GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 401-414

ON THE BUCHSTABER SUBRING IN MSp_{∗}

M. BAKURADZE

Abstract. A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

Introduction

Let Spin(n) and Sp(n) be classical Lie groups. The main result of this paper is the formula which, for any Spin(3)-bundle Λ and Sp(n)-bundle ζ^{n}, expresses the last n Pontrjagin characteristic classes of the symplectic bundle Λ ⊗_{R} ζ^{n} through its first 2n characteristic classes and through characteristic classes of the bundle ζ^{n}. This formula will be given in §1.

We obtain, as a corollary of the above-mentioned formula, some relations in the subring in symplectic cobordisms which is associated with universal Pontrjagin characteristic classes. This ring was introduced in [1] and shown to be generated by the coefficients of formal series given by the characteristic classes of the bundle (ζ_{1} ⊗_{H} ζ_{2}) ⊗_{R} ζ_{3}, where ζ_{i} → BSp(1) are universal Sp(1)-bundles, and by Ray classes. In §2 some relations are established between the coefficients of these formal series. In particular, as we shall see, these relations generalize the formula φ_{i}^{4} = 0 for Ray classes proved in [2].

In §3 some corollaries will be given for the rings of symplectic cobordisms of generalized quaternion groups.

§ 1. On Characteristic Classes of the Bundle Λ ⊗_{R} ζ^{n}

The result of this section is 401

1991 Mathematics Subject Classification. 55N22, 55R12.

Key words and phrases. Symplectic cobordism, characteristic class, transfer.

1072-947X/98/0900-0401$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Proposition 1.1. For any Spin(3)-bundle Λ and Sp(n)-bundle ζ^{n} the formula

pf_{3n−i}(Λ ⊗_{R} ζ^{n}) = ∑_{k=0}^{i} pf_{n−k}(ζ^{n})Δ_{2n−i+k}, i = 0, …, n − 1,

holds, where Δ_{j}, j = n+1, …, 2n, is the determinant of the 2n×2n matrix obtained from the matrix (a_{ql}),

a_{ql} = \begin{cases}pf_{q−l}(ζ^{n}), & l < q ≤ l + n, \\ 1, & q = l, \\ 0, & q < l or q > l + n \end{cases}

by replacing the j-column by the column

( −pf_{1}(ζ^{n}) + pf_{1}(Λ ⊗_{R} ζ^{n}) ; −pf_{2}(ζ^{n}) + pf_{2}(Λ ⊗_{R} ζ^{n}) ; ⋮ ; −pf_{n}(ζ^{n}) + pf_{n}(Λ ⊗_{R} ζ^{n}) ; ⋮ ; pf_{2n}(Λ ⊗_{R} ζ^{n})) .

Proof. Clearly, it is enough to prove the proposition for universal Spin(3)- and Sp(n)-bundles. Denote them also by Λ and ζ^{n}. The proof can be divided into two parts. As we shall see, the first part employs the standard technique of characteristic classes. The second part requires some calculations with transfer. The definition and properties of the transfer can be found in [3, 4, 5].

For the universal Sp(1)-bundle ζ → BSp(1) we shall consider the real bundle ζ ⊗_{H} ζ^{∗}, where ζ^{∗} is the symplectic conjugate of ζ. Here the group Sp(1) acts on H = R^{4} by the conjugates, i.e., by q(h) = qhq^{−1}, q ∈ Sp(1), h ∈ H. This action leaves the real numbers fixed and thus we have the section ζ ⊗_{H} ζ^{∗} = Λ+1 [1], where Λ → BSp(1) = BSpin(3) is the universal Spin(3)-bundle. The spherical bundle of the bundle Λ is BU(1) → BSp(1). The real projectivization of Λ coincides with the bundle π : BN → BSp(1) with the fiber RP^{2}, where N is the normalizer of the unitary group U(1) in Sp(1). Thus the canonical splitting

π^{∗}(Λ) = μ + λ

takes place, where μ is the plane and λ is the linear real bundle. Let π ×1 : BN ×BSp(n) → BSp(1)×BSp(n). Then we have the bundle

(π × 1)^{∗}(Λ ⊗_{R} ζ^{n}) = μ ⊗_{R} ζ^{n} + λ ⊗_{R} ζ^{n}

ON THE BUCHSTABER SUBRING IN MSp_{∗} 403

to which we apply the Whitney formula for Pontrjagin’s symplectic characteristic classes. We obtain

(π × 1)^{∗}(pf_{1}(Λ ⊗_{R} ζ^{n})) = pf_{1}(μ ⊗_{R} ζ^{n}) + pf_{1}(λ ⊗_{R} ζ^{n});
⋮
(π × 1)^{∗}(pf_{2n}(Λ ⊗_{R} ζ^{n})) = pf_{2n}(μ ⊗_{R} ζ^{n}) +

