THE FORMAL GROUP LAW AND TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY

MALKHAZ BAKURADZE

Abstract. Chern classes of transferred bundle for cyclic covering are evaluted in terms of transferred Chern classes by certain property of the formal group law in Morava K-theory

1. Introduction and Statements

This is joint project with S. Priddy in preparation. The author takes the opportunity as a visitor at Max-Planck Institute of Mathematics to give this material in more details with illustrated examples and tables.

A generalized cohomology theory E^{∗}(−) is a homotopy-invariant contravariant fuctor that assings for each integer n to each space X an abelian group E^{n}(X) connected by natural suspension isomorphism Σ : E^{n}(X) ≃ E^{n+1}(ΣX) of abelian groups and satisfies the axioms: (i). E^{n}(−) is half-exact: If X = A ⋃ B and y ∈ E^{n}(A) and z ∈ E^{n}(B) agree in E^{n}(A ⋂ B), there exists x ∈ E^{n}(X) that lifts both y and z. (ii). E^{n}(−) is strongly additive: For any disjoint union X = ∐_{α} X_{α}, the inclusions induce E^{n}(X) ≃ ∏_{α} E^{n}(X_{α}).

The cohomology theory E^{∗}(−) is multiplicative if E^{∗}(X) is naturally a commutative graded ring with unit and suspension isomorphism of E^{∗}(X) modules.

A multiplicative cohomology theory E^{∗} is called complex oriented of 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}). Then by Dold theorem

E^{∗}(CP^{∞}) = E^{∗}[[x]],

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

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

x_{i} = c_{1}(ξ_{i}), and ξ_{i} is the pull back bundle over BU(1)^{n} by the projection BU(1)^{n} → B(1) on the i-th factor.

If E has a line bundle Chern class then the splitting principle says that the diagonal inclusion of U(1)^{n} in U(n) induces the monomorphism and

E^{∗}(BU(n)) = E^{∗}[[c_{1}, ..., c_{n}]],

Date: October, 2002.

1991 Mathematics Subject Classification. Primary 55R12, 55R20; Secondary 55R40 .

Key words and phrases. transfer, formal group law, Chern Class.

The author was supported by Max-Planck Institute of Mathematics.

where c_{i} ∈ E^{2i}BU(n), the i-th Chern class of the universal U(n) bundle over BU(n), restricts to the i-th elementary symmetric function in x_{1}, ..., x_{n}. The following axioms determine the general Chern classes uniquely: (i). Naturality with respect to pull backs; (ii). Normalization, that is, c_{i}(ξ) = 0 for all i > deg(ξ); (iii). Hopf line bundle over CP^{∞} has a Chern class; (iv). The Whitney Formula:

c_{k}(ξ ⊕ η) = c_{k}(ξ) + ∑_{1≤i≤k−1} c_{k−i}(ξ)c_{i}(η) + c_{k}(η).

The formal group law. For computing c_{i}(ξ ⊗η), the Chern class for tensor product of bundles, we need a formula for tensor product of line bundles and Whitney formula. In view of equation (1), there must be

c_{1}(ξ_{1} ⊗ ξ_{2}) = x_{1} + x_{2} + ∑_{i,j} a_{ij}x_{1}^{i}x_{2}^{j} = F (x_{1}, x_{2})

in the universal case, and therefore generally, where

F (x, y) = x + y + ∑_{i,j} a_{ij}x^{i}y^{j}

in formal power series ring E^{∗}[[x, y]].

F (x, y) is called formal group law in E^{∗} and satisfies the identities which obviously follow from the corresponding bundle identities: (i). F (x, y) = F (y, x); (ii). F (F (x, y), z) = F (x, F (y, z)); (iii). F (x, 0) = x.

The universal example of complex-oriented cohomology theory is complex cobordism MU^{∗}(−), accosiated with Thom spectrum MU. By Quillen’s theorem [5] for any complex oriented cohomology theory E^{∗}(−) there is a map of ring spectra g : MU → E, which induces the formal group law, that is, g(α_{ij}^{MU}) = α_{ij}^{E}. The Morava FGL. Let us turn to Morava K-theories. For fixed prime p there is complex oriented cohomology K(s)^{∗}(−) named after J.Morava with the coefficient ring K^{∗} = K^{∗}(pt) = F_{p}[v_{s}, v_{s}^{−1}], with | v_{s} |= 2(1 − p^{s}). The following lemma [3] plays central role in our discussion. Lemma 1.1. For the formal group law in Morava K-theory K(s)^{∗}, s > 1 we have mod y^{p^{2(s−1)}}

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}.

