Formal group laws over Hopf algebras and their application to complex cobordism theory

A.V. Ershov^{∗}

Abstract

It is well known that the formal product in complex cobordism theory describes the relation between first Conner-Floyd classes σ_{1} of two complex line bundles with the σ_{1} of their tensor product. In the present paper we generalize this construction to the case of virtual U-bundles of virtual dimension 1. To describe the relation between first Conner-Floyd classes in this case we need more general concept of a formal group law. In fact, we develop some algebraic technique related to generalized formal group laws with coefficients in a Hopf algebra. The obtained result may be considered as a straightforward solution of the problem, posed by S.P. Novikov: how does one compute the class σ_{1}(ξ⊗η) for two U-bundles (not necessarily U(1))? It is the first generalization of the formal product in complex cobordism theory.

Let Z (resp. Ξ̂) be the universal line bundle over ℂP^{∞} (resp. the universal U-bundle over BU). Then σ_{1}(Ξ̂) = 𝔥(σ_{1}(Z)) for some power series 𝔥(u) ∈ MU^{∗}(BSU)[[u]]. Let ℌ(x, y) be the power series 𝔥(F (𝔥^{−1}(x), 𝔥^{−1}(y))) ∈ MU^{∗}(BSU)[[x, y]], where F (u, v) ∈ MU^{∗}(pt)[[u, v]] is the formal product in complex cobordism theory. Then for any stable equivalence class \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X) of SU-bundles over a finite CW -complex X we can define a formal group law (φ_{ξ}^{∗}ℌ)(x, y) over the ring MU^{∗}(X), where φ_{ξ}: X → BSU is a classifying map for a representative ξ of {ξ}. Suppose ξ has virtual dimension 1. We have shown that σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) = (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})) for virtual U-bundles ξ̂_{1}, ξ̂_{2} over X of virtual dimension 1, corresponding to the virtual SU-bundle ξ (i.e. differ from ξ by tensoring on a line bundle). In general the formal group law (φ_{ξ}^{∗}ℌ)(x, y) has coefficients in MU^{∗}(X), but not MU^{∗}(pt). In particular, for any stable equivalence class {ξ} we obtain a direct summand subring φ_{ξ}^{∗}(K) ≅ MU^{∗}(pt) in MU^{∗}(X), generated by the coefficients of (φ_{ξ}^{∗}ℌ)(x, y).

Contents

1 General properties of formal group laws over Hopf algebras 4 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Logarithms of formal group laws over Hopf algebras . . . . . . . . . . . . . . 9 1.3 The functor, represented by a formal group law over a Hopf algebra . . . . . 14 2 A generalization of the formal product in complex cobordism theory 22

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

2.1 The basic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 The formal group law over H = MU^{∗}(BSU_{⊗}) related to virtual bundles of

virtual dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Formal group laws related to stable equivalence classes of SU-bundles …. 32

3 Appendix: An isomorphism of some H-groups 36

Introduction

It is well known that the set Lin(X) of line bundles over a finite CW -complex X is a group with respect to the tensor product of bundles. Moreover, the assignment

ζ ↦ c_{1}(ζ), Lin(X) → H^{2}(X; ℤ)

is an isomorphism of groups. The first Conner-Floyd class σ_{1} (in other terminology, the first Chern class in complex cobordism theory) also defines an injection

ζ ↦ σ_{1}(ζ), Lin(X) → MU^{2}(X),

whose image consists of so-called “geometric cobordism classes”, but the relation between σ_{1}(ζ_{1}), σ_{1}(ζ_{2}) and σ_{1}(ζ_{1} ⊗ ζ_{2}) is more complicated. More precisely, there exists the formal group law (“formal product in complex cobordism theory”) F (u, v) ∈ MU^{∗}(pt)[[u, v]] such that σ_{1}(ζ_{1} ⊗ ζ_{2}) = F (σ_{1}(ζ_{1}), σ_{1}(ζ_{2})) [6]. Moreover, the formal group law F (u, v) is universal [7].

This paper is devoted to a generalization of the formal product in complex cobordism theory to the case of virtual U-bundles of virtual dimension 1. Let us recall that for a finite CW -complex X the tensor product of virtual bundles of virtual dimension 1 induces on K̃_{ℂ}(X) the structure of a group. We shall denote this group by K̃_{ℂ}(X)_{⊗}. We stress that K̃_{ℂ}(X) consists of stable equivalence classes of virtual bundles over X of virtual dimension 0. Let ξ̂ be a virtual U-bundle over X of virtual dimension 1. By {ξ̂} denote the stable equivalence class in K̃_{ℂ}(X) of 0-dimensional virtual bundle ξ̂−[1], where [1] is the trivial line bundle. The tensor product of virtual bundles of virtual dimension 1 induces the following operation ∘ on the set K̃_{ℂ}(X):

{ξ̂_{1}} ∘ {ξ̂_{2}} = {ξ̂_{1}} + {ξ̂_{2}} + {ξ̂_{1}} · {ξ̂_{2}},

where +, · are the sum and the product in the ring K̃_{ℂ}(X) respectively. By Lin(X)_{0} denote the set {{ζ} | ζ is a geometric line bundle over X} ⊂ K̃_{ℂ}(X) (i.e. every class of Lin(X)_{0} has a representative of the form ζ − [1], where ζ is a geometric line bundle over X). Note that Lin(X)_{0} is a subgroup in K̃_{ℂ}(X)_{⊗}.

Let Ξ̂_{m} be the universal m-dimensional U(m)-bundle over BU(m). If m ≤ n, then Ξ̂_{n}|_{BU(m)} = Ξ̂_{m} ⊕ [n − m], where [k] is the trivial k-dimensional bundle. Due to (Ξ̂_{n} − [n − 1])|_{BU(m)} = Ξ̂_{m} − [m − 1], we can define the universal virtual U-bundle Ξ̂ over BU of virtual dimension 1 such that Ξ̂|_{BU(n)} = Ξ̂_{n} − [n − 1] for all n ∈ ℕ.

The functor X ↦ K̃_{ℂ}(X)_{⊗} is represented by the H-group BU_{⊗}. By ϕ̂ denote the product:

ϕ̂: BU_{⊗} × BU_{⊗} → BU_{⊗} .

It is known that the H-group BU_{⊗} is isomorphic to the product of H-groups BSU_{⊗} ×ℂP^{∞} (see the Appendix). Suppose x := σ_{1}(Ξ̂) ∈ MU^{2}(BU), R := MU^{∗}(pt), H := MU^{∗}(BSU_{⊗}). Then H is a Hopf algebra over R. Clearly the element ϕ̂^{∗}(x) can be expressed as a power series

𝔉(x ⊗ 1, 1 ⊗ x) ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]]  (1)

with the coefficients belonging to the tensor square of the Hopf algebra MU^{∗}(BSU_{⊗}). This power series has some properties, which generalize the corresponding properties of usual formal group laws (see Definition 1). So we will refer to it as a “formal group law over a Hopf algebra”.

The series (1) is closely related to the formal product in complex cobordism theory F (u, v). Put u := σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}), where Z is the universal line bundle over ℂP^{∞}. Due to the isomorphism BU_{⊗} ≅ BSU_{⊗} ×ℂP^{∞} we see that x may be represented as a power series x = 𝔥(u) ∈ H[[u]]. It is not difficult to prove that

𝔉(x ⊗ 1, 1 ⊗ x) = (Δ_{H}𝔥)(F ((𝔥^{−1}(x)) ⊗ 1, 1 ⊗ (𝔥^{−1}(x)))),

i.e. (1) may be considered as a re-parameterization of F (u, v) by 𝔥(u). Thus to obtain the explicit formula for this power series it is sufficient to get the explicit form of the power series 𝔥(u) and to describe the comultiplication Δ_{H} in the Hopf algebra H.

Let ξ be a virtual SU-bundle over X of virtual dimension 1. For stable equivalence class \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X) ⊂ K̃_{ℂ}(X) we construct the formal group law (over the ring MU^{∗}(X)) (φ_{ξ}^{∗}ℌ)(x, y) such that it defines the group structure on the set of first Conner-Floyd classes σ_{1}(ξ̂) of bundles ξ̂, whose stable equivalence classes {ξ̂} belong to the coset {ξ}Lin(X)_{0} ⊂ K̃_{ℂ}(X)_{⊗}. For {ξ} = {[1]} the corresponding formal group law (φ_{ξ}^{∗}ℌ)(x, y) coincides with the formal product in complex cobordism theory F (u, v). Moreover, the formula σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) = (φ_{ξ}^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})) generalizes the corresponding formula σ_{1}(ζ_{1} ⊗ ζ_{2}) = F (σ_{1}(ζ_{1}), σ_{1}(ζ_{2})) for the formal product in complex cobordism theory. Here ξ̂_{1}, ξ̂_{2} are virtual U-bundles of virtual dimension 1 such that {ξ̂_{1}}, {ξ̂_{2}} ∈ {ξ}Lin(X)_{0} ⊂ K̃_{ℂ}(X)_{⊗} and ζ_{1}, ζ_{2} are line bundles over X. 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 a part of this work has been completed. A number of related questions were discussed with Professors V.M. Buchstaber, A.M. Levine, V.M. Manuilov and A.S. Mishchenko and I would like to express my gratitude to them.

1 General properties of formal group laws over Hopf algebras

1.1 Basic definitions

The aim of this section is to define some generalization of the notion of a formal group law. More precisely, we consider an analog of formal group laws with coefficients belonging to a Hopf algebra. In particular, we show that a formal group law over a Hopf algebra may be considered as an extension of a usual formal group law over a ring by a Hopf algebra.

Let (H, μ, η, Δ, ε, S) be a commutative topological Hopf algebra over a ring R (commutative, associative and with identity element 1 ∈ R), where μ: H ˆ⊗_{R}H → H is the multiplication, η: R → H is the unit, Δ: H → H ˆ⊗_{R}H is the diagonal map (comultiplication), ε: H → R is the counit, and S: H → H is the antipode [5].

Definition 1. A power series 𝔉(x⊗1, 1⊗y) ∈ (H ˆ⊗_{R}H)[[x⊗1, 1⊗y]] is called a formal group law over the Hopf algebra (H, μ, η, Δ, ε, S) if the following conditions hold:

1) (associative)

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

2) (identity)

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

3) (formal inverse) There is a power series Θ(x) ∈ H[[x]] such that

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

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

Below we shall deal only with commutative formal group laws.

Remark 2. Note that a formal group law 𝔉(x ⊗ 1, 1 ⊗ y) over a Hopf algebra H over a ring R defines the formal group law (in the usual sense) F (x, y) ∈ R[[x, y]] over the ring R in the following way. Let us consider the series ((ε ⊗ ε)𝔉)(x, y). If we identify R⊗_{R}R with R, we may assume that ((ε ⊗ ε)𝔉)(x, y) = F (x, y) ∈ R[[x, y]]. Note that for any coalgebra H the diagram

\begin{matrix}H & →^{Δ} & H ˆ⊗_{R}H \\ ε ↓ & & ↓ε⊗ε \\ R & →^{≅} & R ⊗ R \end{matrix}  (2)

commutes. Using (2) and 1) of Definition 1, we get F (x, F (y, z)) = F (F (x, y), z). Similarly, the conditions F (x, 0) = x and F (0, y) = y may be verified. It is well known, that the existence of the formal inverse (in the case of usual formal group laws) follows from the proved conditions. However this may be deduced from 3) of Definition 1 in the standard way. Moreover, the formal inverse θ(x) for the formal group law F (x, y) is equal to (ε(Θ))(x).

Therefore we may consider the formal group law 𝔉(x ⊗ 1, 1 ⊗ y) over the Hopf algebra H as an extension of the usual formal group law F (x, y) by the Hopf algebra H. Remark 3. Let us define the homomorphisms

Δ̃: H[[x]] \underrightarrow{Δ̃} H[[x]]ˆ⊗_{R}H[[x]],

ε̃: H[[x]] → R, and S̃: H[[x]] → H[[x]]

in the following way: Δ̃(x) = 𝔉(x ⊗ 1, 1 ⊗ x), ε̃(x) = 0, S̃(x) = Θ(x) and Δ̃ |_{H}= Δ, ε̃ |_{H}= ε, S̃ |_{H}= S. We claim that (H[[x]], μ̃, η̃, Δ̃, ε̃, S̃) is the Hopf algebra (here μ̃, η̃ are evident extensions of μ, η). Indeed, the commutativity of the diagram

H[[x]]  \underrightarrow{Δ̃}  H[[x]]ˆ⊗_{R}H[[x]]
Δ̃ ↓  ↓ id_{H[[x]]} ⊗Δ̃
H[[x]]ˆ⊗_{R}H[[x]] ⟶^{Δ̃⊗id_{H[[x]]}} H[[x]]ˆ⊗_{R}H[[x]]ˆ⊗_{R}H[[x]]

follows from the equations

(id_{H[[x]]} ⊗Δ̃)(𝔉(x ⊗ 1, 1 ⊗ x)) = ((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) = (Δ̃ ⊗ id_{H[[x]]})(𝔉(x ⊗ 1, 1 ⊗ x)).

The commutativity of the diagram

[figure: R⊗_{R}H[[x]]; H[[x]]ˆ⊗_{R}H[[x]]; ε̃⊗id_{H[[x]]}; id_{H[[x]]} ⊗ε̃; H[[x]]⊗_{R}R; H[[x]]; ≅; Δ̃; ≅]

follows from the equations

((id_{H[[x]]} ⊗ε̃) ∘ Δ̃)(x) = ((id_{H} ⊗ε)𝔉)(x ⊗ 1, 1 ⊗ ε̃(x)) = x ⊗ 1,
((ε̃⊗ id_{H[[x]]}) ∘ Δ̃)(x) = ((ε ⊗ id_{H})𝔉)(ε̃(x) ⊗ 1, 1 ⊗ x) = 1 ⊗ x.

The axiom of antipode

(μ̃ ∘ (id_{H[[x]]} ⊗S̃) ∘ Δ̃)(x) = (μ̃ ∘ (S̃ ⊗ id_{H[[x]]}) ∘ Δ̃)(x) = (η̃ ∘ ε̃)(x) = 0

follows from 3) of Definition 1. Let us recall that the Hopf algebra has only one antipode. Therefore the power series Θ(x), satisfies 3) of Definition 1, is a unique.

Let 𝔉_{i}(x ⊗ 1, 1 ⊗ y), i = 1, 2 be formal group laws over the Hopf algebra H.

Definition 4. A homomorphism 𝔥: 𝔉_{1} → 𝔉_{2} of formal group laws over the Hopf algebra H is a power series 𝔥(x) ∈ H[[x]] with constant term belonging to ker ε ⊂ H such that (Δ𝔥)(𝔉_{1}(x⊗1, 1⊗y)) = 𝔉_{2}(𝔥(x)⊗1, 1⊗𝔥(y)). The power series 𝔥(x) defines an isomorphism if it is invertible (hence ε(𝔥′(0)) ∈ R is a unit, where 𝔥′(0) is the coefficient of x). An isomorphism 𝔥 is a strict isomorphism if ε(𝔥′(0)) = 1. Assume that 𝔉_{i}(x ⊗ 1, 1 ⊗ y), i = 1, 2 are extensions of the same formal group law over R. Then a strict isomorphism 𝔥 is called to be a strong isomorphism if (ε𝔥)(x) = x.

