))+∑_{i≥0} δ_{i}^{(k)}σ_{p}^{i}(x, F (x, z), ..., F (x, (p−1)z)) :

ξ_{π} restricts to ∑_{i} ξ⊗θ^{i} on BU(1)×Bπ, thus c_{k}(ξ_{π}) to σ_{k}(x, F (x, z), ..., F (x, (p− 1)z)); by Lemma 5.3 and the fact that z^{p^{s}} = 0, [i]z may be replaced by iz. By Corollary 5.4

σ_{k}(x, F (x, z), ..., F (x, (p − 1)z)) =

− ∑_{0≤i≤p^{s}} λ_{i}^{(k)}σ_{p}^{i}(x, F (x, z), ..., F (x, (p − 1)z)) + p^{−1}\binom{p}{k}x^{k}v_{s}z^{p^{s}−1}.

Then the restriction of (1 × Δ)^{∗} to Kerρ^{∗} is a monomorphism [11]. This proves Proposition 5.2.1) and shows δ_{i}^{(k)} = λ_{i}^{(k)} for 0 ≤ i ≤ p^{s} and zero otherwise. Statement 2) follows from Lemma 4.4. Then 3) follows from the following explicit formula for σ_{1}:

Lemma 5.5. In K(s)^{∗}(Bπ × BU(1)) one has

σ_{1} = v_{s} (z^{p^{s}−1}x + ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}σ_{p}^{p^{i−1}}) .

Proof:

One has

σ_{1} = x + F (x, z) + ... + F (x, (p − 1)z) = x + x + z + v_{s}w_{1}(x^{p^{s−1}}, z^{p^{s−1}})

+... + x + (p − 1)z + v_{s}w_{1}(x^{p^{s−1}}, ((p − 1)z)^{p^{s−1}})

= px + \frac{p(p−1)}{2}z + v_{s} (∑_{i=1}^{p−1} w_{1}(x, iz))^{p^{s−1}} = v_{s} (∑_{i=1}^{p−1} ∑_{j=1}^{p−1} −p^{−1}\binom{p}{j}i^{j}x^{p−j}z^{j})^{p^{s−1}}

= v_{s} (∑_{j=1}^{p−1} − (∑_{i=1}^{p−1} i^{j}) p^{−1}\binom{p}{j}z^{j}x^{p−j})^{p^{s−1}} .

Now ∑_{i=1}^{p−1} i^{j} is an integral linear combination of σ_{k}(1, 2, ..., p − 1) with k ≤ i, hence by it is zero for i < p − 1 and for i = p − 1 it is p − 1.

Thus

(32)

σ_{1} = −v_{s} ((p − 1)p^{−1}\binom{p}{p − 1}z^{p−1}x^{p−(p−1)})^{p^{s−1}} = v_{s}z^{p^{s}−p^{s−1}}x^{p^{s−1}}.

Now since F (x, z)^{p} = x^{p}+z^{p}, one has σ_{p}^{p} = (x(x + z)...(x + (p − 1)z))^{p}. But again we have x(x + z)...(x + (p − 1)z) = x^{p} − xz^{p−1}. Substituting this one obtains

v_{s} (z^{p^{s}−1}x + ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}σ_{p}^{p^{i−1}})

= v_{s} (z^{p^{s}−1}x + z^{p^{s}−p}σ_{p} + ∑_{i=2}^{s−1} z^{p^{s}−p^{i}}(x^{p} − z^{p−1}x)^{p^{i−1}})

= v_{s} (z^{p^{s}−1}x + z^{p^{s}−p}σ_{p} + ∑_{i=2}^{s−1} z^{p^{s}−p^{i}}(x^{p^{i}} − z^{(p−1)p^{i−1}}x^{p^{i−1}})) . But it is straightforward to see that

∑_{i=2}^{s−1} z^{p^{s}−p^{i}}(x^{p^{i}} − z^{(p−1)p^{i−1}}x^{p^{i−1}}) = z^{p^{s}−p^{s−1}}x^{p^{s−1}} − z^{p^{s}−p}x^{p}.

