ON THE ALGEBRAIC COBORDISM SPECTRA MSL AND MSp

IVAN PANIN AND CHARLES WALTER

Abstract. We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T^{∧2}-spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th^{MSp} ∈ MSp^{4,2}(MSp_{2}), a tautological Pontryagin class p_{1}^{MSp} ∈ MSp^{4,2}(HP^{∞}) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment φ ↦ φ(th^{MSp}) identifies the set of homomorphisms of monoids φ: MSp → E in the motivic stable homotopy category SH(S) with the set of tautological Thom elements of symplectic orientations of E. A weaker universality result is obtained for MSL and special linear orientations.

1. Introduction

A dozen years ago Voevodsky [15] constructed the algebraic cobordism spectrum MGL in the motivic stable homotopy category SH(S). This gave a new cohomology theory MGL^{∗,∗} on smooth schemes and on motivic spaces. Later Vezzosi [14] put a commutative monoid structure on MGL. This gave a product to MGL^{∗,∗}. The commutative monoid structure can even be constructed in the symmetric monoidal model category of symmetric T -spectra, with T = A^{1}/(A^{1} − 0) the Morel-Voevodsky object (Panin, Pimenov and Röndigs [10]).

In this paper we construct the algebraic special linear and symplectic cobordism spectra MSL and MSp. The construction of MSL is straightforward although there is one slightly subtle point. We equip each space BSL_{n} and MSL_{n} with an action of GL_{n} which is compatible with the monoid structure BSL_{m} × BSL_{n} → BSL_{m+n} induced by the direct sum of subbundles. This gives an action of the subgroup Σ_{n} ⊂ GL_{n} of permutation matrices. But to define the unit of the monoid structure we need the action on BSL_{n} to have fixed points. The natural action of SL_{n} has fixed points, but the natural action of GL_{n} does not. So we use an embedding Σ_{n} ⊂ Sp_{2n} ⊂ SL_{2n}. This means that our MSL is a commutative monoid in the category of symmetric T^{∧2}-spectra. The categories of symmetric T -spectra and of symmetric T^{∧2}-spectra are both symmetrical monoidal, and their homotopy categories are equivalent symmetric monoidal categories (Theorem 3.2). So a symmetric T^{∧2}-spectrum structure is quite satisfactory, and it seems to be a natural structure for this spectrum.

Cobordism spectra and the cohomology theories they define are expected to have some universal properties among certain classes of cohomology theories. For instance Voevodsky’s and Levine and Morel’s algebraic cobordism theories are universal among oriented cohomology theories [6, 10, 14]. We should therefore expect MSL to have some degree of universality for special linearly oriented theories. Recall that a special linear bundle (E, λ) over X is a pair consisting of a vector bundle E and an isomorphism of line bundles λ: 𝒪_{X} ≅ det E.

Date: October 28, 2010.

The work was supported by Universite de Nice - Sophia-Antipolis , the Presidium of RAS Program “Fundamental Research in modern mathematics”, the joint DFG-RFBR grant 09-01-91333 NNIO-a, and the RFBRgrant 10-01-00551.

A special linear orientation on a cohomology theory A^{∗,∗} is an assignment to every special linear bundle of a Thom class th(E, λ) ∈ A^{2n,n}(E, E − X) = A_{X}^{2n,n}(E) with n = rk E which is functorial, multiplicative, and such that the multiplication maps − ∪ th(E, λ): A^{∗,∗}(X) → A^{∗+2n,∗+n}(E, E − X) are isomorphisms. In the motivic context we generally also require that the Thom class of the trivial line bundle over a point be Σ_{T} 1_{A} ∈ A^{2,1}(T) = A^{2,1}(A^{1}, A^{1} − 0). Hermitian K-theory and Balmer’s derived Witt groups are examples of special linearly oriented theories which are not oriented.

The universality properties we show for MSL are as follows. A morphism of commutative monoids φ: (MSL, μ^{SL}, e^{SL}) → (A, μ, e) in SH(S) determines naturally a special linear orientation on A^{∗,∗} with Thom classes written th^{φ}(E, λ). The compatibility of φ with the monoid structure ensures the multiplicativity of the Thom classes (Theorem 5.5).

Conversely, a special linear orientation on A^{∗,∗} with Thom classes th(E, λ) determines a morphism φ: MSL → A in SH(S) with th^{φ}(E, λ) = th(E, λ) for all (E, λ). This φ is unique modulo a certain subgroup \underleftarrow{lim}^{1} A^{2n−1,n}(MSL_{n}^{⟨n⟩}) ⊂ Hom_{SH(S)}(MSL, A). The obstruction φ ∘ μ^{SL} − μ_{A} ∘ (φ ∧ φ) to having a morphism of monoids lies in a similarly defined subgroup of Hom_{SH(S)}(MSL ∧ MSL, A) (Theorem 5.9).

It would be interesting to know if these obstruction subgroups vanish for Witt groups and hermitian K-theory. The necessary calculations are likely very close to Balmer and Calmès’s computation of Witt groups of Grassmannians [1].

Our MSp is defined similarly with an action of Sp_{2n} on the spaces BSp_{2n} and MSp_{2n}. The actions of the subgroups Σ_{n} ⊂ Sp_{2n} make MSp a commutative monoid in the category of symmetric T^{∧2}-spectra. For MSp we can do much more than for MSL because we have the quaternionic projective bundle theorem [13, Theorem 8.2] for symplectically oriented cohomology theories. Therefore for any symplectically oriented cohomology theory A^{∗,∗} we have Pontryagin classes for symplectic bundles, and we can compute the cohomology of quaternionic Grassmannians [13, §11] and of the spaces BSp_{2r} and MSp_{2r} (§§8–9). Our main result is the following theorem.

Theorem 1.1. Let (A, μ, e) be a commutative monoid in SH(S). Then the following sets are in canonical bijection:

(a) symplectic Thom structures on the bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) such that for the trivial rank 2 bundle A^{2} → pt we have th(A^{2}, ω_{2}) = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}),

(b) Pontryagin structures on (A^{∗,∗}, ∂, ×, 1_{A}) for which p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) ∈ A^{4,2}(HP^{1}, h_{∞}) ⊂ A^{4,2}(HP^{1}) corresponds to −Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}) under the canonical motivic homotopy equivalence (HP^{1}, h_{∞}) ≃ T^{∧2},

(c) Pontryagin classes theories on (A^{∗,∗}, ∂, ×, 1_{A}) with the same normalization condition on p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) as in (b),

(d) symplectic Thom classes theories on (A^{∗,∗}, ∂, ×, 1_{A}) such that for the trivial rank 2 bundle A^{2} → pt we have th(A^{2}, ω_{2}) = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}),

(α) classes ϑ ∈ A^{4,2}(MSp_{2}) with ϑ|_{T^{∧2}} = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}),

(β) classes ϱ ∈ A^{4,2}(HP^{∞}, h_{∞}) with ϱ|_{HP^{1}} ∈ A^{4,2}(HP^{1}, h_{∞}) corresponding to −Σ_{T}^{2} 1_{A} ∈ A^{4,2}(T^{∧2}) under the canonical motivic homotopy equivalence (HP^{1}, h_{∞}) ≅ T^{∧2},

(δ) sequences of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) with ϑ_{r} ∈ A^{4r,2r}(MSp_{2r}) for each r satisfying μ_{rs}^{∗}ϑ_{r+s} = ϑ_{r} × ϑ_{s} for all r, s, and ϑ_{1}|_{T^{∧2}} = Σ_{T}^{2} 1_{A},

(ε) morphisms φ: (MSp, μ^{Sp}, e^{Sp}) → (A, μ, e) of commutative monoids in SH(S).

The bijections are explicit and are given in a series of theorems in the last part of the paper. The presence of (ε) among them is the universality of MSp as a symplectically oriented theory.

The equivalence of (a), (b), (c) and (d) was already shown in [13] in a different axiomatic context. The ability of the motivic language used here to handle tautological classes such as the (α), (β) and (δ) is very useful by itself. But our main new observation is that in the motivic unstable homotopy category H_{•}(S) we have a commutative diagram (Theorem 7.7).

[figure: BSp_{2r}; structure map; quotient; _{MSp_{2r}}; ≅; A^{N} -bundles and excision; BSp_{2r}/BSp_{2r−2}.]

What is surprising about this diagram is that it is the homotopy colimit of diagrams (7.5) of finite-dimensional schemes and their quotient spaces which have a fourth side which is an inclusion of quaternionic Grassmannians of different dimensions which is in no way a motivic equivalence. But in the infinite-dimensional colimit the fourth side becomes A^{1}-homotopic to the identity map of BSp_{2r}, and the picture simplifies significantly.

The fact that this diagram is three-sided instead of four-sided helps us to see more conceptual proofs of two of the trickier points of [13]. One is the construction of the higher-rank symplectic Thom classes and the proof of their multiplicativity. The commutativity of the diagram and the computations of the cohomology of quaternionic Grassmannians imply that given a symplectic Thom structure on A^{∗,∗}, the pullback along the structure map gives an injection A^{∗,∗}(MSp_{2r}) → A^{∗,∗}(BSp_{2r}), and the isomorphism A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom} ≅ A^{∗,∗}(BSp_{2r}) defined by the symplectic Thom structure identifies the image of A^{∗,∗}(MSp_{2r}) with the principal two-sided ideal generated by p_{r}. This makes it easy to define the higher-rank tautological symplectic Thom classes (the ϑ_{r} of (δ)) with the classes of A^{∗,∗}(MSp_{2r}) identified with (−1)^{r}p_{r} ∈ A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom}. Their multiplicativity is also easily established.

The other tricky point of [13] for which the diagram helps is the reconstitution of the symplectic Thom structure from the Pontryagin structure. The tautological rank 2 Thom class is a ϑ ∈ A^{4,2}(MSp_{2}), and it is tempting to identify it (up to sign) with the tautological rank 2 Pontryagin class ϱ ∈ A^{4,2}(BSp_{2}/BSp_{0}) = A^{4,2}(HP^{∞}, h_{∞}) using the horizontal motivic homotopy equivalence. But the Pontryagin is actually (up to sign) the pullback of ϑ along the structure map of the Thom space. For the three-sided diagram this is no problem: in H_{•}(S) the structure map HP^{∞} → MSp_{2} is the composition of the horizontal isomorphism (HP^{∞}, h_{∞}) ≅ MSp_{2} with the pointing map.

It is not difficult to define spectra MO and MSO which resemble formally MGL and our MSL and MSp. However, our proof of even our most basic result about MSL (Theorem 5.5) uses the fact that special linear bundles are locally trivial in the Zariski topology. So we omit MO and MSO.

2. Preliminaries

Let S be a noetherian scheme of finite Krull dimension, and let 𝒮m/S be the category of smooth quasi-projective schemes over S. We will assume that S admits an ample family of line bundles so that for any X in 𝒮m/S there exists an affine bundle Y → X with Y an affine scheme. This condition is used a number of times in this paper, and it was also used in the proof of the symplectic splitting principle in [13, Theorem 10.2].

The category 𝒮m𝒪p/S has objects (X, U) where X is in 𝒮m/S and U ⊂ X is an open subscheme. A morphism f: (X, U) → (X′, U′) in 𝒮m𝒪p/S is a morphism f: X → X′ of S-schemes with f(U) ⊂ U′. We often write X in place of (X, ∅).

A bigraded ring cohomology theory (A^{∗,∗}, ∂, ×, 1) on 𝒮m𝒪p/S is a contravariant functor A^{∗,∗} from 𝒮m𝒪p/S to the category of bigraded abelian groups which satisfies étale excision and A^{1}-homotopy invariance and which has localization long exact sequences

⋯ → A^{∗,∗}(X, U) → A^{∗,∗}(X) → A^{∗,∗}(U) ⟶^{∂} A^{∗+1,∗}(X, U) → · · · .

The × product is assumed to be functorial, bilinear, associative, and compatible with the bigrading with a two-sided unit 1.

In this paper we work mainly with the motivic unstable and stable homotopy categories H_{•}(S) and SH(S). The former is the homotopy category of a model category M_{•}(S) of pointed motivic spaces over S with motivic weak equivalences. There are several versions of this model category with different underlying categories and with different choices of fibrations and cofibrations. See [2, 4, 5, 8, 11, 15, 16] among other papers. It is not essential which of these model category structures is used. However, we give geometric constructions symmetric T - and T^{∧2}-spectra using the the Morel-Voevodsky object T = A^{1}/(A^{1} − 0) itself. So the best adapted model structures is perhaps the flasque motivic model category of [4] which is known to be cellular (so we can apply Hovey’s results [3] in the proof of Theorem 3.2) but for which T is cofibrant.

The category M_{•}(S) is equipped with a symmetric monoidal structure (M_{•}(S), ∧, S^{0}). The smash-product is taken sectionwise, and S^{0} is the constant simplicial zero-sphere. This smashproduct induces a smash-product on H_{•}(S) such that the natural functor M_{•}(S) → H_{•}(S) becomes a strict symmetric monoidal functor (M_{•}(S), ∧, S^{0}) → (H_{•}(S), ∧, S^{0}).

We set T = A^{1}/(A^{1} − 0). A T -spectrum E is a sequence (E_{0}, E_{1}, …) of pointed motivic spaces equipped with a sequence of structure maps σ_{n}: E_{n} ∧ T → E_{n+1} of pointed motivic spaces. A morphism of T -spectra E → E′ is a sequence of maps f_{n}: E_{n} → E′_{n} of pointed motivic spaces which commute with the structure maps. The category of T -spectra Sp(M_{•}(S), T) can be equipped with a motivic stable model structure as in [5, 15, 16]. Its homotopy category is SH(S). This category can be equipped with a structure of a symmetric monoidal category (SH(S), ∧, 1), satisfying the conclusions of [15, Theorem 5.6]. The unit 1 of that monoidal structure is the T -sphere spectrum 𝕊 = (S^{0}, T, T ∧ T, …).

Every T -spectrum E = (E_{0}, E_{1}, …) represents a cohomology theory on the category of pointed motivic spaces M_{•}(S). Namely, let S_{s}^{n} and S_{t}^{n} be as in [15, (16)]. Let S^{p,q} = S_{s}^{p−q}∧S_{t}^{q}. We write

E^{p,q}(A) = Hom_{SH(S)}(Σ_{T}^{∞}A, E ∧ S^{p,q})

as in [15, §6]. There is a canonical element in E^{2n,n}(E_{n}), denoted as

Σ_{T}^{∞}E_{n}(−n) ⟶^{u_{n}} E.

It is represented by the canonical map (∗, …, ∗, E_{n}, E_{n} ∧ T, …) → (E_{0}, E_{1}, …, E_{n}, …) of T -spectra.

A T -ring spectrum is a monoid (E, μ, e) in (SH(S), ∧, 1). The cohomology theory E^{∗,∗} defined by a T -ring spectrum is a ring cohomology theory on M_{•}(S). To see recall the standard isomorphism S^{i,j} ∧ S^{k,l} ≅ S^{i+k,j+l} given by the composition

(S_{s}^{i−j} ∧ S_{t}^{j}) ∧ (S_{s}^{k−l} ∧ S_{t}^{l}) ≅ (S_{s}^{i−j} ∧ S_{s}^{k−l}) ∧ (S_{t}^{j} ∧ S_{t}^{l}) ≅ S_{s}^{i−j+k−l} ∧ S_{t}^{j+l}.