+pf_{2n−1}(μ ⊗_{R} ζ^{n})pf_{1}(λ ⊗_{R} ζ^{n}) + ⋯ + pf_{n}(μ ⊗_{R} ζ^{n})pf_{n}(λ ⊗_{R} ζ^{n});

(π × 1)^{∗}(pf_{2n+1}(Λ ⊗_{R} ζ^{n})) = pf_{2n}(μ ⊗_{R} ζ^{n})pf_{1}(λ ⊗_{R} ζ^{n}) + +pf_{2n−1}(μ ⊗_{R} ζ^{n})pf_{2}(λ ⊗_{R} ζ^{n}) + ⋯ + pf_{n+1}(μ ⊗_{R} ζ^{n})pf_{n}(λ ⊗_{R} ζ^{n});

⋮
(π × 1)^{∗}(pf_{3n}(Λ ⊗_{R} ζ^{n})) = pf_{2n}(μ ⊗_{R} ζ^{n})pf_{n}(λ ⊗_{R} ζ^{n}).

It is obvious that from the first 2n equations we can express the characteristic classes of the bundle μ ⊗_{R} ζ^{n} through the characteristic classes (π × 1)^{∗}(pf_{q}(Λ ⊗_{R} ζ^{n})), q = 1, …, 2n, and the characteristic classes of the bundle λ⊗_{R}ζ^{n}. By substituting these expressions of the classes pf_{j}(μ⊗_{R}ζ^{n}), j = n+1, …, 2n, into the other n equations we obtain the expressions of the characteristic classes (π × 1)^{∗}(pf_{3n−i}(Λ ⊗_{R} ζ^{n})), i = 0, …, n − 1, through the classes (π × 1)^{∗}(pf_{q}(Λ ⊗_{R} ζ^{n})), q = 1, …, 2n, and through the classes of the bundle λ⊗_{R}ζ^{n}. Formally, the expressions for pf_{j}(μ⊗_{R}ζ^{n}) coincide with the determinant Δ_{j} from Proposition 1.1 if in the latter we replace the characteristic classes of the bundle ζ^{n} by the classes of the bundle λ ⊗_{R} ζ^{n}, and the classes pf_{q}(Λ⊗_{R} ζ^{n}) by the classes (π×1)^{∗}(pf_{q}(Λ⊗_{R} ζ^{n})), q = 1, …, 2n.

Thus we have

(π × 1)^{∗}(pf_{3n−i}(Λ ⊗_{R} ζ^{n})) =
= ∑_{k=0}^{i} pf_{n−k}(λ ⊗_{R} ζ^{n})Δ̃_{2n−i+k}, i = 0, …, n − 1,  (1.1)

where Δ̃_{j}, j = n + 1, …, 2n, is the determinant of the 2n × 2n matrix obtained from the matrix (b_{ql}),

b_{ql} = \begin{cases}pf_{q−l}(λ ⊗_{R} ζ^{n}), & l < q ≤ l + n, \\ 1, & q = l, \\ 0, & q < l or q > l + n, \end{cases}

by replacing the jth column by the column

( −pf_{1}(λ ⊗_{R} ζ^{n}) + (π × 1)^{∗}(pf_{1}(Λ ⊗_{R} ζ^{n})) ; −pf_{2}(λ ⊗_{R} ζ^{n}) + (π × 1)^{∗}(pf_{2}(Λ ⊗_{R} ζ^{n})) ; ⋮ ; −pf_{n}(λ ⊗_{R} ζ^{n}) + (π × 1)^{∗}(pf_{n}(Λ ⊗_{R} ζ^{n})) ; ⋮ ; (π × 1)^{∗}(pf_{2n}(Λ ⊗_{R} ζ^{n}))) .

The second part of the proof of Proposition 1.1 is

Proposition 1.2. Let τ(π × 1) be the transfer mapping for the bundle π × 1. Then

τ^{∗}(π × 1)(pf_{i}^{k}(λ ⊗_{R} ζ^{n}) · pf_{j}^{l}(λ ⊗_{R} ζ^{n})) = pf_{i}^{k}(ζ^{n})pf_{j}^{l}(ζ^{n})τ^{∗}(π)(1).

Proof. Let η → BZ_{2} be the universal O(1)-bundle. Since MSp^{∗}(BZ_{2} × BSp(n)) = MSp^{∗}(BZ_{2})[ [pf_{1}(ζ^{n}), …, pf_{n}(ζ^{n})] ], the decomposition

pf_{j}(η ⊗_{R} ζ^{n}) = pf_{j}(ζ^{n}) + ∑_{i_{1}...i_{n}≥0} a_{i_{1}...i_{n}}^{(j)}pf_{1}^{i_{1}}(ζ^{n}) ⋯ pf_{n}^{i_{n}}(ζ^{n}), j = 1, …, n,

takes place for some a_{i_{1}...i_{n}}^{(j)} ∈ M̃Sp^{∗}(BZ_{2}).