Hence one has

(33)

v_{s} (z^{p^{s}−1}x + ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}σ_{p}^{p^{i−1}}) = v_{s} (z^{p^{s}−1}x + z^{p^{s}−p}σ_{p} + z^{p^{s}−p^{s−1}}x^{p^{s−1}} − z^{p^{s}−p}x^{p}) .

Now one has z^{p^{s}−p}F (x, kz) = z^{p^{s}−p}(x+kz +v_{s}w_{1}(x^{p^{s−1}}, (kz)^{p^{s−1}})) = z^{p^{s}−p}(x+kz),

hence z^{p^{s}−p}σ_{p} = z^{p^{s}−p}x(x + z)...(x + (p − 1)z) = z^{p^{s}−p}(x^{p} − z^{p−1}x)

Substituting this into (33) gives v_{s} (z^{p^{s}−1}x + ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}σ_{p}^{p^{i−1}}) =

v_{s}z^{p^{s}−p^{s−1}}x^{p^{s−1}}, which is σ_{1} by (32). □

We now compute some of the elements δ_{i}^{(k)} and δ̃_{i}^{(k)}. First recall from [8], [17] that generators for

π_{∗}BP  ⊂  H_{∗}BP
↕  ↕
Z_{(p)}[v_{1}, v_{2}, ...] ⊂ Z_{(p)}[m_{1}, m_{2}, ...]
|v_{n}| = 2(p^{n} − 1) = |m_{n}|

are given by

v_{n} = pm_{n} − ∑_{i=1}^{n−1} m_{i}v_{n−i}^{p^{i}}.

Given a formal group law over a graded ring R_{∗},

F (x, y) = ∑_{i,j} α_{ij}^{R}x^{i}y^{j} ∈ R_{∗}[[x, y]], α_{ij}^{R} ∈ R_{2(i+j−1)}

there is a ring map g : MU_{∗} ⟶ R_{∗} which induces the formal group law; that is g^{∗}(α_{ij}^{MU}) = α_{ij}^{R}.

We use also the following well known formulas

F (x, y) = exp(logx + logy) and logx = ∑_{n≥0} m_{n}x^{n+1}

for computing the elements δ_{i} in BP theory by the algorithm of Section 3.

Example 1. For δ_{1} ∈ BP^{∗}(BZ/2) = BP^{∗}[[z]]/([2](z)) we have modulo z^{8}:

δ_{1} = v_{1}^{2}z^{2} + (v_{1}^{3} + v_{2})z^{3} + v_{1}z^{4} + (v_{1}^{6} + v_{1}^{3}v_{2})z^{6} + (v_{1}^{4}v_{2} + v_{2}^{2} + v_{3})z^{7}.

Next we give some results of calculations in Morava K-theories, where the formulas are more tractable. In the following examples δ_{i}^{(k)} coincides with the coefficient at σ_{p}^{i} in the expression for σ_{k} from Corollary 5.4, y = z^{p−1}, and δ̃_{i}^{(k)} = k!(p − k)!δ_{i}^{(k)}.

Example 2. p = 3, s = 2.

σ_{1} = v_{2}y^{3}σ_{3} + v_{2}y^{4}x.

σ_{2} = 2 v_{2}^{2}y^{3}σ_{3}^{4} + 2 v_{2}y^{2}σ_{3}^{2} + v_{2}x^{2}y^{4} + 2 y.

Example 3. p = 5, s = 3

σ_{1} = v_{3}y^{25}σ_{5}^{5} + v_{3}y^{30}σ_{5} + v_{3}y^{31}x.

σ_{2} = 4 v_{3}^{2}y^{25}σ_{5}^{30} +4 v_{3}^{2}y^{30}σ_{5}^{26} +3 v_{3}y^{19}σ_{5}^{10} +v_{3}y^{24}σ_{5}^{6} +3 v_{3}y^{29}σ_{5}^{2} + 2 v_{3}y^{31}x^{2}.