For X, Y ∈ M_{•}(k) let α: Σ^{∞}(X) → S^{i,j} ∧E and β: Σ^{∞}(Y) → S^{k,l} ∧E be elements of E^{i,j}(X) and E^{k,l}(Y) respectively. Following Voevodsky [15] define α × β ∈ E^{i+k,j+l}(X ∧ Y) as the composition

Σ^{∞}(X ∧ Y) ≅ Σ^{∞}(X) ∧ Σ^{∞}(Y) ⟶^{α∧β} E ∧ S^{i,j} ∧ E ∧ S^{k,l} ≅ E ∧ E ∧ S^{i+k,j+l} ⟶^{id∧μ} E ∧ S^{i+k,j+l}.

This gives a functorial product which is associative, has a two-sided unit, and takes cofibration sequences to long exact sequences.

A commutative T -ring spectrum is a commutative monoid (E, μ, e) in (SH(S), ∧, 1). To describe the properties of the associated cohomology theory we make some definitions.

Definition 2.1. Let in_{T} : T → T be a morphism of pointed motivic spaces induced by the morphism A^{1} → A^{1} sending t ↦ −t. One has the equality

Hom_{SH(S)}(pt_{+}, pt_{+}) = Hom_{SH(S)}(T, T).

We write in for in_{T} regarded as an element of Hom_{SH(S)}(pt_{+}, pt_{+}). For a commutative monoid (A, m, e) set ϵ = in^{∗}(e) ∈ A^{0,0}(pt_{+}).

Remark 2.2. The morphism A^{2} → A^{2} which sends (t_{1}, t_{2}) ↦ (−t_{1}, −t_{2}) is A^{1}-homotopic to the identity morphism because ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ SL_{2}(ℤ) is a product of elementary matrices. Whence we have ϵ × ϵ = e ∈ A^{0,0}(pt_{+}).

Definition 2.3. A ring cohomology theory on M_{•}(S) is ϵ-commutative if for α ∈ E^{i,j}(X), β ∈ E^{k,l}(Y) one has σ_{X,Y}^{∗} (α ×β) = (−1)^{ik}ϵ^{jl}(β ×α) ∈ E^{i+k,j+l}(X ∧Y) where σ_{X,Y} : X ∧Y → Y ∧ X switches the factors.

Thus ϵ-commutativity is a specific form of bigraded commutativity.

Theorem 2.4 (Morel). Let (E, μ, e) be a commutative monoid in SH(S). Then the data (E^{∗,∗}, ∂, ×, e) is an associative and ϵ-commutative ring cohomology theory on M_{•}(k); e ∈ E^{0,0}(S^{0}) is the two-sided unit of the ring structure.

Note, that if (i, j) = (2m, 2n) or (k, l) = (2m, 2n), then σ_{A,B}^{∗}(α × β) = β × α.

3. Commutative T - and T^{∧2}-monoids

We compare the categories of symmetric T -spectra and symmetric T^{∧2}-spectra. Recall the definition for K = T or K = T^{∧2}.

Definition 3.1. A symmetric K-spectrum E is a sequence of pointed spaces (E_{0}, E_{1}, E_{2}, …) with each E_{n} equipped with an action of the symmetric group Σ_{n} × E_{n} → E_{n} and with a morphism σ_{n}: E_{n}∧K → E_{n+1} such that the induced maps E_{n}∧K^{∧m} → E_{n+m} are (Σ_{n}×Σ_{m})- equivariant for all n and m.

The categories of symmetric T - and T^{∧2}-spectra both have a symmetric monoidal product ∧. They are symmetric monoidal model categories for the stable model structure [3, 5].

Theorem 3.2. The homotopy categories of Sp^{Σ}(M_{•}(S), T) and of Sp^{Σ}(M_{•}(S), T^{∧2}) are equivalent symmetric monoidal categories.

Proof. The proof of this theorem is essentially the same as that given for topological S^{1}- and S^{2}-spectra in [11, Theorem A.44]. The inclusion Sp^{Σ}(M_{•}(S), T) → Sp^{Σ}(Sp^{Σ}(M_{•}(S), T), T^{∧2}) is a Quillen equivalence by [3, Theorem 9.1] because − ∧ T^{∧2} is a Quillen self-equivalence of Sp^{Σ}(M_{•}(S), T). Similarly the inclusion Sp^{Σ}(M_{•}(S), T^{∧2}) → Sp^{Σ}(Sp^{Σ}(M_{•}(S), T^{∧2}), T) is a Quillen equivalence. The two categories of symmetric bispectra are isomorphic with the identical stable model structure by arguments like those used in the proof of [3, Theorem 10.1]. Hovey’s work requires that the model structure have certain properties, but in the flasque model structure [4] these properties hold, and T and T^{∧2} are cofibrant.

The symmetric monoidal structures are the same because (i) the inclusions of the categories of symmetric spectra in the categories of symmetric bispectra are symmetric monoidal functors like any inclusion Σ_{K}^{∞}: 𝒞 → Sp^{Σ}(𝒞, K), and (ii) the symmetric monoidal structures on the two isomorphic categories of symmetric bispectra are the same. □

For natural numbers m, n we denote by c_{m,n} ∈ Σ_{m+n} the (m, n)-shuffle permutation. It acts by c_{m,n}(i) = i + n for 1 ≤ i ≤ m and c_{m,n}(i) = i − m for m + 1 ≤ i ≤ m + n.

Definition 3.3. A commutative K-monoid E in M_{•}(S) is a sequence of pointed motivic spaces (E_{0}, E_{1}, E_{2}, …) with each space equipped with an action Σ_{n} × E_{n} → E_{n} of the symmetric group, plus morphisms

\begin{matrix}e_{0} : & 1_{M_{•}(S)} → E_{0}, \\ e_{1} : & K → E_{1} \\ μ_{mn}: & E_{m} ∧ E_{n} → E_{m+n} \end{matrix}  (3.1)

in M_{•}(S) such that each μ_{mn} is (Σ_{m} × Σ_{n})-equivariant and such that the compositions

\begin{matrix}E_{n} ⟶^{≅} E_{n} ∧ 1_{M_{•}(S)} ⟶^{1∧e_{0}} E_{n} ∧ E_{0} ⟶^{μ_{n0}} E_{n} \\ E_{n} ⟶^{≅} 1_{M_{•}(S)} ∧ E_{n} ⟶^{e_{0}∧1} E_{0} ∧ E_{n} ⟶^{μ_{0n}} E_{n} \end{matrix}  (3.2)

are the identity maps, and the diagrams

\xymatrix{E_{ℓ} ∧ E_{m} ∧ E_{n} \ar[r]^{μ_{ℓm}∧1} \ar[d]_{1∧μ_{mn}} & E_{ℓ+m} ∧ E_{n} \ar[d]^{μ_{ℓ+m,n}} & E_{m} ∧ E_{n} \ar[d]_{switch} \ar[r]^{μ_{mn}} & E_{m+n} \ar[d]_{≅}^{c_{m,n}} \\ E_{ℓ} ∧ E_{m+n} \ar[r]^{μ_{ℓ,m+n}} & E_{ℓ+m+n} & E_{n} ∧ E_{m} \ar[r]^{μ_{nm}} & E_{n+m}}  (3.3)

commute in M_{•}(S) with c_{m,n} the isomorphism given by the action of the (m, n)-shuffle permutation.

Theorem 3.4. Let E be a commutative K-monoid. Define maps σ_{n} as the compositions

σ_{n}: E_{n} ∧ K ⟶^{1∧e_{1}} E_{n} ∧ E_{1} ⟶^{μ_{n1}} E_{n+1}.  (3.4)

Then the spaces (E_{0}, E_{1}, E_{2}, …) equipped with the actions Σ_{n} × E_{n} → E_{n} and the bonding maps σ_{n} form a symmetric K-spectrum E. Moreover, the morphisms μ: E ∧ E → E induced by the μ_{mn} and e: Σ_{K}^{∞}1_{M_{•}(S)} → E composed of the maps e_{n}: K^{∧n} → E_{n} induced by e_{0}, e_{1} and the μ_{mn} make (E, μ, e) a commutative monoid in Sp^{Σ}(M_{•}(S), K).

Sketch of proof. To show that E is a symmetric K-spectrum one has to verify that the induced maps E_{n} ∧ K^{∧j} → E_{n+j} are (Σ_{n} × Σ_{j})-equivariant. To show that the maps μ_{mn} define a morphism E ∧ E → E one has to verify that they are K-linear and K-bilinear in the sense of [5, (4.6)–(4.7)]. One has to verify that each e_{n} is Σ_{n}-equivariant. Finally, one has to verify the commutative monoid axioms. All the verifications are formal, straightforward and left to the reader. □

4. The symmetric T^{∧2}-spectrum MSL

We construct a commutative T -monoid MSL. Each space MSL_{n} comes equipped with an action of GL_{n} such that the multiplication maps μ_{mn} of the monoid structure are (GL_{m} × GL_{n})-equivariant. We then get actions of the Σ_{n} from the embeddings Σ_{n} → GL_{n} given by permutation matrices. The need for an action of GL_{n} with fixed points — necessary for the proper definition of the unit maps — rather than merely of SL_{n} is the delicate part of the construction.

We begin by reviewing the construction of MGL originally done in [15, §6.3] and of its monoid structure given in [10, 14].

For each integer n ≥ 0 let Γ_{n} = 𝒪_{S}^{⊕n} be the trivial rank-n vector bundle. For each integer p ≥ 1 let Gr(n, np) = Gr(n, Γ_{n}^{⊕p}). Let 𝒯GL_{n,np} → Gr(n, np) be the tautological subbundle. The inclusions (1, 0): Γ_{n}^{⊕p} → Γ_{n}^{⊕p} ⊕ Γ_{n} = Γ_{n}^{⊕p+1} induce closed embeddings Gr(n, np) ↪ Gr(n, np + n) and monomorphisms Th 𝒯GL_{n,np} → Th 𝒯GL_{n,np+n} of Thom spaces. We set

BGL_{n} = colim_{p∈ℕ} Gr(n, np),
𝒯GL_{n,n∞} = colim_{p∈ℕ} 𝒯GL_{n,np},
MGL_{n} = colim_{p∈ℕ} Th 𝒯GL_{n,np}.

The diagonal action of GL_{n} = GL(Γ_{n}) on each Gr(n, np) = Gr(n, Γ_{n}^{⊕p}) is compatible with the inclusions over increasing p. Moreover, the 𝒯GL_{n,np} are GL_{n}-equivariant vector bundles. This induces actions

Σ_{n} × MGL_{n} ⊂ GL_{n} × MGL_{n} → MGL_{n} .

Concatenation of bases induces isomorphisms Γ_{m} ⊕ Γ_{n} = Γ_{m+n} which induce (GL_{m} × GL_{n})- equivariant maps

⊕: Gr(m, mp) × Gr(n, np) → Gr(m + n, mp + np)

and therefore (GL_{m} × GL_{n})-equivariant maps

⊕: BGL_{m} × BGL_{n} → BGL_{m+n},
μ_{mn}^{GL}: MGL_{m} ∧ MGL_{n} → MGL_{m+n} .

Finally each Gr(n, np) is pointed by the point corresponding to the trivial rank-n subbundle

Γ_{n} ⟶^{(1,0,...,0)} Γ_{n}^{⊕p}.

In the colimit this gives a S-valued point x_{n} of BGL_{n}, which is fixed by the action of GL_{n}. The Thom space of the fiber of 𝒯(n, n∞) over x_{n} is Γ_{n} ≅ A^{n}, and the inclusion x_{n} ↪ BGL_{n} induces a map of Thom spaces

e_{n}^{GL}: T^{∧n} → MGL_{n} .

Definition 4.1. The algebraic cobordism spectrum MGL is the commutative monoid in the category of symmetric T -spectra associated to the commutative T -monoid composed of the spaces MGL_{n}, the actions Σ_{n} × MGL_{n} → MGL_{n}, the maps e_{0}^{GL}: pt_{+} → MGL_{0} and e_{1}^{GL}: T → MGL_{1} and the maps μ_{mn}^{GL}: MGL_{m} ∧ MGL_{n} → MGL_{m+n}.

We now move on to defining MSL. We begin with the spaces. For n = 0 we have SL_{0} = GL_{0} = {1}. So we set BSL_{0} = pt. The Thom space of a zero vector bundle over a scheme X is the externally pointed space X_{+}. So we set MSL_{0} = pt_{+}.

Now suppose n > 0. Over each Gr(n, np) there is the line bundle 𝒪_{Gr(n,np)}(−1) = det 𝒯GL_{n,np}. Removing the zero section gives a smooth scheme

SGr(n, np) = 𝒪_{Gr(n,np)}(−1) − Gr(n, np).

The projection

π = π_{n,np}: SGr(n, np) → Gr(n, np)

is a principal 𝔾_{m}-bundle. Write

𝒯SL_{n,np} = π^{∗}𝒯GL_{n,np}.

The inclusion SGr(n, np) ↪ 𝒪_{Gr(n,np)}(−1) and the cartesian diagram

[figure: π^{∗}𝒪_{Gr(n,np)}(−1); □; 𝒪_{Gr(n,np)}(−1); SGr(n, np); π; Gr(n, np)]

gives a nowhere vanishing section of π^{∗}𝒪_{Gr(n,np)}(−1) = det 𝒯SL_{n,np}. The corresponding isomorphism λ_{n,np}: 𝒪_{SGr(n,np)} ≅ det 𝒯SL_{n,np} makes (𝒯SL_{n,np}, λ_{n,np}) the tautological special linear bundle over SGr(n, np).

We set

BSL_{n} = colim_{p∈ℕ} SGr(n, np),
𝒯SL_{n,n∞} = colim_{p∈ℕ} 𝒯SL_{n,np},
MSL_{n} = colim_{p∈ℕ} Th 𝒯SL_{n,np}.

We next define the multiplication maps. Morphisms of S-schemes X → SGr(n, np) are in bijection with pairs (f, λ) with f: X → Gr(n, np) a morphism and λ: 𝒪_{X} ≅ det f^{∗}𝒯GL_{n,np} an isomorphism. There are unique maps

(⊕, ⊗): SGr(m, mp) × SGr(n, np) → SGr(m + n, mp + np)

corresponding to the morphisms of representable functors

\begin{matrix}Hom(X, SGr(m, mp)) × Hom(X, SGr(n, np)) & ⟶ & Hom(X, SGr(m + n, mp + np)) \\ ((f, λ), (g, λ_{1})) & ⟼ & (f ⊕ g, λ ⊗ λ_{1}). \end{matrix}

They induce maps

(⊕, ⊗): BSL_{m} × BSL_{n} → BSL_{m+n}
μ_{mn}^{SL}: MSL_{m} ∧ MSL_{n} → MSL_{m+n}.

We now discuss the group actions. Since 𝒯GL_{n,np} is a GL_{n}-equivariant bundle over Gr(n, np), there is an induced action of GL_{n} on the complement of the zero section of the determinant line bundle. This is an action GL_{n} × SGr(n, np) → SGr(n, np). In the colimit this gives an action GL_{n} × BSL_{n} → BSL_{n}. But there is a problem.

The unit maps e_{n}^{GL}: T^{∧n} → MGL_{n} were defined using points x_{n}: pt → BGL_{n} which were fixed under the action of GL_{n}. To define unit maps for a T -monoid MSL we need fixed points for the action of at least Σ_{n} on BSL_{n}, preferably lying over x_{n}. We have a cartesian diagram

[figure: 𝔾_{m}; □; BSL_{n}; pt; x_{n}; BGL_{n}.]

The action of GL_{n} = GL(Γ_{n}) on the fiber Γ_{n} of 𝒯GL_{n,n∞} over the fixed point x_{n} is the standard representation of GL_{n}. So the induced action on the fiber 𝔾_{m} over x_{n} is g · t = det(g)t. Thus there are fixed points for the action of the alternating group 𝔄_{n} ⊂ SL_{n} on BSL_{n} lying over the fixed point x_{n} ∈ BGL_{n}(pt) used to define the unit maps on MGL_{n} but not for the action of Σ_{n} ⊂ GL_{n} (except in characteristic 2).

So we use the embedding Σ_{n} ⊂ Sp_{2n} ⊂ SL_{2n} which sends σ ∈ Σ_{n} to the permutation matrix associated to σ̄ ∈ Σ_{2n} where we have σ̄(2i − 1) = 2σ(i) − 1 and σ̄(2i) = 2σ(i). This gives us an action Σ_{n} × BSL_{2n} → BSL_{2n} which fixes pointwise the fiber over x_{n}.

Therefore we define the spaces of the commutative T^{∧2}-monoid MSL to be the MSL_{2n}. Each is equipped with the action of Σ_{n} × MSL_{2n} → MSL_{2n} induced by the action of SL_{2n}.

We now define the unit maps. Points pt → BSL_{n} lifting the point x_{n}: pt → BGL_{n} are in bijection with isomorphisms λ: 𝒪_{S} ≅ det x_{n}^{∗}𝒯GL_{n,n∞} = Λ^{n}Γ_{n}. Let f_{1}, …, f_{n} be the standard basis of Γ_{n} = 𝒪_{S}^{⊕n}. We let y_{n}: pt → BSL_{n} be the lifting of x_{n} corresponding to λ = f_{1} ∧ ⋯ ∧ f_{n}. The fiber of 𝒯SL_{n,n∞} over y_{n} is Γ_{n} ≅ A^{n}, and we let

e_{n}^{SL}: T^{∧n} → MSL_{n}

be the map of Thom spaces induced by y_{n}. It is SL_{n}-equivariant. Note that e_{0}^{SL}: pt_{+} → MSL_{0} = pt_{+} is the identity.

Having identified the components of the structure MSL, we have to assemble them. It appears as if MSL is a commutative monoid in the category of alternating T -spectra. We do not know how to work in that category. But there is underlying structure.

Definition 4.2. The algebraic special linear cobordism spectrum MSL refers to three related objects.

(a) The commutative monoid in the category of symmetric T^{∧2}-spectra associated to the commutative T^{∧2}-monoid composed of the spaces MSL_{2n}, the actions Σ_{n} × MSL_{2n} → MSL_{2n}, the maps e_{0}^{SL}: pt_{+} → MSL_{0} and e_{2}^{SL}: T^{∧2} → MSL_{2} and the maps μ_{2m,2n}^{SL}: MSL_{2m}∧ MSL_{2n} → MSL_{2m+2n}.

(b) The T -spectrum with spaces MSL_{n}, bonding maps MSL_{n} ∧ T → MSL_{n} ∧ MSL_{1} → MSL_{n+1} induced by e_{1}^{SL} and μ_{n,1}^{SL}, equipped with the morphism of T -spectra e: Σ_{T}^{∞}pt_{+} → MSL and the structural maps μ_{mn}^{SL}.

(c) Their common underlying T^{∧2}-spectrum.

The properties of the commutative monoid structure that we require are given in the following theorem.

Theorem 4.3. The (MSL, μ^{SL}, e^{SL}) is a commutative monoid in SH(S), and the canonical maps u_{n}: Σ_{T}^{∞}MSL_{n}(−n) → MSL and the μ_{mn}^{SL} make the following diagram commute for all m and n

\xymatrix{Σ_{T}^{∞}MSL_{m}(−m) ∧ Σ_{T}^{∞}MSL_{n}(−n) \ar[r]^{Σ_{T}^{∞}μ_{mn}^{SL}} \ar[d]_{u_{m}∧u_{n}} & Σ_{T}^{∞}MSL_{m+n}(−m − n) \ar[d]^{u_{m+n}} \\ MSL ∧ MSL \ar[r]^{μ^{SL}} & MSL.}  (4.1)

Proof. A commutative monoid in Sp^{Σ}(M_{•}(S), T^{∧2}) gives a commutative monoid in SH(S) by Theorem 3.2. When m and n are even, the diagram in Sp^{Σ}(M_{•}(S), T^{∧2}) corresponding to (4.1) commutes by formal arguments. When say m is even and n is odd, the diagram

\xymatrix{Σ_{T}^{∞}MSL_{m}(−m) ∧ Σ_{T}^{∞}MSL_{n}(−n) ∧ Σ_{T}^{∞}T \ar[r]^{Σ_{T}^{∞}μ_{mn}^{SL}∧1} \ar[d]_{1∧Σ_{T}^{∞}σ_{n}} & Σ_{T}^{∞}MSL_{m+n}(−m − n) ∧ Σ_{T}^{∞}T \ar[d]^{Σ_{T}^{∞}σ_{m+1}} \\ Σ_{T}^{∞}MSL_{m}(−m) ∧ Σ_{T}^{∞}MSL_{n+1}(−n) \ar[r]^{Σ_{T}^{∞}μ_{m,n+1}^{SL}} \ar[d]_{u_{m}∧u_{n+1}} & Σ_{T}^{∞}MSL_{m+n+1}(−m − n) \ar[d]^{u_{m+n+1}} \\ MSL ∧ MSL(1) \ar[r]^{μ^{SL}} & MSL(1)}

commutes because m and n+1 are even. One may desuspend. The other cases are similar. □

5. Special linear orientations

We now investigate the relationship between special linear orientations on a ring cohomology theory E, as defined in [12, Definition 3.1] and homomorphisms φ: MSL → A of commutative monoids in SH(S).

A special linear vector bundle over X is a pair (E, λ) with E → X a vector bundle and λ: 𝒪_{X} ≅ det E an isomorphism of line bundles. An isomorphism ϕ: (E, λ) ≅ (E′, λ′) of special linear vector bundles is an isomorphism ϕ: E ≅ E′ of vector bundles such that (det ϕ)∘λ = λ′.

Definition 5.1. A special linear orientation on a bigraded ϵ-commutative ring cohomology theory A^{∗,∗} on 𝒮m𝒪p/S is a rule which assigns to every special linear vector bundle (E, λ) of rank n over an X in 𝒮m/S a class th(E, λ) ∈ A^{2n,n}(E, E − X) satisfying the following conditions:

(1) For an isomorphism f: (E, λ) ≅ (E_{1}, λ_{1}) we have th(E, λ) = f^{∗} th(E_{1}, λ_{1}).

(2) For u: Y → X we have u^{∗} th(E, λ) = th(u^{∗}(E, λ)) in A^{2n,n}(u^{∗}E, u^{∗}E − Y).

(3) The maps − ∪ th(E, λ): A^{∗,∗}(X) → A^{∗+2n,∗+n}(E, E − X) are isomorphisms.

(4) We have

th(E_{1} ⊕ E_{2}, λ_{1} ⊗ λ_{2}) = q_{1}^{∗} th(E_{1}, λ_{1}) ∪ q_{2}^{∗} th(E_{2}, λ_{2}),

where q_{1}, q_{2} are the projections from E_{1} ⊕ E_{2} onto its summands. Moreover, for the zero bundle 0 → pt we have th(0) = 1_{A} ∈ A^{0,0}(pt).

The class th(E, λ) is the Thom class of the special linear bundle, and e(E, λ) = z^{∗} th(E, λ) ∈ A^{2n,n}(X) is its Euler class.

This definition is analogous to the Thom classes theory version of the definition of an orientation [9, Definition 3.32].

For any n the functor − ∧ T^{∧n}: SH(S) → SH(S) is a self-equivalence. So it induces isomorphisms

− ∧ T^{∧n}: Hom_{SH(S)}(X, A ∧ S^{p,q}) ⟶^{≅} Hom_{SH(S)}(X ∧ T^{∧n}, A ∧ S^{p,q} ∧ T^{∧n})

for any X and (p, q) and any cohomology theory on 𝒮m/S defined by a commutative monoid (A, μ, e) in SH(S). We also write these isomorphisms as

Σ_{T}^{n}: A^{p,q}(X) ⟶^{≅} A^{p+2n,q+n}(X × A^{n}, X × (A^{n} − 0))

This isomorphism coincides with − × Σ_{T}^{n} 1_{A}. Thus A^{∗,∗} automatically has Thom classes for trivial bundles: the pullbacks of Σ_{T}^{n} 1_{A}.

Definition 5.2. A special linear orientation on a bigraded ring cohomology theory A^{∗,∗} on 𝒮m𝒪p/S which is representable by a commutative monoid in SH(S) is normalized if

(5) for the trivial line bundle A^{1} → pt we have th(A^{1}, 1) = Σ_{T} 1_{A} ∈ A^{2,1}(A^{1}, A^{1} − 0).

From the multiplicativity and functoriality conditions (4) and (2) in the definition of a special linear orientation one deduces the following result.

Lemma 5.3. Suppose A^{∗,∗} is a bigraded ring cohomology theory on 𝒮m𝒪p/S representable by a commutative monoid in SH(S) with a normalized special linear orientation. For X ∈ 𝒮m/S let (𝒪_{X}^{⊕n}, λ_{n}) be the trivial special linear bundle of rank n over X. Then th(𝒪_{X}^{⊕n}, λ_{n}) is the pullback to X of Σ_{T}^{n} 1_{A}, and

− ∪ th(𝒪_{X}^{⊕n}, λ_{n}): A^{∗,∗}(X) ⟶^{≅} A^{∗+2n,∗+n}(X × A^{n}, X × (A^{n} − 0))

is an isomorphism.

Now suppose φ: MSL → A is a morphism in SH(S). We associate to φ and a special linear bundle (E, λ) of rank n over an X in 𝒮m/S a class th^{φ}(E, λ) defined as follows. By assumption the scheme X admits an ample family of line bundles. So there exists an affine bundle f: Y → X with Y an affine scheme. Then for some p there exist global sections s_{1}, …, s_{np} of f^{∗}E^{∨} generating f^{∗}E^{∨}. The data (f^{∗}E, s_{1}, …, s_{np}) determine a morphism ψ: Y → Gr(n, np), and the data (ψ, f^{∗}λ) determine a morphism ψ̃: Y → SGr(n, np). We have ψ̃^{∗}𝒯SL_{n,np} ≅ f^{∗}E. We deduce maps

Th E ⟵_{∼mot}^{\overline{f}} Th f^{∗}E ≅ Th ψ̃^{∗}𝒯SL_{n,np} ⟶^{\overline{ψ}} Th 𝒯SL_{n,np}  (5.1)

of pointed motivic spaces, which can be composed with the maps

Th 𝒯SL_{n,np} ⟶^{inclusion} MSL_{n} ⟶^{u_{n}} MSL ∧ T^{∧n} ⟶^{1∧φ} A ∧ T^{∧n}.  (5.2)

in SH(S). The composition of (5.1) and (5.2) gives a class

th^{φ}(E, λ) ∈ Hom_{SH(S)}(Th E, A ∧ T^{∧n}) = A^{2n,n}(E, E − X).

Lemma 5.4. The classes th^{φ}(E, λ) depend only on the special linear bundle (E, λ) and the morphism φ: MSL → A in SH(S).

Proof. First suppose f fixed. Let (s_{1}, …, s_{np}) and (t_{1}, …, t_{nq}) be two families of sections generating f^{∗}E^{∨} with p ≥ q. There are A^{1}-homotopies between the morphisms Th f^{∗}E → MSL_{n} in M_{•}(S) defined by the family (s_{1}, …, s_{np}), the family (s_{1}, …, s_{np}, t_{1}, …, t_{nq}), the family (t_{1}, …, t_{nq}, 0, …, 0, t_{1}, …, t_{nq}), and the family (t_{1}, …, t_{nq}). So we get the same morphism Th f^{∗}E → MSL_{n} in H_{•}(S) and the same morphism Th E → A ∧ T^{∧n} in SH(S).

Now suppose given a second affine bundle g: Z → X with Z affine and sections (u_{1}, …, u_{nr}) generating g^{∗}E^{∨}. Let g′: Y ×_{X} Z → Y and f′: Y ×_{X} Z → Z be the projections. The morphisms Th E → MSL_{n} in H_{•}(S) defined by f and (s_{1}, …, s_{np}), by g′f and (g^{′∗}s_{1}, …, g^{′∗}s_{np}), by f′g and (f^{′∗}u_{1}, …, f^{′∗}u_{nr}) and by g and (u_{1}, …, u_{nr}) are then the same. So we again get the same morphism Th E → A ∧ T^{∧n} in SH(S). □

Theorem 5.5. For a homomorphism φ: MSL → A of commutative monoids in SH(S), the classes th^{φ}(E, λ) define a normalized special linear orientation on the bigraded ring cohomology theory A^{∗,∗} on 𝒮m𝒪p(S).

In particular the identity homomorphism induces a normalized special linear orientation on MSL^{∗,∗}.

Proof. The functoriality conditions (1) and (2) follow easily from the construction of the classes th^{φ}(E, λ). The multiplicativity condition (4) holds because of Theorem 4.3 and because φ is a homomorphism of monoids. The normalization condition (5) holds because th^{φ}(A^{1}, 1) and Σ_{T} 1_{A} are both equal to the composition

T ⟶^{e_{1}^{SL}} MSL_{1} ⟶^{u_{1}} MSL ∧ T ⟶^{φ∧1} A ∧ T.

The isomorphism condition (3) holds for trivial special linear bundles because of the normalization condition and Lemma 5.3. It then holds for general special linear bundles by a Mayer-Vietoris argument because special linear bundles are locally trivial in the Zariski topology. □

Now suppose that M and A are (symmetric) T -spectra. Then we have an inverse system of abelian groups

⋯ → A^{2n+2,n+1}(M_{n+1}) ⟶^{α_{n+1}} A^{2n,n}(M_{n}) → ⋯ → A^{0,0}(M_{0})  (5.3)

where the map α_{n} associates to the map v: M_{n+1} → A ∧ T^{∧n+1} in SH(S) the composition

M_{n} ⟶^{σ_{n}^{∗}} Ω_{T} M_{n+1} ⟶^{v} Ω_{T} (A ∧ T^{∧n+1}) ≅ A ∧ T^{∧n}

in SH(S). There is a similar inverse system

⋯ → A^{4n+4,2n+2}(M_{n+1} ∧ M_{n+1}) → A^{4n,2n}(M_{n} ∧ M_{n}) → ⋯ → A^{0,0}(M_{0} ∧ M_{0}). (5.4)

For the following theorem see for example [11, Corollaries 3.4 and 3.5].

Theorem 5.6. For any (symmetric) T - or T^{∧2}-spectra M and A we have exact sequences of abelian groups

0 → \underleftarrow{lim}^{1} A^{2n−1,n}(M_{n}) → Hom_{SH(S)}(M, A) → \underleftarrow{lim} A^{2n,n}(M_{n}) → 0,

0 → \underleftarrow{lim}^{1} A^{4n−1,2n}(M_{n} ∧ M_{n}) → Hom_{SH(S)}(M ∧ M, A) → \underleftarrow{lim} A^{4n,2n}(M_{n} ∧ M_{n}) → 0.

This theorem is actually a special case of the following result [11, Lemma 3.3].

Theorem 5.7. Let E = hocolim_{i∈ℕ} E^{⟨i⟩} be a sequential homotopy colimit of T -spectra. Then for any T -spectrum A and any (p, q) we have an exact sequence of abelian groups

0 → \underleftarrow{lim}^{1}_{i∈ℕ} A^{p−1,q}(E^{⟨i⟩}) → A^{p,q}(E) → \underleftarrow{lim}_{i∈ℕ} A^{p,q}(E^{⟨i⟩}) → 0.

We wish to apply Theorem 5.6 when A is a commutative monoid in SH(S) with a normalized special linear orientation on A^{∗,∗} and when M is a commutative monoid isomorphic to MSL in SH(S). (Note that the exact sequences depend on the levelwise weak equivalence class of M, which is a finer invariant than its isomorphism class in SH(S).) However, the special linear orientation provides Thom classes for special linear bundles over finite-dimensional smooth schemes and not over the infinite-dimensional ind-schemes BSL_{n}. So the orientation does not provide us with classes in the A^{2n,n}(MSL_{n}). But we can solve this problem as follows.

For each n and p write

MSL_{n}^{⟨p⟩} = Th 𝒯SL_{n,np}.

For n = 0 this is MSL_{0}^{⟨p⟩} = pt_{+}. The actions of Σ_{n} on MSL_{n} and the structural maps e_{n}^{SL} and μ_{mn}^{SL} constructed in the previous section are colimits of actions and structural maps

Σ_{n} × MSL_{n}^{⟨p⟩} → MSL_{n}^{⟨p⟩},
e_{n}^{⟨p⟩}: T^{∧n} → MSL_{n}^{⟨p⟩},
μ_{mn}^{⟨p⟩}: MSL_{m}^{⟨p⟩} ∧ MSL_{n}^{⟨p⟩} → MSL_{m+n}^{⟨p⟩}.

We thus get a direct system of commutative T -monoids

MSL^{⟨1⟩} → MSL^{⟨2⟩} → ⋯ → MSL^{⟨p⟩} → ⋯

whose colimit is MSL. We can now define a “diagonal” commutative T -monoid MSL^{fin} with spaces

MSL_{n}^{fin} = MSL_{n}^{⟨n⟩}

with the actions Σ_{n} × MSL_{n}^{⟨n⟩} → MSL_{n}^{⟨n⟩} and unit maps e_{n}^{⟨n⟩} given above and with multiplication maps the compositions

μ_{mn}^{fin} : MSL_{m}^{⟨m⟩} ∧ MSL_{n}^{⟨n⟩} ⟶^{inclusion} MSL_{m}^{⟨m+n⟩} ∧ MSL_{n}^{⟨m+n⟩} ⟶^{μ_{mn}^{⟨m+n⟩}} MSL_{m+n}^{⟨m+n⟩}.

A cofinality argument now gives the nontrivial part of the following result.

Theorem 5.8. The inclusion MSL^{fin} ↪ MSL defines a homomorphism of commutative monoids in the category of symmetric T -spectra which is a motivic stable weak equivalence.

Thus the inclusion becomes an isomorphism of commutative monoids in SH(S). So Theorem 5.6 gives us an exact sequence

0 → \underleftarrow{lim}^{1} A^{2n−1,n}(MSL_{n}^{⟨n⟩}) → Hom_{SH(S)}(MSL, A) → \underleftarrow{lim} A^{2n,n}(MSL_{n}^{⟨n⟩}) → 0  (5.5)

and a similar exact sequence for Hom_{SH(S)}(MSL ∧ MSL, A).

Theorem 5.9. Suppose (A, μ_{A}, e_{A}) is a commutative monoid in SH(S) with a normalized special linear orientation on A^{∗,∗} given by Thom classes th(E, λ). Then there exists a morphism φ: MSL → A in SH(S) such that th^{φ}(E, λ) = th(E, λ) for all special linear bundles over all X in 𝒮m/S. This φ is unique modulo the subgroup

\underleftarrow{lim}^{1} A^{2n−1,n}(MSL_{n}^{⟨n⟩}) ⊂ Hom_{SH(S)}(MSL, A).

It satisfies φ(e_{MSL}) = e_{A}. The obstruction φ∘μ_{MSL}−μ_{A}∘(φ∧φ) to φ being a homomorphism of monoids lies in the subgroup

\underleftarrow{lim}^{1} A^{4n−1,2n}(MSL_{n}^{⟨n⟩} ∧ MSL_{n}^{⟨n⟩}) ⊂ Hom_{SH(S)}(MSL ∧ MSL, A).

Proof. For every n and p the tautological special linear bundle (𝒯SL_{n,np}, λ_{n,np}) over the scheme SGr(n, np) has a Thom class, which we will abbreviate to th_{n,np} ∈ A^{2n,n}(MSL_{n}^{⟨p⟩}). Pullback along the inclusion MSL_{n}^{⟨p−1⟩} ↪ MSL_{n}^{⟨p⟩} sends

th_{n,np} ↦ th_{n,n(p−1)} .  (5.6)

Pullback along the bonding map MSL_{n−1}^{⟨p⟩} ∧ T → MSL_{n}^{⟨p⟩} induced by e_{1}^{⟨p⟩} and μ_{n−1,1}^{⟨p⟩} sends

th_{n,np} ↦ th_{n−1,(n−1)p} × th(A^{1}, 1) = Σ_{T} th_{n−1,(n−1)p} .  (5.7)

So as n and p vary, we get an element

φ̄ = (th_{n,np})_{n,p} ∈ \underleftarrow{lim}_{n,p} A^{2n,n}(MSL_{n}^{⟨p⟩}) = \underleftarrow{lim}_{n} A^{2n,n}(MSL_{n}^{⟨n⟩}).

Let φ ∈ Hom_{SH(S)}(MSL, A) be an element mapping onto φ̄ under the surjection in the exact sequence (5.5).

The image of φ under the composition

Hom_{SH(S)}(MSL, A) → \underleftarrow{lim} A^{2n,n}(MSL_{n}^{⟨n⟩}) → A^{2n,n}(MSL_{n}^{⟨n⟩})

is the composition

Th 𝒯SL_{n,n^{2}} = MSL_{n}^{⟨n⟩} ⟶^{u_{n}} MSL^{fin} ∧ T^{∧n} ⟶^{∼} MSL ∧ T^{∧n} ⟶^{φ∧1} A ∧ T^{∧n}

which is the th^{φ}(𝒯SL_{n,n^{2}}, λ_{n,n^{2}}) defined by (5.1)–(5.2). Thus we have th^{φ}(E, λ) = th(E, λ) for (E, λ) = (𝒯SL_{n,n^{2}}, λ_{n,n^{2}}). The Thom classes for the (𝒯SL_{n,n^{2}}, λ_{n,n^{2}}) determine the Thom classes for all (𝒯SL_{n,np}, λ_{n,np}) by formulas (5.6)–(5.7). These in turn determine the Thom classes for all (E, λ) by formulas (5.1)–(5.2). So we have th^{φ}(E, λ) = th(E, λ) for all special linear bundles.

Similarly for ψ: MSL → A we have th^{ψ}(E, λ) = th^{φ}(E, λ) for all special linear bundles if and only if ψ and φ have the same image in \underleftarrow{lim} A^{2n,n}(MSL_{n}^{⟨n⟩}). This happens if and only if ψ − φ is in the kernel, which is the first \underleftarrow{lim}^{1} of the statement of the theorem.

By construction e_{MSL} is the canonical map Σ_{T}^{∞}pt_{+} = Σ_{T}^{∞}MSL_{0} → MSL. Therefore we have φ(e_{MSL}) = th_{0,0} = th(0) = e_{A} ∈ A^{0,0}(pt) as declared.

By multiplicativity and functoriality we have an equality

th_{n,n^{2}} × th_{n,n^{2}} = th(p_{1}^{∗}𝒯SL_{n,n^{2}} ⊕ p_{2}^{∗}𝒯SL_{n,n^{2}}, p_{1}^{∗}λ_{n,n^{2}} ⊗ p_{2}^{∗}λ_{n,n^{2}}) = μ_{nn}^{fin∗} th_{2n,4n^{2}}

of members of A^{4n,2n}(MSL_{n}^{⟨n⟩} ∧ MSL_{n}^{⟨n⟩}). This equality means that the outer perimeter of the diagram

\xymatrix{Σ_{T}^{∞}MSL_{n}^{⟨n⟩}(−n) ∧ Σ_{T}^{∞}MSL_{n}^{⟨n⟩}(−n) \ar[r]^{μ_{nn}^{fin}} \ar[d]^{u_{n}∧u_{n}} \ar[ddd]_{th_{n,n^{2}} ∧ th_{n,n^{2}}} & Σ_{T}^{∞}MSL_{2n}^{⟨2n⟩}(−2n) \ar[d]^{u_{2n}} \ar[ddd]^{_{th_{2n,4n^{2}}}} \\ MSL^{fin} ∧ MSL^{fin} \ar[r]^{μ_{MSL}^{fin}} \ar[d]^{inclusion} & _{MSL_{fin}} \ar[d]^{inclusion} \\ MSL ∧ MSL \ar[r]^{μ_{MSL}} \ar[d]^{φ∧φ} & MSL \ar[d]^{φ} \\ A ∧ A \ar[r]^{μ_{A}} & A}

commutes. The half-circles commute by the previous calculations, and the top two squares commute. Therefore we have

(φ ∘ μ_{MSL} − μ_{A} ∘ (φ ∧ φ)) ∘ inclusion ∘ (u_{n} ∧ u_{n}) = 0

for all n. So the image of the obstruction class φ ∘ μ_{MSL} − μ_{A} ∘ (φ ∧ φ) under the surjection

Hom_{SH(S)}(MSL ∧ MSL, A) ⟶ \underleftarrow{lim} A^{4n,2n}(MSL_{n}^{⟨n⟩} ∧ MSL_{n}^{⟨n⟩}) ⟶ 0

vanishes. Therefore the obstruction class lies in the kernel, which is the second \underleftarrow{lim}^{1} of the statement of the theorem. □

6. The symmetric T^{∧2}-spectrum MSp

We now define the commutative T^{∧2}-monoid and symmetric T^{∧2}-spectrum MSp.

We write the standard symplectic form on the trivial vector bundle of rank 2n as

ω_{2n} = (\begin{matrix}\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & & & & 0 \\ & . & . & . \\ 0 & & & & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} \end{matrix})

From the symplectic isometry (𝒪_{S}^{⊕2n}, ω_{2n}) ≅ (𝒪_{S}^{⊕2}, ω_{2})^{⊕n} we see that the action of Σ_{n} given by permutations of the n orthogonal direct summands (𝒪_{S}^{⊕2}, ω_{2}) gives an embedding Σ_{n} → Sp_{2n}. Hence Sp_{2n}-actions restrict to Σ_{n}-actions.

In [13] we defined the quaternionic Grassmannian HGr(r, n) as the open subscheme of the Grassmannian Gr(2r, 2n) parametrizing rank 2r subspaces of 𝒪_{S}^{⊕2n} on which the restriction of ω_{2n} is nondegenerate. The restriction of the tautological subbundle over the Grassmannian is the tautological symplectic subbundle 𝒯Sp_{r,n}. It is equipped with the symplectic form ϕ_{r,n} = ω_{2n}|_{𝒯Sp_{r,n}}. For r = 1 we write HP^{n} = HGr(1, n + 1) and HP^{∞} = colim_{n} HP^{n}.

To construct MSp we look at the particular schemes HGr(n, np) = HGr(n, (𝒪_{S}^{⊕2n}, ω_{2n})^{⊕p}). Each has a natural action of Sp_{2n} induced by the diagonal action of Sp_{2n} on the p summands of (𝒪_{S}^{⊕2n}, ω_{2n})^{⊕p}. The vector bundles 𝒯Sp_{n,np} → HGr(n, np) and the inclusions HGr(n, np) → HGr(n, np + n) are Sp_{2n}-equivariant. We set

BSp_{2n} = colim_{p∈ℕ} HGr(n, np),
𝒯Sp_{n,n∞} = colim_{p∈ℕ} 𝒯Sp_{n,np},
MSp_{2n} = colim_{p∈ℕ} Th 𝒯Sp_{n,np}.

As with MGL and MSL the isomorphisms

(𝒪_{S}^{⊕2m}, ω_{2m}) ⊕ (𝒪_{S}^{⊕2n}, ω_{2n}) ≅ (𝒪_{S}^{⊕2m+2n}, ω_{2m+2n})

and the direct sum induce (Sp_{2m} × Sp_{2n})-equivariant maps

⊕: BSp_{2m} × BSp_{2n} → BSp_{2m+2n},

μ_{mn}^{Sp}: MSp_{2m} ∧ MSp_{2n} → MSp_{2m+2n}. (6.1)

Each HGr(n, np) is pointed by the point corresponding to the symplectic subbundle which is the first direct summand (𝒪_{S}^{⊕2n}, ω_{2n}) ⊕ 0^{⊕p−1} ⊂ (𝒪_{S}^{⊕2n}, ω_{2n})^{⊕p}. In the colimit this yields points z_{2n}: pt → BSp_{2n}. The point z_{2n} is fixed by the Sp_{2n}-action. The action of Sp_{2n} on the fiber of 𝒯Sp_{n,n∞} over z_{2n} is the standard representation of Sp_{2n}. The inclusion of the fiber induces an inclusion of Thom spaces

e_{2n}^{Sp}: T^{∧2n} → MSp_{2n}  (6.2)

which is Sp_{2n}-equivariant. The action of the subgroup Σ_{n} ⊂ Sp_{2n} on T^{∧2n} = (T^{∧2})^{∧n} permutes the n factors T^{∧2}.

The spaces MSp_{2n} with the actions and structural maps verify the axioms of a commutative T^{∧2}-monoid.

Definition 6.1. The algebraic symplectic cobordism spectrum MSp is the commutative monoid in the category of symmetric T^{∧2}-spectra associated to the commutative T^{∧2}-monoid composed of the spaces MSp_{2n}, the actions Σ_{n} × MSp_{2n} → MSp_{2n} the maps e_{0}^{Sp}: pt_{+} → MSp_{0} and e_{2}^{Sp}: T^{∧2} → MSp_{2} and the maps μ_{mn}^{Sp}: MSp_{2m} ∧ MSp_{2n} → MSp_{2m+2n}.

This MSp defines a commutative monoid in SH(S) by Theorem 3.2.

7. Quaternionic Grassmannian bundles

We review the geometry of quaternionic projective bundles and Grassmannian bundles studied in [13, §§3–5]. We then translate some of the results into a more motivic language.

Given (E, ϕ) a symplectic bundle of rank 2n over a scheme X and an integer 0 ≤ r ≤ n, there is a quaternionic Grassmannian bundle p: HGr(r, E, ϕ) → X whose fiber over x ∈ X is the quaternionic Grassmannian parametrizing 2r-dimensional subspaces of E_{x} on which ϕ_{x} is nondegenerate. We write 𝒰_{r,E} ⊂ p^{∗}E for the tautological rank 2r subbundle over HGr(r, E, ϕ). Morphisms f: Y → HGr(r, E, ϕ) are in bijection with pairs (g, U) where g: Y → X is a morphism and U ⊂ g^{∗}(E, ϕ) is a symplectic subbundle of rank 2r over Y.

Since 𝒰_{r,E} is a subbundle on which the symplectic form is fiberwise nondegenerate, it has an orthogonal complement such that 𝒰_{r,E} ⊕ 𝒰_{r,E}^{⊥} = p^{∗}E. The symplectic subbundle 𝒰_{r,E}^{⊥} ⊂ p^{∗}E classifies an isomorphism

HGr(r, E, ϕ) ⟶^{≅} HGr(n − r, E, ϕ).  (7.1)

Now let (F, ψ) = (𝒪_{X}^{⊕2}, ω_{2}) ⊕ (E, ϕ). We have a natural embedding

HGr(r, E, ϕ) ↪ HGr(r, F, ψ)  (7.2)

classified by the symplectic subbundle 0 ⊕ 𝒰_{r,E} ⊂ 𝒪_{X}^{⊕2} ⊕ E = F. The normal bundle of this embedding can be naturally identified with the vector bundle N = ℋom(𝒰_{r,E}, 𝒪_{X}^{⊕2}) over HGr(r, E, ϕ). This bundle is a direct sum decomposition N = N^{+} ⊕ N^{−} where

N^{+} = ℋom(𝒰_{r,E}, 𝒪_{X} ⊕ 0),  N^{−} = ℋom(𝒰_{r,E}, 0 ⊕ 𝒪_{X}).

The basic result concerning the geometry of the closed embedding (7.2) is the following.

Theorem 7.1 ([13, Theorem 4.1]). (a) The normal bundle of the embedding (7.2) has a canonical open embedding ν: N ↪ Gr(2r, F). The zero section is sent identically onto HGr(r, E, ϕ).

(b) We have ν(N^{+}) = HGr(r, F, ψ) ∩ Gr_{S}(2r, 𝒪_{X} ⊕ 0 ⊕ E). Consequently ν(N^{+}) ⊂ HGr(r, F, ψ) is a closed subscheme, as is ν(N^{−}) ⊂ HGr(r, F, ψ).

(c) There are natural isomorphisms of vector bundles N^{+} ≅ N^{−} ≅ 𝒰_{r,E}^{∨} ≅ 𝒰_{r,E}.

(d) There is a natural section s_{+} of 𝒰_{r,F} intersecting the zero section transversally in N^{+} and similarly for N^{−}.

(e) Let π_{+} : N^{+} → HGr(r, E, ϕ) be the structural map. Then π_{+}^{∗}(𝒰_{r,E}, ϕ|_{𝒰_{r,E}}) is isometric to (𝒰_{r,F} , ψ|_{𝒰_{r,F}})|_{N^{+}} and similarly for N^{−}.

The second basic result about the geometry of symplectic Grassmannian bundles involves the following embeddings

\xymatrix{HGr(r − 1, E, ϕ) \ar@{^{(}->}[r]^{σ}_{closed} & HGr(r, F, ψ) − ν(N^{+}) \ar@{^{(}->}[r]^{open} & HGr(r, F, ψ).}  (7.3)

The composition is the closed embedding classified by the symplectic subbundle 𝒪^{⊕2} ⊕ 𝒰_{r−1,E} ⊂ 𝒪^{⊕2} ⊕ E = F.

Theorem 7.2 ([13, Theorems 5.1 and 5.2]). There are morphisms over X

HGr(r, F, ψ) − ν(N^{+}) ⟵^{g_{1}} Y_{1} ⟵^{g_{2}} Y_{2} ⟶^{q} HGr(r − 1, E, ϕ)

with g_{1} an A^{2r−1}-bundle, g_{2} an A^{2r−2}-bundle, and q an A^{4n+1}-bundle. Moreover, there is a section s of q such that the composition g_{1}g_{2}s: HGr(r − 1, E, ϕ) → HGr(r, F, ψ) − ν(N^{+}) is the closed embedding σ of (7.3).

The section s appears at the end of the proof of [13, Theorem 5.2]. The statement of the theorem only contains the consequence that σ induces isomorphisms of cohomology groups.

These two theorems have the following consequence.

Theorem 7.3. Let (E, ϕ) be a symplectic bundle of rank 2n over a smooth S-scheme X, and let (F, ψ) = (𝒪_{X}^{⊕2}, ω_{2})⊕(E, ϕ). For 1 ≤ r ≤ n let 𝒰_{r,E} be the tautological symplectic subbundle over HGr(r, E, ϕ). Let HGr(r − 1, E, ϕ) ↪ HGr(r, F, ψ) be the closed embedding of (7.3). Then there is a canonical zigzag of motivic weak equivalences

Th 𝒰_{r,E} ⟶^{∼} HGr(r, F, ψ)/(HGr(r, F, ψ) − ν(N^{+})) ⟵^{∼} HGr(r, F, ψ)/HGr(r − 1, E, ϕ)

inducing an isomorphism in the motivic unstable homotopy category H_{•}(S). These isomorphisms commute with the maps induced by inclusions (E, ϕ) ↪ (E, ϕ)⊕(E_{1}, ϕ_{1}) of symplectic bundles and with the maps induced by base changes Y → X.

Proof. In the geometry of Theorem 7.1 we identify N, N^{+} and N^{−} with their images in Gr(2r, F) under the open embedding ν. Then there are motivic weak equivalences

\xymatrix{N/(N − N^{+}) & (N ∩ HGr(r, F, ψ))/(N ∩ HGr(r, F, ψ) − N^{+}) \ar[l]_{∼}^{excision} \ar[d]_{∼}^{excision} \\ Th 𝒰_{r,E} ≅ N^{−}/(N^{−} − HGr(r, E, ϕ)) \ar[u]_{∼}^{section of vector bundle} \ar@{-->}[ur] \ar[r]_{inclusion} & HGr(r, F, ψ)/(HGr(r, F, ψ) − N^{+})}

The arrows not explicitly labeled ∼ are nevertheless motivic weak equivalences by the 2-out-of-3 axiom. The morphisms g_{1}, g_{2} and q of Theorem 7.2 are affine bundles, so they, the section s, the composition σ, and the map

HGr(r, F, ψ)/HGr(r − 1, E, ϕ) ⟶^{∼} HGr(r, F, ψ)/(HGr(r, F, ψ) − N^{+})

of quotient spaces induced by σ are all motivic weak equivalences.

The functoriality is straightforward. □

Because of (7.1) the quotient appearing in Theorem 7.3 is also isomorphic to

HGr(n − r + 1, F, ψ)/HGr(n − r + 1, E, ϕ).

The case r = n of Theorem 7.3 therefore gives the following result. The corresponding result for ordinary Grassmannian bundles is [8, Proposition 3.2.17(3)].

Theorem 7.4. Suppose that (E, ϕ) is a symplectic bundle of rank 2n over a smooth S-scheme X. Let HP (E, ϕ) ↪ HP (𝒪_{X}^{⊕2} ⊕ E, ω_{2} ⊕ ϕ) be the natural closed embedding. Then we have isomorphisms in H_{•}(S)

Th E ≅ HGr(n, (𝒪_{X}^{⊕2}, ω_{2}) ⊕ (E, ϕ))/HGr(n − 1, E, ϕ) = HP ((𝒪_{X}^{⊕2}, ω_{2}) ⊕ (E, ϕ))/HP (E, ϕ).

For X = pt and trivial E Theorem 7.3 gives the following result.

Theorem 7.5. There are canonical isomorphisms

Th 𝒰_{HGr(r,n)} ≅ HGr(r, 1 + n)/HGr(r − 1, n)

in H_{•}(S). These isomorphisms are compatible with the inclusions of quaternionic projective spaces. Therefore we have commutative diagrams of inclusions and isomorphisms in H_{•}(S)

\xymatrix{T^{∧2r} \ar[r]^{e_{r}^{Sp}} \ar[d]_{≅} & _{MSp_{2r}} \ar[d]^{≅} \\ HP^{r}/HP^{r−1} \ar[r]_{inclusion} & BSp_{2r}/BSp_{2r−2}}  (7.4)

The motivic weak equivalences of Theorem 7.5 fit into a commutative diagram

\xymatrix{HGr(r, n) \ar[rr]^{inclusion} \ar[d]_{section map} &  & HGr(r, 1 + n) \ar[d]^{quotient} \\ Th 𝒰_{HGr(r,n)} \ar[r]^{∼ mot} & Y_{r,n} & HGr(r, 1 + n)/HGr(r − 1, n) \ar[l]_{∼ mot}}  (7.5)

The section map is the structure map of the Thom space induced by a section of the vector bundle. We have a colimit as n → ∞:

\xymatrix{BSp_{2r} = HGr(r, ∞) \ar[rr]^{shift} \ar[d]_{section map} &  & BSp_{2r} = HGr(r, 1 + ∞) \ar[d]^{quotient} \\ _{MSp_{2r}} \ar[r]^{∼ mot} & Y_{r,∞} & BSp_{2r}/BSp_{2r−2} \ar[l]_{∼ mot}}  (7.6)

The shift map is the map induced by the shift endomorphism of (𝒪^{⊕2}, ω_{2})^{⊕∞} acting on sequences of sections of 𝒪 by

(s_{1}, s_{2}, s_{3}, s_{4}, …) ↦ (0, 0, s_{1}, s_{2}, s_{3}, s_{4}, …).

Lemma 7.6. There is an A^{1}-homotopy of symplectic endomorphisms of (𝒪^{⊕2}, ω_{2})^{⊕∞} linking the shift endomorphism of (𝒪^{⊕2}, ω_{2})^{⊕∞} to the identity.

Proof. Any matrix in any Sp_{2r}(ℤ) is a product of elementary symplectic matrices. So there exists an M(t) ∈ Sp_{4}(ℤ[t]) with

M(0) = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) ,  M(1) = (\begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) ,

for example

M(t) = (\begin{matrix}1 − t^{2} & 0 & −2t + 13t^{3} − 14t^{5} + 4t^{7} & 8t^{2} − 12t^{4} + 4t^{6} \\ 0 & 1 − t^{2} & −2t^{2} + 2t^{4} & −t + 2t^{3} \\ t & 0 & 1 − 7t^{2} + 10t^{4} − 4t^{6} & −4t + 8t^{3} − 4t^{5} \\ 0 & 2t − t^{3} & 2t − 4t^{3} + 2t^{5} & 1 − 3t^{2} + 2t^{4} \end{matrix}) .

Consider the sequence of infinite matrices

f_{n}(t) = (\begin{matrix}I_{2n−2} & 0 & 0 \\ 0 & M(t) & 0 \\ 0 & 0 & I_{∞} \end{matrix}) .

Then f_{n}(1) acts on (𝒪^{⊕2}, ω_{2})^{⊕∞} by exchanging the n^{th} and (n + 1)^{st} summands (𝒪^{⊕2}, ω_{2}), while f_{n}(0) is the identity. The infinite product

F (t) = f_{1}(t)f_{2}(t)f_{3}(t) ⋯

is well-defined because the first 2n columns of f_{1}f_{2} ⋯ f_{N} are independent of N for all N ≥ n. Each column of F (t) contains only a finite number of nonzero entries. (More precisely, writing F (t) = (a_{ij}(t)), we have a_{i,2n−1}(t) = a_{i,2n}(t) = 0 for i > 2n + 2.) The endomorphism F (t) is preserves the symplectic form ω_{2}^{⊕∞} because the automorphisms f_{n}(t) all do. Clearly F (0) is the identity. The finite product f_{1}(1)f_{2}(1) ⋯ f_{N}(1) permutes cyclically the first N + 1 summands of (𝒪^{⊕2}, ω_{2})^{∞} and fixes the others. The infinite product F (1) is the shift map. □

Theorem 7.7. In the motivic unstable homotopy category H_{•}(S) we have a commutative diagram

[figure: BSp_{2r}; structure map; quotient; _{MSp_{2r}}; ≅; A^{N} -bundles and excision; BSp_{2r}/BSp_{2r−2}.]

8. The quaternionic projective bundle theorem

The most basic form a symplectic orientation is a symplectic Thom structure [13, Definition 7.1]. The version of the definition for bigraded ϵ-commutative theories is as follows.

Definition 8.1. A symplectic Thom structure on a bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) on 𝒮m𝒪p/S is a rule which assigns to each rank 2 symplectic bundle (E, ϕ) over an X in 𝒮m/S an element th(E, ϕ) ∈ A^{4,2}(E, E −X) with the following properties:

(1) For an isomorphism u: (E, ϕ) ≅ (E_{1}, ϕ_{1}) one has th(E, ϕ) = u^{∗} th(E_{1}, ϕ_{1}).

(2) For a morphism f: Y → X with pullback map f_{E}: f^{∗}E → E one has f_{E}^{∗} th(E, ϕ) = th(f^{∗}E, f^{∗}ϕ).

(3) For the trivial rank 2 bundle A^{2} → pt with the symplectic form ω_{2} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) the map

− × th(A^{2}, ω_{2}): A^{∗,∗}(X) → A^{∗+4,∗+2}(X × A^{2}, X × (A^{2} − 0))

is an isomorphism for all X. The Pontryagin class of (E, ϕ) is p_{1}(E, ϕ) = −z^{∗} th(E, ϕ) ∈ A^{4,2}(X) where z: X → E is the zero section.

The quaternionic projective bundle theorem is proven in [13] using the symplectic Thom structure and not any other version of a symplectic orientation. It is proven first for trivial bundles.

Theorem 8.2 ([13, Theorem 8.1]). Let (A^{∗,∗}, ∂, ×, 1_{A}) be a bigraded ϵ-commutative ring cohomology theory with a symplectic Thom structure. Let (𝒰_{HP^{n}}, ϕ_{HP^{n}}) be the tautological rank 2 symplectic subbundle over HP^{n} and ζ = p_{1}(𝒰_{HP^{n}}, ϕ_{HP^{n}}) its Pontryagin class. Then for any X in 𝒮m/S we have an isomorphism of bigraded rings

A^{∗,∗}(HP^{n} × X) ≅ A^{∗,∗}(X)[ζ]/(ζ^{n+1}).

A Mayer-Vietoris argument gives the more general theorem [13, Theorem 8.2].

Theorem 8.3 (Quaternionic projective bundle theorem). Let (A^{∗,∗}, ∂, ×, 1_{A}) be a bigraded ϵ-commutative ring cohomology theory with a symplectic Thom structure. Let (E, ϕ) be a symplectic bundle of rank 2n over X, let (𝒰, ϕ|_{𝒰}) be the tautological rank 2 symplectic subbundle over the quaternionic projective bundle HP (E, ϕ), and let ζ = p_{1}(𝒰, ϕ|_{𝒰}) be its Pontryagin class. Then we have an isomorphism of bigraded A^{∗,∗}(X)-modules

(1, ζ, …, ζ^{n−1}): A^{∗,∗}(X) ⊕ A^{∗,∗}(X) ⊕ ⋯ ⊕ A^{∗,∗}(X) → A^{∗,∗}(HP (E, ϕ)).

Definition 8.4. Under the hypotheses of Theorem 8.3 there are unique elements p_{i}(E, ϕ) ∈ A^{4i,2i}(X) for i = 1, 2, …, n such that

ζ^{n} − p_{1}(E, ϕ) ∪ ζ^{n−1} + p_{2}(E, ϕ) ∪ ζ^{n−2} − ⋯ + (−1)^{n}p_{n}(E, ϕ) = 0.

The classes p_{i}(E, ϕ) are called the Pontryagin classes of (E, ϕ) with respect to the symplectic Thom structure of the cohomology theory (A, ∂). For i > n one sets p_{i}(E, ϕ) = 0, and one sets p_{0}(E, ϕ) = 1.

For a rank 2 symplectic bundle (E, ϕ) the classes p_{1}(E, ϕ) defined by Definitions 8.1 and 8.4 coincide.

Corollary 8.5. The Pontryagin classes of a trivial symplectic bundle vanish.

