THE SPECIAL LINEAR VERSION OF THE PROJECTIVE BUNDLE THEOREM

ALEXEY ANANYEVSKIY

Abstract. A special linear Grassmann variety SGr(k, n) is the complement to the zero section of the determinant of the tautological vector bundle over Gr(k, n). For a representable ring cohomology theory A(−) with a special linear orientation and invertible stable Hopf map η, including Witt groups and MSL[η^{−1}], we have A(SGr(2, 2n + 1)) = A(pt)[e]/⟨e^{2n}⟩, and A(SGr(2, 2n)) is a truncated polynomial algebra in two variables over A(pt). A splitting principle for such theories is established. We use the computations for the special linear Grassmann varieties to calculate A(BSL_{n}) in terms of the homogeneous power series in certain characteristic classes of the tautological bundle.

1. Introduction.

The basic and most fundamental computation for oriented cohomology theories is the projective bundle theorem (see [Mor1] or [PS, Theorem 3.9]) claiming A(ℙ^{n}) to be a truncated polynomial ring over A(pt) with an explicit basis in terms of the powers of a Chern class. Having this result at hand one can define higher characteristic classes and compute the cohomology of Grassmann varieties and flag varieties. In particular, the fact that cohomology of the full flag variety is a truncated polynomial algebra gives rise to a splitting principle, which states that from a viewpoint of the oriented cohomology theory every vector bundle is in a certain sense a sum of linear bundles. For a representable cohomology theory one can deal with an infinite dimensional Grassmannian which is a model for the classifying space BGL_{n} and obtain even neater answer, the formal power series in the characteristic classes of the tautological vector bundle.

There are analogous computations for symplectically oriented cohomology theories [PW1] with appropriate chosen varieties: quaternionic projective spaces HP^{n} instead of the ordinary ones and symplectic Grassmannian and flag varieties. The answers are essentially the same, algebras of truncated polynomials in characteristic classes.

These computations have a variety of applications, for example theorems of Conner and Floyd’s type [CF] describing the K-theory and hermitian Ktheory as quotients of certain universal cohomology theories [PPR1, PW4].

In the present paper we establish analogous results for the cohomology theories with special linear orientations. The notion of such orientation was introduced in [PW3, Definition 5.1]. At the same preprint there was constructed a universal example of a cohomology theory with a special linear orientation, namely the algebraic special linear cobordisms MSL, [PW3, Definition 4.2]. A more down to earth example is derived Witt groups defined by Balmer [Bal1] and oriented via Koszul complexes [Ne2]. A comprehensive survey on the Witt groups could be found in [Bal2]. Of course, every oriented cohomology theory admits a special linear orientation, but it will turn out that we are not interested in such examples. We will deal with representable cohomology theories and work in the unstable H_{•}(k) and stable SH(k) motivic homotopy categories introduced by Morel and Voevodsky [MV, V]. We recall all the necessary constructions and notions in sections 2-4 as well as provide preliminary calculations with special linear orientations.

Then we need to choose an appropriate version of "projective space" analogous to ℙ^{n} and HP^{n}. Natural candidates are SL_{n+1}/SL_{n} and 𝔸^{n+1} − {0}. There is no difference which one to choose since the first one is an affine bundle over the latter one, so they have the same cohomology. We take 𝔸^{n+1} − {0} since it looks prettier from the geometric point of view. There is a calculation for the Witt groups of this space [BG, Theorem 8.13] claiming that W^{∗}(𝔸^{n+1} − {0}) is a free module of rank two over W^{∗}(pt) with an explicit basis. The fact that it is a free module of rank two is not surprising since 𝔸^{n+1} −{0} is a sphere in the stable homotopy category SH_{•}(k) and W^{∗}(−) is representable [Hor]. The interesting part is the basis. Let T = O^{n+1}/O(−1) be the tautological rank n bundle over 𝔸^{n+1} − {0}. Then for n = 2k the basis consists of the element 1 and the class of a Koszul complex. The latter one is the Euler class e(T ) in the Witt groups. Unfortunately, for the odd n the second term of the basis looks more complicated. Moreover, for the oriented cohomology theory even in the case of n = 2k the corresponding Chern class vanishes, so one can not expect that 1 and e(T ) form a basis for every cohomology theory with a special linear orientation.

Here we introduce another principle. The maximal compact subgroup of SL_{n}(ℝ) is SO_{n}(ℝ), so over ℝ the notion of a special linear orientation of a vector bundle derives to the usual topological orientation of the bundle. The Euler classes of oriented vector bundles in topology behave themselves well only after inverting 2 in the coefficients, so we want to invert in the algebraic setting something analogous to 2. There are two interesting elements in the stable cohomotopy groups π^{∗,∗}(pt) that go to 2 after taking ℝ-points, a usual 2 ∈ π^{0,0}(pt) and the stable Hopf map η ∈ π^{−1,−1}(pt) arising from the morphism 𝔸^{2} − {0} → ℙ^{1}. In general 2 is not invertible in the Witt groups, so we will invert η. Moreover, recall a theorem due to Morel [Mor2] claiming that for a perfect field there is an isomorphism ⊕_{n}π^{n,n}(Spec k)[η^{−1}] ≅ W^{0}(k)[η, η^{−1}], so in a certain sense η is invertible in the Witt groups. In sections 5-6 we do some computations justifying the choice of η.

In this paper we deal mainly with the cohomology theories obtained as follows. Take a commutative monoid (A, m, e : 𝕊 → A) in the stable homotopy category SH(k) and fix a special linear orientation on the cohomology theory A^{∗,∗}(−). The unit e : 𝕊 → A of the monoid (A, m, e) induces a morphism of cohomology theories π^{∗,∗}(−) → A^{∗,∗}(−) making A^{∗,∗}(X) an algebra over the stable cohomotopy groups. Set η = 1, that is

A^{∗}(X) = A^{∗,∗}(X)/⟨1 − η⟩.

It is an easy observation that A^{∗}(−) is still a cohomology theory, see Section 5. For these cohomology theories we have a result analogous to the case of the Witt groups.

Theorem. There is an isomorphism

A^{∗}(𝔸^{2n+1} − {0}) ≅ A^{∗}(pt) ⊕ A^{∗−2n}(pt)e(T ).

The relative version of this statement is Theorem 3 in section 7. Note that there is no similar result for 𝔸^{2n} − {0}.

In the next section we consider another family of varieties, called special linear Grassmannians SGr(2, n) = SL_{n}/P′_{2}, where P′_{2} stands for the derived group of the parabolic subgroup P_{2}, i.e. P′_{2} is the stabilizer of e_{1} ∧ e_{2} in the exterior square of the regular representation of SL_{n}. There are tautological bundles T_{1} and T_{2} over SGr(2, n) of ranks 2 and n − 2 respectively. We have the following theorem which seems to be the correct version of the projective bundle theorem for the special linear orientation.

Theorem. For the special linear Grassmann varieties we have the next isomorphisms.

A^{∗}(SGr(2, 2n)) ≅ ⨁_{i=0}^{2n−2} A^{∗−2i}(pt)e(T_{1})^{i} ⊕ A^{∗−2n+2}(pt)e(T_{2}),

A^{∗}(SGr(2, 2n + 1)) ≅ ⨁_{i=0}^{2n−1} A^{∗−2i}(pt)e(T_{1})^{i}.

Recall that there is a recent computation of the twisted Witt groups of Grassmannians [BC]. The twisted groups are involved since the authors use pushforwards that exist only in the twisted case. We deal with the varieties with a trivialized canonical bundle and closed embeddings with a special linear normal bundle in order to avoid these difficulties. In fact we are interested in the relative computations that could be extended to the Grassmannian bundles, so we look for a basis consisting of characteristic classes rather then pushforwards of certain elements. It turns out that such bases exist only for the special linear flag varieties with all but at most one dimension step being even, i.e. we can handle SGr(1, 7), SF(2, 4, 6) and SF(2, 5, 7) but not SGr(3, 6). Nevertheless it seems that one can construct the basis for the latter case in terms of pushforwards.

Section 9 deals with symmetric polynomials and algebras of coinvariants which appear in section 10 as the cohomology rings of maximal SL_{2} flag varieties,

SF(2n) = SL_{2n}/P′_{2,4,...,2n−2}, SF(2n + 1) = SL_{2n+1}/P′_{2,4,...,2n}.

We obtain an analogue of the splitting principle in Theorem 7 and its relative version.

Theorem. For n ≥ 1 consider

s_{i} = σ_{i}(e_{1}^{2}, e_{2}^{2}, ..., e_{n}^{2}), t = σ_{n}(e_{1}, e_{2}, …, e_{n})

with σ_{i} being the elementary symmetric polynomials in n variables. Then we have the following isomorphisms

(1) A^{∗}(SF(2n)) ≅ A^{∗}(pt)[e^{1}, e^{2}, ..., e^{n}]/⟨s_{1}, s_{2}, …, s_{n−1}, t⟩,

(2) A^{∗}(SF(2n + 1)) ≅ A^{∗}(pt)[e^{1}, e^{2}, ..., e^{n}]/⟨s_{1}, s_{2}, …, s_{n}⟩.

Note that one can substitute the SL_{n}/(SL_{2})^{[n/2]} instead of SF(n). These answers and the choice of commuting SL_{2} in SL_{n} perfectly agree with our principle that SL_{n}(ℝ) stands for SO_{n}(ℝ), since SL_{2}(ℝ) stands for the compact torus S^{1}, and the choice of maximal number of commuting SL_{2} is parallel to the choice of the maximal compact torus. We get the coinvariants for the Weyl groups W (B_{n}) and W (D_{n}) and it is what one gets computing the cohomology of SO_{n}(ℝ)/T.

At the end, in section 11, we assemble the calculations for the special linear Grassmannians and compute in Theorem 8 the cohomology of the classifying spaces in terms of the homogeneous formal power series.

Theorem. We have the following isomorphisms.

A^{∗}(BSL_{2n}) ≅ A^{∗}(pt) [[b_{1}, …, b_{n−1}, e]]_{h} ,
A^{∗}(BSL_{2n+1}) ≅ A^{∗}(pt) [[b_{1}, …, b_{n}]]_{h} .

Finally, we leave for the forthcoming paper [An] the careful proof of the fact that Witt groups arise from the hermitian K-theory in the described above fashion, that is W^{∗}(X) ≅ BO^{∗,∗}(X)/⟨1 − η⟩. We give only a sketch of the proof in Proposition 3. In the same paper we are going to prove the following special linear version of the motivic Conner and Floyd theorem.

Theorem. Let k be a field of characteristic different from 2. Then for all small pointed motivic spaces Y over k there is an isomorphism

MSL^{∗,∗}(Y) ⊗_{MSL^{4∗,2∗}(pt)} W^{2∗}(pt) ≅ W^{∗}(Y).

Another application of the developed technique lies in the field of the equivariant Witt groups and we are going to address it in another paper.

Acknowledgement. The author wishes to express his sincere gratitude to I. Panin for the introduction to the beautiful world of 𝔸^{1}-homotopy theory and numerous discussions on the subject of this paper. Also the author acknowledges support of the RFFI-project 10-01-00551-а.

2. Preliminaries on SH(k) and ring cohomology theories.

Let k be a field of characteristic different from 2 and let Sm/k be the category of smooth varieties over k.

A motivic space over k is a simplicial presheaf on Sm/k. Each X ∈ Sm/k defines an unpointed motivic space Hom_{Sm/k}(−, X) constant in the simplicial direction. We will often write pt for the Spec k regarded as a motivic space.

We use the injective model structure on the category of the pointed motivic spaces M_{•}(k). Inverting the weak motivic equivalences in M_{•}(k) gives the pointed motivic unstable homotopy category H_{•}(k).

Let T = 𝔸^{1}/(A^{1} − {0}) be the Morel-Voevodsky object. A T -spectrum M [Jar] is a sequence of pointed motivic spaces (M_{0}, M_{1}, M_{2}, …) equipped with the structural maps σ_{n}: T ∧ M_{n} → M_{n+1}. A map of T -spectra is a sequence of maps of pointed motivic spaces which is compatible with the structure maps. We write MS(k) for the category of T -spectra. Inverting the stable motivic weak equivalences as in [Jar] gives the motivic stable homotopy category SH(k).

A pointed motivic space X gives rise to a suspension T -spectrum Σ_{T}^{∞}X. Set 𝕊 = Σ_{T}^{∞}(pt_{+}) for the spherical spectrum. Both H_{•}(k) and SH(k) are equipped with symmetric monoidal structures (∧, pt_{+}) and (∧, 𝕊) respectively and

Σ_{T}^{∞}: H_{•}(k) → SH(k)

is a strict symmetric monoidal functor. We will usually omit the subscript T and write Σ^{∞} for this functor.

Recall that there are two spheres in M_{•}(k), the simplicial one S^{1,0} = S_{s}^{1} = Δ^{1}/∂(Δ^{1}) and S^{1,1} = (𝔾_{m}, 1). We write S^{p,q} for (S_{s}^{1})^{∧p−q} ∧ (𝔾_{m}, 1)^{∧q} and Σ^{p,q} for the suspension functor − ∧ S^{p,q}. In the motivic homotopy category there is a canonical isomorphism T ≅ S^{2,1}.

Any T -spectrum A defines a bigraded cohomology theory on the category of pointed motivic spaces. Namely, for a pointed space (X, x) one sets

A^{p,q}(X, x) = Hom_{SH(k)}(Σ^{∞}(X, x), Σ^{p,q}A)

and A^{∗,∗}(X, x) = ⨁_{p,q} A^{p,q}(X, x). In case of j, i − j ≥ 0 one has a canonical suspension isomorphism A^{p,q}(X, x) ≅ A^{p+i,q+j}(Σ^{i,j}(X, x)). For an unpointed space X we set A^{p,q}(X) = A^{p,q}(X_{+}, +) with A^{∗,∗}(X) defined accordingly. Set π^{i,j}(X) = 𝕊^{i,j}(X) to be the stable cohomotopy groups of X.