Let now f : BN → BZ_{2} be the classifying mapping for the bundle λ → BN. Consider the mapping f × 1 : BN × BSp(n) → BZ_{2} × BSp(n). Clearly, (f × 1)^{∗}(η ⊗_{R} ζ^{n}) = λ ⊗_{R} ζ^{n}, and in MSp^{∗}(BN × BSp(n)) we have

pf_{j}(λ ⊗_{R} ζ^{n}) = pf_{j}(ζ^{n}) + ∑_{i_{1}...i_{n}≥0} f^{∗}(a_{i_{1}...i_{n}}^{(j)})pf_{1}^{i_{1}}(ζ^{n}) ⋯ pf_{n}^{i_{n}}(ζ^{n}).

Thus pf_{i}^{k}(λ ⊗_{R} ζ^{n})pf_{j}^{l}(λ ⊗_{R} ζ^{n}) can be rewritten as

pf_{i}^{k}(λ ⊗_{R} ζ^{n})pf_{j}^{l}(λ ⊗_{R} ζ^{n}) =
= pf_{i}^{k}(ζ^{n})pf_{j}^{l}(ζ^{n}) + ∑_{i_{1}...i_{n}≥0} f^{∗}(b_{i_{1}...i_{n}}^{(i,j,k,l)})pf_{1}^{i_{1}}(ζ^{n}) ⋯ pf_{n}^{i_{n}}(ζ^{n}) (1.2)

for some b_{i_{1}...i_{n}}^{(i,j,k,l)} ∈ M̃Sp^{∗}(BZ_{2}).

Next we apply the transfer mapping τ(π × 1) to (1.2). Since τ(π × 1) = τ(π) ∧ τ(1) = τ(π) ∧ 1 it is clear that Proposition 1.2 can be reduced to calculating the homomorphism τ^{∗}(π). Thus we have to prove

Proposition 1.3. τ^{∗}(π)f^{∗}(a) = 0 ∀a ∈ M̃Sp^{∗}(BZ_{2}).

This proposition is proved in [2]. Here we give an outline of another proof (compare with [2]).

Proof. For the real vector bundles Λ, ζ ⊗_{H} ζ^{∗} = Λ+1, ζ_{1}⊗_{H} ζ_{2}^{∗}, ζ_{1}⊗_{H} ζ_{2}^{∗}+1, consider the associated spherical and projective bundles

S(Λ) = BU(1) → BSp(1),  (1.3)
P (Λ) = BN → BSp(1),  (1.4)
S(Λ + 1) → BSp(1),  (1.5)
P (Λ + 1) → BSp(1),  (1.6)
S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) = BSp(1) → BSp(1)^{2},  (1.7)
P (ζ_{1} ⊗_{H} ζ_{2}^{∗}) → BSp(1)^{2},  (1.8)
S(ζ_{1} ⊗_{H} ζ_{2}^{∗} + 1) → BSp(1)^{2},  (1.9)
P (ζ_{1} ⊗_{H} ζ_{2}^{∗} + 1) → BSp(1)^{2}.  (1.10)

Recall from the theory of spherical bundles [6] that the spherical bundle associated with the Whitney sum of the real vector bundle ξ over the space B and the trivial linear vector bundle over B is the suspension ΣS(ξ) in the category of spaces over B. Therefore S(Λ + 1) is the suspension of BU(1) over BSp(1), i.e., the factor-space of the union

BU(1) × I ∪ BSp(1) × I^{•}, I = [−1, 1], I^{•} = {−1, 1},

where the points (x, t) and (projection of x for (1.3), t), x ∈ BU(1), t ∈ I^{•} are identified.

One can easily see that into the above suspension one can embed the disconnected union BU(1) × (−1, 1) ∪ BSp(1) × {−1} ∪ BSp(1) × {1}.

The involution that gives the projectivization P (Λ + 1) acts on S(Λ + 1) in the following manner: on I it changes the sign, on BU(1) it coincides with the involution giving the projectivization P (Λ) and leaves the points BSp(1) motionless.

Thus into P (Λ + 1) we can embed the disconnected union

BSp(1) ∪ P (Λ) ∪ BU(1) × (0, 1).

Taking the above fact into account and applying the double co-set formula we split the transfer homomorphism of bundle (1.5) into three homomorphisms, of which the first one is the transfer homomorphism of bundle (1.3) with the minus sign, while the second and the third ones are induced by the identical mapping of BSp(1) onto itself.

Furthermore, the transfer homomorphism of bundle (1.6) is the sum of three homomorphisms, of which the first one coincides with the transfer homomorphism of bundle (1.3) with the minus sign, the second one coincides with the transfer homomorphism of bundle (1.4), and the third one is induced by the identical mapping of BSp(1) onto itself.

One can proceed in a similar manner in the case of bundles (1.9) and (1.10). Namely, S(ζ_{1} ⊗_{H} ζ_{2}^{∗} + 1) is the suspension of S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) over BSp(1)^{2}, i.e., the factor-space of the union

S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) × I ∪ BSp(1)^{2} × I^{•},

where the points (x, t) and (projection of x by (1.7), t), x ∈ BSp(1), t ∈ I^{•} are identified.