Among the consequences of the quaternionic projective bundle theorem is the symplectic splitting principle [13, Theorem 10.2]. We used it to prove the Cartan sum formula for Pontryagin classes [13, Theorem 10.5].

Theorem 8.6. Let (A^{∗,∗}, ∂, ×, 1_{A}) be a bigraded ϵ-commutative ring cohomology theory with a symplectic Thom structure. Suppose (F, ψ) ≅ (E_{1}, ϕ_{1}) ⊕ (E_{2}, ϕ_{2}) is an orthogonal direct sum of symplectic bundles over an X in 𝒮m/S. Then for all i we have

p_{i}(F, ψ) = p_{i}(E_{1}, ϕ_{1}) + ∑_{j=1}^{i−1} p_{i−j}(E_{1}, ϕ_{1})p_{j}(E_{2}, ϕ_{2}) + p_{i}(E_{2}, ϕ_{2}).  (8.1)

The quaternionic projective bundle theorem also allowed us to compute the cohomology of quaternionic Grassmannians. To explain our results we need to recall a number of facts about symmetric polynomials. They may be found in for example [7, Chap. 1, §§1–3].

Let Λ_{r} ⊂ ℤ[x_{1}, …, x_{r}] be the ring of symmetric polynomials in r variables. Let e_{i} denote the i^{th} elementary symmetric polynomial, and h_{i} the i^{th} complete symmetric polynomial, the sum of all the monomials of degree i. Set e_{0} = h_{0} = 1 and e_{i} = h_{i} = 0 for i < 0 and also e_{i} = 0 for i > r. We have Λ_{r} = ℤ[e_{1}, …, e_{r}]. There is a recurrence relation h_{m} +∑_{i=1}^{r}(−1)^{r}e_{i}h_{m−i} = 0.

Let

Π_{r} = {partitions λ = (λ_{1}, λ_{2}, …, λ_{r}) of length l(λ) ≤ r}.

Write δ = (r − 1, r − 2, …, 1, 0). For λ ∈ Π_{r} let a_{λ+δ} = det(x_{i}^{λ_{j}+r−j})_{1≤i,j≤r}. Then a_{λ+δ} is a skew-symmetric polynomial and therefore divisible by the Vandermonde determinant a_{δ}. The quotient s_{λ} = a_{λ+δ}/a_{δ} is the Schur polynomial for λ. It is symmetric of degree |λ| = ∑ λ_{i}. One has s_{(1^{i})} = e_{i} and s_{(i)} = h_{i}. The a_{λ+δ} with l(λ) ≤ r form a ℤ-basis of the skew-symmetric polynomials in r variables, so the s_{λ} with l(λ) ≤ r form a ℤ-basis of Λ_{r}. Denote by λ′ the partition dual to λ. We have formulas

s_{λ} = det(e_{λ′_{i}−i+j})_{1≤i,j≤m} = det(h_{λ_{i}−i+j})_{1≤i,j,≤r},  (8.2)

for m ≥ l(λ′) and r ≥ l(λ). Set

Π_{r,n−r} = {partitions λ of length l(λ) = λ′_{1} ≤ r and with λ_{1} ≤ n − r}

The set Π_{r,n−r} has \binom{n}{r} members. We will use the following results.

Proposition 8.7. The quotient map

ℤ[e_{1}, …, e_{r}] → ℤ[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n})

sends {s_{λ} | λ ∈ Π_{r} − Π_{r,n−r}} ↦ 0, and it sends {s_{λ} | λ ∈ Π_{r,n−r}} onto a homogeneous ℤ-basis of the quotient ring.

Proof. For λ ∈ Π_{r} − Π_{r,n−r} the first line of the determinant s_{λ} = det(h_{λ_{i}−i+j}) consists of h_{k} with k ≥ n − r + 1. These are all in (h_{n−r+1}, …, h_{n}) because of the recurrence relation satisfied by the h_{k}. So they are sent to 0 in the quotient.

The rank of the quotient as a ℤ-module is ∏ deg h_{i}/ ∏ deg e_{i} = \binom{n}{r}. Since this is the same as the cardinality of Π_{r,n−r}, and since the images of the s_{λ} with λ ∈ Π_{r,n−r} generate the quotient as a ℤ-module, they form a ℤ-basis of the quotient. □

Let the \overline{e}_{i} and the \overline{h}_{i} be, respectively, the elementary and complete symmetric polynomials in r − 1 variables. The natural quotient map sends e_{i} ↦ \overline{e}_{i} for i < r and e_{r} ↦ 0, while it sends h_{i} ↦ \overline{h}_{i} for all i.

Proposition 8.8. The kernel of the quotient map

ℤ[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}) → ℤ[\overline{e}_{1}, …, \overline{e}_{r−1}]/(\overline{h}_{n−r+1}, …, \overline{h}_{n−1}) → 0

is the image of the injection

0 → ℤ[e_{1}, …, e_{r}]/(h_{n−r}, …, h_{n−1}) ⟶^{e_{r}×−} ℤ[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}).

Proof. The kernel is the free ℤ-module with basis {s_{λ} | λ ∈ Π_{r,n−r} − Π_{r−1,n−r}}. These are the λ with λ_{r} ≥ 1 and thus λ′_{1} = r. The formula s_{λ} = det(e_{λ′_{i}−i+j}) shows that for such λ one has s_{λ} = e_{r}s_{μ} with μ = (λ_{1} − 1, …, λ_{r} − 1) ∈ Π_{r,n−r−1}. These s_{μ} form a basis of the ring on the left of the second displayed line. □

Theorem 8.9 ([13, Theorem 11.2]). Let (𝒰_{r,n}, ϕ_{r,n}) be the tautological symplectic bundle of rank 2r on HGr(r, n). Then for any bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) with a symplectic Thom structure and any X in 𝒮m/S the map

A^{∗,∗}(X)[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}) ⟶^{≅} A^{∗,∗}(HGr(r, n) × X)  (8.3)

sending e_{i} ↦ p_{i}(𝒰_{r,n}, ϕ_{r,n}) for all i is an isomorphism of bigraded rings.

Theorem 8.10 ([13, Theorem 11.4]). Let α_{r,n}: HGr(r, n) ↪ HGr(r, n + 1) be the usual inclusion. For any bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) with a symplectic Thom structure and any X in 𝒮m/S the map

(α_{r,n} × 1)^{∗}: A^{∗,∗}(HGr(r, n + 1) × X) → A^{∗,∗}(HGr(r, n) × X)

is a surjection which the isomorphisms (8.3) identify with the natural surjection

A^{∗,∗}(X)[e_{1}, …, e_{r}]/(h_{n−r+2}, …, h_{n}, h_{n+1}) → A^{∗,∗}(X)[e_{1}, …, e_{r}]/(h_{n−r+1}, h_{n−r+2}, …, h_{n}).

Theorem 8.11 ([13, Theorem 11.4]). Let β_{r,n}: HGr(r, n) → HGr(1 + r, 1 + n) be the usual inclusion. For any bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) with a symplectic Thom structure and any X in 𝒮m/S the map

(β_{r,n} × 1)^{∗}: A^{∗,∗}(HGr(1 + r, 1 + n) × X) → A^{∗,∗}(HGr(r, n) × X)

is a surjection which the isomorphisms (8.3) identify with the surjection

A^{∗,∗}(pt)[e_{1}, …, e_{r}, e_{r+1}]/(h_{n−r+1}, …, h_{n}, h_{n+1}) → A^{∗,∗}(pt)[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n})

of A^{∗,∗}(pt)[e_{1}, …, e_{r}]-algebras sending e_{r+1} ↦ 0.

9. The cohomology of BSp_{2r} and MSp_{2r}

Theorem 9.1. Let (A, μ, e) be a commutative T -ring spectrum with a symplectic Thom structure on A^{∗,∗}. Then the isomorphism BSp_{2r} = colim_{n} HGr(r, n) induces isomorphisms

A^{∗,∗}(BSp_{2r}) ⟶^{≅} \underleftarrow{lim}_{n→∞} A^{∗,∗}(HGr(r, n)) ⟵^{≅} A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom}

of bigraded rings. The second isomorphism sends the variable p_{i} to the inverse system of i^{th} Pontryagin classes (p_{i}(𝒰_{r,n}))_{n≥r}.

Here A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom} is the bigraded ring of homogeneous power series. The \underleftarrow{lim} is taken in the category of bigraded rings.

Proof. We have BSp_{2r} = colim_{n} HGr(r, n), so by Theorem 5.7 we have an exact sequence

0 → \underleftarrow{lim}^{1}_{n∈ℕ} A^{∗−1,∗}(HGr(r, n)) → A^{∗,∗}(BSp_{2r}) → \underleftarrow{lim}_{n∈ℕ} A^{∗,∗}(HGr(r, n)) → 0.

The connecting maps are the α_{r,n}^{∗} of Theorem 8.10, which are surjective. So the \underleftarrow{lim}^{1} vanishes, and the first map of the statement of the theorem is an isomorphism.

Let I_{d} ⊂ A^{∗,∗}(pt)[p_{1}, …, p_{r}] be the two-sided ideal generated by the monomials p_{1}^{a_{1}}p_{2}^{a_{2}} … p_{r}^{a_{r}} with ∑ ia_{i} ≥ d. We have inclusions I_{r(n−r)+1} ⊂ (h_{n−r+1}, …, h_{n}) ⊂ I_{n−r+1} because all partitions λ ∈ Π_{r,n−r} have |λ| ≤ r(n − r). So we have

\underleftarrow{lim}_{n} A^{∗,∗}(HGr(r, n)) ≅ \underleftarrow{lim}_{n} A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+1}, …, h_{n})

≅ \underleftarrow{lim}_{d} A^{∗,∗}(pt)[p_{1}, …, p_{r}]/I_{d} = A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom}. □

Theorem 9.2. Let (A, μ, e) be a commutative T -ring spectrum with a symplectic Thom structure on A^{∗,∗}. Then for any r and n the motivic homotopy equivalence

Th 𝒰_{HGr(r,n)} ≅ HGr(r, 1 + n)/HGr(r − 1, n)

of Theorem 7.5 induces isomorphisms

A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+1}, …, h_{n}) ⟶_{≅}^{∪p_{r}} A^{∗+4r,∗+2r}(Th 𝒰_{HGr(r,n)})  (9.1)

of two-sided bigraded modules over A^{∗,∗}(HGr(r, n)) ≅ A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+1}, …, h_{n}). Moreover, the isomorphism MSp_{2r} = colim_{n} Th 𝒰_{HGr(r,n)} induces isomorphisms of bigraded modules over A^{∗,∗}(BSp_{2r}) ≅ A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom}

A^{∗+4r,∗+2r}(MSp_{2r}) ⟶^{≅} \underleftarrow{lim}_{n→∞} A^{∗+4r,∗+2r}(Th 𝒰_{HGr(r,n)}) ⟵^{≅} p_{r}A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]^{hom} (9.2)

Proof. Applying A^{∗,∗} to the cofiber sequence

HGr(r − 1, n) ⟶^{β_{r−1,n}} HGr(r, 1 + n) ⟶ Th 𝒰_{HGr(r,n)}

gives a long exact sequence of cohomology. The map β_{r−1,n} induces a surjection of cohomology groups by Theorem 8.11, so we have A^{∗,∗}(Th 𝒰_{HGr(r,n)}) ≅ ker β_{r−1,n}^{∗}. This kernel is identified by Proposition 8.8, giving the isomorphism (9.1). In principle this is an isomorphism of twosided modules over A^{∗,∗}(HGr(r, 1 + n)) ≅ A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+2}, …, h_{n+1}). But the modules are annihilated by h_{n−r+1}, so they are also two-sided modules over the quotient ring A^{∗,∗}(HGr(r, n)).

The inclusion Th 𝒰_{HGr(r,n)} ↪ Th 𝒰_{HGr(r,n+1)} induces a commutative diagram

\xymatrix{A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+2}, …, h_{n}, h_{n+1}) \ar[r]_{≅}^{∪p_{r}} \ar@{->>}[d] & A^{∗+4r,∗+2r}(Th 𝒰_{HGr(r,n+1)}) \ar@{->>}[d] \\ A^{∗,∗}(pt)[p_{1}, …, p_{r}]/(h_{n−r+1}, h_{n−r+1} …, h_{n}) \ar[r]_{≅}^{∪p_{r}} & A^{∗+4r,∗+2r}(Th 𝒰_{HGr(r,n)}).}

The inverse limit gives the righthand isomorphism of (9.2). The vertical map on the left is surjective, so the vertical map on the right is as well. Therefore the \underleftarrow{lim}^{1} vanishes in the exact sequence

0 → \underleftarrow{lim}^{1}_{n∈ℕ} A^{∗−1,∗}(Th 𝒰_{HGr(r,n)}) → A^{∗,∗}(MSp_{2r}) → \underleftarrow{lim}_{n∈ℕ} A^{∗,∗}(Th 𝒰_{HGr(r,n)}) → 0.

obtained from Theorem 5.7. □

For any r let z_{2r}: BSp_{2r} → Th 𝒰_{BSp_{2r}} = MSp_{2r} be the structure map induced by the zero section of the tautological symplectic bundle 𝒰_{BSp_{2r}} → BSp_{2r}.

Theorem 9.3. Let (A, μ, e) be a commutative T -ring spectrum with a symplectic Thom structure on A^{∗,∗}. The map z_{2r}^{∗}: A^{∗,∗}(MSp_{2r}) → A^{∗,∗}(BSp_{2r}) of two-sided A^{∗,∗}(BSp_{2r})- modules is injective and identifies A^{∗,∗}(MSp_{2r}) with the two-sided principal ideal generated by p_{r} ∈ A^{4r,2r}(BSp_{2r}).