σ_{3} = 2 v_{3}^{3}y^{25}σ_{5}^{55}+2 v_{3}^{3}y^{30}σ_{5}^{51}+v_{3}^{2}y^{19}σ_{5}^{35}+2 v_{3}^{2}y^{24}σ_{5}^{31}+v_{3}^{2}y^{29}σ_{5}^{27}+ 2 v_{3}y^{13}σ_{5}^{15} + v_{3}y^{18}σ_{5}^{11} + v_{3}y^{23}σ_{5}^{7} + 2 v_{3}y^{28}σ_{5}^{3} + 2 v_{3}x^{3}y^{31}.

σ_{4} = 4 v_{3}^{4}y^{25}σ_{5}^{80}+4 v_{3}^{4}y^{30}σ_{5}^{76}+4 v_{3}^{3}y^{19}σ_{5}^{60}+3 v_{3}^{3}y^{24}σ_{5}^{56}+4 v_{3}^{3}y^{29}σ_{5}^{52}+ 4 v_{3}^{2}y^{13}σ_{5}^{40} + 2 v_{3}^{2}y^{18}σ_{5}^{36} + 2 v_{3}^{2}y^{23}σ_{5}^{32} + 4 v_{3}^{2}y^{28}σ_{5}^{28} + 4 v_{3}y^{7}σ_{5}^{20} + v_{3}y^{12}σ_{5}^{16} + 4 v_{3}y^{17}σ_{5}^{12} + v_{3}y^{22}σ_{5}^{8} + 4 v_{3}y^{27}σ_{5}^{4} + v_{3}x^{4}y^{31} + 4 y. □

6. Transfer and K(s)^{∗}(X_{hΣ_{p}}^{p})

Let X be a CW complex whose Morava K-theory K(s)^{∗}(X) is even dimensional and finitely generated as a module over K(s)^{∗}.

In this section we study the transfer homomorphism in this more general context. We extend some results of Hopkins-Kuhn-Ravenel [10] to spaces. We consider the Atiyah-Hirzebruch-Serre (later abbreviated AHS) spectral sequence:

(34)  E_{2}^{∗,∗}(π, X) = H^{∗}(π; K(s)^{∗}X^{p}) ⇒ K(s)^{∗}(X_{hπ}^{p}).

By the Künneth isomorphism

(35)  K(s)^{∗}X^{p} ⟶^{≈} (K(s)^{∗}X)^{⊗p}.

Then K(s)^{∗}X^{p} is a π module where π acts by permuting factors (see [10], Th. 7.3).

An element x ∈ K(s)^{∗}(X) is called good if there is a finite cover Y → X together with an Euler class y ∈ K(s)^{∗}(Y) such that x = T r^{∗}(y) where T r^{∗} : K(s)^{∗}(Y) → K(s)^{∗}(X) is the transfer. The space X is called good if K(s)^{∗}(X) is spanned over K(s)^{∗} by good elements.

Let γ = φ^{∗}(z), where

φ : X_{hπ}^{p} → Bπ

is the projection and let {x_{j}, j ∈ J } be a K(s)^{∗} basis for K(s)^{∗}(X). Hunton [12] has shown that if K(s)^{∗}(X) is concentrated in even dimensions then so is K(s)^{∗}(X_{hπ}^{p}). We adopt the stronger hypothesis that X is good and derive a stronger result, following the argument of [10] Th.7.3 for classifying spaces.

Proposition 6.1. Let X be a good space.

(i) As a K(s)^{∗} module K(s)^{∗}(X_{hπ}^{p}) is free with basis

{ γ^{i} ⊗ (x_{j})^{⊗p} | 0 ≤ i < p^{s}, j ∈ J }

and

{ ∑_{(i_{1},i_{2},...,i_{p})=I} 1 ⊗ x_{i_{1}} ⊗ x_{i_{2}} ⊗ ... ⊗ x_{i_{p}} | I ∈ P_{p} }

where I = {(i_{1}, i_{2}, ..., i_{p})} runs over the set P_{p} of π-equivalence classes of p-tuples of indices i_{j} ∈ J at least two of which are not equal.

