Symmetries in complex cobordism theory, related to stable equivalence classes of SU-bundles

A.V. Ershov^{∗}

Abstract

Let Z (resp. Ξ̂) be the universal line bundle over ℂP^{∞} (resp. the universal Ubundle over BU). Since BU is homotopy equivalent to BSU ×ℂP^{∞}, we have σ_{1}(Ξ̂) = 𝔥(σ_{1}(Z)) for some formal power series 𝔥(u) ∈ MU^{∗}(BSU)[[u]], where σ_{1} denotes the first Conner-Floyd class. Let ℌ(x, y) be the formal group law 𝔥(F (𝔥^{−1}(x), 𝔥^{−1}(y))) ∈ MU^{∗}(BSU)[[x, y]], and K ⊂ MU^{∗}(BSU) the subring, generated by its coefficients, where F (u, v) ∈ MU^{∗}(pt)[[u, v]] is the formal product in complex cobordism theory. Clearly, K ≅ MU^{∗}(pt). Thus for any stable equivalence class \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X) of SU-bundles over a finite CW -complex X we get a formal group law (φ_{ξ}^{∗}ℌ)(x, y) ∈ MU^{∗}(X)[[x, y]], where φ_{ξ}: X → BSU is a classifying map for ξ ∈ {ξ}. Suppose ξ has virtual dimension 1. It was shown in [6] that σ_{1}(ξ ⊗ ζ_{1} ⊗ ζ_{2}) = (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ ⊗ ζ_{1}), σ_{1}(ξ ⊗ ζ_{2})) for any geometric line bundles ζ_{1}, ζ_{2} over X. The coefficients of (φ_{ξ}^{∗}ℌ)(x, y) generate the direct summand subring φ_{ξ}^{∗}(K) ≅ MU^{∗}(pt) in MU^{∗}(X).

In the present paper for a given pair (X, ξ) we construct a ring automorphism 𝔗_{\{ξ\}}: MU[\frac{1}{2}]^{∗}(X) → MU[\frac{1}{2}]^{∗}(X) such that

• it is natural on (X, ξ);

• 𝔗_{{ξ}}(R) = φ_{ξ}^{∗}(K), where R := MU^{∗}(pt) ⊂ MU^{∗}(X);

• 𝔗_{{ξ}}(σ_{1}(ζ)) = σ_{1}(ξ ⊗ ζ) for any line bundle ζ over X.

Furthermore, we prove that the assignment {ξ} ↦ 𝔗_{{ξ}} determines a group homomorphism

\begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} → Aut_{ring}(MU[\frac{1}{2}]^{∗}(X)),

i.e. 𝔗_{{η}} ∘ 𝔗_{{ξ}} = 𝔗_{{η⊗ξ}} for all virtual SU-bundles ξ, η over X of virtual dimension 1.

Contents

1 A natural transformation 𝔗^{ev}: MU^{ev}( ) → H ˆ⊗_{R} MU^{ev}( ) 7

2 A natural transformation K̃ SU( )_{⊗} → Aut_{ring}(MU^{ev}( )) 11

3 An extended natural transformation K̃ SU( )_{⊗} → Aut_{ring}(MU[\frac{1}{2}]^{∗}( )) 17

4 Appendix: A short proof of an isomorphism of some H-groups 21

∗Partially supported by RFFI Grant 02-01-00572.

Introduction

This paper is closely connected to Preprint [6], therefore first let us give a review of its results.

Let BU_{⊗} (resp. BSU_{⊗}) be the space BU (resp. BSU) with the structure of H-group, related to the tensor product of virtual U-bundles (resp. SU-bundles) of virtual dimension 1. By Ĥ (resp. H) denote the Hopf algebra MU^{∗}(BU_{⊗}) (resp. MU^{∗}(BSU_{⊗})) over R := MU^{∗}(pt). Since the H-groups BU_{⊗} and BSU_{⊗} ×ℂP^{∞} are isomorphic (see the Appendix), we have an isomorphism of Hopf algebras Ĥ ≅ H ˆ⊗_{R} R[[u]] = H[[u]]. Clearly, the comultiplication Δ̂ and

the antipode Ŝ in Ĥ are determined as follows:

Δ̂(u) = F (u ⊗ 1, 1 ⊗ u), Δ̂|_{H} = Δ, Ŝ(u) = θ(u), Ŝ|_{H} = S,  (1)

where Δ, S are the comultiplication and the antipode in H and θ(u) ∈ R[[u]] is the formal inverse in the formal product in complex cobordism theory F (u, v).

Let Ξ̂ (resp. Ξ) be the universal U-bundle over BU of virtual dimension 1 (resp. the universal SU-bundle over BSU of virtual dimension 1), and Z the canonical line bundle over ℂP^{∞}. It can be proved that Ξ̂ is identified with Ξ ⊠ Z under the H-group isomorphism (see the Appendix). This implies that the element x := σ_{1}(Ξ̂) ∈ MU^{2}(BU) can be expressed as a formal power series 𝔥(u) ∈ H[[u]] of the variable u := σ_{1}(Z), where σ_{1} denotes the first Conner-Floyd class. Using the splitting principle, one can verify the following explicit form of this power series [6]:

𝔥(u) = 1 ⊗ u + ∑_{i,j≥1} α_{i,j}s̃_{i} ⊗ u^{j} ∈ H ˆ⊗_{R} R[[u]],  (2)

where α_{i,j} ∈ MU^{2(1−i−j)}(pt) are the coefficients of the formal product in complex cobordism theory F (u, v) and s̃_{i} := s_{i}(Ξ) ∈ MU^{2i}(BSU) is the i-th Newton characteristic class in complex cobordism theory of the universal SU-bundle Ξ (recall that s_{1}(Ξ) = σ_{1}(Ξ) = 0). Remark 1. In particular, we see that 𝔥(u) is functionally invertible, (ε𝔥)(u) = u, and 𝔥(0) = 0, where ε: H → R is the counit in the Hopf algebra H. Thus, MU^{∗}(BU) = H[[u]] = H[[x]].

Now consider the formal power series 𝔉(x ⊗ 1, 1 ⊗ x) := Δ̂(x) ∈ (H ˆ⊗_{R} H)[[x ⊗ 1, 1 ⊗ x]] = MU^{∗}(BU × BU) and Θ(x) := ι̂^{∗}(x) ∈ H[[x]], where ι̂: BU → BU is the inversion in BU_{⊗}, i.e. ι̂^{∗}(Ξ̂) = Ξ̂^{−1}. The series 𝔉(x ⊗ 1, 1 ⊗ x) is a formal group law over the Hopf algebra H with the formal inverse Θ(x), i.e. it satisfies the following conditions [6]:

1) (associative)

((id_{H} ⊗Δ)𝔉)(x ⊗ 1 ⊗ 1, 1 ⊗ 𝔉(x ⊗ 1, 1 ⊗ x))
= ((Δ ⊗ id_{H})𝔉)(𝔉(x ⊗ 1, 1 ⊗ x) ⊗ 1, 1 ⊗ 1 ⊗ x).

2) (identity)

((id_{H} ⊗ε)𝔉)(x ⊗ 1, 0) = x ⊗ 1,
((ε ⊗ id_{H})𝔉)(0, 1 ⊗ x) = 1 ⊗ x.

3) (formal inverse)

(μ ∘ (id_{H} ⊗S)𝔉)(x, Θ(x)) = 0 = (μ ∘ (S ⊗ id_{H})𝔉)(Θ(x), x),

where μ: H ˆ⊗_{R} H → H is the multiplication in H.

Moreover, the formal group law 𝔉(x ⊗ 1, 1 ⊗ x) is commutative, i.e. (τ𝔉)(1 ⊗ x, x ⊗ 1) = 𝔉(x ⊗ 1, 1 ⊗ x), where τ: H ˆ⊗_{R} H → H ˆ⊗_{R} H is the switching map τ(a ⊗ b) = b ⊗ a.

Using (1), we can obtain the explicit form of 𝔉(x ⊗ 1, 1 ⊗ x) and Θ(x):

𝔉(x ⊗ 1, 1 ⊗ x) = Δ̂(x) = Δ̂(𝔥(u)) = (Δ𝔥)(F (u ⊗ 1, 1 ⊗ u))
= (Δ𝔥)(F (𝔥^{−1}(x) ⊗ 1, 1 ⊗ 𝔥^{−1}(x)));
Θ(x) = Ŝ(x) = Ŝ(𝔥(u)) = (S𝔥)(θ(u)) = (S𝔥)(θ(𝔥^{−1}(x))).

Let p: BU_{⊗} → BSU_{⊗} be the composition of the isomorphism ϖ: BU → BSU ×ℂP^{∞}, ϖ^{∗}(Ξ ⊠ Z) = Ξ̂ with the projection onto the first factor (note that ϖ = (ϕ ∘ (λ × κ))^{−1}, see the Appendix). Then p is a morphism of H-groups. Let ξ be a virtual vector U-bundle of virtual dimension 1 over a finite CW -complex X, and φ_{ξ}: X → BU its classifying map. Hence the composite map φ\overline{ξ} := p ∘ φ_{ξ}: X → BSU is the classifying map for the corresponding virtual SU-bundle \overline{ξ} of virtual dimension 1 (let us remark that for any ξ there exists a unique line bundle ζ such that c_{1}(ξ ⊗ ζ) = 0, where c_{1} is the first Chern class; clearly \overline{ξ} = ξ ⊗ ζ). By μ_{X} denote the multiplication MU^{∗}(X)⊗_{R} MU^{∗}(X) → MU^{∗}(X).

Theorem 2. For any virtual complex vector bundles ξ, η of virtual dimension 1 over X we have

σ_{1}(ξ ⊗ η) = (μ_{X} ∘ (φ∗ ; \overline{ξ} ⊗ φ∗ ; \overline{η})𝔉)(σ_{1}(ξ), σ_{1}(η)).  (3)

Suppose ξ^{−1} is the virtual bundle of virtual dimension 1 over X such that ξ ⊗ ξ^{−1} = [1], where [1] is the trivial bundle. Then

σ_{1}(ξ^{−1}) = (φ∗ ; \overline{ξ}Θ)(σ_{1}(ξ))

and we have

(μ_{X} ∘ (φ∗ ; \overline{ξ} ⊗ (φ∗ ; \overline{ξ} ∘ S))𝔉)(σ_{1}(ξ), σ_{1}(ξ^{−1})) = 0

(note that φ∗ ; \overline{ξ} ∘ S = φ_{\overline{ξ}^{−1}}^{∗}, where \overline{ξ}^{−1} is the inverse bundle for \overline{ξ}, i.e. \overline{ξ} ⊗ \overline{ξ}^{−1} = [1]).

Remark 3. If ξ, η are geometric line bundles, then (3) reduces to the well-known formula σ_{1}(ξ ⊗η) = F (σ_{1}(ξ), σ_{1}(η)) (note that in the last case φ\overline{ξ}, φ\overline{η} are maps to a point). Moreover, in this case σ_{1}(ξ^{−1}) = (φ∗ ; \overline{ξ}Θ)(σ_{1}(ξ)) = θ(σ_{1}(ξ)). Remark 4. It follows from Remark 1 that the constant term of 𝔉(x ⊗ 1, 1 ⊗ x) equals 0. Therefore, using (3) we see that σ_{1}(ξ ⊗η) belongs to the ideal I = {σ_{1}(ξ), σ_{1}(η)} ⊂ MU^{∗}(X), generated by σ_{1}(ξ) and σ_{1}(η).