Proof. By Theorem 7.7 we have cofiber sequence BSp_{2r−2} ⟶^{i_{2r}} BSp_{2r} ⟶^{z_{2r}} MSp_{2r} yielding a long exact sequence of cohomology groups. By the previous theorems these are isomorphic to (in simplified notation)

⋯ → A^{∗−4r,∗−2r}[[p_{1}, …, p_{r}]] ⟶^{z_{2r}^{∗}} A^{∗,∗}[[p_{1}, …, p_{r}]] ⟶^{i_{2r}^{∗}} A^{∗,∗}[[p_{1}, …, p_{r−1}]] → ⋯ (9.3)

Since i_{2r} is the colimit of the inclusion maps β_{r−1,n}: A(r − 1, n) → A(r, 1 + n) of Theorem 8.11, i_{2r}^{∗} is the quotient by the ideal generated by p_{r}. It is surjective in all bidegrees. It follows that z_{2r}^{∗} is injective in all bidegees and is the inclusion of that ideal. □

The direct sum of symplectic bundle induces compatible monoid structures on the BSp_{2r} and the MSp_{2r}. So the following diagram commutes for all r and s

\xymatrix{BSp_{2r} × BSp_{2s} \ar[r]^{m_{rs}} \ar[d]_{z_{2r}×z_{2s}} & BSp_{2r+2s} \ar[d]^{z_{2r+2s}} \\ MSp_{2r} ∧ MSp_{2s} \ar[r]^{μ_{rs}} & MSp_{2r+2s}.}  (9.4)

Theorem 9.4. Let (A, μ, e) be a commutative T -ring spectrum with a symplectic Thom structure on A^{∗,∗}. Then the isomorphisms

BSp_{2r} × BSp_{2s} = colim_{n}(HGr(r, rn) × HGr(s, sn))
MSp_{2r} ∧ MSp_{2s} = colim_{n}(Th 𝒰_{HGr(r,rn)} ∧ Th 𝒰_{HGr(s,sn)})

induces a commutative diagram of isomorphisms and monomorphisms of two-sided graded A^{∗,∗}(BSp_{2r} × BSp_{2s})-modules

\xymatrix{A^{∗,∗}(MSp_{2r} ∧ MSp_{2s}) \ar[r]^{≅} \ar@{^{(}->}[d]_{(z_{2r}×z_{2s})^{∗}} & \underleftarrow{lim}_{n→∞} A^{∗,∗}(Th 𝒰_{r,rn} ∧ Th 𝒰_{s,sn}) \ar@{^{(}->}[d] & p′_{r}p′′_{s}A^{∗,∗}[[p′_{1}, …, p′_{r}, p′′_{1}, …, p′′_{s}]] \ar[l]_{≅} \ar@{^{(}->}[d]^{inclusion} \\ A^{∗,∗}(BSp_{2r} × BSp_{2s}) \ar[r]^{≅} & \underleftarrow{lim}_{n→∞} A^{∗,∗}(HGr(r, rn) × HGr(s, sn)) & A^{∗,∗}[[p′_{1}, …, p′_{r}, p′′_{1}, …, p′′_{s}]]. \ar[l]_{≅}}

Moreover, these isomorphisms identify the diagram obtained by applying A^{∗,∗} to (9.4) with the diagram of rings and ideals

\xymatrix{p_{r+s}A^{∗,∗}(pt)[[p_{1}, …, p_{r+s}]]^{hom} \ar[r] \ar@{^{(}->}[d] & p′_{r}p′′_{s}A^{∗,∗}(pt)[[p′_{1}, …, p′_{r}, p′′_{1}, …, p′′_{s}]]^{hom} \ar@{^{(}->}[d] \\ A^{∗,∗}(pt)[[p_{1}, …, p_{r+s}]]^{hom} \ar[r] & A^{∗,∗}(pt)[[p′_{1}, …, p′_{r}, p′′_{1}, …, p′′_{s}]]^{hom}}

where the horizontal maps send p_{i} ↦ p′_{i} + ∑_{j=1}^{i−1} p′_{i−j}p′′_{j} + p′′_{i}. Moreover, the horizontal maps of the last diagram are also injective.

Proof. The construction of the first diagram is much the same as in the previous theorems. The second diagram follows. For the last statement of the theorem, let t_{1}, …, t_{r+s} be independent indeterminates of bidegree (2, 1). Then the composition of the bottom horizontal map with the map

A^{∗,∗}(pt)[[p′_{1}, …, p′_{r}, p′′_{1}, …, p′′_{s}]]^{hom} → A^{∗,∗}(pt)[[t_{1}, …, t_{r+s}]]^{hom}

sending p′_{i} ↦ e_{i}(t_{1}, …, t_{r}) and p′′_{j} ↦ e_{j}(t_{r+1}, …, t_{r+s}) is the inclusion of the ring of symmetric homogeneous power series in the ring of homogeneous power series. That is injective. □

The final calculation in this section ought to be that of A^{∗,∗}(MSp). However, we will put this off until Theorem 13.1 because we wish to make the calculation using a symplectic Thom classes theory and not just a symplectic Thom structure.

10. Tautological Thom elements

Suppose that (A, μ, e) is a commutative T -ring spectrum. Let ϑ ∈ A^{4,2}(MSp_{2}).

We associate to ϑ and a symplectic bundle (E, ϕ) of rank 2 over an X in 𝒮m/S a class th^{ϑ}(E, ϕ) defined as follows. By assumption the scheme X admits an ample family of line bundles. So there exists an affine bundle f: Y → X with Y an affine scheme. Then for some p there exist global sections s_{1}, …, s_{p} of f^{∗}E^{∨} generating f^{∗}E^{∨}. There then exist global functions a_{ij} on Y such that f^{∗}ϕ = ∑_{1≤i<j≤p} a_{ij}s_{i} ∧ s_{j}. We set t_{i} = ∑_{j=i+1}^{p} a_{ij}s_{j} so that we have ∑_{i} s_{i} ∧ t_{i} = f^{∗}ϕ. The map (s_{1}, t_{1}, …, s_{p}, t_{p}): f^{∗}E → 𝒪_{Y}^{⊕2p} embeds (f^{∗}E, f^{∗}ϕ) as a symplectic subbundle of (𝒪_{Y}^{⊕2p}, ω_{2p}). So it is classified by a map ψ: Y → HGr(1, p) = HP^{p−1} such that ψ^{∗}(𝒯Sp_{1,p}, ϕ_{1,p}) = f^{∗}(E, ϕ). This gives us maps of (ind)-schemes

X ⟵_{A^{N} -bundle}^{f} Y ⟶^{ψ} HP^{p−1} ⟶^{inclusion} BSp_{2} = HP^{∞}  (10.1)

and of pointed motivic spaces

Th E ⟵_{∼mot}^{\overline{f}} Th f^{∗}E ≅ Th ψ̃^{∗}𝒯Sp_{1,p} ⟶^{\overline{ψ}} Th 𝒯Sp_{1,p}  (10.2)

of pointed motivic spaces, which can be composed with the maps

Th 𝒯Sp_{1,p} ⟶^{inclusion} MSp_{2} ⟶^{ϑ} A ∧ T^{∧2}.  (10.3)

in SH(S). The composition of (10.2) and (10.3) gives a class

th^{ϑ}(E, ϕ) ∈ Hom_{SH(S)}(Th E, A ∧ T^{∧2}) = A^{4,2}(E, E − X).

The following lemma is proven in the same way as Lemma 5.4.

Lemma 10.1. The classes th^{ϑ}(E, ϕ) depend only on the rank 2 symplectic bundle (E, ϕ) and the morphism ϑ: Σ_{T}^{∞}MSp_{2}(−2) → A in SH(S).

Recall the inclusion e_{2}^{Sp}: T^{∧2} → MSp_{2} of (6.2).

Theorem 10.2. Let (A, μ, e) be a commutative T -ring spectrum. Then the map which assigns to a class ϑ as above the family of classes th^{ϑ}(E, ϕ) is a bijection between the sets of

(α) classes ϑ ∈ A^{4,2}(MSp_{2}) with ϑ|_{T^{∧2}} = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}), and

(a) symplectic Thom structures on the bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) such that for the trivial rank 2 bundle A^{2} → pt we have th(A^{2}, ω_{2}) = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}).

Proof. A proof similar to that of Theorem 5.5 shows that for a ϑ as in (α), the family of classes th^{ϑ}(E, ϕ) form a symplectic Thom structure with the stated normalization condition. Note that this uses the fact that all symplectic bundles are locally trivial in the Zariski topology.

Now suppose we have a symplectic Thom structure with the stated normalization condition. For every n the tautological rank 2 symplectic bundle bundle over HP^{n−1} has a Thom class which we will abbreviate as

th_{n} = th(𝒰_{HP^{n−1}}, ϕ_{HP^{n−1}}) ∈ A^{4,2}(Th 𝒰_{HP^{n−1}}).

Pullback along the inclusion Th 𝒰_{HP^{n−1}} → Th 𝒰_{HP^{n}} sends th_{n+1} ↦ th_{n}. So as n varies, we get an element

ϑ̄ = (th_{n})_{n∈ℕ} ∈ \underleftarrow{lim} A^{4,2}(Th 𝒰_{HP^{n−1}}).

We have MSp_{2} = colim Th 𝒰_{HP^{n−1}}, and by Theorem 9.2 the natural map

A^{∗,∗}(MSp_{2}) ⟶^{≅} \underleftarrow{lim} A^{∗,∗}(Th 𝒰_{HP^{n−1}}).

is an isomorphism. Let ϑ ∈ A^{4,2}(MSp_{2}) be the unique class lifting ϑ̄. As in the proof of Theorem 5.9 we have th^{ϑ}(E, ϕ) = th(E, ϕ) for all rank 2 symplectic bundles. Moreover, for ϑ and ξ in A^{4,2}(MSp_{2}) we have th^{ϑ}(E, ϕ) = th^{ξ}(E, ϕ) for all symplectic bundles if and only if ϑ and ξ have the same image in the inverse limit. But that happens only for ϑ = ξ. □

Definition 10.3. The class ϑ ∈ A^{4,2}(MSp_{2}) is the tautological Thom element of the symplectic orientation on A^{∗,∗} whose rank 2 symplectic Thom classes are the th^{ϑ}(E, ϕ).

The canonical morphism u_{2} : Σ_{T}^{∞}MSp_{2}(−2) → MSp which is part of the counit of the adjunction between Σ_{T}^{∞}(−2) and its right adjoint the forgetful functor Ev_{2} defines an element ϑ_{MSp} ∈ MSp^{4,2}(MSp_{2}). It satisfies ϑ_{MSp}|_{T^{∧2}} = Σ_{T}^{2} 1_{MSp} because both elements correspond to the composition u_{2} ∘ e_{2}^{Sp}: T^{∧2} → MSp ∧ T^{∧2}.

Definition 10.4. The standard symplectic Thom structure on MSp^{∗,∗} is the one whose universal Thom element is the ϑ_{MSp} we have just described.

11. Tautological Pontryagin elements

The bigraded version of the definition of a Pontryagin structure [13, Definition 12.1] is as follows.

Definition 11.1. A Pontryagin structure on a bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) on 𝒮m𝒪p/S is a rule which assigns to each rank 2 symplectic bundle (E, ϕ) over an X in 𝒮m/S an element p_{1}(E, ϕ) ∈ A^{4,2}(X) with the following properties:

(1) For (E_{1}, ϕ_{1}) ≅ (E_{2}, ϕ_{2}) we have p_{1}(E_{1}, ϕ_{1}) = p_{1}(E_{2}, ϕ_{2}).

(2) For a morphism f: Y → X we have f^{∗}p_{1}(E, ϕ)) = p_{1}(f^{∗}E, f^{∗}ϕ).

(3) For the tautological rank 2 symplectic subbundle (𝒰_{HP^{1}}, ϕ_{HP^{1}}) on HP^{1} the map

(1, p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}})): A^{∗,∗}(X) ⊕ A^{∗−4,∗−2}(X) → A^{∗,∗}(HP^{1} × X)

is an isomorphism for all X in 𝒮m/S.

(4) For the trivial rank 2 symplectic bundle (A^{2}, ω_{2}) over pt we have p_{1}(A^{2}, ω_{2}) = 0 in A^{4,2}(pt).

The Pontryagin classes associated to a symplectic Thom structure by the formula p_{1}(E, ϕ) = −z^{∗} th(E, ϕ) ∈ A^{4,2}(X) of Definition 8.1 form a Pontryagin structure because of the functoriality of the Thom classes, the quaternionic projective bundle theorem 8.2 and Corollary 8.5.

For r = 1 the diagram (7.4) of morphisms in H_{•}(S) becomes

\xymatrix{T^{∧2} \ar[r]^{e_{1}^{Sp}} \ar[d]_{≅} & _{MSp_{2}} \ar[d]^{≅} \\ (HP^{1}, h_{∞}) \ar[r]_{inclusion} & (HP^{∞}, h_{∞})}  (11.1)

with h_{∞} = pt → HP^{1} a point such that h_{∞}^{∗}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) is a trivial symplectic bundle. We will call the two vertical arrows the canonical motivic homotopy equivalences.

Suppose that (A, μ, e) is a commutative T -ring spectrum. Let ϱ ∈ A^{4,2}(HP^{∞}, h_{∞}). For a rank 2 symplectic bundle (E, ϕ) over X the construction of (10.1) composed with the quotient by the pointing and with ϱ gives us a zigzag

X ⟵_{∼}^{A^{N} -bundle} Y → HP^{∞} → (HP^{∞}, h_{∞}) ⟶^{ϱ} A ∧ T^{∧2}  (11.2)

in which the pullbacks to Y of (E, ϕ) and of (𝒰_{HP^{∞}}, ϕ_{HP^{∞}}) are isomorphic. The composition is a class p_{1}^{ϱ}(E, ϕ) ∈ A^{4,2}(X). This class depends only on (E, ϕ) and ϱ by arguments similar to those of Lemmas 5.4 and 10.1.

Theorem 11.2. Let (A, μ, e) be a commutative T -ring spectrum. Then the map which assigns to a class ϱ as above the family of classes p_{1}^{ϱ}(E, ϕ) is a bijection between the sets of

(β) classes ϱ ∈ A^{4,2}(HP^{∞}, h_{∞}) with ϱ|_{HP^{1}} ∈ A^{4,2}(HP^{1}, h_{∞}) corresponding to −Σ_{T}^{2} 1_{A} ∈ A^{4,2}(T^{∧2}) under the canonical motivic homotopy equivalence (HP^{1}, h_{∞}) ≅ T^{∧2}, and

(b) Pontryagin structures on (A^{∗,∗}, ∂, ×, 1_{A}) for which p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) ∈ A^{4,2}(HP^{1}, h_{∞}) ⊂ A^{4,2}(HP^{1}) corresponds to −Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}) under the canonical motivic homotopy equivalence (HP^{1}, h_{∞}) ≃ T^{∧2}.

The proof is like that of Theorem 10.2. The classes p_{1}^{ϱ}(E, ϕ) satisfy condition (3) of Definition 11.1 because of an argument like Lemma 5.3 and the isomorphism T^{∧2} ≅ (HP^{1}, h_{∞}). They satisfy condition (4) because p_{1}^{ϱ}(A^{2}, ω_{2}) = h_{∞}^{∗}ϱ = 0. The proof that there is a unique ϱ corresponding to each Pontryagin structure invokes the isomorphism A^{∗,∗}(HP^{∞}) ≅ \underleftarrow{lim} A^{∗,∗}(HP^{n}) which is the case r = 1 of Theorem 9.1.

Definition 11.3. The class ϱ ∈ A^{4,2}(HP^{∞}, h_{∞}) is the tautological Pontryagin element of the symplectic orientation on A^{∗,∗} whose rank 2 Pontryagin classes are the p_{1}^{ϱ}(E, ϕ).

Theorem 11.4. Let (A, μ, e) be a commutative T -ring spectrum. Then the canonical motivic homotopy equivalence MSp_{2} ≅ (HP^{∞}, h_{∞}) plus change-of-sign gives a bijection between the sets of

(α) the tautological Thom elements ϑ of Theorem 10.2 and

(β) the tautological Pontryagin elements ϱ of Theorem 11.2.

The composition (a) ↔ (α) ↔ (β) ↔ (b) with the bijections of Theorems 10.2 and 11.2 is the same as the rule which assigns to a symplectic Thom structure with classes th(E, ϕ) the Pontryagin structure with classes p_{1}(E, ϕ) = −z^{∗} th(E, ϕ) for z: X → Th E the structural map of the Thom space.

Proof. The first statement follows from the existence and compatibility of the canonical motivic homotopy equivalences of (11.1). For the second, given a rank 2 symplectic bundle (E, ϕ) on X we have a diagram

\xymatrix{X \ar[d]_{z} & Y \ar[l]_{f}^{∼} \ar[r] \ar[d] & HP^{∞} \ar[r] \ar[d] & (HP^{∞}, h_{∞}) \ar[d]^{ϱ} \ar@{-->}[dl]_{≅} \\ Th E & Th f^{∗}E \ar[l]_{∼} \ar[r] & _{MSp_{2}} \ar[r]^{−ϑ} & A ∧ T^{∧2}}

in which the squares commute by compatibility of the structural maps of Thom spaces with pullbacks, the upper triangle commutes by Theorem 7.7, and the lower triangle commutes because of the rule giving the bijection (α) ↔ (β). We deduce the equality p_{1}^{ϱ}(E, ϕ) = −z^{∗} th^{ϑ}(E, ϕ) in Hom_{SH(S)}(X, A ∧ T^{∧2}) = A^{4,2}(X). □

The bigraded version of the definition of a Pontryagin classes theory [13, Definition 14.1] is as follows.