(ii) X_{hπ}^{p} is good.

Proof: (i) By the Künneth isomorphism,

(36)  K(s)^{∗}(X)^{⊗p} = F ⊕ T

as a π-module, where F is free and T is trivial. Explicitly a K(s)^{∗} basis for T is {(x_{i})^{⊗p}, i ∈ J }, while a K(s)^{∗} basis for F is {x_{i_{1}} ⊗ x_{i_{2}} ⊗ ... ⊗ x_{i_{p}}, i_{j} ∈ J } where not all the factors are equal. Then

H^{∗}(π; F) = F^{π} if ∗ = 0
= 0 if ∗ > 0

and

H^{∗}(π; T) = H^{∗}(Bπ) ⊗ T.

Thus E_{2}^{0,∗}(π, X) = K(s)^{∗}(X_{hπ}^{p})^{π} = F^{π} ⊕ T.

To continue the proof we recall the covering projection

ρ_{π} : Eπ × X^{p} → X_{hπ}^{p},

its associated transfer homorphism

(37)  T r^{∗} = T r_{π}^{∗} : K(s)^{∗}(X^{p}) → K(s)^{∗}(X_{hπ}^{p}),

and induced homomorphism

ρ_{π}^{∗} : K(s)^{∗}(X_{hπ}^{p}) → K(s)^{∗}(X^{p}).

Similar maps are defined for the group Σ_{p}. Then ρ_{π}^{∗}T r^{∗} = N, where N = N_{π} is the trace map.

Thus we have established the following lemma.

Lemma 6.2. If y ∈ K(s)^{∗}(X^{p}) is good then there exists a good element z ∈ K(s)^{∗}(X_{hπ}^{p}) such that ρ_{π}^{∗}(z) = N(y). □

Lemma 6.3. If x ∈ K(s)^{∗}(X) is good then there is a good element z ∈ K(s)^{∗}(X_{hπ}^{p}) such that ρ_{π}^{∗}(z) = x^{⊗p}.

Proof: By assumption there is a finite covering f : Y → X and an Euler class e ∈ K(s)^{∗}(Y) such that x = T r^{∗}(e). Now consider the covering

ϕ = f × ... × f : Y^{p} → X^{p}

which extends to a covering

1 × ϕ : Y_{hπ}^{p} → X_{hπ}^{p}

and yields a map of coverings

[figure: Eπ × Y^{p}; ρ_{π}; Y_{hπ}^{p}; Eπ × X^{p}; 1 × ϕ; ρ_{π}; X_{hπ}^{p}; 1 × ϕ]

The class e^{⊗p} is an Euler class for Y^{p}. Since the transfer is natural and commutes with tensor products we have

ρ_{π}^{∗}T r^{∗}(1⊗e^{⊗p}) = T r^{∗}ρ_{π}^{∗}(1⊗e^{⊗p}) = T r^{∗}(e^{⊗p}) = T r^{∗}(e)⊗...⊗T r^{∗}(e) = x^{⊗p}.

□

Corollary 6.4. E_{2}^{0,∗}(π, X) consists of permanent cycles which are good. □

Thus as differential graded K(s)^{∗} modules, there is an isomorphism of spectral sequences

(E_{r}^{∗,∗}(π, pt) ⊗_{K(s)^{∗}} T) ⊕ F^{π} ⟶^{≈} E_{r}^{∗,∗}(π, X).

Thus it follows that as a K(s)^{∗} algebra, K(s)^{∗}(X_{hπ}^{p}) is generated by K(s)^{∗}(Bπ), T, and F^{π}.

(ii) The proof of [10] Th. 7.3 carries over. This completes the proof of Proposition 6.1. □

Remarks : 1) From the periodicity of the cohomology of a cyclic group [5] Proposition XII, 11.1, we have isomorphisms

H^{t}(π; K(s)^{∗}(X^{p})) ⟶^{·z} H^{t+2}(π; K(s)^{∗}(X^{p}))

for t > 0 and

H^{0}(π; K(s)^{∗}(X^{p}))/Im(N) ⟶^{·z} H^{2}(π; K(s)^{∗}(X^{p})).