Thus we can embed the disconnected union

S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) ∪ BSp(1)^{2} × {1} ∪ BSp(1)^{2} × {−1}

into S(ζ_{1} ⊗_{H} ζ_{2}^{∗} + 1).

The involution acting on S(ζ_{1}⊗_{H} ζ_{2}^{∗}+1) and giving the projectivization of (1.10) acts as follows: on I it changes the sign, on Sp(1)^{2} it acts trivially, and on S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) it coincides with the involution giving the projectivization of (1.8). Thus into P (ζ_{1} ⊗_{H} ζ_{2}^{∗} + 1) we can embed the disconnected union

P (ζ_{1} ⊗_{H} ζ_{2}^{∗}) ∪ BSp(1)^{2} ∪ S(ζ_{1} ⊗_{H} ζ_{2}^{∗}) × (0, 1).

Applying the double coset formula [7], we see that the transfer homomorphism of (1.9) is the sum of three homomorphisms, of which the first one coincides with the transfer homomorphism of bundle (1.7) with the minus sign, the second one is identical, and the third homomorphism is induced by permutation of the factors in Sp(1)^{2}.

Furthermore, the transfer homomorphism of bundle (1.10) is the sum of three homomorphisms, of which the first one corresponds to the transfer homomorphism of bundle (1.8), the second one corresponds to the transfer of bundle (1.7) but with the minus sign, and the third one is identical, since it is induced by the identical mapping of BSp(1)^{2} onto itself.

Note that bundles (1.5) and (1.6) are the pullbacks of bundles (1.7) and (1.8), which enables us to apply the results of calculations for (1.7) and (1.8) to (1.5) and (1.6). Further, as shown in [2] for the proof of Proposition 1.3, it is sufficient to prove an analogous proposition for the projectivization P (ζ^{2}) of the universal Sp(2)-bundle ζ^{2}.

P (ζ^{2}) can be realized as BN, where N is the normalizer of Sp(1)^{2} in Sp(2) or as an orbit space S^{∞} × BSp(1) × BSp(1)/involution, where the involution acts on an infinite sphere antipodally and on BSp(1) × BSp(1) by permuting the factors. Hence we have the obvious inclusion i : BZ_{2} → BN and for the above π : BN → BZ_{2} the composition of i and π is the identity. Moreover, we have the bundle p : BN → BSp(2) with the fiber RP^{4} with the Euler characteristic 1. Let tr be the reduced transfer E(BSp(2)) → E(BSp(2)^{+}) → E(BN^{+}) → E(BN). It is sufficient to construct a splitting of the suspension spectrum E(BN) of BN of the form E(BZ_{2}) ∨ E(BSp(2)) ∨ X = E(BN) and to check that in terms of this splitting the projection E(π) is the projection onto the first summand, and tr is the equivalence on the second term. First, we shall construct another splitting. Consider the Puppe cofibration sequence

BN ⟶^{π} BZ_{2} → BZ_{2} ∪_{π} CBN ⟶^{δ_{π}} ΣBN ⟶^{Σπ} SBZ_{2}

Since the composition πi is the identity, we have the splitting

Σ^{k}BZ_{2} ∨ Σ^{k−1}(BZ_{2} ∪_{π} CBN) ⟶^{Σ^{k}i∨δ_{π}} Σ^{k}BN

for k ≥ 2. Then we define ρ : Σ^{k}BN → Σ^{k}BZ_{2} ∨ Σ^{k}BSp(2) to be the composite

Σ^{k}BN ⟶^{α} Σ^{k}BZ_{2} ∨ Σ^{k−1}(BZ_{2} ∪_{π} CBN) ⟶^{1∨Σ^{k−1}δ_{π}}

Σ^{k}BZ_{2} ∨ Σ^{k}BN ⟶^{1∨Σ^{k}p} Σ^{k}BZ_{2} ∨ Σ^{k}BSp(2).

where α is the homotopy inverse of Σ^{k}i ∨ Σ^{k}δ_{π}.

Consider the Puppe cofibration sequence for ρ and let Y be the cofiber of ρ. The composition of Σ^{k}i∨Σ^{k}tr and ρ induces an Id homomorphism in homologies. This follows from the homology structure of BZ_{2} and BSp(2) and obviously the composition of tr and E(π) induces the zero homomorphism in homologies, i.e., Im tr_{∗} ∈ ker π_{∗} = Im δ_{ρ_{∗}}. Therefore

ρ_{∗}(i ∨ tr)_{∗}(a ⊕ b) = a ⊕ p_{∗}tr_{∗}(b) = a ⊕ b

in the homologies of BZ_{2} ∨ BSp(2) and the stable Whitehead lemma gives the self-homotopy equivalence of Σ^{k+1}E(BZ_{2}) ∨ Σ^{k+1}E(BSp(2)) and the splitting

Σ^{k+1}E(Z_{2}) ∨ Σ^{k+1}E(BSp(2)) ∨ E(Y) = Σ^{k+1}E(BN).

