ON THE RELATION OF SYMPLECTIC ALGEBRAIC COBORDISM TO HERMITIAN K-THEORY

I. PANIN AND C. WALTER

Abstract. We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S) there is a unique morphism φ: MSp → BO of commutative ring T -spectra which sends the Thom class th^{MSp} to the Thom class th^{BO}. Using φ we construct an isomorphism of bigraded ring cohomology theories on the category 𝒮m𝒪p/S

φ̄: MSp^{∗,∗}(X, U) ⊗_{MSp^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt) ≅ BO^{∗,∗}(X, U).

The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSp^{p,q} = MSp_{2q−p}^{[q]}, we have an isomorphism

φ̄: MSp_{∗}^{[∗]}(X, U) ⊗_{MSp_{0}^{[2∗]}(pt)} KO_{0}^{[2∗]}(pt) ≅ KO_{∗}^{[∗]}(X, U),

where the KO_{i}^{[n]}(X, U) are Schlichting’s hermitian K-theory groups.

1. A motivic version of a theorem by Conner and Floyd

Our main result relates symplectic algebraic cobordism to hermitian K-theory. It is an algebraic version of the theorem of Conner and Floyd [2, Theorem 10.2] reconstructing real K-theory using symplectic cobordism. The algebraic version of the reconstruction of complex K-theory using unitary cobordism was done in [5].

In [7] the current authors constructed a commutative ring T -spectrum BO representing hermitian K-theory in the stable homotopy category SH(S) for any regular noetherian separated base scheme S of finite Krull dimension without residue fields of characteristic 2. (These restrictions allowed us to use particularly strong results of Marco Schlichting [9]. We leave it to the expert(s) in negative hermitian K-theory to weaken them.) It has a standard family of Thom classes for special linear vector bundles and hence for symplectic bundles. The symplectic Thom classes can all be derived from a single class th^{BO} ∈ BO^{4,2}(Th 𝒰_{HP^{∞}}) = BO^{4,2}(MSp_{2}), the symplectic Thom orientation.

In [6] we constructed the commutative ring T -spectrum MSp of algebraic symplectic cobordism. It is a commutative monoid in the model category of symmetric T^{∧2}-spectra, just as MSL and Voevodsky’s MGL are commutative monoids in the model category of symmetric T -spectra. The canonical map Σ_{T}^{∞}MSp_{2}(−2) → MSp gives the symplectic Thom orientation th^{MSp} ∈ MSp^{4,2}(MSp_{2}). It is the universal symplectically oriented commutative ring T -spectrum.

Date: September 20, 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.

Therefore there is a unique morphism φ: MSp → BO of commutative monoids in SH(S) with φ(th^{MSp}) = th^{BO}. Our main result is the following theorem. Our notation is that for a motivic space Y and a bigraded cohomology theory we write A^{∗,∗}(Y) = ⨁_{p,q∈ℤ} A^{p,q}(Y) and A^{4∗,2∗}(Y) = ⨁_{i∈ℤ} A^{4i,2i}(Y). A motivic space Y is small if Hom_{SH(S)}(Σ_{T}^{∞}Y, −) commutes with arbitrary coproducts.

Theorem 1.1. Let S be a regular noetherian separated scheme of finite Krull dimension with \frac{1}{2} ∈ Γ(S, 𝒪_{S}). For all small pointed motivic spaces Y over S the map

φ̄: MSp^{∗,∗}(Y) ⊗_{MSp^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt) → BO^{∗,∗}(Y).

induced by φ is an isomorphism.

This has as a consequence the result mentioned in the abstract. For a pair (X, U) consisting of a smooth S-scheme of finite type X and an open subscheme U, there is a quotient pointed motivic space X_{+}/U_{+}. We define MSp^{∗,∗}(X, U) = MSp^{∗,∗}(X_{+}/U_{+}) and BO^{∗,∗}(X, U) = BO^{∗,∗}(X_{+}/U_{+}). There are natural isomorphisms BO^{p,q}(X, U) = KO_{2q−p}^{[q]}(X, U) with the hermitian K-theory of X with supports in X − U as defined by Schlichting [11]. The weight q is the degree of the shift in the duality used for the symmetric bilinear forms on the chain complexes of vector bundles.

For a field k of characteristic not 2 the ring BO^{4∗,2∗}(k) is not large. For all i one has BO^{8i,4i}(k) ≅ GW (k) and BO^{8i+4,4i+2}(k) ≅ ℤ. All members of BO^{0,0}(k) therefore come from composing endomorphisms in SH(k) of the sphere T -spectrum 1 = Σ_{T}^{∞}pt_{+} with the unit e: 1 → BO of the monoid. (See Morel [3, Theorem 4.36] and Cazanave [1] for calculations of the endomorphisms of the sphere T -spectrum.) Consequently φ^{0,0} : MSp^{0,0}(k) → BO^{0,0}(k) is surjective. We do not know what happens in other bidegrees.

This is the fourth in a series of papers about symplectically oriented motivic cohomology theories. All depend on the quaternionic projective bundle theorem proven in the first paper [8].