Proof. As D. Ravenel explained, this result can be derived from the recursive formula for the FGL given in 4.3.8 [6]: For the FGL in Morava K-theory it reads

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)}}, ...)

where w_{i} is a certain homogeneous polynomial of degree p^{i} defined by 4.3.5 and e_{i} = (p^{is} − 1)/(p^{s} − 1). In particular w_{0} = x + y (the first term in the formula above),

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

and w_{i} ∉ (x^{p}, y^{p}).

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 p be a prime. For a given action of cyclic group Z/p on a space X consider the regular covering

ρ : EZ/p × X → EZ/p ×_{Z/p} X.

Let us write for short X_{hZ/p} = EZ/p ×_{Z/p} X. For the permutation action of π = Z/p on X = Y^{p} we have Y_{hπ}^{p} = Eπ ×_{π} Y^{p}. Let t be a generator of π and let N_{π} = 1+t+...+t^{p−1} be the trace map. For a n-plane bundle η^{n} the corresponding classifying map

f : X → BU(n)

gives the classifying map (f, tf, ..., t^{p−1}f) for the bundle N_{π}η^{n} and the map of orbit spaces

f_{η^{n}} : X_{hZ/p} → BU(n)_{hπ}^{p}.

For the covering

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

and the universal U(n) bundle ξ^{n} → BU(n) consider the Atiyah transfer bundle [2]

ξ_{π}^{n} → Eπ ×_{π} BU(n)^{p},

the np-plane bundle

ξ_{π}^{n} = Eπ ×_{π} (ξ^{n})^{×p}.

Then the map f_{η^{n}} classifies the Atiyah transfer bundle for η^{n} and ρ. So by naturality of transfer [1] we can consider ρ_{π} as the universal example.

Recall that by the Künneth isomorphism K(s)^{∗}(BU(n)^{p}) = F ⊕T as a π module, where F is free and T is trivial. Definition 1.2. Let ω_{k}^{(n)} ∈ F be defined modulo kerN_{π} = Im(1 − t^{∗}) by

N_{π}(ω_{k}^{(n)}) = c_{k}(N_{π}(ξ^{n})),

where c_{k} are Chern classes, k ∈ {1, ..., np} \ {p, 2p, ..., np}.

By transfer property T r_{π}^{∗} = T r_{π}^{∗}t^{∗}, where

T r_{π}^{∗} : K(s)^{∗}(Eπ × BU(n)^{p})) → K(s)^{∗}(BU(n)_{hπ}^{p}),

hence T r_{π}^{∗}(ω_{k}^{(n)}) is well defined.

We write ω_{k}(η^{n}) for pullback by above map f_{η^{n}} of orbit spaces.

Recall K(s)^{∗}(BZ/p) = F_{p}[v_{s}, v_{s}^{−1}][z]/(z^{p^{s}}), where z = c_{1}(θ) is the Chern class of the canonical line complex bundle over BZ/p.

Lemma 1.3. We can define the polynomial in 1 + n variables

A_{k}^{(n)}(z, Z_{1}, ..., Z_{n}) ∈ K(s)^{∗}[z, Z_{1}, ..., Z_{n}]

by the equation in K(s)^{∗}B(Z/p × U(n)):

C_{k} − v_{s}z^{p^{s}−1}p^{−1} ∑_{i_{1}+2i_{2}...+ni_{n}=k} \binom{p}{i_{0}, i_{1}, ..., i_{n}}c_{1}^{i_{1}}...c_{n}^{i_{n}} = A_{k}^{(n)}(z, C_{p}, ..., C_{pn})

where C_{i} = c_{i}(ξ^{n} ⊕ θ ⊗ ξ^{n} ⊕ ... ⊕ θ^{p−1} ⊗ ξ^{n}), c_{j} = c_{j}(ξ^{n}) are Chern classes, k ∈ {1, ...np} \ {p, ..., np}.

For example in K(s)^{∗}(Bπ × BU(1)) one has

C_{1} = v_{s} (z^{p^{s}−1}c_{1} + ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}C_{p}^{p^{i−1}}) ,

thus

A_{1}^{(1)}(z, Z_{1}) = v_{s} ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}Z_{1}^{p^{i−1}}.

