TRANSFERRED CHERN CLASSES IN MORAVA K-THEORY

MALKHAZ BAKURADZE AND STEWART PRIDDY

Abstract. Let η be a complex n-plane bundle over the total space of an cyclic covering of prime index p. We show that for k ∈ {1, 2, ..., np}\{p, 2p, ...np} the k-th Chern class of the transferred bundle differs from a certain transferred class ω_{k} of η by a polynomial in the Chern classes c_{p}, ..., c_{np} of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in K(s)^{∗}B(Z/p × U(n)).

1. Statements

Let p be a prime and let π be the cyclic group of order p. For a given action of π on a space X consider the regular covering

ρ : Eπ × X → Eπ ×_{π} X.

Let us write for short X_{hπ} := Eπ ×_{π} X. For the permutation action of π 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)

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

f_{η^{n}} : X_{hπ} → BU(n)_{hπ}^{p}.

For the covering

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

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

ξ_{π}^{n} → BU(n)_{hπ}^{p},

i.e. 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.

Let K(s)^{∗}, s ≥ 1, be the s^{th} Morava K-theory at p. We 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.

Date: February, 2003.

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

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

The first author was supported by Max Planck Institute of Mathematics and CRDF grant GM1 2083.

The second author was partially supported by the NSF.

Definition 1.1. 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 naturality of the transfer 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 the pullback by map f_{η^{n}} of orbit spaces defined above.

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

Lemma 1.2. We can define a polynomial in n + 1 variables

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

uniquely by the equation in K(s)^{∗}B(π × U(n)):

C_{k} − v_{s}z^{p^{s}−1}p^{−1} ∑_{\begin{matrix}i_{1}+2i_{2}...+ni_{n}=k \\ i_{0}+i_{1}+...+i_{n}=p \end{matrix}} \binom{p}{i_{0}, i_{1}, ..., i_{n}}c_{1}^{i_{1}}...c_{n}^{i_{n}} = A_{k}^{(n)}(z^{p−1}, 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, and 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^{p−1}, 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 1.3. Let ρ : X → X/π be the regular cyclic covering of prime index p defined by a free action of π on X and T r^{∗} = T r_{ρ}^{∗} be the associated transfer homomorphism. Let η^{n} → X be a complex n-plane bundle, η_{π}^{n} → X/π the pnplane bundle defined by Atyiah transfer, and ψ → X/π be the complex line bundle associated with ρ. Then

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

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

Example 1.4. ω_{1} = c_{1}(η^{n}) hence if k = 1 we have

c_{1}(η_{π}^{n})) − T r^{∗}(c_{1}(η^{n})) = A_{1}^{(n)}(c_{1}^{p−1}(ψ), c_{p}(η_{π}^{n}), ..., c_{pn}(η_{π}^{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_{π}) − T r^{∗}(c_{1}(η)) = v_{s} ∑_{i=1}^{s−1} z^{p^{s}−p^{i}}c_{p}(η_{π})^{p^{i−1}}.

2. Proofs

For the proof of Lemma 1.2 we need the following property of formal group laws observed in [3].

Lemma 2.1. For the formal group law in Morava K-theory K(s)^{∗}, s > 1 we have mod x^{p^{2(s−1)}} (or modulo 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 [5]: 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 [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}}. □

Remark. We would like to thank Neil Strickland for suggesting an elegant proof of Lemma 1.2 completely within the context of subgroups of formal groups. This material is available from N.P.Strickland@sheffield.ac.uk. We have decided not to include it since it uses rather different techniques and is no shorter than our proof for which we now offer some motivation. The lemma says that modulo image of the transfer

T r^{∗} : K(s)^{∗}B(U(n)) → K(s)^{∗}B(π × U(n))

for the covering π × 1 : Eπ × BU(n) → Bπ × BU(n), the class C_{k} can be written as a polynomial in z^{p−1} and C_{p}, ..., C_{np}, that this expression vanishes when restricted to K(s)^{∗}BU(n), and that the indeterminacy is obtained by applying the transfer to p^{−1}C_{k} after restricting to K(s)^{∗}(BU(n)). This uses Fröbenious reciprocity and the fact that T r^{∗}(1) = [p](z)/z = v_{s}z^{p^{s}−1} in Morava K-theory.