Definition 11.5. A Pontryagin classes theory on a bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) on 𝒮m𝒪p/S is a rule assigning to every symplectic bundle (F, ψ) over every X in 𝒮m/S elements p_{i}(F, ψ) ∈ A^{4i,2i}(X) for all i ≥ 1 satisfying

(1) For (F_{1}, ψ_{1}) ≅ (F_{2}, ψ_{2}) we have p_{i}(F_{1}, ψ_{1}) = p_{i}(F_{2}, ψ_{2}) for all i.

(2) For a morphism f: Y → S we have f^{∗}p_{i}(F, ψ) = p_{i}(f^{∗}E, f^{∗}ψ) for all i.

(3) For the tautological rank 2 symplectic subbundle (𝒰_{HP^{1}}, ϕ_{HP^{1}}) on HP^{1} the maps

(1, p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}})): A^{∗,∗}(X) ⊕ A^{∗−4,∗−2}(X) → A^{∗,∗}(HP^{1} × X)

are isomorphisms for all X.

(4) For the trivial rank 2 symplectic bundle (A^{2}, ω_{2}) over pt we have p_{1}(A^{2}, ω_{2}) = 0 in A^{4,2}(pt).

(5) For an orthogonal direct sum of symplectic bundles (F, ψ) ≅ (F_{1}, ψ_{1}) ⊕ (F_{2}, ψ_{2}) we have p_{i}(F, ψ) = p_{i}(F_{1}, ψ_{1}) + ∑_{j=1}^{i−1} p_{i−j}(F_{1}, ψ_{1})p_{j}(F_{2}, ψ_{2}) + p_{i}(F_{2}, ψ_{2}) for all i.

(6) For (F, ψ) of rank 2r we have p_{i}(F, ψ) = 0 for i > r.

One may also set p_{0}(F, ψ) = 1 and even p_{i}(F, ψ) = 0 for i < 0.

Definition 8.4 associates Pontryagin classes to a symplectic Thom structure on (A^{∗,∗}, ∂, ×, 1_{A}). They form a Pontryagin classes theory because the quaternionic projective bundle Theorems 8.2 and 8.3, Corollary 8.5 and the Cartan sum formula (Theorem 8.6).

Theorem 11.6. Let (A, μ, e) be a commutative T -ring spectrum. Then the forgetful map gives a bijection between the sets of

(c) Pontryagin classes theories on (A^{∗,∗}, ∂, ×, 1_{A}) with the normalization condition on p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) of Theorem 11.2 and

(b) Pontryagin structures on (A^{∗,∗}, ∂, ×, 1_{A}) with the same normalization condition.

The inverse bijection is given by assigning to a Pontryagin structure first the symplectic Thom structure associated to it by Theorem 11.4 and then the the Pontryagin classes theory associated to the symplectic Thom structure by Definition 8.4.

Proof. The chain of associations (b) → (a) → (c) → (b) gives the identity because for rank 2 symplectic bundles the classes p_{1}(E, ϕ) given in Definitions 8.1 and 8.4 coincide.

The chain of associations (c) → (b) → (a) → (c) gives the identity because for a symplectic bundle (F, ψ) of rank 2r on X if we let π: HP (F, ψ) → X be the associated quaternionic projective bundle with rank 2 tautological subbundle (𝒰, ϕ), then from the orthogonal direct sum π^{∗}(F, ψ) = (𝒰, ϕ) ⊕ (𝒰, ϕ)^{⊥} and the axioms we get

0 = (−1)^{r}p_{r}((𝒰, ϕ)^{⊥}) = p_{1}(𝒰, ϕ)^{r} − π^{∗}p_{1}(F, ψ) ∪ p_{1}(𝒰, ϕ)^{r−1} + ⋯ + (−1)^{r}π^{∗}p_{r}(F, ψ).

Hence the Pontryagin classes defined by (c) → (b) → (a) → (c) coincide with the original ones. □

12. Higher rank symplectic Thom classes

The bigraded version of the definition of a symplectic Thom classes theory [13, Definition 14.2] is as follows.

Definition 12.1. A symplectic Thom classes theory on a bigraded ϵ-commutative ring cohomology theory (A^{∗,∗}, ∂, ×, 1_{A}) on 𝒮m𝒪p/S is a rule assigning to every symplectic bundle (F, ψ) over every scheme X in 𝒮m/S an element th(F, ψ) ∈ A^{4r,2r}(F, F − X) with 2r = rk F with the following properties:

(1) For an isomorphism u: (F, ψ) ≅ (F_{1}, ψ_{1}) we have th(F, ψ) = u^{∗} th(F_{1}, ψ_{1}).

(2) For f: Y → X, writing f_{F} : f^{∗}F → F for the pullback, we have f_{F}^{∗} th(F, ψ) = th(f^{∗}F, f^{∗}ψ)) ∈ A^{4r,2r}(f^{∗}F, f^{∗}F − Y).

(3) The maps ∪ th(F, ψ): A^{∗,∗}(X) → A^{∗+4r,∗+2r}(F, F − X) are isomorphisms.

(4) We have th((F_{1}, ψ_{1}) ⊕ (F_{2}, ψ_{2})) = q_{1}^{∗} th(F_{1}, ψ_{1}) ∪ q_{2}^{∗} th(F_{2}, ψ_{2}), where q_{1}, q_{2} are the projections from F_{1} ⊕ F_{2} onto its factors. Moreover, for the zero bundle 0 → pt we have th(0) = 1_{A} ∈ A^{0,0}(pt).

The classes th(F, ψ) are symplectic Thom classes.

Let (A, μ, e) be a commutative T -ring spectrum. Suppose we have a sequence of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) with ϑ_{r} ∈ A^{4r,2r}(MSp_{2r}) for each r. Then for any symplectic bundle (F, ψ) of rank 2r over X one can use ϑ_{r} to define a class th^{ϑ}(F, ψ) by the construction already described in (10.1)–(10.3) for rank 2. For a rank 0 bundle 0_{X} → X we set th(0_{X}) = 1_{X} ∈ A^{0,0}(X). These classes are well-defined by the same argument as in Lemma 5.4 and 10.1.

Recall the inclusion e_{2}^{Sp}: T^{∧2} → MSp_{2} of (6.2) and the monoid maps μ_{rs}: MSp_{2r} ∧ MSp_{2s} → MSp_{2r+2s} of (6.1).

Theorem 12.2. Let (A, μ, e) be a commutative T -ring spectrum. Then the map which assigns to a sequence of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) as above the family of classes th^{ϑ}(F, ψ) is a bijection between the sets of

(δ) sequences of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) with ϑ_{r} ∈ A^{4r,2r}(MSp_{2r}) for each r satisfying μ_{rs}^{∗}ϑ_{r+s} = ϑ_{r} × ϑ_{s} for all r, s, and ϑ_{1}|_{T^{∧2}} = Σ_{T}^{2} 1_{A}, and

(d) symplectic Thom classes theories on (A^{∗,∗}, ∂, ×, 1_{A}) such that for the trivial rank 2 bundle A^{2} → pt we have th(A^{2}, ω_{2}) = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}).

The proof is essentially the same as that of Theorem 10.2. The class ϑ_{r} is the tautological symplectic Thom element of rank 2r.

Recall that for a commutative T -ring spectrum (A, μ, e) with a symplectic Thom structure on (A^{∗,∗}, ∂, ×, 1_{A}) the Thom space structural map z_{r}: BSp_{2r} → MSp_{2r} has the property that z_{r}^{∗}: A^{∗,∗}(MSp_{2r}) → A^{∗,∗}(BSp_{2r}) is injective, and that the isomorphism

A^{∗,∗}(BSp_{2r}) ⟵^{≅} A^{∗,∗}(pt)[[p_{1}, …, p_{r}]]  (12.1)

derived from the symplectic Thom structure identifies the image of z_{r}^{∗} with the two-sided ideal generated by p_{r} (Theorems 9.1, 9.2 and 9.3).

Theorem 12.3. Let (A, μ, e) be a commutative T -ring spectrum. Then the assignment ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) ↦ ϑ_{1} gives a bijection between the sets of

(δ) sequences of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) satisfying the conditions of Theorem 12.2 and (α) the tautological rank 2 Thom elements ϑ of Theorem 10.2. The inverse bijection sends ϑ ↦ ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) where z_{r}^{∗}ϑ_{r} ∈ A^{4r,2r}(BSp_{2r}) is the

element corresponding to (−1)^{r}p_{r} under the isomorphism (12.1) derived from the symplectic Thom structure associated to ϑ by Theorem 10.2.

Proof. Clearly the mapping (δ) → (α) is well-defined.

We will show that (α) → (δ) is well-defined. Suppose ϑ satisfies the conditions of (α). The classes (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) verify the condition μ_{rs}^{∗}ϑ_{r+s} = ϑ_{r} × ϑ_{s} because in Theorem 9.4 the classes in the second diagram verify p_{r+s} ↦ p′_{r}p′′_{s}. The class ϑ_{1} is obtained by the construction corresponding to the assignments (α) → (β) → (α) of Theorem 11.4. So by that theorem we have ϑ_{1} = ϑ. So we have ϑ_{1}|_{T^{∧2}} = Σ_{T}^{2} 1_{A}. Therefore (α) → (δ) is well-defined. In addition this shows that (α) → (δ) → (α) is the identity.

Now suppose given ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) satisfying (δ), and let ϑ′ = (ϑ′_{1}, ϑ′_{2}, ϑ′_{3}, …) be the result obtained by applying (δ) → (α) → (δ). We have already seen that we have ϑ_{1} = ϑ′_{1}. The equalities ϑ_{r} = ϑ′_{r} follow by induction using the injectivity of the maps m_{rs}^{∗} of Theorem 9.4. □

For a symplectic bundle (F, ψ) of rank 2r over X the Pontryagin classes (γ) and the symplectic Thom classes (δ) are related by

p_{r}(F, ψ) = (−1)^{r}z^{∗} th(F, ψ).  (12.2)

13. The universality of MSp

Theorem 13.1. Let (A, μ, e) be a commutative T -ring spectrum with a symplectic Thom classes theory on A^{∗,∗}. Then we have isomorphisms of bigraded rings

A^{∗,∗}(MSp) ⟶^{≅} \underleftarrow{lim} A^{∗+4r,∗+2r}(MSp_{2r}) ⟵^{≅} A^{∗,∗}(pt)[[p_{1}, p_{2}, p_{3}, …]]^{hom},

A^{∗,∗}(MSp ∧ MSp) ⟶^{≅} \underleftarrow{lim} A^{∗+8r,∗+4r}(MSp_{2r} ∧ MSp_{2r}) ⟵^{≅} A^{∗,∗}(pt)[[p′_{1}, p′_{2}, …, p′′_{1}, p′′_{2}, …]]^{hom}.

Proof. By Theorem 12.2 the symplectic Thom classes theory has associated to it a sequence ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) of tautological symplectic Thom classes with the property that ϑ_{r} ∈ A^{4r,2r}(MSp_{2r}) is the Thom class of the tautological symplectic subbundle (𝒰_{BSp_{2r}}, ϕ_{BSp_{2r}}) over BSp_{2r} = HGr(r, ∞) and with ϑ_{1}|_{T^{∧2}} = Σ_{T}^{2} 1_{A}. Set also ϑ_{0} = 1_{A} ∈ A^{0,0}(pt) = A^{0,0}(MSp_{0}).

By Theorem 5.6 the group A^{∗,∗}(MSp) fits into the short exact sequence

0 → \underleftarrow{lim}^{1}A^{∗+4r−1,∗+2r}(MSp_{2r}) → A^{∗,∗}(MSp) → \underleftarrow{lim} A^{∗+4r,∗+2r}(MSp_{2r}) → 0

where the connecting maps in the tower are given by the top line of the commutative diagram

\xymatrix{A^{∗+4r−4,∗+2r−2}(MSp_{2r−2}) & A^{∗+4r,∗+2r}(MSp_{2r−2} ∧ T^{∧2}) \ar[l]_{Σ_{T}^{−2}} & A^{∗+4r,∗+2r}(MSp_{2r}) \ar[l]_{σ^{∗}} \\ A^{∗,∗}(BSp_{2r−2}) \ar[u]_{−∪ϑ_{r−1}}^{≅} & A^{∗,∗}(BSp_{2r−2}) \ar[l]_{id} \ar[u]_{−∪σ^{∗}ϑ_{r}}^{≅} & A^{∗,∗}(BSp_{2r}). \ar[l]_{i_{2r}^{∗}} \ar[u]_{−∪ϑ_{r}}^{≅}}

The map σ^{∗} is the pullback along the bonding map

MSp_{2r−2} ∧ T^{∧2} ⟶^{1×e_{1}} MSp_{2r−2} ∧ MSp_{2} ⟶^{μ_{r−1,1}} MSp_{2r}

of the symmetric T^{∧2}-spectrum. Thus we have σ^{∗}ϑ_{r} = ϑ_{r−1} × Σ_{T}^{2} 1_{A}. The diagram therefore commutes. The vertical maps are isomorphisms by condition (3) of the definition of a symplectic Thom classes theory. The map i_{2r}^{∗} is the surjection A^{∗,∗}[[p_{1}, …, p_{r}]] ↠ A^{∗,∗}[[p_{1}, …, p_{r−1}]] of (9.3). This gives us the second isomorphism of the theorem, while the surjectivity of the connecting maps in the inverse system gives the vanishing of the \underleftarrow{lim}^{1} and the first isomorphism.

The calculations for A^{∗,∗}(MSp ∧ MSp) are similar. □

Let φ: MSp → A be a morphism in SH(S). For each r ≥ 1 let ϑ_{r}^{φ} ∈ A^{4r,2r}(MSp_{2r}) be the composition

Σ_{T}^{∞}MSp_{2r}(−2r) ⟶^{u_{2r}} MSp ⟶^{φ} A,

and let ϑ^{φ} = (ϑ_{1}^{φ}, ϑ_{2}^{φ}, ϑ_{3}^{φ}, …).

Theorem 13.2. Suppose (A, μ, e) is a commutative monoid in (SH(S), ∧, 1). Then the assignment φ ↦ ϑ^{φ} = (ϑ_{1}^{φ}, ϑ_{2}^{φ}, ϑ_{3}^{φ}, …) gives a bijection between the sets of

(ε) morphisms φ: (MSp, μ^{Sp}, e^{Sp}) → (A, μ, e) of commutative monoids in SH(S), and (δ) sequences of classes ϑ = (ϑ_{1}, ϑ_{2}, ϑ_{3}, …) satisfying the conditions of Theorem 12.2.

The proof of this theorem is substantially the same as that of Theorem 5.9. The differences are, first, that the ϑ comes from a unique φ: MSp → A because the map A^{0,0}(MSp) → \underleftarrow{lim} A^{4r,2r}(MSp_{2r}) of Theorem 13.1 is an isomorphism. Second, the obstruction to φ being a morphism of monoids vanishes because A^{0,0}(MSp ∧ MSp) → \underleftarrow{lim} A^{8r,4r}(MSp_{2r} ∧ MSp_{2r}) is also an isomorphism.

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Steklov Institute of Mathematics at St. Petersburg, Russia

Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS), Département de mathématiques, Université de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France