We can regard smooth varieties as unpointed motivic spaces and obtain the groups A^{p,q}(X). Given a closed embedding i: Z → X of varieties we write T h(i) for X/(X − Z). For a vector bundle E → X set T h(E) = E/(E − X) to be the Thom space of E.

A commutative ring T -spectrum is a commutative monoid (A, m, e) in (SH(k), ∧, 𝕊). The cohomology theory defined by a commutative T -spectrum is a ring cohomology theory satisfying a certain bigraded commutativity condition described by Morel.

We recall the essential properties of the cohomology theories represented by a commutative ring T -spectrum A.

(1) Localization: for a closed embedding of varieties i: Z → X with a smooth X and an open complement j: U → X we have a long exact sequence

… ⟶^{∂} A^{∗,∗}(T h(i)) ⟶^{z^{A}} A^{∗,∗}(X) ⟶^{j^{A}} A^{∗,∗}(U) ⟶^{∂} A^{∗+1,∗}(T h(i)) ⟶^{z^{A}} …

It is a special case of the cofiber long exact sequence.

(2) Nisnevich excision: consider a Cartesian square of smooth varieties

\xymatrix{Z′ \ar[r]^{i′} \ar[d]^{f′} & X′ \ar[d]^{f} \\ Z \ar[r]^{i} & X}

where i is a closed embedding, f is etale and f′ is an isomorphism. Then for the induced morphism g: T h(i′) → T h(i) the corresponding morphism g^{A}: A^{∗,∗}(T h(i)) → A^{∗,∗}(T h(i′)) is an isomorphism. It follows from the fact that g is an isomorphism in the homotopy category.

(3) Homotopy invariance: for an 𝔸^{n}-bundle p: E → X over a variety X the induced homomorphism p^{A}: A^{∗,∗}(X) → A^{∗,∗}(E) is an isomorphism.

(4) Mayer-Vietoris: if X = U_{1} ∪ U_{2} is a union of two open subsets U_{1} and U_{2} then there is a natural long exact sequence

… → A^{∗,∗}(X) → A^{∗,∗}(U_{1}) ⊕ A^{∗,∗}(U_{2}) → A^{∗,∗}(U_{1} ∩ U_{2}) → A^{∗+1,∗}(X) → …

(5) Cup-product: for a motivic space Y we have a functorial graded ring structure

∪: A^{∗,∗}(Y) × A^{∗,∗}(Y) → A^{∗,∗}(Y).

Also, for closed subsets i_{1} : Z_{1} → X and i_{2} : Z_{2} → X set i_{12} : Z_{1} ∩ Z_{2} → X, then we have functorial, bilinear and associative cup-product

∪: A^{∗,∗}(T h(i_{1})) × A^{∗,∗}(T h(i_{2})) → A^{∗,∗}(T h(i_{12})).

In particular, setting Z_{1} = X we obtain an A^{∗,∗}(X)-module structure on A^{∗,∗}(T h(i_{2})). All the morphisms in the localization sequence are homomorphisms of A^{∗,∗}(X)-modules.

(6) Module structure over stable cohomotopy groups: for every motivic space Y we have a homomorphism of graded rings π^{∗,∗}(Y) → A^{∗,∗}(Y), which defines a π^{∗,∗}(pt)-module structure on A^{∗,∗}(Y). For a smooth variety X the ring A^{∗,∗}(X) is a graded π^{∗,∗}(pt)-algebra via π^{∗,∗}(pt) → π^{∗,∗}(X) → A^{∗,∗}(X).

(7) Graded ϵ-commutativity [Mor1]: let ϵ ∈ π^{0,0}(pt) be the element corresponding under the suspension isomorphism to the morphism T → T, x ↦ −x. Then for every motivic space X and a ∈ A^{i,j}(X), b ∈ A^{p,q}(X) we have

a ∪ b = (−1)^{ip}ϵ^{jq} ∪ b ∪ a.

Recall that ϵ^{2} = 1.

3. Special linear orientation.

In this section we recall the notion of a special linear orientation introduced in [PW3] and establish some of its basic properties.

Definition 1. A special linear bundle over a variety X is a pair (E, λ) with E → X a vector bundle and λ: det E ⟶^{≃} O_{X} 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 λ′ ∘ (det ϕ) = λ.

Notation 1. Consider a trivialized rank n bundle O_{X}^{n} over a smooth variety X. There is a canonical trivialization det O_{X}^{n} ⟶^{≃} O_{X}. We denote the corresponding special linear bundle by (O_{X}^{n}, 1).

Lemma 1. Let (E, λ) be a special linear bundle over a smooth variety X such that E ≅ O_{X}^{n}. Then there exists an isomorphism of special linear bundles

ϕ: (E, λ) ⟶^{≃} (O_{X}^{n}, 1).

Proof. An exact sequence of algebraic groups

1 → SL_{n} → GL_{n} ⟶^{det} 𝔾_{m} → 1

induces an exact sequence of pointed sets

H^{0}(X, GL_{n}) ⟶^{p} H^{0}(X, 𝔾_{m}) → H^{1}(X, SL_{n}) ⟶^{i} H^{1}(X, GL_{n})

There is a splitting 𝔾_{m} → GL_{n} for det, so p is surjective. Hence we have ker i = {∗} and this means that, up to an isomorphism of special linear bundles, there exists only one trivialization λ: det O_{X}^{n} → O_{X}. □

Lemma 2. Let E_{1} be a subbundle of a vector bundle E over a smooth variety X. Then there are canonical isomorphisms

(1) det E_{1} ⊗ det(E/E_{1}) ≅ det E,

(2) det E^{∨} ≅ (det E)^{∨}.

Proof. These isomorphisms are induced by the corresponding vector space isomorphisms. In the first case we have Λ^{m}V_{1} ⊗ Λ^{n}(V/V_{1}) ⟶^{≃} Λ^{m+n}V with

v_{1} ∧ … ∧ v_{m} ⊗ \overline{w}_{1} ∧ … ∧ \overline{w}_{n} ↦ v_{1} ∧ … ∧ v_{m} ∧ w_{1} ∧ … ∧ w_{n}.

For the second isomorphism consider the perfect pairing

ϕ: Λ^{n}V × Λ^{n}V^{∨} → k

defined by

ϕ(v_{1} ∧ … ∧ v_{n}, f_{1} ∧ … ∧ f_{n}) = ∑_{σ∈S_{n}} sign(σ)f_{σ(1)}(v_{1}) · . . . · f_{σ(n)}(v_{n}). □

Definition 2. Let T = (E, λ_{E}) be a special linear bundle over a smooth variety X. By Lemma 2 there is a canonical trivialization λ_{E^{∨}}: det E^{∨} ⟶^{≃} O_{X}. The special linear bundle T^{∨} = (E^{∨}, λ_{E^{∨}}) is called the dual special linear bundle.

Definition 3. Let A^{∗,∗}(−) be a cohomology theory represented by a T - spectrum A. A (normalized) special linear orientation on A^{∗,∗}(−) is a rule which assigns to every special linear bundle (E, λ) of rank n over a smooth variety X a class th(E, λ) ∈ A^{2n,n}(T h(E)) satisfying the following conditions [PW3, Definition 5.1]:

(1) For an isomorphism f: (E, λ) ⟶^{≃} (E′, λ′) we have th(E, λ) = f^{A}th(E′, λ′).

(2) For a morphism r: Y → X we have r^{A}th(E, λ) = th(r^{∗}(E, λ)) in A^{2n,n}(T h(r^{∗}E)).

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

(4) We have

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

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

(5) (normalization) For the trivial line bundle over a point we have

th(O_{pt}, 1) = Σ^{2,1}1 ∈ A^{2,1}(T). The isomorphism − ∪ th(E, λ) is a Thom isomorphism. The class th(E, λ) is a Thom class of the special linear bundle, and

e(E, λ) = z^{A}th(E, λ) ∈ A^{2n,n}(X)

with natural z: X → T h(E) is its Euler class.

Remark 1. For a rank 2n special linear bundle (E, λ) over a variety X we have th(E, λ) ∈ A^{4n,2n}(T h(E)) and e(E, λ) ∈ A^{4n,2n}(X), so this classes are universally central.

Recall that a symplectic bundle is a special linear bundle in a natural way, so having a special linear orientation we have the Thom classes also for symplectic bundles, thus a cohomology theory with a special linear orientation is also symplectically oriented. We recall the definition of the Pontryagin classes theory [PW1, Definition 14.1] which is equivalent to the symplectic orientation via Thom classes.

Definition 4. Let A^{∗,∗}(−) be a cohomology theory represented by a T - spectrum A. A Pontryagin classes theory on A^{∗,∗}(−) is a rule which assigns to every symplectic bundle (E, ϕ) over every smooth variety X a system of Pontryagin classes p_{i}(E, ϕ) ∈ A^{4i,2i}(X) for all i ≥ 1 satisfying

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

(2) For a morphism r: Y → X and a symplectic bundle (E, ϕ) over X we have r^{A}(p_{i}(E, ϕ)) = p_{i}(r^{∗}(E, ϕ)) for all i.

(3) For the tautological rank 2 symplectic bundle (E, ϕ) over

HP^{1} = Sp_{4}/(Sp_{2} × Sp_{2})

the elements 1, p_{1}(E, ϕ) form a A^{∗,∗}(pt)-basis of A^{∗,∗}(HP^{1}).

(4) For a rank 2 symplectic bundle (V, ϕ) over pt we have p_{1}(V, ϕ) = 0.

(5) For an orthogonal direct sum of symplectic bundles (E, ϕ) ≅ (E_{1}, ϕ_{1}) ⊥ (E_{2}, ϕ_{2}) we have

p_{i}(E, ϕ) = 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})

for all i.

(6) For (E, ϕ) of rank 2r we have p_{i}(E, ϕ) = 0 for i > r. We set p_{∗}(E, ϕ) = 1 + ∑_{j=1}^{∞} p_{i}(E, ϕ)t^{i} to be the total Pontryagin class.

Every oriented cohomology theory possesses a special linear orientation via th(E, λ) = th(E), so one can consider K-theory or algebraic cobordism represented by MGL as examples. We have two main instances of the theories with a special linear orientation but without a general one. The first one is hermitian K-theory [Sch] represented by the spectrum BO [PW2]. The special linear orientation on BO^{∗,∗} via Koszul complexes could be found in [PW2]. The second one is universal in the sense of [PW3, Theorem 5.9] and represented by the algebraic special linear cobordism spectrum MSL [PW3, Definition 4.2].

Notation 2. From now on A^{∗,∗}(−) is a ring cohomology theory represented by a commutative monoid in SH(k) with a fixed special linear orientation.

Lemma 3. Let X be a smooth variety. Then th(O_{X}^{n}, 1) = Σ^{2n,n}1 and th(O_{X}, −1) = Σ^{2,1}ϵ.

Proof. It follows immediately from the conditions (4) and (5) and functoriality. □ Lemma 4. Let (E, λ_{E}) be a special linear bundle over a smooth variety X. Then

e(E, λ_{E}) = ϵ ∪ e(E, −λ_{E}).

Proof. Consider the bundle E ⊕O_{X} and denote the projections onto the summands by q_{1}, q_{2}. We have

(E ⊕ O_{X}, λ_{E} ⊗ 1) = (E ⊕ O_{X}, (−λ_{E}) ⊗ −1),

hence

q_{1}^{∗}th(E, λ_{E}) ∪ q_{2}^{∗}Σ1 = q_{1}^{∗}th(E, −λ_{E}) ∪ q_{2}^{∗}Σϵ.

By the suspension isomorphism we obtain

th(E, λ_{E}) = th(E, −λ_{E}) ∪ ϵ,

hence e(E, λ_{E}) = ϵ ∪ e(E, −λ_{E}). □

Lemma 5. Let T be a rank 2 special linear bundle over a smooth variety X. Then T ≅ T^{∨} and e(T ) = e(T^{∨}).

Proof. Set T = (E, λ_{E}). The trivialization λ_{E}: Λ^{2}E ⟶^{≃} O_{X} defines a symplectic form on E and an isomorphism ϕ: E ⟶^{≃} E^{∨}, thus it is sufficient to check that

λ_{E^{∨}} ∘ det ϕ = λ_{E}.

It could be checked locally, so we can suppose that E ≅ O_{X}^{2} and, in view of Lemma 1, (E, λ_{E}) ≅ (O_{X}^{2}, 1). Fixing a basis {e_{1}, e_{2}} such that e_{1} ∧ e_{2} = 1 and taking the dual basis {e_{1}^{∨}, e_{2}^{∨}} for (O_{X}^{2})^{∨} we have

ϕ(e_{1}) = (e_{1} ∧ −) = e_{2}^{∨}, ϕ(e_{2}) = (e_{2} ∧ −) = −e_{1}^{∨}.

Thus we obtain

det ϕ(e_{1} ∧ e_{2}) = e_{2}^{∨} ∧ (−e_{1}^{∨}) = e_{1}^{∨} ∧ e_{2}^{∨}

and

λ_{E^{∨}} det ϕ(e_{1} ∧ e_{2}) = λ_{E^{∨}}(e_{1}^{∨} ∧ e_{2}^{∨}) = 1.  □

Notation 3. For a vector bundle E we denote by E^{0} the complement to the zero section. For a special linear bundle T = (E, λ) we set T^{0} = E^{0}.

Definition 5. Let T be a rank n special linear bundle over a smooth variety X. The Gysin sequence is a long exact sequence

… ⟶^{∂} A^{∗−2n,∗−n}(X) ⟶^{∪e(T )} A^{∗,∗}(X) → A^{∗,∗}(T^{0}) ⟶^{∂} A^{∗−2n+1,∗−n}(X) → …

obtained from the localization sequence for the zero section X → T via homotopy invariance and Thom isomorphism.

Lemma 6. Let (E, λ_{E}) be a special linear bundle over a smooth variety X. (1) Let λ′_{E} be any other trivialization of det E. Then one has

A^{0,0}(X) ∪ e(E, λ_{E}) = A^{0,0}(X) ∪ e(E, λ′_{E}).

(2) For the dual special linear bundle (E^{∨}, λ_{E^{∨}}) one has

A^{0,0}(X) ∪ e(E, λ_{E}) = A^{0,0}(X) ∪ e(E^{∨}, λ_{E^{∨}}).

Proof. Set n = rank E and denote the projections E^{0} → X and E^{∨0} → X by p and p′ respectively.

(1) Consider the Gysin sequences corresponding to the trivializations λ_{E} and λ′_{E}.

\xymatrix{… \ar[r] & A^{0,0}(X) \ar[r]^{∪e(E,λ_{E})} & A^{2n,n}(X) \ar[r]^{p^{A}} \ar[d]^{=} & A^{2n,n}(E^{0}) \ar[d]^{=} \ar[r] & … \\ … \ar[r] & A^{0,0}(X) \ar[r]^{∪e(E,λ′_{E})} & A^{2n,n}(X) \ar[r]^{p^{A}} & A^{2n,n}(E^{0}) \ar[r] & …}

We have

A^{0,0}(X) ∪ e(E, λ_{E}) = ker p^{A} = A^{0,0}(X) ∪ e(E, λ′_{E}).

(2) Consider

Y = {(v, f) ∈ E ×_{X} E^{∨} | f(v) = 1} .

Projections p_{1} : Y → E^{0} and p_{2} : Y → E^{∨0} have fibres isomorphic to 𝔸^{n−1}, thus

A^{∗,∗}(E^{0}) ≅ A^{∗,∗}(Y) ≅ A^{∗,∗}(E^{∨0})

and there is a canonical isomorphism A^{∗,∗}(E^{0}) ≅ A^{∗,∗}(E^{∨0}) over A^{∗,∗}(X). Now proceed as in the first part and consider the Gysin sequences.

\xymatrix{… \ar[r] & A^{0,0}(X) \ar[r]^{∪e(E,λ_{E})} & A^{2n,n}(X) \ar[r]^{p^{A}} \ar[d]^{=} & A^{2n,n}(E^{0}) \ar[d]^{≅} \ar[r] & … \\ … \ar[r] & A^{0,0}(X) \ar[r]^{∪e(E^{∨},λ_{E^{∨}})} & A^{2n,n}(X) \ar[r]^{p^{′A}} & A^{2n,n}(E^{∨0}) \ar[r] & …}

We have

A^{0,0}(X) ∪ e(E, λ_{E}) = ker p^{A} = ker p^{′A} = A^{0,0}(X) ∪ e(E^{∨}, λ_{E^{∨}}).  □

Lemma 7. Let T be a special linear bundle over a smooth variety X such that there exists a nowhere vanishing section s: X → T . Then e(T ) = 0.

Proof. Set rank T = n and consider the Gysin sequence

… → A^{0,0}(X) ⟶^{∪e(T )} A^{2n,n}(X) ⟶^{j^{A}} A^{2n,n}(T^{0}) → …

The section s induces a splitting s^{A} for j^{A}, thus j^{A} is injective and

e(T ) = 1 ∪ e(T ) = 0.  □

4. Pushforwards along closed embeddings.

In this section we give the construction of the pushforwards along the closed embeddings with special linear normal bundles for a cohomology theory with a special linear orientation. It is quite similar to the construction of such pushforwards for oriented [PS, Ne1] or symplectically oriented [PW1] cohomology theories and twisted Witt groups [Ne2].

Definition 6. Let i: Z → X be a closed embedding of smooth varieties. The deformation space D(Z, X) is obtained as follows.

(1) Consider X × 𝔸^{1}.

(2) Blow-up it along Z × 0.

(3) Remove the blow-up of X × 0 along Z × 0. This construction produces a smooth variety D(Z, X) over 𝔸^{1}. The fiber over 0 is canonically isomorphic to N_{i} while the fiber over 1 is isomorphic to X and we have the corresponding closed embeddings i_{0} : N_{i} → D(Z, X) and i_{1} : X → D(Z, X). There is a closed embedding z: Z × 𝔸^{1} → D(Z, X) such that over 0 it coincides with the zero section s: Z → N_{i} of the normal bundle and over 1 it coincides with the closed embedding i: Z → X. At last, we have a projection p: D(Z, X) → X.

Thus we have homomorphisms of A^{∗,∗}(X)-modules (via p^{A})

\xymatrix{A^{∗,∗}(T h(N_{i})) & A^{∗,∗}(T h(z)) \ar[l]_{i_{0}^{A}} \ar[r]^{i_{1}^{A}} & A^{∗,∗}(T h(i)).}

These homomorphisms are isomorphisms, since in the homotopy category H_{•}(k) we have isomorphisms i_{0} : T h(N_{i}) ≅ T h(z) and i_{1} : T h(i) ≅ T h(z) [MV, Theorem 2.23]. We set

d_{i}^{A} = i_{1}^{A} ∘ (i_{0}^{A})^{−1} : A^{∗,∗}(T h(N_{i})) → A^{∗,∗}(T h(i))

to be the deformation to the normal bundle isomorphism. The functoriality of the deformation space D(Z, X) makes the deformation to the normal bundle isomorphism functorial.

Definition 7. For a closed embedding i: Z → X of smooth varieties a special linear normal bundle is a pair (N_{i}, λ) with N_{i} the normal bundle and λ: det N_{i} ⟶^{≃} O_{Z} an isomorphism of line bundles.

Definition 8. Let i: Z → X be a closed embedding of smooth varieties with a rank n special linear normal bundle (N_{i}, λ). Denote by ĩ_{A} the composition of the Thom and deformation to the normal bundle isomorphisms,

ĩ_{A} = d_{i}^{A} ∘ (− ∪ th(N_{i}, λ)): A^{∗−2n,∗−n}(Z) ⟶^{≃} A^{∗,∗}(T h(i)).

For the inclusion z: X → T h(i) the composition

i_{A} = z^{A} ∘ ĩ_{A}: A^{∗−2n,∗−n}(Z) → A^{∗,∗}(X)

is the pushforward map. Note that in general i_{A} depends on the trivialization of det N_{i}.

Remark 2. We have an analogous definition of the pushforward map for a closed embedding i: Z → X in every cohomology theory possessing a Thom class for the normal bundle N_{i}. In particular, we have pushforwards in the stable cohomotopy groups for closed embeddings with a trivialized normal bundle (N_{i}, θ), where θ: N_{i} ⟶^{≃} O_{Z}^{n} is an isomorphism of vector bundles, since there is a Thom class th(O_{Z}^{n}) = Σ^{2n,n}1 and suspension isomorphism

π^{∗−2n,∗−n}(Z) ⟶^{∪Σ^{2n,n}1} π^{∗,∗}(T h(O_{Z}^{n})).

Notation 4. Let i: Z → X be a closed embedding of smooth varieties with a rank n special linear normal bundle. Then using the notation of pushforward maps the localization sequence boils down to

… ⟶^{∂} A^{∗−2n,∗−n}(Z) ⟶^{i_{A}} A^{∗,∗}(X) ⟶^{j^{A}} A^{∗,∗}(X − Z) ⟶^{∂} A^{∗−2n+1,∗−n}(Z) ⟶^{i_{A}} …

In the rest of this section we sketch some properties of the pushforward maps. The next lemma is similar to [PW1, Proposition 7.4].

Lemma 8. Consider the following pullback diagram with all the involved varieties being smooth.

\xymatrix{X′ = X ×_{Y} Y′ \ar[r]^{i′} \ar[d]^{g′} & Y′ \ar[d]^{g} \\ X \ar[r]^{i} & Y}

Let i, i′ be the closed embeddings with special linear normal bundles (N_{i}, λ) and (N_{i′}, λ′) ≅ (g^{′∗}N_{i}, g^{′∗}λ). Then we have g^{A}ĩ_{A} = ĩ′g^{′A}.

Proof. It follows from the functoriality of the deformation to the normal bundle and the functoriality of Thom classes. □

The next proposition is an analogue of [PW1, Proposition 7.6].

Proposition 1. Let T be a special linear bundle over a smooth variety X with a section s: X → T meeting the zero section r transversally in Y. Then for the inclusion i: Y → X and all b ∈ A^{∗,∗}(X) we have

i_{A}i^{A}(b) = b ∪ e(T ).

Proof. Let z^{A}: A^{∗,∗}(T h(i)) → A^{∗,∗}(X) and \overline{z}^{A}: A^{∗,∗}(T h(T )) → A^{∗,∗}(T ) be the extension of supports maps and let p: T → X be the structure map for the bundle. Consider the following diagram.

\xymatrix{A^{∗,∗}(X) \ar[r]^{r̃_{A}}_{∪th(T )} \ar[d]^{i^{A}} & A^{∗,∗}(T h(T )) \ar[r]^{z^{A}} \ar[d]^{s^{A}} & A^{∗,∗}(T ) \ar[d]^{s^{A}} \ar[d]_{r^{A}} \\ A^{∗,∗}(Y) \ar[r]^{ĩ_{A}} & A^{∗,∗}(T h(i)) \ar[r]^{_{\overline{z}_{A}}} & A^{∗,∗}(X) \ar[u]_{p^{A}}}

The pullbacks along the two section of p are inverses of the same isomorphism p^{A}, so s^{A} = r^{A}. The right-hand square consists of pullbacks thus it is commutative. The left-hand square commutes by Lemma 8. Hence we have

i_{A}i^{A}(b) = \overline{z}^{A}ĩ_{A}i^{A}(b) = r^{A}z^{A}(b ∪ th(T )) = b ∪ e(T ).  □

The pushforward maps are compatible with the compositions of the closed embeddings. The following proposition is similar to [Ne2, Proposition 5.1] and the same reasoning works out, so we omit the proof.

Proposition 2. Let Z ⟶^{i} Y ⟶^{j} X be closed embeddings of smooth varieties with special linear normal bundles (N_{ji}, λ_{ji}), (N_{i}, λ_{i}), (i^{∗}N_{ji}/N_{i}, λ_{j}) such that λ_{i} ⊗ λ_{j} = λ_{ji}. Then

j_{A}i_{A} = (ji)_{A}.

5. Inverting the stable Hopf map.

The Hopf map is the canonical morphism of varieties

H: 𝔸^{2} − {0} → ℙ^{1}

defined via H(x, y) = [x, y]. Pointing 𝔸^{2} − {0} by (1, 1) and ℙ^{1} by [1, 1] and taking the suspension spectra we obtain the corresponding morphism

Σ^{∞}H ∈ Hom_{SH(k)}(Σ^{∞}(𝔸^{2} − {0}), Σ^{∞}ℙ^{1}).

Recall that one has canonical isomorphisms 𝔸^{2} − {0} ⟶^{ϕ} 𝔾_{m} ∧ T and T ≅ ℙ^{1}/𝔸^{1} ≅ ℙ^{1} in H_{•}(k) (see Lemma 10 for the first one, the latter one is given by x ↦ [x : 1]), thus, using the suspension isomorphism, we obtain an element

η ∈ π^{−1,−1}(pt) ≅ Hom_{SH(k)}(Σ^{∞}(𝔸^{2} − {0}), Σ^{∞}ℙ^{1})

such that Σ^{3,2}η = Σ^{∞}H.

Let A^{∗,∗}(−) be a bigraded ring cohomology theory represented by a commutative monoid A ∈ SH(k). Inverting η ∈ A^{−1,−1}(pt) we obtain a new cohomology theory with (2i, i) groups isomorphic to (2i + n, i + n) ones by means of the cup product with η^{−n}. We identify these groups setting η = 1 and obtaining a graded cohomology theory:

\overline{A}^{∗}(X) = A^{∗,∗}(X) ⊗_{A^{∗,∗}(pt)} (A^{∗,∗}(pt)/⟨1 − η⟩) ,
\overline{A}^{i}(X) ≅ (A^{∗,∗}(X) ⊗_{A^{∗,∗}(pt)} A^{∗,∗}(pt)[η^{−1}])^{2i,i} .

For the K-theory represented by BGL [PPR2] this construction gives \overline{BGL}^{∗}(pt) = 0 since we have η ∈ BGL^{−1,−1}(pt) = K_{−1}(pt) = 0. As we will see in Corollary 1 it is always the case that an oriented cohomology theory produces a trivial cohomology theory. Thus we are interested in cohomology theories with a special linear orientation but without a general one. Our running example is hermitian K-theory represented by the spectrum BO that derives to the Witt groups.

Proposition 3. For every smooth variety X we have a natural isomorphism

\overline{BO}^{i}(X) ≅ W^{i}(X).

Proof. We give only a sketch of the proof, for the detailed version see [An]. We have a canonical isomorphism BO^{p,q}(X) ≅ KO_{2q−p}^{[q]}(X) and in case of 2q − p < 0 we have KO_{2q−p}^{[q]}(X) = W^{p−q}(X). Thus for 2q − p < 0 there is a natural isomorphism ϕ: BO^{p,q}(X) ⟶^{≃} W^{p−q}(X). One can show that it is multiplicative, i.e. for every (p, q) and (p′, q′) such that 2q − p, 2q′ − p′ < 0 the diagram

\xymatrix{BO^{p,q}(X) × BO^{p′,q′}(X) \ar[r]^{∪} \ar[d]^{ϕ×ϕ} & BO^{p+p′,q+q′}(X) \ar[d]^{ϕ} \\ W^{p−q}(X) × W^{p′−q′}(X) \ar[r]^{∪} & W^{p+p′−q−q′}(X)}

is commutative. Moreover, for η ∈ BO^{−1,−1}(pt) we have ϕ(η) = 1. Define ψ: BO^{∗,∗}(X) → W^{∗}(X)[η, η^{−1}] in the following way. For α ∈ BO^{p,q}(X) set

ψ(α) = \{ \begin{matrix}ϕ(α)η^{−p+2q}, & 2q − p < 0 \\ ϕ(α ∪ η^{p−2q+1})η^{−p+2q}, & 2q − p ≥ 0. \end{matrix}

Regarding W^{p}(X) as W^{2p,p}(X) and setting on the right-hand side deg η = (−1, −1) we turn ψ into a homomorphism of bigraded algebras. For 2q − p < 0 we know that ψ: BO^{p,q}(X) → W^{p−q}(X)η^{−p+2q} is an isomorphism. Thus inverting η we obtain an isomorphism

ψ̃: BO^{∗,∗}(X)[η^{−1}] ⟶^{≃} W^{∗}(X)[η, η^{−1}].  □

For the stable cohomotopy groups we have the following result by Morel [Mor2].

Theorem 1. There exists a canonical isomorphism \overline{π}^{0}(pt) ⟶^{≃} W^{0}(pt).

Notation 5. From now on A^{∗}(−) denotes a graded ring cohomology theory obtained from a bigraded ring cohomology theory represented by a commutative monoid A ∈ SH(k) with a fixed special linear orientation. Hence we have Thom and Euler classes and all the machinery of theories with a special linear orientation, including the Gysin sequences and pushforwards.

Remark 3. Note that from ϵ-commutativity we have η ∪η = −ϵ∪(η ∪η), thus inverting η we obtain \overline{ϵ} = −1.

Definition 9. Let E be a vector bundle over a smooth variety X. The total Pontryagin class

b_{∗}(E) = p_{∗} (E ⊕ E^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) = ∑ p_{i} (E ⊕ E^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) t^{i}

is called the total Borel class of the vector bundle E. We denote the even Pontryagin classes by b_{i},

b_{i}(E) = p_{2i} (E ⊕ E^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) ,

and refer to them as Borel classes.

We defined Borel classes for arbitrary vector bundles without any additional structure. For special linear bundles there is an interconnection between the Borel classes and the Euler class. The following lemma shows it in the case of rank 2 bundles and the general case would be dealt with in Corollary 4.

Lemma 9. Let T be a rank 2 special linear bundle. Then

b_{∗}(T ) = 1 − e(T )^{2}t^{2}

Proof. Set T = (E, λ). Let ϕ be the symplectic form on E corresponding to λ. There exists an isomorphism [Bal2, Examples 1.1.21, 1.1.22]

(E ⊕ E^{∨}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) ≅ (E ⊕ E, ( \begin{matrix}ϕ & 0 \\ 0 & −ϕ \end{matrix} )) ,

so we have

b_{∗}(E) = p_{∗}(E, ϕ)p_{∗}(E, −ϕ) = (1 + p_{1}(E, ϕ)t)(1 + p_{1}(E, −ϕ)t) =
= (1 + e(E, λ)t)(1 + e(E, −λ)t).

By Lemma 4 and Remark 3 we have e(E, −λ) = −e(E, λ), thus

b_{∗}(E) = (1 + e(T )t)(1 − e(T )t) = 1 − e(T )^{2}t^{2}.  □

6. Preliminary computation in the stable cohomotopy groups.

We are going to do preliminary computations involving π^{∗,∗}. Recall that for this cohomology theory we have canonical Thom classes for the trivialized vector bundles th(O_{X}^{n}) = Σ^{2n,n}1 and pushforwards i_{π} for the closed embeddings with a trivialized normal bundle (N_{i}, θ).

We fix the following notation. For n ≥ 1 let i: 𝔾_{m} → 𝔸^{n+1} − {0} be a closed embedding to the zeroth coordinate with t ↦ (t, 0, …, 0). Identify the normal bundle

N_{i} ≅ U = 𝔾_{m} × 𝔸^{n} ⊂ 𝔸^{n+1} − {0}

with the Zariski neighbourhood U of 𝔾_{m} and consider the trivialization

θ: U ⟶^{≃} O_{𝔾_{m}}^{n} via

θ(t, x_{1}, …, x_{n}) = (t, x_{1}/t, x_{2}, …, x_{n}).

There is a pushforward map

i_{π}: π^{0,0}(𝔾_{m}) → π^{2n,n}(𝔸^{n+1} − {0})

induced by the trivialization θ. Let

∂: π^{2n,n}(𝔸^{n+1} − {0}) → π^{2n+1,n}(T^{∧n+1})

be the connecting homomorphism in the localization sequence for the embedding {0} → 𝔸^{n+1}.

Set X = 𝔸^{n+1} − {0} and let x = (1, 1, 0, …, 0) be a point on X. We need the following well-known result.

Lemma 10. There is a canonical isomorphism in the homotopy category

ϕ: (X, x) ⟶^{≃} (𝔾_{m}, 1) ∧ T^{∧n}

that is a composition ϕ = ϕ_{2}^{−1}ϕ_{1} for isomorphisms

(X, x) ⟶^{ϕ_{1}} X/(X − (𝔾_{m} − {1})) ⟵^{ϕ_{2}} (𝔾_{m}, 1) ∧ T^{∧n},

where ϕ_{1} is induced by the identity map on X and ϕ_{2} is induced by the inclusion 𝔾_{m} × 𝔸^{n} ⊂ X.

Proof. The first map ϕ_{1} is an isomorphism since X − (𝔾_{m} − {1}) is 𝔸^{1}- contractible. The second isomorphism is induced by the excision isomorphism (𝔾_{m}_{+}, +) ∧ T^{∧n} ≅ X/(X − 𝔾_{m}). □

Proposition 4. In the above notation we have ∂i_{π}(1) = Σ^{2n+2,n+1}η.

Proof. From the construction of the pushforward map we have

i_{π}(1) = z^{π}d_{i}^{π}(th(U, θ))

with z^{π} being a support extension and d_{i}^{π} a deformation to the normal bundle isomorphism. Represent i as a composition

i: 𝔾_{m} ⟶^{i_{1}} U ⟶^{i_{2}} X

and let s: T h(i_{1}) ⟶^{≃} T h(i) be the induced isomorphism in the homotopy category. Recall that for the total space of a vector bundle U there is a natural isomorphism [Ne2, proof of Proposition 3.1] D(𝔾_{m}, U) ≅ U ×𝔸^{1} and d_{i_{1}}^{π} = id.

By the functoriality of the deformation construction we have d_{i}^{π} = (s^{π})^{−1}, so we need to compute

∂z^{π}(s^{π})^{−1}(th(U, θ)).

The choice of the point x on X induces a map r: (X_{+}, +) → (X, x) such that r^{π} splits ϕ^{π}Σ^{−1,0}∂, i.e. ϕ^{π}Σ^{−1,0}∂r^{π} = id.

\xymatrix{π^{2n,n}(X) \ar[d]_{∂} \\ π^{2n+1,n}(T^{∧n+1}) \ar[r]_{Σ^{−1,0}} & π^{2n,n}((𝔾_{m}, 1) ∧ T^{∧n}) \ar[r]_{ϕ^{π}} & π^{2n,n}(X, x) \ar@{-->}[ull]_{r^{π}}}

Decomposing z in

z: (X_{+}, +) ⟶^{r} (X, x) ⟶^{z_{1}} T h(i)

we obtain

∂z^{π}(s^{π})^{−1}(th(U, θ)) = Σ^{1,0}(ϕ^{π})^{−1}ϕ^{π}Σ^{−1,0}∂r^{π}z_{1}^{π}(s^{π})^{−1}(th(U, θ)) =

= Σ^{1,0}(ϕ^{π})^{−1}z_{1}^{π}(s^{π})^{−1}(th(U, θ)).

We can represent the Thom class th(U, θ) ∈ π^{2n,n}(T h(i_{1})) by the map

(t, x_{1}, x_{2}, …, x_{n}) ↦ (x_{1}/t, x_{2}, …, x_{n}).

Identifying one copy of T with ℙ^{1}/𝔸^{1} we rewrite the above map in the following way

H̃_{1} : (t, x_{1}, x_{2}, …, x_{n}) ↦ ([t, x_{1}], x_{2}, …, x_{n}).

Consider the following diagram.

\xymatrix{ & (ℙ^{1}/𝔸^{1}) ∧ T^{∧n−1} \\ T h(i_{1}) \ar[ur]^{H̃_{1}} \ar[r]^{s_{1}}_{≃} & ((𝔸^{2} − \{0\})/(𝔸^{2} − 𝔸^{1})) ∧ T^{∧n−1} \ar[u]^{H̃_{2}} \ar[r]^{s_{2}}_{≃} & T h(i) \\ (𝔸^{2} − \{0\}, (1, 1)) ∧ T^{∧n−1} \ar[r]^{ψ_{1}}_{≃} \ar@{-->}[ur]^{j} & X/(X − (𝔾_{m} − \{1\})) \ar[ur]^{j′} & (X, x) \ar[u]^{z_{1}} \ar[l]_{ϕ_{1}}^{≃}}

Here H̃_{2} is defined by the same formula as H̃_{1} and all the other maps are given by the tautological inclusions, i.e. s_{1} is induced by the inclusion U ⊂ (𝔸^{2} − {0}) × 𝔸^{n−1}, s_{2} and ψ_{1} are induced by (𝔸^{2} − {0}) × 𝔸^{n−1} ⊂ X, j′ is given by the identity map on X and j is given by identity map on 𝔸^{2} − {0}. One can easily check that this diagram is commutative.

We can represent z_{1}^{π}(s^{π})^{−1}(th(U, θ)) by the morphism

H̃_{2}jψ_{1}^{−1}ϕ_{1} = (Σ^{2n−2,n−1}H)ψ_{1}^{−1}ϕ_{1}.

At last, there is the following commutative diagram consisting of isomorphisms.

\xymatrix{(𝔸^{2} − \{0\}, (1, 1)) ∧ T^{∧n−1} \ar[r]^{ψ_{1}} \ar[d]_{ψ_{2}} & X/(X − (𝔾_{m} − \{1\})) \\ (𝔸^{2} − \{0\})/(𝔸^{2} − (𝔾_{m} − \{1\})) ∧ T^{∧n−1} \ar@{-->}[ur] & (𝔾_{m}, 1) ∧ T ∧ T^{∧n−1} \ar[u]^{ϕ_{2}} \ar[l]_{ψ_{3}}}

As above, all the maps in the diagram are induced by the tautological inclusions, ψ_{3} is induced by 𝔾_{m} × 𝔸^{1} ⊂ 𝔸^{2} − {0} and ψ_{2} is given by the identity map on 𝔸^{2} − {0}.

Thus we can represent Σ^{1,0}(ϕ^{π})^{−1}z_{1}^{π}(s^{π})^{−1}(th(U, θ)) by the morphism

Σ^{1,0}((Σ^{2n−2,n−1}H)ψ_{1}^{−1}ϕ_{1}ϕ_{1}^{−1}ϕ_{2}) = Σ^{1,0}((Σ^{2n−2,n−1}H)ψ_{1}^{−1}ϕ_{2}) =

= Σ^{1,0}((Σ^{2n−2,n−1}H)ψ_{2}^{−1}ψ_{3}).

It remains to notice that ψ_{2}^{−1}ψ_{3} is a suspension of the canonical isomorphism (𝔾_{m}, 1) ∧ T ≅ (𝔸^{2} − {0}, (1, 1)) we used to define η, so we obtain Σ^{2n+2,n+1}η.

□

7. Complement to the zero section.

In this section we compute the cohomology of the complement to the zero section of a special linear vector bundle. It turns out that there is a good answer in terms of the characteristic classes only in the case of odd rank.

Recall that for a special linear bundle T we denote by T^{0} the complement to the zero section. We start from the following lemma concerning the case of a special linear bundle possessing a section.

Notation 6. We denote an operator of the ∪-product with an element by the symbol of the element.

Lemma 11. Let T be a rank k special linear bundle over a smooth variety X with a nowhere vanishing section s → T . Then for some α ∈ A^{k−1}(X) we have an isomorphism

(1, α): A^{∗}(X) ⊕ A^{∗+1−k}(X) → A^{∗}(T^{0}).

Proof. Consider the Gysin sequence

… → A^{∗−k}(X) ⟶^{0} A^{∗}(X) ⟶^{j^{A}} A^{∗}(T^{0}) ⟶^{∂_{A}} A^{∗−k+1}(X) ⟶^{0} …

The section s induces a splitting s^{A} for j^{A} hence gives a splitting r for ∂_{A}. We have the claim for α = r(1). □

We want to obtain an isomorphism which does not depend on the choice of the section, so we act as in the projective bundle theorem for oriented cohomology theories: take a certain special linear bundle over T^{0} and compute its Euler class.

Definition 10. Let p: E → X be a vector bundle over a smooth variety X. The tautological line subbundle L_{E} of (p^{∗}E)|_{E^{0}} could be trivialized by means of the diagonal section Δ: E^{0} → E^{0} ×_{X} E. Hence, by lemma 2, for a special linear bundle (E, λ) there exists a canonical trivialization

λ_{T_{E}}: det(p^{∗}E|_{E^{0}}/L_{E}) ⟶^{≃} O_{E^{0}}.

We obtain a special linear bundle T_{E} = ((p^{∗}E|_{E^{0}}/L_{E}) , λ_{T_{E}}) over E^{0}.

For the Witt groups there is a result by Balmer and Gille [BG, Theorem 8.13].

Theorem 2. Let (E, λ) = (O_{pt}^{2n+1}, 1) be a trivialized special linear bundle of odd rank over a point with n ≥ 1. Then for e = e(T_{E}) ∈ W^{2n}(E^{0}) we have an isomorphism

(1, e): W^{∗}(pt) ⊕ W^{∗−2n}(pt) ⟶^{≃} W^{∗}(E^{0}).

We can derive an analogous result for A^{∗}(−) from our computation in stable cohomotopy groups.

Lemma 12. Let (E, λ) = (O_{pt}^{2n+1}, 1), n ≥ 1, be a trivialized special linear bundle over a point. Then for e = e(T_{E}) ∈ A^{2n}(E^{0}) we have an isomorphism

(1, e): A^{∗}(pt) ⊕ A^{∗−2n}(pt) ⟶^{≃} A^{∗}(E^{0}).

Proof. Consider the Gysin sequence

… → A^{∗−2n−1}(pt) ⟶^{0} A^{∗}(pt) → A^{∗}(E^{0}) ⟶^{∂_{A}} A^{∗−2n}(pt) ⟶^{0} …

The bundle E is trivial hence e(E, λ) = 0 and the Gysin sequence consists of short exact sequences.

Consider the dual special linear bundle T_{E}^{∨}. Taking the dual trivialization of E^{∨} we obtain

T_{E}^{∨} = {(x_{0}, …, x_{2n}, y_{0}, …, y_{2n}) ∈ E^{0} × E^{∨} | x_{0}y_{0} + ⋯ + x_{2n}y_{2n} = 0}.

There is a section s: E^{0} → T_{E}^{∨} with

s(x_{0}, x_{1}, x_{2}, …, x_{2n−1}, x_{2n}) = (x_{0}, x_{1}, …, x_{2n}, 0, x_{2}, −x_{1}, …, x_{2n}, −x_{2n−1}).

This section meets the zero section in 𝔾_{m} ≅ {(t, 0, …, 0) | t ≠ 0}. Proposition 1 states that e(T_{E}^{∨}) = i_{A}(1) for the inclusion i: 𝔾_{m} → 𝔸^{2n+1} − {0} with the trivialization of det N_{i} arising from the trivialization of det T_{E}^{∨}. Identify N_{i} ≅ T_{E}^{∨}|_{𝔾_{m}} with U = 𝔾_{m} × 𝔸^{2n} ⊂ E^{0} via

(t, 0, …, 0, 0, y_{1} …, y_{2n}) ↦ (t, y_{1}, …, y_{2n}).

The isomorphism λ_{T_{E}^{∨}}: det T_{E}^{∨}|_{𝔾_{m}} ⟶^{≃} O_{𝔾_{m}} arises from the canonical trivialization of E^{∨}|_{𝔾_{m}} and morphism ϕ: E^{∨} → L_{E}^{∨} ≅ O_{𝔾_{m}} with

ϕ(t, y_{0}, y_{1}, …, y_{2n}) = (t, ty_{0}).

Thus over t for y^{i} = (y_{1}^{i}, y_{2}^{i}, …, y_{2n}^{i}) we have

λ_{T_{E}^{∨}}(y^{1} ∧ y^{2} ∧ ⋯ ∧ y^{2n}) = det ( \begin{matrix}1/t & 0 & 0 & … & 0 \\ 0 & y_{1}^{1} & y_{1}^{2} & … & y_{1}^{2n} \\ 0 & y_{2}^{1} & y_{2}^{2} & … & y_{2}^{2n} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & y_{2n}^{1} & y_{2n}^{2} & … & y_{2n}^{2n} \end{matrix} )

and θ(U, λ_{T_{E}^{∨}}) ⟶^{≃} (O_{𝔾_{m}}^{2n}, 1) with θ(t, y_{1}, y_{2}, …, y_{2n}) = (t, y_{1}/t, y_{2}, …, y_{2n}) is

an isomorphism of special linear bundles.

Consider the following diagram with i_{π} being a pushforward in stable cohomotopy groups for the closed embedding i with the trivialization θ of the normal bundle.

\xymatrix{A^{0}(𝔾_{m}) \ar[r]^{i_{A}} & A^{2n}(E^{0}) \ar[r]^{∂_{A}} & A^{0}(pt) \\ \overline{π}^{0}(𝔾_{m}) \ar[r]^{i_{π}} \ar[u] & \overline{π}^{2n}(E^{0}) \ar[u] \ar[r]^{∂_{π}} & \overline{π}^{0}(pt) \ar[u]}

The left-hand side commutes since θ is an isomorphism of special linear bundles. The right-hand side of the diagram consist of the structure morphisms for A and the boundary maps for the Gysin sequences of the inclusion {0} → E hence commutes as well. Proposition 4 states that ∂_{π}i_{π}(1) = 1, hence

∂_{A}(e(T_{E}^{∨})) = ∂_{A}i_{A}(1) = 1

and {1, e(T_{E}^{∨})} forms a basis of A^{∗}(E^{0}) over A^{∗}(pt). There is a nowhere vanishing section of T_{E}^{∨} ⊕ T_{E}^{∨} constructed analogous to s defined above, so

e(T_{E}^{∨})^{2} = e(T_{E}^{∨} ⊕ T_{E}^{∨}) = 0.

By Lemma 6 for some α_{1}, α_{2}, β_{1}, β_{2} ∈ A^{∗}(pt) we have

e = (α_{1} + β_{1} ∪ e(T_{E}^{∨})) ∪ e(T_{E}^{∨}) = α_{1} ∪ e(T_{E}^{∨}),
e(T_{E}^{∨}) = (α_{2} + β_{2} ∪ e(T_{E}^{∨})) ∪ e = α_{2} ∪ α_{1} ∪ e(T_{E}^{∨}).

We already know that {1, e(T_{E}^{∨})} is a basis, then α_{2} ∪ α_{1} = 1 and α_{1} is invertible. Hence {1, α_{1} ∪ e(T_{E}^{∨})} = {1, e} is a basis as well.

□

Corollary 1. Let A^{∗,∗}(−) be a oriented cohomology theory represented by a commutative monoid A ∈ SH(k). Then \overline{A}^{∗}(pt) = 0.

Proof. There is a natural special linear orientation on A^{∗,∗}(−) obtained by setting th(E, λ) = th(E) with the latter Thom class arising from the orientation on A^{∗,∗}(−). Hence for a rank n special linear bundle we have e(E, λ) = c_{n}(E). By the above lemma, for E = O_{pt}^{3} there is an isomorphism

(1, c_{2}(T_{E})): \overline{A}^{∗}(pt) ⊕ \overline{A}^{∗−2}(pt) ⟶^{≃} \overline{A}^{∗}(E^{0}).

Applying the Cartan formula we obtain c_{∗}(O_{E^{0}})c_{∗}(T_{E}) = c_{∗}(O_{E^{0}}^{3}), thus c_{2}(T_{E}) = 0. The above isomorphism yields \overline{A}^{∗}(pt) = 0. □

Having a canonical basis for a trivial bundle we can glue it into a basis in the cohomology of the complement to the zero section of an arbitrary special linear bundle of odd rank.

Theorem 3. Let (E, λ) be a special linear bundle of rank 2n + 1, n ≥ 1, over a smooth variety X. Then for e = e(T_{E}) we have an isomorphism

(1, e): A^{∗}(X) ⊕ A^{∗−2n}(X) → A^{∗}(E^{0}).

Proof. The general case is reduced to the case of the trivial vector bundle E via the usual Mayer-Vietoris arguments. In the latter case we have a commutative diagram of the Gysin sequences

\xymatrix{0 \ar[r] & A^{∗}(X) \ar[r] & A^{∗}(E^{0}) \ar[r]^{∂_{A}} & A^{∗−2n}(X) \ar[r] & 0 \\ 0 \ar[r] & A^{∗}(pt) \ar[r] \ar[u] & A^{∗}(E^{′0}) \ar[r]^{∂_{A}} \ar[u] & A^{∗−2n}(pt) \ar[u]^{p^{A}} \ar[r] & 0}

with E′ = O_{pt}^{2n+1}. By Lemma 12 the element ∂_{A}e (T_{E′}) generates A^{∗−2n}(pt) as a module over A^{∗}(pt), thus for a certain α ∈ A^{∗}(pt) we have α∪∂_{A}e (T_{E′}) = 1. Using E = p^{∗}E′ we obtain

α ∪ ∂_{A}e (T_{E}) = α ∪ p^{A}∂_{A}e (T_{E′}) = 1,

so ∂_{A} (e (T_{E})) generates A^{∗−2n}(X) over A^{∗}(X). Hence (1, e) is an isomorphism. □

Remark 4. In case of rank E = 1 one still has an isomorphism: a special linear bundle of rank one is a trivialized line bundle, hence there is an isomorphism

A^{∗}(X) ⊕ A^{∗}(X) ≅ A^{∗}(E^{0}) = A^{∗}(X × 𝔾_{m})

induced by the isomorphism A^{∗}(pt) ⊕ A^{∗}(pt) ≅ A^{∗}(𝔾_{m}).

Corollary 2. Let T be a special linear bundle of odd rank over a smooth variety X. Then e(T ) = 0.

Proof. Set rank T = 2n + 1 and e = e(T ). Consider the Gysin sequence

… → A^{0}(X) ⟶^{e} A^{2n+1}(X) ⟶^{j^{A}} A^{2n+1}(T^{0}) → A^{1}(X) → …

The above calculations show that j^{A} is injective hence e = 0. □

8. Special linear projective bundle theorem.

Definition 11. For k < n consider the group

P′_{k} = ( \begin{matrix}SL_{k} & ∗ \\ 0 & SL_{n−k} \end{matrix} ) .

The quotient variety SGr(k, n) = SL_{n}/P′_{k} is called a special linear Grassmann variety.

Notation 7. Denote by T_{1} and T_{2} the tautological special linear bundles over SGr(k, n) with rank T_{1} = k and rank T_{2} = n − k.

Remark 5. We have a projection SL_{n}/P′_{k} → SL_{n}/P_{k} identifying the special linear Grassmann variety with the complement to the zero section of the determinant of the tautological vector bundle over the ordinary Grassmann variety Gr(k, n). This yields the following geometrical description of SGr(k, n): fix a vector space V of dimension n. Then

SGr(k, n) = {(U ≤ V, λ ∈ (Λ^{k}U)^{0}) | dim U = k}.

In particular, we have SGr(1, n) ≅ 𝔸^{n} − {0}.

Theorem 4. For the special linear Grassmann varieties we have the following isomorphisms.

(1, e_{1}, ..., e_{1}^{2n−2}, e_{2}): ⨁_{i=0}^{2n−2} A^{∗−2i}(pt) ⊕ A^{∗−2n+2}(pt) → A^{∗}(SGr(2, 2n)),

(1, e_{1}, e_{1}^{2}, ..., e_{1}^{2n−1}): ⨁_{i=0}^{2n−1} A^{∗−2i}(pt) → A^{∗}(SGr(2, 2n + 1)),

with e_{1} = e(T_{1}), e_{2} = e(T_{2}).

Proof. We are going to deal with several special linear Grassmann varieties at once, so we will use T_{i}(r, k) for T_{i} over SGr(r, k) and abbreviate e(T_{i}(r, k)) to e_{i}(r, k) and e(T_{i}(r, k)^{∨}) to e_{i}^{∨}(r, k). The proof is done by induction on the Grassmannian’s dimension.

The base case. We have SGr(2, 3) ≅ SGr(1, 3) ≅ 𝔸^{3} − {0} and under these isomorphisms the bundle T_{1}(2, 3)^{∨} goes to T_{2}(1, 3) which goes to T_{O_{pt}^{3}} in the notation of definition 10. Note that rank T_{1}(2, 3) = 2, thus T_{1}(2, 3) ≅ T_{1}(2, 3)^{∨} and e(T_{1}(2, 3)) = e (T_{1}(2, 3)^{∨}). Hence Lemma 12 gives the claim for SGr(2, 3).

Basic geometry. Fix a vector space V of dimension k + 1, a subspace W ≤ V of codimension one and forms μ_{1} ∈ (Λ^{k+1}V)^{0}, μ_{2} ∈ (Λ^{k}W)^{0}. Then we have the following diagram constructed in the same vein as in the case of ordinary Grassmannians:

\xymatrix{SGr(2, k) \ar[r]^{i} & SGr(2, k + 1) & Y \ar[l]_{j} \ar[d]^{p} \\  &  & SGr(1, k)}

the inclusion i corresponds to the pairs (U, μ ∈ (Λ^{2}U)^{0}) with U ≤ W, dim U = 2; the open complement Y consists of the pairs (U, μ ∈ (Λ^{2}U)^{0}) with dim U = 2, dim U ∩ W = 1; the projection p is given by p(U, μ) = (U ∩ W, μ′) where μ′ is given by the isomorphism (U ∩ W) ⊗ V/W ≅ Λ^{2}U. Here i is a closed embedding, j is an open embedding and p is an 𝔸^{k}-bundle. Take an arbitrary f ∈ V^{∨} such that ker f = W. It gives rise to a constant section of the trivial bundle (O_{SGr(2,k+1)}^{k+1})^{∨} hence a section of T_{1}(2, k + 1)^{∨}. The latter section vanishes exactly over i(SGr(2, k)). Note that we have rank T_{1}(2, k + 1) = 2 hence e_{1}^{∨}(2, k + 1) = e_{1}(2, k + 1).

k=2n-1. Consider the localization sequence.

… → A^{∗−2}(SGr(2, 2n − 1)) ⟶^{i_{A}} A^{∗}(SGr(2, 2n)) ⟶^{j^{A}} A^{∗}(SGr(1, 2n − 1)) → …

Lemma 12 states that {1, e_{2}(1, 2n − 1)} is a basis of A^{∗}(SGr(1, 2n − 1)) over A^{∗}(pt). We have j^{∗}T_{2}(2, 2n) ≅ p^{∗}T_{2}(1, 2n − 1) and

j^{A}(e_{2}(2, 2n)) = e_{2}(1, 2n − 1)

hence j^{A} is a split surjection (over A^{∗}(pt)) with the splitting defined by

1 ↦ 1, e_{2}(1, 2n − 1) ↦ e_{2}(2, 2n).

Then i_{A} is injective. Hence to obtain a basis of A^{∗}(SGr(2, 2n)) it is sufficient to calculate the pushforward for a basis of A^{∗−2}(SGr(2, 2n − 1)) and combine it with {1, e_{2}(2, 2n)}. Using the induction we know that

{1, e_{1}(2, 2n − 1), …, e_{1}(2, 2n − 1)^{2n−3}}

is a basis of A^{∗}(SGr(2, 2n − 1)). We have i^{∗}(T_{1}(2, 2n)) ≅ T_{1}(2, 2n − 1) hence

e_{1}(2, 2n − 1) = i^{A}(e_{1}(2, 2n)).

By Proposition 1 we have

i_{A}(e_{1}(2, 2n − 1)^{l}) = e_{1}(2, 2n)^{l+1}

obtaining the desired basis

{e_{1}(2, 2n), e_{1}(2, 2n)^{2}, …, e_{1}(2, 2n)^{2n−2}, 1, e_{2}(2, 2n)}

of A^{∗}(SGr(2, 2n)) over A^{∗}(pt).

k=2n. Consider the localization sequence.

… ⟶^{∂_{A}} A^{∗−2}(SGr(2, 2n)) ⟶^{i_{A}} A^{∗}(SGr(2, 2n + 1)) ⟶^{j^{A}} A^{∗}(SGr(1, 2n)) ⟶^{∂_{A}} …

Using the induction we know a basis of A^{∗}(SGr(2, 2n)), namely

{1, e_{1}(2, 2n), e_{1}(2, 2n)^{2}, …, e_{1}(2, 2n)^{2n−2}, e_{2}(2, 2n)}

and Lemma 11 gives us a non-canonical basis {1, α} for A^{∗}(SGr(1, 2n)). Examine i_{A}(e_{2}(2, 2n)). It can’t be computed using Proposition 1 since it seems that e_{2}(2, 2n) can not be pullbacked from A^{∗}(SGr(2, 2n+1)), so we use the following argument. Consider a nontrivial vector w ∈ W. It induces constant sections of O_{SGr(2,2n)}^{2n} and O_{SGr(2,2n+1)}^{2n+1} and sections of T_{2}(2, 2n) and T_{2}(2, 2n + 1). The latter sections vanish over SGr(1, 2n − 1) and SGr(1, 2n) respectively. Here SGr(1, 2n−1) corresponds to the vectors in W/⟨w⟩ and SGr(1, 2n) corresponds to the vectors in V/⟨w⟩. Hence we have the following commutative diagram consisting of closed embeddings.

\xymatrix{SGr(1, 2n − 1) \ar[r]^{r′} \ar[d]^{i′} & SGr(2, 2n) \ar[d]^{i} \\ SGr(1, 2n) \ar[r]^{r} & SGr(2, 2n + 1)}

By Proposition 1 we have e_{2}(2, 2n) = r′_{A}(1), so, using Proposition 2, we obtain i_{A}(e_{2}(2, 2n)) = r_{A}i′_{A}(1). Notice that N_{i′} is a trivial bundle of rank one. In fact, there is a section of trivial bundle T_{1}(1, 2n)^{∨} over SGr(1, 2n) constructed using the same element f such that ker f = W and this section meets the zero section exactly at SGr(1, 2n − 1). So we have

i_{A}(e_{2}(2, 2n)) = r_{A}i′_{A}(1) = r_{A}(e_{1}^{∨}(1, 2n)) = r_{A}(0) = 0.

We claim that ker i_{A} = A^{∗}(pt) ∪ e_{2}(2, 2n) and Im j^{A} = A^{∗}(pt) ∪ 1. We have j^{A}(1) = 1, hence ∂_{A}(1) = 0 and

ker i_{A} = Im ∂_{A} = A^{∗}(pt) ∪ ∂_{A}(α).

The localization sequence is exact, so we have

e_{2}(2, 2n) = ∂_{A}(y ∪ α) = y ∪ ∂_{A}(α)

for some y ∈ A^{∗}(pt) and since e_{2}(2, 2n) is an element of the basis, y is not a zero divisor. Consider the presentation of ∂_{A}(α) with respect to the chosen basis:

∂_{A}(α) = x_{0} ∪ 1 + x_{1} ∪ e_{1}(2, 2n) + ⋯ + x_{2n−2} ∪ e_{1}(2, 2n)^{2n−2} + z ∪ e_{2}(2, 2n).

We have y ∪ ∂_{A}(α) = e_{2}(2, 2n), hence y ∪ z = 1 and every y ∪ x_{i} = 0, hence x_{i} = 0. Then ∂_{A}(α) = z ∪ e_{2}(2, 2n) and

ker i_{A} = Im ∂_{A} = A^{∗}(pt) ∪ ∂_{A}(α) = A^{∗}(pt) ∪ e_{2}(2, 2n).

We have

∂_{A}(x_{0} ∪ 1 + x_{1} ∪ α) = x_{1} ∪ ∂_{A}(α) = x_{1} ∪ z ∪ e_{2}(2, 2n),

hence Im j^{A} = ker ∂_{A} = A^{∗}(pt) ∪ 1.

There is an obvious splitting for A^{∗}(SGr(2, 2n+1)) ⟶^{j^{A}} Im j^{A}, 1 ↦ 1. Then calculating by the same vein as in the odd-dimensional case the pushforwards for the basis of Coker ∂_{A}, {e_{1}(2, 2n)^{l}}, and adding to them {1}, we obtain the desired basis of SGr(2, 2n + 1)

{e_{1}(2, 2n + 1), …, e_{1}(2, 2n + 1)^{2n−1}, 1}.  □

Corollary 3. There is an isomorphism:

A^{∗}(pt)[e]/⟨e^{2n−2}⟩ ⟶^{≃} A^{∗}(SGr(2, 2n − 1)),

induced by sending e to e(T_{1}).

Proof. Keep the notations from the proof of the theorem. It is sufficient to show that e_{1}(2, 2n − 1)^{2n−2} = 0.

Consider a vector space V of dimension 2n−1 and a collection of subspaces W_{i} ≤ V of codimension one such that dim ⋂_{i=1}^{2n−2} W_{i} = 1. Every subspace W_{i}

defines a section of (O_{SGr(2,2n−1)}^{2n−1})^{∨} and a section s_{i} of T_{1}(2, 2n − 1)^{∨} in the same vein as in the proof of the theorem. The section of (T_{1}(2, 2n − 1)^{∨})^{⊕2n−2} defined by (s_{1}, …, s_{2n−2}) vanishes nowhere, hence by Lemma 7 we have

e_{1}(2, 2n − 1)^{2n−2} = e((T_{1}(2, 2n − 1)^{∨})^{⊕2n−2}) = 0.

□

Definition 12. Let T be a special linear bundle over a smooth variety X. Then we define the relative special linear Grassmann variety SGr(k, T ) in an obvious way. This variety is a SGr(k, rank T )-bundle over X. Similarly to the above, we denote by T_{1} and T_{2} the tautological special linear bundles over SGr(k, T ).

Theorem 5. Let T be a special linear bundle over a smooth variety X. (1) If rank T = 2n then there is an isomorphism

(1, e_{1}, ..., e_{1}^{2n−2}, e_{2}): ⨁_{i=0}^{2n−2} A^{∗−2i}(X) ⊕ A^{∗−2n+2}(X) ⟶^{≃} A^{∗}(SGr(2, T )),

with e_{1} = e(T_{1}), e_{2} = e(T_{2}).

(2) If rank T = 2n + 1 then there is an isomorphism

(1, e, e^{2}, ..., e^{2n−1}): ⨁_{i=0}^{2n−1} A^{∗−2i}(X) ⟶^{≃} A^{∗}(SGr(2, T )),

with e = e(T_{1}).

Proof. The general case is reduced to the case of the trivial bundle T via the usual Mayer-Vietoris arguments. The latter case follows from Theorem 4. □

9. Symmetric polynomials.

In this section we deal with the polynomials invariant under the action of the Weyl group W (B_{n}) or W (D_{n}) and obtain certain spanning sets for the polynomial rings. Our method is an adaptation of the one used in [Fu, � 10, Proposition 3]. The proof is quite straightforward but a bit messy.

Consider ℤ^{n} and fix a usual basis {e_{1}, …, e_{n}}. Let

W (B_{n}) = {ϕ ∈ Aut(ℤ^{n}) | ϕ(e_{i}) = ±e_{j}}

be the Weyl group of the root system B_{n} and let

W (D_{n}) = {ϕ ∈ Aut(ℤ^{n}) | ϕ(e_{i}) = (−1)^{k_{i}}e_{j}, (−1)^{∑ k_{i}} = 1}

be the Weyl group of the root system D_{n}. Identifying R = ℤ[e_{1}, …, e_{n}] with the symmetric algebra Sym^{∗}((ℤ^{n})^{∨}) in a usual way, we obtain the actions of these Weyl groups on R. Let R_{B} = R^{W (B_{n})} and R_{D} = R^{W (D_{n})} be the algebras of invariants.

For the elementary polynomials σ_{i} ∈ ℤ[x_{1}, …, x_{n}] consider

s_{i} = σ_{i}(e_{1}^{2}, …, e_{n}^{2}), t = σ_{n}(e_{1}, …, e_{n}).

One can easily check that R_{B} = ℤ[s_{1}, …, s_{n}] and R_{D} = ℤ[s_{1}, …, s_{n−1}, t].

In order to compute spanning sets for R over R_{B} and R_{D} we need "decreasing degree" equalities provided by the following lemma.

Lemma 13. There exist homogeneous polynomials g_{i}, h_{i} ∈ R such that

e_{1}^{2n} = ∑_{i=1}^{n} g_{i}s_{i}, e_{1}^{2n−1} = ∑_{i=1}^{n−1} h_{i}s_{i} + h_{n}t.

Proof. Let I_{B} = ⟨s_{1}, …, s_{n}⟩ and I_{D} = ⟨s_{1}, …, s_{n−1}, t⟩ be the ideals generated by the homogeneous invariant polynomials of positive degree. We need to show that e_{1}^{2n} ∈ I_{B} and e_{1}^{2n−1} ∈ I_{D}. Set S_{B} = R/I_{B}, S_{D} = R/I_{D}.

Consider S_{B} [[x]]. Since all the s_{i} belong to I_{B} we have

(1 − ē_{1}^{2}x)(1 − ē_{2}^{2}x)…(1 − ē_{n}^{2}x) = 1,

hence

(1 − ē_{2}^{2}x)(1 − ē_{3}^{2}x)…(1 − ē_{n}^{2}x) = 1 + ē_{1}^{2}x + ē_{1}^{4}x^{2} + …

Comparing the coefficients at x^{n} we obtain ē_{1}^{2n} = 0, thus e_{1}^{2n} ∈ I_{B}. Consider S_{D} [[x]]. As above, we have

(1 − ē_{1}^{2}x^{2})(1 − ē_{2}^{2}x^{2})…(1 − ē_{n}^{2}x^{2}) = 1,

hence

(1 + ē_{1}x)(1 − ē_{2}^{2}x^{2})(1 − ē_{3}^{2}x^{2})…(1 − ē_{n}^{2}x^{2}) = 1 + ē_{1}x + ē_{1}^{2}x^{2} + …

Comparing the coefficients at x^{2n−1} we obtain

ē_{1}^{2n−1} = (−1)^{n−1}ē_{1}ē_{2}^{2}…ē_{n}^{2} = (−1)^{n−1}t̄ē_{2}ē_{3}…ē_{n} = 0,

thus e_{1}^{2n−1} ∈ I_{D}. □

Proposition 5. In the above notation we have the following spanning sets:

(1) B_{1} = {e_{1}^{m_{1}}e_{2}^{m_{2}} … e_{n}^{m_{n}} | 0 ≤ m_{i} ≤ 2n − 2i + 1} spans R over R_{B}.

(2) B_{2} = \{u_{1}u_{2} … u_{n−1} | u_{i} = [ \begin{matrix}e_{i}^{m_{i}}, 0 ≤ m_{i} ≤ 2n − 2i \\ e_{i+1}e_{i+2} … e_{n} \end{matrix} \}

spans R over R_{D}.

Proof. In both cases proceed by induction on n. The base case of n = 1 is clear. Denote by B′_{1} and B′_{2} the spanning sets in R′ = ℤ[e_{2}, …, e_{n}] and let s′_{i}, t′ ∈ R′ be the corresponding invariant polynomials. Note that s_{i} = e_{1}s′_{i−1} + s′_{i} and t = e_{1}t′.

It is sufficient to show that every monomial is a R_{B} (or R_{D}) linear combination of the monomials of lesser total degree and monomials from the corresponding spanning set.

(1) Consider a monomial f = e_{1}^{k_{1}}e_{2}^{k_{2}}…e_{n}^{k_{n}} ∈ R. In case of k_{1} ≥ 2n we can use the preceding lemma and substitute ∑ g_{i}s_{i} for e_{1}^{2n} obtaining

f = ∑ s_{i}g_{i}e_{1}^{k_{1}−2n}e_{2}^{k_{2}}…e_{n}^{k_{n}}

with deg g_{i}e_{1}^{k_{1}−2n}e_{2}^{k_{2}}…e_{n}^{k_{n}} < deg f, so we get the claim. Now suppose that k_{1} < 2n. By the induction we have

e_{2}^{k_{2}}e_{3}^{k_{3}}…e_{n}^{k_{n}} = ∑ α_{j}(s′_{1}, …, s′_{n−1})b′_{j}

for some b′_{j} ∈ B′_{1} and α_{j} ∈ ℤ[x_{1}, …, x_{n−1}]. We can assume that all the summands at the right-hand side are homogeneous of total degree k_{2} + … + k_{n}. Since s_{i} = e_{1}s′_{i−1} + s′_{i} one has

α_{j}(s_{1}, …, s_{n−1}) = α_{j} (s′_{1}, …, s′_{n−1}) + ∑_{l>0} e_{1}^{l}β_{jl}

for some β_{jl} ∈ R_{B}. Thus we obtain

f = ∑_{j} e_{1}^{k_{1}}α_{j}(s_{1}, …, s_{n−1})b′_{j} − ∑_{j,l} e_{1}^{k_{1}+l}β_{jl}b′_{j}.

Note that e_{1}^{k_{1}}b′_{j} ∈ B_{1}, so the first sum is a R_{B}-linear combination of the monomials from the spanning set. If deg β_{jl} > 0 then deg e_{1}^{k_{1}+l}b′_{j} < deg f and it is the case of the linear combination with lesser total degree. At last, in case of deg β_{jl} = 0 there are two variants: if k_{1} + l < 2n then we have e_{1}^{k_{1}+l}b′_{j} ∈ B_{1}, otherwise k_{1}+l ≥ 2n and we can lower the total degree by using the preceding lemma.

(2) Consider a monomial f = e_{1}^{k_{1}}e_{2}^{k_{2}}…e_{n}^{k_{n}} ∈ R. As above, in case of k_{1} ≥ 2n−1 we can use the preceding lemma and lower the total degree, so suppose that k_{1} < 2n − 1. By induction we have

e_{2}^{k_{2}}e_{3}^{k_{3}}…e_{n}^{k_{n}} = ∑ α_{j}(s′_{1}, …, s′_{n−2}, t′)b′_{j}

for some b′_{j} ∈ B′_{2} and α_{j} ∈ ℤ[x_{1}, …, x_{n−1}]. One has t^{′2} = s′_{n−1} hence

α_{j}(s′_{1}, …, s′_{n−2}, t) = α̃_{j}(s′_{1}, …, s′_{n−2}, s′_{n−1}) + t′α̂_{j}(s′_{1}, …, s′_{n−2}, s′_{n−1}).

As above, we can substitute s_{i} into α̃_{j} and α̂_{j} and obtain some β̃_{jl}, β̂_{jl} ∈ R_{D}. Thus we have

f = ∑_{j} e_{1}^{k_{1}} α̃_{j}(s_{1}, …, s_{n−1})b′_{j} + ∑_{j} e_{1}^{k_{1}} α̂_{j}(s_{1}, …, s_{n−1})t′b′_{j}−

− ∑_{j,l} e_{1}^{k_{1}+l}β̃_{jl}b′_{j} − ∑_{j,l} e_{1}^{k_{1}+l}β̂_{jl}t′b′_{j}.

In the first sum we have e_{1}^{k_{1}}b′_{j} ∈ B_{2}. One has t′b′_{j} ∈ B_{2}, so in case of k_{1} = 0 the second sum is a linear combination of the elements from the spanning set, otherwise, if k_{1} ≥ 1, one can lower the total degree by carrying out t = e_{1}t′. The third sum is dealt with like the second one in (1), in case of deg β̃_{jl} = 0 we use that e_{1}^{k_{1}+l}b′_{j} ∈ B_{2} or lower degree using the preceding lemma, otherwise we lower degree by carrying out β̃_{jl}. At last, in the forth sum we lower degree by carrying out t = e_{1}t′.

□

10. A splitting principle.

In this section we assert a splitting principle for the cohomology theories with a special linear orientation and inverted stable Hopf map. The principle states that from the viewpoint of such cohomology theories every special linear bundle is a direct sum of rank 2 special linear bundles and at most one trivial linear bundle.

Definition 13. For k_{1} < k_{2} < ⋯ < k_{m} consider the group

P′_{k_{1},...,k_{m−1}} = ( \begin{matrix}SL_{k_{1}} & ∗ & … & ∗ \\ 0 & SL_{k_{2}−k_{1}} & … & ∗ \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & SL_{k_{m}−k_{m−1}} \end{matrix} )

and define a special linear flag variety as the quotient

SF(k_{1}, …, k_{m}) = SL_{k_{m}}/P′_{k_{1},...,k_{m−1}}.

In particular, we are interested in the following varieties:

SF(2n) = SF(2, 4, ..., 2n), SF(2n + 1) = SF(2, 4, ..., 2n, 2n + 1).

These varieties are called maximal SL_{2} flag varieties.

For every special linear flag variety SF(k_{1}, k_{2}, …, k_{m}) there is an affine variety

S̃F(k_{1}, k_{2}, …, k_{m}) = SL_{k_{m}}/(SL_{k_{1}} × SL_{k_{2}−k_{1}} × ⋯ × SL_{k_{m}−k_{m−1}}).

Note that S̃F(k_{1}, k_{2}, …, k_{m}) is an 𝔸^{r}-bundle over SF(k_{1}, k_{2}, …, k_{m}).

Remark 6. The projection

SF(k_{1}, k_{2}, …, k_{m}) = SL_{n}/P′_{k_{1},...,k_{m}} → SL_{n}/P_{k_{1},...,k_{m}} = F(k_{1}, k_{2}, …, k_{m})

yields the following geometrical description of the special linear flag varieties. Consider a vector space V of dimension k_{m}. Then we have

SF(k_{1}, k_{2}, …, k_{m}) =

= {(V_{1} ≤ ⋯ ≤ V_{m−1} ≤ V, λ_{1}, …, λ_{m−1}) | dim V_{j} = k_{j}, λ_{j} ∈ (Λ^{k_{j}}V_{j})^{0} }

Notation 8. Denote by T_{i} the tautological special linear bundles over SF(k_{1}, k_{2}, …, k_{m}) with rank T_{i} = k_{i} − k_{i−1}.

Definition 14. Let T be a special linear bundle over a smooth variety X. Then we define the relative special linear flag variety SF_{T} (k_{1}, k_{2}, …, k_{m}) with k_{m} = rank T in an obvious way. This variety is a SF(k_{1}, k_{2}, …, k_{m})- bundle over X. We also define relative version of the maximal SL_{2} flag variety, SF(T ), and relative versions for the affine coverings, S̃F_{T} (k_{1}, k_{2}, …, k_{m}) and S̃F(T ).

Theorem 6. Let T be a rank k special linear bundle over a smooth variety X. Then A^{∗}(SF_{T} (2, 4, …, 2n, k)) is a free module over A^{∗}(X) with the following basis:

• k is odd:
{e_{1}^{m_{1}}e_{2}^{m_{2}} … e_{n}^{m_{n}} | 0 ≤ m_{i} ≤ k − 2i} ,
• k is even:

\{u_{1}u_{2} … u_{n} | u_{i} = [ \begin{matrix}e_{i}^{m_{i}}, 0 ≤ m_{i} ≤ k − 2i \\ e_{i+1}e_{i+2} … e_{n+1} \end{matrix} \} ,

where e_{i} = e(T_{i}, λ_{T_{i}}).

Proof. Proceed by induction on n. For n = 1 the claim follows from Theorem 5.

Consider the projection

p: Y = SF_{T} (2, 4, …, 2n, k) → SF_{T} (2, 4, …, 2n − 2, k) = Y_{1}

that forgets about the last subspace. Denote the tautological bundles over Y by T_{i} and the tautological bundles over Y_{1} by T′_{i}.

k is odd. Using an isomorphism Y ≅ SGr(2, T′_{n}) and Theorem 5 we obtain that A^{∗}(Y) is a free module over A^{∗}(Y_{1}) with the basis

B = {1, e_{n}, …, e_{n}^{k−2n}} .

Using the induction we have the following basis for A^{∗}(Y_{1}):

B_{1} = {e_{1}^{′m_{1}}e_{2}^{′m_{2}} … e_{n−1}^{′m_{n−1}} | 0 ≤ m_{i} ≤ k − 2i} ,

with e′_{i} = e(T′_{i}). One has p^{∗}(T′_{i}) ≅ T_{i} and p^{A}(e′_{i}) = e_{i} for i ≤ n−1. Computing the pullback for B_{1} and multiplying it with B we obtain the desired basis.

k is even. This case is completely analogous to the previous one. We have an isomorphism Y ≅ SGr(2, T′_{n}) then by Theorem 5 obtain that A^{∗}(Y) is a free module over A^{∗}(Y_{1}) with the basis

B = \{u_{n} | u_{n} = [ \begin{matrix}e_{n}^{m_{n}}, 0 ≤ m_{n} ≤ k − 2n \\ e_{n+1} \end{matrix} \} .

Using the induction we have the following basis for A^{∗}(Y_{1}):

B_{1} = \{u_{1}u_{2} … u_{n−1} | u_{i} = [ \begin{matrix}e_{i}^{′m_{i}}, 0 ≤ m_{i} ≤ k − 2i \\ e′_{i+1}e′_{i+2} … e′_{n} \end{matrix} \}

with e′_{i} = e(T′_{i}). Note that p^{∗}(T′_{i}) ≅ T_{i} and p^{A}(e′_{i}) = e_{i} for i ≤ n−1. In order to compute p^{A}(e′_{n}) pass to S̃F_{E}(2, 4, …, 2n, k), there we have p^{∗}T′_{n} ≅ T_{n} ⊕ T_{n+1} and p^{A}(e′_{n}) = e_{n}e_{n+1}. Computing the pullback for B_{1} and multiplying it with B we obtain the desired basis of Y. □

Corollary 4. Let T be a rank 2n special linear bundle over a smooth variety X. Then we have

(1) e(T ) = e(T^{∨}),

(2) b_{n}(T ) = (−1)^{n}e(T )^{2}.

Proof. Consider p: S̃F(T ) → X. From the theorem we have that p^{A} is an injection. Also we have that p^{∗}T ≅ ⨁_{i} T_{i} and p^{∗}T^{∨} ≅ ⨁_{i} T_{i}^{∨}. Note that rank T_{i} = 2, hence (T_{i}, λ_{T_{i}}) ≅ (T_{i}^{∨}, λ_{T_{i}^{∨}}) and we obtain p^{∗}T ≅ p^{∗}T^{∨}, so p^{A}e(T ) = p^{A}e(T^{∨}). By Lemma 9 we have

b_{∗}(p^{∗}T ) = ∏(1 − e(T_{i})^{2}t^{2}),

thus p^{A}b_{n}(T ) = (−1)^{n} ∏ e(T_{i})^{2} = (−1)^{n}(p^{A}e(T ))^{2}. □

Having the above corollary at hand we can write down the relations for SGr(2, 2n). Recall that the odd-dimensional case was computed in Corollary 3.

Corollary 5. There is an isomorphism

A^{∗}(pt)[e^{1}, e^{2}]/⟨e_{1}e_{2}, e_{1}^{2n−2} + (−1)^{n}e_{2}^{2}⟩ ⟶^{≃} A^{∗}(SGr(2, 2n)),

induced by sending e_{1} and e_{2} to e(T_{1}) and e(T_{2}) respectively.

Proof. In view of Theorem 4 it is sufficient to show that the relations from the left-hand side hold. Passing to S̃Gr(2, 2n) we obtain

0 = e (O_{S̃Gr(2,2n)}^{2n}) = e (T_{1}) e (T_{2}) .

For the second relation compute the total Borel class:

1 = b_{∗} (O_{S̃Gr(2,2n)}^{2n}) = b_{∗}(T_{1})b_{∗}(T_{2}) = (1 − e(T_{1})^{2}t^{2}) b_{∗}(T_{2}).

Expanding b_{∗}(T_{2}) we have b_{k}(T_{2}) = e(T_{1})^{2k} for k ≤ n − 1. Thus, by the above corollary, e(T_{1})^{2n−2} = (−1)^{n−1}e(T_{2})^{2}. □

We finish this section with an explicit answer for the cohomology of maximal SL_{2} flag variety. Note that the answer looks like the ring of coinvariants for the groups W (B_{n}) and W (D_{n}) rather then W (A_{n−1}), although we deal with the special linear group SL_{n}.

Theorem 7. For n ≥ 1 consider

s_{i} = σ_{i}(e_{1}^{2}, e_{2}^{2}, ..., e_{n}^{2}), t = σ_{n}(e_{1}, e_{2}, …, e_{n})

with σ_{i} being the elementary symmetric polynomials in n variables. Then we have the following isomorphisms

(1) ϕ_{1} : A^{∗}(pt)[e^{1}, e^{2}, ..., e^{n}]/⟨s_{1}, s_{2}, …, s_{n}⟩ ⟶^{≃} A^{∗}(SF(2n + 1)),

(2) ϕ_{2} : A^{∗}(pt)[e^{1}, e^{2}, ..., e^{n}]/⟨s_{1}, s_{2}, …, s_{n−1}, t⟩ ⟶^{≃} A^{∗}(SF(2n)),

induced by sending e_{i} to e(T_{i}).

Proof. First of all we show that the claimed relations on the Euler classes hold. Passing to S̃F(2n + 1) and using Lemma 9 we obtain

b_{∗} (O2n+1 ; S̃F) = ∏_{i=1}^{n+1} b_{∗} (T_{i}) = ∏_{i=1}^{n} (1 − e (T_{i})^{2} t^{2}) ,

hence

ϕ_{1}(s_{i}) = (−1)^{i}b_{i} (O2n+1 ; S̃F) = 0.

In the even case we can do the same calculations in order to obtain

ϕ_{2}(s_{i}) = (−1)^{i}b_{i} (O2n ; S̃F) = 0,

moreover, we have

ϕ_{2}(t) = ∏_{i=1}^{n} e(T_{i}) = e(O2n ; S̃F) = 0.

Hence the homomorphisms ϕ_{1} and ϕ_{2} are well-defined.

To finish the proof note that by Proposition 5 and Theorem 6 the spanning set from the left-hand side goes to the basis of the right-hand side, so ϕ_{1} and ϕ_{2} are isomorphisms. □

11. The cohomology of BSL_{n}.

This section is devoted to the computation of the cohomology ring of the classifying spaces

BSL_{n} = \underrightarrow{lim}_{m∈ℕ} SGr(n, m).

The case of BSL_{2} easily follows from Corollary 3. Then we deal with BSL_{2n} using the calculations for the relative maximal SL_{2} flag varieties, and in the end using certain Gysin sequences compute the cohomology of BSL_{2n+1}.

Recall that A^{∗} is constructed from a representable cohomology theory. In this setting we have the following proposition relating the cohomology groups of a limit space to the limit of the cohomology groups [PPR2, Lemma A.5.10].

Proposition 6. For any sequence of motivic spaces X_{1} ⟶^{i_{1}} X_{2} ⟶^{i_{2}} X_{3} ⟶^{i_{3}} … and any p we have an exact sequence of abelian groups

0 → \underleftarrow{lim}^{1}A^{p−1}(X_{k}) → A^{p}(\underrightarrow{lim} X_{k}) → \underleftarrow{lim} A^{p}(X_{k}) → 0.

Moreover, if all the i_{k}^{A} are surjective, then lim^{1} vanishes and we have

A^{p}(\underrightarrow{lim} X_{k}) ≅ \underleftarrow{lim} A^{p}(X_{k}).

Lemma 14. For every m, n ∈ ℕ the A^{∗}(pt)-algebra A^{∗}(SGr(2n, 2m + 1)) is generated by the classes b_{1}(T_{1}), …, b_{n−1}(T_{1}), e(T_{1}).

Proof. Consider the covering

p: Y = S̃F(2, 4, …, 2n, 2m + 1) → SGr(2n, 2m + 1)

splitting T_{1} into a sum of the rank 2 special linear bundles. Denote the tautological bundles over Y by T′_{1}, T′_{2}, …, T′_{n+1} and the corresponding Euler classes by e_{1}, e_{2}, …, e_{n+1}.

By Theorem 6 applied to T_{1} we have that p^{A} is injective. Also one has p^{∗}T_{1} ≅ ⨁_{i=1}^{n} T′_{i} hence

p^{A}e(T_{1}) = σ_{n}(e_{1}, e_{2}, …, e_{n}) = t

and, by Lemma 9,

p^{A}b_{i}(T_{1}) = σ_{i}(e_{1}^{2}, e_{2}^{2}, …, e_{n}^{2}) = s_{i}

with σ_{i} being the elementary symmetric polynomials.

By Theorem 6 we also have that the set

B = \{u_{1}u_{2} … u_{n−1} | u_{i} = [ \begin{matrix}e_{i}^{m_{i}}, 0 ≤ m_{i} ≤ k − 2i \\ e_{i+1}e_{i+2} … e_{n} \end{matrix} \} ,

forms a basis of A^{∗}(Y) over p^{A}A^{∗}(SGr(2n, 2m + 1)). Note that by the same theorem A^{∗}(Y) is generated as an A^{∗}(pt)-algebra by e_{1}, e_{2} …, e_{n}, thus by Proposition 5 we know that B spans A^{∗}(Y) over A^{∗}(pt)[s_{1}, s_{2}, …, s_{n−1}, t]. Since we have

A^{∗}(pt)[s_{1}, s_{2}, …, s_{n−1}, t] ⊂ p^{A}A^{∗}(SGr(2n, 2m + 1))

it follows that A^{∗}(pt)[s_{1}, s_{2}, …, s_{n−1}, t] = p^{A}A^{∗}(SGr(2n, 2m + 1)). □

Consider the sequence of embeddings

… → SGr(2n, 2m + 1) ⟶^{i_{2m+1}} SGr(2n, 2m + 3) → …

By the above lemma we know that i_{2m+1}^{A} are surjective hence

A^{p}(BSL_{2n}) ≅ \underleftarrow{lim} A^{p}(SGr(2n, 2m + 1)) ≅ \underleftarrow{lim} A^{p}(SGr(2n, m)).

The sequence of tautological special linear bundles T_{1}(2n, m) over SGr(2n, m) gives rise to a bundle T over BSL_{2n}. We have a sequence of embeddings of the Thom spaces

… → T h(T_{1}(2n, 2m + 1)) ⟶^{j_{2m+1}} T h(T_{1}(2n, 2m + 3)) → …

Since all the considered morphisms T_{1}(2n, k) → T_{1}(2n, l) are inclusions there is a canonical isomorphism T h(T ) ≅ \underrightarrow{lim} T_{1}(2n, m). For every k we have an isomorphism

A^{∗−2n}(SGr(2n, k)) ⟶^{∪th(T_{1}(2n,k))} A^{∗}(T h(T_{1}(2n, k))),

so j_{2m+1}^{A} are surjective and

A^{p}(T h(T )) ≅ \underleftarrow{lim} A^{p}(T_{1}(2n, m)).

Notation 9. Let T be the tautological special linear bundle over BSL_{2n}. Denote by b_{i}(T ), e(T ) ∈ A^{∗}(BSL_{2n}) and th(T ) ∈ A^{∗}(T h(T )) the elements corresponding to the sequences of classes of the tautological bundles,

b_{i}(T ) = (…, b_{i}(T_{1}(2n, m)), b_{i}(T_{1}(2n, m + 1)), …),
e(T ) = (…, e(T_{1}(2n, m)), e(T_{1}(2n, m + 1)), …),
th(T ) = (…, th(T_{1}(2n, m)), th(T_{1}(2n, m + 1)), …),

with T_{1}(2n, m) being the tautological special linear bundles over SGr(2n, m).

The above considerations show that we have a Gysin sequence for the tautological bundle over the classifying space BSL_{2n}.

Lemma 15. Let T be the tautological special linear bundle over BSL_{2n}. Then there exists a long exact sequence

… → A^{∗−2n}(BSL_{2n}) ⟶^{∪e} A^{∗}(BSL_{2n}) ⟶^{j^{A}} A^{∗}(BSL_{2n−1}) ⟶^{∂} …

Proof. For the zero section inclusion of motivic spaces BSL_{2n} → T we have the following long exact sequence.

… → A^{∗}(T h(T )) ⟶ A^{∗}(T ) ⟶ A^{∗}(T^{0}) ⟶^{∂} …

The isomorphisms

A^{∗−2n}(SGr(2n, k)) ⟶^{∪th(T_{1}(2n,k))} T h(T_{1}(2n, k)),

induce an isomorphism A^{∗−2n}(BSL_{2n}) ⟶^{∪th(T )} A^{∗}(T h(T )), so we can substitute A^{∗−2n}(BSL_{2n}) for the first term in the above sequence. Using homotopy invariance we exchange T for BSL_{2n}. By the definition of e(T ) the first arrow represents the cup product ∪e(T ).

We have isomorphisms

T^{0} ≅ \underrightarrow{lim} SGr(1, 2n − 1, m) ≅ \underrightarrow{lim} SGr(2n − 1, 1, m).

The sequence of projections

\xymatrix{… \ar[r] & SGr(2n − 1, 1, m) \ar[r] \ar[d] & SGr(2n − 1, 1, m + 1) \ar[r] \ar[d] & … \\ … \ar[r] & SGr(2n − 1, m + 1) \ar[r] & SGr(2n − 1, m + 2) \ar[r] & …}

induces a morphism T^{0} ⟶^{r} BSL_{2n−1}. Note that

SGr(2n − 1, 1, m) ≅ T_{2}(2n − 1, m + 1)^{0},

and T^{0} is an 𝔸^{∞} − {0}-bundle over BSL_{2n−1}, so by [MV, Section 4, Proposition 2.3] r is an isomorphism in homotopy category and we can substitute A^{∗}(BSL_{2n−1}) for the third term in the long exact sequence. □

Definition 15. For a graded ring R^{∗} let R^{∗} [[t]]_{h} be the homogeneous power series ring, i.e. a graded ring with

R^{∗} [[t]]_{h}^{k} = {∑ a_{i}t^{i} | deg a_{i} + i deg t = k} .

Note that R^{∗} [[t]]_{h} = \underleftarrow{lim} R^{∗}[t]/t^{n}, where the limit is taken in the category of graded algebras. For example, considering R^{∗} = ℤ[x] and degrees deg x = 1, deg t = 1 we have

ℤ[x] [[t]]_{h} = ⋃_{k∈ℤ} {∑ p_{i}(x)t^{i} ∈ ℤ[x] [[t]] | deg p_{i} + deg t ≤ k} .

We set R^{∗} [[t_{1}, …, t_{n}]]_{h} = R^{∗} [[t_{1}, …, t_{n−1}]]_{h} [[t_{n}]]_{h}.

Theorem 8. For deg e = 2n, deg b_{i} = 2i we have isomorphisms

A^{∗}(pt) [[b_{1}, …, b_{n−1}, e]]_{h} ⟶^{≃} A^{∗}(BSL_{2n}),

A^{∗}(pt) [[b_{1}, …, b_{n}]]_{h} ⟶^{≃} A^{∗}(BSL_{2n+1}).

Proof. The case of BSL_{2} follows from Corollary 3 and Proposition 6, since for the sequence

… → SGr(2, 2m + 1) → SGr(2, 2m + 3) → …

the pullbacks are surjective and lim^{1} vanishes yielding

A^{∗}(BSL_{2}) ≅ \underleftarrow{lim} A^{∗}(SGr(2, 2m + 1)) = \underleftarrow{lim} A^{∗}(pt)[e]/⟨e^{2m}⟩ = A^{∗}(pt) [[e]]_{h} .

For the even case consider the sequences

\xymatrix{… \ar[r] & SF(T_{1}(2n, 2m + 1)) \ar[r]^{j_{m}} \ar[d]^{p_{m}} & SF(T_{1}(2n, 2m + 3)) \ar[r] \ar[d]^{p_{m+1}} & … \\ … \ar[r] & SGr(2n, 2m + 1) \ar[r]^{i_{m}} & SGr(2n, 2m + 3) \ar[r] & …}

with T_{1}(2n, m) being the tautological rank 2n special linear bundle over SGr(2n, m). We have SF(T_{1}(2n, 2m + 1)) ≅ SF(2, 4, …, 2n, 2m + 1). By Theorem 6 the pullbacks j_{m}^{A} are surjective, so lim^{1} vanishes. By the same theorem A^{∗}(SF(T_{1}(2n, 2m + 1))) is generated by the Euler classes of the tautological bundles and for every polynomial f(e_{1}, e_{2}, …, e_{n}) and 2m+1 > deg f this polynomial is nonzero in A^{∗}(SF(T_{1}(2n, 2m + 1))). Thus we obtain

A^{∗}(\underrightarrow{lim} SF(T_{1}(2n, 2m + 1))) ≅ A^{∗}(pt) [[e_{1}, e_{2}, …, e_{n}]]_{h} ,

deg e_{i} = 2.

On the other hand we know that A^{∗}(BSL_{2n}) ≅ \underleftarrow{lim} A^{∗}(SGr(2n, 2m + 1)). By Lemma 14 and Theorem 6 p_{m}^{A}A^{∗}(SGr(2n, 2m + 1)) is the subalgebra of A^{∗}(SF(T_{1}(2n, 2m + 1))) generated by

p_{m}^{A}b_{i}(T_{1}) = s_{i} = σ_{i}(e_{1}^{2}, e_{2}^{2}, …, e_{n}^{2}), p_{m}^{A}e(T ) = t = σ_{n}(e_{1}, e_{2}, …, e_{n}).

Passing to the limit we obtain

p^{∗}A^{∗}(BSL_{2n}) = A^{∗}(pt) [[s_{1}, …, s_{n−1}, t]]_{h} ⊂ A^{∗}(pt) [[e_{1}, e_{2}, …, e_{n}]]_{h} ,

where s_{i} = p^{∗}b_{i}(T ), t = p^{∗}e(T ).

For the odd case consider the Gysin sequence from Lemma 15 for BSL_{2n+2}. By the above calculations e(T ) is not a zero divisor, so the the map ∪e(T ) is injective and we have a short exact sequence

0 → A^{∗−2n−2}(BSL_{2n+2}) ⟶^{∪e(T )} A^{∗}(BSL_{2n+2}) → A^{∗}(BSL_{2n+1}) → 0.

Identifying A^{∗}(BSL_{2n+2}) with the homogeneous power series and removing e we obtain the desired result. □

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[PS] I. Panin (after I. Panin and A. Smirnov), Oriented cohomology theories of algebraic varieties, K-Theory 30 (2003), 265-314.

[PW1] I. Panin and C. Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, arXiv:1011.0649.

[PW2] I. Panin and C. Walter, On the motivic commutative spectrum BO, arXiv:1011.0650.

[PW3] I. Panin and C. Walter, On the algebraic cobordism spectra MSL and MSp, arXiv:1011.0651.

[PW4] I. Panin and C. Walter, On the relation of the symplectic algebraic cobordism to hermitian K-theory, arXiv:1011.0652.

[Sch] M. Schlichting, Hermitian K-theory of exact categories, J. K-theory 5 (2010), no. 1, 105 - 165

[V] V. Voevodsky, 𝔸^{1}-homotopy theory, Doc. Math., Extra Vol. I (1998), pp. 579–604