Proof of Lemma 1.2. We begin by considering the case n = 1. Let σ_{i} be the i-th symmetric functions in p variables. Then

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

where x = c_{1}(ξ^{1}), z = c_{1}(θ) are Chern classes in K(s)^{∗}B(π×U(1)) = K(s)^{∗}[[z, x]]/(z^{p^{s}}) Consider the equation

C_{k} = − ∑_{0≤i≤p^{s}} λ_{ki}C_{p}^{i} + p^{−1}\binom{p}{k}x^{k}z^{p^{s}−1}; 1 ≤ k ≤ p − 1.

We want to prove that such λ_{ki} exist uniquely as elements in K(s)^{∗}[[z]]/(z^{p^{s}}) and to compute these elements as polynomials in z^{p−1}. Then A_{k}^{(1)}(z^{p−1}, C_{p}) = ∑_{0≤i≤p^{s}} λ_{ki}C_{p}^{i}.

C_{k} is the Chern class of the bundle ξ ⊗ (1 + θ + θ^{2} + ... + θ^{p−1}) and can be written as a series in the Chern classes of ξ, that is x, and the Chern classes of 1 + θ + θ^{2} + ... + θ^{p−1}. But the Chern classes of the latter bundle are elementary symmetric functions in z, 2z, ..., (p − 1)z, the Chern classes of θ, θ^{2}, ..., θ^{p−1} all of which vanish except (p − 1)-th class which is −z^{p−1}. Hence we can write the classes C_{k} as series in x and z^{p−1}. Lemma 2.1 enables us to write C_{k} as explicit polynomials in x and z^{p−1}. Now noting that C_{p} = x^{p} mod z^{p−1} we obtain from the above equation a system of linear equations in variables λ_{kj} by equating the coefficients at x^{i}, i ≥ 0. Vanishing also implies λ_{k0} = 0, k = 1, ..., p − 2 and λ_{p−10} = c_{p−1}(1 + θ + ... + θ^{p−1}) = −z^{p−1}.

Then equating the coefficients at x^{p},...,x^{p^{s+1}} in the above equation after rewriting it in terms of x and z^{p−1} as above, we have a system of p^{s} linear equations in p^{s} variables λ_{ki}, i = 1, ..., p^{s} . The determinant of this system is invertible since the diagonal coefficients are invertible and all other coefficients lie in the (nilpotent) augmentation ideal. Thus the elements λ_{ki} are uniquely defined.

Of course equating the coefficients at x^{i} for i ≠ p, 2p, ..., p^{s+1} will produce other equations in λ_{kj}, j = 1, ..., p^{s}. But these equations are derived from the old equations above. These additional equations make the matrix upper triangular. This defines A_{k}^{(1)}.

In the general n case we proceed analogously noting that C_{ip} = c_{i}^{p} mod z^{p−1}, i = 1, ..., n. Our additional claim again is that the A_{k}^{(n)} are polynomials, that is, only a finite number of elements λ_{k,i_{1},...,i_{n}} are nontrivial. Here we need the splitting principle and Lemma 2.1 to express explicitly the elements C_{i}, i = 1, ..., np in terms of polynomials in z^{p−1} and c_{1}, ..., c_{n}. For k ∈ {1, ..., np} \ {p, ..., np}, let

A_{k}^{(n)}(z^{p−1}, C_{p}, ..., C_{pn}) = ∑_{0≤i_{1},...,i_{n}≤p^{s}} λ_{k,i_{1},...,i_{n}}C_{p}^{i_{1}}...C_{np}^{i_{n}}.