Note that ε(𝔥): (ε ⊗ ε)(𝔉_{1}) → (ε ⊗ ε)(𝔉_{2}) is the homomorphism of the formal group laws over R (where ε is the counit of the Hopf algebra H). We say that the homomorphism 𝔥 covers the homomorphism ε(𝔥). In particular, a strong isomorphism covers the identity isomorphism of the formal group law over R. Moreover, ε(𝔥) is an isomorphism (a strict isomorphism) if 𝔥 is an isomorphism (a strict isomorphism respectively).

Let us consider some examples of defined objects.

Example 5. (Trivial extension) Let F (x, y) be a formal group law (in the usual sense) over a ring R, and (H, μ, η, Δ, ε, S) be a Hopf algebra over the same ring R. Then 𝔉(x⊗1, 1⊗y) = ((η ⊗η)F)(x⊗1, 1⊗y) ∈ (H ˆ⊗_{R}H)[[x⊗1, 1⊗y]] is the formal group law over the Hopf algebra H (recall that we identify R⊗_{R}R with R).

Example 6. A formal group law of the form 𝔠 + x ⊗ 1 + 1 ⊗ y ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ y]] over Hopf algebra H is an extension of the “linear group” x ⊗ 1 + 1 ⊗ y ∈ R[[x ⊗ 1, 1 ⊗ y]]. It follows from Definition 1 (axioms 1)-3)) that 𝔠 is a 2-cocycle in the cobar complex of H such that (id ⊗ε)𝔠 = 0 = (ε ⊗ id)𝔠, τ(𝔠) = 𝔠 (the last condition follows from the commutativity of considered formal group laws). Below we shall show that a formal group law 𝔠+x⊗1+1⊗y ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ y]] over H is determined up to isomorphism by the cohomology class of 2-cocycle 𝔠 (see Lemma 15).

By 𝔊(x, y) denote the power series ((μ ∘ (id_{H} ⊗S))𝔉)(x, y) ∈ H[[x, y]].

One can check by a straightforward computation that H[[x, y]] is a Hopf algebra with respect to a comultiplication

Δ̂: H[[x, y]] → H[[x, y]]ˆ⊗_{R}H[[x, y]]

and an antipode

Ŝ: H[[x, y]] → H[[x, y]]

defined as follows:

Δ̂|_{H} = Δ, Δ̂(x) = 𝔉(x ⊗ 1, 1 ⊗ x), Δ̂(y) = ((S ⊗ S)𝔉)(y ⊗ 1, 1 ⊗ y)  (3)

and

Ŝ|_{H} = S, Ŝ(x) = Θ(x), Ŝ(y) = (SΘ)(y).

Let us define a ring homomorphism (η): R[[z]] → H[[x, y]] as follows:

(η)|_{R} = η: R → H ⊂ H[[x, y]], (η)(z) = 𝔊(x, y).

We consider the ring R[[z]] as a Hopf algebra over R with a comultiplication, induced by z ↦ F (z ⊗ 1, 1 ⊗ z) and an antipode, induced by z ↦ θ(z), where F (x, y) ∈ R[[x, y]] is the formal group law ((ε ⊗ ε)𝔉)(x, y) and θ(x) is its formal inverse.

Proposition 7. The ring homomorphism (η): R[[z]] → H[[x, y]] is a homomorphism of Hopf algebras.

Proof. At first we must prove that

Δ̂(𝔊(x, y)) = (Δ𝔊)(𝔉(x ⊗ 1, 1 ⊗ x), ((S ⊗ S)𝔉)(y ⊗ 1, 1 ⊗ y))
= F (𝔊(x, y) ⊗ 1, 1 ⊗ 𝔊(x, y)).  (4)

By S̃′ denote the homomorphism H[[x]] → H[[x]] such that S̃′|_{H} = S: H → H and S̃′(x) = x. Let us consider the following composite homomorphism of Hopf algebras:

H[[z]] \underrightarrow{Δ̃} H[[x]]ˆ⊗_{R}H[[y]] →^{id ⊗S̃′} H[[x]]ˆ⊗_{R}H[[y]] →^{(μ)} H[[x, y]],

where Δ̃(z) = 𝔉(x ⊗ 1, 1 ⊗ y), Δ̃|_{H} = Δ, (μ) is the homomorphism such that

(μ) |_{H ˆ⊗_{R}H}= μ: H ˆ⊗_{R}H → H

and

(μ)(x ⊗ 1) = x, (μ)(1 ⊗ y) = y.

It follows from the axiom of antipode

μ ∘ (id_{H} ⊗S) ∘ Δ = η ∘ ε

that

(μ) ∘ (id_{H[[x]]} ⊗S̃′) ∘ Δ̃ |_{H}= η ∘ ε.

Hence there exists the homomorphism of Hopf algebras

(η)′: R[[z]] → H[[x, y]]

such that the following diagram

\xymatrix{H[[z]] \ar[r]^{Δ̃} \ar[dr]_{(ε)} & H[[x]]ˆ⊗_{R}H[[y]] \ar[r]^{id ⊗S̃′} & H[[x]]ˆ⊗_{R}H[[y]] \ar[d]^{(μ)} \\  & R[[z]] \ar[r]^{(η)′} & H[[x, y]]}

commutes, where (ε) is the homomorphism such that (ε) |_{H}= ε, (ε)(z) = z. Since

(η)′(z) = 𝔊(x, y),

we obtain (η)′ = (η). This completes the proof of (4) that

Δ_{R[[z]]}(z) = F (z ⊗ 1, 1 ⊗ z),

where F (x, y) ∈ R[[x, y]] is the formal group law as above.

It remains to prove that

(η)(θ(z)) = Ŝ(𝔊(x, y)) = (S𝔊)(Θ(x), (SΘ)(y)).

Applying the operator μ̂ ∘ (id_{H[[x,y]]} ⊗Ŝ) (where μ̂: H[[x, y]]ˆ⊗_{R}H[[x, y]] → H[[x, y]] is the obvious multiplication) to the both sides of (4) and using (μ̂∘(id_{H[[x,y]]} ⊗Ŝ)∘Δ̂)(𝔊(x, y)) = 0, we get

0 = F (𝔊(x, y), (S𝔊)(Θ(x), (SΘ)(y))).

Therefore, (S𝔊)(Θ(x), (SΘ)(y)) = θ(𝔊(x, y)). □

By g(x) ∈ (R⊗_{ℤ}ℚ)[[x]] denote the logarithm of the formal group law F (x, y) = ((ε ⊗ ε)𝔉)(x, y).

Corollary 8. The element exp(g(𝔊(x, y))) ∈ (H ˆ⊗_{ℤ}ℚ)[[x, y]] is a grouplike element of the Hopf algebra (H ˆ⊗_{ℤ}ℚ)[[x, y]].

Proof. Let Δ̂_{ℚ} be the comultiplication in the Hopf algebra (H ˆ⊗_{ℤ}ℚ)[[x, y]], corresponding to (3). We have:

Δ̂_{ℚ}(exp(g(𝔊(x, y)))) = exp(g(F (𝔊(x, y) ⊗ 1, 1 ⊗ 𝔊(x, y))))
= exp(g(𝔊(x, y)) ⊗ 1 + 1 ⊗ g(𝔊(x, y))) = exp(g(𝔊(x, y))) ⊗ exp(g(𝔊(x, y))).

Moreover, we have

Ŝ(exp(g(𝔊(x, y)))) = exp(g(𝔊(x, y)))^{−1},

i.e. Ŝ(g(𝔊(x, y))) = g((S𝔊)(Θ(x), (SΘ)(y))) = −g(𝔊(x, y)). To prove this, we shall show that

F (𝔊(x, y), (S𝔊)(Θ(x), (SΘ)(y))) = 0.

Indeed, applying the operator μ̂ ∘ (id_{H[[x,y]]} ⊗Ŝ) to the both sides of (4), we get:

F (𝔊(x, y), (S𝔊)(Θ(x), (SΘ)(y)))
= ((η ∘ ε)𝔊)((μ ∘ (id_{H} ⊗S)𝔉)(x, Θ(x)), (μ ∘ (S ⊗ id_{H})𝔉)(y, (SΘ)(y)))
= F (0, 0) = 0.

Now the required equality follows from

0 = g(0) = g(F (𝔊(x, y), (S𝔊)(Θ(x), (SΘ)(y))))
= g(𝔊(x, y)) + g((S𝔊)(Θ(x), (SΘ)(y))). □

1.2 Logarithms of formal group laws over Hopf algebras

In this section we generalize some useful construction from the case of formal group laws over rings to the case of formal group laws over Hopf algebras. More precisely, we show that there exists an analog of the logarithm in the general case. We also establish some condition of triviality of a formal group law over a Hopf algebra as an extension of a formal group law over a ring by a Hopf algebra.

In this section by (H, μ, η, Δ, ε, S) we denote a commutative Hopf algebra over a ring R without torsion. In addition, suppose that H is complete with respect to the augmentation ideal ker(ε) ⊂ H. Note that this assumption implies that for any a ∈ H, ε(a) = 1 ∈ R there exists a^{−1} ∈ H. By 𝔉(x ⊗ 1, 1 ⊗ x) denote a formal group law over the Hopf algebra H. By H_{ℚ} denote the Hopf algebra H ˆ⊗_{ℤ}ℚ over the ring R_{ℚ} := R⊗_{ℤ}ℚ. Clearly, that the formal group law 𝔉(x ⊗ 1, 1 ⊗ y) may be considered as a formal group law over the Hopf algebra H_{ℚ}. We shall write μ, η, … instead of μ_{ℚ}, η_{ℚ}, ….

The main tool of this section is the following result.

Theorem 9. Let 𝔉(x ⊗ 1, 1 ⊗ x) be a commutative formal group law over H, considered as a formal group law over H_{ℚ}. Then there exists a strict isomorphism of formal group laws over H_{ℚ} from 𝔉(x ⊗ 1, 1 ⊗ x) to a commutative formal group law of the form 𝔠 + x ⊗ 1 + 1 ⊗ x,

where 𝔠 ∈ H_{ℚ} ˆ⊗_{R_{ℚ}}H_{ℚ}.

We recall that the definitions of homomorphism, isomorphism and strict isomorphism of formal group laws over a Hopf algebra were given in the previous section. Below we shall use notation of Section 1.1.

To prove the theorem, we need the following lemma.

Lemma 10. A series of the form 𝔠+x⊗1+1⊗x ∈ (H_{ℚ} ˆ⊗_{R_{ℚ}}H_{ℚ})[[x⊗1, 1⊗x]] is a commutative formal group law over the Hopf algebra H_{ℚ} if and only if the following conditions hold:

(i) (id ⊗Δ)𝔠 + 1 ⊗ 𝔠 − (Δ ⊗ id)𝔠 − 𝔠 ⊗ 1 = 0;

(ii) (id ⊗ε)𝔠 = 0 = (ε ⊗ id)𝔠;

(iii) τ(𝔠) = 𝔠, where τ is the switching map τ(a ⊗ b) = b ⊗ a.

(Note that condition (i) means that 𝔠 is a 2-cocykle in the cobar complex of the Hopf algebra H_{ℚ}.) Proof of the lemma. Conditions (i), (ii), and (iii) are equivalent to the associativity axiom, the identity axiom and the commutativity axiom respectively. Let us show that the series Θ(x) := −(μ ∘ (id ⊗S))𝔠 − x is a formal inverse. Indeed,

(μ ∘ (id ⊗S))𝔠 + x + Θ(x) = (μ ∘ (id ⊗S))𝔠 + x − (μ ∘ (id ⊗S))𝔠 − x = 0.

The symmetric condition follows from the equality (μ ∘ (id ⊗S))𝔠 = (μ ∘ (S ⊗ id))𝔠. □

Proof of the theorem. By definition, put

ω̃(x) := (id ⊗ε̃)\frac{∂𝔉(x, z)}{∂z} ∈ H[[x]],

where ε̃: H[[z]] → R is the map such that ε̃ |_{H}= ε: H → R, ε̃(z) = 0. Recall that Δ̃: H[[x]] → H[[x]]ˆ⊗_{R}H[[x]] = (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]] is the homomorphism such that Δ̃ |_{H}= Δ, Δ̃(x) = 𝔉(x ⊗ 1, 1 ⊗ x). We have:

(Δω̃)(𝔉(x ⊗ 1, 1 ⊗ x)) = Δ̃(ω̃(x)) = (Δ̃ ∘ (id ⊗ε̃) ∘ \frac{∂}{∂z})(𝔉(x, z))

= ((id ⊗ id ⊗ε̃) ∘ (Δ̃ ⊗ id) ∘ \frac{∂}{∂z})(𝔉(x, z))

= ((id ⊗ id ⊗ε̃) ∘ \frac{∂}{∂z} ∘ (Δ̃ ⊗ id))(𝔉(x, z))

= ((id ⊗ id ⊗ε̃) ∘ \frac{∂}{∂z} ) ((Δ ⊗ id)𝔉)(𝔉(x ⊗ 1, 1 ⊗ x), z)

= ((id ⊗ id ⊗ε̃) ∘ \frac{∂}{∂z} ) ((id ⊗Δ)𝔉)(x ⊗ 1, 𝔉(1 ⊗ x, z))

= (id ⊗ id ⊗ε̃) (\frac{∂((id ⊗Δ)𝔉)(x ⊗ 1, 𝔉(1 ⊗ x, z))}{∂𝔉(1 ⊗ x, z)}) · 1 ⊗ ((id ⊗ε̃)\frac{∂𝔉(1 ⊗ x, z)}{∂z})

= \frac{∂𝔉(x ⊗ 1, 1 ⊗ x)}{∂(1 ⊗ x)} · (1 ⊗ ω̃)(1 ⊗ x).

Therefore, we have

(Δω̃)(𝔉(x ⊗ 1, 1 ⊗ x)) = \frac{∂𝔉(x ⊗ 1, 1 ⊗ x)}{∂(1 ⊗ x)} · (1 ⊗ ω̃)(1 ⊗ x).  (5)

If

𝔉(x, z) = ∑_{i,j≥0} A_{i,j}x^{i}z^{j} (A_{i,j} ∈ H ˆ⊗_{R}H),

then

ω̃(x) = (id ⊗ε̃) ∑_{i,j} A_{i,j}x^{i}jz^{j−1} = (id ⊗ε)A_{0,1} + ∑_{i≥1}((id ⊗ε)A_{i,1})x^{i},

where (ε ∘ (id ⊗ε))A_{0,1} = 1 ≠ 0. Therefore

\frac{1}{ω̃(x)} ∈ H[[x]] and

Δ̃ (\frac{1}{ω̃(x)}) = \frac{1}{Δ̃(ω̃(x))} = \frac{1}{(Δω̃)(𝔉(x ⊗ 1, 1 ⊗ x))} ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]].

Therefore equality (5) may be rewritten in the form

\frac{d(1 ⊗ x)}{(1 ⊗ ω̃)(1 ⊗ x)} = \frac{d𝔉(x ⊗ 1, 1 ⊗ x)}{(Δω̃)(𝔉(x ⊗ 1, 1 ⊗ x))}.  (6)

It is clear that

\frac{1}{ω̃(x)} = b_{0} + b_{1}x + …,

where b_{i} ∈ H, ε(b_{0}) = 1. By 𝔤(x) denote the power series

∫_{0}^{x} \frac{dt}{ω̃(t)} ∈ H_{ℚ}[[x]].