Then using the polynomials A_{k}^{(n)} we evaluate the transferred classes ω_{k}(η^{n}) for

regular coverings:

Theorem 1.4. Let ρ : X → X/_{Z/p} be the regular cyclic covering of prime index p defined by a free action of Z/p on X and T r^{∗} = T r_{ρ}^{∗} be the associated transfer homomorphism. Let η^{n} → X be a complex n-plane bundle, η_{Z/p}^{n} → X/_{Z/p} the pnplane bundle defined by Atyiah transfer, and ψ → X/_{Z/p} be the line complex bundle associated with ρ. Then

c_{k}(η_{Z/p}^{n})) − T r^{∗}(ω_{k}(η^{n})) = A_{k}^{(n)}(c_{1}(ψ), c_{p}(η_{Z/p}^{n}), ..., c_{pn}(η_{Z/p}^{n})),

where k ∈ {1, ..., np} \ {p, ..., np}.

Example 1.5. ω_{1} = c_{1}(η^{n}) hence if k = 1 we have

c_{1}(η_{Z/p}^{n})) − T r^{∗}(c_{1}(η^{n})) = A_{1}^{(n)}(c_{1}(ψ), c_{p}(η_{Z/p}^{n}), ..., c_{pn}(η_{Z/p}^{n})).

If in addition n = 1 then by above example we have for the line bundle η → X and transferred Chern class

c_{1}(η_{Z/p}) − T r^{∗}(c_{1}(η)) = v_{s} ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}c_{p}(η_{Z/p})^{p^{i−1}}.

2. Proofs

Proof of Lemma 1.3. It is clear that we have to prove the Lemma in case n = 1 and then apply the splitting principle. In case n = 1, let σ_{k} be the k-th symmetric function in p variables. Lemma 2.1 implies that in K(s)^{∗}B(π × U(1)) = K(s)^{∗}[[z, x]]/(z^{p^{s}}) the difference

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

for 1 ≤ k ≤ p − 1 is a polynomial in z^{p−1} and σ_{p} = σ_{p}(x, F (x, z), ..., F (x, (p − 1)z)). By definition the classes C_{k} = σ_{k}, and therefore define A_{k}^{(1)}, as polynomials in z^{p−1} and σ_{p}.

Next we consider the case n = 2 in detail. Let us apply the splitting principle. Consider Bπ × BU(1) × BU(1) and write formally

ξ^{2} = ξ + η and

x = c_{1}(ξ),
X_{i} = c_{i}(ξ + ξ ⊗ θ + ... + ξ ⊗ θ^{p−1}),
y = c_{1}(η),
Y_{i} = c_{i}(η + η ⊗ θ + ... + η ⊗ θ^{p−1}),

i = 1, ..., p.

Then from the case n = 1 we have:

X_{k} = σ_{k}(x, F (x, z), ..., F (x, (p − 1)z)) = p^{−1}\binom{p}{k}x^{k}v_{s}z^{p^{s−1}} + A_{k}^{(1)}(z^{p−1}, X_{p}) (∗)

and

Y_{k} = σ_{k}(y, F (y, z), ..., F (y, (p − 1)z)) = p^{−1}\binom{p}{k}y^{k}v_{s}z^{p^{s−1}} + A_{k}^{(1)}(z^{p−1}, Y_{p}) (∗∗)

for k = 1, ..., p − 1.

Then the classes C_{i} = c_{i}(ξ^{2} +ξ^{2} ⊗θ +...+ξ^{2} ⊗θ^{p−1}) we can express by X_{i} and Y_{i} using Whitney formula for the sum of bundles C_{i} is symmetric in classes X_{i} and Y_{i} and the above formulas (*), (**) for the elements X_{i} and Y_{i} defines the equation:

C_{k} = ∑_{i+2j=k} δ_{ij}c_{1}^{i}c_{2}^{j} + A^{(2)}(z, C_{p}, C_{2}p)

We want to see that

δ_{i,j} = v_{s}z^{p^{s}−1}p^{−1} ∑_{i+2j=k} \binom{p}{i_{0}, i, j}

For this we have to compute the contribution of this procedure to coefficients δ_{ij}.

