����������� ������ 53 (2001), 1–13

UDK 515.143.5

���������� ������ ��� research paper

THE GENERALIZED COHOMOLOGY THEORIES, BRUMFIEL-MADSEN FORMULA AND TOPOLOGICAL CONSTRUCTION OF BGG-TYPE OPERATORS

Cenap Özel

Abstract. In this work, we investigate the topological construction of BGG-type operators, giving details about complex orientable theories, Becker-Gottlieb transfer and a formula of Brumfiel-Madsen. We generalize the BGG operators on the Morava K-theory and the others 𝔽_{p}-generalized cohomology theories.

1. Introduction

In this work, we will discuss the generalized complex-oriented cohomology and homology theories of the flag space G/B, and the classical BGG and Kac operators will be constructed topologically using the transfer map for compact fiber bundles.

In order to do this, we will give some topological notations.

In the third section, we will discuss the Becker-Gottlieb map and transfer map for a fiber bundle π: E → B with the fiber F, which is a compact differentiable G-manifold for a compact Lie group G.

In the fourth section, we will examine the Brumfiel-Madsen formula for the transfer map.

In the last section, we will give the results of this work.

2. Topological preliminaries

The reference for this section is [1].

2.1. Generalities on generalized cohomology. A generalized cohomology theory h^{∗}( ) is a contravariant functor from topological spaces to graded abelian groups which satisfies all the Eilenberg-Steenrod axioms except the dimension axiom. That is, the coefficients h^{∗} = h^{∗}(pt) need not be concentrated in a single degree. We will always assume that h^{∗} is multiplicative, and that the associated ring structure is commutative in the graded sense. Then for a topological space X, h^{∗}(X) is a h^{∗}-module. The first example is ordinary cohomology with coefficients in ℤ.

AMS Subject Classification: 55 R 10, 55 R 12

Keywords and phrases: Generalized cohomology theory, Bernstein-Gelfand-Gelfand operators.

We take H^{i}(X) = H^{i}(X, ℤ) = [X, K(ℤ, i)], where K(ℤ, i) is an Eilenberg-Maclane space, and [X, Y] denotes homotopy classes of based maps from X to Y for X and Y topological spaces with based points.

For a generalized theory h^{∗}( ), there is a spectral sequence which computes h^{∗}(X) in terms of H^{∗}(X; h^{∗}). This spectral sequence is called the Atiyah-Hirzebruch spectral sequence, and details can be found in [1].

Theorem 2.1. There is a spectral sequence with E_{2} term H^{p}(X, h^{q}(pt)) ⟹ h^{p+q}(X). The differential d_{r} is of bi-degree (r, 1 − r).

Corollary 2.2. Suppose that X has no odd dimensional cells and h^{q}(pt) = 0 for q odd. Then the Atiyah-Hirzebruch spectral sequence collapses at the E_{2} term.

Now we define reduced cohomology. Let i: pt → X be the inclusion of a point and π: X → pt be the collapsing map. Then π ∘ i = id, so i^{∗} ∘ π^{∗} = id on h^{∗}(pt). Let h̃^{∗}(X) = ker i^{∗} be the reduced cohomology of X. Then, as a h^{∗}-module,

h^{∗}(X) = h̃^{∗}(X) ⊕ h^{∗}.

2.2. Classifying spaces. In this section, we give some facts about the construction of universal bundles and classifying spaces of groups. The general reference for this section is [12]. Let G be a compact Lie group. There is a universal space EG with a free right G-action and homotopy groups π_{i}(EG) = 0 for all i > 0. Moreover, EG is a limit of Stiefel manifolds with the inductive limit topology. For example, for G = U(n), the unitary group,

EU(n) = lim_{m→∞} V_{n}(ℂ^{n+m}),

where V_{n}(ℂ^{n+m}) = \frac{U(n+m)}{U(m)} is a Stiefel manifold. The classifying space BG is defined as EG/G. For G = U(n),

BG ≅ lim_{m→∞} G_{n}(ℂ^{n+m}),

the Grassmannian manifold of n-planes.

We have the universal bundle (EG, p, BG) where EG →^{p} BG is the obvious projection map. Then BG has the following universal property.