Equality (6) implies

𝔠′ + (1 ⊗ 𝔤)(1 ⊗ x) = (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x)),  (7)

where 𝔠′ is independent of 1 ⊗ x. Applying id ⊗ε̃ to the both sides of (7), we get

(id ⊗ε̃)𝔠′ = (((id ⊗ε) ∘ Δ)𝔤)(x ⊗ 1) = (𝔤 ⊗ 1)(x ⊗ 1).

Applying ε̃⊗ id to the both sides of (7), we get

(ε̃⊗ id)𝔠′ + (1 ⊗ 𝔤)(1 ⊗ x) = (1 ⊗ 𝔤)(1 ⊗ x).

Hence

(Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x)) = 𝔠 + (𝔤 ⊗ 1)(x ⊗ 1) + (1 ⊗ 𝔤)(1 ⊗ x),  (8)

where 𝔠′ = (𝔤 ⊗ 1)(x ⊗ 1) + 𝔠, 𝔠 ∈ H_{ℚ} ˆ⊗_{R_{ℚ}}H_{ℚ} and (id ⊗ε)𝔠 = 0 = (ε ⊗ id)𝔠.

Since 𝔤(0) = 0 and ε(𝔤′(0)) = 1, the power series 𝔤(x) is invertible (i.e. there is the power series 𝔤^{−1}(x) ∈ H_{ℚ}[[x]] such that 𝔤(𝔤^{−1}(x)) = x = 𝔤^{−1}(𝔤(x))). Therefore it defines a strict isomorphism of formal group laws over H_{ℚ}.

To complete the proof we must verify condition (i) of the previous lemma. For this purpose we apply id ⊗Δ̃ and Δ̃ ⊗ id to the both sides of (8). We have:

(((id ⊗Δ) ∘ Δ)𝔤)((id ⊗Δ)𝔉(x ⊗ 1 ⊗ 1, 1 ⊗ 𝔉(x ⊗ 1, 1 ⊗ x)))
= (id ⊗Δ)𝔠 + 𝔤(x) ⊗ 1 ⊗ 1 + 1 ⊗ (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x))
= (((Δ ⊗ id) ∘ Δ)𝔤)((Δ ⊗ id)𝔉(𝔉(x ⊗ 1, 1 ⊗ x) ⊗ 1, 1 ⊗ 1 ⊗ x))
= (Δ ⊗ id)𝔠 + (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x)) ⊗ 1 + 1 ⊗ 1 ⊗ 𝔤(x),

i. e.

(id ⊗Δ)𝔠 + 𝔤(x) ⊗ 1 ⊗ 1 + 1 ⊗ 𝔠 + 1 ⊗ 𝔤(x) ⊗ 1 + 1 ⊗ 1 ⊗ 𝔤(x)
= (Δ ⊗ id)𝔠 + 𝔠 ⊗ 1 + 𝔤(x) ⊗ 1 ⊗ 1 + 1 ⊗ 𝔤(x) ⊗ 1 + 1 ⊗ 1 ⊗ 𝔤(x).

This completes the proof. □ Remark 11. Note that to any formal group law 𝔉(x ⊗ 1, 1 ⊗ x) over H we assign some 2-cocycle 𝔠 in the cobar complex of the coalgebra H_{ℚ}. Remark 12. Note that (ε𝔤)(x) ∈ R_{ℚ}[[x]] is the logarithm of the formal group law ((ε ⊗ ε)𝔉)(x, y) = F (x, y) ∈ R[[x, y]] over the ring R [4].

Remark 13. Since 𝔤(x) = b_{0}x + b_{1}x^{2} + … and ε(b_{0}) = 1, there exists the power series (Δ𝔤)^{−1}(x) = (Δ(𝔤^{−1}))(x) ∈ (H_{ℚ} ˆ⊗_{R_{ℚ}}H_{ℚ})[[x]]. Using (8), we get

𝔉(x ⊗ 1, 1 ⊗ x) = (Δ𝔤)^{−1}(𝔠 + 𝔤(x) ⊗ 1 + 1 ⊗ 𝔤(x)).  (9)

Corollary 14. Suppose (μ∘(id_{H} ⊗S))𝔠 = 0, where 𝔠 is the cocycle in (8); then there exists a strong isomorphism 𝔣 from 𝔉(x⊗1, 1⊗y) to the trivial extension of ((ε⊗ε)𝔉)(x, y) = F (x, y) by H. Moreover, 𝔣(0) = 0.

Proof. Recall that by 𝔊(x, y) we denoted the power series (μ ∘ (id_{H} ⊗S)𝔉)(x, y). It follows from (9) under the condition (μ ∘ (id_{H} ⊗S))𝔠 = 0 that 𝔊(x, y) = g^{−1}(𝔤(x) + (S𝔤)(y)), where g(x) = (ε𝔤)(x) is the logarithm of F (x, y), see Remark 12 above. Since 𝔤(0) = 0, we have 𝔊(x, 0) = g^{−1}(𝔤(x)) ∈ H[[x]]. The straightforward computation shows, that the power series 𝔣(x) = g^{−1}(𝔤(x)) defines a homomorphism from 𝔉(x ⊗ 1, 1 ⊗ y) to the trivial extension of F (x, y) by H. Since 𝔣(0) = 0, ε(𝔣′(0)) = ε(𝔤′(0)) = 1 ∈ R, we see that 𝔣(x) is invertible. Moreover, (ε𝔣)(x) = g^{−1}(g(x)) = x. □

Lemma 15. If cohomology classes [𝔠_{1}] and [𝔠_{2}] are equal, then the “linear” formal group laws 𝔠_{i} + x ⊗ 1 + 1 ⊗ x, i = 1, 2 over H_{ℚ} are isomorphic.

Proof. Suppose [𝔠_{1}] = [𝔠_{2}]; then there exists λ ∈ H_{ℚ} such that ε(λ) = 0 and 𝔠_{2} − 𝔠_{1} = Δλ − λ ⊗ 1 − 1 ⊗ λ. Hence Δλ + 𝔠_{1} + x ⊗ 1 + 1 ⊗ x = 𝔠_{2} + (λ + x) ⊗ 1 + 1 ⊗ (λ + x). Therefore 𝔱(x) := λ + x is an isomorphism from 𝔠_{1} + x ⊗ 1 + 1 ⊗ x to 𝔠_{2} + x ⊗ 1 + 1 ⊗ x. □

We can make more precise the statement of the previous theorem.

Corollary 16. If 2-cocycle 𝔠 in (8) is a coboundary, then there is a strict isomorphism from 𝔉(x⊗1, 1⊗x), considered as a formal group law over H_{ℚ}, to the “trivial” group x⊗1+1⊗x.

Proof. We have

(Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x)) = 𝔠 + 𝔤(x) ⊗ 1 + 1 ⊗ 𝔤(x),

where 𝔠 = λ ⊗ 1 − Δλ + 1 ⊗ λ. Let us consider the homomorphism 𝔥(x) := λ + 𝔤(x). It follows from the completeness of the Hopf algebra H with respect to the augmentation ideal ker(ε) ⊂ H that there exists 𝔥^{−1}(x) = 𝔤^{−1}(x − λ) ∈ H_{ℚ}[[x]] (recall that ε(λ) = 0). Clearly ε(𝔥′(0)) = 1. Hence the homomorphism 𝔥(x) is a strict isomorphism. We have:

(Δ𝔥)(𝔉(x ⊗ 1, 1 ⊗ x)) = Δλ + (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x))
= Δλ + 𝔠 + 𝔤(x) ⊗ 1 + 1 ⊗ 𝔤(x) = Δλ − λ ⊗ 1 − 1 ⊗ λ + 𝔠 + 𝔥(x) ⊗ 1 + 1 ⊗ 𝔥(x)
= 𝔥(x) ⊗ 1 + 1 ⊗ 𝔥(x). □

Theorem 17. Let 𝔉(x⊗1, 1⊗y) ∈ (H ˆ⊗_{R}H)[[x⊗1, 1⊗y]] be a formal group law over the Hopf algebra H and let F (x, y) := ((ε ⊗ ε)𝔉)(x, y) ∈ R[[x, y]] be the corresponding formal group law over R. Assume that 2-dimensional cohomology of the cobar complex of the Hopf algebra H_{ℚ} equals 0. Then there exists a strong isomorphism 𝔣 from 𝔉(x ⊗ 1, 1 ⊗ y), considered as a formal group law over the Hopf algebra H_{ℚ}, to the trivial extension of F (x, y) by H_{ℚ}.

Proof. According to the previous corollary, there is an invertible power series 𝔥(x) ∈ H_{ℚ}[[x]] such that

(Δ𝔥)(𝔉(x ⊗ 1, 1 ⊗ y)) = 𝔥(x) ⊗ 1 + 1 ⊗ 𝔥(y)  (10)

and (ε𝔥)(x) = g(x), where g(x) ∈ R_{ℚ}[[x]] is the logarithm of the formal group law F (x, y). Let 𝔣(z) ∈ H_{ℚ}[[z]] be the power series g^{−1}(𝔥(z)). We have:

(Δ𝔣)(𝔉(x ⊗ 1, 1 ⊗ y)) = g^{−1}((Δ𝔥)(𝔉(x ⊗ 1, 1 ⊗ y))) = g^{−1}(𝔥(x) ⊗ 1 + 1 ⊗ 𝔥(y))
= g^{−1}(g(𝔣(x)) ⊗ 1 + 1 ⊗ g(𝔣(y))) = F (𝔣(x) ⊗ 1, 1 ⊗ 𝔣(y)).

Clearly, (ε𝔣)(x) = x. □ Remark 18. Let us remark that under the hypothesis of the previous theorem the formal group law 𝔉(x ⊗ 1, 1 ⊗ y) over the Hopf algebra H has the formal inverse of the form Θ(x) = (S𝔣)^{−1}(θ(𝔣(x))), where θ(x) ∈ R[[x]] is the formal inverse for the formal group law F (x, y). Indeed, we have:

((μ ∘ (id_{H} ⊗S))(Δ𝔣)^{−1})(F (𝔣(x), (S𝔣)((S𝔣)^{−1}(θ(𝔣(x))))))
= F (𝔣(x), θ(𝔣(x))) = F (u, θ(u)) = 0.

Example 19. Let us consider a formal group law of the form γ + αx ⊗ 1 + (τα)1 ⊗ x, where α, γ ∈ H ˆ⊗_{R}H and τ is the switching map τ(a ⊗ b) = b ⊗ a. The axioms of associativity, identity and commutativity imply

(id ⊗ε)γ = 0 = (ε ⊗ id)γ, (id ⊗ε)α = 1 = (ε ⊗ id)(τα),

(id ⊗ε)(τα) = (ε ⊗ id)α, (11)

(id ⊗Δ)γ + (id ⊗Δ)(τα) · 1 ⊗ γ = (Δ ⊗ id)γ + (Δ ⊗ id)α · γ ⊗ 1,  (12)

(id ⊗Δ)α = (Δ ⊗ id)α · α ⊗ 1,

(id ⊗Δ)(τα) · 1 ⊗ α = (Δ ⊗ id)α · (τα) ⊗ 1,
(id ⊗Δ)(τα) · 1 ⊗ (τα) = (Δ ⊗ id)(τα).

(13)

The logarithm is the series

𝔤(x) = \frac{x}{(id ⊗ε)(τα)} = \frac{x}{(ε ⊗ id)α} .

We shall verify the equality

(Δ𝔤)(γ + αx ⊗ 1 + (τα)1 ⊗ x) = \frac{γ}{Δ(ε ⊗ id)α} + \frac{x}{(id ⊗ε)(τα)} ⊗ 1 + 1 ⊗ \frac{x}{(id ⊗ε)(τα)},

using (11)-(13). Since

(ε ⊗ id ⊗ id)(id ⊗Δ)α = Δ(ε ⊗ id)α, (ε ⊗ id ⊗ id)(Δ ⊗ id)α = α,

(ε ⊗ id ⊗ id)(α ⊗ 1) = ((ε ⊗ id)α) ⊗ 1,

applying ε ⊗ id ⊗ id to the both sides of the first equality (13), we get

Δ(ε ⊗ id)α = α · (((ε ⊗ id)α) ⊗ 1).

Since

(id ⊗ id ⊗ε)(Δ ⊗ id)(τα) = Δ(id ⊗ε)(τα), (id ⊗ id ⊗ε)(id ⊗Δ)(τα) = τα,
(id ⊗ id ⊗ε)(1 ⊗ (τα)) = 1 ⊗ ((id ⊗ε)(τα)),

applying id ⊗ id ⊗ε to the both sides of the third equality (13), we obtain

Δ(id ⊗ε)(τα) = (τα) · (1 ⊗ ((id ⊗ε)(τα)).

Now we must check that

𝔠 = \frac{γ}{Δ(ε ⊗ id)α}

is a cocycle. To show this, we use (12), which is equivalent to

(id ⊗Δ)(Δ(ε ⊗ id)α · 𝔠) + (id ⊗Δ)(τα) · 1 ⊗ (Δ(ε ⊗ id)α · 𝔠)
= (Δ ⊗ id)(Δ(ε ⊗ id)α · 𝔠) + (Δ ⊗ id)α · (Δ(ε ⊗ id)α · 𝔠) ⊗ 1.

It is easy to prove that

(id ⊗Δ)(τα) · 1 ⊗ (Δ(ε ⊗ id)α) = (id ⊗Δ)(Δ(ε ⊗ id)α)
= (Δ ⊗ id)(Δ(ε ⊗ id)α) = (Δ ⊗ id)α · (Δ(ε ⊗ id)α) ⊗ 1.

1.3 The functor, represented by a formal group law over a Hopf algebra

It is well known that a formal group law over a ring R defines a group structure on the nilradical 𝔫(A) of any associative commutative unitary R-algebra A [3]. We show that under some assumption of triviality a formal group law over a Hopf algebra defines a set of group structures on 𝔫(A), which are different in general case. The technique, developed in this section, will be the main tool of proofs in Section 2.3.

To avoid a confusion, in this section we shall write μ_{H}, ε_{H} instead of μ, ε respectively. Let R be an associative commutative ring with identity element 1 ∈ R. By 𝔄𝔩𝔤_{R} denote

the category of associative commutative unitary algebras over R. By 𝔫(A) we denote the nil-radical of A ∈ 𝔄𝔩𝔤_{R}. Any commutative formal group law F (x, y) ∈ R[[x, y]] defines a group operation +_{F} on the set 𝔫(A) by α +_{F} β := F (α, β). This group we shall denote by 𝔫(A)_{F} .

Let H be a commutative topological Hopf algebra over R. As in Section 1.2 we suppose H is complete with respect to the augmentation ideal ker(ε_{H}) ⊂ H. Let 𝔉(x ⊗ 1, 1 ⊗ y) ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ y]] be a formal group law over H such that it is an extension of F (x, y) by H (i.e. ((ε_{H} ⊗ ε_{H})𝔉)(x, y) = F (x, y), where ε_{H}: H → R is the counit in H).