Thus multiplication by z is also injective on T at the E_{2} term.

2) ρ_{π}^{∗}T r^{∗} = N, thus modulo ker(ρ_{π}^{∗}) we have

(38) T r^{∗}(x_{i_{1}} ⊗ x_{i_{2}} ⊗ ... ⊗ x_{i_{p}}) = ∑_{σ∈π} 1 ⊗ x_{σ(i_{1})} ⊗ x_{σ(i_{2})} ⊗ ... ⊗ x_{σ(i_{p})}.

Note that if the i_{j} in (38) are equal, the right hand side is zero. However

T r^{∗}(x_{j}^{⊗p}) = 1 ⊗ x_{j}^{⊗p} · T r^{∗}(1).

We now turn to K(s)^{∗}(X_{hΣ_{p}}^{p}).

Let c = φ^{∗}(y) where φ : EΣ_{p} ×_{Σ_{p}} X^{p} → BΣ_{p} is the projection.

Proposition 6.5. Let X be a good space. As a K(s)^{∗} module K(s)^{∗}(X_{hΣ_{p}}^{p}) is free with basis

{ c^{i} ⊗ (x_{j})^{⊗p} | 0 ≤ i < m_{s}, j ∈ J }

and

{ ∑_{(i_{1},i_{2},...,i_{p})=I} 1 ⊗ x_{i_{1}} ⊗ x_{i_{2}} ⊗ ... ⊗ x_{i_{p}} | I ∈ E_{p} }

where I = {(i_{1}, i_{2}, ..., i_{p})} runs over the set E_{p} of Σ_{p}-equivalence classes of p-tuples of indices i_{j} ∈ J at least two of which are not equal.

Proof: Since |W | is prime to p, the result follows from the AHS spectral sequence, as in the proof of Proposition 6.1.

7. Applications

7.1 π ≀ (Z/p^{n}).

We now turn to G_{n} = π ≀ (Z/p^{n}) where π = Z/p. Then BG_{n} = X_{hπ}^{p} for X = BZ/p^{n}. Consider the AHS spectral sequence for

B(Z/p^{n})^{p} → BG_{n} →^{φ} Bπ.

Then

E_{2}^{p,q} = H^{∗}(π; K^{∗}(s)(B(Z/p^{n})^{p})),

where

K^{∗}(s)(B(Z/p^{n})^{p}) = (K(s)^{∗}[z]/(z^{p^{ns}}))^{⊗p} = F ⊕ T,

where F and T as in (36) above are free (resp. trivial) π modules. Let γ = φ^{∗}(z) where K(s)^{∗}(Bπ) = K(s)^{∗}[z]/(z^{p^{s}}) as above.

Proposition 7.1. As a K(s)^{∗} module K(s)^{∗}(BG_{n}) is free with basis

{γ^{i} ⊗ (z^{j})^{⊗p} 0 ≤ i < p^{s}, 0 ≤ j < p^{ns}}

and

{ ∑_{(i_{1},i_{2},...,i_{p})=I} 1 ⊗ z^{i_{1}} ⊗ z^{i_{2}} ⊗ ... ⊗ z^{i_{p}} | I ∈ P_{p}(n) },

where I = {(i_{1}, i_{2}, ..., i_{p})} runs over the set P_{p}(n) of π-equivalence classes of p-tuples of integers {0 ≤ i_{j} < p^{ns}} at least two of which are not equal.

Proof: This spectral sequence computation is exactly analogous to that of Proposition 6.1. □

Remarks: (i) For X = CP^{∞}, Proposition 7.1 gives another derivation of K(s)^{∗}(X_{hπ}^{p}). Since CP^{∞}_{p}^{∧} = [colim_{n} B(Z/p^{n})]_{p}^{∧}, we have K(s)^{∗}(X_{hπ}^{p}) = lim_{n} K(s)^{∗}(BG_{n}). (ii) G_{n} is good for K(s)^{∗} by [10] Th. 7.3.