Theorem 2.3. Let P →^{ρ} B be a right principal G-bundle. Then there exists a unique (up to homotopy) classifying map f: B → BG such that f^{∗}(EG) ≅ P as principal G-bundles over B.

As a consequence,

Corollary 2.4. BG is well-defined up to homotopy and classifies induced vector bundles.

Let P →^{ρ} B be a right principal G-bundle. Then, if F is a finite dimensional representation of G, E = P ×_{G} F is the associated vector bundle over B with structure group G, where

E = P ×_{G} F = P × F/ ∼

is the space obtained as the quotient of the product space P × F by the relation

(x, y) ∼ (xt, t^{−1}y), t ∈ G, x ∈ P, y ∈ F.

Theorem 2.5. Let E → B be a vector bundle associated to the fibre F with structure group G. Then there exists f: B → BG with f^{∗}(EG×_{G} F) ≅ E as vector bundles over B.

Consider the special case of the classifying space for a complex line bundle. The appropriate structure group is U(1), so the appropriate classifying space is BU(1). By the above construction,

BU(1) = lim_{m→∞} ℂP^{m} = ℂP^{∞}.

We know from [12] that H^{∗}(BU(1), ℤ) = ℤ[x], where ℤ[x] is the graded ring of polynomials in one variable with coefficients in ℤ and deg x = 2. Let T = ∏_{i=1}^{l} U(1) be a torus. Then,

BT = ∏_{i=1}^{l} BU(1),

and since H^{∗}(BU(1), ℤ) is torsion-free, by the Künneth formula, we have

H^{∗}(BT, ℤ) ≅ ⨂_{i=1}^{l} H^{∗}(BU(1), ℤ) ≅ ℤ[x_{1},…, x_{l}],

where ℤ[x_{1},…, x_{l}] is the graded ring of polynomials in l variables with coefficients in the ring ℤ.

2.3. Complex orientable cohomology theories. We follow [1] in this discussion.

Let i: ℂP^{1} → ℂP^{∞} = BU(1) be the inclusion.

Definition 2.6. We say that the multiplicative cohomology theory h^{∗} is complex oriented if there exists a class x ∈ h̃^{∗}(ℂP^{∞}) such that i^{∗}(x) is a generator of h̃^{∗}(ℂP^{1}) over the ring h^{∗}(pt). Such a class x is called a complex orientation.

h̃^{∗}(ℂP^{1}) ≅ h̃^{∗}(𝕊^{2}) is generated by one element over h^{∗}(pt).

As an example, if h^{∗} = H^{∗}, then x can be taken as a ring generator of H^{∗}(ℂP^{∞}, ℤ), so x ∈ H^{2}(ℂP^{∞}, ℤ). ℂP^{∞} has a universal line bundle L_{λ} given as follows. Let e^{λ} be the one-dimensional representation of 𝕋 = U(1) given by

e^{λ}(e^{iθ}) · v = e^{iθ} ·v,

where λ ∈ Lie(T) is a fundamental weight. Then, for a complex orientable theory h^{∗} with orientation given by x, the first Chern class is given by x = c_{1}(L_{λ}), where L_{λ} is the line bundle associated to e^{λ}. Let T be an l-dimensional torus.

Theorem 2.7. [1] With the above notation, we have isomorphisms of graded h^{∗}-algebras

h^{∗}(ℂP^{∞}) ≅ h^{∗}(pt)[[x]],
h^{∗}(BT) ≅ h^{∗}(pt)[[x_{1},…, x_{l}]],
h^{∗}(ℂP^{n}) ≅ h^{∗}(pt)[[x]]/(x^{n+1}),
h^{∗}( ∏_{i=1}^{l} ℂP^{n_{i}}) ≅ h^{∗}(BT)/(x_{1}^{n_{1}+1},…, x_{l}^{n_{l}+1}).

Now let π: L → X be a line bundle over X. Then L induces a classifying map θ: X → ℂP^{∞}. Then the first Chern class of L is c_{1}(L) = θ^{∗}(x). Next we define the top Chern class of a vector bundle.

Definition 2.8. Let π: E → X be a vector bundle. If there is a space Y and a map f: Y → X such that f^{∗}: h^{∗}(X) → h^{∗}(Y) is injective and f^{∗}(E) ≅ ⨁ L_{i}, where L_{i} are line bundles on Y, f is called a splitting map for π.

