Witt Groups of Complex Cellular Varieties

Marcus Zibrowius

Received: December 30, 2010 Revised: May 10, 2011

Communicated by Max Karoubi

Abstract. We show that the Grothendieck-Witt and Witt groups of smooth complex cellular varieties can be identified with their topological KO-groups. As an application, we deduce the values of the Witt groups of all irreducible hermitian symmetric spaces, including smooth complex quadrics, spinor varieties and symplectic Grassmannians.

2010 Mathematics Subject Classification: Primary 11E81; Secondary 19G99, 19L99, 32M15 Keywords and Phrases: Witt groups, KO-theory, projective homogeneous varieties, hermitian symmetric spaces

Contents

Introduction 466

1 Preliminaries 467 1.1 Witt groups and hermitian K-theory . . . . . . . . . . . . . . . 467 1.2 KO-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 1.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 1.4 𝔸^{1}-homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . 475 1.5 Representing algebraic and hermitian K-theory . . . . . . . . . 476

2 The comparison maps 479 2.1 Twisting by line bundles . . . . . . . . . . . . . . . . . . . . . . 481 2.2 The comparison for cellular varieties . . . . . . . . . . . . . . . 483

3 The Atiyah-Hirzebruch spectral sequence 488 3.1 The Atiyah-Hirzebruch spectral sequence for cellular varieties . 489 3.2 The Atiyah-Hirzebruch spectral sequence for Thom spaces … 491 4 Examples 493

4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 4.2 Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 495 4.3 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 4.4 Maximal symplectic Grassmannians . . . . . . . . . . . . . . . 499 4.5 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 4.6 Spinor varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 4.7 Exceptional hermitian symmetric spaces . . . . . . . . . . . . . 507

Introduction

The purpose of this paper is to demonstrate that the Grothendieck-Witt and Witt groups of complex projective homogeneous varieties can be computed in a purely topological way. That is, we show in Theorem 2.5 how to identify them with the topological KO-groups of these varieties, and we illustrate this with a series of known and new examples. Our theorem holds more generally for any smooth complex cellular variety. By this we mean a smooth complex variety X with a filtration by closed subvarieties ∅ = Z_{0} ⊂ Z_{1} ⊂ Z_{2} ⋯ ⊂ Z_{N} = X such that the complement of Z_{k} in Z_{k+1} is an open “cell” isomorphic to 𝔸^{n_{k}} for some n_{k}. Let us put our result into perspective. It is well-known that for such cellular X we have an isomorphism

K_{0}(X) ⟶^{≅} K^{0}(X(ℂ))

between the algebraic K-group of X and the complex K-group of the underlying topological space X(ℂ). In fact, both sides are easy to compute: they decompose as direct sums of the K-groups of the cells, each of which is isomorphic to ℤ. Such decompositions are characteristic of oriented cohomology theories. Witt groups, however, are strictly non-oriented, and this makes computations much harder. It is true that the Witt groups of complex varieties decompose into copies of ℤ/2, the Witt group of ℂ, but even in the cellular case there is no general understanding of how many copies to expect. Nonetheless, we can prove our theorem by an induction over the number of cells of X. The main issue is to define the map from Witt groups to the relevant KO-groups in such a way that it respects various exact sequences. The basic idea is clear: the Witt group W^{0}(X) classifies vector bundles equipped with non-degenerate symmetric forms, and in topology symmetric complex vector bundles are in one-to-one correspondence with real vector bundles, classified by KO^{0}(X). More precisely, we have two natural maps:

GW^{0}(X) → KO^{0}(X(ℂ))
W^{0}(X) → \frac{KO^{0}(X(ℂ))}{K^{0}(X(ℂ))}

Here, GW^{0}(X) is the Grothendieck-Witt group of X, and in the second line K^{0}(X) is mapped to KO^{0}(X) by sending a complex vector bundle to the underlying real bundle. It is possible to extend these maps to shifted groups and groups with support in a concrete and “elementary” way, as was done in [Zib09]. The method advocated here is to rely instead on a result in 𝔸^{1}-homotopy theory: the representability of hermitian K-theory by a spectrum whose complex realization is the usual topological KO-spectrum. Currently, our only reference is a draft paper of Morel [Mor06], but the result is well-known to the experts and a full published account will undoubtedly become available in due course. In the unstable homotopy category at least, the statement is immediate from Schlichting and Tripathi’s recent description of a geometric representing space for hermitian K-theory (see Section 1.5). The structure of the paper is as follows: In the first section we assemble the basic definitions, reviewing some representability results along the way before finally stating in 1.9 the results in 𝔸^{1}-homotopy theory that we ultimately take as our starting point. Our main result, Theorem 2.5, is stated and proved in the second section. Section 3 reviews mostly well-known facts about the Atiyah-Hirzebruch spectral sequence, on which the computations of examples in the final section rely.

1 Preliminaries

1.1 Witt groups and hermitian K-theory

From a modern point of view, the theory of Witt groups represents a Ktheoretic approach to the study of quadratic forms. We briefly run through some of the basic definitions. Recall that the algebraic K-group K_{0}(X) of a scheme X can be defined as the free abelian group on isomorphism classes of vector bundles over X modulo the following relation: for any short exact sequence of vector bundles

0 → E → F → G → 0

over X we have [F] = [E] + [G] in K_{0}(X). In particular, as far as K_{0}(X) is concerned, we may pretend that all exact sequences of vector bundles over X split. Now let (E, ϵ) be a symmetric vector bundle, by which we mean a vector bundle E equipped with a non-degenerate symmetric bilinear form ϵ. We may view ϵ as an isomorphism from E to its dual bundle E^{∨}, in which case its symmetry may be expressed by saying that ϵ and ϵ^{∨} agree under the canonical identification of the double-dual (E^{∨})^{∨} with E. Two symmetric vector bundles (E, ϵ) and (F, ϕ) are isometric if there is an isomorphism of vector bundles i: E → F compatible with the symmetries, i. e. such that i^{∨}ϕi = ϵ. The orthogonal sum of two symmetric bundles has the obvious definition (E, ϵ) ⊥ (F, ϕ) := (E ⊕ F, ϵ ⊕ ϕ). Any vector bundle E gives rise to a symmetric bundle H(E) := (E ⊕ E^{∨}, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )), the hyperbolic bundle associated with E. These hyperbolic bundles are the simplest members of a wider class of so-called metabolic bundles: symmetric bundles (M, μ) which contain a subbundle j: L → M of half their own rank on which μ vanishes. In other words, (M, μ) is metabolic if it fits into a short exact sequence of the form

0 → L ⟶^{j} M ⟶^{j^{∨}μ} L^{∨} → 0

The subbundle L is then called a Lagrangian of M. If the sequence splits, (M, μ) is isometric to H(L), at least in any characteristic other than two. This motivates the definition of the Grothendieck-Witt group.

Definition 1.1 ([Wal03a,Sch10a]). The Grothendieck-Witt group GW^{0}(X) of a scheme X is the free abelian group on isometry classes of symmetric vector bundles over X modulo the following two relations:

• [(E, ϵ) ⊥ (G, γ)] = [(E, ϵ)] + [(G, γ)]

• [(M, μ)] = [H(L)] for any metabolic bundle (M, μ) with Lagrangian L

The Witt group W^{0}(X) is defined similarly, except that the second relation reads [(M, μ)] = 0. Equivalently, we may define W^{0}(X) by the exact sequence

K_{0}(X) ⟶^{H} GW^{0}(X) ⟶ W^{0}(X) → 0

Shifted Witt groups. The groups above can be defined more generally in the context of exact or triangulated categories with dualities. The previous definitions are then recovered by considering the category of vector bundles over X or its bounded derived category. However, the abstract point of view allows for greater flexibility. In particular, a number of useful variants of Witt groups can be introduced by passing to related categories or dualities. For example, if we take a line bundle L over X and replace the usual duality E^{∨} := Hom(E, O_{X}) on vector bundles by Hom(−, L) we obtain “twisted” Witt groups W^{0}(X; L). On the bounded derived category, we can consider dualities that involve shifting complexes, leading to the definition of “shifted” Witt groups W^{i}(X). This approach, pioneered by Paul Balmer in [Bal00, Bal01a], elevates the theory of Witt groups into the realm of cohomology theories. We illustrate the meaning and significance of these remarks with a few of the key properties of the theory, concentrating on the case when X is a smooth scheme over a field of characteristic not equal to two. The interested but unacquainted reader may prefer to consult [Bal01b] or [Bal05].

• For any line bundle L over X and any integer i, we have a Witt group

W^{i}(X; L)

This is the i^{th} Witt group of X “with coefficients in L”, or “twisted by L”. When L is trivial it is frequently dropped from the notation.

• The Witt groups are four-periodic in i and “two-periodic in L” in the sense that, for any i and any line bundles L and M over X, we have canonical isomorphisms

W^{i}(X; L) ≅ W^{i+4}(X; L)
W^{i}(X; L) ≅ W^{i}(X; L ⊗ M^{⊗2})

• More generally, for any closed subset Z of X we have Witt groups “with support on Z”, written W_{Z}^{i}(X; L). For Z = X these agree with W^{i}(X; L).

• We have long exact “localization sequences” relating the Witt groups of X and X − Z, which can be arranged as 12-term exact loops by periodicity.

Balmer’s approach already works on the level of Grothendieck-Witt groups, as shown in [Wal03a]. In this context, the localization sequences take the form

GW_{Z}^{i}(X) → GW^{i}(X) → GW^{i}(X − Z) →

W_{Z}^{i+1}(X) → W^{i+1}(X) → W^{i+1}(X − Z) → W_{Z}^{i+2}(X) → ⋯ (1)

continuing to the right with shifted Witt groups of X, and similarly for arbitrary twists L [Wal03a, Theorem 2.4]. However, if one wishes to continue the sequences to the left, one has to revert to the methods of higher algebraic K-theory.

Hermitian K-theory. Recall that the higher algebraic K-groups of a scheme X can be defined as the homotopy groups of a topological space K(X) associated with X. If one replaces K(X) by an appropriate spectrum one can similarly define groups K_{n}(X) in all degrees n ∈ ℤ. On a smooth scheme X, however, the groups in negative degrees vanish. An analogous construction of hermitian K-theory is developed in [Sch10b]. Given a scheme X and a line bundle L over X, Schlichting constructs a family of spectra 𝔾W^{i}(X; L) from which hermitian K-groups can be defined as

GW_{n}^{i}(X; L) := π_{n}(𝔾W^{i}(X; L))

In degree n = 0, one recovers Balmer and Walter’s Grothendieck-Witt groups, and the Witt groups appear as hermitian K-groups in negative degrees. To be precise, for any smooth scheme X over a field of characteristic not equal to two one has the following natural identifications:

GW_{0}^{i}(X; L) ≅ GW^{i}(X; L)  (2)
GW_{n}^{i}(X; L) ≅ W^{i−n}(X; L) for n < 0  (3)

For affine varieties, the identifications of Witt groups may be found in [Hor05]: see Proposition A.4 and Corollary A.5. For a general smooth scheme X, we can pass to a vector bundle torsor T over X such that T is affine [Jou73, Lemma 1.5; Hor05, Lemma 2.1].^{1} Both Balmer’s Witt groups and Schlichting’s hermitian K-groups are homotopy invariant in the sense that the groups of T may naturally be identified with those of X. This is proved for Witt groups in [Gil03, Corollary 4.2] and may be deduced for hermitian K-theory from the Mayer-Vietoris sequences established in [Sch10b, Theorem 1]. The identifications also hold more generally for hermitian K-groups with support GW_{n,Z}^{i}(X) [Sch]. They will be used implicitly throughout. For completeness, we mention how the 4-periodic notation used here translates into the traditional notation in terms of KO- and U-theory, as used for example in [Hor05]. Namely, we have

GW_{n}^{i}(X) = \begin{cases}KO_{n}(X) & for i ≡ 0 mod 4 \\ U_{n}(X) & for i ≡ −1 \\ _{−}KO_{n}(X) & for i ≡ −2 \\ _{−}U_{n}(X) & for i ≡ −3 \end{cases}

(This notation will not be used elsewhere in this paper.)

1.2 KO-theory

We now turn to the corresponding theories in topology. To ensure that the definitions given here are consistent with the literature, we restrict our attention to finite-dimensional CW complexes.^{2} Since we are ultimately only interested in topological spaces that arise as complex varieties, this is not a problem. The definitions of K_{0} and GW^{0} given above applied to complex vector bundles over a finite-dimensional CW complex X yield its complex and real topological K-groups K^{0}(X) and KO^{0}(X). Since short exact sequences of vector bundles over X always split, the definitions may even be simplified:

Definition 1.2. For a finite-dimensional CW complex X, the complex Kgroup K^{0}(X) is the free abelian group on isomorphism classes of complex vector bundles over X modulo the relation [E ⊕G] = [E]+[G]. Likewise, the KO-group KO^{0}(X) is the free abelian group on isometry classes of symmetric complex vector bundles over X modulo the relation [(E, ϵ) ⊥ (G, γ)] = [(E, ϵ)] + [(G, γ)].

There is a more common description of KO^{0}(X) as the K-group of real vector bundles. The equivalence with the definition given here can be traced back to the fact that the orthogonal group O(n) is a maximal compact subgroup of both GL_{n}(ℝ) and O_{n}(ℂ), but also seen very concretely along the following lines. We say that a complex bilinear form ϵ on a real vector bundle F is real if ϵ: F ⊗ F → ℂ factors through ℝ.

1This step is known as Jouanolou’s trick.

2The key property we need is that any vector bundle over a finite-dimensional CW complex has a stable inverse. See the proof of Theorem 1.5.

Documenta Mathematica 16 (2011) 465–511

Lemma 1.3. Let (E, ϵ) be a symmetric complex vector bundle. There exists a unique real subbundle ℜ(E, ϵ) ⊂ E such that ℜ(E, ϵ)⊗_{ℝ} ℂ = E and such that the restriction of ϵ to ℜ(E, ϵ) is real and positive definite. Concretely, a fibre of ℜ(E, ϵ) is given by the real span of any orthonormal basis of the corresponding fibre of E.

Corollary 1.4. For any CW complex X, the monoid of isomorphism classes of real vector bundles over X is isomorphic to the monoid of isometry classes of symmetric complex vector bundles over X (with respect to the operations ⊕ and ⊥, respectively).

Proof of Lemma 1.3. In the case of a vector bundle over a point we may assume without loss of generality that

(E, ϵ) = (ℂ^{r}, ( ..._{0  1}^{1  0} ) )

Clearly, the subspace ℝ^{r} ⊂ ℂ^{r} has the required properties. Uniqueness follows from elementary linear algebra. If (E, ϵ) is an arbitrary symmetric complex vector bundle over a space X, then any point of X has some neighbourhood over which (E, ϵ) can be trivialized in the form above. We know how to define ℜ(E, ϵ) over each such neighbourhood, and by uniqueness these local bundles can be glued together.

Proof of Corollary 1.4. A map in one direction is given by sending a symmetric complex vector bundle (E, ϵ) to ℜ(E, ϵ). Conversely, with any real vector bundle E over X we may associate a symmetric complex vector bundle (E ⊗_{ℝ} ℂ, σ_{ℂ}), where σ_{ℂ} is the ℂ-linear extension of some inner product σ on E. Since σ is defined uniquely up to isometry, so is (E ⊗_{ℝ} ℂ, σ_{ℂ}). See [MH73, Chapter V, § 2] for a proof that avoids the uniqueness part of the preceding lemma.

Representing topological K-groups. A standard construction of the cohomology theories associated with K^{0} and KO^{0} is based on the fact that these functors are representable in the homotopy category H of topological spaces. The starting point is the homotopy classification of vector bundles: Let us write Gr_{r,n} for the Grassmannian Gr(r, ℂ^{r+n}) of complex r-bundles in ℂ^{r+n}, and let Gr_{r} be the union of Gr_{r,n} ⊂ Gr_{r,n+1} ⊂ ⋯ under the obvious inclusions. Denote by U_{r,n} and U_{r} the universal r-bundles over these spaces. For any connected paracompact Hausdorff space X we have a one-to-one correspondence between the set Vect_{r}(X) of isomorphism classes of rank r complex vector bundles over X and homotopy classes of maps from X to Gr_{r}: a homotopy class [f] in H(X, Gr_{r}) corresponds to the pullback of U_{r} along f [Hus94, Chapter 3, Theorem 7.2]. To describe K^{0}(X), we need to pass to Gr, the union of the Gr_{r} under the embeddings Gr_{r} ↪ Gr_{r+1} that send a complex r-plane W to ℂ ⊕ W.

Theorem 1.5. For finite-dimensional CW complexes X we have natural isomorphisms

K^{0}(X) ≅ H(X, ℤ × Gr)  (4)

such that, for X = Gr_{r,n}, the class [U_{r,n}] + (d − r)[ℂ] in K^{0}(Gr_{r,n}) corresponds to the inclusion Gr_{r,n} ↪ {d} × Gr_{r,n} ↪ ℤ × Gr.

Proof. The theorem is of course well-known, see for example [Ada95, page 204]. To deduce it from the homotopy classification of vector bundles, we note first that any CW complex is paracompact and Hausdorff [Hat09, Proposition 1.20]. Moreover, we may assume that X is connected. The product ℤ × Gr can be viewed as the colimit of the inductive system

∐_{d≥0}{d} × Gr_{d} ↪ ∐_{d≥−1}{d} × Gr_{d+1} ↪ ∐_{d≥−2}{d} × Gr_{d+2} ↪ ⋯ ⊂ ℤ × Gr

Any continuous map from X to ℤ × Gr factors though one of the components colim_{n}({d}×Gr_{n}). By cellular approximation, it is in fact homotopic to a map that factors through {d} × Gr_{n} for some n. Thus,

H(X, ℤ × Gr) ≅ ∐_{d∈ℤ} colim_{n} Vect_{n}(X)

where the colimit is taken over the maps Vect_{n}(X) → Vect_{n+1}(X) sending a vector bundle E to ℂ ⊕ E. We define a map from the coproduct to K^{0}(X) by sending a vector bundle E in the d^{th} component to the class [E]+(d−rank E)[ℂ] in K^{0}(X). To see that this is a bijection, we use the fact that every vector bundle E over a finite-dimensional CW complex has a stable inverse: a vector bundle E^{⊥} over X such that E ⊕ E^{⊥} is a trivial bundle [Hus94, Chapter 3, Proposition 5.8].

If we replace the complex Grassmannians by real Grassmannians ℝGr_{r,n}, we obtain the analogous statement that KO^{0} can be represented by ℤ × ℝGr. Equivalently, but more in the spirit of Definition 1.2, we could work with the following spaces:

Definition 1.6. Let (V, ν) be a symmetric complex vector space, and let Gr(r, V) be the Grassmannian of complex k-planes in V. The “non-degenerate Grassmannian”

Gr^{nd}(r, (V, ν))

is the open subspace of Gr(r, V) given by r-planes T for which the restriction ν|_{T} is non-degenerate.

Complexification induces an inclusion of ℝGr(k, ℜ(V, ν)) into Gr^{nd}(r, (V, ν)), which, by Lemma 1.7 below, is a homotopy equivalence. So let Gr_{r,n}^{nd} abbreviate Gr^{nd}(r, ℍ^{r+n}), where ℍ is the hyperbolic plane (ℂ^{2}, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )), and let U_{r,n}^{nd} denote the restriction of the universal bundle over Gr(r, ℂ^{2r+2n}) to Gr_{r,n}^{nd}. Then colimits Gr_{r}^{nd} and Gr^{nd} can be defined in the same way as for the usual Grassmannians, and, for finite-dimensional CW complexes X, we obtain natural isomorphisms

KO^{0}(X) ≅ H(X, ℤ × Gr^{nd})  (5)

Here, for even (d−r), the inclusion Gr_{r,n}^{nd} ↪ {d}×Gr_{r,n}^{nd} ↪ ℤ×Gr^{nd} corresponds to the class of [U_{r,n}^{nd}] + \frac{d−r}{2} [ℍ] in GW^{0}(Gr_{r,n}^{nd}).

Lemma 1.7. For any symmetric complex vector space (V, ν), the following inclusion is a homotopy equivalence:

ℝGr(k, ℜ(V, ν)) ↪^{j} Gr^{nd}(k, (V, ν))
U ↦ U ⊗_{ℝ} ℂ

Proof. Consider the projection π: V = ℜ(V, ν) ⊕ iℜ(V, ν) ↠ ℜ(V, ν). We define a retract r of j by sending a complex k-plane T ∈ Gr^{nd}(k, (V, ν)) to π(ℜ(T, ν|_{T})) ⊂ ℜ(V, ν). This is indeed a linear subspace of real dimension k: since ν is positive definite on ℜ(T, ν|_{T}) but negative definite on iℜ(V, ν), the intersection ℜ(T, ν|_{T}) ∩ iℜ(V, ν) is trivial. More generally, we can define a family of endomorphisms of V parametrized by t ∈ [0, 1] by

π_{t}: ℜ(V, ν) ⊕ iℜ(V, ν) ↠ ℜ(V, ν) ⊕ iℜ(V, ν)
(x, y) ↦ (x, ty)

This family interpolates between the identity π_{1} and the projection π_{0}, which we can identify with π. We claim that

π_{t}(ℜ(T, ν|_{T})) ⊂ V

is a real linear subspace of dimension k on which ν is real and positive definite. The claim concerning the dimension has already been verified in the case t = 0 and follows for non-zero t from the fact that π_{t} is an isomorphism. Now take a non-zero vector v ∈ π_{t}(ℜ(T, ν|_{T})) and write it as v = x + tiy, where x, y ∈ ℜ(V, ν) and x + iy ∈ ℜ(T, ν|_{T}). Since ν(x, x), ν(y, y) and ν(x + iy, x + iy) are all real we deduce that ν(x, y) = 0; it follows that ν(v, v) is real as well. Moreover, since ν(x + iy, x + iy) is positive we have ν(x, x) > ν(y, y), so that ν(v, v) > (1 − t^{2})ν(y, y). In particular, ν(v, v) > 0 for all t ∈ [0, 1], as claimed. It follows that T ↦ π_{t}(ℜ(T, ν|_{T})) ⊗_{ℝ} ℂ defines a homotopy from j ∘ r to the identity on Gr^{nd}(k, (V, ν)).

K-spectra and cohomology theories. The infinite Grassmannian Gr can be identified with the classifying space BU of the infinite unitary group. Consequently, K^{0} can be represented by ℤ × BU, which by Bott periodicity is equivalent to its own two-fold loopspace Ω^{2}(ℤ × BU). This can be used to construct a 2-periodic Ω-spectrum 𝕂^{top} in the stable homotopy category SH whose even terms are all given by ℤ × BU. Similarly, ℝGr is equivalent to the classifying space BO of the infinite orthogonal group, and Bott periodicity in this case says that ℤ × BO is equivalent to Ω^{8}(ℤ × BO). Thus, one obtains a spectrum 𝕂O^{top} in SH which is 8-periodic. The associated cohomology theories are given by

K^{i}(X) := SH(Σ^{∞}(X_{+}), S^{i} ∧ 𝕂^{top})
KO^{i}(X) := SH(Σ^{∞}(X_{+}), S^{i} ∧ 𝕂O^{top})

where X_{+} denotes the union of X and a disjoint base point, and Σ^{∞} is the functor assigning to a pointed space its suspension spectrum. We refer the reader to [Ada95, III.2] for background and details. For convenience and later reference, we include here the values of the theories on a point. Since we are in fact dealing with multiplicative theories, these can be summarized in the form of coefficient rings:

K^{∗}(point) = ℤ[g, g^{−1}]  (6)
KO^{∗}(point) = ℤ[η, α, λ, λ^{−1}]/(2η, η^{3}, ηα, α^{2} − 4λ)  (7)

where g is of degree −2 and η, α and λ have degrees −1, −4 and −8, respectively [Bot69, pages 66–74^{3}].

1.3 Comparison

Now suppose X is a smooth complex variety. We write X(ℂ) for the set of complex points of X equipped with the analytic topology. If E is a vector bundle over X then E(ℂ) has the structure of a complex vector bundle over X(ℂ), so that we obtain natural maps

K_{0}(X) → K^{0}(X(ℂ))  (8)
GW^{0}(X) → KO^{0}(X(ℂ))  (9)

and an induced map

W^{0}(X) →\frac{KO^{0}(X(ℂ))}{K^{0}(X(ℂ))}  (10)

We now wish to extend these maps to be defined on GW^{i}(X) and W^{i}(X) for arbitrary i, and also on groups with support and twisted groups. Let us comment on some “elementary” constructions that are possible before outlining the approach that we will ultimately follow here.

3The multiplicative relations among the generators are given on page 74, but unfortunately the relation ηα = 0 is missing. This omission seems to have pervaded much of the literature, and I am indebted to Ian Grojnowski for pointing out the same mistake in an earlier version of this paper. Of course, the relation follows from the fact that KO^{−5}(point) = 0.

Documenta Mathematica 16 (2011) 465–511

Firstly, one way to extend the maps to the groups GW^{i}(X) and W^{i}(X) is to use the multiplicative structure of the theories together with Walter’s results on projective bundles [Wal03b]. Namely, for any variety X one has isomorphisms

GW^{i}(X × ℙ^{1}) ≅ GW^{i}(X) ⊕ GW^{i−1}(X)
KO^{2i}(X(ℂ) × S^{2}) ≅ KO^{2i}(X(ℂ)) ⊕ KO^{2i−2}(X(ℂ))

This allows an inductive definition of comparison maps, at least for all negative i. Basic properties of these maps, for example compatibility with the periodicities of Grothendieck-Witt and KO-groups, can be checked by direct calculations. It is less clear how to obtain maps on Witt groups with restricted supports. One possibility, pursued in [Zib09], is to work on the level of complexes of vector bundles and adapt a construction of classes in relative K-groups described in [Seg68] to the case of KO-theory. However, it remains unclear to the author how to see in this approach that the resulting maps are compatible with the boundary morphisms in localization sequences. Theorem 2.5 below could in fact be proved without knowing that the comparison maps respect the boundary morphisms in localization sequences in general. However, 𝔸^{1}-homotopy theory provides an alternative construction of a comparison map for which this property immediately follows from the construction, and which in any case is so compellingly elegant that it would be difficult to argue in favour of any other approach.

1.4 𝔸^{1}-homotopy theory

Theorem 1.5 describing K^{0} in terms of homotopy classes of maps to Grassmannians has an analogue in algebraic geometry, in the context of 𝔸^{1}-homotopy theory. Developed mainly by Morel and Voevodsky, the theory provides a general framework for a homotopy theory of schemes emulating the situation for topological spaces. The authoritative reference is [MV99]; closely related texts by the same authors are [Voe98], [Mor99] and [Mor04]. See [DLØ^{+}07] for a textbook introduction and [Dug01] for an enlightening perspective on one of the main ideas. We summarize the main points relevant for us in just a few sentences. The category Sm_{k} of smooth schemes over a field k can be embedded into some larger category of “spaces” Spc_{k} which is closed under small limits and colimits, and which can be equipped with a model structure. The 𝔸^{1}-homotopy category H(k) over k is the homotopy category associated with this model category. In fact, there are several possible choices for Spc_{k} and many possible model structures yielding the same homotopy category H(k). One possibility is to consider the category of simplicial presheaves over Sm_{k}, or the category of simplicial sheaves with respect to the Nisnevich topology. Both categories contain Sm_{k} as full subcategories via the Yoneda embedding, and they also contain simplicial sets viewed as constant (pre)sheaves. One may thus apply a general recipe for equipping the category of simplicial (pre)sheaves over a site with a model structure (see [Jar87]). In a crucial last step, one forces the affine line 𝔸^{1} to become contractible by localizing with respect to the set of all projections 𝔸^{1} × X ↠ X. As in topology, we also have a pointed version H_{•}(k) of H(k). Remarkably, these categories contain several distinct “circles”: the simplicial circle S^{1}, the “Tate circle” 𝔾_{m} = 𝔸^{1} − 0 (pointed at 1) and the projective line ℙ^{1} (pointed at ∞). These are related by the intriguing formula ℙ^{1} = S^{1} ∧ 𝔾_{m}. A common notational convention which we will follow is to define

S^{p,q} := (S^{1})^{∧(p−q)} ∧ 𝔾_{m}^{∧q}

for any p ≥ q. In particular, we then have S^{1} = S^{1,0}, 𝔾_{m} = S^{1,1} and ℙ^{1} = S^{2,1}. One can take the theory one step further by passing to the stable homotopy category SH(k), a triangulated category in which the suspension functors S^{p,q} ∧ − become invertible. This category is usually constructed using ℙ^{1}-spectra. The triangulated shift functor is given by suspension with the simplicial sphere S^{1,0}. Finally and crucially, the analogy with topology can be made precise: when we take our ground field k to be the complex numbers, or more general any subfield of ℂ, we have a complex realization functor

H(k) → H  (11)

that sends a smooth scheme X to its set of complex points X(ℂ) equipped with the analytic topology. There is also a pointed realization functor and, moreover, a triangulated functor of the stable homotopy categories

SH(k) → SH  (12)

which takes Σ^{∞}(X_{+}) to Σ^{∞}(X(ℂ)_{+}) for any smooth scheme X [Rio06, Théorème I.123; Rio07a, Théorème 5.26].

1.5 Representing algebraic and hermitian K-theory

Grassmannians of r-planes in k^{n+r} can be constructed as smooth projective varieties over any field k. Viewing them as objects in Spc_{k}, we can form their colimits Gr_{r} and Gr in the same way as in topology. The following analogue of Theorem 1.5 is established in [MV99, § 4]; see Théorème III.3 and Assertion III.4 in [Rio06].

Theorem 1.8. For smooth schemes X over k we have natural isomorphisms

K_{0}(X) ≅ H(k)(X, ℤ × Gr)  (13)

such that the inclusion of Gr_{r,n} ↪ {d} × Gr_{r,n} ↪ ℤ × Gr corresponds to the class [U_{r,n}] + (d − r)[O] in K_{0}(Gr_{r,n}).

An analogous result for hermitian K-theory has recently been obtained by Schlichting and Tripathi^{4}: Let Gr_{r,n}^{nd} denote the “non-degenerate Grassmannians” defined as open subvarieties of Gr_{r,r+2n} as above, and let Gr_{r}^{nd} and Gr^{nd} be the respective colimits. Then for smooth schemes over k we have natural isomorphisms

GW^{0}(X) ≅ H(k)(X, ℤ × Gr^{nd})  (14)

It follows from the construction that, when (d − r) is even, the inclusion of Gr_{r,n}^{nd} ↪ \{d\} × Gr_{r,n}^{nd} ↪ ℤ × Gr^{nd} corresponds to the class of [U_{r,n}^{nd}] + \frac{d−r}{2} [ℍ] in GW^{0}(Gr_{r,n}^{nd}), where U_{r,n}^{nd} is the universal symmetric bundle over Gr_{r,n}^{nd}. The fact that hermitian K-theory is representable in H(k) has been known for longer, see [Hor05]. One of the advantages of having a geometric description of a representing space, however, is that one can easily see what its complex realization is. In particular, this gives us an alternative way to define the comparison maps. For any smooth complex scheme X we have the following commutative squares, in which the left vertical arrows are the comparison maps (8) and (9), the right vertical arrows are induced by the complex realization functor (11).

[figure: K_{0}(X); ≅; H(ℂ)(X, ℤ × Gr); K^{0}(X(ℂ)); ≅; H(X(ℂ), ℤ × Gr); GW^{0}(X); ≅; H(ℂ)(X, ℤ × Gr^{nd}); KO^{0}(X(ℂ)); ≅; H(X(ℂ), ℤ × Gr^{nd})]

Some of the results quoted here are in fact known in a much greater generality. Firstly, higher algebraic and hermitian K-groups of X are obtained by passing to suspensions of X in (13) and (14). Even better, algebraic and hermitian K-theory are representable in the stable 𝔸^{1}-homotopy category SH(k). Let us make the statement a little more precise by fixing some notation. Given a spectrum 𝔼 in SH(k), we obtain a bigraded reduced cohomology theory Ẽ^{∗,∗} on H_{•}(k) and a corresponding unreduced theory E^{∗,∗} on H(k) by setting

Ẽ^{p,q}(X) := SH(k)(Σ^{∞}X , S^{p,q} ∧ 𝔼)  for X ∈ H_{•}(k)
E^{p,q}(X) := Ẽ^{p,q}(X_{+})  for X ∈ H(k)

A spectrum 𝕂 representing algebraic K-theory was first constructed in [Voe98, § 6.2]; see [Rio06] or [Rio07b] for some further discussion. It is (2,1)-periodic, meaning that in SH(k) we have an isomorphism

S^{2,1} ∧ 𝕂 →^{≅} 𝕂

4Talk “Geometric representation of hermitian K-theory in 𝔸^{1}-homotopy theory” at the Workshop “Geometric Aspects of Motivic Homotopy Theory”, 6.–10. September 2010 at the Hausdorff Center for Mathematics, Bonn

Documenta Mathematica 16 (2011) 465–511

Thus, the bigrading of the corresponding cohomology theory K^{p,q} is slightly artificial. The identification with the usual notation for algebraic K-theory is given by

K^{p,q}(X) = K_{2q−p}(X)  (15)

For hermitian K-theory we have an (8,4)-periodic spectrum 𝕂O, and the corresponding cohomology groups KO^{p,q} are honestly bigraded. The translation into the notation used for hermitian K-groups in Section 1.1 is given by

KO^{p,q}(X) = GW_{2q−p}^{q}(X)  (16)

We will refer to the number 2q − p as the degree of the group KO^{p,q}(X). The relation with Balmer’s Witt groups obtained by combining (16) and (2) is illustrated by the following table:

KO^{p,q} p = 0  1  2  3  4  5  6  7
q = 0 GW^{0} W^{1} W^{2} W^{3} W^{0} W^{1} W^{2} W^{3}
q = 1 GW_{2}^{1} GW_{1}^{1} GW^{1} W^{2} W^{3} W^{0} W^{1} W^{2}
q = 2 GW_{4}^{2} GW_{3}^{2} GW_{2}^{2} GW_{1}^{2} GW^{2} W^{3} W^{0} W^{1}
q = 3 GW_{6}^{3} GW_{5}^{3} GW_{4}^{3} GW_{3}^{3} GW_{2}^{3} GW_{1}^{3} GW^{3} W^{0}

As for the representing spaces in the unstable homotopy category, it is known that the complex realizations of 𝕂O and 𝕂 represent real and complex topological K-theory. This is well-documented in the latter case, see for example [Rio06, Proposition VI.12]. For 𝕂O, our references are slightly thin. Since the emphasis in this article is on showing how such a result in 𝔸^{1}-homotopy theory can be used for some concrete computations, we will at this point succumb to an “axiomatic” approach — the key statements we will be using are as follows:

Standing assumptions 1.9. There exist spectra 𝕂 and 𝕂O in SH(ℂ) representing algebraic K-theory and hermitian K-theory in the sense described above, such that:

(a) The complex realization functor (12) takes 𝕂 to 𝕂^{top} and 𝕂O to 𝕂O^{top}.

(b) We have an exact triangle in SH(ℂ) of the form

𝕂O ∧ S^{1,1} →^{η} 𝕂O → 𝕂 → S^{1,0} ∧ …  (17)

which corresponds to the usual triangle in SH.

These results are announced in [Mor06]. Independent constructions of spectra representing hermitian K-theory can be found in [Hor05] and in a recent preprint of Panin and Walter [PW10].

2 The comparison maps

It follows immediately from 1.9 that complex realization induces comparison maps

k̃^{p,q}:  K̃^{p,q}(X) → K̃^{p}(X (ℂ))
k̃_{h}^{p,q}:  K̃O^{p,q}(X) → K̃O^{p}(X (ℂ))

and hence comparison maps k^{p,q} and k_{h}^{p,q} on K- and hermitian K-groups. In particular, in degrees 0 and −1 we have maps

k^{0,0} :  K_{0}(X) → K^{0}(X(ℂ))
gw^{q} := k_{h}^{2q,q}:  GW^{q}(X) → KO^{2q}(X(ℂ))
w^{q} := k_{h}^{2q−1,q−1} : W^{q}(X) → KO^{2q−1}(X(ℂ))

for any smooth complex scheme X. Some good properties of these maps follow directly from the construction:

• They commute with pullbacks along morphisms of smooth schemes.

• They are compatible with suspension isomorphisms.

• They are compatible with the periodicity isomorphisms, so we can identify k_{h}^{p,q} with k_{h}^{p+8,q+4} (and hence w^{q} with w^{q+4} and gw^{q} with gw^{q+4}).

It is also clear that they are compatible with long exact sequences arising from exact triangles in SH(ℂ). This will be particularly useful in the following two cases.

Localization sequences. Given a smooth closed subscheme Z of a smooth scheme X, we have an exact triangle

Σ^{∞}(X − Z)_{+} → Σ^{∞}X_{+} → Σ^{∞}(\frac{X}{X−Z} ) → S^{1,0} ∧ …

in SH(ℂ). It induces long exact “localization sequences” for cohomology theories. For example, for hermitian K-theory we obtain sequences of the form

⋯ → K̃O^{p,q} (\frac{X}{X−Z} ) → KO^{p,q}(X) → KO^{p,q}(X − Z) → K̃O^{p+1,q} (\frac{X}{X−Z} ) → …

(18)

The comparison maps commute with all maps appearing in this sequence and the corresponding sequence of topological KO-groups. The space X/(X − Z) depends only on the normal bundle N of Z in X. To make this precise, we introduce the Thom space of a vector bundle E over an arbitrary smooth scheme Z, defined as the homotopy quotient of E by the complement of the zero section:

Thom_{Z}(E) := E/(E − Z)

Using a geometric construction known as deformation to the normal bundle, Morel and Voevodsky show in Theorem 2.23 of [MV99, Chapter 3] that X/(X − Z) is canonically isomorphic to Thom_{Z}(N) in the unstable pointed 𝔸^{1}-homotopy category. Thus, sequence (18) can be rewritten in the following form:

… → K̃O^{p,q}(Thom_{Z} N) → KO^{p,q}(X) → KO^{p,q}(X − Z) →
K̃O^{p+1,q}(Thom_{Z} N) → KO^{p+1,q}(X) → KO^{p+1,q}(X − Z) → …

Karoubi/Bott sequences. The KO- and K-groups of a topological space X fit into a long exact sequence known as the Bott sequence [Bot69, pages 75 and 112^{5}; BG10, 4.I.B]. It has the form

… → KO^{2i−1}X → KO^{2i−2}X → K^{0}X → KO^{2i}X → KO^{2i−1}X → K^{1}X
→ KO^{2i+1}X → KO^{2i}X → K^{0}X → KO^{2i+2}X → KO^{2i+1}X → …

(19)

The maps from KO- to K-groups are essentially given by complexification (or, depending on our choice of description of KO-groups, by forgetting the symmetric structure of a complex symmetric bundle), and the maps from K- to KO-groups are given by sending a complex vector bundle to its underlying real bundle (or to the associated hyperbolic bundle). The maps between KO-groups are given by multiplication with the generator η of KO^{−1}(point) (see (7)). This long exact sequence is induced by the triangle described in 1.9. The sequence arising from the corresponding triangle (17) in the stable 𝔸^{1}-homotopy category is known as the Karoubi sequence. The comparison maps induce a commutative ladder diagram that allows us to compare the two. Near degree zero, this takes the following form:

\xymatrix{… \ar[r] & KO^{2i−1,i}X \ar[r] \ar[d] & GW^{i−1}X \ar[r] \ar[d] & K_{0}X \ar[r] \ar[d] & GW^{i}X \ar[r] \ar[d] & W^{i}X \ar[r] \ar[d] & 0 \ar[r] \ar[d] & … \\ … \ar[r] & KO^{2i−1}X \ar[r] & KO^{2i−2}X \ar[r] & K^{0}X \ar[r] & KO^{2i}X \ar[r] & KO^{2i−1}X \ar[r] & K^{1}X \ar[r] & …}

(20)

As a consequence, the comparison maps w^{i} factor as

W^{i}(X) → \frac{KO^{2i}(X)}{K^{0}(X)} → KO^{2i−1}(X)

For cellular varieties, or more generally for spaces for which the odd topological K-groups vanish, the second map in this factorization is an isomorphism.

5Unfortunately, there are misprints on both pages. In particular, the central group in the diagram on page 112 should be K^{0}.

Documenta Mathematica 16 (2011) 465–511

Groups with restricted support. Comparing the localization sequences (1) and (18), we see that the groups K̃O^{p,q}(\frac{X}{X−Z}) play the role of hermitian K-groups of X supported on Z. This should be viewed as part of any representability statement, see for example [PW10, Theorem 6.5]. Alternatively, a formal identification of the groups in degrees zero and below using only the minimal assumptions we have stated could be achieved as follows:

Lemma 2.1. Let Z be a smooth closed subvariety of a smooth quasi-projective variety X. We have the following isomorphisms:

K̃O^{2q,q}(\frac{X}{X−Z}) ≅ GW_{Z}^{q}(X)
K̃O^{p,q}(\frac{X}{X−Z}) ≅ W_{Z}^{p−q}(X) for 2q − p < 0

Proof. Consider Z = Z × {0} as a subvariety of X × 𝔸^{1}. Its open complement (X × 𝔸^{1}) − Z contains X = X × {1} as a retract. Thus, the projection from X ×𝔸^{1} onto X induces a splitting of the localization sequences associated with (X × 𝔸^{1} − Z) ↪ X × 𝔸^{1}, and we have

GW_{Z}^{i+1}(X × 𝔸^{1}) ≅ coker (GW_{1}^{i+1}(X × 𝔸^{1}) ↪ GW_{1}^{i+1}(X × 𝔸^{1} − Z))
K̃O^{2i+2,i+1}( \frac{X×𝔸^{1}}{X×𝔸^{1}−Z}) ≅ coker (KO^{2i+1,i+1}(X × 𝔸^{1}) ↪ KO^{2i+1,i+1}(X × 𝔸^{1} − Z))

By (16), we can identify the groups appearing on the right, so we obtain an induced isomorphism of the cokernels. The quotient X × 𝔸^{1}/(X × 𝔸^{1} − Z) can be identified with the suspension of X/(X − Z) by S^{2,1}, so we have an isomorphism

K̃O^{2i+2,i+1}(\frac{X×𝔸^{1}}{X×𝔸^{1}−Z}) ≅ K̃O^{2i,i}(\frac{X}{X−Z})

On the other hand, we have analogous isomorphisms

GW_{Z}^{i+1}(X × 𝔸^{1}) ≅ GW_{Z}^{i}(X)

for Grothendieck-Witt and Witt groups. For Witt groups, this is a special case of Theorem 2.5 in [Nen07], the case when Z = X being Theorem 8.2 in [BG05]. The corresponding isomorphisms of Grothendieck-Witt groups may be deduced via Karoubi induction. The proof in lower degrees is analogous.

2.1 Twisting by line bundles

As described in Section 1.1, there is a natural notion of Witt groups twisted by line bundles. In the homotopy theoretic approach, such a twist can be encoded by passing to the Thom space of the bundle.

Definition 2.2. For a vector bundle E of constant rank r over a smooth scheme X, we define the hermitian K-groups of X with coefficients in E by

KO^{p,q}(X; E) := K̃O^{p+2r,q+r}(Thom E)

Likewise, for any complex vector bundle of rank r over a topological space X, we define

KO^{p}(X; E) := K̃O^{p+2r}(Thom E)

When E is a trivial bundle, its Thom space is just a suspension of X, so that KO^{p,q}(X; E) and KO^{p,q}(X) agree.

Lemma 2.3. For any vector bundle E over a smooth quasi-projective variety X, we have isomorphisms

KO^{2q,q}(X; E) ≅ GW^{q}(X; det E)
KO^{p,q}(X; E) ≅ W^{p−q}(X; det E) for 2q − p < 0

Proof. This follows from Lemma 2.1 and Nenashev’s Thom isomorphisms for Witt groups: for any vector bundle E of rank r there is a canonical Thom class in W_{X}^{r}(E) which induces an isomorphism W^{i}(X; det E) ≅ W_{X}^{i+r}(E) by multiplication [Nen07, Theorem 2.5]. This Thom class actually comes from a class in GW_{X}^{r}(E), and, as in the proof of Lemma 2.1, we can deduce that it induces an analogous isomorphism on Grothendieck-Witt groups via Karoubi induction.

Remark. The isomorphisms of Lemmas 2.1 and 2.3 are constructed here in a rather ad hoc fashion, and we have taken little care in recording their precise form. Whenever we give an argument concerning the comparison maps on “twisted groups” in the following, we do all constructions on the level of representable groups of Thom spaces. The identifications with the usual twisted groups are only needed to identify the final output of concrete calculations as in Section 4.

It follows similarly from Thom isomorphisms in topology that the groups KO(X; E) only depend on the determinant line bundle of E:

Lemma 2.4. For complex vector bundles E and F on a topological space X with identical first Chern class modulo 2, we have

KO^{p}(X; E) ≅ KO^{p}(X; F)

Proof. A complex vector bundle E whose first Chern class vanishes modulo 2 has a spin structure and is therefore oriented with respect to KO-theory [ABS64, § 12]. That is, we have a Thom isomorphism

KO^{p} X ⟶^{≅} K̃O^{p+2r}(Thom E)

Now suppose c_{1}(E) ≡ c_{1}(F) mod 2. We may view E ⊕ E ⊕ F both as a vector bundle over E and as a vector bundle over F, and by assumption it is oriented with respect to KO-theory in both cases. Thus, both groups in the lemma can be identified with KO^{p}(X; E ⊕ E ⊕ F).

Remark 2.1. In general the identifications of Lemma 2.4 are non-canonical. Given a spin structure on a real vector bundle, the constructions in [ABS64] do yield a canonical Thom class, but there may be several different spin structures on the same bundle. Still, canonical identifications exist in many cases. For example, there is a canonical spin structure on the square of any complex line bundle, yielding canonical identifications

KO^{p}(X; L) ≅ KO^{p}(X; L ⊗ M^{⊗2})

for any two complex line bundles L and M over X. Moreover, as different spin structures on a spin bundle over X are classified by the singular cohomology group H^{1}(X; ℤ/2), all spin structures arising in the context of complex cellular varieties below will be unique.

2.2 The comparison for cellular varieties

Theorem 2.5. For a smooth cellular complex variety X, the following comparison maps are isomorphisms:

K_{0}(X) ⟶^{≅} K^{0}(X(ℂ))
gw^{q}: GW^{q}(X) ⟶^{≅} KO^{2q}(X(ℂ))
w^{q}: W^{q}(X) ⟶^{≅} KO^{2q−1}(X(ℂ))

This remains true for twisted groups (see Section 2.1).

As indicated in the introduction, the first isomorphism is well-known and almost self-evident, given that both K_{0}(X) and K^{0}(X(ℂ)) are free abelian of rank equal to the number of cells of X. In particular, both the algebraic group K_{0}(ℂ) and the topological K-group K^{0}(point) are isomorphic to the integers, generated by the trivial line bundle, and the comparison map is the obvious isomorphism. Let us begin the proof of the theorem by also considering the other two maps first in the case when X is just a point Spec(ℂ). We can easily see that the corresponding groups are isomorphic by direct comparison:

KO^{p,q}(ℂ) p = 0 1 2 3  4  5  6  7
q = 0  ℤ ℤ ℤ  0 0 0 ℤ/2 0  0  0
q = 1  . . . . . . 0 0  0 ℤ/2 0  0
q = 2  . . . . . . . . . . . . ℤ ℤ ℤ  0 ℤ/2 0
q = 3  . . . . . . . . . . . . . . . . . . ℤ ℤ ℤ/2 ℤ ℤ ℤ/2
KO^{p}(point)  ℤ ℤ ℤ  0 0 0 ℤ ℤ ℤ  0 ℤ ℤ ℤ/2 ℤ ℤ ℤ/2

Table 1: (Grothendieck-)Witt and KO-groups of a point.

To see that the isomorphisms are given by our comparison maps, we can use the comparison of the Karoubi and Bott sequences. First, setting i = 0 in Diagram (20), we see that gw^{0} and w^{0} are isomorphisms on a point. As W^{0}(ℂ) is the only non-trivial Witt group of a point, it follows that w^{q} is an isomorphism on a point in general, so that we have

\xymatrix{… \ar[r] & GW^{q−1} \ar[r] \ar[d]^{_{gw_{q−1}}} & K_{0} \ar[r] \ar[d]^{≅} & GW^{q} \ar[r] \ar[d]^{_{gw_{q}}} & W^{q} \ar[r] \ar[d]^{≅} & 0 \ar[r] \ar[d] & … \\ … \ar[r] & _{KO_{2q−2}} \ar[r] & _{K_{0}} \ar[r] & _{KO_{2q}} \ar[r] & _{KO_{2q−1}} \ar[r] & 0 \ar[r] & …}

Given the periodicity of the Grothendieck-Witt groups, repeated applications of the Five Lemma now show that gw^{q} is an isomorphism on a point for all values of q. (This strategy of proof is known as “Karoubi induction”.)

We now treat the hermitian case in general. The case of algebraic/complex Ktheory could be dealt with similarly, or deduced from the hermitian case using triangle (17). It will be helpful to consider not only the maps gw^{q} = k_{h}^{2q,q} and w^{q+1} = k_{h}^{2q+1,q} in degrees 0 and −1, respectively, but also the maps k_{h}^{2q−1,q} in degree 1 and the maps k_{h}^{2q+2,q} in degree −2. We will prove the following extended statement:

Theorem 2.6. For a smooth cellular variety X, the hermitian comparison maps in degrees 1, 0, −1 and −2 have the properties indicated:

KO^{2q−1,q}(X) ↠ KO^{2q−1}(X(ℂ))
KO^{2q,q}(X) →^{≅} KO^{2q}(X(ℂ))
KO^{2q+1,q}(X) →^{≅} KO^{2q+1}(X(ℂ))
KO^{2q+2,q}(X) ↣ KO^{2q+2}(X(ℂ))

The analogous statements for twisted groups are also true.

Remark 2.2. The map in degree 1 is not an isomorphism even when X is a point. For example, it is known that KO^{−1,0}(ℂ) = ℤ/2 (see [Kar05, Example 18]), from which we may deduce via the Karoubi sequence that KO^{1,1}(ℂ) ≅ ℂ^{∗}. In particular, KO^{1,1}(ℂ) cannot be isomorphic to KO^{1}(point) = 0. The map in degree −2 can be identified with the inclusion of the 2-torsion subgroup of KO^{2q+2}(X(ℂ)) into KO^{2q+2}(X(ℂ)) for any cellular variety X. This follows from the theorem and the description of the KO-groups of cellular varieties given in Lemma 3.1. In degrees less than −2, the comparison map is necessarily zero. The problem is that while η: W^{p−q}(X) → W^{p−q}(X) is an isomorphism in all negative degrees, the topological η is nilpotent (η^{3} = 0).

The proof of Theorem 2.6 will proceed by induction over the number of cells of X and occupy the remainder of this section. To begin the induction, we need to consider the case of only one cell, which immediately reduces to the case of a point by homotopy invariance. In this case, degrees 0 and −1 have already been dealt with above. In degrees 1 and −2, on the other hand, most of the statements are trivial, and we only need to look at a few particular cases, which we postpone to the end of the proof. Spheres. Assuming the theorem to be true for a point, the compatibility of the comparison maps with suspensions immediately shows that the theorem is also true for the reduced cohomology of the spheres (ℙ^{1})^{∧d} = S^{2d,d}. To be precise, the following maps in degrees 1, 0, −1 and −2 have the properties indicated:

K̃O^{2q−1,q}(S^{2d,d}) ↠ K̃O^{2q−1}(S^{2d})
K̃O^{2q,q}(S^{2d,d}) →^{≅} K̃O^{2q}(S^{2d})
K̃O^{2q+1,q}(S^{2d,d}) →^{≅} K̃O^{2q+1}(S^{2d})
K̃O^{2q+2,q}(S^{2d,d}) ↣ K̃O^{2q+2}(S^{2d})

Cellular varieties. Now let X be a smooth cellular variety. By definition, X has a filtration by closed subvarieties ∅ = Z_{0} ⊂ Z_{1} ⊂ Z_{2} ⋯ ⊂ Z_{N} = X such that the open complement of Z_{k} in Z_{k+1} is isomorphic to 𝔸^{n_{k}} for some n_{k}. In general, the subvarieties Z_{k} will not be smooth. Their complements U_{k} := X − Z_{k} in X, however, are always smooth as they are open in X. So we obtain an alternative filtration X = U_{0} ⊃ U_{1} ⊃ U_{2} ⋯ ⊃ U_{N} = ∅ of X by smooth open subvarieties U_{k}. Each U_{k} contains a closed cell C_{k} ≅ 𝔸^{n_{k}} with open complement U_{k+1}. Our inductive hypothesis is that we have already proved the theorem for U_{k+1}, and we now want to prove it for U_{k}. We can use the following exact triangle in SH(ℂ):

Σ^{∞}(U_{k+1})_{+} → Σ^{∞}(U_{k})_{+} → Σ^{∞}Thom(N_{C_{k}\U_{k}}) → S^{1,0} ∧ …

As C_{k} is a cell, the Quillen-Suslin theorem tells us that the normal bundle N_{C_{k}\U_{k}} of C_{k} in U_{k} has to be trivial. Thus, Thom(N_{C_{k}\U_{k}}) is 𝔸^{1}-weakly equivalent to S^{2d,d}, where d is the codimension of C_{k} in U_{k}. Figure 1 displays the comparison between the long exact cohomology sequences induced by this triangle. The inductive step is completed by applying the Five Lemma to each dotted map in the diagram. The twisted case. To obtain the theorem in the case of coefficients in a vector bundle E over X, we replace the exact triangle above by the triangle

Σ^{∞} Thom(E_{|U_{k+1}}) → Σ^{∞} Thom(E_{|U_{k}}) → Σ^{∞} Thom(E_{|C_{k}} ⊕ N_{C_{k}\U_{k}}) → S^{1,0} ∧ …

The existence of this exact triangle is shown in the next lemma. The Thom space on the right is again just a sphere, so we can proceed as in the untwisted case.

[figure: ···; K̃O^{2q−1,q}(S^{2d,d}); K̃O^{2q−1}(S^{2d}); KO^{2q−1,q}(U_{k}); k_{h}^{2q−1,q}; KO^{2q−1}(U_{k}); KO^{2q−1,q}(U_{k+1}); KO^{2q−1}(U_{k+1}); K̃O^{2q,q}(S^{2d,d}); ≅; K̃O^{2q}(S^{2d}); KO^{2q,q}(U_{k}); k_{h}^{2q,q}; KO^{2q}(U_{k}); KO^{2q,q}(U_{k+1}); ≅; KO^{2q}(U_{k+1}); K̃O^{2q+1,q}(S^{2d,d}); ≅; K̃O^{2q+1}(S^{2d}); KO^{2q+1,q}(U_{k}); k_{h}^{2q+1,q}; KO^{2q+1}(U_{k}); KO^{2q+1,q}(U_{k+1}); ≅; KO^{2q+1}(U_{k+1}); K̃O^{2q+2,q}(S^{2d,d}); K̃O^{2q+2}(S^{2d}); KO^{2q+2,q}(U_{k}); k_{h}^{2q+2,q}; KO^{2q+2}(U_{k}); KO^{2q+2,q}(U_{k+1}); KO^{2q+2}(U_{k+1}); ···]

Figure 1: The inductive step.

Documenta Mathematica 16 (2011) 465–511

Lemma 2.7. Given a smooth subvariety Z of a smooth variety X with complement U, and given any vector bundle E over X, we have an exact triangle

Σ^{∞} Thom(E_{|U}) → Σ^{∞} Thom E → Σ^{∞} Thom(E_{|Z} ⊕ N_{Z\X}) → S^{1,0} ∧ …

Proof. From the Thom isomorphism theorem we know that the Thom space of a vector bundle over a smooth base is 𝔸^{1}-weakly equivalent to the quotient of the vector bundle by the complement of the zero section. Consider the closed embeddings U ↪ (E −Z), X ↪ E and Z ↪ E. Computing the normal bundles, we obtain

(E − Z)/(E − X) ≅ Thom_{U}(E_{|U})
E/(E − X) ≅ Thom_{X} E
E/(E − Z) ≅ Thom_{Z}(E_{|Z} ⊕ N_{Z\X})

The claim follows by passing to the stable homotopy category and applying the octahedral axiom to the composition of the embeddings (E − X) ⊆(E − Z) ⊆ E.

Remaining details concerning a point. To finish the proof of Theorem 2.6, we now return to the maps of degrees 1 and −2 in the case of a point, which we skipped above. First, let us deal with degree 1. The odd KO-groups of a point are all trivial except for KO^{−1}, so k_{h}^{2q−1,q} is trivially a surjection unless q ≡ 0 mod 4. In that case, surjectivity of k_{h}^{−1,0} is clear from the following diagram:

[figure: …; KO^{−1,0}; k_{h}^{−1,0}; GW^{−1}; ≅; K_{0}; ≅; …; …; _{KO_{−1}}; =; _{KO_{−2}}; _{K_{0}}; …; …; _{KO_{−1,0}}; k_{h}^{−1,0}; ℤ/2; 0; ≅; ℤ; ≅; …; …; ℤ/2; ≅; ℤ/2; 0; ℤ; …]

Lastly, we consider what happens in degree −2. Again, three out of four cases are trivial as KO^{2q+2,q} = W^{q+2} is zero unless q ≡ 2 mod 4. For the non-trivial case, consider the map η appearing in triangle (17). As the negative algebraic K-groups of ℂ are zero, η yields automorphisms of W^{p−q} in negative degrees. In topology, the corresponding maps are given by multiplication with a generator η of KO^{−1}, and η^{2} generates KO^{−2}. So the commutative square

\xymatrix{W^{0} \ar[r]^{≅} \ar[d]^{≅} & W^{0} \ar[d]^{k_{h}^{0,−2}} \\ _{KO_{−1}} \ar[r]^{η}_{≅} & _{KO_{−2}}}

shows that k_{h}^{0,−2} is an injection (in fact, an isomorphism), as claimed. This completes the proof of Theorem 2.6.

Remark 2.3. We indicate briefly how Theorem 2.5 can alternatively be obtained by working only with the maps in degrees 0 and −1 that can be defined by more elementary means. The basic strategy — comparing the localization sequences arising from the inclusion of a closed cell C_{k} into the union of “higher” cells U_{k} — still works. But we cannot deduce that the comparison maps are isomorphisms on U_{k} from the fact that they are isomorphisms on U_{k+1} because the parts of the sequences that we can actually compare are now too short. We can, however, still deduce that the maps in degree 0 with domains the Grothendieck-Witt groups of U_{k} are surjective, and that the maps in degree −1 with domains the Witt groups of U_{k} are injective. The inductive step can then be completed with the help of the Bott/Karoubi sequences. This argument works even without assuming that the comparison maps are compatible with the boundary maps in localization sequences in general: in the relevant cases the cohomology groups involved are so simple that this property can be checked by hand.

3 The Atiyah-Hirzebruch spectral sequence

We now aim to prepare the ground for the discussion of the KO-theory of some examples in the next section. The main computational tool will be the Atiyah-Hirzebruch spectral sequence, which in topology exists for any generalized cohomology theory and any finite-dimensional CW complex X [Ada95, III.7; Koc96, Theorem 4.2.7]. For KO-theory, it has the form

E_{2}^{p,q} = H^{p}(X; KO^{q}(point)) ⇒ KO^{p+q}(X)

with differential d_{r} of bidegree (r, −r + 1). The E_{2}-page is thus concentrated in the half-plane p ≥ 0 and 8-periodic in q: we have the integral cohomology of X in rows q ≡ 0 and q ≡ −4 mod 8, its cohomology with ℤ/2-coefficients in rows q ≡ −1 and q ≡ −2, and all other rows are zero. The differential d_{2} is given by Sq^{2} ∘π_{2} and Sq^{2} on rows q ≡ 0 and q ≡ −1, respectively, where

Sq^{2} : H^{∗}(X; ℤ/2) → H^{∗+2}(X; ℤ/2)

is the second Steenrod square and π_{2} is mod-2 reduction [Fuj67, 1.3].

The spectral sequence is multiplicative [Koc96, Proposition 4.2.9]. That is, the multiplication on the E_{2}-page induced by the cup product on singular cohomology and the ring structure of KO^{∗}(point) (see (7)) descends to a multiplication on all subsequent pages, such that the multiplication on the E_{∞}-page is compatible with the multiplication on KO^{∗}(X). In particular, each page is a module over KO^{∗}(point). The differentials of the spectral sequence are derivations, i. e. they satisfy a Leibniz rule.

3.1 The Atiyah-Hirzebruch spectral sequence for cellular varieties

For cellular varieties, or more generally for CW complexes with only evendimensional cells, the spectral sequence becomes simple enough to make some general deductions. We summarize some lemmas of Hoggar and Kono and Hara.

Lemma 3.1. [Hog69, 2.1 and 2.2] Let X be a CW complex with only evendimensional cells. Then:

• The ranks of the free parts of KO^{0}X and KO^{4}X are equal to the number t_{0} of cells of X of dimension a multiple of 4.

• The ranks of the free parts of KO^{2}X and KO^{6}X are equal to the number t_{1} of cells of X of dimension 2 modulo 4.

• The groups of odd degrees are two-torsion, i. e. KO^{2i−1}X = (ℤ/2)^{s_{i}} for some s_{i}.

• KO^{2i}X is isomorphic to the direct sum of its free part and KO^{2i+1}X.

Table 3 in Section 4.1 summarizes these statements.

Proof. The cohomology of X is free on generators given by the cells and concentrated in even degrees. The first two statements thus follow easily from the Atiyah-Hirzebruch spectral sequence for KO-theory (e. g. after tensoring with ℚ). On the other hand, we see from the Atiyah-Hirzebruch spectral sequence for complex K-theory that K^{0}(X) is a free abelian group on the cells while K^{1}(X) is zero. The last two statements thus become consequences of the Bott sequence (19).

The free part of KO^{∗} is thus very simple. In good cases, the spectral sequence also provides a nice description of the 2-torsion. To see this, note that Sq^{2} Sq^{2} = Sq^{3} Sq^{1} must vanish when the cohomology of X with ℤ/2-coefficients is concentrated in even degrees. So we can view (H^{∗}(X; ℤ/2), Sq^{2}) as a differential graded algebra over ℤ/2. To lighten the notation, we will write

H^{∗}(X, Sq^{2}) := H^{∗}(H^{∗}(X; ℤ/2), Sq^{2}) for the cohomology of this algebra. We keep the same grading as before, so that it is concentrated in even degrees. The row q ≡ −1 on the E_{3}-page is given by H^{∗}(X, Sq^{2}) · η, where η is the generator of KO^{−1}(point). Since it is the only row that contributes to KO^{∗} in odd degrees, we arrive at the following lemma, which will be central to our computations.

Lemma 3.2. Let X be as above. If the Atiyah-Hirzebruch spectral sequence of KO^{∗}(X) degenerates on the E_{3}-page, then

KO^{2i−1}(X) ≅ ⨁_{k} H^{2i+8k}(X, Sq^{2})

In all the examples we consider below, the spectral sequence does indeed degenerate at this stage. However, showing that it does can be tricky. One step in the right direction is the following observation of Kono and Hara [KH91, Proposition 1].

Lemma 3.3. Let X be as above. If the differentials d_{3}, d_{4}, …, d_{r−1} are trivial and d_{r} is non-trivial, then r ≡ 2 mod 8. In other words, the first non-trivial differential after d_{2} can only appear on a page E_{r} with page number r ≡ 2 mod 8. Such a differential is non-zero only on rows q ≡ 0 and q ≡ −1 mod 8. If it is non-zero on some x in row q ≡ 0, then it is also non-zero on ηx in row q ≡ −1. Conversely, if it is non-zero on some y in row q ≡ −1, there exists some x in row q ≡ 0 such that y = xη and d_{r} is non-zero on x.

Proof. We see from the spectral sequence of a point that d_{r}η = 0 for all differentials. Thus, multiplication by η gives a map of bidegree (0, −1) on the spectral sequence that commutes with the differentials. On the E_{2}-page this map is mod-2 reduction from row q ≡ 0 to row q ≡ −1 and the identity between rows q ≡ −1 and q ≡ −2. It follows that on the E_{3}-page multiplication by η induces a surjection from row q ≡ 0 to row q ≡ −1 and an injection of row q ≡ −1 into row q ≡ −2. This implies all statements above.

We derive a corollary that we will use to deduce that the spectral sequence collapses for certain Thom spaces:

Corollary 3.4. Suppose we have a continuous map p: X → T of CW complexes with only even-dimensional cells. Suppose further that the Atiyah-Hirzebruch spectral sequence for KO^{∗}(X) collapses on the E_{3}-page, and that p^{∗} induces an injection in row q ≡ −1:

p^{∗}: H^{∗}(T, Sq^{2}) ↪ H^{∗}(X, Sq^{2})

Then the spectral sequence for KO^{∗}(T) also collapses at this stage.

Proof. Write d_{r} for the first non-trivial higher differential, so r ≡ 2 mod 8. Then, for any element x in row q ≡ 0, we have p^{∗}(d_{r}x) = d_{r}p^{∗}(x) = 0 since the spectral sequence for X collapses. From our assumption on p^{∗} we can deduce that d_{r}x = 0. By the preceding lemma, this is all we need to show.

3.2 The Atiyah-Hirzebruch spectral sequence for Thom spaces

In order to compute twisted KO-groups, we need to apply the Atiyah-Hirzebruch spectral sequence of KO-theory to Thom spaces. So let X be a finite-dimensional CW complex, and let π: E → X be a vector bundle of constant rank over X. Though we will be mainly interested in the case when E is complex, we may more generally assume here that E is any real vector bundle which is oriented. Then the Thom isomorphism for singular cohomology tells us that the reduced cohomology of the Thom space Thom E is additively isomorphic to the cohomology of X itself, apart from a shift in degrees by r := rank_{ℝ} E. The isomorphism is given by multiplication with a Thom class θ in H̃^{r}(Thom E; ℤ):

\begin{matrix}H^{∗}(X; ℤ) & ⟶^{≅} H̃^{∗+r}(Thom E; ℤ) \\ x & ↦ π^{∗}(x) · θ \end{matrix}

Similarly, the reduction of θ modulo two induces an isomorphism of the respective singular cohomology groups with ℤ/2-coefficients. Thus, apart from a shift of columns, the entries on the E_{2}-page of the spectral sequence for K̃O^{∗}(Thom E) are identical to those on the E_{2}-page for KO^{∗}(X). However, the differentials may differ.

Lemma 3.5. Let E →^{π} X be a complex vector bundle of constant rank over a topological space X, with Thom class θ as above. The second Steenrod square on H̃^{∗}(Thom E; ℤ/2) is given by “ Sq^{2} + c_{1}(E)”, where c_{1}(E) is the first Chern class of E modulo two. That is,

Sq^{2}(π^{∗}x · θ) = π^{∗} (Sq^{2}(x) + c_{1}(E)x) · θ

for any x ∈ H^{∗}(X; ℤ/2). More generally, if E is a real oriented vector bundle, the second Steenrod square on the cohomology of its Thom space is given by “ Sq^{2} + w_{2}(E)”, where w_{2} is the second Stiefel-Whitney class of E.

Proof. This is a special case of an identity of Thom, which he in fact used to define Stiefel-Whitney classes:

Sq^{i}(π^{∗}x · θ) = π^{∗} (Sq^{i}(x) + w_{i}(E)x) · θ

See [MS74, page 91].

When X is a CW complex with cells only in even dimensions, the operation Sq^{2}+c_{1} can be viewed as a differential on H^{∗}(X; ℤ/2) for any c_{1} in H^{2}(X; ℤ/2). Extending our previous notation, we denote the cohomology with respect to this differential by

H^{∗}(X, Sq^{2} + c_{1}) := H^{∗}(H^{∗}(X; ℤ/2), Sq^{2} + c_{1})  (21)

Corollary 3.6 (of Lemmas 3.2 and 3.5). If the Atiyah-Hirzebruch spectral sequence of K̃O^{∗}(Thom E) degenerates on the E_{3}-page, then

KO^{2i−1}(X; E) ≅ ⨁_{k} H^{2i+8k}(X, Sq^{2} + c_{1}E)

It is true more generally that the differentials in the spectral sequence for K̃O^{∗}(Thom E) depend only on the second Stiefel-Whitney class of E. This follows from the observation that the Atiyah-Hirzebruch spectral sequence is compatible with Thom isomorphisms, as is made more precise by the next lemma: Fix a vector bundle E of constant rank r over a finite-dimensional CW complex X. Suppose E is oriented with respect to ordinary cohomology and let θ ∈ H̃^{∗}(Thom E; ℤ) be a Thom class.

Lemma 3.7. If E is oriented with respect to KO^{∗}, then θ survives to the E_{∞}- page of the Atiyah-Hirzebruch spectral sequence computing K̃O^{∗}(Thom E), and the Thom isomorphism for H^{∗} extends to an isomorphism of spectral sequences. That is, for each page right multiplication with the class of θ in Ẽ_{s}^{r,0}(Thom E) gives an isomorphism of E_{s}^{∗,∗}(X)-modules

E_{s}^{∗,∗}(X) ⟶_{≅}^{·θ} Ẽ_{s}^{∗+r,∗}(Thom E)

Moreover, any lift of θ ∈ Ẽ_{∞}^{r,0}(Thom E) to K̃O^{r}(Thom E) defines a Thom class of E with respect to KO^{∗}. The isomorphism of the E_{∞}-pages of the spectral sequences is induced by the Thom isomorphism given by multiplication with any such class.

Proof. We may assume without loss of generality that X is connected. Fix a point x on X. The inclusion of the fibre over x into E induces a map i_{x}: S^{r} ↪ Thom E. By assumption, the pullback i_{x}^{∗} on ordinary cohomology maps θ to a generator of H̃^{r}(S^{r}), and the pullback on K̃O^{∗} gives a surjection

K̃O^{∗}(Thom E) ↠^{i_{x}^{∗}} K̃O^{r}(S^{r})

Consider the pullback along i_{x} on the E_{∞}-pages of the spectral sequences for S^{r} and Thom E. Since we can identify Ẽ_{∞}^{r,0}(Thom E) with a quotient of K̃O^{r}(Thom E) and Ẽ_{∞}^{r,0}(S^{r}) with K̃O^{r}(S^{r}), we must have a surjection

i_{x}^{∗}: Ẽ_{∞}^{r,0}(Thom E) ↠ Ẽ_{∞}^{r,0}(S^{r})

On the other hand, the behaviour of i_{x}^{∗} on Ẽ_{∞}^{r,0} is determined by its behaviour on H̃^{r}, whence we can only have such a surjection if θ survives to the Ẽ_{∞}- page of Thom E. Thus, all differentials vanish on θ, and if multiplication by θ induces an isomorphism from E_{s}^{∗,∗}(X) to Ẽ_{s}^{∗+r,∗} on page s, it also induces an isomorphism on the next page. Lastly, consider any lift of θ to an element Θ of K̃O^{r}(Thom E). It is clear by construction that right multiplication with Θ gives an isomorphism from E_{∞}(X) to Ẽ_{∞}(Thom E), and thus it also gives an isomorphism from KO^{∗}(X) to K̃O^{∗}(Thom E). Thus, Θ is a Thom class for E with respect to KO^{∗}.

Lemma 3.7 allows the following strengthening of Lemma 2.4:

Corollary 3.8. For complex vector bundles E and F over X with identical first Chern class modulo 2, the spectral sequences computing K̃O^{∗}(Thom E) and K̃O^{∗}(Thom F) can be identified up to a possible shift of columns when E and F have different ranks.

4 Examples

We now turn to the study of projective homogeneous varieties, that is, varieties of the form G/P for some complex simple linear algebraic group G with a parabolic subgroup P. Any such variety has a cell decomposition [BGG73, Proposition 5.1], so that our comparison theorem applies. As far as we are only interested in the topology of G/P, we may alternatively view it as a homogeneous space for the compact real Lie group G^{c} corresponding to G:

Proposition 4.1. Let P be a parabolic subgroup of a simple complex algebraic group G. Then we have a diffeomorphism

G/P ≅ G^{c}/K

where K is a compact subgroup of maximal rank in a maximal compact subgroup G^{c} of G. More precisely, K is a maximal compact subgroup of a Levi subgroup of P.

Proof. The Iwasawa decomposition for G viewed as a real Lie group implies that we have a diffeomorphism G ≅ G^{c} · P [GOV94, Ch. 6, Prop. 1.7], inducing a diffeomorphism of quotients as claimed for K = G^{c} ∩ P. Since G^{c} ↪ G is a homotopy equivalence, so is the inclusion G^{c} ∩ P ↪ P. On the other hand, if L is a Levi subgroup of P then P = U ⋊ L, where U is unipotent and hence contractible. So the inclusion L ↪ P is also a homotopy equivalence. It follows that any maximal compact subgroup L^{c} of L is also maximal compact in P, and conversely that any maximal compact subgroup of P will be contained as a maximal compact subgroup in some Levi subgroup of P. We may therefore assume that K ⊂ L^{c} ⊂ L ⊂ P and conclude that K ↪ L^{c} is a homotopy equivalence. Since both groups are compact, it follows that in fact K ≅ L^{c}.

The KO-theory of homogeneous varieties has been studied intensively. In particular, the papers [KH91] and [KH92] of Kono and Hara provide complete computations of the (untwisted) KO-theory of all compact irreducible hermitian symmetric spaces, which we list in Table 2. For the convenience of the

G/P  G  Diagram of P  G^{c}/K

Grassmannians

(AIII) Gr_{m,n} SL_{m+n} ∘ ⋯ ∘ • ∘ ⋯ ∘_{1}  _{n}  _{n+m–1} \frac{U(m + n)}{U(m) × U(n)}

Maximal symplectic Grassmannians (CI)

X_{n} Sp_{2n} ∘ ∘ ⋯ ∘ ∘ ^{<} • Sp(n)/U(n)

Projective ; quadrics of ; dimension ; n ≥ 3 (BDI)  Q^{n}  SO_{n+2}  [figure: •; ∘; ⋯; ∘; >; ∘; (n odd); ∘; •; ∘; ⋯; ∘; (n even); ∘] \frac{SO(n + 2)}{SO(n) × SO(2)}

Spinor varieties (DIII)

S_{n}

SO_{2n}

[figure: ∘; ∘; ∘; ⋯; ∘; •]

SO(2n)/U(n)

Exceptional hermitian symmetric spaces:  EIII  E_{6}  ∘ ∘ ∘ ∘ ∘ •  \frac{E_{6}^{c}}{Spin(10) · S^{1}}
(Spin(10)∩S^{1}=ℤ/4)
EVII  E_{7}  ∘ ∘ ∘ ∘ ∘ ∘ •  \frac{E_{7}^{c}}{E_{6}^{c} · S^{1}}
(E_{6}^{c}∩S^{1}=ℤ/3)

Table 2: List of irreducible compact hermitian symmetric spaces. The symbols AIII, CI, … refer to E. Cartan’s classification. In the description of Gr_{m,n} we use U(m + n) instead of G^{c} = SU(m + n).

reader, we indicate how each of these arises as a quotient of a simple complex algebraic group G by a parabolic subgroup P, describing the latter in terms of marked nodes on the Dynkin diagram of G as in [FH91, § 23.3]. The last column gives an alternative description of each space as a quotient of a compact real Lie group. On the following pages, we will run through this list of examples and, in each case, extend Kono and Hara’s computations to include KO-groups twisted by a line bundle. Since each of these spaces is a “Grassmannian” in the sense that the parabolic subgroup P in G is maximal, its Picard group is free abelian on a single generator. Thus, there is exactly one non-trivial twist that we need to consider. In most cases, we — reassuringly — recover results for Witt groups that are already known. In a few other cases, we consider our results new. The untwisted KO-theory of complete flag varieties is also known in all three classical cases thanks to Kishimoto, Kono and Ohsita. We do not reproduce their result here but instead refer the reader directly to [KKO04]. By a recent result of Calmès and Fasel, all Witt groups with non-trivial twists vanish for these varieties [CF11].

4.1 Notation

Topologically, a cellular variety is a CW complex with cells only in even (real) dimensions. For such a CW complex X the KO-groups can be written in the form displayed in Table 3 below. This was shown in Section 3.1 in the case when the twist L is trivial, and the general case follows: if X is a CW complex with only even-dimensional cells, so is the Thom space of any complex vector bundle over X [MS74, Lemma 18.1]. In the following examples, results on KO^{∗} will be displayed by listing the values of the t_{i} and s_{i}. Since the t_{i} are just given by counting cells, and since the numbers of odd- and even-dimensional cells of a Thom space Thom_{X}E only depend on X and the rank of E, the t_{i} are in fact independent of L. The s_{i}, on the other hand, certainly will depend on the twist, and we will sometimes acknowledge this by writing s_{i}(L).

KO^{6}(X; L) = ℤ^{t_{1}} ⊕ (ℤ/2)^{s_{0}} = GW^{3}(X; L)
KO^{7}(X; L) =  (ℤ/2)^{s_{0}} = W^{0}(X; L)
KO^{0}(X; L) = ℤ^{t_{0}} ⊕ (ℤ/2)^{s_{1}} = GW^{0}(X; L)
KO^{1}(X; L) =  (ℤ/2)^{s_{1}} = W^{1}(X; L)
KO^{2}(X; L) = ℤ^{t_{1}} ⊕ (ℤ/2)^{s_{2}} = GW^{1}(X; L)
KO^{3}(X; L) =  (ℤ/2)^{s_{2}} = W^{2}(X; L)
KO^{4}(X; L) = ℤ^{t_{0}} ⊕ (ℤ/2)^{s_{3}} = GW^{2}(X; L)
KO^{5}(X; L) =  (ℤ/2)^{s_{3}} = W^{3}(X; L)

Table 3: Notational conventions in the examples. Only the s_{i} depend on L.

4.2 Projective spaces

Complex projective spaces are perhaps the simplest examples for which Theorem 2.5 asserts something non-trivial, so we describe the results here separately before turning to complex Grassmannians in general. The computations of the Witt groups of projective spaces were certainly landmark events in the history of the theory. In 1980, Arason was able to show that the Witt group W^{0}(ℙ^{n}) of ℙ^{n} over a field k agrees with the Witt group of k [Ara80]. The shifted Witt groups of projective spaces, and more generally of arbitrary projective bundles, were first computed by Walter in [Wal03b]. Quite recently, Nenashev deduced the same results via different methods [Nen09].

In the topological world, complete computations of KO^{i}(ℂℙ^{n}) were first published in a 1967 paper by Fujii [Fuj67]. It is not difficult to deduce the values of the twisted groups KO^{i}(ℂℙ^{n}; O(1)) from these: the Thom space Thom(O_{ℂℙ}^{n}(1)) can be identified with ℂℙ^{n+1}, so

KO^{i}(ℂℙ^{n}; O(1)) = K̃O^{i+2}(Thom(O(1)))
= K̃O^{i+2}(ℂℙ^{n+1})

Alternatively, we could do all required computations directly following the methods outlined in Section 3. The result, in any case, is displayed in Table 4, coinciding with the known results for the (Grothendieck-)Witt groups.

KO^{∗}(ℂℙ^{n}; L)  L ≡ O  L ≡ O(1)
t_{0}  t_{1}  s_{0} s_{1} s_{2} s_{3} s_{0} s_{1} s_{2} s_{3}
n ≡ 0 mod 4 (n/2) + 1  n/2  1 0 0 0 1 0 0 0
n ≡ 1  (n + 1)/2 (n + 1)/2 1 1 0 0 0 0 0 0
n ≡ 2 (n/2) + 1 n/2 1 0 0 0 0 0 1 0
n ≡ 3  (n + 1)/2 (n + 1)/2 1 0 0 1 0 0 0 0

Table 4: KO-groups of projective spaces

4.3 Grassmannians

We now consider the Grassmannians Gr_{m,n} of complex m-planes in ℂ^{m+n}. Again both the Witt groups and the untwisted KO-groups are already known: the latter by Kono and Hara [KH91], the former by the work of Balmer and Calmès [BC08]. A detailed comparison of the two sets of results in the untwisted case has been carried out by Yagita [Yag09]. We provide here an alternative topological computation of the twisted groups. Balmer and Calmès state their result by describing an additive basis of the total Witt group of Gr_{m,n} in terms of certain “even Young diagrams”. This is probably the most elegant approach, but needs some space to explain. We will stick instead to the tabular exposition used in the other examples. Let O(1) be a generator of Pic(Gr_{m,n}), say the dual of the determinant line bundle of the universal m-bundle over Gr_{m,n}. The result is displayed in Table 5. Our computation is based on the following geometric observation. Let U_{m,n} and U_{m,n}^{⊥} be the universal m-bundle and the orthogonal n-bundle on Gr_{m,n}, so that U ⊕ U^{⊥} = O^{⊕(m+n)}. We have various natural inclusions between the Grassmannians of different dimensions, of which we fix two:

Gr_{m,n−1} ↪ Gr_{m,n} via the inclusion of the first m + n − 1 coordinates into ℂ^{m+n}

Gr_{m−1,n} ↪ Gr_{m,n} by sending an (m − 1)-plane Λ to the m-plane Λ ⊕ ⟨e_{m+n}⟩, where e_{1}, e_{2}, …, e_{m+n} are the canonical basis vectors of ℂ^{m+n}

KO^{∗}(Gr_{m,n}; L)  L ≡ O  L ≡ O(1)
t_{0}  t_{1}  s_{0} s_{1} s_{2} s_{3} s_{0} s_{1} s_{2} s_{3}

m and n odd s.t.

m ≡ n

\frac{a}{2} \frac{a}{2} b b 0 0 0 0 0 0

m and n odd s.t.

m ≢ n

\frac{a}{2} \frac{a}{2} b 0 0 b 0 0 0 0

m ≡ n ≡ 0

m ≡ 0 and n odd n ≡ 0 and m odd

\frac{a + b}{2} \frac{a − b}{2} b 0 0 0 b 0 0 0

m ≡ n ≡ 2

m ≡ 2 and n odd n ≡ 2 and m odd

\frac{a + b}{2} \frac{a − b}{2} b 0 0 0 0 0 b 0

m ≡ 0 and n ≡ 2  \frac{a + b}{2} \frac{a − b}{2} b 0 0 0 b_{1} 0 b_{2} 0
m ≡ 2 and n ≡ 0  \frac{a + b}{2} \frac{a − b}{2} b 0 0 0 b_{2} 0 b_{1} 0

All equivalences (≡) are modulo 4. For the values of a and b = b_{1} + b_{2}, put k := ⌊^{m}/2⌋ and l := ⌊^{n}/2⌋. Then

a := ( \begin{matrix}m+n \\ m \end{matrix} )  b := ( \begin{matrix}k+l \\ k \end{matrix} )  b_{1} := ( \begin{matrix}k+l−1 \\ k \end{matrix} )  b_{2} := ( \begin{matrix}k+l−1 \\ k−1 \end{matrix} )

Table 5: KO-groups of Grassmannians

Lemma 4.2. The normal bundle of Gr_{m,n−1} in Gr_{m,n} is the dual U_{m,n−1}^{∨} of the universal m-bundle. Similarly, the normal bundle of Gr_{m−1,n} in Gr_{m,n} is given by U_{m−1,n}^{⊥}. In both cases, the embeddings of the subspaces extend to embeddings of their normal bundles, such that one subspace is the closed complement of the normal bundle of the other.

This gives us two cofibration sequences of pointed spaces:

Gr_{m−1,n}_{+} ↪^{i} Gr_{m,n}_{+} ↠^{p} Thom(U_{m,n−1}^{∨})  (22)
Gr_{m,n−1}_{+} ↪^{i} Gr_{m,n}_{+} ↠^{p} Thom(U_{m−1,n}^{⊥})  (23)

These sequences are the key to relating the untwisted KO-groups to the twisted ones. Following the notation in [KH91], we write A_{m,n} for the cohomology of Gr_{m,n} with ℤ/2-coefficients, denoting by a_{i} and b_{i} the Chern classes of U and U^{⊥}, respectively, and by a and b the total Chern classes 1 + a_{1} + ⋯ + a_{m} and 1 + b_{1} + ⋯ + b_{n}:

A_{m,n} = \frac{ℤ/2 [a_{1}, a_{2}, …, a_{m}, b_{1}, b_{2}, … b_{n}]}{a · b = 1}

We write d for the differential given by the second Steenrod square Sq^{2}, and d′ for Sq^{2} + a_{1}. To describe the cohomology of A_{m,n} with respect to these differentials, it is convenient to introduce the algebra

B_{k,l} = \frac{ℤ/2 [a_{2}^{2}, a_{4}^{2}, …, a_{2k}^{2}, b_{2}^{2}, b_{4}^{2}, …, b_{2l}^{2}]}{(1 + a_{2}^{2} + ⋯ + a_{2k}^{2})(1 + b_{2}^{2} + ⋯ + b_{2l}^{2}) = 1}

Note that this subquotient of A_{2k,2l} is isomorphic to A_{k,l} up to a “dilatation” in grading. Proposition 2 in [KH91] tells us that

H^{∗}(A_{m,n}, d) = \begin{cases}B_{k,l} & if (m, n) = {(2k, 2l), (2k + 1, 2l) or \atop (2k, 2l + 1)} \\ B_{k,l} ⊕ B_{k,l} · a_{m}b_{n−1} & if (m, n) = (2k + 1, 2l + 1) \end{cases}

Here, the algebra structure in the case where both m and n are odd is determined by (a_{m}b_{n−1})^{2} = 0.

Lemma 4.3. The cohomology of A_{m,n} with respect to the twisted differential d′ is as follows:

H^{∗}(A_{m,n}, d′) = \begin{cases}B_{k,l−1} · a_{m} ⊕ B_{k−1,l} · b_{n} & if (m, n) = (2k, 2l) \\ B_{k,l} · a_{m} & if (m, n) = (2k, 2l + 1) \\ B_{k,l} · b_{n} & if (m, n) = (2k + 1, 2l) \\ 0 & if (m, n) = (2k + 1, 2l + 1) \end{cases}

Proof. Let us shift the dimensions in the cofibration sequences (22) and (23) in such a way that we have the Thom spaces of U_{m,n}^{∨} and U_{m,n}^{⊥} on the right. Since the cohomologies of the spaces involved are concentrated in even degrees, the associated long exact sequence of cohomology groups falls apart into short exact sequences. Reassembling these, we obtain two short exact sequences of differential (A_{m,n+1}, d)- and (A_{m+1,n}, d)-modules, respectively:

0 → (A_{m,n}, d′) · θ^{∨} ⟶^{p^{∗}}(A_{m,n+1}, d) ⟶^{i^{∗}} (A_{m−1,n+1}, d) → 0  (24)
0 → (A_{m,n}, d′) · θ^{⊥} ⟶^{p^{∗}}(A_{m+1,n}, d) ⟶^{i^{∗}} (A_{m+1,n−1}, d) → 0  (25)

Here, θ^{∨} and θ^{⊥} are the respective Thom classes of U_{m,n}^{∨} and U_{m,n}^{⊥}. The map i^{∗} in the first row is the obvious quotient map annihilating a_{m}. Its kernel, the image of A_{m,n} under multiplication by a_{m}, is generated as an A_{m,n+1}-module by its unique element in degree 2m, and thus we must have p^{∗}(θ^{∨}) = a_{m}. Likewise, in the second row we have p^{∗}(θ^{⊥}) = b_{n}.

The lemma can be deduced from here case by case. For example, when both m and n are even, i^{∗} maps H^{∗}(A_{m,n+1}, d) = B_{k,l} to the first summand of H^{∗}(A_{m−1,n+1}, d) = B_{k−1,l} ⊕ B_{k−1,l} · a_{m−1}b_{n} by annihilating a_{m}^{2}. We know by comparison with the short exact sequences for the A_{m,n} that the kernel of this map is B_{k,l−1} mapping to B_{k,l} under multiplication by a_{m}^{2}. Thus, we obtain a short exact sequence

0 → B_{k−1,l} · a_{m−1}b_{n} ⟶^{∂} H^{∗}(A_{m,n}, d′) · θ^{∨} ⟶^{p^{∗}} B_{k,l−1} · a_{m}^{2} → 0  (26)

For the Steenrod square Sq^{2} of the top Chern class a_{m} of U, we have Sq^{2}(a_{m}) = a_{1}a_{m}. This can be checked, for example, by expressing a_{m} as the product of the Chern roots of U. Consequently, d′(a_{m}) = 0. Together with the fact that H^{∗}(A_{m,n}, d′) is a module over H^{∗}(A_{m,n+1}, d), this shows that we can define a splitting of p^{∗} by sending a_{m}^{2} to a_{m}θ^{∨}. Thus, H^{∗}(A_{m,n}, d′) contains B_{k,l−1} · a_{m} as a direct summand. If instead of working with sequence (24) we work with sequence (25), we see that H^{∗}(A_{m,n}, d′) also contains a direct summand B_{k−1,l} · b_{n}. These two summands intersect trivially, and a dimension count shows that together they encompass all of H^{∗}(A_{m,n}, d′). Alternatively, one may check explicitly that the boundary map ∂ above sends a_{m−1}b_{n} to b_{n}θ. The other cases are simpler.

Lemma 4.4. The Atiyah-Hirzebruch spectral sequence for K̃O^{∗}(Thom U_{m,n}^{∨}) collapses at the E_{3}-page.

Proof. By Proposition 4 of [KH91] we know that the spectral sequence for KO^{∗}(Gr_{m,n}) collapses as this stage, for any m and n. Now, if both m and n are even, we have

(B_{k,l−1} · a_{m} ⊕ B_{k−1,l} · b_{n}) · θ

in the (−1)^{st} row of the E_{3}-pages of the spectral sequences for Thom U^{∨} and Thom U^{⊥}, where θ = θ^{∨} or θ^{⊥}, respectively. In the case of U^{∨} we see from (26) that p^{∗} maps the second summand injectively to the E_{3}-page of the spectral sequence for KO^{∗}(Gr_{m,n+1}). Similarly, in the case of U^{⊥}, the first summand is mapped injectively to the E_{3}-page of KO^{∗}(Gr_{m+1,n}). Since the spectral sequences for Thom U^{∨} and Thom U^{⊥} can be identified via Corollary 3.8, we can argue as in Corollary 3.4 to see that they must collapse at this stage. Again, the cases when at least one of m, n is odd are similar but simpler.

We may now apply Corollary 3.6. The entries of Table 5 that do not appear in [KH91], i. e. those of the last four columns, follow from Lemma 4.3 by noting that B_{k,l} is concentrated in degrees 8i and of dimension dim B_{k,l} = dim A_{k,l} =

( \begin{matrix}k+l \\ k \end{matrix} ).

4.4 Maximal symplectic Grassmannians

The Grassmannian of isotropic n-planes in ℂ^{2n} with respect to a nondegenerate skew-symmetric bilinear form is given by X_{n} = Sp(n)/U(n). The universal bundle U on the usual Grassmannian Gr(n, 2n) restricts to the universal bundle on X_{n}, and so does the orthogonal complement bundle U^{⊥}. We will continue to denote these restrictions by the same letters. Thus, U ⊕ U^{⊥} ≅ ℂ^{2n} on X_{n}, and the fibres of U are orthogonal to those of U^{⊥} with respect to the standard hermitian metric on ℂ^{2n}. The determinant line bundles of U and U^{⊥} give dual generators O(1) and O(−1) of the Picard group of X_{n}.

Theorem 4.5. The additive structure of KO^{∗}(X_{n}; L) is as follows:

t_{0}  t_{1}  s_{i}(O)  s_{i}(O(1))
n even 2^{n−1} 2^{n−1} ρ(\frac{n}{2}, i) ρ(\frac{n}{2}, i − n)
n odd 2^{n−1} 2^{n−1} ρ(\frac{n+1}{2}, i)  0

Here, for any i ∈ ℤ/4 we write ρ(n, i) for the dimension of the i-graded piece of a ℤ/4-graded exterior algebra Λ_{ℤ/2}(g_{1}, g_{2}, …, g_{n}) on n homogeneous generators g_{1}, g_{2}, …, g_{n} of degree 1, i. e.

ρ(n, i) = ∑_{\begin{matrix}d≡i \\ mod 4 \end{matrix}} \binom{n}{d}

A table of the values of ρ(n, i) can be found in [KH92, Proposition 4.1].

It turns out to be convenient to work with the vector bundle U^{⊥} ⊕ O for the computation of the twisted groups KO^{∗}(X_{n}; O(1)). Namely, we have the following analogue of Lemma 4.2.

Lemma 4.6. There is an open embedding of the bundle U^{⊥} ⊕ O over the symplectic Grassmannian X_{n} into the symplectic Grassmannian X_{n+1} whose closed complement is again isomorphic to X_{n}.

Proof. To fix notation, let e_{1}, e_{2} be the first two canonical basis vectors of ℂ^{2n+2}, and embed ℂ^{2n} into ℂ^{2n+2} via the remaining coordinates. Assuming X_{n} is defined in terms of a skew-symmetric form Q_{2n}, define X_{n+1} with respect to the form

Q_{2n+2} := (\begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & Q_{2n} \end{matrix})

Then we have embeddings i_{1} and i_{2} of X_{n} into X_{n+1} sending an n-plane Λ ⊂ ℂ^{2n} to e_{1} ⊕ Λ or e_{2} ⊕ Λ in ℂ^{2n+2}, respectively. We extend i_{1} to an embedding of U^{⊥} ⊕ O by sending an n-plane Λ ∈ X_{n} together with a vector v in Λ^{⊥} ⊂ ℂ^{2n} and a complex scalar z to the graph Γ_{Λ,v,z} ⊂ ℂ^{2n+2} of the linear map

(\begin{matrix}z & Q_{2n}(−, v) \\ v & 0 \end{matrix}) : ⟨e_{1}⟩ ⊕ Λ → ⟨e_{2}⟩ ⊕ Λ^{⊥}

To avoid confusion, we emphasize that v is orthogonal to Λ with respect to a hermitian metric on ℂ^{2n}. The value of Q_{2n}(−, v), on the other hand, may well be non-zero on Λ. Consider the above embedding of U^{⊥} ⊕ O together with the embedding i_{2}:

U^{⊥} ⊕ O ^{⟶} X_{n+1}  X_{n}^{⟵}^{i_{2}} (Λ, v, z) _{↦} Γ_{Λ,v,z} ⟨e_{2}⟩ ⊕ Λ  Λ_{↤}

To see that the two embeddings are complementary, take an arbitrary (n + 1)- plane W in X_{n+1}. If e_{2} ∈ W then we can consider a basis

e_{2}, ( \begin{matrix}a_{1} \\ 0 \\ v_{1} \end{matrix} ) , …, ( \begin{matrix}a_{n} \\ 0 \\ v_{n} \end{matrix} )

of W, and the fact that Q_{2n+2} vanishes on W implies that all a_{i} are zero. Thus W can be identified with i_{2}(⟨v_{1}, …, v_{n}⟩). If, on the other hand, e_{2} is not contained in W then we must have a vector of the form ^{t}(1, z′, v′) in W, for some z′ ∈ ℂ and v′ ∈ ℂ^{2n}. Extend this vector to a basis of W of the form

( \begin{matrix}1 \\ z′ \\ v′ \end{matrix} ) , ( \begin{matrix}0 \\ b_{1} \\ v_{1} \end{matrix} ) , …, ( \begin{matrix}0 \\ b_{n} \\ v_{n} \end{matrix} )

and let Λ := ⟨v_{1}, …, v_{n}⟩. The condition that Q_{2n+2} vanishes on W implies that Q vanishes on Λ and that b_{i} = Q_{2n}(v_{i}, v′) for each i. In particular, Λ is ndimensional. Moreover, we can replace the first vector of our basis by a vector t(1, z, v) with v ∈ Λ^{⊥}, by subtracting appropriate multiples of the remaining basis vectors. Since Q vanishes on Λ we have Q_{2n}(v_{i}, v′) = Q_{2n}(v_{i}, v) and our new basis has the form

( \begin{matrix}1 \\ z \\ v \end{matrix} ) , ( \begin{matrix}0 \\ Q(v_{1},v) \\ v_{1} \end{matrix} ) , …, ( \begin{matrix}0 \\ Q(v_{n},v) \\ v_{n} \end{matrix} )

This shows that W = Γ_{Λ,v,z}.

Corollary 4.7. We have a cofibration sequence

X_{n}_{+} ↪^{i} X_{n+1}_{+} ↠^{p} Thom_{X_{n}}(U^{⊥} ⊕ O)

The associated long exact cohomology sequence splits into a short exact sequence of H^{∗}(X_{n+1})-modules since all cohomology here is concentrated in even degrees:

0 → H̃^{∗}(Thom_{X_{n}}(U^{⊥} ⊕ O)) →^{p^{∗}} H^{∗}(X_{n+1}) →^{i^{∗}} H^{∗}(X_{n}) → 0  (27)

Lemma 4.8. Let c_{i} denote the i^{th} Chern classes of U over X_{n}. We have

H^{∗}(X_{n}, Sq^{2}) = \begin{cases}Λ(a_{1}, a_{5}, a_{9}, …, a_{4m−3}) & if n = 2m \\ Λ(a_{1}, a_{5}, a_{9}, …, a_{4m−3}, a_{4m+1}) & if n = 2m + 1 \end{cases}

H^{∗}(X_{n}, Sq^{2} + c_{1}) = \begin{cases}Λ(a_{1}, a_{5}, …, a_{4m−3}) · c_{2m} & if n = 2m \\ 0 & if n is odd \end{cases}

for certain generators a_{i} of degree 2i.

Proof. Consider the short exact sequence (27). The mod-2 cohomology of X_{n} is an exterior algebra on the Chern classes c_{i} of U,

H^{∗}(X_{n}; ℤ/2) = Λ(c_{1}, c_{2}, …, c_{n})

and i^{∗} is given by sending c_{n+1} to zero. Thus, p^{∗} is the unique morphism of H^{∗}(X_{n+1}; ℤ/2)-modules that sends the Thom class θ of U^{⊥} ⊕ O to c_{n+1}. This short exact sequence induces a long exact sequence of cohomology groups with respect to the Steenrod square Sq^{2}. The algebra H^{∗}(X_{n}, Sq^{2}) was computed in [KH92, 2–2], with the result displayed above, so we already know two thirds of this sequence. Explicitly, we have a_{4i+1} = c_{2i}c_{2i+1},^{6} so i^{∗} is the obvious surjection sending a_{i} to a_{i} (or to zero). Thus, the long exact sequence once again splits. If n = 2m we obtain a short exact sequence

0 → H^{∗}(X_{2m}, Sq^{2} +c_{1})·θ →^{p^{∗}} Λ(a_{1}, …, a_{4m−3}, a_{4m+1}) →^{i^{∗}} Λ(a_{1}, …, a_{4m−3}) → 0

We see that H^{∗}(X_{2m}, Sq^{2} + c_{1}) · θ is isomorphic to Λ(a_{1}, …, a_{4m−3}) · a_{4m+1} as a Λ(a_{1}, …, a_{4m+1})-module. It is thus generated by a single element, which is the unique element of degree 8m + 2. Since p^{∗}(c_{2m}θ) = a_{4m+1}, the class of c_{2m}θ is the element we are looking for, and the result displayed above follows. If, on the other hand, n is odd, then i^{∗} is an isomorphism and H^{∗}(X_{n}, Sq^{2} + c_{1}) must be trivial.

We see from the proof that p^{∗} induces an injection of H^{∗}(X_{n}, Sq^{2} + c_{1}) · θ into H^{∗}(X_{n}, Sq^{2}). Since we already know from [KH92, Theorem 2.1] that the Atiyah-Hirzebruch spectral sequence for KO^{∗}(X_{n}) collapses, we can apply Corollary 3.4 to deduce that the spectral sequence for K̃O^{∗}(Thom_{X_{n}}(U^{⊥} ⊕ O)) collapses at the E_{3}-page as well. This completes the proof of Theorem 4.5.

4.5 Quadrics

We next consider smooth complex quadrics Q^{n} in ℙ^{n+1}. As far as we are aware, the first complete results on (shifted) Witt groups of split quadrics were due to Walter: they are mentioned together with the results for projective bundles in [Wal03a] as the main applications of that paper. Unfortunately, they seem to have remained unpublished. Partial results are also included in Yagita’s preprint [Yag04], see Corollary 8.3. More recently, Nenashev obtained almost complete results by considering the localization sequences arising from the inclusion of a linear subspace of maximal dimension [Nen09]. Calmès informs me that the geometric description of the boundary map given in [BC09] can be used to show that these localization sequences split in general, yielding a complete computation. The calculation described here is completely independent of these results.

6In [KH92] the generators are written as c_{2i}c′_{2i+1} with c′_{2i+1} = c_{2i+1} + c_{1}c_{2i}.

Documenta Mathematica 16 (2011) 465–511

KO^{∗}(Q^{n}; L)  L ≡ O  L ≡ O(1)
t_{0}  t_{1}  s_{0} s_{1} s_{2} s_{3} s_{0} s_{1} s_{2} s_{3}
n ≡ 0 mod 8 (n/2) + 2  n/2  2 0 0 0 2 0 0 0
n ≡ 1  (n + 1)/2 (n + 1)/2 1 1 0 0 1 1 0 0
n ≡ 2  (n/2) + 1 (n/2) + 1 1 2 1 0 0 0 0 0
n ≡ 3  (n + 1)/2 (n + 1)/2 1 1 0 0 0 0 1 1
n ≡ 4 (n/2) + 2 n/2 2 0 0 0 0 0 2 0
n ≡ 5  (n + 1)/2 (n + 1)/2 1 0 0 1 0 1 1 0
n ≡ 6  (n/2) + 1 (n/2) + 1 1 0 1 2 0 0 0 0
n ≡ 7  (n + 1)/2 (n + 1)/2 1 0 0 1 1 0 0 1

Table 6: KO-groups of projective quadrics (n ≥ 3)

For n ≥ 3 the Picard group of Q^{n} is free abelian on a single generator given by the restriction of the universal line bundle O(1) over ℙ^{n+1}. We will use the same notation O(1) for this restriction.

Theorem 4.9. The KO-theory of a smooth complex quadric Q^{n} of dimension n ≥ 3 is as described in Table 6.

Untwisted KO-groups. Before turning to KO^{∗}(Q^{n}; O(1)) we review the initial steps in the computation of the untwisted KO-groups. The integral cohomology of Q^{n} is well-known: If n is even, write n = 2m. We have a class x in H^{2}(Q^{n}) given by a hyperplane section, and two classes a and b in H^{n}(Q^{n}) represented by linear subspaces of Q of maximal dimension. These three classes generate the cohomology multiplicatively, modulo the relations

x^{m} = a + b  x^{m+1} = 2ax
ab = \begin{cases}0 & if n ≡ 0 \\ ax^{m} & if n ≡ 2 \end{cases}  a^{2} = b^{2} = \begin{cases}ax^{m} & if n ≡ 0 mod 4 \\ 0 & if n ≡ 2 mod 4 \end{cases}

Additive generators can thus be given as follows:

\begin{matrix}d & 0 & 2 & 4 & … & n − 2 & n & n + 2 & n + 4 & … & 2n \\ H^{d}(Q^{n}) & 1 & x & x^{2} & … & x^{m−1} & a, b & ax & ax^{2} & … & ax^{m} \end{matrix}

If n is odd, write n = 2m + 1. Then similarly multiplicative generators are given by the class of a hyperplane section x in H^{2}(Q^{n}) and the class of a linear subspace a in H^{n+1}(Q^{n}) modulo the relations x^{m+1} = 2a and a^{2} = 0.

\begin{matrix}d & 0 & 2 & 4 & … & n − 1 & n + 1 & n + 3 & n + 5 & … & 2n \\ H^{d}(Q^{n}) & 1 & x & x^{2} & … & x^{m} & a & ax & ax^{2} & … & ax^{m} \end{matrix}

The action of the Steenrod square on H^{∗}(Q^{n}; ℤ/2) is also well-known; see for example [Ish92, Theorem 1.4 and Corollary 1.5] or [EKM08, § 78]:

Sq^{2}(x) = x^{2}
Sq^{2}(a) = \begin{cases}ax & if n ≡ 0 or 3 mod 4 \\ 0 & if n ≡ 1 or 2 \end{cases}
Sq^{2}(b) = Sq^{2}(a) (for even n)

As before, we write H^{∗}(Q^{n}, Sq^{2}) for the cohomology of H^{∗}(Q^{n}; ℤ/2) with respect to the differential Sq^{2}.

Lemma 4.10. Write n = 2m or n = 2m+1 as above. The following table gives a complete list of the additive generators of H^{∗}(Q^{n}, Sq^{2}).

d 0 … n − 1 n n + 1 … 2n
H^{d}(Q^{n}, Sq^{2}) 1  ax^{m} if n ≡ 0 mod 4
1  a  if n ≡ 1
1  a, b  ab if n ≡ 2
1  x^{m}  if n ≡ 3

The results of Kono and Hara on KO^{∗}(Q) follow from here provided there are no non-trivial higher differentials in the Atiyah-Hirzebruch spectral sequence. This is fairly clear in all cases except for the case n ≡ 2 mod 4. In that case, the class a + b = x^{m} can be pulled back from Q^{n+1}, and therefore all higher differentials must vanish on a + b. But one has to work harder to see that all higher differentials vanish on a (or b). Kono and Hara proceed by relating the KO-theory of Q^{n} to that of the spinor variety S_{\frac{n}{2} +1} discussed in Section 4.6.

Twisted KO-groups. We now compute KO^{∗}(Q^{n}; O(1)). Let θ ∈ H^{2}(Thom_{Q^{n}}O(1)) be the Thom class of O(1), so that multiplication by θ maps the cohomology of Q^{n} isomorphically to the reduced cohomology of Thom_{Q^{n}}O(1). The Steenrod square on H̃^{∗}(Thom_{Q^{n}}O(1); ℤ/2) is determined by Lemma 3.5: for any y ∈ H^{∗}(Q^{n}; ℤ/2) we have Sq^{2}(y · θ) = (Sq^{2} y + xy) · θ. We thus arrive at

Lemma 4.11. The following table gives a complete list of the additive generators of H̃^{∗}(Thom_{Q^{n}}O(1), Sq^{2}).

d … n + 1 n + 2 n + 3 … 2n + 2
H̃^{d}(…)  aθ, bθ  if n ≡ 0 mod 4
x^{m}θ  ax^{m}θ if n ≡ 1
if n ≡ 2
aθ  ax^{m}θ if n ≡ 3

We claim that all higher differentials in the Atiyah-Hirzebruch spectral sequence for K̃O^{∗}(Thom_{Q^{n}}O(1)) vanish. For even n this is clear. But for n = 8k + 1 the differential d_{8k+2} might a priori take x^{m}θ to ax^{m}θ, and for n = 8k + 3 the differential d_{8k+2} might take aθ to ax^{m}θ. We therefore need some geometric considerations. Namely, the Thom space Thom_{Q^{n}}O(1) can be identified with the projective cone over Q^{n} embedded in ℙ^{n+2}. This projective cone can be realized as the intersection of a smooth quadric Q^{n+2} ⊂ ℙ^{n+3} with its projective tangent space at the vertex of the cone [Har92, p. 283]. Thus, we can consider the following inclusions:

Q^{n} ↪^{j} Thom_{Q^{n}}O(1) ↪^{i} Q^{n+2}

The composition is the inclusion of the intersection of Q^{n+2} with two transversal hyperplanes.

Lemma 4.12. All higher differentials ( d_{k} with k > 2) in the Atiyah-Hirzebruch spectral sequence for KO^{∗}(Thom_{Q^{n}}O(1)) vanish.

Proof. We need only consider the cases when n is odd. Write n = 2m + 1. When n ≡ 1 mod 4 we claim that i^{∗} maps x^{m+1} in H^{n+1}(Q^{n+2}, Sq^{2}) to x^{m}θ in H^{n+1}(Thom_{Q^{n}}O(1), Sq^{2}). Indeed, j^{∗}i^{∗} maps the class of the hyperplane section x in H^{2}(Q^{n+2}) to the class of the hyperplane section x in H^{2}(Q^{n}). So i^{∗}x in H^{2}(Thom_{Q^{n}}O(1)) must be non-zero, hence equal to θ modulo 2. It follows that i^{∗}(x^{m+1}) = θ^{m+1}. Since θ^{2} = Sq^{2}(θ) = xθ, we have θ^{m+1} = x^{m}θ, proving the claim. As we already know that all higher differentials vanish on H^{∗}(Q^{n+2}, Sq^{2}), we may now deduce that they also vanish on H^{∗}(Thom_{Q^{n}}O(1), Sq^{2}). When n ≡ 3 mod 4 we claim that i^{∗} maps a in H^{n+3}(Q^{n+2}, Sq^{2}) to aθ in H^{n+3}(Thom_{Q^{n}}O(1), Sq^{2}). Indeed, a represents a linear subspace of codimension m + 2 in Q^{n+2} and is thus mapped to the class of a linear subspace of the same codimension in Q^{n}: j^{∗}i^{∗}(a) = ax in H^{n+3}(Q^{n}). Thus, i^{∗}(a) is nonzero in H^{n+3}(Thom_{Q^{n}}O(1)), equal to aθ modulo 2. Again, this implies that all higher differentials vanish on H^{∗}(Thom_{Q^{n}}O(1), Sq^{2}) since they vanish on H^{∗}(Q^{n+2}, Sq^{2}).

The additive structure of KO^{∗}(Q^{n}; O(1)) thus follows directly from the result for H^{d}(Q^{n}, Sq^{2} + x) = H̃^{d+2}(Thom_{Q^{n}}O(1)) displayed in Lemma 4.11 via Corollary 3.6.

4.6 Spinor varieties

Let Gr_{SO}(n, N) be the Grassmannian of n-planes in ℂ^{N} isotropic with respect to a fixed non-degenerate symmetric bilinear form, or, equivalently, the Fano variety of projective (n − 1)-planes contained in the quadric Q^{N−2}. For each N > 2n, this is an irreducible homogeneous variety. In particular, for N = 2n+ 1 we obtain the spinor variety S_{n+1} = Gr_{SO}(n, 2n+1). The variety Gr_{SO}(n, 2n) falls apart into two connected components, both of which are isomorphic to S_{n}. This is reflected by the fact that we can equivalently identify S_{n} with SO(2n − 1)/U(n − 1) or SO(2n)/U(n). As for all Grassmannians, the Picard group of S_{n} is isomorphic to ℤ; we fix a line bundle S which generates it. The KO-theory twisted by S vanishes:

Theorem 4.13. For all n ≥ 2 the additive structure of KO^{∗}(S_{n}; L) is as follows:

t_{0}  t_{1}  s_{i}(O)  s_{i}(S)
n ≡ 2 mod 4 2^{n−2} 2^{n−2} ρ(\frac{n}{2}, 1 − i)  0
otherwise  2^{n−2} 2^{n−2} ρ(⌊\frac{n}{2}⌋, −i)  0

Here, the values ρ(n, i) are defined as in Theorem 4.5.

Proof. The cohomology of S_{n} with ℤ/2-coefficients has simple generators e_{2}, e_{4}, …, e_{2n−2}, i. e. it is additively generated by products of distinct elements of this list. Its multiplicative structure is determined by the rule e_{2i}^{2} = e_{4i}, and the second Steenrod square is given by Sq^{2}(e_{2i}) = ie_{2i+2} [Ish92, Proposition 1.1]. In both formulae it is of course understood that e_{2j} = 0 for j ≥ n. What we need to show is that for all n ≥ 2 we have

H^{∗}(S_{n}, Sq^{2} + e_{2}) = 0

Let us abbreviate H^{∗}(S_{n}, Sq^{2} + e_{2}) to (H_{n}, d′). We claim that we have the following short exact sequence of differential ℤ/2-modules:

0 → (H_{n}, d′) →^{·e_{2n}} (H_{n+1}, d′) → (H_{n}, d′) → 0  (28)

This can be checked by a direct calculation. Alternatively, it can be deduced from the geometric considerations below. Namely, it follows from the cofibration sequence of Corollary 4.15 that we have such an exact sequence of ℤ/2-modules with maps respecting the differentials given by Sq^{2} on all three modules. Since they also commute with multiplication by e_{2}, they likewise respect the differential d′ = Sq^{2} + e_{2}. The long exact cohomology sequence associated with (28) allows us to argue by induction: if H^{∗}(H_{n}, d′) = 0 then also H^{∗}(H_{n+1}, d′) = 0. Since we can see by hand that H^{∗}(H_{2}, d′) = 0, this completes the proof.

We close with a geometric interpretation of the exact sequence (28), via an analogue of Lemmas 4.2 and 4.6. Let us write U for the universal bundle over S_{n}, i. e. for the restriction of the universal bundle over Gr(n − 1, 2n − 1) to S_{n}, and U^{⊥} for the restriction of the orthogonal complement bundle, so that U ⊕ U^{⊥} is the trivial (2n − 1)-bundle over S_{n}. As in Section 4.4, we emphasize that under these conventions the fibres of U and U^{⊥} are perpendicular with respect to a hermitian metric on ℂ^{2n−1} — they are not orthogonal with respect to the chosen symmetric form.

Lemma 4.14. The spinor variety S_{n} embeds into the spinor variety S_{n+1} with normal bundle U^{⊥} such that the embedding extends to an embedding of this bundle. The closed complement of U^{⊥} in S_{n+1} is again isomorphic to S_{n}.

Corollary 4.15. We have a cofibration sequence

S_{n}_{+} ↪^{i} S_{n+1}_{+} ↠^{p} Thom_{S_{n}} U^{⊥}

Note however that, unlike in the symplectic case, the first Chern classes of U and U^{⊥} pull back to twice a generator of the Picard group of S_{n}. For example, the embedding of S_{2} into Gr(1, 3) can be identified with the embedding of the one-dimensional smooth quadric into the projective plane, of degree 2, and the higher dimensional cases can be reduced to this example. Thus, c_{1}(U) and c_{1}(U^{⊥}) are trivial in Pic(S_{n})/2.

proof of Lemma 4.14. The proof is similar to the proof of Lemma 4.6. Let e_{1}, e_{2} be the first two canonical basis vectors of ℂ^{2n+1}, and let ℂ^{2n−1} be embedded into ℂ^{2n+1} via the remaining coordinates. Let S_{n} be defined in terms of a symmetric form Q on ℂ^{2n−1}, and define S_{n+1} in terms of

Q_{2n+1} := (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & Q \end{matrix})

Let i_{1} and i_{2} be the embeddings of S_{n} into S_{n+1} sending an (n − 1)-plane Λ ⊂ ℂ^{2n−1} to e_{1} ⊕ Λ or e_{2} ⊕ Λ in ℂ^{2n+1}, respectively. Given an (n − 1)-plane Λ ∈ S_{n} together with a vector v in Λ^{⊥} ⊂ ℂ^{2n−1}, consider the linear map

(\begin{matrix}−\frac{1}{2}Q(v, v) & −Q(−, v) \\ v & 0 \end{matrix}) : ⟨e_{1}⟩ ⊕ Λ → ⟨e_{2}⟩ ⊕ Λ^{⊥}

Sending (Λ, v) to the graph of this function defines an open embedding of U^{⊥} whose closed complement is the image of i_{2}.

4.7 Exceptional hermitian symmetric spaces

Lastly, we turn to the exceptional hermitian symmetric spaces EIII and EVII. We write O(1) for a generator of the Picard group in both cases.

Theorem 4.16. The KO-groups of the exceptional hermitian symmetric spaces EIII and EVII are as follows:

L ≡ O  L ≡ O(1)
t_{0} t_{1} s_{0} s_{1} s_{2} s_{3} s_{0} s_{1} s_{2} s_{3}
KO^{∗}(EIII; L) 15 12 3 0 0 0 3 0 0 0
KO^{∗}(EVII; L) 28 28 1 3 3 1 0 0 0 0

Proof. The untwisted KO-groups have been computed in [KH92], the main difficulty as always being to prove that the Atiyah-Hirzebruch spectral sequence collapses. For the twisted groups, however, there are no problems. We quote from § 3 of said paper that the cohomologies of the spaces in question can be written as

H^{∗}(EIII; ℤ/2) = ℤ/2[t, u]/(u^{2}t, u^{3} + t^{12})
H^{∗}(EVII; ℤ/2) = ℤ/2[t, v, w]/(t^{14}, v^{2}, w^{2})

with t of degree 2 in both cases, and u, v and w of degrees 8, 10 and 18, respectively. The Steenrod squares are determined by Sq^{2} u = ut and Sq^{2} v = Sq^{2} w = 0. Thus, we find

H^{∗}(EIII, Sq^{2} + t) = ℤ/2 · u ⊕ ℤ/2 · u^{2} ⊕ ℤ/2 · u^{3}
H^{∗}(EVII, Sq^{2} + t) = 0

By Lemma 3.3 the Atiyah-Hirzebruch spectral sequence for EIII must collapse. This gives the result displayed above.

Acknowledgements

The present paper is based on research carried out in the context of the author’s PhD studies, which have been generously funded by the Engineering and Physical Sciences Research Council (EPSRC). I wish to thank my supervisor Burt Totaro for his commitment, time and patience, for directing me towards the problems and ideas discussed here, and for his continuing support. Special thanks are moreover due to Baptiste Calmès and Marco Schlichting for highly helpful conversations and explanations of their respective work. Finally, I am grateful to Jens Hornbostel for constructive comments on an earlier version of this paper, and to the anonymous referee for careful proofreading.

References

[Ada95] Frank Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. Reprint of the 1974 original.

[Ara80] Jón Kristinn Arason, Der Wittring projektiver Räume, Math. Ann. 253 (1980), no. 3, 205–212.

[ABS64] Michael Atiyah, Raoul Bott, and Arnold Shapiro, Clifford modules, Topology 3 (1964), no. suppl. 1, 3–38.

[Bal00] Paul Balmer, Triangular Witt groups. I. The 12-term localization exact sequence, K-Theory 19 (2000), no. 4, 311–363.

[Bal01a] , Triangular Witt groups. II. From usual to derived, Math. Z. 236 (2001), no. 2, 351–382.

[Bal01b] , Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture, K-Theory 23 (2001), no. 1, 15–30.

Witt Groups of Complex Cellular Varieties 509

[Bal05] , Witt groups, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 539–576.

[BC09] Paul Balmer and Baptiste Calmès, Geometric description of the connecting homomorphism for Witt groups, Doc. Math. 14 (2009), 525–550.

[BC08] , Witt groups of Grassmann varieties (2008), available at

www.math.uiuc.edu/K-theory/0903/. Preprint.

[BG05] Paul Balmer and Stefan Gille, Koszul complexes and symmetric forms over the punctured affine space, Proc. London Math. Soc. (3) 91 (2005), no. 2, 273–299.

[BGG73] Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, Schubert cells, and the cohomology of the spaces G/P, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian); English transl., Russian Math. Surveys 28 (1973), no. 3, 1–26.

[Bot69] Raoul Bott, Lectures on K(X), Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[BG10] Robert Bruner and John Greenlees, Connective real K-theory of finite groups, Mathematical Surveys and Monographs, vol. 169, American Mathematical Society, Providence, RI, 2010.

[CF11] Baptiste Calmès and Jean Fasel, Trivial Witt groups of flag varieties (2011),

available at arxiv.org/abs/1103.4412. Preprint.

[Dug01] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144– 176.

[DLØ^{+}07] Bjørn Ian Dundas, Marc Levine, Paul Arne Østvær, Oliver Röndigs, and Vladimir Voevodsky, Motivic homotopy theory, Universitext, Springer-Verlag, Berlin, 2007. Lectures from the Summer School held in Nordfjordeid, August 2002.

[EKM08] Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008.

[Fuj67] Michikazu Fujii, K_{0}-groups of projective spaces, Osaka J. Math. 4 (1967), 141– 149.

[FH91] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991.

[Gil03] Stefan Gille, Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), no. 2, 211–233.

[Har92] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course.

[Hat09] Allen Hatcher, Vector bundles and K-theory, 2009. Version 2.1, available at

www.math.cornell.edu/~hatcher/VBKT/VBpage.html.

[Hog69] Stuart Hoggar, On KO theory of Grassmannians, Quart. J. Math. Oxford Ser. (2) 20 (1969), 447–463.

[Hor05] Jens Hornbostel, 𝔸^{1}-representability of Hermitian K-theory and Witt groups, Topology 44 (2005), no. 3, 661–687.

[Hus94] Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994.

[Ish92] Kiminao Ishitoya, Squaring operations in the Hermitian symmetric spaces, J. Math. Kyoto Univ. 32 (1992), no. 1, 235–244.

[Jar87] John Frederick Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87.

[Jou73] Jean-Pierre Jouanolou, Une suite exacte de Mayer-Vietoris en K-théorie algébrique, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 293–316. Lecture Notes in Math., Vol. 341.

510 Marcus Zibrowius

[Kar05] Max Karoubi, Bott periodicity in topological, algebraic and Hermitian K-theory, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 111–137.

[KKO04] Daisuke Kishimoto, Akira Kono, and Akihiro Ohsita, KO-theory of flag manifolds, J. Math. Kyoto Univ. 44 (2004), no. 1, 217–227.

[Koc96] Stanley Kochman, Bordism, stable homotopy and Adams spectral sequences, Fields Institute Monographs, vol. 7, American Mathematical Society, Providence, RI, 1996.

[KH91] Akira Kono and Shin-ichiro Hara, KO-theory of complex Grassmannians, J. Math. Kyoto Univ. 31 (1991), no. 3, 827–833.

[KH92] , KO-theory of Hermitian symmetric spaces, Hokkaido Math. J. 21 (1992), no. 1, 103–116.

[MH73] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73.

[MS74] John Milnor and James Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.

[Mor99] Fabien Morel, Théorie homotopique des schémas, Astérisque 256 (1999).

[MV99] Fabien Morel and Vladimir Voevodsky, 𝔸^{1}-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001).

[Mor04] Fabien Morel, An introduction to 𝔸^{1}-homotopy theory, Contemporary developments in algebraic K-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 357–441.

[Mor06]  ,  Rational  stable  splitting  of  Grassmanians  and
rational  motivic  sphere  spectrum  (2006),  available  at

www.mathematik.uni-muenchen.de/~morel/preprint.html. Draft.

[Nen07] Alexander Nenashev, Gysin maps in Balmer-Witt theory, J. Pure Appl. Algebra 211 (2007), no. 1, 203–221.

[Nen09] , On the Witt groups of projective bundles and split quadrics: geometric reasoning, J. K-Theory 3 (2009), no. 3, 533–546.

[GOV94] Vladimir Gorbatsevich, Arkadiĭ Onishchik, and Èrnest Vinberg, Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994.

[PW10] Ivan Panin and Charles Walter, On the motivic commutative ring spectrum BO

(2010), available at www.math.uiuc.edu/K-theory/0978/. Preprint.

[Rio06] Joël Riou, Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas, Ph. D. Thesis, l’Université Paris 7 – Denis Diderot,

2006, www.math.u-psud.fr/~riou/these/these.pdf.

[Rio07a] , Catégorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math. France 135 (2007), no. 4, 495–547.

[Rio07b] , Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas, C. R. Math. Acad. Sci. Paris 344 (2007), no. 1, 27–32.

[Sch10a] Marco Schlichting, Hermitian K-theory of exact categories, J. K-Theory 5 (2010), no. 1, 105–165.

[Sch10b] , The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math. 179 (2010), no. 2, 349–433.

[Sch] , Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem. In preparation.

[Seg68] Graeme Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151.

Witt Groups of Complex Cellular Varieties 511

[Voe98] Vladimir Voevodsky, 𝔸^{1}-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604.

[Wal03a] Charles Walter, Grothendieck-Witt groups of triangulated categories (2003), avail-

able at www.math.uiuc.edu/K-theory/0643/. Preprint.

[Wal03b] , Grothendieck-Witt groups of projective bundles (2003), available at

www.math.uiuc.edu/K-theory/0644/. Preprint.

[Yag04] Nobuaki Yagita, Coniveau spectral sequence and motivic cohomology of quadrics and classifying spaces (2004), available at www.math.uiuc.edu/K-theory/0709/. Preprint.

[Yag09] , Note on Witt group and KO-theory of complex Grassmannians (2009),

available at www.math.uni-bielefeld.de/LAG/man/325.html. Preprint.

[Zib09] Marcus Zibrowius, Comparing Grothendieck-Witt groups of a complex variety to its real topological K-groups (2009), available at

www.dspace.cam.ac.uk/handle/1810/236624. Smith-Knight/Rayleigh-Knight

Essay.

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Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 512

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