After applying Σ^{−k−1}, we obtain the above splitting as needed. Now we can calculate the transfer homomorphism for (1.10). Next, applying the double co-set formula and the above arguments, we calculate the transfer homomorphism for (1.6). After that we repeat the procedure and calculate the transfer homomorphism for (1.4), which completes the proof of Proposition 1.3.

We are now ready to complete the proof of Proposition 1.1. By Proposition 1.3 we have

τ^{∗}(π)(f^{∗}(b_{i_{1}...i_{n}}^{(i,j,k,l)})) = 0,

where b_{i_{1}...i_{n}}^{(i,j,k,l)} ∈ M̃Sp^{∗}(BZ_{2}) from (1.2).

By these equalities we find from (1.2) that

(π × 1)^{∗}(pf_{i}^{k}(λ ⊗_{R} ζ^{n})pf_{j}^{l}(λ ⊗_{R} ζ^{n})) = (pf_{i}^{k}(ζ^{n})pf_{j}^{l}(ζ^{n}))τ^{∗}(π)(1),

which proves Proposition 1.2.

Next, by Proposition 1.2 we find from (1.1) that

τ^{∗}(π × 1)((π × 1)^{∗}(pf_{3n−i}(Λ ⊗_{R} ζ^{n}))) =

= ∑_{k=0}^{i} pf_{n−k}(ζ^{n})τ^{∗}(π × 1)(Δ̃_{2n−i+k}) = ∑_{k=0}^{i} pf_{n−k}(ζ^{n})Δ_{2n−i+k})τ^{∗}(π)(1).

and, on the other hand,

τ^{∗}(π × 1)((π × 1)^{∗}(pf_{3n−i}(Λ ⊗_{R} ζ^{n}))) =
= pf_{3n−i}(Λ ⊗_{R} ζ^{n})τ^{∗}(π × 1)(1) = pf_{3n−i}(Λ ⊗_{R} ζ^{n})τ^{∗}(π)(1),

Since τ^{∗}(π)(1) is invertible, this completes the proof of Proposition 1.1.

§ 2. On the Buchstaber Subring in MSp^{∗}

In [1], V. M. Buchstaber introduced the subring in symplectic cobordisms associated with universal Pontrjagin characteristic classes. This ring is defined as a subring in symplectic cobordisms, consisting of the coefficients of formal series given by Pontrjagin characteristic classes of all symplectic bundles over the space Y (q, n), where

Y (q, n) = S^{q} ∧ ( ∏^{n} BSp(1)), q ≥ 0, n ≥ 1,

if we fix an additive isomorphism MSp^{∗}(Y (q, n)) = MSp^{∗}[ [x_{1}, …, x_{n}] ], where x_{i} = pf_{1}(ζ_{i}), ζ_{i} → BSp(1) are the universal Sp(1)-bundles.

In the same paper, it was shown that this ring is generated by coefficients of the series pf_{j}((ζ_{1} ⊗_{H} ζ_{2}^{∗})⊗_{R} ζ_{3}), j = 1, …, 4, from MSp^{∗}(∏ BSp(1)) and by the coefficients of pf_{1}(η ⊗_{R} ζ) from MSp^{∗}(S^{1} ∧ BSp(1)), i.e., by the Ray classes θ_{i}.

As one knows, of the Ray classes θ_{i} the classes θ_{1}, θ_{2i} are indecomposable and have order 2 [8]; θ_{2i−1} = 0, i > 1 [9], so that most of the triple products of Ray classes are nonzero [10], [11].

We shall show that the formula θ_{i}^{4} = 0 ∀i ≥ 1 proved in [2] is a corollary of Proposition 1.1. Indeed, in a particular case with Λ as a universal Spin(3)- bundle and ζ^{n} as a trivial Sp(1)-bundle, Proposition 1.1 implies pf_{3}(Λ ⊗_{R} H) = 0. But, as shown in [2], this formula is equivalent to the formula φ_{i} = 0. Note also that in [2] the formula φ_{i}φ_{j}φ_{k}φ_{l} = 0 ∀i, j, k, l ≥ 1 is shown, where φ_{i} = θ_{2i}.

Now let us consider another particular case of Proposition 1.1. Let Λ be again the universal Spin(3)-bundle and ζ the universal Sp(1)-bundle. Then (1.1) takes the form

pf_{3}(Λ ⊗_{R} ζ) = pf_{1}^{3}(ζ) − pf_{1}^{2}(ζ)pf_{1}(Λ ⊗_{R} ζ) + pf_{1}(ζ)pf_{2}(Λ ⊗_{R} ζ). (2.1)

Using the notation of §1, we write

pf_{i}(x, y) = pf_{i}((ζ_{1} ⊗_{H} ζ_{1}^{∗}) ⊗_{R} ζ_{2}),
Λ + 1 = ζ_{1} ⊗_{H} ζ_{1}^{∗}, ζ_{2} = ζ, x = pf_{1}(ζ_{1}), y = pf_{1}(ζ_{2}).