From [12],

Theorem 2.9. If π: E → X is a vector bundle, there exists a splitting map of π.

Then,

Definition 2.10. The top Chern class c_{n}(E) where dim E = n, which also will be referred as the Euler class χ(E), is defined by the formula

f^{∗}(c_{n}(E)) = ∏_{i} c_{1}(L_{i}),

where f is a splitting map for π.

2.4. Formal group laws. Let 𝔽 be a commutative ring with unit.

Definition 2.11. A formal group law over 𝔽 is a power series F (x, y) over 𝔽 that satisfies the following conditions:

1. F (x, 0) = F (0, x) = x,

2. F (x, y) = F (y, x),

3. F (F (x, y), z) = F (x, F (y, z)),

4. there exists a series i(x) such that F (x, i(x)) = 0.

From [17], we have

Theorem 2.12. In a complex oriented theory, for line bundles L, M, we have

c_{1}(L ⊗ M) = F (c_{1}(L), c_{1}(M))

where F is a formal group law over the coefficient ring h^{∗}.

A line bundle L over a space X is equivalent to a homotopy class of maps f_{L}: X → ℂP^{∞}. Let L and M be two line bundles. Then we have

f_{L} × f_{M}: X → ℂP^{∞} × ℂP^{∞}.

ℂP^{∞} has an H-space structure m: ℂP^{∞} × ℂP^{∞} → ℂP^{∞}. The homotopy class of m ∘ (f_{L} × f_{M}) is then equivalent to the tensor product L ⊗ M. There is an induced map m^{∗}: h^{∗}(ℂP^{∞}) → h^{∗}(ℂP^{∞} × ℂP^{∞}). Since, h^{∗}(ℂP^{∞}) ≅ h^{∗}(pt)[[x]] and h^{∗}(ℂP^{∞} × ℂP^{∞}) ≅ h^{∗}(pt)[[x_{1}, x_{2}]], m^{∗} has the form,

m^{∗}(x) = ∑ a_{ij}x_{1}^{i}x_{2}^{j} = F (x_{1}, x_{2}).

Then c_{1}(L ⊗ M) = F (c_{1}(L), c_{1}(M)). As an example, if L and M are line bundles, we see in ordinary cohomology H^{∗}( ) that

c_{1}(L ⊗ M) = c_{1}(L) + c_{1}(M).

The complex cobordism MU^{∗} is the universal cohomology theory with respect to push-forwards. From [1],

Theorem 2.13. The complex cobordism ring MU^{∗} is ℤ[x_{1}, x_{2},…] where dim x_{i} = 2i. The formal group law of MU^{∗} is the Lazard’s universal formal group law.

3. The Becker-Gottlieb map and transfer.

The general reference for this section is [2].

Let π: E → B be a fiber bundle with the fiber F, which is a compact differentiable G-manifold for a compact Lie group G. For any cohomology theory h^{∗} we have the induced map π^{∗}: h^{∗}(B) → h^{∗}(E). A transfer map is a backwards map h^{∗}(E) → h^{∗}(B). Here, we will give a technique for producing a transfer map.

Definition 3.1. Let ξ → B be a vector bundle. Let D(ξ) = {x ∈ ξ : |x| ⩽ 1} and S(ξ) = {x ∈ ξ : |x| = 1} be the disk and sphere bundles respectively. Then, Bξ = D(ξ)/S(ξ) is called the Thom space of the vector bundle ξ.

Now we give the useful propositions from [12],

Proposition 3.2. If ξ → B is a trivial n dimensional vector bundle, then the Thom space Bξ = Σ^{n}B^{+}, where B^{+} is the union of B with a point.

Proposition 3.3. If ξ and η are two vector bundles over B, then Bξ ∧ Bη = B(ξ ⊕ η).

We define transfer for the map from the fiber F to a point. We can embed F equivariantly into a real G-representation V of dimension r such that r ≫ dim F. Let N → F be the normal bundle of the embedding. By the tubular neighbourhood theorem, we can identify the normal bundle N with a neighbourhood U of F by a diffeomorphism φ. There is an associated Pontryagin-Thom collapsing map c: S_{V} → F_{N}, where S_{V} is the one point compactification of V, defined by

c(x) = \{ \begin{matrix}base point of F^{N} & if x ∉ U, \\ φ(x) & if x ∈ U. \end{matrix}

Let T (F) be the tangent bundle of F. Then we can identify T (F) ⊕ N with the trivial bundle F × V. There is an inclusion i: N → N ⊕ T (F) ≅ F × V and hence we have an inclusion of Thom spaces i: F_{N} → S_{V} ∧ F^{+}.