By analogy with Section 6 we have

Lemma 7.2. i) Im(T r^{∗}) · γ = 0. ii) T r^{∗}(1) = v_{s}γ^{p^{s}−1}. iii) If y ∈ T then T r^{∗}(y) = y · T r^{∗}(1).

Finally we consider the case p = 2 where c = γ. If n = 1, G_{n} ≈ D_{8}, the dihedral group of order 8; the rings K(s)^{∗}(BD_{2^{k}}) were determined by Schuster [20],[21]. In general we have a partial result

Proposition 7.3. Let p = 2. As a K(s)^{∗}(Bπ) algebra, K(s)^{∗}(BG_{n}) is generated by c̃_{1}, c_{2} subject to the relation c̃_{1} · c = 0 and the relations c̃_{1}^{2^{ns}} = c_{2}^{2^{ns}} = 0 modulo terms divisible by c.

Proof: c̃_{1} · c = 0 by Lemma 7.2. The other relations hold in the E_{∞} term of the spectral sequence. The only possible extensions are those on the fiber, involving c̃_{1}, c_{2}. □

Similar results hold for Σ_{p} ≀ Z/p and Σ_{p} ≀ Σ_{p} for p odd.

7.2 p-groups with cyclic subgroup of index p.

In this section we consider the class of p-groups with a (necessarily normal) cyclic subgroup of index p. It is known [3], Th. 4.1, Chpt. IV, that every p-group of this form is isomorphic to one of the groups:

(a) Z/q (q = p^{n}, n ≥ 1).

(b) Z/q × Z/p (q = p^{n}, n ≥ 1).

(c) Z/q ⋊ Z/p (q = p^{n}, n ≥ 2), where the canonical generator of Z/p acts on Z/q as multiplication by 1 + p^{n−1}. This group is called the modular group if p ≥ 3 and the quasi-dihedral group if p = 2, n ≥ 4.

For p = 2 there are three additional families.

(d) Dihedral 2-groups D_{2m} = Z/m ⋊ Z/2, (m ≥ 2), where the generator of Z/2 acts on Z/m as multiplication by −1. If m = 2^{n}, D_{2m} is a 2-group. Note that D_{4} belongs to (b) and D_{8} belongs to (c).

(e) Generalized quaternion 2-groups. Let H be the algebra of quaternions R ⊕ Ri ⊕ Rj ⊕ Rk. For m ≥ 2 the generalized quaternion group Q_{4m} is defined as the subgroup of the multiplicative group H^{∗} generated by x = e^{πi/m} and y = j. Z/2m generated by x is normal and has index 2. If m is a power of 2, Q_{4m} is a 2-group. In the extension 0 → Z/2m → Q_{4m} → Z/2 → 0 the generator of Z/2 acts on Z/2m as −1. In particular Q_{8} is the group of quaternions {±1, ±i, ±j, ±k}.

(f) Semi-dihedral groups. Z/q ⋊ Z/2 (q = p^{n}, n ≥ 3), where the generator of Z/2 acts on Z/q as multiplication by −1 + 2^{n−1}.

Consider now the task of computing the stable Euler class, T r_{G}^{∗}(1), for the universal G-covering EG → BG.

For the case (a) there is the well known formula of Quillen (1)

T r_{Z/q}^{∗}(1) = [q]_{F} (z)/z

in MU^{∗}(BZ/q) = MU^{∗}[[z]]/([q]_{F} (z)).

For the case (b) the answer follows from transfer property (ii): T r_{G} = T r_{Z/q} ∧ T r_{Z/p};

In cases (d),(e) and (f) T r_{G}^{∗} is the composition of two transfers T r_{Z/q}^{∗} and T r_{C,G}^{∗} : MU^{∗}(BC) → MU^{∗}(BG), where C is the corresponding cyclic subgroup. So we have to compute T r_{C,G}^{∗}(z^{i}), i ≥ 1 and we can apply our results for BZ/2 ≀ U(1), namely Remark 3.7.

Similarly for the case (c), T r_{G} is the composition T r_{Z/q,G}T r_{Z/q} and we can apply Corollary 3.6.