Set ℌ(x, y) := 𝔥(F (𝔥^{−1}(x), 𝔥^{−1}(y))) ∈ H[[x, y]]. Obviously, ℌ(x, y) is a formal group law over the ring H, with the formal inverse ℑ(x) := 𝔥(θ(𝔥^{−1}(x))) ∈ H[[x]]. Moreover, the formal group law ℌ(x, y) is isomorphic to F (u, v) (although 𝔥(u) is not a strict isomorphism of formal group laws, it is not difficult to find a strict one).

Write ℌ(x, y) = x+y+∑_{i,j≥1} α̃_{i,j}x^{i}y^{j}. Recall that the formal group law F (u, v) ∈ R[[u, v]] is the universal one and the ring R is the Lasard ring [4]. Hence there is a unique ring homomorphism λ: R → H such that λ(α_{i,j}) = α̃_{i,j}, i, j ≥ 1. Clearly, ε(α̃_{i,j}) = α_{i,j}, i, j ≥ 1. By P denote the composite homomorphism λ ∘ ε: H → H .

Let K ⊂ H be the subring, generated by {α̃_{i,j}, i, j ≥ 1}.

Theorem 5. λ determines the canonical ring isomorphism λ: R →^{≅} K ⊂ H, α_{i,j} ↦^{λ} α̃_{i,j}. Furthermore, P is a projector (i.e. P^{2} = P and H = im P ⊕ ker P) and im P = K ⊂ H.

Moreover, it can be shown that K ≠ R as subrings in H.

Thus we get the formal group law ℌ(x, y) ∈ K[[x, y]] with the formal inverse ℑ(x) ∈ K[[x]] and the ring isomorphism λ: R → K, realizing the isomorphism of formal group laws F → ℌ.

Now let us fix a virtual SU-bundle ξ of virtual dimension 1 over a finite CW -complex X. We say that a virtual bundle ξ̂ of the form ξ ⊗ ζ, where ζ is a geometric line bundle over X, is a lift of ξ. By L(ξ) ⊂ K_{ℂ}(X) denote the set of stable equivalence classes of lifts of ξ.

Proposition 6. The set L(ξ) is an Abelian group with respect to the operation, induced by the following operation on lifts:

ξ̂_{1} ∘ ξ̂_{2} := ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}.

Furthermore, for all ξ the corresponding groups L(ξ) are isomorphic to each other (in particular, all of them are isomorphic to the group H^{2}(X; ℤ) of line bundles over X with respect to the tensor product).

Theorem 7. Let ξ̂_{1}, ξ̂_{2} be two lifts of ξ, and φ_{ξ}: X → BSU a classifying map for ξ. Then

σ_{1}(ξ̂_{1} ∘ ξ̂_{2}) = (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})).  (4)

Moreover, σ_{1}(ξ̂^{−1}) = (φ_{ξ}^{∗}ℑ)(σ_{1}(ξ̂)), where ξ̂^{−1} is a lift of ξ, whose stable equivalence class is the inverse element for the class {ξ̂} of ξ̂ with respect to the operation on L(ξ) (note that the class of ξ is the identity element in L(ξ)). The assignment ξ̂ ↦ σ_{1}(ξ̂) determines a group isomorphism from L(ξ) to the group {σ_{1}(ξ̂) | ξ̂ is a lift of ξ} ⊂ MU^{2}(X) with respect to the operation σ_{1}(ξ̂_{1}) ∘ σ_{1}(ξ̂_{2}) := (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})).

Remark 8. Note that (4) can be written as follows:

σ_{1}(ξ ⊗ ζ_{1} ⊗ ζ_{2}) = (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ ⊗ ζ_{1}), σ_{1}(ξ ⊗ ζ_{2})),

where ξ̂_{1} = ξ ⊗ ζ_{1}, ξ̂_{2} = ξ ⊗ ζ_{2} for line bundles ζ_{1}, ζ_{2}. In the case ξ = [1] it shows that we recur to the relation σ_{1}(ζ_{1} ⊗ ζ_{2}) = F (σ_{1}(ζ_{1}), σ_{1}(ζ_{2})).

By MU^{ev}(X) denote the subring ⊕_{n∈ℤ}MU^{2n}(X) ⊂ MU^{∗}(X). Thus the formal group law ℌ_{ξ}(x, y) := (φ_{ξ}^{∗}ℌ)(x, y) ∈ MU^{ev}(X)[[x, y]] describes the relation between first Conner-Floyd classes of lifts of the fixed SU-bundle ξ. Recall that by K we denote the subring, generated by the coefficients α̃_{i,j} of ℌ(x, y). Hence the coefficients φ_{ξ}^{∗}(α̃_{i,j}) ∈ MU^{2(1−i−j)}(X) of ℌ_{ξ}(x, y) generate the subring K_{ξ} := φ_{ξ}^{∗}(K) ⊂ MU^{ev}(X), which is a direct summand in MU^{ev}(X), isomorphic to R. Note that K_{ξ} is equipped with the ring isomorphism λ_{ξ} := φ_{ξ}^{∗}∘λ: R → K_{ξ} .

Thus we have the following situation: for any \{ξ\} ∈ { \atop KSU}(X) we get the formal group law ℌ_{ξ}(x, y) over K_{ξ} ⊂ MU^{ev}(X), which determines the group structure on the set {σ_{1}(ξ̂) | ξ̂ is a lift of ξ} ⊂ MU^{2}(X). Note that for all {ξ} the corresponding formal group laws ℌ_{ξ}(x, y), the corresponding subrings K_{ξ}, and the corresponding sets {σ_{1}(ξ̂) | ξ̂ is a lift of ξ} are isomorphic to each other. Note also that having a pair (K_{ξ}, λ_{ξ}) one can restore the corresponding formal group law ℌ_{ξ}(x, y). Indeed, ℌ_{ξ}(x, y) = (λ_{ξ∗}F)(x, y).

We say that the triple, consisting of the subring K_{ξ} ⊂ MU^{ev}(X), the ring isomorphism λ_{ξ}: R → K_{ξ}, and the set {σ_{1}(ξ̂) | ξ̂ is a lift of ξ} ⊂ MU^{2}(X) is a data, corresponding to the stable equivalence class \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X).

The aim of the present paper is to give some explanation of the described results. We shall show that for any pair (X, {η}) there exists a natural ring automorphism 𝔗_{{η}}^{ev}: MU^{ev}(X) → MU^{ev}(X) such that for any \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X) with a representative ξ of virtual dimension 1 it transform the data corresponding to {ξ} isomorphically to the data, corresponding to {η⊗ξ}. More precisely, K_{η⊗ξ} = 𝔗_{{η}}^{ev}(K_{ξ}), λ_{η⊗ξ} = 𝔗_{{η}}^{ev} ∘ λ_{ξ}, σ_{1}(η ⊗ ξ̂) = 𝔗_{{η}}^{ev}(σ_{1}(ξ̂)) for any lift ξ̂ of ξ.

Let \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} be the set \begin{matrix} \\ K & SU \end{matrix}(X) with respect to the group structure, induced by the tensor product of virtual bundles of virtual dimension 1, i.e. \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} = [X, BSU_{⊗}]. It will be proved that the assignment \{ξ\} ↦ 𝔗_{\{ξ\}}^{ev} determines a group homomorphism \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} → Aut_{ring}(MU^{ev}(X)), i.e. 𝔗_{{η}}^{ev} ∘ 𝔗_{{ξ}}^{ev} = 𝔗_{{η⊗ξ}}^{ev} for all virtual SU-bundles ξ, η over X of virtual dimension 1.

Now let us introduce an algebraic view on some results of Sections 2, 3. Let 𝔄𝔩𝔤_{R} be a subcategory of the category of associative commutative unitary algebras over R, equipped with the tensor product, i.e. for any two objects C, C′ ∈ 𝔄𝔩𝔤_{R} we have C ⊗_{R} C′ ∈ 𝔄𝔩𝔤_{R}. In particular, 𝔄𝔩𝔤_{R} can be the category of S-modules, where S is a Hopf algebra over ℤ. Let H ∈ 𝔄𝔩𝔤_{R} be a Hopf algebra over R. Assume that for any C ∈ 𝔄𝔩𝔤_{R} (in particular, for C = R) there is a natural homomorphism Δ_{C} : C → H ⊗_{R} C (i.e. Δ_{C}′ ∘ f = (id_{H} ⊗f) ∘ Δ_{C} for f ∈ Mor_{𝔄𝔩𝔤_{R}}(C, C′)) such that the diagrams

\xymatrix{R ⊗_{R} C & H ⊗_{R} C \ar[l]_{ε⊗id_{C}} & C \ar[r]^{Δ_{C}} \ar[d]_{Δ_{C}} & H ⊗_{R} C \ar[d]^{Δ_{_{H} ⊗_{R}_{C}}} \\  & C, \ar[ul]^{≅} \ar[u]_{Δ_{C}} & _{H ⊗_{R} C} \ar[r]_{Δ⊗id_{C}} & _{H ⊗_{R} H ⊗_{R} C}}

commute, where Δ: H → H ⊗_{R} H, ε: H → R are the comultiplication and the counit map in H respectively (let us remark that (i) Δ_{H ⊗_{R} C} : H ⊗_{R} C → H ⊗_{R} H ⊗_{R} C makes sense because of H ⊗_{R} C ∈ 𝔄𝔩𝔤_{R}, and (ii) Δ_{C} induces an isomorphism from C onto a direct summand in H ⊗_{R} C. Indeed, it follows from the first diagram above that Δ_{C} ∘ (ε ⊗ id_{C}) is a projector onto im(Δ_{C})). Then the assignment

Hom_{R}(H, C) ∋ φ ↦ μ_{C} ∘ (φ ⊗ id_{C}) ∘ Δ_{C},

where μ_{C} : C ⊗_{R} C → C is the multiplication in C, determines a group homomorphism

𝔊: Hom_{R}(H, C) → Aut_{ring}(C),

which is natural on C (here Hom_{R}(H, C) is a group with respect to the convolution ⋆: φ⋆ψ := μ_{C} ∘ (φ ⊗ ψ) ∘ Δ). The assertion that 𝔊 is a group homomorphism follows from the diagram of Corollary 21. The commutative diagram

\xymatrix{C \ar[r]^{Δ_{C}} \ar[d]_{f} & H ⊗_{R} C \ar[d]^{id_{H} ⊗f} \ar[r]^{φ⊗id_{C}} & C ⊗_{R} C \ar[d]^{f⊗f} \ar[r]^{μ_{C}} & C \ar[d]^{f} \\ _{C_{′}} \ar[r]_{Δ_{_{C}′}} & _{H ⊗_{R} C_{′}} \ar[r]_{f_{∗}(φ)⊗id_{_{C}′}} & _{C_{′} ⊗_{R} C_{′}} \ar[r]_{μ_{_{C}′}} & C′,}

where f ∈ Mor_{𝔄𝔩𝔤_{R}}(C, C′), φ ∈ Hom_{R}(H, C) and f_{∗}(φ) := f ∘ φ, shows that 𝔊(f_{∗}(φ)) ∘ f = f ∘ 𝔊(φ), i.e. that 𝔊 is natural. We do not assert that 𝔊(φ) is an automorphism of the R-algebra C (it depends on the homomorphism Δ_{R} : R → H ⊗_{R} R = H).