Let us remark that it follows from 2) of Definition 1 that (id_{H} ⊗ε_{H})A_{0,0} = 0 = (ε_{H} ⊗ id_{H})A_{0,0}, where 𝔉(x ⊗ 1, 1 ⊗ y) = ∑_{i,j} A_{i,j}x^{i} ⊗ y^{j}. The completeness of the Hopf algebra H with respect to the augmentation ideal ker(ε_{H}) implies that (μ_{A} ∘ (φ ⊗ ψ))A_{0,0} ∈ 𝔫(A) for all φ, ψ ∈ Hom_{R}(H, A). Hence (μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β) ∈ 𝔫(A) for any α, β ∈ 𝔫(A), where μ_{A}: A⊗_{R}A → A is the multiplication in the algebra A.

Define the operation ∘ on the set 𝔫(A) × Hom_{R}(H; A) as follows:

( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}β \\ ψ \end{matrix} ) := ( \begin{matrix}(μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β) \\ φ ⋆ ψ \end{matrix} )  (14)

for α, β ∈ 𝔫(A), φ, ψ ∈ Hom_{R}(H, A), where

φ ⋆ ψ = μ_{A} ∘ (φ ⊗ ψ) ∘ Δ ∈ Hom_{R}(H, A)

is the convolution of φ and ψ, Δ: H → H ˆ⊗_{R}H is the comultiplication in H.

Proposition 20. The set 𝔫(A) × Hom_{R}(H; A) is an Abelian group with respect to the operation ∘.

Proof. Associativity of (14) follows from 1) of Definition 1. Indeed,

( \begin{matrix}α \\ φ \end{matrix} ) ∘ (( \begin{matrix}β \\ ψ \end{matrix} ) ∘ ( \begin{matrix}γ \\ χ \end{matrix} )) = ( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}(μ_{A} ∘ (ψ ⊗ χ)𝔉)(β, γ) \\ ψ ⋆ χ \end{matrix} )

= ( \begin{matrix}(μ_{A} ∘ (φ ⊗ (ψ ⋆ χ))𝔉)(α, (μ_{A} ∘ (ψ ⊗ χ)𝔉)(β, γ)) \\ φ ⋆ ψ ⋆ χ \end{matrix} ) .

Similarly,

(( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}β \\ ψ \end{matrix} )) ∘ ( \begin{matrix}γ \\ χ \end{matrix} ) = ( \begin{matrix}(μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β) \\ φ ⋆ ψ \end{matrix} ) ∘ ( \begin{matrix}γ \\ χ \end{matrix} )

= ( \begin{matrix}(μ_{A} ∘ ((φ ⋆ ψ) ⊗ χ)𝔉)((μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β), γ) \\ φ ⋆ ψ ⋆ χ \end{matrix} ) .

It is not difficult to show that

(μ_{A} ∘ (φ ⊗ (ψ ⋆ χ))𝔉)(α, (μ_{A} ∘ (ψ ⊗ χ)𝔉)(β, γ))

is the result of application of

μ_{A} ∘ (id_{A} ⊗μ_{A}) ∘ (φ ⊗ ψ ⊗ χ)

to

((id_{H} ⊗Δ)𝔉)(x ⊗ 1 ⊗ 1, 1 ⊗ 𝔉(y ⊗ 1, 1 ⊗ z))

and substitution of α, β, γ for x, y, z respectively. Similarly,

(μ_{A} ∘ ((φ ⋆ ψ) ⊗ χ)𝔉)((μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β), γ)

is the result of application of

μ_{A} ∘ (μ_{A} ⊗ id_{A}) ∘ (φ ⊗ ψ ⊗ χ)

to

((Δ ⊗ id_{H})𝔉)(𝔉(x ⊗ 1, 1 ⊗ y) ⊗ 1, 1 ⊗ 1 ⊗ z)

and substitution of α, β, γ for x, y, z respectively.

By ε denote the composition η_{A} ∘ ε_{H}: H → A, where η_{A}: R → A is the unit in A. It follows from 2) of Definition 1 that

( \begin{matrix}0 \\ ε \end{matrix} )

is the identity element in the group.

Moreover, for any

( \begin{matrix}α \\ φ \end{matrix} ) ∈ 𝔫(A) × Hom_{R}(H; A)

there exists the inverse element

( \begin{matrix}α \\ φ \end{matrix} )^{−1} = ( \begin{matrix}(φΘ)(α) \\ φ ∘ S \end{matrix} ) ,  (15)

where Θ(x) ∈ H[[x]] is the formal inverse in the formal group law 𝔉(x ⊗ 1, 1 ⊗ y) and S: H → H is the antipode in H. Indeed,

(μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, (φΘ)(α)) = φ((μ_{H} ∘ (id_{H} ⊗S)𝔉)(x, Θ(x))|_{x=α})
= φ(0) = 0

(we used 3) of Definition 1).

Commutativity follows from the commutativity of 𝔉(x⊗1, 1⊗y) (recall that we assumed H to be commutative). □

The group 𝔫(A)×Hom_{R}(H; A) with respect to the just defined operation will be denoted by G_{𝔉} . We have the following exact sequence of groups:

0 → 𝔫(A)_{F} →^{i} G_{𝔉} →^{j} Hom_{R}(H, A) → 0,  (16)

where

i(α) = ( \begin{matrix}α \\ ε \end{matrix} ) , j ( \begin{matrix}α \\ φ \end{matrix} ) = φ

(recall that ε is the identity element in the group (with respect to the convolution) Hom_{R}(H, A)).

Proposition 21. Suppose there exists a strong isomorphism from 𝔉(x ⊗ 1, 1 ⊗ y) to the trivial extension of F (x, y) by H (compare with Corollary 14); then sequence (16) splits.

Proof. Recall that proposition’s assumption means that there is an invertible power series 𝔣(x) ∈ H[[x]] such that (Δ𝔣)(𝔉(x ⊗ 1, 1 ⊗ y)) = F (𝔣(x) ⊗ 1, 1 ⊗ 𝔣(y)) and (ε_{H}𝔣)(x) = x. To prove the proposition we shall show that

σ: G_{𝔉} → 𝔫(A)_{F} , σ ( \begin{matrix}α \\ φ \end{matrix} ) = (φ𝔣)(α)

is a splitting homomorphism.

1) σ is a group homomorphism:

σ (( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}β \\ ψ \end{matrix} )) = σ ( \begin{matrix}(μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β) \\ φ ⋆ ψ \end{matrix} )

= ((φ ⋆ ψ)𝔣)((μ_{A} ∘ (φ ⊗ ψ)𝔉)(α, β)).  (17)

Applying μ_{A} ∘ (φ ⊗ ψ) to the both sides of the identity

(Δ𝔣)(𝔉(x ⊗ 1, 1 ⊗ y)) = F (𝔣(x) ⊗ 1, 1 ⊗ 𝔣(y))

and substituting of α, β for x, y respectively, we obtain that (17) equals F ((φ𝔣)(α), (ψ𝔣)(β)).

2) σ is a section of i, i.e. σ ∘ i = id_{𝔫(A)_{F}} . Indeed,

(σ ∘ i)(α) = σ ( \begin{matrix}α \\ ε \end{matrix} ) = (ε𝔣)(α) = α. □

Note that for any α ∈ 𝔫(A) and φ ∈ Hom_{R}(H, A) the element

((ε_{H} ⊗ φ)𝔉)(α, 0) ∈ 𝔫(A)

may be considered as a shift of α by φ (let us remark that we use the identity μ_{A} ∘ (ε ⊗ φ) = ε_{H} ⊗ φ, which follows from the unital multiplication axiom μ_{A} ∘ (η_{A} ⊗ id_{A}): R ⊗ A ≅ A for the algebra A). Clearly that the function

⟨α⟩: Hom_{R}(H, A) → 𝔫(A), ⟨α⟩(φ) := ((ε_{H} ⊗ φ)𝔉)(α, 0)

is a section of the projection π_{2} : 𝔫(A) × Hom_{R}(H, A) → Hom_{R}(H, A) such that ⟨α⟩(ε) = α. Therefore we obtain the map

𝔫(A) → Mor_{Set}(Hom_{R}(H, A); 𝔫(A)), α ↦ ⟨α⟩.

It follows from 2) of Definition 1 that for any φ ∈ Hom_{R}(H, A) ⟨0⟩(φ) = 0.

If 𝔉(x ⊗ 1, 1 ⊗ y) is the trivial extension (see Example 5 above), then ⟨α⟩(φ) = α for all φ ∈ Hom_{R}(H, A).

The set of sections {⟨α⟩: α ∈ 𝔫(A)} is a group with respect to the operation ⟨α⟩⟨β⟩ := ⟨F (α, β)⟩ (obviously that it is isomorphic to 𝔫(A)_{F}).

Now let φ ∈ Hom_{R}(H, A) be a fixed element. Since

\{( \begin{matrix}α \\ φ \end{matrix} ) : α ∈ 𝔫(A), φ ∈ Hom_{R}(H, A)\}

is a group, the shift

( \begin{matrix}α \\ ε \end{matrix} ) ↦ ( \begin{matrix}α \\ ε \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ \end{matrix} ) = ( \begin{matrix}((ε_{H} ⊗ φ)𝔉)(α, 0) \\ φ \end{matrix} )

is a bijection from the set (more precisely, subgroup)

\{( \begin{matrix}α \\ ε \end{matrix} ) : α ∈ 𝔫(A)\} to the (co)set \{( \begin{matrix}β \\ φ \end{matrix} ) : β ∈ 𝔫(A)\} .

Therefore for any φ ∈ Hom_{R}(H, A) the map ρ_{φ}: α ↦ ⟨α⟩(φ) = ((ε_{H} ⊗ φ)𝔉)(α, 0) defines a bijection 𝔫(A) → 𝔫(A).

Now we assume that (μ_{H} ∘ (id_{H} ⊗S))𝔠 = 0, where 𝔠 as in Corollary 14. It follows from our assumption that

( \begin{matrix}0 \\ φ ∘ S \end{matrix} ) = ( \begin{matrix}0 \\ φ \end{matrix} )^{−1} .

Given φ let us define a new operation ∘_{φ} on the set 𝔫(A) as follows:

( \begin{matrix}α ∘_{φ} β \\ φ \end{matrix} ) := ((( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} )) ∘ (( \begin{matrix}β \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} ))) ∘ ( \begin{matrix}0 \\ φ \end{matrix} ) . (18)

The operation ∘_{φ} is compatible with the group structure on the set of sections {⟨α⟩: α ∈ 𝔫(A)} in the following sense.

Proposition 22. If α = ⟨α′⟩(φ), β = ⟨β′⟩(φ) for some α′, β′ ∈ 𝔫(A) (obviously, α′ = (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, 0), β′ = (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(β, 0)), then α ∘_{φ} β = (⟨α′⟩⟨β′⟩)(φ).

Proof. The proof is trivial. □ This proposition justifies the definition (18) of the operation ∘_{φ}.

Note that

((( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} )) ∘ (( \begin{matrix}β \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} ))) ∘ ( \begin{matrix}0 \\ φ \end{matrix} )

= ( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}β \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} ) = ( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}(μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(β, 0) \\ ε \end{matrix} )

= ( \begin{matrix}((φ ⊗ ε_{H})𝔉)(α, (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(β, 0)) \\ φ \end{matrix} ) ,  (19)

i.e. the product α ∘_{φ} β may be represented as follows

((φ ⊗ ε_{H})𝔉)(α, (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(β, 0)).

For φ = ε the operation ∘_{ε} on 𝔫(A) coincides with +_{F}, i.e. the set 𝔫(A) with respect to the operation, corresponding to ε, is the group 𝔫(A)_{F}.

Proposition 23. For any φ ∈ Hom_{R}(H, A) the corresponding operation ∘_{φ} induces the structure of an Abelian group on the set 𝔫(A).

Proof. It follows from Proposition 20 and from the definition that the operation ∘_{φ} is associative and commutative. We claim that 0 ∈ 𝔫(A) is its identity element. Indeed, it follows from the assumption (μ_{H} ∘ (id_{H} ⊗S))𝔠 = 0 that (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(0, 0) = 0. Then using the identity axiom 2) of Definition 1, we get

α ∘_{φ} 0 = ((φ ⊗ ε_{H})𝔉)(α, (μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(0, 0)) = ((φ ⊗ ε_{H})𝔉)(α, 0) = α.

To complete the proof we must verify that for any α ∈ 𝔫(A) there is the inverse element β = (α)_{φ}^{−1} ∈ 𝔫(A) (with respect to the operation ∘_{φ}). It follows from the definition of ∘_{φ} that

( \begin{matrix}α \\ φ \end{matrix} ) ∘ ( \begin{matrix}β \\ φ \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ ∘ S \end{matrix} ) = ( \begin{matrix}0 \\ φ \end{matrix} ) .

Therefore

( \begin{matrix}(μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, 0) \\ ε \end{matrix} ) ∘ ( \begin{matrix}β \\ φ \end{matrix} ) = ( \begin{matrix}0 \\ φ \end{matrix} ) .

Using (15), we get

( \begin{matrix}β \\ φ \end{matrix} ) = ( \begin{matrix}θ((μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, 0)) \\ ε \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ \end{matrix} )

= ( \begin{matrix}((ε_{H} ⊗ φ)𝔉)(θ((μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, 0)), 0) \\ φ \end{matrix} ) ,

where θ(x) = (ε_{H}Θ)(x) ∈ R[[x]] is the formal inverse in the formal group law F (x, y). Hence,

(α)_{φ}^{−1} = ((ε_{H} ⊗ φ)𝔉)(θ((μ_{A} ∘ (φ ⊗ (φ ∘ S))𝔉)(α, 0)), 0). □  (20)

Using the compatibility of the operation ∘_{φ} with the group structure on the set of sections {⟨α⟩: α ∈ 𝔫(A)}, one can establish a canonical isomorphism between (𝔫(A), ∘_{φ}) and 𝔫(A)_{F} = (𝔫(A), ∘_{ε}).

Although for all φ the corresponding group structures on 𝔫(A) are isomorphic to each other (in particular, all of them are isomorphic to the group, corresponding to ε, i.e. 𝔫(A)_{F}), in general they do not coincide. Moreover, we have a homomorphism from the group Hom_{R}(H, A) to the “group of group structures” on the set 𝔫(A).

Hence a nontrivial extension 𝔉(x ⊗ 1, 1 ⊗ y) of a formal group law F (x, y) in general defines a set of isomorphic group structures on the nil-radical 𝔫(A).

By ℌ(x, y) ∈ H[[x, y]] denote the power series ((id_{H} ⊗ε_{H})𝔉)(x, 𝔊(y, 0)), where 𝔊(x, y) = (μ_{H} ∘ (id_{H} ⊗S)𝔉)(x, y) as above.

Lemma 24. The power series ℌ(x, y) may be obtained by applying

(μ_{H} ∘ (id_{H} ⊗μ_{H}) ∘ (id_{H} ⊗ id_{H} ⊗S))  (21)

to the power series

((id_{H} ⊗Δ)𝔉)(x ⊗ 1 ⊗ 1, 1 ⊗ 𝔉(y ⊗ 1, 1 ⊗ z))  (22)

and putting z = 0.

Proof. Lemma’s assertion may be verified by a straightforward computation. Indeed, the result of application of (21) to (22) is equal to

((id_{H} ⊗ε_{H})𝔉)(x, (μ_{H} ∘ (id_{H} ⊗S)𝔉)(y, 0)) = ((id_{H} ⊗ε_{H})𝔉)(x, 𝔊(y, 0)). □

Note that (ε_{H}ℌ)(x, y) = F (x, y).

By ℑ(x) ∈ H[[x]] denote the power series ((ε_{H} ⊗ id_{H})𝔉)(θ(𝔊(x, 0)), 0), where θ(x) is the formal inverse for F (x, y).

Lemma 25. The power series ℑ(x) may be obtained by applying

(μ_{H} ∘ (μ_{H} ⊗ id_{H}) ∘ ((id_{H} ⊗S) ⊗ id_{H}))

to the power series

((Δ ⊗ id_{H})𝔉)(θ(𝔉(x ⊗ 1, 1 ⊗ y)) ⊗ 1, 1 ⊗ 1 ⊗ z)

and putting y = z = 0.

Proof. Straightforward computation. □

Note that (ε_{H}ℑ)(x) = θ(x).

Proposition 26. Let A be an R-algebra and let φ be an R-homomorphism H → A. Then

α ∘_{φ} β = (φℌ)(α, β)

for all α, β ∈ 𝔫(A) and (α)_{φ}^{−1} = (φℑ)(α) for any α ∈ 𝔫(A).

Proof follows from the comparison of (19) and (20) with the lemmas above.□

Remark 27. Let us remember that the set Hom_{R}(H, A) is a group with respect to the convolution ⋆: φ ⋆ ψ = μ_{A} ∘ (φ ⊗ ψ) ∘ Δ ∈ Hom_{R}(H, A) for all φ, ψ ∈ Hom_{R}(H, A). The convolution induces a group operation ⋄ on the set of formal group laws {(φℌ)(x, y) ∈ A[[x, y]] | φ ∈ Hom_{R}(H, A)} in the following way:

((φℌ)(x, y)) ⋄ ((ψℌ)(x, y)) := ((φ ⋆ ψ)ℌ)(x, y).

Note that the series (εℌ)(x, y) = (η_{A}F)(x, y) is the identity element in considered group and ((φ ∘ S)ℌ)(x, y) = (φ^{−1}ℌ)(x, y) is the inverse element for the given one (φℌ)(x, y).

Let us recall that we have assumed that (μ_{H} ∘ (id_{H} ⊗S))𝔠 = 0. It follows from Corollary 14 that

(Δ𝔣)(𝔉(x ⊗ 1, 1 ⊗ y)) = F (𝔣(x) ⊗ 1, 1 ⊗ 𝔣(y)))

for an invertible power series 𝔣(x) ∈ H[[x]] with zero constant term such that (ε_{H}𝔣)(x) = x.

Proposition 28. We have:

ℌ(x, y) = 𝔣^{−1}(F (𝔣(x), 𝔣(y))), ℑ(x) = 𝔣^{−1}(θ(𝔣(x))).

The power series ℌ(x, y) is a formal group law (in the usual sense) over the ring H with the formal inverse ℑ(x).

Remark 29. Note that the formal group law ℌ(x, y) is isomorphic to F (x, y), considered as a formal group law over H ⊃ R. Proof. We have:

𝔊(x, y) = (ε_{H}𝔣^{−1})(F (𝔣(x), (S𝔣)(y))) = F (𝔣(x), (S𝔣)(y)),

hence

𝔊(y, 0) = F (𝔣(y), 0) = 𝔣(y).

Also we have:

((id_{H} ⊗ε_{H})𝔉)(x, y) = 𝔣^{−1}(F (𝔣(x), y)).

By substituting of 𝔊(y, 0) for y in the latter formula, we obtain

ℌ(x, y) = 𝔣^{−1}(F (𝔣(x), 𝔣(y))),

as required.

Similarly,

((ε_{H} ⊗ id_{H})𝔉)(x, y) = 𝔣^{−1}(F (x, 𝔣(y))).

Therefore,

ℑ(x) = 𝔣^{−1}(F (θ(𝔣(x)), 0) = 𝔣^{−1}(θ(𝔣(x))). □

2 A generalization of the formal product in complex cobordism theory

2.1 The basic formula

By BU_{⊗}, BSU_{⊗} we denote the classifying spaces BU, BSU with H-group structures related to the tensor product of virtual bundles of virtual dimension 1. Note that the H-group BU_{⊗} (resp. BSU_{⊗}) represents the homotopy functor X ↦ K̃_{ℂ}(X)_{⊗} (resp. X ↦ \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗}) to the category of Abelian groups. 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).

It is known that there is an isomorphism of H-groups

BU_{⊗} ≅ BSU_{⊗} ×ℂP^{∞},  (23)

where the space ℂP^{∞} is considered with respect to the usual H-group structure, related to the tensor product of line bundles. The proof of (23) will be given in the Appendix. Isomorphism of H-groups (23) corresponds to the following isomorphism of bundles

Ξ̂ ≅ Ξ ⊠ Z,

where Z is the canonical (universal) line bundle over ℂP^{∞} and ⊠ denotes the exterior tensor product of bundles.

We shall use the notation σ_{r} for the r-th Conner-Floyd characteristic class [2]. By s_{r} we denote the r-th Newton characteristic class in complex cobordism theory, corresponding to the symmetric polynomial u_{1}^{r} + … + u_{n}^{r} (note that σ_{1} = s_{1}).

Let R be the cobordism ring MU^{∗}(pt) and let H be the Hopf algebra MU^{∗}(BSU_{⊗}) over R. Put x = σ_{1}(Ξ̂) ∈ MU^{2}(BU), u = σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}). It is well known that MU^{∗}(BU) = R[[σ_{1}(Ξ̂), σ_{2}(Ξ̂), …]] and H = MU^{∗}(BSU) = R[[σ_{2}(Ξ), σ_{3}(Ξ), …]]. On the other hand, it follows from (23) that MU^{∗}(BU) = H[[u]]. Hence x = 𝔥(u) for some power series 𝔥(u) ∈ H[[u]]. The main goal of this section is to prove the explicit formula for this power series.

By definition, put s̃_{i} := s_{i}(Ξ) ∈ MU^{2i}(BSU) (clearly s̃_{1} = 0).

Proposition 30.

𝔥(u) = 1 ⊗ u + ∑_{i,j≥1} α_{i,j}s̃_{i} ⊗ u^{j},  (24)

where α_{i,j} ∈ R are the coefficients of the formal product F (u, v) = u + v + ∑ α_{i,j}u^{i}v^{j} in complex cobordism theory [1].

Proof. Let Ξ̂_{k} (Ξ_{k}) be the universal k-dimensional U(k)-bundle over BU(k) (the universal kdimensional SU(k)-bundle over BSU(k) respectively). Consider the restriction of the virtual bundle Ξ_{k} ⊗ [k] − [k(k − 1)] (where by [n] we denote the trivial n-dimensional bundle) to the 2k-skeleton BSU(k)^{(2k)}. There exists a (unique up to isomorphism) geometric k-dimensional representative Ξ_{k}^{•} of the stable equivalence class {Ξ_{k} ⊗ [k] − [k(k − 1)]} ∈ K_{ℂ}(BSU(k)^{(2k)}).

Let

λ_{k}: BSU(k)^{(2k)} → BU(k)

be a classifying map for Ξ_{k}^{•} (i.e. λ_{k}^{∗}(Ξ̂_{k}) = Ξ_{k}^{•}). It follows easily that

λ_{k}^{∗}(s_{i}(Ξ̂_{k})) = s_{i}(λ_{k}^{∗}(Ξ̂_{k})) = ks̃_{i}^{(k)},

where s̃_{i}^{(k)} := s_{i}(Ξ_{k} − [k − 1]) ∈ MU^{2i}(BSU(k)^{(2k)}).

Now let us apply the splitting principle to the tensor product Ξ̂_{k} ⊠ Z :

[figure: Ξ̂_{k} ⊠ Z; (Z_{1} ⊕ … ⊕ Z_{k}) ⊠ Z; BU(k) × ℂP^{∞}; \underbrace{ℂP^{∞} × … × ℂP^{∞}}_{k times} × ℂP^{∞}.; μ̂_{k}×id_{ℂP}]

We have:

(μ̂_{k} × id_{ℂP})^{∗}(Ξ̂_{k} ⊠ Z) = (Z_{1} ⊕ … ⊕ Z_{k}) ⊠ Z = (Z_{1} ⊠ Z) ⊕ … ⊕ (Z_{k} ⊠ Z) := Θ_{k},

s_{r}(Θ_{k}) = s_{1}(Z_{1} ⊠ Z)^{r} + … + s_{1}(Z_{k} ⊠ Z)^{r} = F (v_{1} ⊗ 1, 1 ⊗ u)^{r} + … + F (v_{k} ⊗ 1, 1 ⊗ u)^{r}.

In particular,

s_{1}(Θ_{k}) = v_{1} ⊗ 1 + 1 ⊗ u + ∑_{i,j≥1} α_{i,j}v_{1}^{i} ⊗ u^{j} + … + v_{k} ⊗ 1 + 1 ⊗ u + ∑_{i,j≥1} α_{i,j}v_{k}^{i} ⊗ u^{j}

= (μ̂_{k} × id_{ℂP})^{∗}(s_{1}(Ξ̂_{k}) ⊗ 1 + k ⊗ u + ∑_{i,j≥1} α_{i,j}s_{i}(Ξ̂_{k}) ⊗ u^{j}),

hence

s_{1}(Ξ̂_{k} ⊠ Z) = s_{1}(Ξ̂_{k}) ⊗ 1 + k ⊗ u + ∑_{i,j≥1} α_{i,j}s_{i}(Ξ̂_{k}) ⊗ u^{j}.

By x^{(k)} denote the first Conner-Floyd class

σ_{1}((Ξ_{k} − [k − 1]) ⊠ Z) = s_{1}((Ξ_{k} − [k − 1]) ⊠ Z) ∈ MU^{2}(BSU(k)^{(2k)} × ℂP^{∞}).

Consider the following diagram:

[figure: Ξ_{k}^{•} ⊠ Z; Ξ̂_{k} ⊠ Z; BSU(k)^{(2k)} × ℂP^{∞}; λ_{k}×id_{ℂP}; BU(k) × ℂP^{∞},]

where λ_{k} is the classifying map for Ξ_{k}^{•}, as above. We have:

s_{1}(Ξ_{k}^{•} ⊠ Z) = λ_{k}^{∗}(s_{1}(Ξ̂_{k})) ⊗ 1 + k ⊗ u + ∑_{i,j≥1} α_{i,j}λ_{k}^{∗}(s_{i}(Ξ̂_{k})) ⊗ u^{j}

= k ⊗ u + k ∑_{i,j≥1} α_{i,j}s̃_{i}^{(k)} ⊗ u^{j} = k(1 ⊗ u + ∑_{i,j≥1} α_{i,j}s̃_{i}^{(k)} ⊗ u^{j})

(let us recall that λ_{k}^{∗}(s_{1}(Ξ̂_{k})) = s_{1}(Ξ_{k}^{•}) = 0, because Ξ_{k}^{•} is an SU(k)-bundle). Since

Ξ_{k}^{•} ⊠ Z = ((Ξ_{k} − [k − 1]) ⊗ [k]) ⊠ Z = ((Ξ_{k} − [k − 1]) ⊠ Z) ⊗ [k],

we get

x^{(k)} = 1 ⊗ u + ∑_{i,j≥1} α_{i,j}s̃_{i}^{(k)} ⊗ u^{j}.  (25)

Using the isomorphism of the bundles Ξ̂ ≅ Ξ ⊠ Z, it can easily be checked that the sequence

{x^{(k)}}_{k∈ℕ}, x^{(k)} ∈ MU^{2}(BSU(k)^{(2k)} × ℂP^{∞})

defines the element x = σ_{1}(Ξ̂) ∈ MU^{2}(BSU ×ℂP^{∞}) = MU^{2}(BU). Therefore the required formula (24) may be obtained as the limit of (25), as k tends to ∞. □ Remark 31. Let us remark that the series 𝔥(u) is invertible, i.e. there exists the series 𝔥^{−1}(x) ∈ H[[x]] such that 𝔥^{−1}(𝔥(u)) = u, 𝔥(𝔥^{−1}(x)) = x. Moreover, (ε_{H}𝔥)(u) = u.

The remainder of this section is devoted to the description of the comultiplication in the Hopf algebra Ĥ := MU^{∗}(BU_{⊗}) (consequently, in the Hopf algebra H = MU^{∗}(BSU_{⊗})).

Consider the following diagram

\xymatrix{(ℂP^{∞})^{k} × (ℂP^{∞})^{l} \ar[r]^{Ψ_{k,l}} \ar[d]_{μ̂_{k}×μ̂_{l}} & (ℂP^{∞})^{kl} \ar[d]^{μ̂_{kl}} \\ BU(k) × BU(l) \ar[r]^{Φ̂_{k,l}} & BU(kl),}

where the map Φ̂_{k,l} such that Φ̂_{k,l}^{∗}(Ξ̂_{kl}) = Ξ̂_{k} ⊠ Ξ̂_{l}. We have the corresponding diagram for characteristic classes:

\xymatrix{∑_{i,j} F (u_{i} ⊗ 1, 1 ⊗ v_{j})^{r} & ∑_{i,j} w_{ij}^{r} \ar[l]_{Ψ_{k,l}^{∗}} \\ s_{r}(Ξ̂_{k} ⊠ Ξ̂_{l}) = Φ̂_{k,l}^{∗}(s_{r}(Ξ̂_{kl})) \ar[u]^{μ_{k}^{∗}⊗μ_{l}^{∗}} & s_{r}(Ξ̂_{kl}), \ar[l]_{Φ̂_{k,l}^{∗}} \ar[u]_{μ_{kl}^{∗}}}

where 1 ≤ i ≤ k, 1 ≤ j ≤ l, r ≥ 1. Clearly the power series

∑_{1≤i≤k,1≤j≤l} F (u_{i} ⊗ 1, 1 ⊗ v_{j})^{r}

may be expressed in terms of s_{m}(u_{1}, …, u_{k}) := u_{1}^{m} + … + u_{k}^{m} and s_{n}(v_{1}, …, v_{l}) := v_{1}^{n} + … + v_{l}^{n}, m, n ≥ 1. Let us define γ_{m,n}^{(r)} ∈ R as follows:

F (u, v)^{r} = u^{r} + v^{r} + ∑_{m,n≥1, m+n≥r} γ_{m,n}^{(r)}u^{m}v^{n}.

In particular, γ_{m,n}^{(1)} = α_{m,n} ∈ R are the coefficients of the formal product in complex cobordism theory. It is easily shown that

Φ̂_{k,l}^{∗}(s_{r}(Ξ̂_{kl})) = s_{r}(Ξ̂_{k}) ⊗ l + k ⊗ s_{r}(Ξ̂_{l}) + ∑_{m,n≥1} γ_{m,n}^{(r)}s_{m}(Ξ̂_{k}) ⊗ s_{n}(Ξ̂_{l}).  (26)

In particular, if r = 1, we have:

Φ̂_{k,l}^{∗}(s_{1}(Ξ̂_{kl})) = s_{1}(Ξ̂_{k}) ⊗ l + k ⊗ s_{1}(Ξ̂_{l}) + ∑_{m,n≥1} α_{m,n}s_{m}(Ξ̂_{k}) ⊗ s_{n}(Ξ̂_{l}).  (27)

Let Ξ̂_{n}^{•} be the geometric representative of the stable equivalence class

{Ξ̂_{n} ⊗ [n] − [n(n − 1)]} ∈ K_{ℂ}(BU(n)^{(2n)}).