2. Preliminaries

Let S be a Noetherian separated scheme of finite Krull dimension. We will be dealing with hermitian K-theory, and we prefer avoiding the subtleties of negative K-theory, so we will assume as we did in [7] that S is regular and that \frac{1}{2} ∈ Γ(S, 𝒪_{S}). Let 𝒮m/S be the category of smooth S-schemes of finite type. Let 𝒮m𝒪p/S be the category whose objects are pairs (X, U) with X ∈ 𝒮m/S and U ⊂ X an open subscheme and whose arrows f: (X, U) → (X′, U′) are morphisms f: X → X′ of S-schemes with f(U) ⊂ U′. Note that all X in 𝒮m/S have an ample family of line bundles.

A motivic space over S is a simplicial presheaf on 𝒮m/S. We will often write pt for the base scheme regarded as a motivic space over itself. Inverting the motivic weak equivalences in the category of pointed motivic spaces gives the pointed motivic unstable homotopy category H_{•}(S).

Let T = A^{1}/(A^{1} − 0) be the Morel-Voevodsky object. A T -spectrum M is a sequence of pointed motivic spaces (M_{0}, M_{1}, M_{2}, …) equipped with structural maps σ_{n}: M_{n}∧T → M_{n+1}. Inverting the stable motivic weak equivalences gives the motivic stable homotopy category SH(S). A pointed motivic space X has a T -suspension spectrum Σ_{T}^{∞}X. For any T -spectrum M there are canonical maps of spectra

u_{n}: Σ_{T}^{∞}M_{n}(−n) → M.  (1)

Both H_{•}(S) and SH(S) are equipped with closed symmetric monoidal structures, and Σ_{T}^{∞}: H_{•}(S) → SH(S) is a strict symmetric monoidal functor. The symmetric monoidal structure (∧, 1_{S} = Σ_{T}^{∞}pt_{+}) on the homotopy category SH(S) can be constructed on the model category level using symmetric T -spectra.

Any T -spectrum A defines a cohomology theory on the category of pointed motivic spaces. Namely, for a pointed space (X, x) one sets A^{p,q}(X, x) = Hom_{H_{•}(S)}(Σ_{T}^{∞}(X, x), Σ^{p,q}(A)) and A^{∗,∗}(X, x) = ⨁_{p,q∈ℤ} A^{p,q}(X, x). We write (somewhat inconsistently)

A^{4∗,2∗}(X, x) = ⨁_{i∈ℤ} A^{4i,2i}(X, x).

For an unpointed space X we set A^{p,q}(X) = A^{p,q}(X_{+}, +), with A^{∗,∗}(X) and A^{4∗,2∗}(X) defined accordingly. We will not always write the pointings explicitly.

Each Y ∈ 𝒮m/S defines an unpointed motivic space which is constant in the simplicial direction Hom_{𝒮m/S}(−, Y). So we regard smooth S-schemes as motivic spaces and set A^{p,q}(Y) = A^{p,q}(Y_{+}, +). Given a monomorphism U ↪ Y of smooth S-schemes, we write A^{p,q}(Y, U) = A^{p,q}(Y_{+}/U_{+}, U_{+}/U_{+}).

A commutative ring T -spectrum is a commutative monoid (A, μ, e) in (SH(S), ∧, 1).

The cohomology theory A^{∗,∗} defined by a commutative ring T -spectrum is a ring cohomology theory satisfying a certain bigraded commutativity condition described by Morel. Namely, let ε ∈ A^{0,0}(pt) be the element such that Σ_{T}^{2} ε ∈ Hom_{SH(S)}(T ∧T, T ∧T) is the map exchanging the two factors T. Then for α ∈ A^{p,q}(X, x) and β ∈ A^{p′,q′}(X, x) we have α ∪ β = (−1)^{pp′}ε^{qq′}β ∪ α. In particular, A^{4∗,2∗}(X, x) is contained in the center of A^{∗,∗}(X, x).

We work in this text with the algebraic cobordism T -spectrum MSp of [6, §6] and the hermitian K-theory T -spectrum BO of [7, §8]. The spectrum MSp is a commutative ring T -spectrum because it is naturally a commutative monoid in the category of symmetric T^{∧2}- spectra. The T -spectrum BO has a commutative monoid structure as shown in [7, Theorem 1.3].

3. The first Pontryagin class p_{1}(E, ϕ)

Let V be a vector bundle over a smooth S-scheme X with zero section z: X ↪ V. The Thom space of V is the quotient motivic space Th V = V/(V − z(X)). It is pointed by the image of V − z(X). It comes with a canonical structure map z: X_{+} → Th V induced by the zero section. For the trivial bundle A^{n} → pt one has Th A^{n} = T^{∧n}.

We write H for the trivial rank 2 symplectic bundle (𝒪^{⊕2}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )). The orthogonal direct sum H^{⊕n} is the trivial symplectic bundle of rank 2n.

The most basic form a symplectic orientation is a symplectic Thom structure [8, Definition 7.1]. We will use the following version of the definition.

Definition 3.1. Let (A, μ, e) be a symmetric ring T -spectrum. A symplectic Thom structure on the cohomology theory A^{∗,∗} 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}(Th E) = 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 rank 2 trivial symplectic bundle H over pt the map

− × th(H): 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 sign in the Pontryagin class is simply conventional. It is chosen so that if A^{∗,∗} is an oriented cohomology theory with an additive formal group law, then the Chern and Pontryagin classes satisfy the traditional formula p_{i}(E, ϕ) = (−1)^{i}c_{2i}(E).

From Mayer-Vietoris one sees that for any rank 2 symplectic bundle

∪ th(E, ϕ): A^{∗,∗}(X) ⟶^{≅} A^{∗,∗}(E, E − X)

is an isomorphism.

The quaternionic Grassmannian HGr(r, n) = HGr(r, H^{⊕n}) is defined as the open subscheme of Gr(2r, 2n) = Gr(2r, H^{⊕n}) parametrizing subspaces of dimension 2r of the fibers of H^{⊕n} on which the symplectic form of H^{⊕n} is nondegenerate. We write 𝒰_{HGr(r,n)} for the restriction to HGr(r, n) of the tautological subbundle of Gr(2r, 2n). The symplectic form of H^{⊕n} restricts to a symplectic form on 𝒰_{HGr(r,n)} which we denote by ϕ_{HGr(r,n)}. The pair (𝒰_{HGr(r,n)}, ϕ_{HGr(r,n)}) is the tautological symplectic subbundle of rank 2r on HGr(r, n).

More generally, given a symplectic bundle (E, ϕ) of rank 2n over X, the quaternionic Grassmannian bundle HGr(r, E, ϕ) is the open subscheme of the Grassmannian bundle Gr(2r, E) parametrizing subspaces of dimension 2r of the fibers of E on which ϕ is nondegenerate.

For r = 1 we have quaternionic projective spaces and bundles HP^{n} = HGr(1, n + 1) and HP (E, ϕ) = HGr(1, E, ϕ).

The quaternionic projective bundle theorem is proven in [8] using the symplectic Thom structure and not any other version of a symplectic orientation. It is proven first for trivial bundles.

Theorem 3.2 ([8, Theorem 8.1]). Let (A, μ, e) be a commutative ring T -spectrum with a symplectic Thom structure on A^{∗,∗}. Let (𝒰_{HP^{n}}, ϕ_{HP^{n}}) be the tautological rank 2 symplectic subbundle over HP^{n} and t = p_{1}(𝒰_{HP^{n}}, ϕ_{HP^{n}}) ∈ A^{4,2}(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)[t]/(t^{n+1}).

A Mayer-Vietoris argument gives the more general theorem [8, Theorem 8.2].

Theorem 3.3 (Quaternionic projective bundle theorem). Let (A, μ, e) be a commutative ring T -spectrum with a symplectic Thom structure on A^{∗,∗}. 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 t = p_{1}(𝒰, ϕ|_{𝒰}) be its Pontryagin class. Then we have an isomorphism of bigraded A^{∗,∗}(X)-modules

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

Definition 3.4. Under the hypotheses of Theorem 3.3 there are unique elements p_{i}(E, ϕ) ∈ A^{4i,2i}(X) for i = 1, 2, …, n such that

t^{n} − p_{1}(E, ϕ) ∪ t^{n−1} + p_{2}(E, ϕ) ∪ t^{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.

Corollary 3.5. The Pontryagin classes of a trivial symplectic bundle vanish: p_{i}(H^{⊕n}) = 0.

The Cartan sum formula holds for Pontryagin classes [8, Theorem 10.5]. In particular:

Theorem 3.6. Let (A, μ, e) be a commutative ring T -spectrum with a symplectic Thom structure on A^{∗,∗}. Let (E, ϕ) and (F, ψ) be symplectic bundles over X. Then we have

p_{1}((E, ϕ) ⊕ (F, ψ)) = p_{1}(E, ϕ) + p_{1}(F, ψ).  (2)

We also have the following result [8, Proposition 8.5].

Proposition 3.7. Suppose that (E, ϕ) is a symplectic bundle over X with a totally isotropic subbundle L ⊂ E. Then for all i we have

p_{i}(E, ϕ) = p_{i} ((L^{⊥}/L, \overline{ϕ}) ⊕ (L ⊕ L^{∨}, ( \begin{matrix}0 & 1_{L^{∨}} \\ −1_{L} & 0 \end{matrix} ))) .

This is because there is an A^{1}-deformation between the two symplectic bundles.

Definition 3.8. The Grothendieck-Witt group of symplectic bundles GW^{−}(X) is the abelian group of formal differences [E, ϕ] − [F, ψ] of symplectic vector bundles over X modulo three relations:

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

(2) For an orthogonal direct sum one has [(E, ϕ) ⊕ (E_{1}, ϕ_{1})] = [E, ϕ] + [E_{1}, ϕ_{1}].

(3) If (E, ϕ) is a symplectic bundle over X with a totally isotropic subbundle L ⊂ E, then

we have [E, ϕ] = [L^{⊥}/L, \overline{ϕ}] + [L ⊕ L^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )].

The Grothendieck-Witt group of orthogonal bundles GW^{+}(X) is defined analogously.

Theorem 3.9. Let (A, μ, e) be a commutative ring T -spectrum with a symplectic Thom structure on A^{∗,∗}. Then the associated first Pontryagin class induces a well-defined additive map

p_{1} : GW^{−}(X) → A^{4,2}(X)

which is functorial in X.

In [10] Schlichting constructed hermitian K-theory spaces for exact categories. This gives hermitian K-theory spaces KO(X) and KSp(X) for orthogonal and symplectic bundles on schemes. Their π_{0} are GW^{+}(X) and GW^{−}(X) respectively. In [11] he constructed Hermitian K-theory spaces KO^{[m]}(X, U) for complexes of vector bundles on X acyclic on the open subscheme U equipped with a nondegenerate symmetric bilinear form for the duality shifted by m. For an even integer 2n an orthogonal bundle (U, ψ) gives a chain complex U[2n] equipped with a nondegenerate symmetric bilinear form ψ[4n]: U[2n] ⊗_{𝒪_{X}} U[2n] → 𝒪_{X}[4n] in the symmetric monoidal category D^{b}(V B_{X}). For an odd integer 2n + 1 a symplectic bundle (E, ϕ) gives a chain complex E[2n+1] equipped with a nondegenerate symmetric bilinear form ϕ[4n+2]: E[2n+1] ⊗_{𝒪_{X}} E[2n+1] → 𝒪_{X}[4n+2]. These functors induce homotopy equivalences of spaces KO(X) → KO^{[4n]}(X) and KSp(X) → KO^{[4n+2]}(X) [11, Proposition 6].

The simplicial presheaves X ↦ KO^{[n]}(X) are pointed motivic spaces. Dévissage gives schemewise weak equivalences KO^{[n]}(X) → KO^{[n+1]}(X × A^{1}, X × (A^{1} − 0)) which are adjoint to maps KO^{[n]} × T → KO^{[n+1]}. These are the structural maps of a T -spectrum (KO^{[0]}, KO^{[1]}, KO^{[2]}, …) of which our BO is a fibrant replacement [7, §§7–8]. One has KO_{i}^{[n]}(X, U) = BO^{4n−i,2n}(X_{+}/U_{+}) for all i ≥ 0 and n. Hence BO^{4n,2n}(X_{+}/U_{+}) is the Grothendieck-Witt group for the usual duality shifted by n of symmetric chain complexes of vector bundles on X which are acyclic on U.

Definition 3.10. The right isomorphisms are

unsign.trans_{4n}: GW^{+}(X) ⟶^{≅} KO_{0}^{[4n]}(X) = BO^{8n,4n}(X)
[U, ψ] ⟼ [U[2n], ψ[4n]]

and

sign.trans_{4n+2} : GW^{−}(X) ⟶^{≅} KO_{0}^{[4n+2]}(X) = BO^{8n+4,4n+2}(X)
[E, ϕ] ⟼ −[E[2n + 1], ϕ[4n + 2]]

The sign in sign.trans_{4n+2} is chosen so that it commutes with the forgetful maps to K_{0}(X), where we have [E] = −[E[2n + 1]]. Most authors of papers on Witt groups do not use this sign because Witt groups do not have forgetful maps to K_{0}(X).

Definition 3.11. The periodicity elements β_{8} ∈ BO^{8,4}(pt) and β_{8}^{−1} ∈ BO^{−8,−4}(pt) correspond to the unit 1 = [𝒪_{X}, 1] ∈ GW^{+}(X) under the isomorphisms BO^{8,4}(pt) ≅ GW^{+}(pt) ≅ BO^{−8,−4}(pt) of Definition 3.10.

We have the composition

p_{1}^{A}: BO^{4,2}(X) ⟵_{≅}^{sign.trans_{2}} GW^{−}(X) ⟶^{p_{1}} A^{4,2}(X)  (3)

The Thom classes for hermitian K-theory are constructed by the same method that Nenashev used for Witt groups [4, §2]. Suppose we have an SL_{n}-bundle (E, λ) consisting of a vector bundle π: E → X of rank n and λ: 𝒪_{X} ≅ det E an isomorphism of line bundles. The pullback π^{∗}E = E ⊕ E → E has a canonical section Δ_{E}, the diagonal. There is a Koszul complex

K(E) = (0 → Λ^{n}π^{∗}E^{∨} → Λ^{n−1}π^{∗}E^{∨} → ⋯ → Λ^{2}π^{∗}E^{∨} → E^{∨} → 𝒪_{E} → 0)

in which each boundary map the contraction with Δ_{E}. It is a locally free resolution of the coherent sheaf z_{∗}𝒪_{X} on E. There is a canonical isomorphism Θ(E, λ): K(E) → K(E)^{∨}[n] induced by λ which is symmetric for the shifted duality.

Definition 3.12. In the standard special linear Thom structure on BO the Thom class of the special linear bundle (E, λ) of rank n is

th^{BO}(E, λ) = [K(E), Θ(E, λ)] ∈ KO_{0}^{[n]}(E, E − X) = BO^{2n,n}(E, E − X)

In the standard symplectic Thom structure on BO the Thom class of the symplectic bundle (E, ϕ) of rank 2r is

th^{BO}(E, ϕ) = th^{BO}(E, λ_{ϕ}) ∈ BO^{4r,2r}(E, E − X)

for λ_{ϕ} = (Pf ϕ)^{−1} where Pf ϕ ∈ Γ(X, det E^{∨}) denotes the Pfaffian of ϕ ∈ Γ(X, Λ^{2}E^{∨}).

The corresponding first Pontryagin class of a rank 2 symplectic bundle is therefore

p_{1}^{BO}(E, ϕ) = −[K(E), Θ(E, λ_{ϕ})]|_{X} ∈ BO^{4,2}(X).

A short calculation shows that this is the class which corresponds to [E, ϕ] − [H] ∈ GW^{−}(X) under the isomorphism sign.trans_{2}. The symplectic splitting principle [8, Theorem 10.2] and Theorem 3.6 now give the next proposition.

Proposition 3.13. Let (E, ϕ) be a symplectic bundle of rank 2r on X. Then p_{1}^{BO}(E, ϕ) ∈ BO^{4,2}(X) is the class which corresponds to [E, ϕ] − r[H] ∈ GW^{−}(X) under the isomorphism sign.trans_{2}.

Let X = ⨆ X_{i} be the connected components of X. We consider the elements and functions

1_{X_{i}} ∈ BO^{0,0}(X),  rk_{X_{i}}: BO^{4,2}(X) → ℤ,  h ∈ BO^{4,2}(pt).  (4)

The first is the central idempotent which is the image of the unit 1_{X_{i}} ∈ BO^{0,0}(X_{i}). The second is the rank function on the Grothendieck-Witt group KO_{0}^{[2]}(X) of bounded chain complexes of vector bundles. The third is the class corresponding to [H] ∈ GW^{−}(pt) under the right isomorphism sign.trans_{2} : GW^{−}(pt) ≅ BO^{4,2}(pt).

Let p_{1}^{BO} : BO^{4,2}(X) → BO^{4,2}(X) be the map of (3).

Corollary 3.14. For all α ∈ BO^{4,2}(X) we have α = p_{1}^{BO}(α) + h ∏_{i} \frac{1}{2}(rk_{X_{i}}α)1_{X_{i}}.

4. Symplectically oriented commutative ring T -spectra

Embed H^{⊕n} ⊂ H^{⊕∞} as the direct sum of the first n summands. The ensuing filtration H ⊂ H^{⊕2} ⊂ H^{⊕3} ⊂ ⋯ for each r a direct system of schemes

pt = HGr(r, r) ↪ HGr(r, r + 1) ↪ HGr(r, r + 2) ↪ · · · .

The ind-scheme and motivic space

BSp_{2r} = HGr(r, ∞) = colim_{n≥r} HGr(r, n)

is pointed by h_{r}: pt = HGr(r, r) ↪ BSp_{2r}. Each HGr(r, n) has a tautological symplectic subbundle (𝒰_{HGr(r,n)}, ϕ_{HGr(r,n)}), and their colimit is an ind-scheme 𝒰_{BSp_{2r}} which is a vector bundle over the ind-scheme BSp_{2r}. It has a Thom space Th 𝒰_{BSp_{2r}} just like for ordinary schemes. We write

MSp_{2r} = Th 𝒰_{BSp_{2r}} = Th 𝒰_{HGr(r,∞)} = colim_{n≥r} Th 𝒰_{HGr(r,n)}.

We refer the reader to [6, §6] for the complete construction of MSp as a commutative monoid in the category of symmetric T^{∧2}-spectra. The unit comes from the pointings h_{r}: pt ↪ BSp_{2r}, which induce canonical inclusions of Thom spaces

e_{r}: T^{∧2r} ↪ MSp_{2r}.

Let (A, μ, e) be a commutative ring T -spectrum. The unit of the monoid defines the unit element 1_{A} ∈ A^{0,0}(pt_{+}). Applying the T -suspension isomorphism twice gives an element Σ_{T}^{2} 1_{A} ∈ A^{4,2}(T^{∧2}) = A^{4,2}(Th A^{2}).

Definition 4.1. A symplectic Thom orientation on a commutative ring T -spectrum (A, μ, e) is an element th ∈ A^{4,2}(MSp_{2}) = A^{4,2}(Th 𝒰_{HP^{∞}}) with th |_{T^{∧2}} = Σ_{T}^{2} 1_{A} ∈ A^{4,2}(T^{∧2}).

The element th should be regarded as the symplectic Thom class of the tautological quaternionic line bundle 𝒰_{HP^{∞}} over HP^{∞}.

Example 4.2. The standard symplectic Thom orientation on algebraic symplectic cobordism is the element th^{MSp} = u_{2} ∈ MSp^{4,2}(MSp_{2}) corresponding to the canonical map u_{2} : Σ_{T}^{∞}MSp_{2}(−2) → MSp described in (1).

The main theorem of [6] gives seven other structures containing the same information as a symplectic Thom orientation. First:

Theorem 4.3 ([6, Theorem 10.2]). Let (A, μ, e) be a commutative monoid in SH(S). There is a canonical bijection between the sets of

(a) symplectic Thom structures on the ring cohomology theory A^{∗,∗} such that for the trivial rank 2 symplectic bundle H over pt we have th(H) = Σ_{T}^{2} 1_{A} in A^{4,2}(T^{∧2}), and

(α) symplectic Thom orientations on (A, μ, e).

Thus a symplectic Thom orientation determines Thom and Pontryagin classes for all symplectic bundles. Lemma 4.4. In the standard special linear and symplectic Thom structures on BO we have th(A^{1}, 1) = Σ_{T} 1_{BO} and th(H) = Σ_{T}^{2} 1_{BO}.

Proof. The structural maps KO^{[n]} ∧ T → KO^{[n+1]} of the spectrum are by definition [7, §8] adjoint to maps KO^{[n]} → Hom_{•}(T, KO^{[n+1]}) which are fibrant replacements of maps of simplicial presheaves

(− ⊠ (K(𝒪), Θ(𝒪, 1)))_{∗}: KO^{[n]}(−) → KO^{[n+1]}(− ∧ T)

which act on the homotopy groups as − ∪ [K(𝒪), Θ(𝒪, 1)] = − ∪ th(A^{1}, 1). So we have Σ_{T} 1_{BO} = th(A^{1}, 1). It then follows that we have th(H) = th(A^{1}, 1)^{∪2} = Σ_{T}^{2} 1_{BO}. □

The standard symplectic Thom structure on BO thus satisfies the normalization condition of Theorem 4.3. It corresponds to the standard symplectic Thom orientation on hermitian K-theory th^{BO} ∈ BO^{4,2}(MSp_{2}). It is given by the formulas of Definition 3.12 for (E, ϕ) = (𝒰_{HP^{∞}}, ϕ_{HP^{∞}}) tautological subbundle on HP^{∞} = BSp_{2}.

A symplectically oriented commutative T -ring spectrum is a pair (A, ϑ) with A a commutative monoid in SH(S) and ϑ a symplectic Thom orientation on A. We could write the associated Thom and Pontryagin classes as th^{ϑ}(E, ϕ) and p_{i}^{ϑ}(E, ϕ).

A morphism of symplectically oriented commutative T -ring spectra φ: (A, ϑ) → (B, ϖ) is a morphism of commutative monoids with φ(ϑ) = ϖ. For such a φ one has φ(th^{ϑ}(E, ϕ)) = th^{ϖ}(E, ϕ) and φ(p_{i}^{ϑ}(E, ϕ)) = p_{i}^{ϖ}(E, ϕ) for all symplectic bundles. Theorem 4.5 (Universality of MSp). Let (A, μ, e) be a commutative monoid in SH(S). The assignments φ ↦ φ(th^{MSp}) gives a bijection between the sets of

(ε) morphisms φ: (MSp, μ_{MSp}, e_{MSp}) → (A, μ, e) of commutative monoids in SH(S), and (α) symplectic Thom orientations on (A, μ, e). This is [6, Theorems 12.3, 13.2]. Thus (MSp, th_{MSp}) is the universal symplectically ori-

ented commutative T -ring spectrum.

Let φ: (A, ϑ) → (B, ϖ) be a morphism of symplectically oriented commutative T -ring spectra. For a space X the isomorphisms X ∧ pt_{+} ≅ X ≅ pt_{+} ∧ X make A^{∗,∗}(X) into a two-sided module over the ring A^{∗,∗}(pt) and into a bigraded-commutative algebra over the commutative ring A^{4∗,2∗}(pt). The morphism φ induces morphisms of graded rings

\begin{matrix}φ̄_{X}: A^{∗,∗}(X) ⊗_{A^{4∗,2∗}(pt)} B^{4∗,2∗}(pt) → B^{∗,∗}(X) \\ φ̄_{X}: A^{4∗,2∗}(X) ⊗_{A^{4∗,2∗}(pt)} B^{4∗,2∗}(pt) → B^{4∗,2∗}(X) \end{matrix}  (5)

which are natural in X, with the pullbacks acting on the left side of the ⊗. Theorem 4.6 (Weak quaternionic cellularity of MSp_{2r}). Let φ: (A, ϑ) → (B, ϖ) be a morphism of symplectically oriented commutative T -ring spectra. Then for all r the natural morphism of graded rings

φ̄_{MSp_{2r}}: A^{4∗,2∗}(MSp_{2r}) ⊗_{A^{4∗,2∗}(pt)} B^{4∗,2∗}(pt) → B^{4∗,2∗}(MSp_{2r})

is an isomorphism.

Proof. Let t_{1}, …, t_{r} be independent indeterminates with t_{i} of bidegree (4i, 2i). By [6, Theorems 9.1, 9.2, 9.3] there is a commutative diagram of isomorphisms

\xymatrix{A^{∗,∗}(pt)[[t_{1}, …, t_{r}]]^{hom} \ar[r]^{t_{i}↦p_{i}^{ϑ}(𝒰_{BSp_{2r}} ,ϕ_{BSp_{2r}})}_{≅} \ar[d]_{×t_{r}}^{≅} & A^{∗,∗}(BSp_{2r}) \ar[d]^{∪ th^{ϑ}(𝒰_{BSp_{2r}} ,ϕ_{BSp_{2r}})}_{≅} \\ t_{r}A^{∗,∗}(pt)[[t_{1}, …, t_{r}]]^{hom} \ar[r]_{≅} & A^{∗+4r,∗+2r}(MSp_{2r})}

The notation on the left refers to homogeneous formal power series. There is a similar diagram for (B, ϖ). The maps φ: A^{∗,∗} → B^{∗,∗} commute with the maps of the two diagrams because φ sends the Thom and Pontryagin classes of (A, ϑ) onto the Thom and Pontryagin classes of (B, ϖ). The morphism φ̄_{MSp_{2r}}^{4∗,2∗} is an isomorphism because

t_{r}A^{4∗,2∗}(pt)[[t_{1}, …, t_{r}]]^{hom} ⊗_{A^{4∗,2∗}(pt)} B^{4∗,2∗}(pt) → t_{r}B^{4∗,2∗}(pt)[[t_{1}, …, t_{r}]]^{hom}

is an isomorphism. □

5. Where the class p_{1} takes the place of honour

We suppose that (U, u) → (BO, th^{BO}) is a morphism of symplectically oriented commutative ring T -spectra. We set

Ū^{∗,∗}(X) = U^{∗,∗}(X) ⊗_{U^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt),
Ū^{4∗,2∗}(X) = U^{4∗,2∗}(X) ⊗_{U^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt),

and we write φ̄_{X} for the morphisms of (5).

Theorem 5.1. Let (U, u) → (BO, th^{BO}) be a morphism of symplectically oriented commutative ring T -spectra. Suppose there exists an N such that for all n ≥ N the maps φ̄_{U_{2n}}: Ū^{4i,2i}(U_{2n}) → BO^{4i,2i}(U_{2n}) are isomorphisms for all i. Then for all small pointed motivic spaces X and all (p, q) the homomorphism φ̄_{X}: Ū^{p,q}(X) → BO^{p,q}(X) is an isomorphism.

Before turning to the theorem itself, we prove a series of lemmas. The first three demonstrate the significance of the first Pontryagin class for this problem.

Lemma 5.2. The functorial map φ̄_{X}: Ū^{4,2}(X) → BO^{4,2}(X) has a section s_{X} which is functorial in X.

Proof. Write HGr = colim_{r} HGr(r, ∞). According to Theorem [7, Theorem 10.1, (11.1)] there is an isomorphism à la Morel-Voevodsky τ̄: (ℤ × HGr, (0, x_{0})) ≅ KSp in H_{•}(S) such that the restrictions are

τ̄|_{{i}×HGr(n,2n)} = [𝒰_{HGr(n,2n)}, ϕ_{HGr(n,2n)}] + (i − n)[H]

in KSp_{0}(HGr(n, 2n)) = GW^{−}(HGr(n, 2n)). Composing with the isomorphisms in H_{•}(S)

(ℤ × HGr, (0, x_{0})) ⟶^{τ̄} KSp ⟶^{trans_{1}} KO^{[2]} ⟶^{−1} KO^{[2]}.

where the trans_{1} comes from the translation functor (ℱ, ϕ) ↦ (ℱ[1], ϕ[2]), and the −1 is the inverse operation of the H-space structure. It gives us an element

τ_{2} ∈ KO_{0}^{[2]}(ℤ × HGr, (0, x_{0})) = BO^{4,2}(ℤ × HGr, (0, x_{0}))

corresponding to the composition. By Corollary 3.14 we have

τ_{2}|_{{i}×HGr(n,2n)} = p_{1}(𝒰_{HGr(n,2n)}, ϕ_{HGr(n,2n)}) + ih.

For any symplectically oriented cohomology theory A^{∗,∗} we have [7, (9.3)]

A^{∗,∗}(ℤ × HGr) = (A^{∗,∗}(pt)[[p_{1}, p_{2}, p_{3}, …]]^{hom})^{×ℤ}.

For such a theory let

\frac{1}{2}rk^{A} = (i1_{HGr})_{i∈ℤ} ∈ A^{0,0}(ℤ × HGr),  p_{1}^{A} = (p_{1})_{i∈ℤ} ∈ A^{4,2}(ℤ × HGr)

Then τ_{2} = p_{1}^{BO} + \frac{1}{2}rk^{BO}h. Consider the element

s = p_{1}^{U} ⊗ 1_{BO} + \frac{1}{2}rk^{U} ⊗ h ∈ Ū^{4,2}(ℤ × HGr).

Clearly one has φ̄(s) = τ_{2}. The element s may be regarded as a morphism of functors Hom_{H_{•}(S)}(−, ℤ × HGr) → Ū^{4,2}(−) by the Yoneda lemma. The composite map

Hom_{H_{•}(S)}(−, ℤ × HGr) ⟶^{s} Ū^{4,2}(−) ⟶^{φ̄} BO^{4,2}(−)

coincides with a functor transformation given by the adjoint Σ_{T}^{∞}(ℤ × HGr)(−2) → BO of the motivic weak equivalence τ_{2} : ℤ × HGr → KO^{[2]}. Thus for every pointed motivic space X the map

s_{X}: BO^{4,2}(X) = Hom_{H_{•}(S)}(X, KO^{[2]}) = Hom_{H_{•}(S)}(X, ℤ × HGr) ⟶^{s} Ū^{4,2}(X)

is a section of the map φ̄_{X}: Ū^{4,2}(X) → BO^{4,2}(X) which is natural in X. □

Lemma 5.3. For any integer i the functorial map φ̄_{X}: Ū^{8i+4,4i+2}(X) → BO^{8i+4,4i+2}(X) has a section t_{X} which is functorial in X.

Proof. We have BO^{8∗+4,4∗+2} = BO^{4,2}[β_{8}, β_{8}^{−1}] for the periodicity element β_{8} ∈ BO^{8,4}(pt) of Definition 3.11. So any element of BO^{8∗+4,4∗+2}(X) may be written uniquely in the form a∪β_{8}^{i} with a ∈ BO^{4,2}(X) and i ∈ ℤ. We define

t_{X}(a ∪ β_{8}^{i}) = s_{X}(a) ∪ (1_{U} ⊗ β_{8}^{i}) ∈ Ū^{8∗+4,4∗+2}(X).

Then t_{X} is a section of φ̄_{X} which is natural in X. □

Lemma 5.4. If X is a small pointed motivic space and i is an integer, then for any α ∈ Ū^{4i,2i}(X) there exists an n ≥ 0 with t_{X∧T^{∧2n}} ∘ φ̄_{X∧T^{∧2n}}(Σ_{T}^{2n}α) = Σ_{T}^{2n}α.

Proof. We may assume that α = a ⊗ b with a ∈ U^{4d,2d}(X) and b ∈ BO^{4i−4d,2i−2d}(pt). For a small motivic space X there is a canonical isomorphism [12, Theorem 5.2]

U^{4d,2d}(X) = colim_{m} Hom_{H_{•}(S)}(X ∧ T^{∧m}, U_{2d+m}).

This isomorphism implies that there exists an integer n ≥ 0 such that Σ_{T}^{2n}a = f^{∗}[u_{2d+2n}] for an appropriate map f: X ∧ T^{∧2n} → U_{2d+2n} in H_{•}(S). We may assume that d + n ≥ N and that n + i is odd.

We have [u_{2d+2n}] ⊗ b ∈ Ū^{4n+4i,2n+2i}(U_{2d+2n}). By hypothesis

φ̄_{U_{2d+2n}}: Ū^{4n+4i,2n+2i}(U_{2d+2n}) → BO^{4n+4i,2n+2i}(U_{2d+2n})

is an isomorphism. So its section t_{U_{2d+2n}} is the inverse isomorphism. Hence we have

(t_{U_{2d+2n}} ∘ φ̄_{U_{2d+2n}})([u_{2d+2n}] ⊗ b) = [u_{2d+2n}] ⊗ b.

Then by the functoriality of Ū, t and φ̄ we have

Σ_{T}^{2n}α = f^{∗}([u_{2d+2n}]⊗b) = f^{∗} ∘t_{U_{2d+2n}} ∘φ̄_{U_{2d+2n}}([u_{2d+2n}]⊗b) = t_{X∧T^{∧2n}} ∘φ̄_{X∧T^{∧2n}}(Σ_{T}^{2n}α). □

Lemma 5.5. Suppose for some (p, q) that the homomorphism φ̄_{X}: Ū^{p,q}(X) → BO^{p,q}(X) is an isomorphism for all small pointed motivic spaces X. Then the same holds for (p − 1, q) and (p − 1, q − 1).

Proof. For (p − 1, q) this is because the suspension Σ_{S^{1}} induces isomorphisms U^{p−1,q}(X) ≅ U^{p,q}(X ∧ S^{1}) and similar isomorphisms for Ū and BO, and these are compatible with φ and φ̄. For (p − 1, q − 1) use the suspension Σ_{𝔾_{m}}. □

Proof of Theorem 5.1. First suppose (p, q) = (8i + 4, 4i + 2) for some i. Then for any small motivic space X the map φ_{X}: Ū^{8i+4,4i+2}(X) → BO^{8i+4,4i+2}(X) is surjective because it has the section t_{X} of Lemma 5.3. To show it injective, we suppose α is in its kernel. The suspension Σ_{T} is compatible with φ and φ̄, so we have φ̄_{X∧T^{∧2n}}(Σ_{T}^{2n}α) = Σ_{T}^{2n}φ_{X}(α) = 0. By Lemma 5.4 we therefore also have Σ_{T}^{2n}α = 0. But Σ_{T}^{2n} induces an isomorphism of cohomology groups. So we have α = 0. Thus φ̄_{X}: Ū^{p,q}(X) → BO^{p,q}(X) is an isomorphism for all small motivic spaces X for (p, q) = (8i + 4, 4i + 2).

The result for other values of (p, q) follows from Lemma 5.5 and a numerical argument. □

6. Last details

Proof of Theorem 1.1. By the universality of the symplectically oriented commutative ring T -spectrum (MSp, th^{MSp}) (Theorem 4.5) there is a unique morphism φ: MSp → BO of commutative ring T -spectra with φ(th^{MSp}) = th^{BO}. It induces the morphisms of (5):

φ̄_{X}: MSp^{∗,∗}(X) ⊗_{MSp^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt) → BO^{∗,∗}(X),
φ̄_{X}: MSp^{4∗,2∗}(X) ⊗_{MSp^{4∗,2∗}(pt)} BO^{4∗,2∗}(pt) → BO^{4∗,2∗}(X).

The second morphism, with the bidegrees (4i, 2i) only, is an isomorphism for X = MSp_{2r} for all r by Theorem 4.6. So all the hypotheses of Theorem 5.1 hold with (U, u) = (MSp, th^{MSp}). The conclusions of Theorem 5.1 imply Theorem 1.1. □

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12 I. PANIN AND C. WALTER

Steklov Institute of Mathematics at St. Petersburg, Russia Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS), Département de mathématiques, Univer-

sité de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France