Now let us describe roughly a construction of the natural homomorphism Δ_{C} : C → H ⊗_{R} C under the following additional assumption: 𝔄𝔩𝔤_{R} is a category of S-modules, where S is a Hopf algebra over ℤ. That is, the Hopf algebra S acts on any C ∈ 𝔄𝔩𝔤_{R} (in particular, on R) and on the tensor product C ⊗_{R} C′, C, C′ ∈ 𝔄𝔩𝔤_{R} the action is defined as follows: S(α ⊗ β) = ∑_{i} S′_{i}(α) ⊗ S′′_{i}(β), for any α ∈ C, β ∈ C′, S ∈ S, where Δ_{S}S = ∑_{i} S′_{i} ⊗ S′′_{i} and Δ_{S} : S → S ⊗_{ℤ}S is the comultiplication in S.

Assume that there is an element 𝔗 = ∑_{i≥0} γ_{i} ⊗ S_{i} ∈ H ⊗_{ℤ}S such that

• γ_{0} ⊗ S_{0} = 1_{H} ⊗ 1_{S}, ε(γ_{i}) = 0, i ≥ 1 (⇒ (ε ⊗ id_{S})𝔗 = 1 ⊗ S_{0} = 1 ⊗ 1_{S} ∈ R ⊗_{ℤ}S);

• for any C ∈ 𝔄𝔩𝔤_{R} 𝔗 defines a ring homomorphism 𝔗_{C} : C → H ⊗_{R} C by setting 𝔗_{C}(α) := 𝔗(α) = ∑_{i} γ_{i} ⊗ S_{i}(α) ∈ H ⊗_{R} C for any α ∈ C;

• the identity (id_{H} ⊗Δ_{S})𝔗 ∘ 𝔗 = (Δ ⊗ id_{S})𝔗 ∈ H ⊗_{R} H ⊗_{ℤ}S holds. More explicitly,

∑_{i} γ_{i} ⊗ (Δ_{S}S_{i}(∑_{j} γ_{j} ⊗ S_{j})) = ∑_{k} Δ(γ_{k}) ⊗ S_{k}.

Then, setting Δ_{C} = 𝔗_{C} for any C ∈ 𝔄𝔩𝔤_{R}, we obtain a natural transformation with the required properties (i.e. (ε ⊗ id_{C}) ∘ Δ_{C} = id_{C} and Δ_{H ⊗_{R} C} ∘ Δ_{C} = (Δ ⊗ id_{C}) ∘ Δ_{C}, see the diagrams above).

Roughly, below we shall deal with the following situation: 𝔄𝔩𝔤_{R} is the category whose objects are cobordism rings MU^{∗}(X) with morphisms MU^{∗}(X) → MU^{∗}(Y), induced by continuous maps Y → X; H = MU^{∗}(BSU_{⊗}), S is Landweber-Novikov’s algebra, and the tensor product ⊗ must be replaced by the completed graded tensor product ˆ⊗. Acknowledgments The author is grateful to Professor E.V. Troitsky for constant attention to this work. I would like to thank the Max Planck Institute for Mathematics in Bonn for its kind support and hospitality while this work has been completed. A number of related questions were discussed with Professors V.G. Gorbounov and S.V. Konyagin and I would like to express my gratitude to them.

1 A natural transformation 𝔗^{ev}: MU^{ev}( ) → H ˆ⊗_{R} MU^{ev}( )

The main tool of this paper is a multiplicative natural transformation 𝔗^{ev}: MU^{ev}(X) → H ˆ⊗_{R} MU^{ev}(X), which is uniquely determined by the following condition: for X = ℂP^{∞} and

u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}) 𝔗^{ev}(u) = 𝔥(u) = σ_{1}(Ξ̂) ∈ H ˆ⊗_{R} MU^{∗}(ℂP^{∞}) = MU^{∗}(BU). Let us remember that Landweber-Novikov’s operations act on the “geometric cobordism” u in the following way: S_{ω}u = 0 if l(ω) ≥ 2 and S_{(k)}u = u^{k+1}, where l( ) is the length of a partition (if ω = (k_{1}, …, k_{l}), then l(ω) = l) [3]. The idea of the construction of the natural transformation 𝔗^{ev} is based on the following observation. It follows from the just described action of S_{ω} on u that the element σ_{1}(Ξ̂) = σ_{1}(Ξ ⊗ Z) = 1 ⊗ u + ∑_{i,j≥1} α_{i,j}s̃_{i} ⊗ u^{j} can be obtained by application of 1 ⊗ S_{0} + ∑_{i,j≥1} α_{i,j}s̃_{i} ⊗ S_{j−1} ∈ H ˆ⊗_{ℤ}S (here S is Landweber-Novikov’s algebra, see below) to the element u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}). Thus, the goal is to extend this transformation to a multiplicative transformation of whole cobordism theory.

Now we recall some results on Landweber-Novikov’s operations and Landweber-Novikov’s algebra (see [2], [3], [5]).

Let s_{ω} ∈ MU^{2|ω|}(BU) be the characteristic class in complex cobordism theory, corresponding to the partition ω = (k_{1}, …, k_{l}), where |ω| = ∑_{1≤q≤l} k_{q}. Let 𝔱: MU^{∗}(BU) → MU^{∗}(MU) be the stable Thom isomorphism; then 𝔱(s_{ω}) = S_{ω} ∈ MU^{2|ω|}(MU) is Landweber-Novikov’s operation, corresponding to the partition ω. Landweber-Novikov’s algebra S is a 2ℤ^{+}-graded Hopf algebra over ℤ, S = ⊕_{j≥0} S^{2j}, with the basis {S_{ω}}, S_{ω} ∈ S^{2|ω|}, where ω is either a partition (k_{1}, …, k_{l}), k_{q} ∈ ℤ^{+} or the symbol of an empty set (note that S_{∅} = S_{(0)} = 1 ∈ S^{0}). The comultiplication Δ_{S} in S is defined as follows:

Δ_{S}(S_{ω}) = ∑_{ω′∪ω′′=ω} S_{ω′} ⊗ S_{ω′′}.

Furthermore, S acts on any cobordism ring MU^{∗}(X), turning it into Milnor’s module:

S_{ω}(αβ) = ∑_{ω′∪ω′′=ω} S_{ω′}(α)S_{ω′′}(β).

For a CW -complex X by MU^{ev}(X) we denote the subalgebra ⊕_{n∈ℤ}MU^{2n}(X) ⊂ MU^{∗}(X). Suppose α ∈ MU^{2n}(X); then S_{ω}(α) ∈ MU^{2(n+|ω|)}(X). Therefore S preserves the subset MU^{ev}(X) ⊂ MU^{∗}(X).

Let A be a graded topological algebra over R with an augmentation ε_{A} : A → R . Assume that A is complete with respect to the augmentation ideal ker(ε_{A}). For example, we can take A = MU^{∗}(BSU) or A = MU^{∗}(X), where X is a finite CW -complex. Note that in the last case the topology on A is discrete. By u we denote the element σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}), where Z is the canonical line bundle over ℂP^{∞}. Recall that MU^{∗}(ℂP^{∞}) = R[[u]].

Theorem 9. Given a formal power series 𝔣(u) = ∑_{i≥1} a_{i} ⊗ u^{i} ∈ A ˆ⊗_{R} MU^{∗}(ℂP^{∞}) such that

(i) deg a_{i} = 2 − 2i;

(ii) ε_{A}(a_{1}) = 1, ε_{A}(a_{i}) = 0, i ≥ 2

there is a unique multiplicative natural transformation 𝔗_{𝔣}^{ev}: MU^{ev}(X) → A ˆ⊗_{R} MU^{ev}(X) such that 𝔗_{𝔣}^{ev}(u) = 𝔣(u) for X = ℂP^{∞} and u = σ_{1}(Z). Furthermore, 𝔗_{𝔣}^{ev} induces a ring isomorphism from MU^{ev}(X) onto the direct summand 𝔗_{𝔣}^{ev}(MU^{ev}(X)) ⊂ A ˆ⊗_{R} MU^{ev}(X).

Proof. Consider the symmetric formal power series of N variables u_{1}, …, u_{N}:

∏_{1≤j≤N} (∑_{i≥0} α_{i} ⊗ (u_{j})^{i}) ∈ A ˆ⊗_{R} MU^{∗}((ℂP^{∞})^{N}),

where α_{i} = \frac{a_{i+1}}{a_{1}} , i ≥ 0, deg(α_{i}) = −2i (since ε_{A}(a_{1}) = 1 and A is complete with respect to ker(ε_{A}), we see that there is a_{1}^{−1} ∈ A). Let us remark that this power series has degree 0 as an element of the tensor product of graded rings.

Let

∑_{ω} α_{ω}^{(N)} ⊗ s_{ω} ∈ A ˆ⊗_{R} MU^{∗}(BU(N))

be the element such that

(id_{A} ⊗μ̂_{N}^{∗})(∑_{ω} α_{ω}^{(N)} ⊗ s_{ω}) = ∏_{1≤j≤N} (∑_{i≥0} α_{i} ⊗ (u_{j})^{i}) ,

where id_{A} ⊗μ̂_{N}^{∗}: A ˆ⊗_{R} MU^{∗}(BU(N)) → A ˆ⊗_{R} MU^{∗}((ℂP^{∞})^{N}) is the homomorphism, induced

by the “splitting map” μ̂_{N}: (ℂP^{∞})^{N} → BU(N) and α_{ω}^{(N)} ∈ A, deg(α_{ω}^{(N)}) = −2|ω|. Note that α_{ω}^{(N)} = 0 if l(ω) ≥ N +1 and α_{ω}^{(N)} = α_{k_{1}} … α_{k_{l}} if l(ω) ≤ N (in particular, α_{ω}^{(N)} is independed of N under the last condition), where l( ) denotes the length of a partition, l(k_{1}, …, k_{l}) = l. Therefore, passing to the limit N → ∞, we get the element ∑_{ω} α_{ω} ⊗ s_{ω} ∈ A ˆ⊗_{R} MU^{∗}(BU) of degree 0.

Let 𝔱: MU^{∗}(BU) → MU^{∗}(MU) be the stable Thom isomorphism. Applying the operator id_{A} ⊗𝔱 to ∑_{ω} α_{ω} ⊗ s_{ω}, we obtain the element 𝔗_{𝔣,0} := ∑_{ω} α_{ω} ⊗ S_{ω} ∈ A ˆ⊗_{R} MU^{∗}(MU) of degree 0, where S_{ω} = 𝔱(s_{ω}) are Landweber-Novikov’s operations. Note that 𝔗_{𝔣,0} can be considered as an element of A ˆ⊗_{ℤ}S, where S is Landweber-Novikov’s algebra.

For any n ∈ ℤ set 𝔗_{𝔣,2n} := (a_{1}^{n} ⊗ 1)𝔗_{𝔣,0} = ∑_{ω}(a_{1}^{n}α_{ω}) ⊗ S_{ω} ∈ A ˆ⊗_{ℤ}S . We define the natural transformation 𝔗_{𝔣}^{ev} in the following way. Given a homogeneous element α ∈ MU^{2n}(X) of degree 2n we set 𝔗_{𝔣}^{ev}(α) := 𝔗_{𝔣,2n}(α) = ∑_{ω}(a_{1}^{n}α_{ω}) ⊗ S_{ω}(α) ∈ (A ˆ⊗_{R} MU^{ev}(X))^{(2n)}.