Definition 3.4. The transfer τ to a point is the composition τ = i ∘ c.

Let π: E → B be a fiber bundle associated to the principle G-bundle p: P → B. Then the transfer to a point gives a map

Id ×τ: P ×_{G} S_{V} → P ×_{G} (F × V)^{+}.

When we collapse the section at ∞ to a point, which is equivalent to taking Thom spaces, we get a map t: Bξ → Bπ^{∗}(ξ) where ξ is a vector bundle associated to the representation V. Then there is a map t ∧ Id: Bξ ∧ B\overline{ξ} → Bπ^{∗}(ξ) ∧ B\overline{ξ} where \overline{ξ} is the complementary bundle of of ξ. If we restrict to the diagonal Δ in B × B, we have transfer map

τ(π): Σ^{m}B^{+} → Σ^{m}E^{+}.

4. The Brumfiel-Madsen formula for transfer

The general reference for this section is [6].

Let G be a compact connected semi-simple Lie group with maximal torus T. Let H be a closed connected subgroup of G containing the maximal torus T. Let W_{G} and W_{H} be the Weyl groups of G and H respectively. Suppose that P → B is a principal G-bundle. We have associated bundles

π_{1} : E_{1} = P ×_{G} G/T → B
π_{2} : E_{2} = P ×_{G} G/H → B.

Then there is a fibration π: E_{1} → E_{2} with the fiber H/T. Since the Weyl group W_{G} acts on G/T, W_{G} also acts on E_{1}. The Weyl group W_{H} of H also acts on E_{1} over E_{2}. Thus, cosets w ∈ W_{G}/W_{H} define maps π ∘ w on E_{1}.

Theorem 4.1. We have

π_{1}^{∗} ∘ τ(π_{2})^{∗} = ∑_{w∈W_{G}/W_{H}} w ∘ π^{∗}.

Corollary 4.2. If we choose H = T, we get

π_{1}^{∗} ∘ τ(π_{1})^{∗} = ∑_{w∈W_{G}} w.

Although Brumfiel and Madsen were the first to assert that Theorem 4.1 is true, their proof was wrong. In [10] Feshbach, and in [15] Lewis, May and Steinberger have given different proofs of Theorem 4.1. Since EG is the universal space for G, we have the principle bundle EG → BG.

Corollary 4.3. Let BT → BG be the fiber bundle with the fiber G/T. Then

π^{∗} ∘ τ(π)^{∗} = ∑_{w∈W_{G}} w.

For a compact semi-simple Lie group G, any root α defines a subgroup M_{α} = K_{α} · T such that the complexified Lie algebra m_{α} contains the root spaces g_{α} and g_{−α} where K_{α} already has been defined in [13]. The induced fiber bundle π_{i}: BT → BM_{i} has fiber M_{i}/T ≅ SU_{2}/T ≅ ℂP^{1}. Then

Corollary 4.4.

π_{i}^{∗} ∘ τ(π_{i})^{∗} = 1 + r_{α_{i}},

if r_{α_{i}} is the reflection to corresponding to the simple root α_{i}.

5. The transfer and the Gysin homomorphism

Let ξ: E → X be a vector bundle and h^{∗} be the complex oriented theory. Then there is the associated Thom class u ∈ h^{∗}(Bξ). From [4], we have

Theorem 5.1. The Thom map Φ: h^{∗}(X) → h^{∗}(Xξ) given by Φ(x) = u·π^{∗}(x) is an isomorphism.

Let π: E → B be a fiber bundle with compact smooth f-dimensional fiber F. Suppose that the tangent bundle T F → F is a complex vector bundle. Then we have the Gysin homomorphism π_{∗}: h^{k}(E) → h^{k−f}(B). Since the tangent bundle T (F) has complex structure, so does the tangent space along the fibers T_{π}. Hence, in the complex orientable theory h^{∗}, T_{π} has an Euler class, so χ(T_{π}) = c_{n}(T_{π}).

Theorem 5.2. (see [2]) The transfer τ(π)^{∗}: h^{k}(E) → h^{k}(B) is given by τ(π)^{∗}(x) = π_{∗}(x · χ(T_{π})).

Let L_{α} be the line bundle on BT associated to the character e^{α} where α is a root. We want to determine when its characteristic classes are not zero divisors. We know that the characters e^{α} do not usually generate the representation ring R(T). Let λ_{i} be the fundamental weight corresponding to the simple root α_{i} such that λ_{i}(h_{α_{i}}) = 1, where h_{α_{i}} is the coroot. Then

Theorem 5.3. [12] These e^{λ_{i}} generate the representation ring R(T).

By Theorem 2.7,

h^{∗}(BT) ≅ h^{∗}(pt)[[c_{1}(L_{λ_{1}}), ⋯, c_{1}(L_{λ_{l}})]]

where l is the rank of the compact Lie group G. Since χ(L_{λ_{i}}) are generators of h^{∗}(BT), the χ(L_{λ_{i}}) are not zero-divisors in h^{∗}(BT). This implies that χ(L_{λ_{i}}) is not nilpotent. We know that for any weight λ ∈ h^{∗}, λ can be written as λ = ∑_{i=1}^{l} n_{i}λ_{i}, where n_{i} is the multiplicity number. Using the formal group law in h^{∗}, the Euler class χ(L_{λ}) of the line bundle L_{λ} in h^{∗} is equal to

∑_{i=1}^{l} n_{i}c_{1}(L_{λ_{i}}) + higher order terms.

If n_{i} is not a zero-divisor in h^{∗}(pt), then χ(L_{λ}) is not a zero-divisor in h^{∗}(BT). If the weight λ is a root corresponding to the adjoint representation, the multiplicity numbers n_{i} in the sum are the Cartan integers. By an examination of the Cartan matrices, we have

Proposition 5.4. If p ⩾ 3 is a prime, there is some n_{i} such that p does not divide n_{i}.

Corollary 5.5. If h^{∗}(pt) has no 2-torsion and 3-torsion, then the Euler class χ(L_{α_{i}}) is not a zero-divisor for any simple root α_{i}.

Since every root is the image of a simple root by an element of the Weyl group W_{G} and the Weyl group acts by automorphism on h^{∗}(BT), we have

Corollary 5.6. If h^{∗}(pt) has no 2-torsion and 3-torsion, then the Euler class χ(L_{α}) is not a zero-divisor for any root α.

Now we want to give the Brumfiel-Madsen formula for the Gysin map of the fibration π: BT → BG with the fiber G/T.

We need a complex structure on G/T. We know that the smooth manifold G/T is diffeomorphic to the complexified space G_{ℂ}/B where B is a Borel group. Then we can determine the tangent bundle of the fiber G_{ℂ}/B. The tangent bundle T (G_{ℂ}/B) is isomorphic to G×_{T} g/b,where mathbfg is the complexified Lie algebra of G and b is the Borel subalgebra of g. Using the adjoint representation of T, we have

g = b ⊕ ⨁_{α∈Δ^{+}} g_{−α},

where Δ^{+} is the set of positive roots corresponding to B. Thus

g/b = ⨁_{α∈Δ^{+}} g_{−α}.

Therefore the tangent bundle along the fiber G/T is

T_{π} = EG ×_{T} g/b ≅ ⨁_{α∈Δ^{+}} L_{−α},

where L_{−α} is as above. We know that

χ^{n}(T_{π}) = ∏_{α∈Δ^{+}} c_{1}(L_{−α}),

where ∏ is the cup product in any complex orientable theory h^{∗}. By Theorem 5.2, we have

π^{∗} ∘ τ(π)^{∗}(x) = π^{∗} ∘ π_{∗}(x · χ(T_{π}))

for x ∈ h^{∗}(BT). Since χ(T_{π}) is a product of the non-zero divisors in h^{∗}(BT), we have

Theorem 5.7. (see [4]) For x ∈ h^{∗}(BT),

π^{∗} ∘ π_{∗}(x) = ∑_{w∈W} w (\frac{x}{∏ χ(L_{−α})}) ,

here the right hand side is in a localization h^{∗}(BT)[\frac{1}{∏ χ(L_{−α})}].

But since the left hand side preserves the subring h^{∗}(BT), it may be regarded as an identity on h^{∗}(BT).

Corollary 5.8. If χ(L_{−α}) is a non-zero divisor, for the fibration π_{i}: BT → BM_{i} with the fiber M_{i}/T,

D_{i}(x) = π_{i}^{∗} ∘ π_{i}_{∗}(x) = (1 + r_{i}) (\frac{x}{χ(L_{−α})}) .

Let h^{∗} be the ordinary cohomology with complex coefficients. From chapter 1 of [16], we know that there is an isomorphism Θ: h^{∗} → H^{2}(BT, ℂ) given by λ → χ(L_{λ}), where h^{∗} is the dual Cartan subalgebra of semi-simple Lie algebra.

The isomorphism Θ extends to an inclusion of the symmetric algebra R = S(h^{∗}) into H^{∗}(BT, C). Then

H^{∗}(BT, ℂ) ≅ ℂ[λ_{1},…, λ_{l}]

under the identification χ(L_{λ_{i}}) = λ_{i}.

In [3], Bernstein-Gelfand-Gelfand introduced certain operators

\frac{1}{α}(r_{i} − 1): H^{k}(BT) → H^{k−2}(BT)

where r_{i} is the simple reflection associated to the simple root α_{i}.

When G = M_{i},

Corollary 5.9.

D_{i} = \frac{1}{α}(r_{i} − 1)

is just the classical BGG operator.

If we apply Theorem 5.7 to K-theory, for G = M_{i}, the formula D_{i} = π_{i}^{∗} ∘ π_{i}_{∗} in K-theory gives the Demazure operator. We map the representation ring R(T) to K(BT) by mapping e^{λ} to [L(λ)], the class of the line bundle defined by λ. In K-theory, we can take χL(λ) = [1]−[L(λ)], [1] is the class of the trivial line bundle. In the case where G = M_{i} is rank one, D_{i} is the Demazure operator.It has the form

D_{i} = \frac{1}{1 − e^{−α_{i}}} (1 − e^{−α_{i}} r_{i}).

Now, we will apply this result to BP -theory and Morava K-theory. In order to do this, we will give some definitions. The BP -theory began with Brown and Peterson [5] who showed that after localization at any prime p, the MU spectrum splits into an infinite wedge suspension of identical smaller spectra subsequently called BP. We have

BP^{∗} = ℤ_{p}[v_{1}, v_{2},…,]

where ℤ_{p} denotes the integers localized at p and dim v_{n} = 2(p^{n} − 1).

Definition 5.10. A formal group law over a torsion free 𝔽_{p}-algebra is p-typical if its logarithm has the form ∑_{i⩾0} λ_{i}x^{p^{i}} with λ_{0} = 1.

This definition works even when the 𝔽_{p}-algebra R has torsion. A formal group law F over a 𝔽_{p}-algebra is p-typical if f_{q}(x) = 0 for all primes q ≠ p.

By a Cartier Theorem [7], we have

Theorem 5.11. Every formal group law over a 𝔽_{p}-algebra is canonically strictly isomorphic to a p-typical one.

Let F be a formal group law over commutative ring with unit R.

Definition 5.12. For each n, the n-series [n](x) of F is given by

[1](x) = x,
[n](x) = F (x, [n − 1](x)) for n > 1,
[−n](x) = i([n](x)).

Of particular interest is the p-series, where p is a prime. In characteristic p it always has leading term ax^{q} where q = p^{h} for some integer h. This leads to the following.

Definition 5.13. Let F (x, y) be a formal group law over an 𝔽_{p}-algebra. If [p](x) has the form

[p](x) = ax^{p^{h}} + higher terms

with a invertible, then we say that F has height h at p. If [p](x) = 0 then the height is infinity.

Now we give the construction the universal p-typical formal group law.

Theorem 5.14. Let V = 𝔽_{p}[v_{1}, v_{2},…,] with dim v_{n} = 2(p^{n} − 1). Then there is a universal p-typical formal group law F over V.

Theorem 5.15. The height of a formal group law is an isomorphism invariant.

Suppose that h^{∗} is an 𝔽_{p}-algebra and the formal group law F has the height h. Since the elements x = χ(L_{λ_{i}}) ∈ h^{∗}(BT) are non-zero divisors, [p](x) has the form

[p](x) = ax^{p^{h}} + higher terms, (a is a unit.)

This leads us to mod p K-theory and the Morava K-theories. The reference for these cohomology theories is [17]. We give some imporant special cases of Corollaries 6 and 7.

Theorem 5.16. For any prime p, in K(n)^{∗}(BT), the Euler class χ(L_{α_{i}}) is not a zero divisor for any simple root α_{i}.

Theorem 5.17. For any prime p, in K(n)^{∗}(BT), the Euler class χ(L_{α}) is not a zero divisor for any root α.

Let π: BT → BG is a fiber bundle with the fiber G/T.

Theorem 5.18. For x ∈ K(n)^{∗}(BT),

π^{∗} ∘ π_{∗}(x) = ∑_{w∈W} w (\frac{x}{∏ χ(L_{−α})}) ,

here the right hand side is in a localization K(n)^{∗}(BT)[\frac{1}{∏ χ(L_{−α})}].

Corollary 5.19. Let π_{i}: BT → BM_{i} be a fiber bundle with the fiber M_{i}/T. For x ∈ K(n)^{∗}(BT),

D_{i}(x) = π_{i}^{∗} ∘ π_{i}_{∗}(x) = (1 + r_{i}) (\frac{x}{χ(L_{−α})}) .

Of course, these results can be generalized to a 𝔽_{p}-algebra h^{∗} which has a formal group law F with the height n.

In this section, so far we have concentrated our attention on BT. Now, we will give some interesting results about the flag variety G/T. Since the cohomology of G/T vanishes in odd degrees, Corollary 1 gives

Corrolay 5.20. Let h^{∗} be any complex oriented cohomology theory. Then the Atiyah-Hirzebruch spectral sequence for G/T collapses at the E_{2}-term.

Let π_{i}: BT → BM_{i}. Since G/T is a T -principal bundle, there is a classifying map θ: G/T → BT. Similarly there is a classifying map θ_{i}: G/M_{i} → BM_{i}. The following diagram is a cartesian square.

G/T ⟶^{θ} BT
↓^{p_{i}}  ↓^{π_{i}}
G/M_{i} ⟶^{θ_{i}} BM_{i}

Let C_{i} = p_{i}^{∗} ∘p_{i}_{∗}. Then θ^{∗} ∘D_{i} = C_{i} ∘θ^{∗}. The following theorem gives a topological description of the operator C_{i}. From [9],

Theorem 5.21. If h^{∗}(pt) contains ℤ[\frac{1}{|W_{G}|}], then θ^{∗} is surjective.

Definition 5.22. For i = 1,…, l, let D_{i} be the linear operator associated to the simple root α_{i}. Then we say that D_{i} satisfy braid relations if

(D_{i}D_{j}D_{i})^{m_{ij}} = (D_{j}D_{i}D_{j})^{m_{ij}},

where m_{ij} is the number of factors in each side for all pairs i and j.

For any Kac-Moody group, since T and M_{i} are still finite dimensional compact groups, we can define D_{i} operators on h^{∗}(BT).

Theorem 5.23. Let G be a Kac-Moody group and let h^{∗} be torsion-free. Then the operators D_{i} satisfy braid relations if and only if the formal group law is polynomial.

Proof. There are three cases to consider. These cases are when two nonorthogonal roots α_{i} and α_{j} have m_{ij} = 3, 4, or 6. In the finite dimensional case, the reference for case m_{ij} = 3 is [4] and for the remaining two cases the reference is [11]. In the affine case, it can be done similar way.

This theorem tells that the ordinary cohomology and K-theory satisfy braid relations but cobordism and elliptic cohomology and another complex oriented cohomology theories do not satisfy.

By Theorems 5.21 and 5.23, we have

Theorem 5.24. The operators C_{i} satisfy braid relations for ordinary cohomology and K-theory.

Now we will give our result about the infinite dimensional flag variety. Let G be an affine Kac-Moody group and K be the unitary form of G. For every simple root α_{i}, let M_{i} = K_{i} · T. We have a principal M_{i}-bundle K → K/M_{i}, and the associated fiber bundle K/T → K/M_{i} with fiber M_{i}/T. M_{i}/T is diffeomorphic to the complex projective space ℂP^{1}.