This task is trivial for wreath products Z/p ≀ Z/p^{n} since T r_{Z/p^{n}}^{∗}(1) is symmetric in z_{1}, ..., z_{p} in the ring

MU^{∗}((BZ/p^{n})^{p}) = MU^{∗}[[z_{1}, ...z_{p}]]/([p^{n}](z_{1}), ..., [p^{n}](z_{p}))

and hence invariant under the Z/p action. So in this case we need only Quillen’s formula.

Finally we note that if G is the modular group of case (c), Brunetti [4] has completely computed the ring K(s)^{∗}(BG). The relations are quite simple but the generators are technically complicated. In a future paper we plan to use transferred Chern classes to give a more natural presentation.

7.3 Other examples.

Consider the semi-direct products G = (Z/p)^{n} ⋊ Z/p where the generator α of Z/p acts on H_{n} = Z/p[T]/(T^{n}) by 1 − α = T, 1 ≤ n ≤ p. Then every Z/p[Z/p]-module is a direct sum of the modules H_{n}. As shown by Yagita [24] and Kriz [14], these semi-direct products are good in the sense of Hopkins-Kuhn-Ravenel.

We recall

K(s)^{∗}(B(Z/p)^{n}) = K(s)^{∗}[[z_{1}, ..., z_{n}]]/(z_{i}^{p^{s}}),

where z_{i} is the Euler class of a faithful complex line bundle θ_{i} on the i-th factor. Then Z/p acts on K(s)^{∗}[z_{1}, ..., z_{n}]/(z_{i}^{p^{s}}) by

α : z_{i} → F_{K(s)}(z_{i}, z_{i+1}), z_{n+1} := 0,

where F_{K(s)} denotes the formal group law for Morava K-theory.

Our aim is to show how to compute the stable Euler classes in terms of characteristic classes and the formal group law.

The transfer T r^{∗} : K(s)^{∗}EG → K(s)^{∗}BG is the composition of two transfers

T r_{1}^{∗} : K(s)^{∗}E((Z/p)^{n}) → K(s)^{∗}B((Z/p)^{n})

and

T r_{2}^{∗} : K(s)^{∗}B((Z/p)^{n}) → K(s)^{∗}BG.

Recall also that

T r_{1}^{∗}(1) = z_{1}^{p^{s}−1} ⋯ z_{n}^{p^{s}−1}.

It is easy to see that in K(s)^{∗}((BZ/p)^{n}) we have

e^{p^{s}−1}(α^{i_{1}}(θ_{1})) ⋯ e^{p^{s}−1}(α^{i_{n}}(θ_{1})) = z_{1}^{p^{s}−1} ⋯ z_{n}^{p^{s}−1},

where e is the Euler class and 1 ≤ i_{1} < ... < i_{n} ≤ p. Then recall the elements ω_{n} from Theorem 3.1 and let ω_{n}(l) be the sum of the same monomials after raising to the power l. Since ω_{n}(l) consist of p^{−1}\binom{p}{n} summands and p^{−1}\binom{p}{n} = (−1)^{n}/n mod p, we have that in K(s)^{∗}((BZ/p)^{n})

η_{π}^{∗}(ω_{n}(p^{s} − 1)) = (−1)^{n}z_{1}^{p^{s}−1} ⋯ z_{n}^{p^{s}−1}/n,

where the map η_{π}, defined in Section 2, sends ξ_{i} = t^{i−1}ξ_{1} to α^{i−1}θ_{1}. Hence

T r_{G}^{∗}(1) = T r_{2}^{∗}(T r_{1}^{∗}(1)) = T r_{2}^{∗}((−1)^{n}nη_{π}^{∗}(ω_{n}(p^{s} − 1))) =
(−1)^{n}nT r_{2}^{∗}(η_{π}^{∗}(ω_{n}(p^{s} − 1))),

and we have to apply Corollary 3.6.

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Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia

E-mail address: maxo@rmi.acnet.ge

Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

E-mail address: priddy@math.northwestern.edu