Let α ∈ MU^{2m}(X), β ∈ MU^{2n}(X) be two homogeneous elements. Let us show that 𝔗_{𝔣}^{ev}(αβ) = 𝔗_{𝔣}^{ev}(α)𝔗_{𝔣}^{ev}(β). We have:

𝔗_{𝔣}^{ev}(αβ) = 𝔗_{𝔣,2(m+n)}(αβ) = ∑_{ω}(a_{1}^{m+n}α_{ω}) ⊗ S_{ω}(αβ)

= ∑_{ω}(a_{1}^{m+n}α_{ω} ⊗ ∑_{ω′∪ω′′=ω} S_{ω′}(α)S_{ω′′}(β))

= ∑_{ω} ∑_{ω′∪ω′′=ω}(a_{1}^{m}α_{ω′})(a_{1}^{n}α_{ω′′}) ⊗ S_{ω′}(α)S_{ω′′}(β)

= (∑_{ω′} a_{1}^{m}α_{ω′} ⊗ S_{ω′}(α))(∑_{ω′′} a_{1}^{n}α_{ω′′} ⊗ S_{ω′′}(β))

= 𝔗_{𝔣,2m}(α)𝔗_{𝔣,2n}(β) = 𝔗_{𝔣}^{ev}(α)𝔗_{𝔣}^{ev}(β),

where we used the following well-known identity for Landweber-Novikov’s operations

S_{ω}(αβ) = ∑_{ω′∪ω′′=ω} S_{ω′}(α)S_{ω′′}(β)

and the following obvious identity: α_{ω} = α_{ω′}α_{ω′′} for any partitions ω′, ω′′ such that ω = ω′ ∪ ω′′. This implies the multiplicativity of 𝔗_{𝔣}^{ev} for any α, β, not necessarily homogeneous.

Now by using S_{ω}u = 0 if l(ω) > 1 and S_{(k)}u = u^{k+1}, the required property of 𝔗_{𝔣}^{ev} : 𝔗_{𝔣}^{ev}(u) = 𝔣(u) = ∑_{i≥1} a_{i} ⊗ u^{i} for u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}) can easily be verified.

The naturality of 𝔗_{𝔣}^{ev} is clear. Since the representation of Landweber-Novikov’s algebra S is asymptotically precise on the product of copies of “geometric cobordism” u_{1} … u_{N}, N → ∞ it follows that 𝔗_{𝔣}^{ev} is unique. More explicitly, take X = (ℂP^{∞})^{N} and set u_{N} := u_{1} … u_{N} ∈ MU^{2N}(X) = R[[u_{1}, …, u_{N}]]. For ω = (k_{1}, …, k_{l}), l(ω) = l ≤ N let s_{ω} be the smallest symmetric polynomial of u_{1}, …, u_{N} which contains u_{1}^{k_{1}} … u_{l}^{k_{l}}. We have S_{ω}(u_{N}) = s_{ω}u_{N}. Hence

𝔗_{𝔣}^{ev}(u_{N}) = ∑_{ω} a_{1}^{N}α_{ω}^{(N)} ⊗ S_{ω}(u_{N})

= ∑_{ω} a_{1}^{N}α_{ω}^{(N)} ⊗ s_{ω}u_{N} ∈ A ˆ⊗_{R} R[[u_{1}, …, u_{N}]] ≅ A[[u_{1}, …, u_{N}]].

Thus the element a_{1}^{N}α_{ω} ∈ A, l(ω) ≤ N (⇒ α_{ω}, l(ω) ≤ N) is uniquely determined as the coefficient of s_{ω}u_{N} in the formal power series 𝔗_{𝔣}^{ev}(u_{N}) ∈ A[[u_{1}, …, u_{N}]].

We now prove that 𝔗_{𝔣}^{ev} induces a ring isomorphism from MU^{ev}(X) onto the direct summand 𝔗_{𝔣}^{ev}(MU^{ev}(X)) ⊂ A ˆ⊗_{R} MU^{ev}(X). It can easily be checked that for any n ∈ ℤ (ε_{A} ⊗ id_{S})𝔗_{𝔣,2n} = S_{∅} = S_{(0)} = 1. This implies that the following composition:

(ε_{A} ⊗ id_{MU^{ev}(X)}) ∘ 𝔗_{𝔣}^{ev}: MU^{ev}(X) → MU^{ev}(X)

is the identity map id_{MU^{ev}(X)} . Therefore 𝔗_{𝔣}^{ev} is a monomorphism. By P we denote the composite homomorphism 𝔗_{𝔣}^{ev} ∘ (ε_{A} ⊗ id_{MU^{ev}(X)}): A ˆ⊗_{R} MU^{ev}(X) → A ˆ⊗_{R} MU^{ev}(X). It is easy to see that P is a projector onto the subring 𝔗_{𝔣}^{ev}(MU^{ev}(X)) ⊂ A ˆ⊗_{R} MU^{ev}(X). In particular, P is an idempotent. □

Corollary 10. Suppose X is a CW -complex such that MU^{2n+1}(X) = 0 for any n ∈ ℤ. Then 𝔗_{𝔣}^{ev} is a transformation MU^{∗}(X) → A ˆ⊗_{R} MU^{∗}(X) which determines a ring isomorphism from MU^{∗}(X) onto the direct summand 𝔗_{𝔣}^{ev}(MU^{∗}(X)) ⊂ A ˆ⊗_{R} MU^{∗}(X).

Since MU^{2n+1}(pt) = 0, MU^{2n+1}(ℂP^{∞}) = 0 ∀n ∈ ℤ we see that 𝔗_{𝔣}^{ev}(MU^{∗}(pt)) ⊂ A ⊗_{R} MU^{∗}(pt) = A and 𝔗_{𝔣}^{ev}(MU^{∗}(ℂP^{∞})) ⊂ A ˆ⊗_{R} MU^{∗}(ℂP^{∞}) = A[[u]].

Let F (u, v) ∈ MU^{∗}(ℂP^{∞}×ℂP^{∞}) = R[[u, v]] be the formal product in complex cobordism theory. Clearly, 𝔉(x, y) := 𝔗_{𝔣}^{ev}(F (u, v)) ∈ A[[x, y]] = A[[u, v]] is a formal group law, where x := 𝔣(u), y := 𝔣(v). By the universality of F (u, v) we have the ring homomorphism λ: R → A such that (λ_{∗}F)(x, y) = 𝔉(x, y).

Proposition 11. For X = pt the homomorphism 𝔗_{𝔣}^{ev}: MU^{∗}(pt) → A coincides with the homomorphism λ: R → A, determined by the formal group law 𝔉(x, y) ∈ A[[x, y]].

Proof. Since the transformation 𝔗_{𝔣}^{ev} is multiplicative, we have:

𝔗_{𝔣}^{ev}(F (u, v)) = 𝔗_{𝔣}^{ev}(u) + 𝔗_{𝔣}^{ev}(v) + ∑_{i,j≥1} 𝔗_{𝔣}^{ev}(α_{i,j})𝔗_{𝔣}^{ev}(u)^{i}𝔗_{𝔣}^{ev}(v)^{j}

= x + y + ∑_{i,j≥1} 𝔗_{𝔣}^{ev}(α_{i,j})x^{i}y^{j} = x + y + ∑_{i,j≥1} λ(α_{i,j})x^{i}y^{j}.

Now we must only note that α_{i,j}, i, j ≥ 1 generate R . □

From the other hand, for the geometric cobordism w := F (u, v) ∈ MU^{2}(ℂP^{∞} × ℂP^{∞}) we have:

𝔗_{𝔣}^{ev}(w) = 𝔣(w) = 𝔣(F (u, v)) = 𝔣(F (𝔣^{−1}(x), 𝔣^{−1}(y))) = x + y + ∑_{i,j≥1} α̃_{i,j}x^{i}y^{j}.

It gives us another description of 𝔉(x, y).

Since (ε_{A}𝔣)(u) = u, we see that ε_{A}(α̃_{i,j}) = ε_{A}(λ(α_{i,j})) = α_{i,j}. Hence ε_{A} ∘ λ = id_{R} . This implies that P := λ ∘ ε_{A} : A → A is a projector onto im λ and λ induces a ring isomorphism R ≅ im λ ⊂ A . Set K := im λ. In particular, K is a direct summand in A, generated by the coefficients of 𝔉(x, y).

By H = (H, μ, η, Δ, ε, S) we denote the Hopf algebra MU^{∗}(BSU_{⊗}), where μ: H ˆ⊗_{R} H → H is the multiplication, η: R → H is the unit (map), Δ: H → H ˆ⊗_{R} H is the comultiplication, ε: H → R is the counit (map), and S: H → H is the antipode. It is a graded commutative topological Hopf algebra over R. It is well known that MU^{∗}(BSU) = R[[σ_{2}(Ξ), σ_{3}(Ξ), …]] as an R-algebra. Clearly, we can take H as A.

Corollary 12. There is a unique multiplicative natural transformation 𝔗^{ev} :=

𝔗_{𝔥}^{ev}: MU^{ev}(X) → H ˆ⊗_{R} MU^{ev}(X) such that 𝔗^{ev}(u) = 𝔥(u) = 1 ⊗ u + ∑_{k,i≥1} α_{k,i}s̃_{k} ⊗ u^{i} ∈

H ˆ⊗_{R} MU^{∗}(ℂP^{∞}) for X = ℂP^{∞} and u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}).

Proof. We have: a_{1} = 1 + ∑_{k≥1} α_{k,1}s̃_{k}, a_{i} = ∑_{k≥1} α_{k,i}s̃_{k}, i ≥ 2. We see that the formal power series 𝔣(u) = 𝔥(u) satisfies the conditions of Theorem 9. □

Remark 13. Let us remember that H ˆ⊗_{R} MU^{∗}(ℂP^{∞}) = MU^{∗}(BSU) ˆ⊗_{R} MU^{∗}(ℂP^{∞}) =

MU^{∗}(BU) and 𝔥(u) = σ_{1}(Ξ̂) ∈ MU^{2}(BU), where u = σ_{1}(Z) as above. Therefore 𝔗^{ev}(σ_{1}(Z)) = σ_{1}(Ξ̂) ∈ MU^{2}(BU). Clearly, in the case 𝔣(u) = 𝔥(u) the formal group law ℌ(x, y) := 𝔗^{ev}(F (u, v)) ∈ K[[x, y]] ⊂ H[[x, y]] coincides with the one, defined in Subsection 2.3 of [6].

Corollary 14. For X = pt the homomorphism 𝔗^{ev}: MU^{∗}(pt) → H ⊗_{R} MU^{∗}(pt) = H coincides with the homomorphism λ: R → H, determined by ℌ(x, y) ∈ K[[x, y]].

Proof follows from Proposition 11. Another version of the proof. By R_{ℚ} (resp. H_{ℚ}) denote R ⊗_{ℤ}ℚ (resp. H ˆ⊗_{ℤ}ℚ). It is known [1] that S_{ω}(g(u)) = 0 for any ω ≠ ∅, where g(u) ∈ R_{ℚ}[[u]] is the logarithm of the formal product in complex cobordism theory. Hence 𝔗^{ev}(g(u)) = a_{1} ⊗ g(u). From the other hand, the elements g(u) ∈ R_{ℚ}[[u]] ⊂ H_{ℚ}[[x]] and 𝔤(x) ∈ H_{ℚ}[[x]] coincide (as elements of H_{ℚ}[[x]]), where 𝔤(x) ∈ H_{ℚ}[[x]] is the formal power series, which has been defined in [6]. It is easily shown that the formal power series a_{1}𝔤(x) ∈ H_{ℚ}[[x]] is the logarithm of the formal group law ℌ(x, y). This (together with the multiplicativity of 𝔗^{ev}) implies the required assertion. □ Remark 15. Actually we obtain the following explicit form of the homomorphism λ: R → H. Suppose α ∈ MU^{−2n}(pt); then S_{ω}(α) ∈ MU^{2(|ω|−n)}(pt). Recall that MU^{m}(pt) = 0 if m > 0. Therefore we have:

λ(α) = ∑_{ω, |ω|≤n} a_{1}^{−n}α_{ω} ⊗ S_{ω}(α) ∈ H ⊗_{R} R = H .

2 A natural transformation K̃ SU( )_{⊗} → Aut_{ring}(MU^{ev}( ))

Let X be a finite CW -complex. Let ξ be a virtual SU-bundle over X of virtual dimension 1 and let \{ξ\} ∈ { \atop KSU}(X) be its stable equivalence class. By 𝔗_{\{ξ\}}^{ev} denote the composite homomorphism

MU^{ev}(X) →^{𝔗^{ev}} H ⊗_{R} MU^{ev}(X) ⟶^{φ_{ξ}^{∗}⊗id_{MU^{ev}(X)}} MU^{ev}(X) ⊗_{R} MU^{ev}(X) →^{μ_{X}^{ev}} MU^{ev}(X),

where the homomorphism φ_{ξ}^{∗}: H → MU^{ev}(X) is induced by a classifying map φ_{ξ}: X → BSU for the bundle ξ and μ_{X}^{ev}: MU^{ev}(X) ⊗_{R} MU^{ev}(X) → MU^{ev}(X) is the multiplication, restricted to the subalgebra MU^{ev}(X) ⊂ MU^{∗}(X). Note that for X such that MU^{2n+1}(X) = 0 ∀n ∈ ℤ 𝔗_{{ξ}}^{ev} is a homomorphism MU^{∗}(X) → MU^{∗}(X).

Theorem 16. For any stable equivalence class of SU-bundles {ξ} over X the homomorphism 𝔗_{{ξ}}^{ev}: MU^{ev}(X) → MU^{ev}(X) is a ring isomorphism such that

(i) it is natural on (X, ξ);

(ii) 𝔗_{{ξ}}^{ev}(R) = φ_{ξ}^{∗}(K); (iii) for any line bundle ζ over X we have 𝔗_{{ξ}}^{ev}(σ_{1}(ζ)) = σ_{1}(ξ⊗ζ), where ξ is a representative

of {ξ} of virtual dimension 1 as above.

(Let us remember that K ⊂ H is the subring, generated by the coefficients of the formal group law ℌ(x, y) ∈ H[[x, y]]).

Proof. (i) Suppose (Y, η) is another pair with a morphism f: (Y, η) → (X, ξ) (i.e. f is a continuous map Y → X such that f^{∗}(ξ) = η). We claim that f^{∗} ∘ 𝔗_{{ξ}}^{ev} = 𝔗_{{η}}^{ev} ∘ f^{∗}. Indeed, since φ_{η} = φ_{ξ} ∘ f: Y → BSU is a classifying map for the bundle η, we see that the following digram

\xymatrix{MU^{ev}(X) \ar[d]_{f^{∗}} \ar[r]^{𝔗^{ev}} & H ⊗_{R} MU^{ev}(X) \ar[d]_{id_{H} ⊗f^{∗}} \ar[r]^{φ_{ξ}^{∗}⊗id} & MU^{ev}(X) ⊗_{R} MU^{ev}(X) \ar[d]_{f^{∗}⊗f^{∗}} \ar[r]^{μ_{X}^{ev}} & MU^{ev}(X) \ar[d]^{f^{∗}} \\ MU^{ev}(Y) \ar[r]^{𝔗^{ev}} & H ⊗_{R} MU^{ev}(Y) \ar[r]^{φ_{η}^{∗}⊗id} & MU^{ev}(Y) ⊗_{R} MU^{ev}(Y) \ar[r]^{μ_{Y}^{ev}} & MU^{ev}(Y)}

commutes.

Assertion (ii) follows easily from the naturality of 𝔗^{ev} and Proposition 11.

(iii) Because of the naturality of 𝔗^{ev}, we have the following commutative diagram:

\xymatrix{MU^{2}(ℂP^{∞}) \ar[d]_{φ_{ζ}^{∗}} \ar[r]^{𝔗_{2}} & H ˆ⊗_{R} MU^{ev}(ℂP^{∞}) \ar[d]^{id_{H} ⊗φ_{ζ}^{∗}} \\ MU^{2}(X) \ar[r]^{𝔗_{2}} & H ⊗_{R} MU^{ev}(X),}

where φ_{ζ}: X → ℂP^{∞} is a classifying map for the line bundle ζ. Since 𝔗^{ev}(u) = 𝔥(u) = σ_{1}(Ξ̂), we have: 𝔗^{ev}(σ_{1}(ζ)) = 𝔥(σ_{1}(ζ)) = 1⊗σ_{1}(ζ)+∑_{i,j≥1} α_{i,j}s̃_{i}⊗σ_{1}(ζ)^{j} ∈ H ⊗_{R} MU^{ev}(X). Therefore

𝔗_{{ξ}}^{ev}(σ_{1}(ζ)) = σ_{1}(ζ) + ∑_{i,j≥1} α_{i,j}φ_{ξ}^{∗}(s̃_{i})σ_{1}(ζ)^{j} = (φ_{ξ}^{∗}𝔥)(σ_{1}(ζ)) = φ_{ξ⊗ζ}^{∗}(σ_{1}(Ξ̂)) = σ_{1}(ξ ⊗ ζ), where φ_{ξ⊗ζ}: X → BU is a classifying map for the U-bundle ξ ⊗ ζ over X. □ Remark 17. We do not assert that 𝔗_{{ξ}}^{ev}(σ_{i}(ζ)) = σ_{i}(ξ ⊗ ζ) for i ≥ 2. Otherwise, using Theorem 9 and σ_{i}(ζ) = 0 for i ≥ 2 for any line bundle ζ, we obtain σ_{i}(ξ ⊗ ζ) = 0, i ≥ 2, but clearly this is not true in general.

For example, using formulas, obtained in [6], one can easily verify that

s_{r}(Ξ ⊗ Z) = s_{r}(Ξ) ⊗ 1 + 1 ⊗ s_{r}(Z) + ∑_{m,n≥1} γ_{m,n}^{(r)}s_{m}(Ξ) ⊗ s_{n}(Z)

= s_{r}(Ξ) ⊗ 1 + 1 ⊗ u^{r} + ∑_{m,n≥1} γ_{m,n}^{(r)}s_{m}(Ξ) ⊗ u^{n},  (5)

where the coefficients γ_{m,n}^{(r)} ∈ MU^{2(r−m−n)}(pt) are determined by the equation F (u, v)^{r} = u^{r}+v^{r}+∑_{m,n≥1, m+n≥r} γ_{m,n}^{(r)}u^{m}v^{n}. On the other hand, 𝔗_{{ξ}}^{ev}(s_{r}(Z)) = 𝔗_{{ξ}}^{ev}(u^{r}) = s_{1}(Ξ⊗Z)^{r} = 𝔥(u)^{r}, but it is clear that 𝔥(u)^{r} does not coincide with expression (5).

Actually, we do not use the Hopf algebra structure on H = MU^{∗}(BSU_{⊗}) up to now. But it gives us, for example, the following identity:

(φ_{η⊗ξ}^{∗}ℌ)(x, y) = ((φ_{η}^{∗} ⋆ φ_{ξ}^{∗})ℌ)(x, y)

for two virtual SU-bundles ξ, η over X of virtual dimension 1, where ⋆ denotes the convolution in Hom_{R}(H, MU^{∗}(X)), φ_{η}^{∗} ⋆ φ_{ξ}^{∗} = μ_{MU^{∗}(X)} ∘ (φ_{η} ⊗ φ_{ξ}) ∘ Δ: H → MU^{∗}(X).

Let us define the composite multiplicative natural transformation

(Δ ⊗ id_{S})𝔗^{ev} = (Δ ⊗ id_{MU^{ev}( )}) ∘ 𝔗^{ev}: MU^{ev}( ) → H ˆ⊗_{R} H ˆ⊗_{R} MU^{ev}( ),

where Δ: H → H ˆ⊗_{R} H is the comultiplication in H. Let us remember that the transformation 𝔗^{ev} was defined by means of the elements 𝔗_{2n}^{ev} = ∑_{ω} a_{1}^{n}α_{ω} ⊗ S_{ω} ∈ H ˆ⊗_{ℤ}S, n ∈ ℤ. Similarly, (Δ ⊗ id_{S})𝔗^{ev} can be defined by means of the elements (Δ ⊗ id_{S})𝔗_{2n}^{ev} = ∑_{ω} Δ(a_{1}^{n}α_{ω}) ⊗ S_{ω} ∈

H ˆ⊗_{R} H ˆ⊗_{ℤ}S, n ∈ ℤ.

Since Landweber-Novikov’s algebra S acts on H = MU^{∗}(BSU_{⊗}), we can define the composition ((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev}, where Δ_{S} : S → S ⊗_{ℤ}S is the comultiplication in S [2], [3]. More precisely, the multiplicative natural transformation

((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev}: MU^{ev}( ) → H ˆ⊗_{R} H ˆ⊗_{R} MU^{ev}( )

is given by the elements

((id_{H} ⊗Δ_{S})𝔗_{2n}^{ev}) ∘ 𝔗_{2n}^{ev} = (∑_{ω} a_{1}^{n}α_{ω} ⊗ Δ_{S}(S_{ω})) ∘ (∑_{τ} a_{1}^{n}α_{τ} ⊗ S_{τ})

= ∑_{ω}(a_{1}^{n}α_{ω} ⊗ ∑_{ω′∪ω′′=ω, τ} S_{ω′}(a_{1}^{n}α_{τ}) ⊗ S_{ω′′}S_{τ}) ∈ H ˆ⊗_{R} H ˆ⊗_{ℤ}S, n ∈ ℤ,

where we used the following well-known fact [5]: for every pair of partitions ω′, ω′′, the composition S_{ω′′}S_{ω′} is an integral linear combination of elements S_{ω}.

Theorem 18. ((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev} and (Δ ⊗ id_{S})𝔗^{ev} coincide as a natural transformations

MU^{ev}( ) → H ˆ⊗_{R} H ˆ⊗_{R} MU^{ev}( ).

Remark 19. Now let us introduce some additional notation. For a CW -complex X by C we denote the R-algebra MU^{ev}(X) and by Δ_{C} we denote the homomorphism 𝔗^{ev}: C → H ˆ⊗_{R} C . Using the comultiplication in the Hopf algebra S we can define the homomorphism

Δ_{H ˆ⊗_{R} C} : H ˆ⊗_{R} C → H ˆ⊗_{R} H ˆ⊗_{R} C,

corresponding to (id_{H} ⊗Δ_{S})𝔗^{ev} and such that the following diagram

\xymatrix{H ˆ⊗_{R} C \ar[r]^{Δ_{_{H} ˆ⊗ ; R_{C}}} \ar[d]_{m_{BSU, X}^{ev}} & H ˆ⊗_{R} H ˆ⊗_{R} C \ar[d]^{id_{H} ⊗m_{BSU, X}^{ev}} \\ D \ar[r]^{Δ_{D}} & _{H ˆ⊗_{R} D}}

commutes,  where  D  is  the  R-algebra  MU^{ev}(BSU ×X)  and

m_{BSU, X}^{ev}: MU^{∗}(BSU) ˆ⊗_{R} MU^{ev}(X) → MU^{ev}(BSU ×X) is the canonical homomorphism. We see that theorem’s statement is equivalent to the commutativity of the following diagram (for any X):

\xymatrix{C \ar[r]^{Δ_{C}} \ar[d]_{Δ_{C}} & H ˆ⊗_{R} C \ar[d]^{Δ_{_{H} ˆ⊗ ; R_{C}}} \\ _{H ˆ⊗_{R} C} \ar[r]_{Δ⊗id_{C}} & H ˆ⊗_{R} H ˆ⊗_{R} C .}

Together with the commutative diagram

\xymatrix{R ⊗_{R} C & H ˆ⊗_{R} C \ar[l]_{ε⊗id_{C}} \\  & C, \ar[ul]^{≅} \ar[u]_{Δ_{C}}}

where ε: H → R is the counit in H, this shows that Δ_{C} defines a structure on C which is similar to the structure of a comodule over the Hopf algebra H = MU^{∗}(BSU_{⊗}).

Recall that by Ξ̂ (resp. Ξ) we denote the universal virtual U-bundle over BU of virtual dimension 1 (the universal virtual SU-bundle over BSU of virtual dimension 1 respectively). Theorem’s proof is based on the following lemma.

Lemma 20. For u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}) as above we have:

(((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev})(u) = ((Δ ⊗ id_{S})𝔗^{ev})(u) = σ_{1}(Ξ ⊠ Ξ̂) ∈ MU^{2}(BSU × BU).

Proof of the theorem. Theorem’s assertion follows from the lemma. Indeed, applying Theorem 9 for A = H ˆ⊗_{R} H, 𝔣(u) = (((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev})(u) = ((Δ ⊗ id_{S})𝔗^{ev})(u) = (Δ𝔥)(u) ∈ (H ˆ⊗_{R} H)[[u]] = A[[u]] we see that the multiplicative natural transformations ((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev} and (Δ ⊗ id_{S})𝔗^{ev}: MU^{ev}( ) → A ˆ⊗_{R} MU^{ev}( ) coincide. □

Proof of the lemma. Let us remember that the elements s_{1}(Ξ̂) = σ_{1}(Ξ̂) ∈ MU^{2}(BU) and 𝔗^{ev}(u) = 𝔥(u) ∈ H[[u]] coincide, if we identify MU^{∗}(BU) with H[[u]], u = σ_{1}(Z). From Cartan’s formula it follows that the following diagram

[figure: MU^{∗}(BSU ×ℂP^{∞}); (id_{H} ⊗Δ_{S})𝔗^{ev}; ≅; H ˆ⊗_{R} MU^{∗}(BSU ×ℂP^{∞}); ≅; MU^{∗}(BU); 𝔗^{ev}; H ˆ⊗_{R} MU^{∗}(BU)]

commutes. Hence (((id_{H} ⊗Δ_{S})𝔗^{ev}) ∘ 𝔗^{ev})(u) = 𝔗^{ev}(σ_{1}(Ξ̂)) ∈ H ˆ⊗_{R} MU^{∗}(BU) =

H ˆ⊗_{R} H ˆ⊗_{R} MU^{∗}(ℂP^{∞}). Therefore at first we must compute the element 𝔗^{ev}(σ_{1}(Ξ̂)).

Let Ξ̂_{m} be the universal m-dimensional U(m)-bundle over BU(m). Consider the diagram

\xymatrix{Ξ̂_{m} \ar[d] & Z_{1} ⊕ … ⊕ Z_{m} \ar[l] \ar[d] \\ BU(m) & ℂP^{∞} × … × ℂP^{∞}, \ar[l]_{μ̂_{m}}}

where μ̂_{m}: (ℂP^{∞})^{m} → BU(m) is the “splitting map”, μ̂_{m}^{∗}(Ξ̂_{m}) = Z_{1} ⊕ … ⊕ Z_{m}. Using the splitting principle, we have:

μ̂_{m}^{∗}(s_{1}(Ξ̂_{m})) = σ_{1}(Z_{1}) + … + σ_{1}(Z_{m}) = u_{1} + … + u_{m}.

Because of the naturality of 𝔗^{ev}, we have:

(id_{H} ⊗μ̂_{m}^{∗})(𝔗^{ev}(s_{1}(Ξ̂_{m}))) = 𝔥(u_{1}) + … + 𝔥(u_{m}) = 1 ⊗ ( ∑_{1≤i≤m} u_{i}) + ∑_{k,l≥1} α_{k,l}s̃_{k} ⊗ ( ∑_{1≤i≤m} u_{i}^{l}) .

Therefore we get:

𝔗^{ev}(s_{1}(Ξ̂_{m})) = 1 ⊗ s_{1}(Ξ̂_{m}) + ∑_{k,l≥1} α_{k,l}s̃_{k} ⊗ s_{l}(Ξ̂_{m}).

Now, passing to the limit as m → ∞, we obtain:

𝔗^{ev}(s_{1}(Ξ̂)) = 1 ⊗ s_{1}(Ξ̂) + ∑_{k,l≥1} α_{k,l}s̃_{k} ⊗ s_{l}(Ξ̂).  (6)

It was proved in [6] that (6) coincides with σ_{1}(Ξ ⊠ Ξ̂) ∈ MU^{2}(BSU × BU), i.e. with σ_{1}(Ξ ⊠ Ξ ⊠ Z) ∈ MU^{2}(BSU × BSU ×ℂP^{∞}).

From the other hand, we have ϕ^{∗}(Ξ) = Ξ ⊠ Ξ, where ϕ: BSU_{⊗} × BSU_{⊗} → BSU_{⊗} is the multiplication in the H-group BSU_{⊗} and Δ = ϕ^{∗}: H → H ˆ⊗_{R} H. Therefore we obtain ((Δ ⊗ id_{S})𝔗^{ev})(u) = σ_{1}(Ξ ⊠ Ξ ⊠ Z) (let us remark that the element ((Δ ⊗ id_{S})𝔗^{ev})(u) can also be expressed as (Δ𝔥)(u) ∈ H ˆ⊗_{R} H ˆ⊗_{R} MU^{∗}(ℂP^{∞}) = MU^{∗}(BSU × BU)). □

Recall that by \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} we denote the group [X, BSU_{⊗}] of stable equivalence classes of SU-bundles over X with respect to the operation, induced by the tensor product of virtual bundles of virtual dimension 1.

Corollary 21. For any virtual SU-bundles ξ, η over X of virtual dimension 1 we have 𝔗_{\{η\}}^{ev} ∘ 𝔗_{\{ξ\}}^{ev} = 𝔗_{\{η⊗ξ\}}^{ev}. In particular, we obtain the group homomorphism \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} → Aut_{ring}(MU^{ev}(X)), {ξ} ↦ 𝔗_{{ξ}}^{ev}, which is natural on X.

Proof. We shall use the following notation: C := MU^{ev}(X), Δ_{C} := 𝔗^{ev}: C → H ˆ⊗_{R} C, φ :=

φ_{ξ}^{∗}, ψ := φ_{η}^{∗}: H → C, τ_{φ} := μ_{C} ∘ (φ ⊗ id_{C}), τ_{ψ} := μ_{C} ∘ (ψ ⊗ id_{C}): H ˆ⊗_{R} C → C, 𝔗_{φ}^{ev} := 𝔗_{{ξ}}^{ev} =

τ_{φ} ∘ Δ_{C}, 𝔗_{ψ}^{ev} := 𝔗_{{η}}^{ev} = τ_{ψ} ∘ Δ_{C}. Because of φ_{η⊗ξ}^{∗} = φ_{η}^{∗} ⋆ φ_{ξ}^{∗}, we see that 𝔗_{{η⊗ξ}}^{ev} = 𝔗_{ψ⋆φ}^{ev} in the just introduced notation, where ⋆ is the convolution in Hom_{R}(H, C).

It can easily be checked that the diagram

\xymatrix{C \ar[r]^{Δ_{C}} \ar[dr]_{𝔗_{φ}^{ev}} & H ˆ⊗_{R} C \ar[d]^{τ_{φ}} \ar[r]^{Δ_{_{H} ˆ⊗ ; R_{C}}} & H ˆ⊗_{R} H ˆ⊗_{R} C \ar[r]^{ψ⊗φ⊗id_{C}} \ar[d]^{id_{H} ⊗τ_{φ}} & C ˆ⊗_{R} C ˆ⊗_{R} C \ar[d]^{μ_{C}⊗id_{C}} \\  & C \ar[r]^{Δ_{C}} \ar[dr]_{𝔗_{ψ}^{ev}} & _{H ˆ⊗_{R} C} \ar[d]^{τ_{ψ}} & _{C ˆ⊗_{R} C} \ar[dl]^{μ_{C}} \\  &  & C}

commutes. For example, the commutativity of the square easily follows from the commutative diagram

\xymatrix{H ˆ⊗_{R} C \ar[d]_{φ⊗id_{C}} \ar[r]^{Δ_{_{H} ˆ⊗ ; R_{C}}} & H ˆ⊗_{R} H ˆ⊗_{R} C \ar[d]^{id_{H} ⊗φ⊗id_{C}} \\ _{C ˆ⊗_{R} C} \ar[r]^{Δ_{_{C} ˆ⊗ ; R_{C}}} \ar[d]_{μ_{C}} & _{H ˆ⊗_{R} C ˆ⊗_{R} C} \ar[d]^{id_{H} ⊗μ_{C}} \\ C \ar[r]_{Δ_{C}} & H ˆ⊗_{R} C .}

Then, using Remark 19, we have:

𝔗_{ψ}^{ev} ∘ 𝔗_{φ}^{ev} = μ_{C} ∘ (μ_{C} ⊗ id_{C}) ∘ (ψ ⊗ φ ⊗ id_{C}) ∘ Δ_{H ˆ⊗_{R} C} ∘ Δ_{C}

= μ_{C} ∘ (μ_{C} ⊗ id_{C}) ∘ (ψ ⊗ φ ⊗ id_{C}) ∘ (Δ ⊗ id_{C}) ∘ Δ_{C} = 𝔗_{ψ⋆φ}^{ev}. □

Recall that by K_{ξ} we denote the subring φ_{ξ}^{∗}(K) ⊂ MU^{ev}(X).

Corollary 22. Suppose ξ, η are virtual SU-bundles over X of virtual dimension 1. Then 𝔗_{η}^{ev}(σ_{1}(ξ ⊗ ζ)) = σ_{1}(η ⊗ ξ ⊗ ζ) for any line bundle ζ over X. Furthermore, 𝔗_{η}^{ev}(K_{ξ}) = K_{η⊗ξ} .