Let k = 1, ..., p − 1. By the Whitney formila

C_{k} = X_{k} + Y_{k} + ...

but it follows from (*), (**) that the other terms X_{i}Y_{j} are irrelevant since z is nilpotent and these terms have no contribution to δ_{ij}.(Of course they are relevant to A^{(2)}(z, C_{p}, C_{2p}) but we are looking at deltas only)

Then as above the contribution of X_{k} is

p^{−1}\binom{p}{k}x^{k}v_{s}z^{p^{s−1}}

and the total contribution will

p^{−1}\binom{p}{k}(x^{k} + y^{k})v_{s}z^{p^{s−1}}

hence for 1 ≤ k ≤ p − 1 the result follows from the following : Lemma(***). Let

x^{k} + y^{k} = ∑_{i+2j=k} β_{ij}c_{1}^{i}c_{2}^{j}

where c_{1} = x + y, c_{2} = xy then modulo p

p^{−1}\binom{p}{k}β_{ij} = p^{−1} ∑_{i+2j=k} \binom{p}{i_{0}, i, j}.

To complete the proof in case n = 2 we have to consider C_{k} for k = p+1, ...2p−1. Again

C_{p+l} = X_{l}Y_{p} + X_{p}Y_{l} + ...

l = 1, ..., p − 1, the other terms are irrelevant. Also since Y_{p} = y^{p}+ terms divisible by z the contribution of X_{l}Y_{p} is

p^{−1}v_{s}z^{p^{s}−1}\binom{p}{l}x^{l}y^{p}

hence we have the total contribution for C_{p+l}

p^{−1}v_{s}z^{p^{s}−1}\binom{p}{l}(x^{l}y^{p} + x^{p}y^{l}). (IV ∗)

Then

x^{l}y^{p} + x^{p}y^{l} = x^{l}y^{l}(x^{p−l} + y^{p−l}) = c_{2}^{l}(x^{p−l} + y^{p−l}).

By above Lemma(***) we have that if

c_{2}^{l}(x^{p−l} + y^{p−l}) = c_{2}^{l} ∑_{i+2j=p−l} γ_{ij}c_{1}^{i}c_{2}^{j}

then modulo p

p^{−1}\binom{p}{p − l}γ_{ij} = p^{−1} ∑_{i_{1}+2j=p−l} \binom{p}{i_{0}, i, j}.

Here in binomial coefficient we can replase p − l by l. Also since in multinomial coefficient l = p − i − 2j we can replace j by j + l = p − i − j = i_{0}. Thus for above (IV*) we have

p^{−1}v_{s}z^{p^{s}−1}\binom{p}{l}(x^{l}y^{p} + x^{p}y^{l}) = p^{−1}v_{s}z^{p^{s}−1} ∑_{i_{1}+2j=p−l} \binom{p}{i_{0}, i, j}c_{1}^{i}c_{2}^{i}c_{2}^{l}

= p^{−1}v_{s}z^{p^{s}−1} ∑_{i_{1}+2(j+l)=p+l} \binom{p}{i_{0}, i, j}c_{1}^{i}c_{2}^{j+l}.

This completes the froof in case n = 2. The general case is completely analogous. Initially the guess was another. Namely the formula from Lemma 1.3 should

follow from similar but less tractable formula in BP^{∗}. In BP ∗ should be

C_{k} − T r^{∗}(1)p^{−1} ∑_{i_{1}+2i_{2}...+ni_{n}=k} \binom{p}{i_{0}, i_{1}, ..., i_{n}}c_{1}^{i_{1}}...c_{n}^{i_{n}}

on the left side since C_{k} pulls back to c_{k}(ξ^{n} + ... + ξ^{n}), (p times.) in BP^{∗}(BZ/p × BU(n)) and the total Chern class 1 + C_{1} + ... + C_{np} pulls back to (1 + c_{1} + ... + c_{n})^{p} So C_{k} pulls back to

pp^{−1} ∑_{i_{1}+2i_{2}...+ni_{n}=k} \binom{p}{i_{0}, i_{1}, ..., i_{n}}c_{1}^{i_{1}}...c_{n}^{i_{n}}

so is

T r^{∗}(1)p^{−1} ∑_{i_{1}+2i_{2}...+ni_{n}=k} \binom{p}{i_{0}, i_{1}, ..., i_{n}}c_{1}^{i_{1}}...c_{n}^{i_{n}}.

Proof of Theorem 1.4. Consider the homotopy orbit space BU(n)_{hπ}^{p} = Eπ ×_{π} BU(n)^{p} as the universal example. The diagonal map BU(n) → BU(n)^{p} induces the inclusion

i : Bπ × BU(n) → Eπ ×_{π} BU(n)^{p}.

We use a result from [4] (Proposition 4.2) which implies that since BU(n) is a unitary-like space (i.e. K(s)^{∗}BU(n) has no nilpotent elements) the map (i ∨ ρ_{π})^{∗} is a monomorphism. Since ρ_{π}^{∗}T r^{∗} = N_{π}, the difference

c_{k}(η_{π}^{n}) − T r^{∗}(ω_{k}^{(n)})

belongs to kerρ_{π}^{∗} and hence is detected by i^{∗}. The result now follows from Lemma 1.3.

3. Some exapmles and tables

First recall from [6] 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}}.

We use the above formula with the following well known formulas

F (x, y) = exp(logx + logy),

logx = ∑_{n≥0} m_{n}x^{n+1}

for computing α_{ij} = α_{ij}^{BP}.

The coefficients of formal group law in BP theory α_{11} = −v_{1}; α_{12} = v_{1}^{2}; α_{13} = −2v_{1}^{3} − 2v_{2}; α_{22} = −4v_{1}^{3} − 3v_{2}; α_{14} = 3v_{1}^{4} + 4v_{1}v_{2}; α_{23} = 10v_{1}^{4} + 11v_{1}v_{2}; α_{15} = −4v_{1}^{5} + 6v_{1}^{2}v_{2}; α_{24} = −21v_{1}^{5} − 28v_{1}^{2}v_{2}; α_{33} = −34v_{1}^{5} − 43v_{1}^{2}v_{2}; α_{16} = 6v_{1}^{6} + 12v_{1}^{3}v_{2} + 4v_{2}^{2}; α_{25} = 43v_{1}^{6} + 75v_{1}^{3}v_{2} + 18v_{2}^{2}; α_{34} = 101v_{1}^{6} + 164v_{1}^{3}v_{2} + 34v_{2}^{2}; α_{17} = −10v_{1}^{7} − 24v_{1}^{4}v_{2} − 14v_{1}v_{2}^{2} − 4v_{3}; α_{26} = −88v_{1}^{7} − 190v_{1}^{4}v_{2} − 89v_{1}v_{2}^{2} − 14v_{3}; α_{35} = −275v_{1}^{7} − 551v_{1}^{4}v_{2} − 226v_{1}v_{2}^{2} − 28v_{3}; α_{44} = −394v_{1}^{7} − 765v_{1}^{4}v_{2} − 302v_{1}v_{2}^{2} − 35v_{3}; α_{18} = 15v_{1}^{8} + 40v_{1}^{5}v_{2} + 28v_{1}^{2}v_{2}^{2} + 8v_{1}v_{3}; α_{27} = 169v_{1}^{8} + 420v_{1}^{5}v_{2} + 257v_{1}^{2}v_{2}^{2} + 46v_{1}v_{3}; α_{36} = 680v_{1}^{8} + 1586v_{1}^{5}v_{2} + 879v_{1}^{2}v_{2}^{2} + 126v_{1}v_{3}; α_{45} = 1303v_{1}^{8} + 2933v_{1}^{5}v_{2} + 1543v_{1}^{2}v_{2}^{2} + 203v_{1}v_{3}; α_{19} = −22v_{1}^{9} − 66v_{1}^{6}v_{2} − 58v_{1}^{3}v_{2}^{2} − 12v_{1}^{2}v_{3} − v_{2}^{3}; α_{28} = −312v_{1}^{9} − 880v_{1}^{6}v_{2} − 688v_{1}^{3}v_{2}^{2} − 104v_{1}^{2}v_{3} − 72v_{2}^{3}; α_{37} = −1573v_{1}^{9} − 4192v_{1}^{6}v_{2} − 3001v_{1}^{3}v_{2}^{2} − 382v_{1}^{2}v_{3} − 260v_{2}^{3}; α_{46} = −3861v_{1}^{9} − 9900v_{1}^{6}v_{2} − 6707v_{1}^{3}v_{2}^{2} − 791v_{1}^{2}v_{3} − 523v_{2}^{3}; α_{55} = −5156v_{1}^{9} − 13042v_{1}^{6}v_{2} − 8671v_{1}^{3}v_{2}^{2} − 1001v_{1}^{2}v_{3} − 654v_{2}^{3}.

Example n = 2

Let α_{ij} = α_{ij}^{G(2)} then using the map BP → G(n), BP^{∗} = Z_{(p)}[v_{1}, v_{2}, ...] , G(n)^{∗} = Z_{(p)}[v_{n}.v_{n}^{−1}], which satisfies v_{r} → v_{r} if r = n, and v_{r} → 0, otherwise

α_{13} = −2v_{2}; α_{22} = −3v_{2}; α_{16} = 4v_{2}^{2}; α_{25} = 18v_{2}^{2}; α_{34} = 34v_{2}^{2}; and otherwise α_{ij} = 0, for i + j = 8.

Below are the initial segments of FGL of Morava K-theory K(s)^{∗}.

p = 2, s = 2:

x + y + v_{2}x^{2}y^{2} +v_{2}^{3} (x^{6}y^{4} + x^{4}y^{6}) +v_{2}^{5} (y^{12}x^{4} + y^{4}x^{12}) +v_{2}^{7} (y^{10}x^{12} + y^{12}x^{10} + y^{14}x^{8} + y^{8}x^{14}) +v_{2}^{9} (y^{20}x^{8} + y^{8}x^{20}) +v_{2}^{11} (y^{8}x^{26} + y^{10}x^{24} + y^{24}x^{10} + y^{26}x^{8}) +v_{2}^{13} (y^{12}x^{28} + y^{28}x^{12}) +v_{2}^{15} (y^{30}x^{16} + y^{16}x^{30} + y^{18}x^{28} + y^{20}x^{26} + y^{22}x^{24} + y^{24}x^{22} + y^{26}x^{20} + y^{28}x^{18}) +v_{2}^{17} (y^{20}x^{32} + y^{32}x^{20}) +v_{2}^{19} (y^{16}x^{42} + y^{18}x^{40} + y^{40}x^{18} + y^{42}x^{16}) +v_{2}^{21} (y^{8}x^{56} + y^{20}x^{44} + y^{40}x^{24} + y^{56}x^{8} + y^{24}x^{40} + y^{44}x^{20}) +v_{2}^{23} (y^{16}x^{54} + y^{18}x^{52} + y^{20}x^{50} + y^{22}x^{48} + y^{48}x^{22} + y^{50}x^{20} + y^{52}x^{18} + y^{54}x^{16}) +v_{2}^{25} (y^{28}x^{48} + y^{16}x^{60} + y^{48}x^{28} + y^{60}x^{16}) +v_{2}^{27} (y^{24}x^{58} + y^{26}x^{56} + y^{56}x^{26} + y^{58}x^{24}) +v_{2}^{29} (y^{16}x^{72} + y^{28}x^{60} + y^{32}x^{56} + y^{40}x^{48} + y^{48}x^{40} + y^{60}x^{28} + y^{56}x^{32} + y^{72}x^{16})

+v_{2}^{31} (y^{32}x^{62} + y^{34}x^{60} + y^{38}x^{56} + y^{36}x^{58} + y^{40}x^{54} + y^{42}x^{52} + y^{44}x^{50} + y^{48}x^{46} +y^{46}x^{48} + y^{50}x^{44} + y^{52}x^{42} + y^{54}x^{40} + y^{56}x^{38} + y^{58}x^{36} + y^{60}x^{34} + y^{62}x^{32}) +...

p = 2, s = 3:

x + y + v_{3}x^{4}y^{4} +v_{3}^{5} (y^{20}x^{16} + y^{16}x^{20}) +v_{3}^{9} (y^{16}x^{48} + y^{48}x^{16}) +v_{3}^{21} (y^{80}x^{68} + y^{84}x^{64} + y^{64}x^{84} + y^{68}x^{80}) +v_{3}^{25} (y^{112}x^{64} + y^{64}x^{112}) +...

p = 2, s = 4:

x + y + v_{4}y^{8}x^{8} +v_{4}^{9} (y^{72}x^{64} + y^{64}x^{72}) +v_{4}^{17} (y^{64}x^{192} + y^{192}x^{64}) +v_{4}^{73} (y^{520}x^{576} + y^{512}x^{584} + y^{584}x^{512} + y^{576}x^{520}) +...

p = 2, s = 5:

x + y + v_{5}y^{16}x^{16} +v_{5}^{17} (y^{272}x^{256} + y^{256}x^{272}) +v_{5}^{33} (y^{768}x^{256} + y^{256}x^{768}) +...

p = 3, s = 2:

x + y + v_{2} (2 x^{6}y^{3} + 2 x^{3}y^{6}) +v_{2}^{4} (2 y^{12}x^{21} + y^{9}x^{24} + y^{15}x^{18} + y^{18}x^{15} + y^{24}x^{9} + 2 y^{21}x^{12}) +v_{2}^{7} (2 y^{21}x^{36} + 2 y^{18}x^{39} + 2 y^{36}x^{21} + y^{30}x^{27} + y^{27}x^{30} + 2 y^{39}x^{18}) +v_{2}^{10} (2 y^{45}x^{36} + 2 y^{36}x^{45} + 2 y^{18}x^{63} + 2 y^{9}x^{72} + 2 y^{72}x^{9} + 2 y^{63}x^{18}) +v_{2}^{13} (y^{48}x^{57} + 2 y^{27}x^{78} + y^{57}x^{48} + 2 y^{33}x^{72} + y^{30}x^{75} + 2 y^{60}x^{45} + y^{36}x^{69} +2 y^{39}x^{66} + 2 y^{66}x^{39} + 2 y^{45}x^{60} + y^{42}x^{63} + 2 y^{51}x^{54} + 2 y^{54}x^{51} + y^{75}x^{30} +2 y^{78}x^{27} + y^{63}x^{42} + y^{69}x^{36} + 2 y^{72}x^{33} +v_{2}^{16} (2 y^{90}x^{39} + 2 y^{93}x^{36} + 2 y^{39}x^{90} + 2 y^{36}x^{93} + y^{63}x^{66} + y^{66}x^{63}) +v_{2}^{19} (2 y^{36}x^{117} + y^{27}x^{126} + y^{81}x^{72} + y^{72}x^{81} + 2 y^{54}x^{99} + 2 y^{99}x^{54} + y^{126}x^{27} + 2 y^{117}x^{36}) +v_{2}^{22} (y^{69}x^{108} + y^{63}x^{114} + 2 y^{57}x^{120} + y^{60}x^{117} + 2 y^{66}x^{111} + y^{54}x^{123} + y^{123}x^{54} +2 y^{81}x^{96} + y^{108}x^{69} + 2 y^{96}x^{81} + 2 y^{111}x^{66} + y^{114}x^{63} + y^{117}x^{60} + 2 y^{120}x^{57} +y^{93}x^{84} + y^{84}x^{93} + 2 y^{87}x^{90} + 2 y^{90}x^{87}) +v_{2}^{25} (2 y^{57}x^{144} + 2 y^{54}x^{147} + 2 y^{75}x^{126} + 2 y^{72}x^{129} + y^{84}x^{117} + y^{81}x^{120} +y^{99}x^{102} + 2 y^{93}x^{108} + 2 y^{90}x^{111} + y^{102}x^{99} + 2 y^{111}x^{90} + 2 y^{108}x^{93} + y^{117}x^{84} +y^{120}x^{81} + 2 y^{129}x^{72} + 2 y^{126}x^{75} + 2 y^{144}x^{57} + 2 y^{147}x^{54}) +v_{2}^{28} (2 y^{54}x^{171} + 2 y^{90}x^{135} + 2 y^{135}x^{90} + 2 y^{171}x^{54}) +...

p = 3, s = 3:

x + y + v_{3} (2 x^{9}y^{18} + 2 x^{18}y^{9}) +v_{3}^{10} (y^{81}x^{180} + 2 y^{90}x^{171} + y^{99}x^{162} + y^{162}x^{99} + 2 y^{171}x^{90} + y^{180}x^{81}) +v_{3}^{19} (2 y^{171}x^{324} + 2 y^{162}x^{333} + y^{243}x^{252} + 2 y^{333}x^{162} + y^{252}x^{243} + 2 y^{324}x^{171}) +v_{3}^{28} (2 y^{567}x^{162} + 2 y^{162}x^{567} + 2 y^{648}x^{81} + 2 y^{81}x^{648} + 2 y^{324}x^{405} + 2 y^{405}x^{324}) +...

p = 7, s = 2:

x + y + v_{2} (4 y^{35}x^{14} + 6 y^{42}x^{7} + 4 y^{14}x^{35} + 2 y^{28}x^{21} + 2 y^{21}x^{28} + 6 y^{7}x^{42}) +v_{2}^{8} (4 y^{105}x^{280} + 3 y^{98}x^{287} + 3 y^{126}x^{259} + 4 y^{119}x^{266} + 2 y^{154}x^{231} + 3 y^{140}x^{245} +5 y^{147}x^{238} + 4 y^{133}x^{252} + y^{91}x^{294} + 2 y^{182}x^{203} + 5 y^{175}x^{210} + 2 y^{168}x^{217} +5 y^{161}x^{224} + 2 y^{217}x^{168} + 5 y^{210}x^{175} + 2 y^{203}x^{182} + 5 y^{196}x^{189} + 5 y^{189}x^{196} +4 y^{252}x^{133} + 3 y^{245}x^{140} + 5 y^{238}x^{147} + 2 y^{231}x^{154} + 5 y^{224}x^{161} +3 y^{273}x^{112} + 4 y^{266}x^{119} + 3 y^{259}x^{126} + 6 y^{315}x^{70} + y^{308}x^{77} + 6 y^{301}x^{84} +y^{294}x^{91} + 3 y^{287}x^{98} + 4 y^{280}x^{105} + y^{336}x^{49} + 6 y^{329}x^{56} + y^{322}x^{63} +y^{49}x^{336} + y^{63}x^{322} + 6 y^{56}x^{329} + 6 y^{70}x^{315} + y^{77}x^{308} + 6 y^{84}x^{301} + 3 y^{112}x^{273}) +v_{2}^{15} (4 y^{413}x^{308} + 6 y^{427}x^{294} + 2 y^{420}x^{301} + 2 y^{448}x^{273} + 6 y^{441}x^{280} + 4 y^{98}x^{623} +4 y^{462}x^{259} + 4 y^{455}x^{266} + 4 y^{308}x^{413} + 6 y^{490}x^{231} + 5 y^{119}x^{602} + 6 y^{476}x^{245} +4 y^{315}x^{406} + 6 y^{126}x^{595} + 2 y^{469}x^{252} + 2 y^{322}x^{399} + 2 y^{518}x^{203} + 4 y^{511}x^{210} +4 y^{504}x^{217} + 2 y^{497}x^{224} + 2 y^{553}x^{168} + y^{546}x^{175} + 3 y^{539}x^{182} + 6 y^{525}x^{196} +5 y^{609}x^{112} + 5 y^{602}x^{119} + 6 y^{595}x^{126} + 4 y^{588}x^{133} + 3 y^{574}x^{147} + 4 y^{133}x^{588} +y^{567}x^{154} + 2 y^{560}x^{161} + 3 y^{147}x^{574} + 4 y^{623}x^{98} + 6 y^{616}x^{105} +6 y^{329}x^{392} + y^{154}x^{567} + 2 y^{168}x^{553} + 2 y^{161}x^{560} + 3 y^{182}x^{539} + y^{175}x^{546} +6 y^{196}x^{525} + 4 y^{210}x^{511} + 2 y^{203}x^{518} + 4 y^{217}x^{504} + 6 y^{231}x^{490} + 2 y^{224}x^{497} +6 y^{392}x^{329} + 2 y^{252}x^{469} + 4 y^{259}x^{462} + 6 y^{245}x^{476} + 4 y^{406}x^{315} + 2 y^{399}x^{322} +2 y^{273}x^{448} + 2 y^{301}x^{420} + 4 y^{266}x^{455} + 6 y^{280}x^{441} + 6 y^{294}x^{427} + 6 y^{105}x^{616} +5 y^{112}x^{609} + ...

Examples of the polynomials A_{k}^{1} from the Theorem 1.4 Below are some A_{k}^{1}(z, Z) from Lemma 1.3, y = z^{p−1}, σ_{p} = Z

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} + x^{2}v_{2}y^{4} + 2 y 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 x^{3}v_{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 y^{7}v_{3}σ_{5}^{20} + y^{12}v_{3}σ_{5}^{16} + 4 y^{17}v_{3}σ_{5}^{12} + y^{22}v_{3}σ_{5}^{8} + 4 y^{27}v_{3}σ_{5}^{4} + x^{4}v_{3}y^{31} + 4 y

References

1. J.F. Adams, Infinite Loop Spaces., Annals of Mathematics Studies, Princeton University Press, Princeton, (1978).

2. M. F. Atiyah, Characters and cohomology of finite groups., Math. of the I.H.E.S 9 (1961), 23–64.

3. M. Bakuradze, S. Priddy, Transfer and complex oriented cohomology rings., to appear, (2002).

4. J. R. Hunton, The Morava K-theory of wreath products., Math. Proc. Camb. Phil. Soc. 107 (1990), 309–318.

5. D.G. Quillen, On the formal group laws of oriented and unoriented cobordism theory., Bull. Amer. Math. Soc., 75 (1969), 1293–1298.

6. D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres., Academic press (1986).

Razmadze Mathematical Institute, Tibilsi 380093, Republic of Georgia

Current address: Max-Planck-Institut Für Mathematik, Bonn

  E-mail address: bakuradz@mpim-bonn.mpg.de