We have

Corollary 2.1. 4y^{3} − 3y^{2}pf_{1}(x, y) + 2ypf_{2}(x, y) − pf_{3}(x, y) = 0.

Proof. By the Whitney formula for characteristic classes we have

pf_{1}(x, y) = pf_{1}(Λ ⊗_{R} ζ) + y,
pf_{2}(x, y) = pf_{2}(Λ ⊗_{R} ζ) + ypf_{1}(Λ ⊗_{R} ζ),
pf_{3}(x, y) = pf_{3}(Λ ⊗_{R} ζ) + ypf_{2}(Λ ⊗_{R} ζ).

Therefore

pf_{3}(Λ ⊗_{R} ζ) = pf_{3}(x, y) − ypf_{2}(x, y) + y^{2}pf_{1}(x, y) − y^{3}.

On the other hand, taking into account that by (2.1)

pf_{3}(Λ ⊗_{R} ζ) = y^{3} − y^{2}(pf_{1}(x, y) − y) + y(pf_{2}(x, y) − y(pf_{1}(x, y) − y)) =
= 3y^{3} − 2y^{2}pf_{1}(x, y) + ypf_{2}(x, y)

and equating these expansions for pf_{3}(Λ ⊗_{R} ζ), we obtain Corollary 2.1.

Clearly, the coefficients of series pf_{i}(x, y) belong to the Buchstaber ring, since these series are obtained from the series

pf_{i}(x_{1}, x_{2}, x_{3}) = pf_{i}(ζ_{1} ⊗_{H} ζ_{2}^{∗}) ⊗_{R} ζ_{3})

if x_{1} = x_{2} = x and x_{3} = y.

On the other hand, the arguments used in proving Corollary 2.1 imply that the coefficients of the series pf_{i}(x, y) are expressed through the coefficients of the series pf_{i}(Λ ⊗_{R} ζ) and vice versa. Thus, by equating the coefficients of the same monomes in the (2.1) or, with equal success, in Corollary 2.1 we obtain relations between the generatrices of the Buchstaber ring.

We identify MU^{∗}(BSp(1) × BSp(1) with the subring MU^{∗}(BU(1) × BU(1)) by the monomorphism (p × p)^{∗}, where p : BU(1) → BSp(1) is the canonical bundle (it the spherical bundle Spin(3) of the bundle Λ → BSp(1)).

Let η → BU(1) be the universal U(1)-bundle. Since

p!(Λ) = \overline{η}_{1}^{2} + R, p!(ζ) = η_{2} + \overline{η}_{2},

we have

(p × p)!(Λ ⊗_{R} ζ) = (η_{1}^{2} + \overline{η}_{1}^{2} + ℂ) ⊗_{C} (η_{2} + \overline{η}_{2}) =
= (η_{1}^{2} + \overline{η}_{1}^{2}) ⊗_{C} (η_{2} + \overline{η}_{2}) + (η_{2} + \overline{η}_{2}).  (2.2)

Recall from [1] that for the homomorphism μ_{Sp}^{U} : MSp^{∗}(·) → MU^{∗}(·) we have

μ_{Sp}^{U}(pf_{1}((η_{1} + \overline{η}_{1}) ⊗_{C} (η_{2} + \overline{η}_{2}))) = Θ_{1}(e(η_{1} + \overline{η}_{1}), e(η_{2} + \overline{η}_{2})),
μ_{Sp}^{U}(pf_{2}((η_{1} + \overline{η}_{1}) ⊗_{C} (η_{2} + \overline{η}_{2}))) = Θ_{2}(e(η_{1} + \overline{η}_{1}), e(η_{2} + \overline{η}_{2})),

where Θ_{1}, Θ_{2} are the coefficients of a two-valued formal group in cobordisms, and e is Euler’s class in complex cobordisms.

Therefore

μ_{Sp}^{U}(pf_{1}(η_{1}^{2} + \overline{η}_{1}^{2})) = μ_{Sp}^{U}(pf_{1}(η_{1}^{2} + \overline{η}_{1}^{2} + ℂ)) =

= μ_{Sp}^{U}(pf_{1}((η_{1} + \overline{η}_{1}) ⊗_{C} (η_{1} + \overline{η}_{1}))) = Θ_{1}(e(η_{1} + \overline{η}_{1}), e(η_{1} + \overline{η}_{1}));

Further,

μ_{Sp}^{U}(pf_{1}((η_{1}^{2} + \overline{η}_{1}^{2}) ⊗_{C} (η_{2} + \overline{η}_{2}))) = Θ_{1}(e(η_{1}^{2} + \overline{η}_{1}^{2}), e(η_{2} + \overline{η}_{2})) =

= Θ_{1}(Θ_{1}(e(η_{1} + \overline{η}_{1}), e(η_{1} + \overline{η}_{1})), e(η_{1} + \overline{η}_{2}))),
μ_{Sp}^{U}(pf_{2}((η_{1}^{2} + \overline{η}_{1}^{2}) ⊗_{C} (η_{2} + \overline{η}_{2}))) =
= Θ_{2}(Θ_{1}(e(η_{1} + \overline{η}_{1}), e(η_{1} + \overline{η}_{1})), e(η_{1} + \overline{η}_{2}))).