For a pair (X, ξ) consisting of a CW -complex X and of a virtual SU-bundle ξ over X of virtual dimension 1 we can also define the natural transformation

𝔗_{X,{ξ}}^{ev} := (φ_{ξ}^{∗} ⊗ id_{S})𝔗^{ev}: MU^{ev}(Z) → MU^{ev}(X) ˆ⊗_{R} MU^{ev}(Z),

where Z is an arbitrary CW -complex. In a different way 𝔗_{X,{ξ}}^{ev} can be defined by application of Theorem 9 for A = MU^{ev}(X) and 𝔣(u) = (φ_{ξ}^{∗}𝔥)(u) ∈ A[[u]]. If we take Z = ℂP^{∞}×ℂP^{∞}, then the formal power series 𝔗_{X,{ξ}}^{ev}(F (u, v)) = (φ_{ξ}^{∗}ℌ)(x, y) ∈ MU^{ev}(X)[[x, y]] ⊂ MU^{∗}(X)[[x, y]] describes the relation between first Conner-Floyd classes of lifts ξ̂ of the bundle ξ (see [6]), where F (u, v) is the formal product in complex cobordism theory and x = 𝔗_{X,{ξ}}^{ev}(u) = (φ_{ξ}^{∗}𝔥)(u), y = 𝔗_{X,{ξ}}^{ev}(v) = (φ_{ξ}^{∗}𝔥)(v) (recall that a virtual U-bundle ξ̂ of virtual dimension 1 over X is a lift of ξ if ξ̂= ξ ⊗ζ for some geometric line bundle ζ over X).

Let ℜ_{X,{ξ}}^{ev} be the natural transformation

MU^{ev}(Z) →^{𝔗_{X,{ξ}}^{ev}} MU^{ev}(X) ˆ⊗_{R} MU^{ev}(Z) →^{m_{X,Z}^{ev}} MU^{ev}(X × Z),

where m_{X, Z}^{ev} is the canonical homomorphism.

Let (Y, η) be another pair. Using the commutativity of the diagram

[figure: H ˆ⊗_{R} MU(Z); φ_{ξ}^{∗}⊗id; (id_{H} ⊗Δ_{S})𝔗^{ev}; H ˆ⊗_{R} H ˆ⊗_{R} MU(Z); φ_{η}^{∗}⊗φ_{ξ}^{∗}⊗id; MU(Z); 𝔗^{ev}; 𝔗_{X,{ξ}}^{ev}; ℜ_{X,{ξ}}^{ev}; MU(X) ˆ⊗_{R} MU(Z); m_{X,Z}^{ev}; (φ_{η}^{∗}⊗Δ_{S})𝔗^{ev}; MU(Y) ˆ⊗_{R} MU(X) ˆ⊗_{R} MU(Z); m_{Y,X}^{ev}⊗id; id ⊗m_{X,Z}^{ev}; MU(X×Z); ℜ_{Y,{η}}^{ev}; (φ_{η}^{∗}⊗id_{S})𝔗^{ev}; MU(Y) ˆ⊗_{R} MU(X×Z); m_{Y,X×Z}^{ev}; MU(Y ×X) ˆ⊗_{R} MU(Z); m_{Y ×X,Z}^{ev}; MU(Y ×X×Z)]

(here we write MU( ) instead of MU^{ev}( )) together with Theorem 18 and the commutative diagram

\xymatrix{H \ar[r]^{Δ} \ar[d]_{φ_{η⊠ξ}^{∗}} & H ˆ⊗_{R} H \ar[d]^{φ_{η}^{∗}⊗φ_{ξ}^{∗}} \\ MU^{ev}(Y × X) & MU^{ev}(Y) ˆ⊗_{R} MU^{ev}(X) \ar[l]^{m_{Y,X}^{ev}}}

one can easily prove the following proposition.

Proposition 23. For any pairs (X, ξ) and (Y, η) as above we have

ℜ_{Y,{η}}^{ev} ∘ ℜ_{X,{ξ}}^{ev} = ℜ_{Y ×X,{η⊠ξ}}^{ev},

where η ⊠ ξ is the exterior tensor product of bundles (which is a bundle over Y × X).

3 An extended natural transformation K̃ SU( )_{⊗} → Aut_{ring}(MU[\frac{1}{2}]^{∗}( ))

Now we want to extend the natural transformation 𝔗^{ev}: MU^{ev}( ) → H ˆ⊗_{R} MU^{ev}( ) to whole cobordism ring MU^{∗}( ). Let A be a graded topological algebra over R as in the statement of Theorem 9. By MU[\frac{1}{2}]^{∗}( ) denote the localization in 2 (i.e. 2 is invertible) of complex cobordism theory. Set A[\frac{1}{2}] := A ˆ⊗_{ℤ}ℤ[\frac{1}{2}].

The following theorem generalizes Theorem 9.

Theorem 24. Given a formal power series 𝔣(u) = ∑_{i≥1} a_{i} ⊗u^{i} ∈ A ˆ⊗_{R} MU^{∗}(ℂP^{∞}) such that

(i) deg a_{i} = 2 − 2i;

(ii) ε_{A}(a_{1}) = 1, ε_{A}(a_{i}) = 0, i ≥ 2

there is a multiplicative natural transformation 𝔗_{𝔣}: MU^{∗}(X) → A[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X) such that a) 𝔗_{𝔣}(u) = 𝔣(u) for X = ℂP^{∞} and u ∈ MU^{2}(ℂP^{∞}) as above;

b) its restriction 𝔗_{𝔣}|_{MU^{ev}(X)} : MU^{ev}(X) → A[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X) can be decomposed into the composition of the natural transformation 𝔗_{𝔣}^{ev}: MU^{ev}(X) → A ˆ⊗_{R} MU^{ev}(X) and of the homomorphism, induced by the canonical homomorphism A → A[\frac{1}{2}].

Furthermore, using the universal property of the localization, we obtain the natural ring homomorphism 𝔗_{𝔣}[\frac{1}{2}]: MU[\frac{1}{2}]^{∗}(X) → A[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X) which is a ring isomorphism onto a direct summand in A[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X).

In order to prove the theorem we need the following lemma.

Lemma 25. For any element a ∈ A such that ε_{A}(a) = 1 there is κ ∈ A[\frac{1}{2}] such that κ^{2} = a.

Proof of the lemma. We have:

r(t) := \sqrt{1 + t} = 1 + \frac{t}{2} + ∑_{n≥2}(−1)^{n−1} \frac{(2n − 3)!!}{2^{n}n!}t^{n} ∈ ℤ[\frac{1}{2}][[t]],

where (2n − 3)!! = ∏_{k=1}^{n−1}(2k − 1). In other words, the minimal subring in ℚ, containing the coefficients of r(t), is ℤ[\frac{1}{2}]. Now note that a = 1 + b, where b ∈ ker(ε_{A}). Finally, using the completeness of A with respect to the ideal ker(ε_{A}), we see that r(b) ∈ A[\frac{1}{2}] is the element such that r(b)^{2} = a. □ Proof of the theorem. The proof is very similar to the one of Theorem 9 and we give only a sketch.

Note that a_{1} satisfies the condition of the previous lemma. Therefore we have the element κ ∈ A[\frac{1}{2}] such that κ^{2} = a_{1}. For any n ∈ ℤ we set 𝔗_{𝔣,n} := (κ^{n} ⊗ 1)𝔗_{𝔣,0} = ∑_{ω}(κ^{n}α_{ω}) ⊗ S_{ω} ∈ A[\frac{1}{2}] ˆ⊗_{ℤ}S . Given a homogeneous element α ∈ MU^{n}(X) we define the value 𝔗_{𝔣}(α) as 𝔗_{𝔣,n}(α) = ∑_{ω}(κ^{n}α_{ω}) ⊗ S_{ω}(α) ∈ A[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X). Clearly, if n = 2m, m ∈ ℤ, then 𝔗_{𝔣}(α) = 𝔗_{𝔣}^{ev}(α).

The multiplicativity and the naturality of the transformation 𝔗_{𝔣} can be verified in just the same way as in Theorem 9. □

Corollary 26. There is a unique multiplicative natural transformation 𝔗 :=

𝔗_{𝔥}: MU^{∗}(X) → H[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(X) such that 1) 𝔗(u) = 𝔥(u) = 1 ⊗ u + ∑_{k,i≥1} α_{k,i}s̃_{k} ⊗ u^{i} ∈

H[\frac{1}{2}] ˆ⊗_{R} MU^{∗}(ℂP^{∞}) for X = ℂP^{∞}, u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}) and 2) the composition (ε[\frac{1}{2}] ⊗ id_{MU^{∗}(X)}) ∘ 𝔗 coincides with the localization homomorphism MU^{∗}(X) → MU[\frac{1}{2}]^{∗}(X) (an equivalent condition: the composition (ε[\frac{1}{2}] ⊗ id_{MU^{∗}(X)}) ∘ 𝔗[\frac{1}{2}] is the identity map id_{MU[ \frac{1}{2} ](X)}).

Let X be a finite CW -complex. Let ξ be a virtual SU-bundle over X of virtual dimension 1 and let \{ξ\} ∈ { \atop KSU}(X) be its stable equivalence class. By 𝔗_{\{ξ\}} denote the composite homomorphism

MU^{∗}(X) →^{𝔗} H[\frac{1}{2}] ⊗_{R} MU^{∗}(X) ⟶^{φ_{ξ}^{∗}⊗id_{MU^{∗}(X)}} MU[\frac{1}{2}]^{∗}(X) ⊗_{R} MU^{∗}(X) →^{μ_{X}} MU[\frac{1}{2}]^{∗}(X),

where φ_{ξ}^{∗}: H[\frac{1}{2}] → MU[\frac{1}{2}]^{∗}(X) is induced by a classifying map φ_{ξ}: X → BSU for the bundle ξ and μ_{X}: MU[\frac{1}{2}]^{∗}(X) ⊗_{R} MU^{∗}(X) → MU[\frac{1}{2}]^{∗}(X) is the multiplication. By 𝔗_{\{ξ\}}[\frac{1}{2}] denote the corresponding homomorphism MU[\frac{1}{2}]^{∗}(X) → MU[\frac{1}{2}]^{∗}(X).

Theorem 27. For any stable equivalence class of SU-bundles {ξ} over X we have:

(i) 𝔗_{{ξ}}(R) = φ_{ξ}^{∗}(K);

(ii) for any line bundle ζ over X 𝔗_{{ξ}}(σ_{1}(ζ)) = σ_{1}(ξ ⊗ ζ), where ξ is a representative of {ξ} as above (note that here we abuse notation by letting σ_{1}(ξ ⊗ ζ) denote the element in MU[\frac{1}{2}]^{∗}(X)).

Moreover, the homomorphism 𝔗_{\{ξ\}}[\frac{1}{2}]: MU[\frac{1}{2}]^{∗}(X) → MU[\frac{1}{2}]^{∗}(X) is a ring automorphism.