We define λ_{k,0...0} = c_{k}(n + nθ + ... + nθ^{p−1}) again by looking at reductions to K(s)^{∗}(Bπ). The other n(p^{s}+1)−1 elements λ_{k,i_{1},...,i_{n}} can be defined as the solution of a system of n(p^{s} + 1) − 1 linear equations with an invertible determinant. This system is obtained from the equation after using Lemma 2.1 to rewrite it in terms of z^{p−1} and c_{1}, ...c_{n} and equating coefficients at c_{1}^{pi_{1}}...c_{n}^{pi_{n}}. The solution defines λ_{k,i_{1},...,i_{n}}, 0 ≤ i_{j} ≤ p^{s}. Again the additional equations in these elements arise from the coefficients of other monomials and are not new. The desired polynomials are thus uniquely defined. □

Proof of Theorem 1.3. 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 1.2. □

Note we can replace the cyclic group by the symmetric group Σ_{p} and use the polynomials A_{k}^{(n)} to evaluate the disparity or “gap” between Chern class c_{k}(ξ_{Σ_{p}}^{n}) and ImT r_{Σ_{p}}^{∗}, for ρ_{Σ_{p}} : EΣ_{p} × U(n)^{p} → BU(n)_{hΣ_{p}}^{p}.

Namely the Euler characteristic of the coset space Σ_{p} ≀ U(n)/π ≀ U(n) = (p − 1)! is prime to p. Hence the inclusion ρ_{π,Σ_{p}} : π → Σ_{p} induces a monomorphism

K(s)^{∗}(B(Σ_{p} ≀ U(n))) → K(s)^{∗}(B(π ≀ U(n))).

Hence Hunton’s result above holds for BU(n)_{hΣ_{p}}^{p}.

Now let ς_{k}^{(n)} ∈ F be defined modulo kerN_{Σ_{p}} by N_{Σ_{p}}(ς_{k}^{(n)}) = c_{k}(N_{π}(ξ^{n})). Again T r_{Σ_{p}}^{∗}(ς_{k}^{(n)}) is well-defined: as above KerN_{π} = Im(1 − t^{∗}) therefore

a ∈ F ⋂ kerN_{Σ_{p}} ⇒ ∑_{g∈Σ_{p}/π} g^{∗}a ∈ Im(1 − t^{∗}) ⇒ T r_{Σ_{p}}^{∗}( ∑_{g∈Σ_{p}/π} g^{∗}a) = 0

⇒ (p − 1)!T r_{Σ_{p}}^{∗}(a) = 0 ⇒ T r_{Σ_{p}}^{∗}(a) = 0.

Then for the np-plane bundle over BU(n)_{hΣ_{p}}^{p}

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

the difference

c_{k}(ξ_{Σ_{p}}^{n}) − T r_{Σ_{p}}^{∗}(ς_{k}^{(n)})

belongs to kerρ_{Σ_{p}}^{∗} and hence is detected by the polynomials

A_{k}^{(n)}(y, c_{p}(ξ_{Σ_{p}}^{n}), ..., c_{np}(ξ_{Σ_{p}}^{n})),

where

y ∈ K(s)^{∗}(B(Σ_{p})) = K(s)^{∗}[[y]]/(y^{m_{s}})

|y| = 2(p − 1), m_{s} = [(p^{s} − 1/(p − 1))] + 1.

Thus we have:

Theorem 2.2.

c_{k}(ξ_{Σ_{p}}^{n}) − T r_{Σ_{p}}^{∗}(ς_{k}^{(n)}) = A_{k}^{(n)}(y, c_{p}(ξ_{Σ_{p}}^{n}), ..., c_{n}p(ξ_{Σ_{p}}^{n})),

for k ∈ {1, ..., np} \ {p, ..., np}.

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, (2003).

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

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

Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia Current address: Max-Planck-Institut Für Mathematik , Bonn

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

Northwestern University, Evanston, IL 60208, USA

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