By (2.2) we have

μ_{Sp}^{U}(pf_{1}(Λ ⊗_{R} ζ)) = Θ_{1}(Θ_{1}(x, x), y)) + y,
μ_{Sp}^{U}(pf_{2}(Λ ⊗_{R} ζ)) = Θ_{2}(Θ_{1}(x, x), y)) + Θ_{1}(Θ_{1}(x, x), y) · y,
μ_{Sp}^{U}(pf_{3}(Λ ⊗_{R} ζ)) = Θ_{2}(Θ_{1}(x, x), y)) · y.

Thus relation (2.1) in complex cobordisms takes the tautological form

y^{3} − y^{2}(Θ_{1}(Θ_{1}(x, x), y)) + y) + y(Θ_{2}(Θ_{1}(x, x), y) + Θ_{1}(x, x), y) · y) =

= yΘ_{2}(Θ_{1}(x, x), y).

In symplectic cobordisms, relations obtained from (2.1) are not trivial in the dimensions where there are elements from the kernel MSp^{4∗} → MU^{4∗}.

§ 3. On Rings of Symplectic Cobordisms Classifying Spaces of Generalized Quaternion Groups

Let, as above, N be the normalizer of the unitary group U(1) in the symplectic group Sp(1) consisting of U(1) and jU(1), where j is the quaternion unit. Define, over BN, the real two-dimensional bundles η^{(2m)} by the representation [2]

z → z^{2m}, j → (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

The real linear bundle λ → BN from §1 is defined by the representation z → 1, j → −1, where z ∈ U(1), m = 1, 2, . . ..

Let, further, H_{m} be a generalized quaternion group. For m = 2 the group H_{2} = {±1, ±i, ±j, ±k} is a group of quaternions. In the general case H_{m} is defined as a subgroup in Sp(1) generated by two elements

α = exp (πi/2^{m−1}) and β = j.

Define, over BH_{m}, the real linear bundles η_{1}^{(m)} and η_{2}^{(m)} by the representations

η_{1}^{(m)} : α → 1, β → −1,
η_{2}^{(m)} : α → −1, β → 1.

Let, further, ρ(K, L) : BK → BL denote the mapping of the classifying spaces induced by the embedding of the Lie group K into L. We have

Proposition 3.1. (3.1.a) The bundle ρ(H_{m}, N) is the projectivization of the bundle η^{(2k)}, k = 2^{m−2}, and

ρ^{∗}(H_{m}, N)(η^{(2k)}) = η_{2}^{(m)} + η_{1}^{(m)} ⊗_{R} η_{2}^{(m)}, ρ^{∗}(H_{m}, N)(λ) = η_{1}^{(m)}.

(3.1.b) For k = 2^{n+1}, n ≥ 1, the real bundle η^{(2)} + η^{(k)} → BN is MSp^{∗}- orientated.

(3.1.c) the real bundle η^{(k)} + 3λ is MSp^{∗}-orientated and has the zero symplectic Euler class.

Proof. To prove (3.1.a), note that from the definition of the bundle η^{(2m)} it follows that when N acts on the circle S^{1} for the spherical bundle S(η^{(2m)}), the isotropy group is isomorphic to the group consisting of the elements z = πi/m and j. When m = 2^{n−1}, this group coincides with H_{n}. Clearly, in that case ρ(H_{n+1}, N) will be the projective bundle. The remainder of the proof follows from the definitions.

(3.1.b) This can be proved in two ways. Firstly, we can calculate Shtiefel-Whitney characteristic classes ω_{i} of the bundle η^{(2)} + η^{(k)}, k = 2^{n+1}, using the result of [13]

H^{∗}(BN, Z_{2}) = H^{∗}(RP^{2}, Z_{2}) ⊗_{Z_{2}} H^{∗}(BSp(1), Z_{2}).

This follows from the fact that the Serre spectral sequence for the bundle BN → BSp(1) with fiber RP^{2} is trivial.

When the fiber RP^{2} is embedded into BN, the bundle η^{(k)} passes to ν+1, where ν → RP^{2} is a nontrivial linear bundle; the bundle η^{(2)} passes to the tangent bundle τ → RP^{2}. But, as is well known, τ +1 = 3ν. The definitions of η^{(2)} and η^{(k)} imply that det(η^{(2)}) = det(η^{(k)}) = λ and therefore

ω_{2}(η^{(2)}) = ω_{1}^{2}(η^{(2)}), ω_{1}(η^{(k)}) = ω_{1}(η^{(2)}), ω_{2}(η^{(k)}) = 0.