Note that the pair (MU[\frac{1}{2}]^{∗}(X), 𝔗_{\{ξ\}}[\frac{1}{2}]) is natural on (X, ξ). Indeed, suppose (Y, η) is another pair with a morphism f: (Y, η) → (X, ξ) (i.e. f is a continuous map Y → X such that f^{∗}(ξ) = η). We claim that f^{∗} ∘ 𝔗_{{ξ}}^{X} = 𝔗_{{η}}^{Y} ∘ f^{∗}, where to avoid a confusion we denoted 𝔗_{{ξ}}, 𝔗_{{η}} by 𝔗_{{ξ}}^{X}, 𝔗_{{η}}^{Y} respectively. Indeed, since φ_{η} = φ_{ξ} ∘ f: Y → BSU is a classifying map for the bundle η, we see that the diagram

\xymatrix{MU^{∗}(X) \ar[d]_{f^{∗}} \ar[r]^{𝔗} & H[\frac{1}{2}] ⊗_{R} MU^{∗}(X) \ar[d]_{id_{H[ \frac{1}{2} ]} ⊗f^{∗}} \ar[r]^{φ_{ξ}^{∗}⊗id} & MU[\frac{1}{2}]^{∗}(X) ⊗_{R} MU^{∗}(X) \ar[d]_{f^{∗}⊗f^{∗}} \ar[r]^{μ_{X}} & MU[\frac{1}{2}]^{∗}(X) \ar[d]^{f^{∗}} \\ MU^{∗}(Y) \ar[r]^{𝔗} & H[\frac{1}{2}] ⊗_{R} MU^{∗}(Y) \ar[r]^{φ_{η}^{∗}⊗id} & MU[\frac{1}{2}]^{∗}(Y) ⊗_{R} MU^{∗}(Y) \ar[r]^{μ_{Y}} & MU[\frac{1}{2}]^{∗}(Y)}

commutes. This implies that the pair (MU[\frac{1}{2}]^{∗}(X), 𝔗_{\{ξ\}}[\frac{1}{2}]) is natural on (X, ξ).

Let us remember that the transformation 𝔗 was defined by means of the elements 𝔗_{n} ∈ H[\frac{1}{2}] ˆ⊗_{ℤ}S, n ∈ ℤ. Therefore the elements (Δ[\frac{1}{2}] ⊗ id_{S})𝔗_{n} ∈ H[\frac{1}{2}] ˆ⊗_{R[ \frac{1}{2} ]} H[\frac{1}{2}] ˆ⊗_{ℤ}S, n ∈ ℤ (where

Δ[\frac{1}{2}]: H[\frac{1}{2}] → H[\frac{1}{2}] ˆ⊗_{R[ \frac{1}{2} ]} H[\frac{1}{2}] is the extension of Δ to H[\frac{1}{2}]) determine the multiplicative

natural transformation

(Δ[\frac{1}{2}] ⊗ id_{S})𝔗: MU^{∗}( ) → H[\frac{1}{2}] ˆ⊗_{R[ \frac{1}{2} ]} H[\frac{1}{2}] ˆ⊗_{R} MU^{∗}( ).

Since Landweber-Novikov’s algebra acts on H and therefore on H[\frac{1}{2}], we can define the composition ((id_{H[ \frac{1}{2} ]} ⊗Δ_{S})𝔗) ∘ 𝔗. More explicitly, we have

((id_{H[ \frac{1}{2} ]} ⊗Δ_{S})𝔗_{n}) ∘ 𝔗_{n} = (∑_{ω} κ^{n}α_{ω} ⊗ Δ_{S}(S_{ω})) ∘ (∑_{τ} κ^{n}α_{τ} ⊗ S_{τ})

= ∑_{ω}(κ^{n}α_{ω} ⊗ ∑_{ω′∪ω′′=ω, τ} S_{ω′}(κ^{n}α_{τ}) ⊗ S_{ω′′}S_{τ}) ∈ H[\frac{1}{2}] ˆ⊗_{R[ \frac{1}{2} ]} H[\frac{1}{2}] ˆ⊗_{ℤ}S,

where we used the following well-known fact [5]: for every pair of partitions ω′, ω′′, the composition S_{ω′′}S_{ω′} is an integral linear combination of elements S_{ω}.

Theorem 28. ((id_{H} ⊗Δ_{S})𝔗) ∘ 𝔗 and (Δ[\frac{1}{2}] ⊗ id_{S})𝔗 coincide as a natural transformations

MU^{∗}( ) → H[\frac{1}{2}] ˆ⊗_{R[ \frac{1}{2} ]} H[\frac{1}{2}] ˆ⊗_{R} MU^{∗}( ).

Corollary 29. For any virtual SU-bundles ξ, η over X of virtual dimension 1 𝔗_{\{η\}}[\frac{1}{2}] ∘ 𝔗_{\{ξ\}}[\frac{1}{2}] = 𝔗_{\{η⊗ξ\}}[\frac{1}{2}]: MU[\frac{1}{2}]^{∗}(X) → MU[\frac{1}{2}]^{∗}(X). In particular, we obtain a group homomorphism \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} → Aut_{ring}(MU[\frac{1}{2}]^{∗}(X)), \{ξ\} ↦ 𝔗_{\{ξ\}}[\frac{1}{2}], which is natural on X.

Remark 30. It can be reasonable to treat the functor

(X, ξ) ↦ \{K_{ξ} -algebra MU[\frac{1}{2}]^{∗}(X)\}

as an extension of the functor of (localized) complex cobordism theory X ↦ MU[\frac{1}{2}]^{∗}(X) to the homotopy category of pairs (X, ξ). Using the isomorphism 𝔗_{\{ξ\}}[\frac{1}{2}], we can define relevant cohomology operations S_{ω}^{\{ξ\}} acting on K_{ξ} -algebra MU[\frac{1}{2}]^{∗}(X). More precisely, set S_{ω}^{\{ξ\}} := 𝔗_{\{ξ\}}[\frac{1}{2}] ∘ S_{ω} ∘ (𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}, where S_{ω} is usual Landweber-Novikov’s operation. Clearly, S_{ω}^{\{ξ\}} preserves K_{ξ} ⊂ MU[\frac{1}{2}]^{∗}(X). Furthermore, such operations S_{ω}^{\{ξ\}} form an algebra S^{\{ξ\}}, which is isomorphic to Landweber-Novikov’s algebra S . Moreover, K_{ξ} -algebra MU[\frac{1}{2}]^{∗}(X) is Milnor’s module over S^{{ξ}}. Indeed, we have:

S_{ω}^{\{ξ\}}(αβ) = (𝔗_{\{ξ\}}[\frac{1}{2}] ∘ S_{ω})((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(αβ))
= (𝔗_{\{ξ\}}[\frac{1}{2}] ∘ S_{ω})((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(α)(𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(β))

= 𝔗_{\{ξ\}}[\frac{1}{2}]( ∑_{ω′∪ω′′=ω} S_{ω′}((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(α))S_{ω′′}((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(β))

= ∑_{ω′∪ω′′=ω} 𝔗_{\{ξ\}}[\frac{1}{2}](S_{ω′}((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(α)))𝔗_{\{ξ\}}[\frac{1}{2}](S_{ω′′}((𝔗_{\{ξ\}}[\frac{1}{2}])^{−1}(β)))

= ∑_{ω′∪ω′′=ω} S_{ω′}^{{ξ}}(α)S_{ω′′}^{{ξ}}(β).

For a pair (X, ξ) consisting of a CW -complex X and of a virtual SU-bundle ξ over X of virtual dimension 1 we can also define the natural transformation

𝔗_{X,\{ξ\}} := (φ_{ξ}^{∗} ⊗ id_{S})𝔗: MU^{∗}(Z) → MU[\frac{1}{2}]^{∗}(X) ˆ⊗_{R} MU^{∗}(Z),

where Z is an arbitrary CW -complex. In a different way 𝔗_{X,{ξ}} can be defined by application of Theorem 24 for A = MU^{∗}(X) and 𝔣(u) = (φ_{ξ}^{∗}𝔥)(u) ∈ A[[u]].

Let ℜ_{X,{ξ}} be the composite natural transformation

MU[\frac{1}{2}]^{∗}(Z) →^{𝔗_{X,\{ξ\}}[ \frac{1}{2} ]} MU[\frac{1}{2}]^{∗}(X) ˆ⊗_{R[ \frac{1}{2} ]} MU[\frac{1}{2}]^{∗}(Z) →^{m_{X,Z}} MU[\frac{1}{2}]^{∗}(X × Z),

where m_{X, Z} is the canonical homomorphism.

Corollary 31. For any pairs (X, ξ) and (Y, η) as above we have

ℜ_{Y,{η}} ∘ ℜ_{X,{ξ}} = ℜ_{Y ×X,{η⊠ξ}},

where η ⊠ ξ is the exterior tensor product of bundles.

Remark 32. Let us remember that the spectrum (MU, u) (where u ∈ MU^{2}(ℂP^{∞}) as above) is the universal ℂ-oriented spectrum [5]. Therefore all previous constructions can be extended to any ℂ-oriented spectrum (E, u^{E}). Clearly, in the case (H(ℤ), u^{H}) (where u^{H} is the generator of H^{2}(ℂP^{∞}; ℤ) = H^{2}(K(ℤ , 2); ℤ) ≅ ℤ, corresponding to 1 ∈ ℤ) our theory is trivial. But this is also true in the case of complex K-theory (K_{ℂ}, u^{K^{ℂ}}). Indeed, it is well known [5] that the formal product in (K_{ℂ}, u^{K^{ℂ}}) has the form F^{K^{ℂ}}(x, y) = x + y + σ^{−1}xy, where σ ∈ K̃_{ℂ}^{2}(S^{0}) is the Bott element. Since Ξ is an SU-bundle, we have s̃_{1}^{K_{ℂ}} := s_{1}^{K_{ℂ}}(Ξ) = 0, where s_{1}^{K_{ℂ}} is the first Newton characteristic class in the theory (K_{ℂ}, u^{K^{ℂ}}). Now using the universality of (MU, u) and the formula (2) we see that σ_{1}^{K_{ℂ}}(Ξ̂) = σ_{1}^{K_{ℂ}}(Z) = u^{K^{ℂ}}.

4 Appendix: A short proof of an isomorphism of some H-groups

Lemma 33. There exists an isomorphism of H-groups: BSU_{⊗} ×ℂP^{∞} ≅ BU_{⊗} .

Proof. Let λ: BSU → BU be a classifying map for the universal SU-bundle Ξ, κ: ℂP^{∞} → BU a classifying map for the universal line bundle Z. Consider the composite map

BSU ×ℂP^{∞} ⟶^{λ×κ} BU × BU →^{ϕ} BU,

where ϕ is the multiplication in the H-group BU. Clearly, ϕ ∘ (λ × κ) is a classifying map for the virtual bundle Ξ ⊠ Z (recall that Ξ assumed to be of virtual dimension 1). Now it follows easily that ϕ ∘ (λ × κ) is a homotopy equivalence BSU ×ℂP^{∞} ≃ BU and moreover it is an isomorphism of H-groups. □

References

[1] B.I. Botvinnik, V.M. Buchstaber, S.P. Novikov, S.A. Yuzvinskiĭ Algebraic aspects of multiplication theory in complex cobordisms. Russian Math. Surveys 55 (2000), no. 4, 613–633.

[2] P.S. Landweber Cobordism operations and Hopf algebras. Trans. Amer. Math. Soc. 129, 94-110 (1967).

[3] S.P. Novikov The Methods of Algebraic Topology from the viewpoint of Cobordism Theory. Math. USSR Izvestija 1 (4) 827-913 (1967).

[4] D. Quillen On the formal group law of unoriented and complex cobordism theory.— Bull. Amer. Math. Soc., 75:6 (1969), 1293–1298.

[5] Yu.B. Rudyak On Thom spectra, orientability, and cobordism. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[6] A.V. Ershov Formal group laws over Hopf algebras and their application to complex cobordism theory. MPI-Preprint, 2002 (39).