Let λ̂_{n}: BU(n)^{(2n)} → BU(n) be the classifying map of Ξ̂_{n}^{•}, i.e. λ̂_{n}^{∗}(Ξ̂_{n}) = Ξ̂_{n}^{•}. Clearly,

λ̂_{n}^{∗}(s_{r}(Ξ̂_{n})) = s_{r}(Ξ̂_{n}^{•}) = ns_{r}(Ξ̂_{n} − [n − 1]), r ≥ 0.  (28)

The bundle Ξ̂_{n}^{•} over BU(n)^{(2n)} has the following universal property.

Lemma 32. Let X be a finite CW -complex, dim(X) ≤ 2n. Let η_{n} be an n-dimensional complex vector bundle over X such that {η_{n}} = n{θ_{1}} ∈ K_{ℂ}(X), where θ_{1} is a virtual bundle of virtual dimension 1. Then there exists a (unique up to homotopy) map τ: X → BU(n)^{(2n)} such that η_{n} = τ^{∗}(Ξ̂_{n}^{•}).

Proof. The proof is trivial. □

Applying the previous lemma to the bundle Ξ̂_{k}^{•} ⊠ Ξ̂_{l}^{•} over BU(k)^{(2k)} × BU(l)^{(2l)}, we get a classifying map

ϕ̂_{k,l}: BU(k)^{(2k)} × BU(l)^{(2l)} → BU(kl)^{(2kl)}

such that ϕ̂_{k,l}^{∗}(Ξ̂_{kl}^{•}) = Ξ̂_{k}^{•} ⊠ Ξ̂_{l}^{•} and the following diagram

\xymatrix{BU(k) × BU(l) \ar[r]^{Φ̂_{k,l}} & BU(kl) \\ BU(k)^{(2k)} × BU(l)^{(2l)} \ar[r]^{ϕ̂_{k,l}} \ar[u]^{λ̂_{k}×λ̂_{l}} & BU(kl)^{(2kl)} \ar[u]_{λ̂_{kl}}}

commutes (up to homotopy). Using (26), (28) and the previous diagram, we get:

ϕ̂_{k,l}^{∗}(s_{r}(Ξ̂_{kl} − [kl − 1])) = s_{r}(Ξ̂_{k} − [k − 1]) ⊗ 1 + 1 ⊗ s_{r}(Ξ̂_{l} − [l − 1])

+ ∑_{m,n≥1} γ_{m,n}^{(r)}s_{m}(Ξ̂_{k} − [k − 1]) ⊗ s_{n}(Ξ̂_{l} − [l − 1]).

Thus, passing to the limit as k, l → ∞, we get:

ΔĤ(s_{r}(Ξ̂)) = s_{r}(Ξ̂) ⊗ 1 + 1 ⊗ s_{r}(Ξ̂) + ∑_{m,n≥1} γ_{m,n}^{(r)}s_{m}(Ξ̂) ⊗ s_{n}(Ξ̂), r ≥ 1.  (29)

In particular, using (27), we obtain:

ϕ̂_{k,l}^{∗}(s_{1}(Ξ̂_{kl} − [kl − 1])) = s_{1}(Ξ̂_{k} − [k − 1]) ⊗ 1 + 1 ⊗ s_{1}(Ξ̂_{l} − [l − 1])

+ ∑_{m,n≥1} α_{m,n}s_{m}(Ξ̂_{k} − [k − 1]) ⊗ s_{n}(Ξ̂_{l} − [l − 1]),

i.e. (passing to the limit as k, l → ∞)

ΔĤ(x) = x ⊗ 1 + 1 ⊗ x + ∑_{m,n≥1} α_{m,n}s_{m}(Ξ̂) ⊗ s_{n}(Ξ̂)

(compare with (32)).

Since the rings MU^{∗}(BSU), MU^{∗}(BU) have no torsion and MU^{∗}(BSU)ˆ⊗_{ℤ}ℚ = R_{ℚ}[[s_{2}(Ξ), s_{3}(Ξ), …]], MU^{∗}(BU)ˆ⊗_{ℤ}ℚ = R_{ℚ}[[s_{1}(Ξ̂), s_{2}(Ξ̂), …]], formula (29) gives the complete description of the comultiplication of considered Hopf algebras.

2.2 The formal group law over H = MU^{∗}(BSU_{⊗}) related to virtual bundles of virtual dimension 1

The aim of this section is to obtain a generalization of the formal product in complex cobordism theory from the case of complex line bundles to the case of all complex vector bundles of virtual dimension 1.

Since BU_{⊗} is an H-group, we have the product

ϕ̂: BU_{⊗} × BU_{⊗} → BU_{⊗}

and the inversion ι̂: BU_{⊗} → BU_{⊗}.

As above, by H we denote the Hopf algebra MU^{∗}(BSU_{⊗}) over the ring R = MU^{∗}(pt).

Proposition 33.

MU^{∗}(BU) = H[[x]], MU^{∗}(BU × BU) = (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]],  (30)

where x = σ_{1}(Ξ̂) as above. Moreover,

ϕ̂^{∗}(x) = 𝔉(x ⊗ 1, 1 ⊗ x) ∈ (H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]]  (31)

is a commutative formal group law over the Hopf algebra H with the formal inverse Θ(x) = ι̂^{∗}(x) ∈ H[[x]].

Proof. It follows from (23) that MU^{∗}(BU) = H[[u]]. On the other hand, the explicit formula

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

obtained in the previous section (see also Remark 31), shows that H[[u]] = H[[x]]. This verifies equalities (30). Hence the element ϕ̂^{∗}(x) is the power series 𝔉(x ⊗ 1, 1 ⊗ x) ∈

(H ˆ⊗_{R}H)[[x ⊗ 1, 1 ⊗ x]].

To verify the associative axiom 1) of Definition 1 note that the maps ϕ̂ ∘ (ϕ̂ ⊗ id_{BU}) and ϕ̂ ∘ (id_{BU} ⊗ϕ̂) are classifying maps for the bundle Ξ̂ ⊠ Ξ̂ ⊠ Ξ̂ over BU × BU × BU and hence they are homotopic. Therefore

((Δ_{H} ⊗ id_{H})𝔉)(𝔉(x ⊗ 1, 1 ⊗ x) ⊗ 1, 1 ⊗ 1 ⊗ x)

= (ϕ̂ ⊗ id_{BU})^{∗} ∘ ϕ̂^{∗}(x) = (id_{BU} ⊗ϕ̂)^{∗} ∘ ϕ̂^{∗}(x)
= ((id_{H} ⊗Δ_{H})𝔉)(x ⊗ 1 ⊗ 1, 1 ⊗ 𝔉(x ⊗ 1, 1 ⊗ x)).

To verify the identity axiom 2) of Definition 1 note that

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

is the image of x under the composite map i_{1}^{∗} ∘ ϕ̂^{∗}, i_{2}^{∗} ∘ ϕ̂^{∗} respectively, where i_{k}: BU → BU × BU by i_{1}(y) = (y, ∗), i_{2}(y) = (∗, y). Clearly that both ϕ̂ ∘ i_{1} and ϕ̂ ∘ i_{2} are homotopic to the identity map.

To verify the formal inversion axiom 3) of Definition 1 note that

ϕ̂ ∘ (id_{BU} ×ι̂) ∘ diag_{BU} ≃ ∗, ϕ̂ ∘ (ι̂× id_{BU}) ∘ diag_{BU} ≃ ∗,

where diag_{BU} : BU → BU × BU is the diagonal map. Therefore

diag_{BU}^{∗} ∘(id_{BU} ×ι̂)^{∗} ∘ ϕ̂^{∗}(x) = ((μ ∘ (id_{H} ⊗S))𝔉)(x, Θ(x)) = 0
= ((μ ∘ (S ⊗ id_{H}))𝔉)(Θ(x), x) = diag_{BU}^{∗} ∘(ι̂× id_{BU})^{∗} ∘ ϕ̂^{∗}(x). □

The following proposition asserts that the formal group law 𝔉(x ⊗ 1, 1 ⊗ x) is strongly isomorphic to the trivial extension of the formal product F (u, v) in complex cobordism theory by the Hopf algebra H. Moreover, the explicit form of the change of variable is given.

Proposition 34.

𝔉(x ⊗ 1, 1 ⊗ x) = (Δ_{H}𝔥)(F ((𝔥^{−1}(x)) ⊗ 1, 1 ⊗ (𝔥^{−1}(x)))).  (32)

Moreover Θ(x) = (S_{H}𝔥)(θ(𝔥^{−1}(x))), where Θ(x) ∈ H[[x]] is the formal inverse for the formal group law 𝔉(x ⊗ 1, 1 ⊗ x).

Remark 35. It follows from (24) that the formal group law 𝔉(x ⊗ 1, 1 ⊗ x) is strongly isomorphic to the trivial extension of F (u, v) by H. Indeed, (ε_{H}𝔥)(u) = u, in particular, ε_{H}(𝔥′(0)) = ε_{H}(1 + ∑_{i≥2} α_{i,1}s̃_{i}) = 1. Proof. Note that (23) implies that the Hopf algebra MU^{∗}(BU_{⊗}) is isomorphic to the tensor product of Hopf algebras

H[[u]] = H ˆ⊗_{R}R[[u]],  (33)

i.e. the comultiplication

Δ̂: H[[u]] → (H ˆ⊗_{R}H)[[u ⊗ 1, 1 ⊗ u]]

is following: Δ̂|_{H} = Δ_{H}, Δ̂(u) = F (u ⊗ 1, 1 ⊗ u), where F (u, v) is the formal product in complex cobordism theory; and the antipode

Ŝ: H[[u]] → H[[u]]

is following: Ŝ|_{H} = S_{H}, Ŝ(u) = θ(u), where θ(u) is the formal inverse in the formal group law F (u, v).

Using isomorphism (33), we get:

ϕ̂^{∗}(x) = Δ̂(x) = Δ̂(𝔥(u)) = (Δ_{H}𝔥)(F (u ⊗ 1, 1 ⊗ u))
= (Δ_{H}𝔥)(F ((𝔥^{−1}(x)) ⊗ 1, 1 ⊗ (𝔥^{−1}(x)))).

Using (33) again, we get:

ι̂^{∗}(x) = Θ(x) = (S_{H}𝔥)(θ(u)) = (S_{H}𝔥)(θ(𝔥^{−1}(x))). □

Remark 36. Note that ((ε_{H} ⊗ ε_{H})𝔉)(u, v) = F (u, v) and (ε_{H}Θ)(u) = θ(u) (see Remark 31). Remark 37. Now we mention briefly the way to obtain another extensions of the formal product F (u, v) by H. To this purpose we can consider cohomology operations. For the Adams operation Ψ^{k} let us define the endomorphism ρ_{k} of the Hopf algebra H by setting ρ_{k}(σ)(Ξ) = σ(Ψ^{k}Ξ) for any σ ∈ H, where Ξ is the universal SU-bundle over BSU. Put x_{k} := σ_{1}((Ψ^{k}Ξ) ⊠ Z) ∈ MU^{2}(BSU ×ℂP^{∞}) = MU^{2}(BU). Clearly

x_{k} = (ρ_{k}𝔥)(u) = u + ∑_{i,j≥1} α_{i,j}ρ_{k}(s̃_{i})u^{j}.

By 𝔥_{k}(u) denote the power series (ρ_{k}𝔥)(u) ∈ H[[u]]. If we replace 𝔥(u) by 𝔥_{k}(u) in (32), we get some other extension

𝔉_{k}(x_{k} ⊗ 1, 1 ⊗ x_{k}) := (Δ_{H}𝔥_{k})(F ((𝔥_{k}^{−1}(x_{k})) ⊗ 1, 1 ⊗ (𝔥_{k}^{−1}(x_{k}))))
∈ (H ˆ⊗_{R}H)[[x_{k} ⊗ 1, 1 ⊗ x_{k}]] = MU^{∗}(BU)ˆ⊗_{R} MU^{∗}(BU)

of the formal group law F (u, v) by the Hopf algebra H (note that 𝔉_{1}(x_{1} ⊗ 1, 1 ⊗ x_{1}) = 𝔉(x ⊗ 1, 1 ⊗ x)). We claim that 𝔉_{k}(x_{k} ⊗ 1, 1 ⊗ x_{k}) = ((ρ_{k} ⊗ ρ_{k})𝔉)(x_{k} ⊗ 1, 1 ⊗ x_{k}). Indeed, it follows from the equality (ρ_{k} ⊗ ρ_{k}) ∘ Δ_{H} = Δ_{H} ∘ ρ_{k}.

Applying the endomorphisms of the Hopf algebra H, corresponding to the operations Ξ ↦ Ξ^{k}, k ∈ ℤ, to the coefficients of the power series 𝔥(u), we obtain another extensions.

Let g(u) = u + ∑_{j≥1} β_{j}u^{j+1} ∈ R_{ℚ}[[u]] be the logarithm of the formal product F (u, v) in complex cobordism theory [6]. By \overline{𝔤}(x) denote the power series g(𝔥^{−1}(x)) ∈ H_{ℚ}[[x]]. Note that g(u) = \overline{𝔤}(x) as elements of MU^{2}(BU)ˆ⊗_{ℤ}ℚ, because of x = 𝔥(u).

Proposition 38. The power series \overline{𝔤}(x) is the logarithm of the formal group law 𝔉(x⊗1, 1⊗ x) over the Hopf algebra H in the sense of Theorem 9.

Proof. Let us recall that in Theorem 9 the power series

𝔤(x) := ∫_{0}^{x} \frac{dt}{ω̃(t)} ∈ H_{ℚ}[[x]]

was considered, where

ω̃(x) := (id ⊗ε̃)\frac{∂𝔉(x ⊗ 1, 1 ⊗ z)}{∂(1 ⊗ z)} ∈ H[[x]]

(here ε̃: H[[z]] → R is the homomorphism such that ε̃ |_{H}= ε_{H}: H → R, ε̃(z) = 0). We must prove that 𝔤(x) = \overline{𝔤}(x). We have:

\frac{∂𝔉(x ⊗ 1, 1 ⊗ z)}{∂(1 ⊗ z)} = \frac{d(Δ_{H}𝔥)(F (𝔥^{−1}(x) ⊗ 1, 1 ⊗ 𝔥^{−1}(z)))}{dF (𝔥^{−1}(x) ⊗ 1, 1 ⊗ 𝔥^{−1}(z))}·

\frac{∂F (𝔥^{−1}(x) ⊗ 1, 1 ⊗ 𝔥^{−1}(z))}{∂(1 ⊗ 𝔥^{−1}(z))} · (1 ⊗ \frac{d(𝔥^{−1}(z))}{dz}) .

Applying the operator (id ⊗ε̃) to the both sides of the previous equation, we get:

ω̃(x) = 𝔥′(𝔥^{−1}(x)) · ω(𝔥^{−1}(x)) · \frac{1}{ε_{H}(𝔥′(0))}

(we used 𝔥^{−1}(0) = 0), where ω(y) := \frac{∂F (y,t)}{∂t}|_{t=0}. Recall that ε_{H}(𝔥′(0)) = 1. Therefore we obtain:

𝔤′(x) = \frac{dg(𝔥^{−1}(x))}{dx}.

Using 𝔤(0) = 0 = g(𝔥^{−1}(0)), we obtain the required identity \overline{𝔤}(x) = 𝔤(x). □

Due to this coincidence, we shall use the notation 𝔤(x) instead of \overline{𝔤}(x) for the series g(𝔥^{−1}(x)). Reminder. We have proved in Theorem 9 that the power series 𝔤(x) has the following properties:

(i) (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x)) = 𝔠 + 𝔤(x) ⊗ 1 + 1 ⊗ 𝔤(x);

(ii) 𝔤(0) = 0;

(iii) (ε_{H}𝔤)(u) = g(u).

One can check the property (i) by the direct computation, using 𝔤(x) = g(𝔥^{−1}(x)) = g(u) ∈ MU^{2}(BU)ˆ⊗_{ℤ}ℚ. Indeed, we have:

Δ̂(g(u)) = g(F (u ⊗ 1, 1 ⊗ u)) = g(u) ⊗ 1 + 1 ⊗ g(u)
= 𝔤(x) ⊗ 1 + 1 ⊗ 𝔤(x) = Δ̂(𝔤(x))
= (Δ𝔤)(𝔉(x ⊗ 1, 1 ⊗ x))

(in particular we see that the cocycle 𝔠 equals 0). The property 𝔤(0) = 0 easily follows from the explicit formula (24) for 𝔥(u) and g(0) = 0. □

Now let us return to isomorphism (23). Let p: BU_{⊗} → BSU_{⊗} be the projection onto the first factor. 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. Let φ_{ξ}: X → BU be a classifying map for ξ. 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 39. 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)).  (34)

Let ξ^{−1} be the inverse vector bundle for ξ in sense of H-group structure BU_{⊗} (i.e. ξ⊗ξ^{−1} = [1] is the trivial bundle). Then

σ_{1}(ξ^{−1}) = (φ∗ ; \overline{ξ}Θ)(σ_{1}(ξ))  (35)

and we have

(μ_{X} ∘ (φ∗ ; \overline{ξ} ⊗ (φ∗ ; \overline{ξ} ∘ S_{H}))𝔉)(σ_{1}(ξ), σ_{1}(ξ^{−1})) = 0  (36)

(note that φ∗ ; \overline{ξ} ∘ S_{H} = φ_{\overline{ξ}^{−1}}^{∗}, \overline{ξ} ⊗ \overline{ξ}^{−1} = [1] is the trivial bundle).

Remark 40. If ξ, η are line bundles (i.e. geometric), then (34) reduces to the well-known formula σ_{1}(ξ ⊗ η) = F (σ_{1}(ξ), σ_{1}(η)), where F (u, v) ∈ R[[u, v]] is the formal product in complex cobordism theory (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)) (see Remark 36). Proof. The proof is based on the following observation. Since the diagram

\xymatrix{ & _{BU_{⊗}} \ar[d]^{p} \\ X \ar[ur]^{φ_{ξ}} \ar[r]_{φ\overline{ξ}} & _{BSU_{⊗}}}

commutes, φ_{ξ}^{∗}|_{H} = φ∗ ; \overline{ξ}: H → MU^{∗}(X) (we consider H as a Hopf subalgebra in MU^{∗}(BU_{⊗})). More precisely, consider the following commutative diagram:

\xymatrix{ & BU_{⊗} × BU_{⊗} \ar[d]^{p×p} \ar[r]^{ϕ̂} & _{BU_{⊗}} \ar[d]^{p} \\  & BSU_{⊗} × BSU_{⊗} \ar[r]_{ϕ} & _{BSU_{⊗}} \\ X × X \ar[uur]^{φ_{ξ}×φ_{η}} \ar[ur]_{φ\overline{ξ}×φ\overline{η}} & X, \ar[l]^{_{diag_{X}}} \ar[uur]^{†} \ar[ur]_{φ_{\overline{ξ}⊗\overline{η}}}}

where by † we denote the classifying map φ_{ξ⊗η} and ϕ is the product in the H-group BSU_{⊗}. We have:

σ_{1}(ξ ⊗ η) = φ_{ξ⊗η}^{∗}(x) = (μ_{X} ∘ (φ_{ξ}^{∗} ⊗ φ_{η}^{∗}) ∘ ϕ̂^{∗})(x)

= (μ_{X} ∘ (φ_{ξ}^{∗} ⊗ φ_{η}^{∗}))(𝔉(x ⊗ 1, 1 ⊗ x)) = (μ_{X} ∘ (φ∗ ; \overline{ξ} ⊗ φ∗ ; \overline{η})𝔉)(φ_{ξ}^{∗}(x), φ_{η}^{∗}(x))

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

because the coefficients of the power series 𝔉(x ⊗ 1, 1 ⊗ x) belong to H ˆ⊗_{R}H.

To verify (35), consider the following commutative diagram:

\xymatrix{ & _{BU_{⊗}} \ar[d]^{p} \ar[r]^{ι̂} & _{BU_{⊗}} \ar[d]^{p} \\  & _{BSU_{⊗}} \ar[r]_{ι} & _{BSU_{⊗}} \\ X \ar[uur]^{φ_{ξ}} \ar[ur]_{φ\overline{ξ}} & X, \ar[l]^{_{id_{X}}} \ar[uur]^{‡} \ar[ur]_{φ_{\overline{ξ}−1}}}

where by ‡ we denote the classifying map φ_{ξ^{−1}} and ι is the inversion in the H-group BSU_{⊗}. We have:

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

Now (36) follows from axiom 3) of Definition 1. □ Remark 41. Let us note the analogy between the formulas (34), (14) and (35), (15).

Let (ξ, η) be a couple of virtual U-bundles over X of virtual dimension 1 such that the stable equivalence class {ξ ⊗ η} ∈ K_{ℂ}(X) has a geometric line bundle as a representative. One can extend the stable equivalence relation to the set of such couples over X. The set of stable equivalence classes \{(ξ, η)\} of couples is a group (isomorphic to \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} ×Lin(X)× Lin(X) = K̃_{ℂ}(X)_{⊗}×Lin(X)) with respect to the operation ∘, induced by the tensor product: {(ξ_{1}, η_{1})}∘{(ξ_{2}, η_{2})} := {(ξ_{1}⊗ξ_{2}, η_{1}⊗η_{2})}. For two virtual U-bundles ϑ_{1}, ϑ_{2} over X of virtual dimension 1 by σ_{1}(ϑ_{1}) ⊙ σ_{1}(ϑ_{2}) denote (μ_{X} ∘ (φ_{\overline{ϑ}_{1}}^{∗} ⊗ φ_{\overline{ϑ}_{2}}^{∗})𝔉)(σ_{1}(ϑ_{1}), σ_{1}(ϑ_{2})) ∈ MU^{2}(X). Using (4), one can prove the following proposition.

Proposition 42.

((φ_{\overline{ξ}_{1}}^{∗} ⋆ φ_{\overline{ξ}_{2}}^{∗})𝔊)(σ_{1}(ξ_{1}) ⊙ σ_{1}(ξ_{2}), σ_{1}(η_{1}) ⊙ σ_{1}(η_{2}))
= F ((φ_{\overline{ξ}_{1}}^{∗}𝔊)(σ_{1}(ξ_{1}), σ_{1}(η_{1})), (φ_{\overline{ξ}_{2}}^{∗}𝔊)(σ_{1}(ξ_{2}), σ_{1}(η_{2}))),

where φ_{\overline{ξ}_{1}}^{∗} ⋆ φ_{\overline{ξ}_{2}}^{∗} := μ_{X} ∘ (φ_{\overline{ξ}_{1}}^{∗} ⊗ φ_{\overline{ξ}_{2}}^{∗}) ∘ Δ_{H}. Note that for a couple (ξ, η) we have (φ∗ ; \overline{ξ}𝔊)(σ_{1}(ξ), σ_{1}(η)) = σ_{1}(ζ), where ζ is the line bundle over X such that ζ = ξ⊗η. Moreover, φ∗ ; \overline{ξ} ∘ S_{H} = φ∗ ; \overline{η}: H → MU^{∗}(X), where S_{H} is the antipode in the Hopf algebra H.

Remark 43. Now let us give a “topological interpretation” of the series 𝔊(x, y). Let us consider the bundle Ξ^{−1}⊠Z over BSU ×ℂP^{∞} ≅ BU, where Ξ^{−1} is the virtual SU-bundle over BSU of virtual dimension 1 such that Ξ ⊗ Ξ^{−1} = [1]. Suppose y := σ_{1}(Ξ^{−1} ⊠ Z) ∈ MU^{2}(BU). Then y = (S_{H}𝔥)(u), where u = σ_{1}(Z) as above. Let τ: BSU ×ℂP^{∞} × ℂP^{∞} → ℂP^{∞} be a classifying map for the line bundle [1] ⊠ Z ⊠ Z, i.e. τ^{∗}(Z) = [1] ⊠ Z ⊠ Z. Clearly τ^{∗}(w) = F (u, v), where w := σ_{1}(Z) ∈ MU^{2}(ℂP^{∞}), u := σ_{1}([1]⊠Z⊠[1]), v := σ_{1}([1]⊠[1]⊠Z). But the bundle [1] ⊠ Z ⊠ Z can be represented as the tensor product (Ξ ⊠ Z ⊠ [1]) ⊗ (Ξ^{−1} ⊠ [1] ⊠ Z). This corresponds to the representation τ^{∗}(w) as a power series of x, y. More precisely, τ^{∗}(w) = F (u, v) = F (𝔥^{−1}(x), (S_{H}𝔥)^{−1}(y)) = 𝔊(x, y) (compare with (4)).

2.3 Formal group laws related to stable equivalence classes of SUbundles

In this section we obtain another generalization of the formal product in complex cobordism theory. More precisely, for any stable equivalence class of SU-bundles \{ξ\} ∈ \begin{matrix} \\ K & SU \end{matrix}(X) we construct the formal group law (φ_{ξ}^{∗}ℌ)(x, y) such that it defines the group structure on the set of first Conner-Floyd classes σ_{1}(ξ̂) of bundles ξ̂, whose stable equivalence classes {ξ̂} ∈ K̃_{ℂ}(X) belong to the coset {ξ}Lin(X)_{0} ⊂ K̃_{ℂ}(X)_{⊗}, where Lin(X)_{0} is the subgroup in K̃_{ℂ}(X)_{⊗}, corresponding to line bundles over X (see the Introduction), and the symbol ⊗ in K̃_{ℂ}(X)_{⊗} denotes the group structure on K̃_{ℂ}(X), induced by the tensor product of virtual bundles of virtual dimension 1. It follows from isomorphism (23) that K̃_{ℂ}(X)_{⊗} ≅ \begin{matrix} \\ K & SU \end{matrix}(X)_{⊗} × Lin(X). Moreover, for {ξ} = {[1]} the corresponding formal group law (φ_{ξ}^{∗}ℌ)(u, v) coincides with the formal product in complex cobordism theory F (u, v). It is well known that this power series describes the relation between first Conner-Floyd classes of two line bundles ζ_{i}, i = 1, 2 and the first Conner-Floyd class of their tensor product ζ_{1} ⊗ ζ_{2}, more precisely, σ_{1}(ζ_{1} ⊗ ζ_{2}) = F (σ_{1}(ζ_{1}), σ_{1}(ζ_{2})) [6]. Below we shall obtain a generalization of this formula.

By ℌ(x, y) denote the power series 𝔥(F (𝔥^{−1}(x), 𝔥^{−1}(y))) ∈ H[[x, y]], where 𝔥(u) ∈ H[[u]] as in (24). Obviously ℌ(x, y) is a formal group law (in the usual sense) over the ring H, with the formal inverse ℑ(x) := 𝔥(θ(𝔥^{−1}(x))). Moreover, the formal group law ℌ(x, y) is isomorphic (not strictly, because of 𝔥′(0) ≠ 1) to the formal product in complex cobordism theory F (u, v).

The power series ℌ(x, y) may be obtained in the following way. Recall that

MU^{∗}(BU) = H[[u]] = H ˆ⊗_{R}R[[u]].

Let us define a homomorphism κ: H[[w]] → H[[u, v]] as follows: κ|_{H} = id_{H}, κ(w) = F (u, v) = u+v +∑_{i,j≥1} α_{i,j}u^{i}v^{j}. Instead of w, u, v we take another generators z = 𝔥(w), x = 𝔥(u), and y = 𝔥(v) respectively. Then

κ(z) = κ(𝔥(w)) = 𝔥(κ(w)) = 𝔥(F (u, v)) = 𝔥(F (𝔥^{−1}(x), 𝔥^{−1}(y))) = ℌ(x, y).

Define a homomorphism ς: H[[u]] → H[[u]] as follows: ς|_{H} = id_{H}, ς(u) = θ(u). We have: ς(x) = ς(𝔥(u)) = 𝔥(ς(u)) = 𝔥(θ(𝔥^{−1}(x))) = ℑ(x).

Put ℌ(x, y) = x + y + ∑_{i,j≥1} α̃_{i,j}x^{i}y^{j}. Let us recall that the formal group law F (u, v) ∈ R[[u, v]] is the universal one and the ring R is the Lasard ring [7]. Hence there exists a unique ring homomorphism λ: R → H such that λ(α_{i,j}) = α̃_{i,j}, i, j ≥ 1. Clearly, ε_{H}(α̃_{i,j}) = α_{i,j}, i, j ≥ 1. By P denote the composite homomorphism λ ∘ ε_{H}: H → H.

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

Theorem 44. λ defines the canonical ring isomorphism λ: R → K ⊂ H, α_{i,j} ↦^{λ} α̃_{i,j}. P is a projector (i.e. P^{2} = P and H = im P ⊕ ker P) and im P = K ⊂ H.

Proof. The proof is trivial. Note that ε_{H} ∘ λ = id_{R} . □

Proposition 45. The subrings K, R ⊂ H do not coincide.

Proof. It is sufficient to prove that α̃_{i,j} ≠ α_{i,j} for some i, j ≥ 1. Indeed, we have: ε_{H}|_{R} = id_{R} . Therefore if α̃_{i,j} ≠ ε_{H}(α̃_{i,j}) = α_{i,j}, then α̃_{i,j} ∉ R.

To prove this proposition we need some more information on the logarithm g_{ℌ}(x) (more precisely, about its coefficients) of the formal group law ℌ(x, y). By c denote the element 1 + ∑_{i≥2} α_{i,1}s̃_{i} ∈ H. We claim that

g_{ℌ}(x) = cg_{F} (𝔥^{−1}(x))  (37)

(where g_{F} (u) is the logarithm of the formal group law F (u, v)). Indeed, it can easily be checked by a direct computation that g_{ℌ}(ℌ(x, y)) = g_{ℌ}(x) + g_{ℌ}(y) and g′_{ℌ}(0) = 1.

Assume that F (x, y) = ℌ(x, y), i.e. α_{i,j} = α̃_{i,j} for all i, j ≥ 1. This implies g_{F} (x) = g_{ℌ}(x). Let us recall that 𝔥^{−1}(x) = u. Substituting u for 𝔥^{−1}(x) in (37), we get:

g_{F} (x) = cg_{F} (u),

where x = 𝔥(u). Hence

𝔥(u) + β_{1}(𝔥(u))^{2} ≡ c(u + β_{1}u^{2}) mod u^{3}

(recall that g_{F} (u) = u + ∑_{i≥1} β_{i}u^{i+1}). Let us compare the coefficients of u^{2}. Using (24), we get:

∑_{i≥2} α_{i,2}s̃_{i} + β_{1} ∑_{i≥2} α_{i,1}s̃_{i} + β_{1}(∑_{i≥2} α_{i,1}s̃_{i})^{2} = 0.  (38)

It is well known that the characteristic classes s̃_{i} := s_{i}(Ξ), i ≥ 2, are algebraically independent. Therefore the equation (38) contradicts well-known properties of the universal formal group law (see, for example, [1]). □

Since κ(z) = ℌ(x, y) ∈ K[[x, y]], we have:

H[[z]] ⊃ K[[z]] ⟶^{κ|_{K[[z]]}} K[[x, y]] ⊂ H[[x, y]],

i.e. κ(K[[z]]) ⊂ K[[x, y]]. It follows from the existence and the uniqueness of a formal inverse in a formal group law that ℑ(x) ∈ K[[x]] ⊂ H[[x]]. Therefore the subring K[[x]] ⊂ H[[x]] may be considered as an analog of the ring MU^{∗}(ℂP^{∞}) = R[[u]], where the formal product in complex cobordism theory “lives”.

Now let X be a finite CW -complex. Let us fix a virtual SU-bundle ξ over X of virtual dimension 1. Let φ := φ_{ξ}: X → BSU be its classifying map. Then we have the formal group law (φ^{∗}ℌ)(x, y) ∈ MU^{∗}(X)[[x, y]] over the ring MU^{∗}(X). Its coefficients φ^{∗}(α̃_{i,j}), i, j ≥ 1 generate the subring φ^{∗}(K) ⊂ MU^{∗}(X), which is isomorphic to the ring R. Moreover, it is a direct summand in MU^{∗}(X).

We say that a virtual U-bundle ξ̂ over X of virtual dimension 1 is a lift of ξ, if \overline{ξ̂} = ξ. Let us recall that \overline{ξ̂} is the virtual SU-bundle of virtual dimension 1, classifying by the map p ∘ φξ̂, where φξ̂: X → BU is a classifying map for ξ̂ and p: BU_{⊗} → BSU_{⊗} is the canonical homomorphism of H-groups. Let L(ξ) ⊂ K_{ℂ}(X) be the set of stable equivalence classes of lifts of the bundle ξ. Consider the following operation ∘

ξ̂_{1} ∘ ξ̂_{2} := ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}  (39)

on the set of lifts of fixed SU-bundle ξ.

Proposition 46. The set L(ξ) is an Abelian group with respect to the operation, induced by the operation (39) on lifts. Moreover, for all ξ the corresponding groups L(ξ) are isomorphic to each other (in particular, they are isomorphic to the group H^{2}(X; ℤ) of line bundles over X with respect to the tensor product).

Proof. The proof is trivial. □

Theorem 47. Let ξ̂_{1}, ξ̂_{2} be two lifts of ξ. Then

σ_{1}(ξ̂_{1} ∘ ξ̂_{2}) = σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) = (φ^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})).  (40)

Moreover, σ_{1}(ξ̂^{−1}) = (φ^{∗}ℑ)(σ_{1}(ξ̂)), where ξ̂^{−1} is the lift of ξ such that its stable equivalence class {ξ̂^{−1}} ∈ L(ξ) is the inverse element for {ξ̂} in the group L(ξ) (i.e. {ξ̂⊗ ξ̂^{−1} ⊗ ξ^{−1}} = {ξ}). The assignment ξ̂ ↦ σ_{1}(ξ̂) defines an isomorphism from the group L(ξ) to the group {σ_{1}(ξ̂) | {ξ̂} ∈ L(ξ)} ⊂ MU^{2}(X) with respect to the operation σ_{1}(ξ̂_{1}) ∘ σ_{1}(ξ̂_{2}) := (φ^{∗}ℌ)(σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})).

Remark 48. We stress that the bundle ξ̂^{−1} in the statement of the theorem in general does not coincide with the inverse bundle for ξ̂ in the sense of H-group structure on BU_{⊗}, i.e. in general {ξ̂⊗ ξ̂^{−1}} ≠ {[1]} ∈ K_{ℂ}(X). Remark 49. Note, that (40) may be considered as a generalization of the formal product in complex cobordism theory. Indeed, the following conditions are equivalent:

(i) the bundle ξ is (stable equivalent to) the trivial bundle [1];

(ii) its classifying map φ is (homotopic to) the map into a point;

(iii) the bundles ξ̂_{1}, ξ̂_{2} are geometric line bundles (more precisely, their stable equivalence classes have line bundles as representatives).

The condition (i) implies σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) = σ_{1}(ξ̂_{1} ⊗ ξ̂_{2}). It follows from (ii) that

(φ^{∗}ℌ)(u, v) = (ε_{H}ℌ)(u, v) = F (u, v).

Therefore in the case {ξ} = {[1]} we obtain the well-known formula σ_{1}(ξ̂_{1} ⊗ ξ̂_{2}) = F (σ_{1}(ξ̂_{1}), σ_{1}(ξ̂_{2})) for first Conner-Floyd classes of line bundles. Proof. If ζ is a line bundle over X, then we say that the element σ_{1}(ζ) ∈ MU^{2}(X) is a geometric cobordism.

It follows from the isomorphism (23) that for any lift ξ̂ of ξ there is a unique line bundle ζ such that ξ̂= ζ ⊗ ξ. Applying Theorem 39, we get:

σ_{1}(ξ̂) = (μ_{X} ∘ (ε ⊗ φ^{∗})𝔉)(σ_{1}(ζ), 0),  (41)

where ε = η_{X} ∘ ε_{H}: H → MU^{∗}(X), here η_{X}: R → MU^{∗}(X) is the unit of the Ralgebra MU^{∗}(X) (in other words, ε = φ_{[1]}^{∗}, where φ_{[1]} : X → BSU is a classifying map for the trivial bundle). Hence σ_{1}(ξ̂) is the shift (see page 17) of the geometric cobordism σ_{1}(ζ) by φ^{∗} = φ_{ξ}^{∗} ∈ Hom_{R}(H, MU^{∗}(X)). Moreover, we have a bijection from the set {σ_{1}(ζ) | ζ is a line bundle over X} to the set {σ_{1}(ξ̂) | ξ̂ is a lift of ξ}.

Using the notation of Section 1.3, we have:

( \begin{matrix}σ_{1}(ξ_{1}) \\ φ_{\overline{ξ}_{1}}^{∗} \end{matrix} ) ∘ ( \begin{matrix}σ_{1}(ξ_{2}) \\ φ_{\overline{ξ}_{2}}^{∗} \end{matrix} ) = ( \begin{matrix}(μ_{X} ∘ (φ_{\overline{ξ}_{1}}^{∗} ⊗ φ_{\overline{ξ}_{2}}^{∗})𝔉)(σ_{1}(ξ_{1}), σ_{1}(ξ_{2})) \\ φ_{\overline{ξ}_{1}}^{∗} ⋆ φ_{\overline{ξ}_{2}}^{∗} \end{matrix} )

for two virtual U-bundles ξ_{1}, ξ_{2} over X of virtual dimension 1 with the corresponding SU-bundles \overline{ξ}_{1}, \overline{ξ}_{2} (compare with (14)). It follows from Theorem 39 that (μ_{X} ∘ (φ_{\overline{ξ}_{1}}^{∗} ⊗ φ_{\overline{ξ}_{2}}^{∗})𝔉)(σ_{1}(ξ_{1}), σ_{1}(ξ_{2})) = σ_{1}(ξ_{1} ⊗ ξ_{2}). Therefore for two lifts ξ̂_{1}, ξ̂_{2} of the virtual SU-bundle ξ as above, using σ_{1}(ξ^{−1}) = 0 and φ_{ξ^{−1}}^{∗} = φ_{ξ}^{∗} ∘ S_{H}, we obtain:

( \begin{matrix}σ_{1}(ξ̂_{1}) \\ φ_{ξ}^{∗} \end{matrix} ) ∘ ( \begin{matrix}σ_{1}(ξ̂_{2}) \\ φ_{ξ}^{∗} \end{matrix} ) ∘ ( \begin{matrix}0 \\ φ_{ξ}^{∗} ∘ S_{H} \end{matrix} ) = ( \begin{matrix}σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) \\ φ_{ξ}^{∗} \end{matrix} )

(compare with (19)), hence σ_{1}(ξ̂_{1} ⊗ ξ̂_{2} ⊗ ξ^{−1}) = σ_{1}(ξ̂_{1}) ∘_{φ_{ξ}^{∗}} σ_{1}(ξ̂_{2}) (see (18)). Applying the results of Section 1.3 (Propositions 22, 23, 26, and 28), we obtain the required assertion. Note that it follows from Proposition 38 that the condition (μ_{H} ∘ (id_{H} ⊗S))𝔠 = 0 (see page 18) is satisfied. □ Remark 50. Note that σ_{1}(ξ̂) = (μ_{X} ∘ (ε ⊗ φ_{ξ}^{∗})𝔉)(σ_{1}(ζ), 0) = (φ_{ξ}^{∗}𝔥)(σ_{1}(ζ)).

Thus we have the following situation: for any stable equivalence class \{ξ\} ∈ { \atop KSU}(X) we get the formal group law (φ_{ξ}^{∗}ℌ)(x, y) over the ring φ_{ξ}^{∗}(K) ⊂ MU^{∗}(X) (recall that the ring φ_{ξ}^{∗}(K) is a direct summand in MU^{∗}(X)). This formal group law defines the group structure on the set {σ_{1}(ξ̂) | ξ̂ is a lift of ξ} of first Conner-Floyd classes of lifts of ξ (here ξ is a representative of the class {ξ} as above). Note that for all {ξ} the corresponding formal group laws are isomorphic to each other (and the corresponding rings φ_{ξ}^{∗}(K) are isomorphic to each other). If {ξ} = {[1]} is the stable equivalence class of the trivial bundle, then (φ_{[1]}^{∗}ℌ)(u, v) = F (u, v) is the usual formal product in complex cobordism theory, the corresponding ring φ_{[1]}^{∗}(K) is R = MU^{∗}(pt), and the set {σ_{1}(ˆ[1]) | ˆ[1] is a lift of [1]} is the set of geometric cobordisms {σ_{1}(ζ) | ζ is a line bundle} in MU^{2}(X). Note also that we have (φ_{ξ⊗η}^{∗}ℌ)(x, y) = ((φ_{ξ}^{∗} ⋆ φ_{η}^{∗})ℌ)(x, y) for virtual SU-bundles ξ, η over X of virtual dimension 1, where ⋆ denotes the convolution μ_{X} ∘ (φ_{ξ}^{∗} ⊗ φ_{η}^{∗}) ∘ Δ_{H} of φ_{ξ}^{∗}, φ_{η}^{∗} in Hom_{R}(H, MU^{∗}(X)) (compare with Remark 27).

3 Appendix: An isomorphism of some H-groups

Lemma 51. There exists an isomorphism of H-groups: BU_{⊗} ≅ BSU_{⊗} ×ℂP^{∞}.

Proof. Put

G_{n,2n} = U(2n)/ U(n) × U(n), G_{n,2n}^{+} = U(2n)/ U(n) × SU(n).

Recall that BU = lim_{\overrightarrow{n}} G_{n,2n} and BSU = lim_{\overrightarrow{n}} G_{n,2n}^{+} .

Below we shall use the notation of Section 2.1.

By ξ̂_{n} denote the restriction Ξ̂_{n}|_{G_{n,2n}} of the universal U(n)-bundle Ξ̂_{n} to the subspace G_{n,2n} ⊂ BU(n). Let λ_{n}: G_{n,2n} → ℂP^{∞} be the classifying map for c_{1}(ξ̂_{n}) ∈ H^{2}(G_{n,2n}; ℤ). If i_{n}^{n+1} : G_{n,2n} ↪ G_{n+1,2n+2} is the standard embedding, then λ_{n} = λ_{n+1} ∘i_{n}^{n+1}. Hence we get the map λ := lim_{\overrightarrow{n}} λ_{n}: BU → ℂP^{∞}. Consider the virtual bundle ω_{n} := (ξ̂_{n} −[n−1])⊗λ_{n}^{∗}(\overline{Z}) over

G_{n,2n} of virtual dimension 1, where \overline{Z} = Z^{−1}. We claim that it is an SU-bundle. Indeed, c_{1}(ω_{n}) = c_{1}(ξ̂_{n} − [n − 1]) − c_{1}(λ_{n}^{∗}(Z)) = 0. Let κ_{n}: G_{n,2n} → BSU be the classifying map for the bundle ω_{n}, i.e. κ_{n}^{∗}(Ξ) = ω_{n}. It can easily be checked that (i_{n}^{n+1})^{∗}(ω_{n+1}) = ω_{n}. Therefore κ := lim_{\overrightarrow{n}} κ_{n}: BU → BSU is the classifying map for the virtual SU-bundle ω over BU such

that ω|_{G_{n,2n}} = ω_{n} for all n ≥ 0.

Now let us consider the standard embedding j_{n}: G_{n,2n}^{+} ↪ G_{n,2n} . It is easy to prove that j_{n}^{∗}(ξ̂_{n}) = ξ_{n}, where ξ_{n} := Ξ_{n}|_{G}_{n,2n}^{+} (here we consider G_{n,2n}^{+} as a subspace in BSU(n)). Hence

j_{n}^{∗}(ω_{n}) = j_{n}^{∗}((ξ̂_{n} − [n − 1]) ⊗ λ_{n}^{∗}(\overline{Z})) = ξ_{n} − [n − 1]

(where we used j_{n}^{∗}(λ_{n}^{∗}(\overline{Z})) = [1]). Therefore, (κ_{n} ∘ j_{n})^{∗}(Ξ) = ξ_{n} − [n − 1]. This implies that

lim_{\overrightarrow{n}}(κ_{n} ∘ j_{n}): lim_{\overrightarrow{n}} G_{n,2n}^{+} = BSU → BSU  (42)

is a homotopy equivalence. Applying Whitehead’s theorem (using (42)), we get that the map κ × λ: BU → BSU ×ℂP^{∞} is a homotopy equivalence.

Now we consider κ_{n} × λ_{n}: G_{n,2n} → BSU ×ℂP^{∞}. We have:

(κ_{n} × λ_{n})^{∗}(Ξ ⊠ Z) = ω_{n} ⊗ λ_{n}^{∗}(Z) = (ξ̂_{n} − [n − 1]) ⊗ λ_{n}^{∗}(\overline{Z}) ⊗ λ_{n}^{∗}(Z) = ξ̂_{n} − [n − 1].

Therefore κ × λ is the homotopy equivalence such that (κ × λ)^{∗}(Ξ ⊗ Z) = Ξ̂. Now it is easy to prove that κ × λ induces the required isomorphism of H-groups. □

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