Theorem 5.25. Let π_{i}: K/T → K/M_{i} be the fiber bundle with the compact fiber ℂP^{1} and let 𝔽 be a commutative ring with unit. For x ∈ H^{∗}(K/T, 𝔽),

O_{i}(x) = π_{i}^{∗} ∘ π_{i}_{∗}(x) = −(1 + r_{i}) (\frac{x}{ε^{r_{i}}} ) ,

here the right hand side is in the localization H^{∗}(BT)[\frac{1}{∏ χ(L_{−α})}]. In fact O_{i} is the Kac operator which was introduced in [13].

Proof. By the Brumfiel-Madsen formula and Theorem [10], we have the following identity

π_{i}^{∗} ∘ τ(π_{i})^{∗}(x) = π_{i}^{∗} ∘ π_{i}_{∗}(ψ(−χ_{i}) · x) = (1 + r_{i})(x),

where r_{i} is the simple reflection associated to α_{i} and χ_{i} is the fundamental weight corresponding to the simple root α_{i}. Let x ∈ H^{∗}(K/T, 𝔽). We know from [14] that ψ(χ_{i}) = ε^{r_{i}} where ψ: S(h^{∗}) → H^{∗}(K/T, 𝔽). In H^{∗}(K/T, 𝔽), we know that the element ε^{r_{i}} is a non zero-divisor, so we can define the local ring H^{∗}(K/T, 𝔽)[\frac{1}{ε^{r_{i}}}].

Then, we have the following identity in the local ring H^{∗}(K/T, 𝔽)[\frac{1}{ε^{r_{i}}}],

π_{i}^{∗} ∘ π_{i}_{∗}(x) = −(1 + r_{i}) (\frac{x}{ε^{r_{i}}} ) .

Since the left hand side of the identity is an element of H^{∗}(K/T, 𝔽), we are done.

We know from [13] that the Kac operators satisfy braid relations for all affine Kac-Moody groups.

REFERENCES

[1] J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 1974.

[2] J. C. Becker, D. H. Gottlieb, The transfer map and fiber bundles, Topology 14 (1975), 1–12.

[3] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys 28 (1973), 1–26.

[4] P. Bressler, S. Evens, The Schubert calculus, braid relations, and generalized cohomology, Trans. Amer. Math. Soc. 317 (1990), 799–811.

[5] E. H. Brown, F. P. Peterson, A spectrum whose ℤ_{p} cohomology is the algebra of reduced p-th powers, Topology 5 (1966), 149–154.

[6] G. Brumfiel, I. Madsen, Evaluation of the transfer and universal surgery classes, Invent. Math. 32 (1976), 133–169.

[7] P. Cartier, Modules associes a un groupe formel commutatif, Courbes typiques, C. R. Acad. Sci. Paris Ser. A 265 (1967), 129–132.

[8] A. Dold, The fixed point transfer for fiber preserving maps, Math. Zeit. 148 (1976), 215–244.

[9] S. Evens, The transfer for compact Lie groups, induced representations, and braid relations, Thesis MIT, 1988.

[10] M. Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251 (1979), 139–169.

[11] E. Gutkin, Representations of Hecke algebras, Trans. Amer. Math. Soc. 309 (1988), 269–277.

[12] D. Husemoller, Fibre Bundles, Springer-Verlag, 1975.

[13] V. G. Kac, Constructing groups associated to infinite dimensional Lie algebras, In: Infinite dimensional Groups with applications, Math. Sci. Res. Inst. Publ. 4 (1985), 167–217.

[14] B. Kostant, S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Advances in Math. 62 (1986), 187–237.

[15] L. G. Lewis, J. P. May, M. Steinberger, Equivarant Stable Homotopy Theory, Lecture Notes in Mathematics 1213, Springer-Verlag, 1986.

[16] C. Ozel, On the Complex Cobordism of Flag Varieties Associated to Loop Grpoups, PhD Thesis, University of Glasgow, 1998.

[17] D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Princeton University.

(received 17.03.1999, in revised form 01.12.2001)

AIBU Golkoy Kampusu, Bolu 14280, Turkey E-mail: cenap@ibu.edu.tr