Thus ω_{1}(η^{(2)} + η^{(k)}) = ω_{2}(η^{(2)} + η^{(k)}) = 0 and hence η^{(2)} + η^{(k)} is a Spin(4)-bundle. On the other hand, each Spin(4) bundle is MSp-orientated. Indeed (see, e.g., [14]), for Spin(4)-bundle ψ, to the class of KO-orientation from KO^{4}(T ψ), where T ψ is the Thom space, there corresponds some symplectic bundle over T ψ provided that the isomorphism KO^{4}(T ψ) ≈ KS^{0}(T ψ). The first symplectic Pontrjagin class can be taken as the class of MSp-orientation. This completes the proof.

The second way is as follows. Let ξ → BU(1) be the universal U(1)- bundle. Consider the 2-covering BU(1) → BN and bundles ξ_{!}^{2i+1} → BN, the images of bundles ξ^{2i+1} for the Atiyah transfer of this covering [15]. The bundle ξ_{!} is symplectic, since ξ_{!} = π^{∗}(ζ), where as above π : BN → BSp(1) and ζ → BSp(1) is the universal Sp(1)-bundle. We shall show that ξ_{!}^{2i+1}, i ≥ 1, are also Sp(1)-bundles. Since Sp(1) = SU(2), it is enough to show that ξ_{!}^{2i+1} has the trivial determinant, i.e., that the first Chern class c_{1}(ξ_{!}^{2i+1}) = 0 in cohomologies with integral coefficients. This can be done immediately after writing the corresponding representations for ξ_{!}^{2i+1} [12], but we shall do this in terms of transfer and characteristic Chern classes. For the transfer homomorphism τ^{∗} of the covering BU(1) → BN we have the formula

τ^{∗}(c_{1}(ξ^{2i+1})) = c_{1}(1_{!}) + c_{1}(ξ_{!}^{2i+1})

in integral cohomologies. This follows from the corresponding formula for the case of complex cobordisms [16]. By virtue of equalities c_{1}(ξ^{2i+1}) = (2i + 1)c_{1}(ξ) and τ^{∗}(c_{1}(ξ)) = c_{1}(1_{!}) we now have

τ^{∗}(c_{1}(ξ_{!}^{2i+1})) = (2i + 1)τ^{∗}(c_{1}(ξ)) = (2i + 1)c_{1}(1_{!}) = c_{1}(1_{!}),

i.e., c_{1}(ξ_{!}^{2i+1}) = 0. Further,

η^{(2)} + η^{(k)} = ξ_{!}^{k/2+1} ⊗_{H} ξ_{!}^{k/2−1},

but the tensor product of Sp(1)-bundles over H is MSp-orientated, since BSpin(4) = BSp(1)^{2}. This proves 3.1.b.

(3.1.c) The bundle η^{(k)} +3λ is a difference between MSp-orientated bundles η^{(2)} + η^{(k)} + 4λ and η^{(2)} + λ and hence is MSp-orientated. On the other hand, this bundle is a Spin(5)-bundle and each Spin(5)-bundle has the zero symplectic Euler class [17].

Using this lemma, one can show some relations in rings of symplectic cobordisms of generalized quaternion groups. We have

Proposition 3.2. The relation

pf_{1}(η_{1}^{(m)} ⊗_{R} H) · pf_{1}(η_{2}^{(m)} ⊗_{R} H) · pf_{1}(η_{1}^{(m)} ⊗_{R} η_{2}^{(m)} ⊗_{R} H) = 0

holds in the ring MSp^{∗}(BH_{m}), m ≥ 2.

Proof. For m > 2 we have

ρ^{∗}(H_{m}, N)(η^{(m)} ⊗_{R} H + λ ⊗_{R} H) =
= η_{1}^{(m)} ⊗_{R} H + η_{2}^{(m)} ⊗_{R} H + η_{1}^{(m)} ⊗_{R} η_{2}^{(m)} ⊗_{R} H.

But the real bundle η^{(m)} ⊗_{R} H + λ ⊗_{R} H coincides with the sum of the MSp-orientated bundles η^{(m)} + 3λ and 3η^{(m)} + λ. By Proposition (3.1.c) the first bundle has the zero symplectic Euler class. Therefore the Euler classes of both sides are equal to zero in the above-indicated splitting. This proves the proposition for m > 2.

For the case m = 2, i.e., for the group of quaternions, the above reasoning does not hold, since the bundle η^{(2)} + 3λ is not MSp-orientated. But according to §1, for the universal Spin(3)-bundle Λ → BSp(1) the Euler class e(Λ ⊗_{R} H) = pf_{3}(Λ ⊗_{R} H) = 0. On the other hand, ρ^{∗}(H_{2}, Sp(1)) = η_{1}^{(2)} + η_{2}^{(2)} + η_{1}^{(2)} ⊗_{R} η_{2}^{(2)} so that the proposition is valid in this case too.

Acknowledgment

The research described in this publication was made possible in part by Grant RVJ 200 from the International Science Foundation.

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(Received 15.12.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia