Witt Groups of Complex Varieties
Marcus Zibrowius
Trinity Hall
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
April 2011

Declaration of Originality
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration. No part of this dissertation has
been submitted for any other qualification.
i
ii
Acknowledgements
First and foremost, I thank my supervisor Burt Totaro, for his initiative to
adopt me as a PhD student, and for his continuous guidance and support. No
line of this thesis would have been written without him.
Discussions with other mathematicians have greatly enriched my studies. I
would particularly like to thank Carl McTague and Julian Holstein for their
efforts in establishing a weekly Geometry Tea. Special thanks are moreover
due to Baptiste Calme`s and Marco Schlichting, both of whom I wish I had
talked to more often and much earlier. Finally, I thank Julia Goedecke for
persistent encouragement, as well as last-minute proofreading services.
My PhD studies were generously funded by the Engineering and Physical
Sciences Research Council. I gratefully acknowledge further financial support
from the Cambridge Philosopical Society.
iii
iv
Abstract
The thesisWitt Groups of Complex Varieties studies and compares two related
cohomology theories that arise in the areas of algebraic geometry and topology:
the algebraic theory of Witt groups, and real topological K-theory. Specifically,
we introduce comparison maps from the Grothendieck-Witt and Witt groups
of a smooth complex variety to the KO-groups of the underlying topological
space and analyse their behaviour.
We focus on two particularly favourable situations. Firstly, we explicitly com-
pute the Witt groups of smooth complex curves and surfaces. Using the
theory of Stiefel-Whitney classes, we obtain a satisfactory description of the
comparison maps in these low-dimensional cases. Secondly, we show that the
comparison maps are isomorphisms for smooth cellular varieties. This result
applies in particular to projective homogeneous spaces. By extending known
computations in topology, we obtain an additive description of the Witt groups
of all projective homogeneous varieties that fall within the class of hermitian
symmetric spaces.
v
Contents
Introduction 1
I Two Cohomology Theories 3
I.1 Witt groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1a Symmetric bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1b Symmetric complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1c Witt groups of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1d Hermitian K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
I.2 KO-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2a Real versus symmetric bundles . . . . . . . . . . . . . . . . . . . . . . 15
2b Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2c Generalized cohomology theories . . . . . . . . . . . . . . . . . . . . . 21
2d K- and KO-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2e The Atiyah-Hirzebruch spectral sequence . . . . . . . . . . . . . . . . 27
II Comparison Maps 31
II.1 An elementary approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
II.2 A homotopy-theoretic approach . . . . . . . . . . . . . . . . . . . . . . . . 36
2a A1-homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2b Representing algebraic and hermitian K-theory . . . . . . . . . . . . . 37
2c The comparison maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2d Comparison of the coefficient groups . . . . . . . . . . . . . . . . . . . 44
2e Comparison with Z/2-coefficients . . . . . . . . . . . . . . . . . . . . 45
III Curves and Surfaces 47
III.1 Stiefel-Whitney classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
III.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2a Grothendieck-Witt groups of curves . . . . . . . . . . . . . . . . . . . 54
2b KO-groups of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
III.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3a Witt groups of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3b KO/K-groups of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 65
III.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4a The classical Witt group . . . . . . . . . . . . . . . . . . . . . . . . . 72
4b Shifted Witt groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4c Comparison with Z/2-coefficients . . . . . . . . . . . . . . . . . . . . 77
vi
IV Cellular Varieties 81
IV.1 The comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
IV.2 The Atiyah-Hirzebruch spectral sequence for cellular varieties . . . . . . . 86
IV.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3a Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3b Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3c Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3d Maximal symplectic Grassmannians . . . . . . . . . . . . . . . . . . . 95
3e Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3f Spinor varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3g Exceptional hermitian symmetric spaces . . . . . . . . . . . . . . . . 103
vii
viii
Introduction
In this thesis, we study and compare two related cohomology theories that arise in the
areas of algebraic geometry and topology: the algebraic theory of Witt groups, and real
topological K-theory. Specifically, we introduce comparison maps from the Grothendieck-
Witt and Witt groups of a smooth complex variety to the KO-groups of the underlying
topological space and analyse their behaviour. Satisfactory results are obtained in low
dimensions and for cellular varieties.
The set-up is analogous to the situation one finds in K-theory. Given a smooth complex
variety X, we have on the one hand the algebraic K-group K0(X) and on the other the
complex topological K-group K0(X). One is defined in terms of algebraic vector bundles
over X, the other in terms of complex continuous vector bundles with respect to the
analytic topology on X. Moreover, we have a natural map
k : K0(X)→ K0(X)
that sends an algebraic vector bundle to the underlying continuous bundle. Of course, a
given continuous vector bundle may have several different algebraic structures. For exam-
ple, the bundles O(1)⊕O(−1) and O⊕O over the projective line P1 are both topologically
trivial but they are non-isomorphic as algebraic vector bundles. In this particular case,
they nevertheless represent the same equivalence class in the algebraic K-group K0(X),
and in fact the comparison map k is an isomorphism for X = P1. But more serious prob-
lems already appear on curves of positive genus. Over any such curve, we have an infinite
family of line bundles of degree zero, all of which are topologically trivial, but all of which
define distinct elements in K0(X). It may also happen that a continuous complex vector
bundle has no algebraic structure at all. This occurs, for example, over projective surfaces
of positive geometric genus. In general, the comparison map k is neither surjective nor
injective.
The idea behind Witt groups is to study not simply vector bundles but vector bundles
endowed with the additional structure of a symmetric bilinear form. In this context, the
analogues of the K-groups are given by the Grothendieck-Witt group GW0(X) and the
real topological K-group KO0(X). Again, we have a comparison map
gw0 : GW0(X)→ KO0(X)
The Witt group W0(X) may be defined as the cokernel of a natural map from K0(X) to
GW0(X). Writing (KO0/K)(X) for the corresponding cokernel in topology, we obtain an
1
Witt groups of complex varities
induced comparison map
w0 : W0(X)→ (KO0/K)(X)
These two comparison maps and their generalizations will be our main objects of study.
In Chapter I we collect background material. Precise definitions of the objects dis-
cussed so far can be found there, along with an introduction to Balmer and Walter’s shifted
Grothendieck-Witt and Witt groups GWi(X) and Wi(X). In particular, we explain in
what sense these groups constitute a cohomology theory.
The aim of Chapter II is to generalize the comparison with topology to these shifted
groups, so that we obtain maps
gw i : GWi(X)→ KO2i(X)
w i : Wi(X)→ (KO2i/K)(X)
Two different approaches are discussed. The first is more elementary but has some short-
comings. The second, more elegant approach requires the context of A1-homotopy theory.
A drawback of our construction is that it relies on some recent results that have not
hitherto been properly published. Nonetheless, we believe this is ultimately the more
promising route to follow, not least because it places the comparison maps in their natural
context. It is this approach that we will focus on.
After these preliminaries, we show in Chapter III that the comparison maps behave
well in low dimensions. In particular, the maps w i on the Witt groups are isomorphisms
for all smooth complex curves. For a surface X, they are isomorphisms if and only if every
continuous complex line bundle over X is algebraic. For example, this is the case for all
projective surfaces of geometric genus zero. As a first step towards the proof, we explicitly
compute the Witt groups of arbitrary smooth complex curves and surfaces. The behaviour
of w0 is then analysed using the theory of Stiefel-Whitney classes and the usual comparison
theorem for e´tale cohomology with finite coefficients. For the comparison maps on shifted
groups, more work is required, and the advantages of the homotopy-theoretic approach
to their construction become essential. To round off this chapter, we briefly discuss how
the results are related to more general comparison theorems involving Grothendieck-Witt
groups with torsion coefficients.
In Chapter IV, we show that the comparison maps gw i and w i are isomorphisms for
smooth cellular varieties. This result is illustrated with a series of examples to which it
applies. Namely, by extending known computations in topology, we obtain a complete
additive description of the Witt groups of all projective homogeneous varieties that fall
into the class of hermitian symmetric spaces, such as complex Grassmannians, projective
quadrics and spinor varieties.
2
Chapter I
Two Cohomology Theories
In this chapter, we introduce the two main cohomology theories that we will be dealing
with: the theory of Witt groups and KO-theory.
Witt groups originated in the study of quadratic forms over fields, but the concept
was later extended to rings, varieties and schemes by Knebusch [Kne77]. They may be
thought of as variants of K-groups: in the same sense that the K-group of a variety X
classifies vector bundles over it, there is a Grothendieck-Witt group GW0(X) classifying
vector bundles equipped with symmetric forms, of which the Witt group is a quotient.
Thus, we begin our discussion by introducing such symmetric bundles and spelling out
the necessary definitions. The construction of cohomology theories based on Witt groups is
fairly recent. Roughly ten years ago, Balmer gave a purely algebraic construction of a Witt
cohomology theory by introducing Witt groups of triangulated categories [Bal00,Bal01a].
Rather than giving a detailed account of the general theory, we concentrate here on those
aspects relevant in the context of regular schemes. More comprehensive introductions may
be found in [Bal01b] and [Bal05]. We also briefly touch on a more general cohomology
theory known as hermitian K-theory, which parallels algebraic K-theory more closely. This
theory is currently being developed mainly by Schlichting [HS04,Sch10a,Sch10b,Sch].
The corresponding cohomology theory in topology, known as real topological K-theory
or simply as KO-theory, has been well-known and studied since the early work of Atiyah.
Its construction, which parallels that of complex topological K-theory perfectly, is recalled
in the second half of this chapter. We close with a description of the Atiyah-Hirzebruch
spectral sequence, a standard tool in topology that allows us to relate the KO-groups of a
space to its singular cohomology.
1 Witt groups
The algebraic K-group K0(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
3
Chapter I. Two Cohomology Theories
over X, we have [F] = [E] + [G] in K0(X). In particular, as far as K0(X) is concerned, we
may pretend that all exact sequences of vector bundles over X split. Witt groups can be
defined similarly, using the notion of symmetric bundles.
1a Symmetric bundles
Let X be a scheme over Z[12 ]. A symmetric bundle (E, ε) over X is a vector bundle
1 E over
X equipped with a non-degenerate symmetric bilinear form ε : E⊗ E→ O. Alternatively,
we may view ε as an isomorphism from E to its dual bundle E∨. In this case, the symmetry
of ε is encoded by the fact that it agrees with its dual ε∨ under the canonical identification
ω of E with (E∨)∨. That is, the following triangle commutes:
E ε
'.VVVV
VVVV
∼=ω

E∨
(E∨)∨ ε∨
18iiiiii
(1)
Two symmetric 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).
1.1 Example (Symmetric line bundles). Let Pic(X)[2] be the subgroup of line bun-
dles of order ≤ 2 in the Picard group Pic(X). Any line bundle L ∈ Pic(X)[2] defines a
symmetric bundle over X. When X is a projective variety over an algebraically closed
field, all symmetric line bundles arise in this way. In general, the set of isometry classes
of symmetric line bundles over X is described by H1et(X;Z/2), the first e´tale cohomology
group of X with coefficients in Z/2 = O(1). The Kummer sequence exhibits this group as
an extension of Pic(X)[2]:
0→ O
∗(X)
O∗(X)2
→ H1et(X;Z/2)→ Pic(X)[2]→ 0
The additional contribution comes from symmetric line bundles of the form (O, ϕ), where ϕ
is some invertible regular function which has no globally defined square root. For example,
the trivial line bundle over the punctured disk A1−0 carries a non-trivial symmetric form
given by multiplication with the standard coordinate function.
1.2 Example (Hyperbolic bundles). Any vector bundle E gives rise to a symmetric
bundle H(E) := (E⊕ E∨, ( 0 1ω 0 )) over X, the hyperbolic bundle associated with E.
Hyperbolic bundles are the simplest members of the wider class of metabolic bundles. A
metabolic bundle is a symmetric bundle (M, µ) which contains a Lagrangian, a subbundle
1By convention, we identify a vector bundle with its sheaf of sections. Thus, the common notation O
for the sheaf of regular functions on X will be used to denote the trivial line bundle over X.
4
1 Witt groups
j : N ↪→M of half the rank ofM on which the symmetric form µ vanishes. In other words,
(M, µ) is metabolic if it fits into a short exact sequence of the form
0→ N j−−→M j
∨µ−−→ N∨ → 0 (2)
The sequence splits if and only if (M, µ) is isometric to H(N) [Bal05, Example 1.1.21].
This motivates the definition of the Grothendieck-Witt group.
1.3 Definition. [Kne77, § 4] The Grothendieck-Witt group GW0(X) of a scheme X over
Z[12 ] is the free abelian group on isometry classes of symmetric bundles over X modulo
the following two relations:
• [(E, ε) ⊥ (G, γ)] = [(E, ε)] + [(G, γ)]
• [(M,µ)] = [H(N)] for any metabolic bundle (M,µ) with Lagrangian N
The Witt group W0(X) is defined similarly, except that the second relation reads
[(M,µ)] = 0. Equivalently, we may define W0(X) by the exact sequence
K0(X)
H−→ GW0(X) −→W0(X)→ 0
In addition to the hyperbolic map H appearing here, we have a forgetful map F in the
opposite direction, sending symmetric bundles to their underlying vector bundles:
GW0(X) F−→ K0(X)
The most basic invariant of a vector bundle is its rank. For a connected scheme X, it
induces the following well-defined homomorphisms on the above groups:
K0(X)
rk−→ Z
GW0(X) rk−→ Z
W0(X) rk−→ Z/2
Of course, on the Witt group the rank is only well-defined modulo two, since arbitrary
metabolic bundles are equivalent to zero.
1.4 Example. If k is a field (of characteristic not 2), we simply write K0(k), GW0(k)
and W0(k) for the corresponding groups of Spec(k). Since short exact sequences of vector
spaces always split, these groups may be defined more directly: if Vect(k) and Bil(k) denote
the monoids of vector spaces and of symmetric forms over k, then K0(k) and GW0(k) are
simply the Grothendieck groups of Vect(k) and Bil(k), respectively, and W0(k) may be
identified with the quotient Bil(k)/(N ·H), where H is the hyperbolic form H = ( 0 11 0 ).
Since vector spaces are uniquely determined by their rank, the rank homomorphism
on the K-group yields an isomorphism K0(k)
∼=→ Z. In contrast, the rank homomorphisms
5
Chapter I. Two Cohomology Theories
on the Grothendieck-Witt and Witt groups are not isomorphisms in general unless k is
algebraically closed. For example, the Witt group of R can be identified with Z via the
map that sends a real symmetric form to its signature.
Variants. Returning to the world of schemes, we may more generally consider vector
bundles equipped with non-degenerate symmetric bilinear forms with values in any fixed
line bundle L over X. Such vector bundles may be viewed as symmetric bundles with
respect to the twisted duality
E∨L := Hom(E,L)
Under this interpretation, all notions introduced above immediately generalize, leading to
the definition of twisted groups GW0(X;L) and W0(X;L). If E is symmetric with respect
to the twisted duality ∨L⊗L, then E⊗ L∨ is symmetric with respect to the usual duality.
Thus, these groups depend only on the class of L in Pic(X)/2.
We could also work with −ω in place of the usual double-dual identification, leading
to the notion of anti -symmetric bundles. The corresponding Grothendieck-Witt and Witt
groups are denoted by GW2(X;L) and W2(X;L). The choice of notation will become
clear in the next section, where shifted groups GWi(X;L) and Wi(X;L) are introduced
for arbitrary integers i.
Witt groups of exact categories
The situation may be formalized by introducing the notion of an exact category with
duality (A,∨, ω). An exact category is essentially an additive category A equipped with
an abstract notion of short exact sequences, and a duality is determined by an endofunctor
∨ : Aop → A together with a natural isomorphism
ω : idA
∼=−−→ ∨ ◦ ∨
which one refers to as double-dual identification. These structures are required to satisfy
certain compatibility conditions. Precise definitions may be found in [Sch10a, § 2] or
[Bal05, Definitions 1.1.1 and 1.1.13]. The notions of symmetric and metabolic objects
over an arbitrary exact category with duality (A,∨, ω) are completely analogous to the
notions for bundles given above, and they can be used to define the Grothendieck-Witt
and Witt groups GW0(A,∨, ω) and W0(A,∨, ω).
From this point of view, GW0(X) and W0(X) are simply the Grothendieck-Witt and
Witt groups of the exact category Vect(X) of vector bundles over X equipped with its
usual duality and double-dual identification. For the twisted groups, we use the same
category equipped with the twisted duality ∨L and the canonical identification ωL of E
6
1 Witt groups
with (E∨L)∨L , while for the groups of anti-symmetric bundles we use ∨L and −ωL:
GW0(X;L) := GW0(Vect(X),∨L, ωL)
GW2(X;L) := GW0(Vect(X),∨L,−ωL)
1b Symmetric complexes
The notions of duality and symmetry may be generalized to complexes. If we fix an exact
category with duality (A,∨, ω), then the usual dual of a complex
E• : · · · → E2 d2−→ E1 d1−→ E0 d0−→ E−1 d−1−→ E−2 → · · ·
over A is defined term-by-term, i. e. E∨• is given by
E∨• : · · · → E∨−2
d∨−1−→ E∨−1
d∨0−→ E∨0
d∨1−→ E∨1
d∨2−→ E∨2 → · · ·
with E∨0 placed in degree zero. Likewise, the dual of a morphism ϕ• : E• → F• is given by
(ϕ∨)l := ϕ∨−l.
Alternative dualities may be defined by composing with the shift functor, as follows.
By convention, the shifted complex E•[1] is obtained from E• by moving all terms by one
position to the left and changing the signs of the differentials, while a morphism ϕ is
shifted without any sign changes. Thus, (ϕ[1])l = ϕl−1, and E•[1] takes the form
E•[1] : · · · → E1 −d1−→ E0 −d0−→E−1 −d−1−→ E−2 −d−2−→ E−3 → . . .
with E−1 in degree zero [Wei94, 1.2.8]. Shifts [i] for arbitrary integers i are obtained by
iterating this construction or its inverse. Thus, for any integer i, we can define a shifted
duality ∨i on complexes by
E∨i• := (E
∨
• )[i]
The double-dual identification ω on A induces a canonical identification of E• with (E∨i• )∨i ,
which we denote by ωi.
1.5 Definition. An i-symmetric complex (E•, ε) is a bounded complex E• together with
a quasi-isomorphism
ε : E•
'−−→ E∨i•
which is symmetric with respect to ∨i and the double-dual identification (−1)
i(i+1)
2 ·ωi. In
other words, ε satisfies the condition
ε = (−1) i(i+1)2 · ε∨i ◦ ωi
Because the double-dual identification is modified by a sign, this definition generalizes
7
Chapter I. Two Cohomology Theories
at the same time the notions of symmetric and anti-symmetric objects: while symmetric
objects over A may be viewed as 0-symmetric complexes concentrated in degree zero, anti-
symmetric objects may be viewed as 2-symmetric complexes concentrated in degree one.
Given any i-symmetric complex (E•, ε), its two-fold shift (E•[2], ε[2]) is (i+4)-symmetric.
Thus, the essential aspects of i-symmetry depend only on the value of i modulo 4.
Simple examples of i-symmetric complexes for arbitrary i are given by hyperbolic
complexes, i. e. complexes of the form H i(E) := (E ⊕ E∨i ,
(
0 1
(−1)
i(i+1)
2 ωi 0
)
). A less trivial
example is the following.
1.6 Example (A 1-symmetric complex over P1). Consider the complex of vector
bundles O(−1) ·x−→ O over the projective line P1 with coordinates [x : y]. Place O in degree
zero. Multiplication by y induces a symmetric quasi-isomorphism with the dual complex
shifted one to the left, so that we obtain a 1-symmetric complex
Ψ0 :=

O(−1) ·x ,2
·y

O
·(−y)

O
·(−x)
,2 O(1)
 (3)
Analogues of this 1-symmetric complex over projective curves of higher genus are described
in Remark III.2.2.
Witt groups of triangulated categories
The category Chb(A) of bounded complexes over A equipped with the exact structure
inherited from A and any of the dualities defined above is again an exact category with
duality. However, i-symmetric complexes are not symmetric objects over (Chb(A),∨i,±ωi)
in the above sense, since the symmetries are only required to be quasi-isomorphisms. On
the other hand, the derived category Db(A) obtained from Chb(A) by formally inverting
all quasi-isomorphisms is no longer exact but rather triangulated.
One is thus led to develop the above theory in the context of triangulated categories.
The definition of the K-group of a triangulated category is straightfoward: one simply
replaces the short exact sequence in the definition of K0 by an exact triangle. For any
exact category A, we then have K0(Db(A)) ∼= K0(A). So this is a perfect generalization.
Witt groups of triangulated categories are developed by Balmer in [Bal00] and [Bal01a].
Naturally enough, the idea is to replace the short exact sequence (2) defining metabolic
objects by an appropriate exact triangle. Once the notions of a triangulated category with
duality (D,∨, ω) and its Witt group W(D,∨, ω) are settled, the Witt groups of i-symmetric
complexes over an exact category with duality (A,∨, ω) may be defined as
Wi(A,∨, ω) := W(Db(A),∨i, (−1)
i(i+1)
2 ωi)
8
1 Witt groups
These groups are known as the shifted Witt groups of (A,∨, ω). They are 4-periodic in i,
in the sense that we have canonical isomorphisms Wi(A,∨, ω) = Wi+4(A,∨, ω). For i = 0
and i = 2, we recover the Witt groups of symmetric and anti-symmetric objects over A
defined above, at least when 2 is invertible in A (i. e. when all homomorphism groups of
A are uniquely 2-divisible) [Bal01a, Theorem 4.3; BW02, Theorem 1.4].
As Walter notes in [Wal03a], Balmer’s approach already works on the level of Grothen-
dieck-Witt groups, so that we also have shifted groups GWi(A,∨, ω). Just like the usual
Grothendieck-Witt groups, these are equipped with hyperbolic and forgetful maps. They
appear in the following exact sequences [Wal03a, Theorem 2.6]:
GWi−1(A,∨, ω) F−→ K0(A) H
i−→ GWi(A,∨, ω) −→W0(A,∨, ω)→ 0
Not only do these generalize the exact sequence of Definition 1.3 but they also extend it
by one term to the left.
1c Witt groups of schemes
The shifted Witt groups of a schemeX are, of course, defined in terms of the exact category
(Vect(X),∨, ω), or more generally in terms of (Vect(X),∨L, ωL) for any line bundle L over
X. The essence of the previous sections may be summarized as follows.
• For any scheme X over Z[12 ], any line bundle L over X and any integer i, we have
groups
GWi(X;L) := GWi(Vect(X),∨L, ωL)
Wi(X;L) := Wi(Vect(X),∨L, ωL)
which generalize the Witt groups of symmetric and anti-symmetric bundles discussed
earlier [Bal01a, Theorem 4.7]. They are referred to as shifted (Grothendieck-)Witt
groups of X with coefficients in L, or as such groups twisted by L. When L is trivial,
it is frequently dropped from the notation.
• The groups are four-periodic in i and two-periodic in L. That is, for any X, any i
and arbitrary line bundles L and M over X, we have canonical isomorphisms
GWi(X;L) ∼= GWi+4(X;L)
GWi(X;L) ∼= GWi(X;L⊗M⊗2)
and similarly for the Witt groups.
• We have hyperbolic maps H iL : K0(X) → GWi(X;L) and forgetful maps F in the
9
Chapter I. Two Cohomology Theories
opposite direction. They fit into exact sequences of the form
GWi−1(X;L) F−→ K0(X)
HiL−→ GWi(X;L) −→Wi(X;L)→ 0 (4)
These sequences are known as Karoubi sequences.
• It follows easily from the definitions that all these constructions are natural. That
is, for any morphism of schemes f : X ′ → X and any line bundle L over X, we have
induced maps
K0(X)
f∗−→ K0(X ′)
GWi(X;L)
f∗−→ GWi(X ′; f∗L)
Wi(X;L)
f∗−→Wi(X ′; f∗L)
which allow us to view K0(−), GWi(−;−) and Wi(−;−) as contravariant functors
from schemes with line bundles to abelian groups. The maps f∗ commute with the
maps in the Karoubi sequences.
1.7 Example (Witt groups of a geometric point). Let p = Spec(k), where k is an
algebraically closed field of characteristic not 2. Its Grothendieck-Witt and Witt groups
are as follows:
GW0(p) = Z W0(p) = Z/2
GW1(p) = 0 W1(p) = 0
GW2(p) = Z W2(p) = 0
GW3(p) = Z/2 W3(p) = 0
The classical Grothendieck-Witt group GW0(p) is generated by the symmetric bundle
(O, id), while GW2(p) is generated by the anti-symmetric bundle (O ⊕ O, ( 0 1−1 0 )). The
group GW3(p) is generated by the hyperbolic bundleH3(O). The hyperbolic bundleH1(O)
is trivial in GW1(p) because it shares its Lagrangian O with the following 1-symmetric
metabolic complex, which is exact:
O
id ,2
id
O
− id

O
− id
,2 O
It follows from the naturality property that the groups of an arbitrary schemeX contain
the groups of a point as direct summands. Namely, the inclusion of a point j : p ↪→ X and
the projection of X onto p induce decompositions
K0(X) = Z ⊕ K˜0(X)
GWi(X) = GWi(p)⊕ G˜Wi(X)
Wi(X) = Wi(p) ⊕ W˜i(X)
10
1 Witt groups
in which K˜0(X), G˜Wi(X) and W˜i(X) denote the kernels of j∗ on the respective groups.
When X is connected, these kernels are independent of the choice of p, and we refer to
them as the reduced groups of X. Over an algebraically closed field, the pullback maps
j∗ on K0(X), GW0(X) and W0(X) can be identified with the rank homomorphisms.
1.8 Example (Witt groups of P1 [Ara80,Wal03b]). For the projective line P1 over k,
where k is as in the previous example, one finds that
GW0(P1) = [Z]⊕ Z/2 W0(P1) = [Z/2]
GW1(P1) = Z W1(P1) = Z/2
GW2(P1) = [Z] W2(P1) = 0
GW3(P1) = [Z/2]⊕ Z W3(P1) = 0
Here, the summands in square brackets are those generated by constant bundles, i. e. those
that disappear when passing to reduced groups. The groups GW1(P1) and W1(P1) are
generated by the 1-symmetric complex Ψ0 given in Example 1.6. Grothendieck-Witt and
Witt groups of projective spaces of arbitrary dimensions are discussed in IV.3b.
In general, when working over an algebraically closed ground field, the Witt groups are
always 2-torsion: 2Ψ = 0 for any Ψ ∈ Wi(X;L). This is a consequence of the following
well-known lemma.
1.9 Lemma. Let X be a scheme over an algebraically closed field k of characteristic not
2. Then, for any Ψ ∈ GWi(X;L), we have H iL(F (Ψ)) ∼= 2Ψ.
Proof. We may assume that Ψ is the class of some i-symmetric complex (E•, ε). Using
the assumption that char(k) 6= 2, we define an isometry between the hyperbolic complex
H iL(E•) and the direct sum (E•, ε) ⊕ (E•,−ε) by 12
(
1 ε−1
1 −ε−1
)
. Since k contains a square
root of −1, the symmetric complex (E•,−ε) is isometric to (E•, ε).
Witt groups of regular schemes
One of the key features of Balmer’s triangulated approach is that it elevates the theory of
Witt groups into the realm of cohomology theories. This assertion is filled with meaning
by the following paragraphs. By convention, a regular scheme will be a regular, noetherian,
separated scheme over Z[12 ]. In order to avoid the regularity assumption, we would have to
work with Witt groups defined in terms of coherent sheaves rather than vector bundles.
Localization sequences. Let X be a regular scheme, in the above sense. Then, for any
open subscheme U ⊂ X, we have a long exact sequence relating the Witt groups of X to
those of U . These sequences are known as localization sequences. If we denote the closed
complement of U by Z and the open inclusion by j : U ↪→ X, the assoicated sequence can
11
Chapter I. Two Cohomology Theories
be written as follows:
· · · →WiZ(X;L)→Wi(X;L) j
∗
→Wi(U ;L|U )→Wi+1Z (X;L)→ · · · (5)
The groups WiZ(X;L) appearing here are the Witt groups of X with support on Z.
The terminology comes from the fact that they are defined in terms of complexes whose
cohomology is supported on Z.
The groups with support on Z = X agree with the usual Witt groups of X. When Z is
a smooth closed subvariety of a smooth quasi-projective variety X, we have a de´vissage or
Thom isomorphism relating the Witt groups of X with support on Z to the Witt groups
of Z. The precise form of these isomorphisms depends not only on the codimension c, but
also on the normal bundle N of Z in X [Nen07, § 4]:
Wi−c(Z;L|Z ⊗ detN)
∼=−→WiZ(X;L) (6)
Thus, in the above situation, the localization sequence may be rewritten purely in terms
of the Witt groups of X, U and Z.
By periodicity, the localization sequences may be arranged as exact polygons with
twelve vertices. We also have localization sequences involving Grothendieck-Witt groups,
of the following form [Wal03a, Theorem 2.4]:
GWiZ(X)→ GWi(X)→ GWi(U)
→Wi+1Z (X)→Wi+1(X)→Wi+1(U)→Wi+2Z (X)→ · · ·
(7)
These sequences agree with the localization sequences for Witt groups from the fourth
term onwards, and may of course again be defined for arbitrary twists L. However, if one
wishes to continue the sequences to the left, one has to revert to the methods of higher
algebraic K-theory. This is discussed very briefly in Section 1d.
Excision. Let f : X ′ → X be a morphism of regular schemes. Given a closed subscheme
Z ⊂ X, write Z ′ for its preimage X ′×X Z under f . If f is flat and maps Z ′ isomorphically
to Z, then f induces isomorphisms of Witt groups with support [Bal01b, Corollary 2.3]:
f∗ : WiZ(X)
∼=−→WiZ′(X ′) (8)
In particular, if Z is contained in an open subscheme U of X, then WiZ(X) ∼= WiZ(U).
Mayer-Vietoris sequences. The existence of localization sequences and the excision
property imply the exactness of Mayer-Vietoris sequences. That is, given a covering of a
regular scheme X by two open subschemes U and V , we have a long exact sequence of the
following form [Bal01b, Theorem 2.5]:
· · · →Wi(X)→Wi(U)⊕Wi(V )→Wi(U ∩ V )→Wi+1(X)→ · · ·
12
1 Witt groups
Homotopy invariance. If E is the total space of a vector bundle over a regular scheme
X, then the projection pi : E → X induces isomorphisms of Witt groups:
pi∗ : Wi(X)
∼=−→Wi(E)
More generally, this holds for any flat affine morphism of regular schemes pi : E → X
whose fibres are affine spaces [Gil03, Corollary 4.2].
The properties of Witt groups discussed also hold on the level of Grothendieck-Witt
groups. This may be deduced in each case from the corresponding properties of K-
groups and the Karoubi sequences (4), using Karoubi induction (c. f. the proof of Propo-
sition II.1.3 in the next chapter).
Multiplication. Finally, we mention the multiplicative structure on shifted Witt groups.
In the same way that the tensor product of vector bundles or complexes over X induces
a ring structure on K0(X), we have a ring structure on GW0(X) and W0(X) induced by
the tensor product of symmetric bundles. More generally, in [GN03] Gille and Nenashev
develop pairings between the shifted Witt groups of X, of the following form:
? : WiZ(X;L)⊗WjZ′(X;M)→Wi+jZ∩Z′(X;L⊗M)
Again, this product may be lifted to Grothendieck-Witt groups [Wal03a, end of § 2]. It is
graded-commutative in the sense that
Ψi ?Ψj = (OX ,− id)ij ?Ψj ?Ψi
for Ψi ∈ WiZ(X;L) and Ψj ∈ WjZ′(X;M). Over an algebraically closed field, (OX ,− id)
is isometric to the trivial symmetric bundle (OX , id), acting as the unit, so the product
becomes honestly commutative.
As usual, this “internal” product gives rise to an “external” product between the (Gro-
thendieck-)Witt groups of two different schemes X and Y : if L and M are line bundles
over X and Y , respectively, we have a cross product
× : WiZ(X;L)⊗WjW (Y ;M)→Wi+jZ×W (X × Y ;pi∗XL⊗ pi∗YM)
Here, piX and piY denote the respective projections from X × Y to X and Y . For
Ψ ∈WiZ(X;L) and Φ ∈WjW (Y ;M), the product is defined as Ψ×Φ := pi∗X(Ψ) ? pi∗Y (Φ).
13
Chapter I. Two Cohomology Theories
1d Hermitian K-theory
The algebraic K-group of a scheme fits into a family of higher algebraic K-groups Kn(X).
In general, however, there seems to be no purely algebraic description of the higher groups.
Rather, following Quillen [Qui73], one defines the higher K-groups of a scheme X as
the homotopy groups of a topological space K(X) associated with X. By working with
an appropriate spectrum in place of K(X), one may further define groups Kn(X) in all
degrees n ∈ Z. For a regular scheme, however, the groups in negative degrees vanish
[TT90, Proposition 6.8].
An analogous construction for Grothendieck-Witt groups, usually referred to as (higher)
hermitian K-theory, is developed in [Sch10b, Section 10]. Given a scheme X and a line
bundle L over it, Schlichting constructs a family of spectra GWi(X;L) from which her-
mitian K-groups can be defined as
GWin(X;L) := pin(GWi(X;L))
These groups satisfy many properties analogous to those discussed in the case of Witt
groups above. The localization sequences now have the following form [Sch10b, Theo-
rem 14]:
· · · → GWin,Z(X)→ GWin(X)→ GWin(U)→ GWin−1,Z(X)→ · · · (9)
In degree n = 0, one recovers Walter’s Grothendieck-Witt groups, while Balmer’s Witt
groups appear as hermitian K-groups in negative degrees. More precisely, for any regular
scheme X, we have natural identifications
GWi0(X;L) ∼= GWi(X;L)
GWin(X;L) ∼= Wi−n(X;L) for n < 0
(10)
These identifications are due to appear in full generality in [Sch]. For affine varieties,
the identifications of Witt groups may be found in [Hor05]: see Proposition A.4 and
Corollary A.5. For a general regular 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 By homotopy
invariance, the Witt groups of T may be identified with those of X, and the same is true
for the hermitian K-groups: in this case, homotopy invariance follows from homotopy
invariance in the affine case, as shown in [Hor05, Corollary 1.12], and the Mayer-Vietoris
sequences established in [Sch10b, Theorem 1].
1This step is known as Jouanolou’s trick.
14
2 KO-theory
2 KO-theory
We now turn to the corresponding cohomology theories in topology. These are known as
complex and real topological K-theory, or simply as K- and KO-theory. To ensure that
the definitions given here are consistent with the literature, we restrict our attention to
finite-dimensional CW complexes.1 Since we are ultimately only interested in topological
spaces that arise as complex varieties, this is not a problem.
So let X be such a CW complex. If we imitate the definitions of K0(X) and GW0(X)
given in Section 1a, using continuous complex vector bundles over X in place of algebraic
vector bundles, we obtain the complex and real topological K-groups K0(X) and KO0(X).
However, since short exact sequences of vector bundles over CW complexes always split,
the definitions may be simplified:
2.1 Definition. For a finite-dimensional CW complex X, the complex topological K-
group K0(X) is the free abelian group on isomorphism classes of continuous complex
vector bundles over X modulo the relation [E ⊕ G] = [E] + [G]. Similarly, the KO-group
KO0(X) is the free abelian group on isometry classes of continuous complex vector bun-
dles equipped with non-degenerate symmetric bilinear forms over X, modulo the relation
[(E, ε) ⊥ (G, γ)] = [(E, ε)] + [(G, γ)].
In the following, continuous vector bundles will be referred to simply as vector bundles,
and a vector bundle equipped with a non-degenerate symmetric bilinear form will be
referred to as a symmetric bundle.
As in the algebraic case, we may view K0 and KO0 as contravariant functors, defined
in this case on the category of finite-dimensional CW complexes and continuous maps
between them. Also, we again have natural transformations from K0 to KO0 and vice
versa, corresponding to the hyperbolic and the forgetful map. These are traditionally
referred to as “realification” and “complexification”, and written as
r : K0(X)→ KO0(X)
c : KO0(X)→ K0(X)
The terminology is explained in the next section.
2a Real versus symmetric bundles
There is a more common description of KO0(X) as the K-group of continuous real vector
bundles over X. 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 GLn(R) and
On(C). Alternatively, this equivalence may be seen concretely along the following lines.
1The key property we need is that any vector bundle over a finite-dimensional CW complex has a
stable inverse. See the proof of Theorem 2.7.
15
Chapter I. Two Cohomology Theories
We say that a complex bilinear form ε on a (real or complex) vector bundle F is real if
ε : F ⊗ F → C factors through R.
2.2 Lemma. Any complex symmetric bundle (E, ε) has a unique real subbundle <(E, ε) ⊂
E such that <(E, ε)⊗RC = E and such that the restriction of ε to <(E, ε) is real and positive
definite. Concretely, a fibre of <(E, ε) is given by the real span of any orthonormal basis
of the corresponding fibre of E.
To prove this lemma, we need the following fact:
2.3 Lemma. Any complex symmetric bundle (F, ϕ) over a topological space X is locally
trivial. That is, any point of X has an open neighbourhood over which (F, ϕ) is isometric
to the trivial symmetric bundle (CrkF, id).
Proof. Let p be an arbitrary point of X. To prove the claim, we may assume without loss
of generality that F is trivial as a complex vector bundle, and that the fibre of (F, ϕ) at p
is the trivial symmetric space (
Cr,
(
1 0. . .
0 1
))
Let e1, . . . , er be the sections of the trivial bundle Cr that provide the standard basis
at each fibre. Since ϕ is orthonormal at p, we may find an open neighbourhood of p on
which the deviation of ϕ from orthonormality is arbitrarily small. That is, for any positive
 ∈ R, we may find an open neighbourhood of p over which
‖ϕ(ei, ej)− δij‖ <  (∗)
for all i and j, where ‖·‖ denotes the standard hermitian metric and δij is the Kronecker
delta. The lemma follows from the existence of a Gram-Schmidt process for such “almost
orthonormal” symmetric bilinear forms. More precisely, we have the following statement:
Let ϕ be a non-degenerate symmetric bilinear form on an r-dimensional com-
plex vector space with basis e1, . . . , er. Suppose that (∗) holds for some suffi-
ciently small . Then there exists another basis e˜1, . . . , e˜r which is orthonormal
with respect to ϕ, such that each e˜i may be expressed by a formula depending
continuously on the coefficients ϕ(ei, ej) of ϕ.
For lack of reference, we include the following proof of this statement. We take “suffi-
ciently small” to mean that  ≤ 1
2r+1r!
. Let
√· denote the continuous endofunction of
{z ∈ C | <(z) > 0} that assigns to a complex number with positive real part that square
root whose real part is also positive. The orthonormal basis e˜1, . . . , e˜r is constructed
inductively, starting with
e˜1 :=
e1√
ϕ(e1, e1)
By definition, ϕ(e˜1, e˜1) = 1. For l > 1, we have ‖ϕ(e˜1, el)‖ < 2.
16
2 KO-theory
Now, suppose we have already constructed e˜1, . . . , e˜i satisfying the following two con-
ditions:
ϕ(e˜j , e˜k) = δjk for all j, k ≤ i (C1)
‖ϕ(e˜j , el)‖ < 2ii! ·  for all j ≤ i and l > i (C2)
Set e′i+1 := ei+1 −
∑i
k=1 ϕ(ei+1, e˜k)e˜k. Then ϕ(e
′
i+1, e˜k) = 0 for all k ≤ i, and
ϕ(e′i+1, e
′
i+1) = ϕ(ei+1, ei+1)−
i∑
k=1
ϕ(ei+1, e˜k)2
A crude estimation shows that
‖ϕ(e′i+1, e′i+1)− 1‖ < ‖ϕ(ei+1, ei+1)− 1‖+
i∑
k=1
‖ϕ(ei+1, e˜k)‖2 < + i(2ii!)2 < 1/2
In particular, ϕ(e′i+1, e
′
i+1) has positive real part, so that we may normalize e
′
i+1 by setting
e˜i+1 :=
e′i+1√
ϕ(e′i+1, e
′
i+1)
Then ϕ(e˜j , e˜k) = δij for all k, j ≤ i+ 1, as desired. In other words, (C1) holds with i
replaced by i+ 1. The same is true of condition (C2). Indeed, if we fix l > i+ 1, then for
any j ≤ i we have ‖ϕ(e˜j , el)‖ < 2ii! < 2i+1(i+ 1)!, and for j = i+ 1 we find that
‖ϕ(e˜i+1, el)‖ ≤ 1
‖
√
ϕ(e′i+1, e
′
i+1)‖
·
(
‖ϕ(ei+1, el)‖+
i∑
k=1
‖ϕ(ei+1, e˜k)‖ · ‖ϕ(e˜k, el)‖
)
< 2 · (+ i(2ii!)2) < 2 · (2ii!+ i(2ii!)) = 2i+1(i+ 1)!
Proof of Lemma 2.2. In the case of a vector bundle over a point we may assume as above
that (E, ε) is isometric to (Cr, id). Clearly, the subspace Rr ⊂ Cr has the required proper-
ties. To see uniqueness, suppose W is another r-dimensional real subspace of Cr such that
ε|W is real and positive definite. Pick an orthonormal basis of W with respect to ε|W ,
e1 + if1, e2 + if2, . . . , er + ifr
where ej and fj are vectors in Rr and i is the imaginary unit in C. Then we see that
ε(ej , fk) = 0 for all j, k (D1)
ε(ej , ek) = ε(fj , fk) for all j 6= k (D2)
ε(ek, ek) = ε(fk, fk) + 1 for all k (D3)
17
Chapter I. Two Cohomology Theories
It follows from (D2) and (D3) that the vectors e1, . . . , er are linearly independent. Indeed,
if
∑
j λjej = 0 for certain λj ∈ R, then
0 = ε (
∑
λjej ,
∑
λjej) = ε(
∑
λjfj ,
∑
λjfj) +
∑
λ2j
Since ε is positive definite on Rr, all λj must be zero. On the other hand, (D1) implies that
the real spans of e1, . . . , er and f1, . . . , fr in Rr are orthogonal. Thus, f1 = · · · = fr = 0,
and W = Rr.
If (E, ε) is an arbitrary complex symmetric bundle over a space X, then by Lemma 2.3
any point of X has some neighbourhood over which (E, ε) can be trivialized in the form
above. We know how to define <(E, ε) over each such neighbourhood, and by uniqueness
these local bundles can be glued together.
2.4 Corollary. For any CW complex X, the mapping (E, ε) 7→ <(E, ε) defines an iso-
morphism between the monoid of isometry classes of complex symmetric bundles over X
and the monoid of isomorphism classes of real vector bundles over X.
Proof. Given a real vector bundle E overX, we choose an inner product σ on E and consider
the complex symmetric bundle (E ⊗R C, σC), where σC denotes the C-linear extension of
σ to E⊗R C. Since σ is defined uniquely up to isometry, so is (E⊗R C, σC), and we obtain
a well-defined inverse to the map above. An alternative proof that avoids the uniqueness
part of the preceding lemma is given in [MH73, Chapter V, § 2].
It follows that the definitions of KO0(X) in terms of complex symmetric bundles and
in terms of real bundles agree. Moreover, we see from the concrete description of the
correspondence between these types of bundles given in the corollary and its proof that
the hyperbolic map K0(X)→ KO0(X) corresponds to the map sending a complex vector
bundle to its underlying real bundle, whereas the forgetful map from KO0(X) to K0(X)
corresponds to the map that sends a real vector bundle F to its complexification F ⊗R C.
This explains the terminology and the notation used for these maps in topology.
Real versus non-degenerate Grassmannians
The correspondence between complex symmetric and real bundles is reflected by the fact
that their projectivizations, or more generally their Grassmannian bundles, are homotopy
equivalent. This will be used in the next section to describe a representing space for
KO-theory, and it will also be relevant in the discussion of Stiefel-Whitney classes in
Section III.1.
For any real or complex vector bundle E over a topological spaceX, we have Grassman-
nian bundles RGr(k,E) or Gr(k,E) over X whose fibres are given by the Grassmannians
of real or complex k-planes in the corresponding fibres of E. There are universal k-bundles
18
2 KO-theory
over these spaces, which we denote by UE. For complex symmetric bundles, Grassmanni-
ans may be defined as follows.
2.5 Definition. The non-degenerate Grassmannian Grnd(k, (E, ε)) associated with a com-
plex symmetric bundle (E, ε) is the open subbundle of the complex Grassmannian bundle
Gr(k,E) given in each fibre by those k-planes T for which the restriction ε|T is non-
degenerate. The universal symmetric bundle U(E,ε) is defined as the restriction of the
universal bundle over Grnd(k,E) endowed with the symmetric form induced by ε.
The non-degenerate Grassmannian Grnd(k, (E, ε)) contains the Grassmannian of k-
planes of the real bundle <(E, ε). An inclusion
j : RGr(k,<(E, ε)) ↪→ Gr(k, (E, ε)) (11)
is given by sending k-dimensional subspaces T in the fibres of <(E, ε) to their complexifi-
cations T ⊗R C. In fact, we will see in the next lemma that this inclusion is a homotopy
equivalence. Moreover, the restriction of the universal bundle U(E,ε) to RGr(k,<(E, ε))
corresponds to the universal real bundle over this space in the sense that
<(j∗U(E,ε)) = U<(E,ε) (12)
2.6 Lemma. For any complex symmetric bundle (E, ε), the inclusion j defined in (11) is
a homotopy equivalence. A homotopy inverse is provided by a retract of Gr(k, (E, ε)) onto
RGr(k,<(E, ε)).
Proof. Consider the projection pi : E = <(E, ε)⊕ i<(E, ε) <(E, ε).
We define a retract r of j by sending a complex k-plane T in a fibre of E to the subspace
pi(<(T, ε|T )) in the corresponding fibre of <(E, ε). This is indeed a linear subspace of real
dimension k: since ε is positive definite on <(T, ε|T ) but negative definite on i<(E, ε), the
intersection <(T, ε|T ) ∩ i<(E, ε) is trivial.
More generally, we define a family of endomorphisms of E parametrized by t ∈ [0, 1]
by
pit : <(E, ε)⊕ i<(E, ε) <(E, ε)⊕ i<(E, ε)
(x, y) 7→ (x, ty)
This family interpolates between the identity pi1 and the projection pi0, which we can
identify with pi. We claim that for any k-plane T in a fibre of E the image
pit(<(T, ε|T )) ⊂ E
is a real linear subspace of dimension k on which ε is real and positive definite. The
claim concerning the dimension has already been verified in the case t = 0 and follows
19
Chapter I. Two Cohomology Theories
for non-zero t from the fact that pit is an isomorphism. Now take a non-zero vector
v ∈ pit(<(T, ε|T )) and write it as v = x+ tiy, where x, y ∈ <(E, ε) and x+ iy ∈ <(T, ε|T ).
Since ε(x, x), ε(y, y) and ε(x + iy, x + iy) are all real we deduce that ε(x, y) = 0; it
follows that ε(v, v) is real as well. Moreover, since ε(x + iy, x + iy) is positive we have
ε(x, x) > ε(y, y), so that ε(v, v) > (1− t2)ε(y, y). In particular, ε(v, v) > 0 for all t ∈ [0, 1],
as claimed.
It follows that T 7→ pit(<(T, ε|T )) ⊗R C defines a homotopy from j ◦ r to the identity
on Grnd(k, (E, ε)).
2b Representability
The homotopy classification of vector bundles implies the representability of the functors
K0 and KO0 in the homotopy category H of topological spaces. More precisely, there
are representable functors on H which agree with K0 and KO0 on all finite-dimensional
CW complexes. We first recall how this works in the complex case. For KO0, we have
two equivalent representing spaces, corresponding to the characterizations in terms of real
vector bundles and complex symmetric bundles, respectively.
Let us write Grr,n for the Grassmannian Gr(r,Cr+n) of complex r-bundles in Cr+n,
and let Grr be the union of Grr,n ⊂ Grr,n+1 ⊂ · · · under the obvious inclusions. Denote
the universal r-bundles over these spaces by Ur,n and Ur. For any connected paracompact
Hausdorff space X, we have a one-to-one correspondence between the set Vectr(X) of
isomorphism classes of rank r complex vector bundles over X and homotopy classes of
maps from X to Grr: a homotopy class [f ] in H(X,Grr) corresponds to the pullback of
Ur along f [Hus94, Chapter 3, Theorem 7.2].
To describe K0(X), we need to pass to Gr, the union of the Grr under the embeddings
Grr ↪→ Grr+1 that send a complex r-plane W to C⊕W .
2.7 Theorem. For finite-dimensional CW complexes X, we have natural identifications
K0(X) ∼= H(X,Z×Gr) (13)
such that, for X = Grr,n, the class [Ur,n] + (d − r)[C] in K0(Grr,n) corresponds to the
inclusion Grr,n ↪→ {d} ×Grr,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} ×Grd ↪→
∐
d≥−1
{d} ×Grd+1 ↪→
∐
d≥−2
{d} ×Grd+2 ↪→ · · · ⊂ Z×Gr
20
2 KO-theory
Any continuous map from X to Z × Gr must factor through one of the components
colimn({d} × Grn). By cellular approximation, it is in fact homotopic to a map that
factors through {d} ×Grn for some n. Thus,
H(X,Z×Gr) ∼=
∐
d∈Z
colimnVectn(X)
where the colimit is taken over the maps Vectn(X)→ Vectn+1(X) sending a vector bundle
E to C⊕E. We define a map from the coproduct to K0(X) by sending a vector bundle E in
the dth component to the class [E]+(d−rkE)[C] in K0(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 RGrr,n, we obtain the
analogous statement that KO0 can be represented by Z×RGr. Equivalently, but more in
the spirit of Definition 2.1, we could work with the non-degenerate Grassmannians defined
in the previous section.
So let Grndr,n abbreviate Gr
nd(r,Hr+n), where H is the hyperbolic plane (C2, ( 0 11 0 )), and
let Undr,n denote the restriction of the universal bundle over Gr(r,C2r+2n) to Grndr,n. Then
colimits Grndr and Gr
nd can be defined in the same way as for the usual Grassmannians.
By (12) and Lemma 2.6 the homotopy classification of real vector bundles is equivalent
to the homotopy classification of complex symmetric bundles in the sense that we have
commutative diagrams(
isomorphism classes of real vector
bundles of rank r over X
)
H(X,RGrr)
∼=lr
(
isometry classes of complex symmetric
bundles of rank r over X
)< ∼=
LR
H(X,Grndr )
∼=lr
j∗ ∼=
LR
For finite-dimensional CW complexes X, we obtain natural identifications
KO0(X) ∼= H(X,Z×Grnd) (14)
Here, for even (d− r), the inclusion Grndr,n ↪→ {d} ×Grndr,n ↪→ Z×Grnd corresponds to the
class of [Undr,n] +
d−r
2 [H] in GW
0(Grndr,n).
2c Generalized cohomology theories
The representable functors K0 and KO0 may be embedded into sequences of functors Ki
and KOi defined for all integers i. These constituted the first examples of generalized coho-
21
Chapter I. Two Cohomology Theories
mology theories in topology. By definition, a generalized cohomology theory is a sequence
of functors Ei from topological spaces to abelian groups satisfying all the axioms of a
cohomology theory formulated by Eilenberg and Steenrod except for one: the dimension
axiom. That is, it is not required that Ei(point) vanishes for i 6= 0. Indeed, as we will see,
the values of Ki(X) and KOi(X) are periodic in i, and in particular, there are non-zero
groups Ki(point) and KOi(point) in arbitrarily high degrees.
Here, we describe the general method of defining generalized cohomology theories via
spectra. The construction of spectra Ktop and KOtop representing topological K- and
KO-theory is sketched at the beginning of the next section. The classical book [Ada95]
may be consulted both for general background on spectra and for details concerning the
particular case of K-theory; see in particular Example 2.2.
Generalized cohomology theories arise as representable functors on the stable homotopy
category SH, the homotopy category of spectra. More precisely, a generalized cohomology
theory E∗ may be obtained from a spectrum E as follows. Recall that SH is a triangulated
category in which the shift functor is given by suspension. We will denote the suspension
by S1 ∧ −. Given any spectrum X, one defines the cohomology groups of X with respect
to E by
E˜i(X) := SH(X, Si ∧ E)
Since suspension is invertible, these groups are defined for all integers i. Using the functor
Σ∞ : H• → SH from the pointed to the stable homotopy category that assigns to a pointed
space (X,x) its suspension spectrum, the definition may be specialized to pointed spaces.
That is, we define the (reduced) generalized cohomology theory on H• corresponding to
E by
E˜i(X,x) := Ei(Σ∞(X,x))
If X is connected, we often simply write E˜i(X) for E˜i(X,x), where x is an arbitrary point
of X. Finally, the cohomology groups Ei(X) of an unpointed space are defined as the
reduced groups of X+, the union of X with a disjoint base point:
Ei(X) := E˜i(X+)
For connected X, we have a canonical decomposition Ei(X) = Ei(point)⊕ E˜i(X).
The fact that the functors E∗ define a generalized cohomology theory follows directly
from the construction. We emphasize two properties:
• For the reduced theory E˜∗, we have suspension isomorphisms
σn : E˜i−n(X,x) ∼= E˜i(Sn ∧ (X,x)) (15)
• Exact triangles in SH give rise to long exact sequences of cohomology groups. For
22
2 KO-theory
example, any continuous map f : X ′ → X fits into an exact triangle involving its
mapping cone C(f):
Σ∞(X ′+)
Σ∞(f+)
,2 Σ∞(X+)
qxjjjj
jj
Σ∞(C(f))
fm
This triangle induces the long exact sequence
· · · → E˜i(C(f))→ Ei(X) f
∗
−→ Ei(X ′)→ E˜i+1(C(f))→ · · ·
2d K- and KO-theory
We now specialize the preceding discussion to the cases of K- and KO-theory. After sketch-
ing the constructions of the corresponding spectra, we discuss some general consequences
along with more specific aspects of these two theories.
The key ingredient in the construction of a K-theory spectrum Ktop is Bott periodicity.
One observes that the infinite Grassmannian Gr can be identified with the classifying space
BU of the infinite unitary group, so that K0 is represented by Z×BU. By Bott periodicity,
this space is equivalent to its own two-fold loop space Ω2(Z×BU). Thus, one may define
a 2-periodic Ω-spectrum Ktop whose even terms are all given by Z× BU. Similarly, RGr
is equivalent to the classifying space BO of the infinite orthogonal group. In this case,
Bott periodicity says that Z×BO is equivalent to Ω8(Z×BO), so one obtains a spectrum
KOtop which is 8-periodic. Following the definition in the previous section, we obtain 2-
and 8-periodic cohomology theories on reduced spaces:
K˜i(X,x) := SH(Σ∞(X,x), Si ∧Ktop)
K˜Oi(X,x) := SH(Σ∞(X,x), Si ∧KOtop)
The groups K˜i(X) and Ki(X) of unpointed spaces and the corresponding KO-groups are
defined exactly as above.
It follows from the construction that the functors K2i and KO8i agree with the functors
K0 and KO0 that we started with. If X is a connected finite-dimensional CW complex,
then K˜0(X) may be identified with the subgroup of K0(X) given by virtual bundles of
rank zero, i. e. by those elements [E]− [F] of K0(X) for which rkE−rkF = 0. Moreover, the
suspension isomorphisms allow us to express all the lower K-groups K−i(X) and KO−i(X)
in terms of K˜0 and K˜O0. That is, for all i ≥ 0 we have the following isomorphisms:
K−i(X) ∼= K˜0(Si ∧ (X+))
KO−i(X) ∼= K˜O0(Si ∧ (X+))
23
Chapter I. Two Cohomology Theories
Thus, whenX is a finite-dimensional CW complex, its K- and KO-groups may be described
in terms of vector bundles over suspensions of X.
2.8 Example (K-groups of a point). The topological K-group of a point is the free
abelian group generated by the trivial complex line bundle. Thus, K2i(point) ∼= Z for all
integers i. On the other hand, the preceding isomorphisms imply that
K−2(point) ∼= K˜0(S2)
An explicit generator of K˜0(S2) is given by [τ ]− [C], where τ denotes the Hopf bundle over
S2. If we identify S2 with the complex projective line CP1, the Hopf bundle corresponds
to the tautological or universal bundle O(−1).
Long exact cohomology sequences. As we have seen, any continuous map f : X ′ → X
gives rise to long exact sequences of K- and KO-groups. By periodicity, these may be
arranged as exact polygons with 6 and 24 vertices, respectively. We examine two particular
cases.
• If f : A→ X is a cofibration, for example an inclusion of a closed subcomplex A into
a CW complex X, the mapping cone C(f) is homotopy equivalent to the quotient
space X/A. The long exact sequences then take the form in which they appear
most often in topology. The groups K˜Oi(X/A) are usually denoted KOi(X,A), and
similarly for the K-groups.
• Suppose X is a smooth manifold with a smooth submanifold Z. Let f : U ↪→ X
denote the inclusion of the open complement of Z into X. Then the mapping cone
C(f) is the homotopy quotient X/hU , which may be realized as the quotient of X
by the closed complement of a tubular neighbourhood of Z in X. It follows that
C(f) is homotopy equivalent to the Thom space of the normal bundle N of Z in X.
Thus, the long exact sequences have the form
· · · → K˜Oi(ThomZ N)→ KOi(X)→ KOi(U)→ K˜Oi+1(ThomZ N)→ · · · (16)
Alternatively, we will sometimes denote the groups K˜i(X/hU) and K˜O
i(X/hU) by
KiZ(X) and KO
i
Z(X), in analogy with the notation used in algebraic geometry.
Multiplication. The spectra Ktop and KOtop are ring spectra. For the associated coho-
mology theories, this means that we have multiplication maps
K˜Oi(X,x)⊗ K˜Oj(Y, y) ·→ K˜Oi+j((X,x) ∧ (Y, y))
KOi(X)⊗KOj(Y ) ×−→ KOi+j(X × Y )
on KO-groups and similarly for K-groups. These products generalize the products that
may be defined in terms of vector bundles for finite-dimensional CW complexes [Gra75,
24
2 KO-theory
Theorem 29.14]. They are natural and respect the suspension isomorphisms (15) in the
sense that σn(x · y) = σn(x) · y. In particular, σn itself may be expressed as multiplication
with σn(1), where 1 denotes the unit in K0(point) or KO0(point), respectively. The
periodicity isomorphisms Ki(X) ∼= Ki−2(X) and KOi(X) ∼= KOi−8(X) are induced by
multiplication with generators g ∈ K−2(point) and λ ∈ KO−8(point). The generator g is
described in Example 2.8.
Coefficient rings. The groups Ei(point) of a generalized cohomology theory E∗ are
known as its coefficient groups. For K-theory, we simply have
K0(point) ∼= Z
K1(point) = 0
The coefficient groups of KO-theory are more complicated [Bot69, page 66]:
KO0(point) = Z KO7(point) = Z/2
KO2(point) = 0 KO1(point) = 0
KO4(point) = Z KO3(point) = 0
KO6(point) = Z/2 KO5(point) = 0
(17)
We have arranged them in a slightly unusual fashion so as to facilitate the comparison with
Example 1.7. Since our theories are multiplicative, their coefficient groups may moreover
be assembled to coefficient rings E∗(point) = ⊕Ei(point). These can be written as follows
[Bot69, page 741]:
K∗(point) = Z
[
g, g−1
]
(18)
KO∗(point) = Z
[
η, α, λ, λ−1
]/
(2η, η3, ηα, α2 − 4λ) (19)
Here, g and λ are the generators of K−2(point) and KO−8(point) inducing the periodicity
isomorphisms, as mentioned above. The generators η and α are of degrees −1 and −4,
respectively.
The Bott sequence. The spectra Ktop and KOtop fit into an exact triangle of the form
KOtop ∧ S1 η ,2 KOtop
szmmm
mmm
m
Ktop
fm
This triangle induces long exact sequences known as Bott sequences [Bot69, pages 75 and
1122; BG10, 4.I.B]. Concretely, the Bott sequence of a topological space X looks as follows:
1Unfortunately, the relation ηα = 0 is missing here, and this omission seems to have pervaded much
of the literature. Of course, the relation follows from the fact that KO−5(point) = 0.
2There are misprints on both pages. In particular, the central group in the diagram on page 112 should
be K0.
25
Chapter I. Two Cohomology Theories
. . .→ KO2i−1X → KO2i−2X → K2i−2X → KO2iX → KO2i−1X → K2i−1X
→ KO2i+1X → KO2iX → K2iX → KO2i+2X → KO2i+1X → . . .
(20)
The maps appearing in this sequence may be described explicitly:
• The maps KOjX → KjX are the complexification maps c.
• The maps Kj−2X → KOjX are the composites of the periodicity isomorphisms
Kj−2X ∼= Kj X and the realification maps r.
• The maps between KO-groups are given by multiplication with η ∈ KO−1(point).
Twisted KO-groups
It is possible to define twisted KO-groups KO0(X;L) for complex line bundles L over
X similarly to the way this is done for Grothendieck-Witt groups. It turns out, how-
ever, that these groups may alternatively be expressed as the usual (reduced) KO-groups
K˜O2(ThomL) of the Thom space of L [AR76, 3.8]. Here, we take this identification as
our definition. More generally, we define KO-groups twisted by arbitrary complex bundles
E over X as follows.
2.9 Definition. For any complex vector bundle E of constant rank r over a topological
space X, we define
KOp(X;E) := K˜Op+2r(ThomE)
When E is a trivial bundle, its Thom space is just a suspension of X, so that KOp(X;E)
agrees with the usual KO-group KOp(X). Moreover, the Thom isomorphisms for KO-
theory of [ABS64] show that the groups KO(X;E) only depend on the determinant line
bundle of E:
2.10 Lemma. For complex vector bundles E and F over a topological space X with iden-
tical first Chern class modulo two, we have
KOp(X;E) ∼= KOp(X;F)
Proof. A complex vector bundle E whose first Chern class vanishes modulo two has a spin
structure and is therefore oriented with respect to KO-theory [ABS64, § 12]. That is, we
have a Thom isomorphism
KOpX
∼=−→ K˜Op+2r(ThomE)
Now suppose c1(E) ≡ c1(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
26
2 KO-theory
to KO-theory in both cases. Thus, both groups in the lemma can be identified with
KOp(X;E⊕ E⊕ F).
Remark. In general, the identifications of Lemma 2.10 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
KOp(X;L) ∼= KOp(X;L⊗M⊗2)
for any two complex line bundles L and M over X. In general, different spin structures
on a spin bundle over X are classified by H1(X;Z/2).
2e The Atiyah-Hirzebruch spectral sequence
In topology, there is a standard computational tool for generalized cohomology theories
known as the Atiyah-Hirzebruch spectral sequence (AHSS). Given a generalized cohomo-
logy theory E∗ and a finite-dimensional CW complex X, this spectral sequence takes the
form [Ada95, III.7; Koc96, Theorem 4.2.7]
Ep,q2 = H
p(X;Eq(point))⇒ Ep+q(X)
It is concentrated in the half-plane p ≥ 0 and has differentials dr of bidegree (r,−r + 1).
In good cases, it enables us to compute the cohomology E∗(X) from the coefficient groups
of E∗ and the singular cohomology of X.
The spectral sequence is natural with respect to continuous maps of finite-dimensional
CW complexes. Moreover, if E∗ is a multiplicative cohomology theory represented by a
ring spectrum, then the spectral sequence is also multiplicative [Koc96, Proposition 4.2.9].
That is, the multiplication on the E2-page induced by the cup product on singular coho-
mology and the ring structure of E∗(point) descends to a multiplication on all subsequent
pages, such that the multiplication on the E∞-page is compatible with the multiplica-
tion on E∗(X). In particular, each page is a module over E∗(point). In this case, the
differentials of the spectral sequence are derivations, i. e. they satisfy a Leibniz rule.
In some situations, it is more natural to consider the spectral sequence for the reduced
cohomology theory E˜∗ associated with E∗. Then the spectral sequence is written as
E˜p,q2 = H˜
p(X;Eq(point))⇒ E˜p+q(X)
27
Chapter I. Two Cohomology Theories
The AHSS for K- and KO-theory
Specializing to the cases of K- and KO-theory, we obtain the following two multiplicative
spectral sequences.
Ep,q2 = H
p(X; Kq(point))⇒ Kp+q(X)
Ep,q2 = H
p(X; KOq(point))⇒ KOp+q(X)
For K-theory, the spectral sequence has the singular cohomology of X with integral coef-
ficients in all even rows, while the odd rows vanish. For KO-theory, the spectral sequence
is 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.
RLq
H0(X;Z) H1(X;Z) H2(X;Z)
Sq2◦piYY
YYYYY
)/YYYYYY
H3(X;Z) H4(X;Z)
p
,2
H0(X;Z/2) H1(X;Z/2) H2(X;Z/2)
Sq2
YYYYYY
)/YYYYY
Y
H3(X;Z/2) H4(X;Z/2)
H0(X;Z/2) H1(X;Z/2) H2(X;Z/2) H3(X;Z/2) H4(X;Z/2)
0 0 0 0 0
H0(X;Z) H1(X;Z) H2(X;Z) H3(X;Z) H4(X;Z)
0 0 0 0 0
...
...
...
...
...
Figure 1: The E2-page of the Atiyah-Hirzebruch spectral sequence computing KO∗(X)
The low-order differentials of these spectral sequences may be described as follows. Let
pi : H∗(X;Z)→ H∗(X;Z/2)
Sq2 : H∗(X;Z/2)→ H∗+2(X;Z/2)
β : H∗(X;Z/2)→ H∗+1(X;Z)
denote reduction modulo two, the second Steenrod square, and the Bockstein homo-
morphism, respectively. For K-theory, the differentials on the E3-page of the Atiyah-
Hirzebruch spectral sequence are given by the composition β◦Sq2◦pi [HJJS08, Chapter 21,
§ 5]. For KO-theory, the differentials d∗,02 and d∗,−12 on the E2-page are given by Sq2 ◦pi
and Sq2, respectively. The differential d∗,−23 on the E3-page can be identified with β ◦ Sq2
[Fuj67, 1.3].
28
2 KO-theory
The AHSS for Thom spaces
In order to compute twisted KO-groups as discussed in Section 2d, 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 pi : 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 ThomE is additively isomorphic to the cohomology of X itself, apart from a shift
in degrees by r := rkRE. The isomorphism is given by multiplication with a Thom class θ
in H˜r(ThomE;Z):
H∗(X;Z)
∼=−→ H˜∗+r(ThomE;Z)
x 7→ pi∗(x) · θ
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 E2-page of the spectral sequence for K˜O∗(ThomE) are identical to those on the
E2-page for KO∗(X). However, the differentials may differ.
2.11 Lemma. Let E pi→ 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 H˜∗(ThomE;Z/2)
is given by “ Sq2 + c1(E)”, where c1(E) is the first Chern class of E modulo two. That is,
Sq2(pi∗x · θ) = pi∗ (Sq2(x) + c1(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 “ Sq2 + w2(E)”, where
w2 is the second Stiefel-Whitney class of E.
Proof. This is a special case of the following well-known identity of Thom [MS74, page 91]:
Sqi(pi∗x · θ) = pi∗ (Sqi(x) + wi(E)x) · θ
The higher differentials in the spectral sequence for K˜O∗(ThomE) also 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, which is made more
precise by the next lemma.
2.12 Lemma. Let E be a real oriented vector bundle over a finite-dimensional CW com-
plex X, of constant rank r. Suppose E is oriented, and let θ be a Thom class as above.
If E is moreover oriented with respect to KO-theory, then θ survives to the E∞-page
of the Atiyah-Hirzebruch spectral sequence computing K˜O∗(ThomE), and the Thom iso-
morphism for H∗ extends to an isomorphism of spectral sequences. That is, for each
29
Chapter I. Two Cohomology Theories
page right multiplication with the class of θ in E˜r,0s (ThomE) gives an isomorphism of
E∗,∗s (X)-modules
E∗,∗s (X)
·θ−→∼= E˜
∗+r,∗
s (ThomE)
Moreover, any lift of θ ∈ E˜r,0∞ (ThomE) to K˜Or(ThomE) defines a Thom class of E
with respect to KO-theory. 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 ix : Sr ↪→ ThomE. By assumption,
the pullback i∗x on ordinary cohomology maps θ to a generator of H˜r(Sr), and the pullback
on K˜O∗ gives a surjection
K˜O∗(ThomE)
i∗x K˜Or(Sr)
Consider the pullback along ix on the E∞-pages of the spectral sequences for Sr and
ThomE. Since we can identify E˜r,0∞ (ThomE) with a quotient of K˜Or(ThomE) and E˜r,0∞ (Sr)
with K˜Or(Sr), we must have a surjection
i∗x : E˜
r,0
∞ (ThomE) E˜r,0∞ (Sr)
On the other hand, the behaviour of i∗x on E˜
r,0∞ is determined by its behaviour on H˜r,
whence we can only have such a surjection if θ survives to the E˜∞-page of ThomE.
Thus, all differentials vanish on θ, and if multiplication by θ induces an isomorphism from
E∗,∗s (X) to E˜∗+r,∗s on page s, it also induces an isomorphism on the next page. Lastly,
consider any lift of θ to an element Θ of K˜Or(ThomE). It is clear by construction that
right multiplication with Θ gives an isomorphism from E∞(X) to E˜∞(ThomE), and thus
it also gives an isomorphism from KO∗(X) to K˜O∗(ThomE). Thus, Θ is a Thom class for
E with respect to KO-theory.
Lemma 2.12 allows the following strengthening of Lemma 2.10:
2.13 Corollary. For complex vector bundles E and F over X with identical first Chern
class modulo two, the spectral sequences computing K˜O∗(ThomE) and K˜O∗(ThomF) can
be identified up to a possible shift of columns when E and F have different ranks.
30
Chapter II
Comparison Maps
If X is a smooth complex variety, then we can consider both the algebraic and the topo-
logical cohomology theories discussed in the previous chapter. That is, on the one hand
we can study the algebraic K-group and the (Grothendieck-)Witt groups of X viewed as
a variety, and on the other hand we can study the topological complex and real K-groups
of X viewed as a topological space, i. e. the K-groups of the set X(C) of complex points
of X equipped with the analytic topology. It follows directly from the definitions of these
groups that we have natural maps
k : K0(X)→K0(X(C)) (1)
gw0 : GW0(X)→KO0(X(C)) (2)
If we write (KO0/K)(X(C)) for the cokernel of the realification map from K0(X(C)) to
KO0(X(C)), we moreover have an induced map
w0 : W0(X)→(KO0/K)(X(C)) (3)
Our aim in this chapter is to extend these maps to be defined on shifted groups GWi(X)
and Wi(X) for arbitrary integers i in such a way that they are compatible with as much
structure of the respective cohomology theories as possible. Two different approaches will
be discussed.
The first approach is the more elementary one. The idea is to use the multiplicative
structure of the theories to define the maps on shifted Grothendieck-Witt groups in such a
way that many of their properties can be checked by explicit calculations. This approach
has previously been detailed in [Zib09]. We refrain from reproducing the full details here,
but only sketch the key steps in the argument. The results obtained in [Zib09] only
just suffice to prove the comparison theorem for cellular varieties included in Chapter IV
of this thesis. However, as we will explain, the fundamental question of whether the
comparison maps are in general compatible with the boundary maps appearing in long
exact localization sequences remains open.
The idea of the second approach is to use a construction of Grothendieck-Witt groups,
or more generally of hermitian K-theory, that resembles the homotopy-theoretic definition
of KO-theory so closely that all compatibility issues disappear. More precisely, our second
31
Chapter II. Comparison Maps
definition of the comparison maps will rely on the following theorem from A1-homotopy
theory:
Hermitian K-theory is representable in the stable A1-homotopy category by
a spectrum whose complex realization in the usual stable homotopy category
represents KO-theory.
Unfortunately, although this result seems to be well-known among experts, it has not yet
been properly published. We therefore highlight both this and a related statement that
we will be using in the form of Assumption 2.2. Proofs of both parts of the assumption
are announced by Morel in [Mor06]. To add further credibility to the first statement,
we explain in Section 2b how the analogous claim in the unstable homotopy category
may be deduced from recent results of Schlichting and Tripathi. A brief summary of
those superficial aspects of A1-homotopy theory that we will constantly be referring to is
included in Section 2a.
1 An elementary approach
Recall from I.2d that the suspension isomorphisms in KO-theory are given by multiplica-
tion with generators of the KO-groups of spheres. That is, if e˜n denotes the generator of
K˜On(Sn) corresponding to the unit in KO0(point) under suspension, then the suspension
isomorphism for an arbitrary pointed space (X,x) is given by multiplication with e˜n:
K˜Oi−n(X,x)
∼=−−→ K˜Oi(Sn ∧ (X,x))
Similarly, for an unpointed space X, we have isomorphisms
KOi−n(X)
∼=−−→ K˜Oi(Sn ∧ (X+))
KOi(X)⊕KOi−n(X) ∼=−−→ KOi(Sn ×X)
Explicitly, if we write en for the element of KOn(Sn) corresponding to e˜n and pi for the
projection Sn×X  X, then the lower map is given by (x, y) 7→ pi∗x+ en× y. Analogous
isomorphisms exist for Grothendieck-Witt groups:
1.1 Lemma. Let X be a smooth variety with some closed subset Z. Let Ψ0 be the gener-
ator of GW1(P1) described in Example I.1.6. We have isomorphisms
GWi−1Z (X)
∼=−−→ GWiZ(X × P1)
GWi−1Z (X)
∼=−−→ GWiZ(X × A1)
given by multiplication with the restriction of Ψ0 to GW1{0}(P
1) and its further restriction
32
1 An elementary approach
to GW1{0}(A
1), respectively. Moreover, we have an isomorphism
GWiZ(X)⊕GWi−1Z (X)
∼=−−→ GWiZ×P1(X × P1)
given by (x, y) 7→ pi∗x+ y×Ψ0, where pi denotes the projection X ×P1  X. All of these
isomorphisms also hold with Witt groups in place of Grothendieck-Witt groups.
Proof. For Witt groups, the first isomorphism is a special case of Theorem 2.5 in [Nen07],
the case when Z = X being Theorem 8.2 in [BG05]. The second isomorphism follows via
excision. The decomposition of Wi(X × P1) into Wi(X) ⊕Wi−1(X) is a special case of
Theorem 1.5 in [Wal03b]. It may be deduced from the second isomorphism by considering
the localization sequences associated with the inclusion of X×A1 into X×P1, since these
sequences split. A decomposition of WiZ(X × P1) can be obtained analogously, using the
more general localization sequences of triples discussed in Theorem 1.6 of [Bal01b]. Lastly,
since the corresponding isomorphisms also hold for K0, with F (Ψ0) ∈ K0(P1) in place of
Ψ0, the corresponding isomorphisms of Grothendieck-Witt groups may be deduced via
Karoubi induction (c. f. the proof of Lemma 1.3 below and Lemma 2.3 in [Zib09]).
Now let X be a smooth complex variety. Then the two isomorphisms
GWi(X)⊕GWi−1(X) ∼=−−−→ GWi(X × P1)
KO2i(X(C))⊕KO2i−2(X(C)) ∼=−→KO2i(X(C)× S2)
can be used to define comparison maps
gw i : GWi(X)→ KO2i(X(C))
inductively for all non-positive i. Explicitly, if gw i is already defined for some i ≤ 0,
then for x ∈ GWi−1(X) we define gw i−1(x) to be the unique element in KO2i−2(X(C))
satisfying1
gw i−1(x)× e2 = gw i(x× (−Ψ0))
The following properties of these maps can be checked via explicit calculations. Firstly, it
is clear from the construction that they are natural with respect to morphisms of smooth
varieties, and that they respect the multiplicative structures of Grothendieck-Witt groups
and KO-theory. Moreover, they are compatible with the Karoubi and Bott sequences (see
1We use −Ψ0 instead of Ψ0 because the image of e2 under the complexification map KO2(S2) c−→ K2(S2)
and the periodicity isomorphism K2(S2)
∼=−→ K0(S2) agrees with the image of −Ψ0 under the forgetful map
GW1(P1) F−→ K0(X): we have k(F (−Ψ)) = k(O(−1) − O) = [τ ] − [C] = σ−2(g) = c(e2) × g. See
Examples I.1.6 and I.2.8.
33
Chapter II. Comparison Maps
(I.4) and (I.20)) in the sense that we have induced maps
w i : Wi(X)→ (KO2i/K)(X(C))
and commutative diagrams of the following form:
GWi−1(X) F ,2
gw i−1

K0(X)
Hi ,2
k

GWi(X) ,2
gw i

Wi(X) ,2
w i

0
KO2i−2(X(C)) g
i−1◦c
,2 K0(X(C)) r◦g
−i
,2 KO2i(X(C)) ,2 (KO2i/K)(X(C)) ,2 0
To obtain comparison maps in positive degrees, one checks next that the maps already
defined respect the periodicity isomorphisms. This can again be done by expressing these
isomorphisms in a multiplicative way. We include a short proof to convey the flavour of
these arguments.
1.2 Lemma. The maps gw i : GWi(X) → KO2i(X(C)) defined above respect the periodi-
city isomorphisms GWi(X) ∼= GWi−4(X) and KO2i(X(C)) ∼= KO2i−8(X(C)).
Proof. Recall from Section I.2d that the periodicity isomorphism in KO-theory is given
by multiplication with a generator λ ∈ KO−8(point) whose sign is fixed by the relation
α2 = 4λ for a generator α ∈ KO−4(point). The periodicity isomorphism for Grothendieck-
Witt groups is induced by shifting complexes two positions to the right. This isomorphism
may be interpreted as cross product with the element Λ := [O[−2], id] of GW−4(point),
where O[−2] denotes the complex consisting of the trivial line bundle in degree−2. To show
that gw−4 maps Λ to λ, we use the following square from the diagram above, evaluated
on a point:
K0(point)
k

H−4 ,2 GW−4(point)
gw−4

K0(point)
r◦g4
,2 KO−8(point)
By Lemma I.1.9, we have H−4(O) = H−4(F (Λ)) = 2Λ, and similarly r(g4) = 2λ. Thus,
gw−4 must map Λ to λ.
We see similarly that the comparison maps gw i and w i, now defined for all shifts i ∈ Z,
induce isomorphisms between the Grothendieck-Witt and KO-groups of a point.
1.3 Proposition. The comparison maps are isomorphism on a point:
gw i : GWi(point)
∼=−−→ KO2i(point)
w i : Wi(point)
∼=−−→ (KO2i/K)(point)
34
1 An elementary approach
Proof. We can easily see that the corresponding groups of a point p are isomorphic by
direct comparison: since K1(p) vanishes, the Bott sequence (I.20) implies that we can
identify the quotients (KO2i/K)(p) with the odd KO-groups KO2i−1(p), so we can read
off the values of all groups from Example I.1.7 and (I.17). Moreover, it is clear that the
maps k, gw0 and w0 are isomorphisms. It follows by periodicity that gw i and w i are
isomorphisms for all i divisible by four. For all other values of i, the groups Wi(p) are
trivial, so w i is an isomorphism on a point in general. To see that the maps gw i are
also isomorphisms for arbitrary i, we can use the comparison of the Karoubi and Bott
sequences:
. . . ,2 GWi−1(p) ,2
gw i−1

K0(p) ,2
∼=

GWi(p) ,2
gw i

Wi(p) ,2
∼=

0 ,2

. . .
. . . ,2 KO2i−2(p) ,2 K0(p) ,2 KO2i(p) ,2 KO2i−1(p) ,2 0 ,2 . . .
Given the periodicity of the Grothendieck-Witt groups, the claim concerning the maps gw i
follows from repeated applications of the Five Lemma. This strategy of proof is known as
“Karoubi induction”.
As a next step, we would like to compare the localization sequence (I.7) of a smooth
variety X with a smooth closed subvariety Z to the corresponding localization sequence
(I.16) in KO-theory. Actually, this only makes sense for small parts of these sequences,
and we have to compose the maps w i defined on Witt groups with multiplication by
η ∈ KO−1(point):
GWiZ(X) ,2
?

GWi(X) ,2
gw i


GWi(U) ∂ ,2
gw i

Wi+1Z (X) ,2
?

Wi+1(X) ,2
η◦w i+1


Wi+1(U)
η◦w i+1

KO2iZ (X) ,2 KO
2i(X) ,2 KO2i(U) ∂ ,2 KO2i+1Z (X) ,2 KO
2i+1(X) ,2 KO2i+1(U)
Here, we have written X for X(C) in the second row. The symbol “” marks the two
squares that commute by naturality, and the question is how to fill in the dotted arrows
such that all squares commute. Using Thom isomorphisms, we could replace the groups
with support on Z by the actual cohomology groups of Z, and then we could use the
comparison maps already defined for the dotted arrows. But we do not know a way of
proving that the resulting squares are commutative without using comparison maps on
groups with restricted support at some point.
Unfortunately, we are similarly unaware of any straight-forward way of defining com-
parison maps on groups GW0Z(X) with support Z 6= X. The solution in [Zib09] is to
adapt a known construction of classes in relative K-groups detailed in [Seg68], in terms
of complexes of vector bundles whose cohomology is supported on an open subspace. To
construct a comparison map on GW0Z(X), we first choose a suitable open neighbourhood of
35
Chapter II. Comparison Maps
Z in X and define a map to a KO-group of complexes with support on this neighbourhood.
In a second step, we construct maps from such KO-groups defined in terms of complexes to
the usual relative KO-groups. We do not know whether these maps are isomorphisms but,
in any case, we obtain a comparison map gw0 : GW0Z(X)→ KO0Z(X(C)) by composition.
Finally, we need to check that this map is independent of the choices made.
Once the construction of comparison maps on GW0Z(X) is settled, maps on shifted
groups GWiZ(X) can be defined in exactly the same way as before. Moreover, these
constructions also work for groups twisted by line bundles. This allows us to check, again
by explicit calculations, that the comparison maps are compatible with Nenashev’s Thom
isomorphisms, and to conclude that two further squares in the above diagram commute.
Lastly, we are left with the square involving the boundary maps. This square remains
problematic. What is needed is either an algebraic description of the boundary morphism
in KO-theory, or a homotopy-theoretic description of the boundary morphism for (Gro-
thendieck-)Witt groups. The latter idea led us to consider a totally different approach to
the comparison problem altogether, which we describe in the next section.
2 A homotopy-theoretic approach
A homotopy theory of schemes emulating the situation for topological spaces has been
developed over the past 20 years mainly by Morel and Voevodsky. Today, it is known as
either A1-homotopy theory or motivic homotopy theory. The authoritative reference is
[MV99]. Closely related texts by the same authors are [Voe98], [Mor99] and [Mor04]. A
textbook introduction is provided by [DLØ+07]. Below, we summarize the main points
relevant for us in just a few sentences.
2a A1-homotopy theory
In short, the A1-homotopy category is constructed as follows. Consider the category Smk
of smooth separated schemes of finite type over a field k. One of the problems with this
category that arises if one tries to imitate topological arguments is that it does not have
colimits: given two schemes X and Y with a common subscheme A, there is in general no
smooth candidate for the union of X and Y “glued along A”. However, the category Smk
can be embedded into some larger category Spck of “spaces over k” which is closed under
small limits and colimits, and which can even be equipped with a model structure. The
A1-homotopy category H(k) over k is the homotopy category associated with this model
category.
In fact, there are several possible choices for Spck and many possible model structures
yielding the same homotopy category H(k). One possibility is to consider the category of
simplicial presheaves over Smk, or the category of simplicial sheaves with respect to the
Nisnevich topology. Both categories contain Smk as a full subcategory via the Yoneda
36
2 A homotopy-theoretic approach
embedding, and they also contain simplicial sets viewed as constant (pre)sheaves. One
may then 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 A1
to become contractible by localizing with respect to the set of all projections A1×X  X.
As in topology, we also have a pointed version H•(k) of H(k). Remarkably, these
two categories contain several distinct “circles”: the simplicial circle S1, the “Tate circle”
Gm = A1 − 0 (pointed at 1) and the projective line P1 (pointed at ∞). They are related
by the intriguing formula P1 = S1 ∧Gm. A common notational convention which we will
follow is to define
Sp,q := (S1)∧(p−q) ∧G∧qm
for any p ≥ q. In particular, we then have S1 = S1,0, Gm = S1,1 and P1 = S2,1.
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 Sp,q ∧ − become inverti-
ble. This category is usually constructed using P1-spectra. The triangulated shift functor
is given by suspension with the simplicial sphere S1,0.
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 (4)
that sends a smooth schemeX to its set of complex pointsX(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 (5)
which takes Σ∞(X+) to Σ∞(X(C)+) for any smooth scheme X [Rio06, The´ore`me I.123;
Rio07a, The´ore`me 5.26].
2b Representing algebraic and hermitian K-theory
In the context of A1-homotopy theory, Theorem I.2.7 describing K0 in terms of homotopy
classes of maps to Grassmannians has an algebraic analogue. Grassmannians of r-planes
in kn+r can be constructed as smooth projective varieties over any field k. Viewing them
as objects in Spck, we can form their colimits Grr and Gr in the same way as in topology.
The following theorem is established in [MV99, § 4] and spelt out explicitly in [Rio06,
The´ore`me III.3 and Assertion III.4].
37
Chapter II. Comparison Maps
2.1 Theorem. For smooth schemes X over k we have natural identifications
K0(X) ∼= H(k)(X,Z×Gr) (6)
such that the inclusion Grr,n ↪→ {d} × Grr,n ↪→ Z × Gr corresponds to the class [Ur,n] +
(d− r)[O] in K0(Grr,n).
An analogous result for hermitian K-theory has recently been obtained by Schlichting
and Tripathi1: Let Grndr,n denote the “non-degenerate Grassmannians” defined as open
subvarieties of Grr,r+2n as above, and let Grndr and Gr
nd be the respective colimits. Then
for smooth schemes over k we have natural identifications
GW0(X) ∼= H(k)(X,Z×Grnd) (7)
It follows from the construction that, when (d − r) is even, the inclusion of Grndr,n ↪→
{d} ×Grndr,n ↪→ Z×Grnd corresponds to the class of [Undr,n] + d−r2 [H] in GW0(Grndr,n), where
Undr,n is the universal symmetric bundle over Gr
nd
r,n.
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 (1) and (2), and the right vertical arrows are induced by the
complex realization functor (4).
K0(X) ∼=

H(C)(X,Z×Gr)

K0(X(C))∼=H(X(C),Z×Gr)
GW0(X) ∼=

H(C)(X,Z×Grnd)

KO0(X(C))∼=H(X(C),Z×Grnd)
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 (6) and (7). Even better, algebraic and hermitian K-theory are representable in the
stable A1-homotopy category SH(k). Let us make the statement a little more precise by
fixing some notation. With any spectrum E in SH(k) one may associate a cohomology
theory in exactly the same way as explained in Section I.2c, the only difference being that
the theory obtained is now bigraded. Explicitly, a reduced cohomology theory E˜∗,∗ on
1Talk “Geometric representation of hermitian K-theory in A1-homotopy theory” at the Workshop
“Geometric Aspects of Motivic Homotopy Theory”, 6.–10. September 2010 at the Hausdorff Center for
Mathematics, Bonn
38
2 A homotopy-theoretic approach
H•(k) and a corresponding unreduced theory E∗,∗ on H(k) are defined by setting
E˜p,q(X) := SH(k)(Σ∞X, Sp,q ∧ E) for X ∈ H•(k)
Ep,q(X) := E˜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
S2,1 ∧K ∼=→ K
Thus, the bigrading of the corresponding cohomology theory Kp,q is slightly artificial. The
identification with the usual notation for algebraic K-theory is given by
Kp,q(X) = K2q−p(X) (8)
For hermitian K-theory we have an (8,4)-periodic spectrum KO, and the corresponding
cohomology groups KOp,q are honestly bigraded. The translation into the notation used
for hermitian K-groups in Section I.1d is given by
KOp,q(X) = GWq2q−p(X) (9)
We will refer to the number 2q− p as the degree of the group KOp,q(X). The relation
with Balmer’s Witt groups obtained by combining (9) and (I.10) is illustrated by the
following table:
KOp,q p = 0 1 2 3 4 5 6 7
q = 0 GW0 W1 W2 W3 W0 W1 W2 W3
q = 1 GW12 GW
1
1 GW
1 W2 W3 W0 W1 W2
q = 2 GW24 GW
2
3 GW
2
2 GW
2
1 GW
2 W3 W0 W1
q = 3 GW36 GW
3
5 GW
3
4 GW
3
3 GW
3
2 GW
3
1 GW
3 W0
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].
However, for KO our references are slightly thin. Rather than shedding any further light
on this problem, we will at this point succumb to an “axiomatic approach”. For clarity,
we make a record of the precise properties of the spectra KO and K that we will be using.
2.2 Standing assumptions. There exist spectra K and KO in SH(C) representing alge-
braic K-theory and hermitian K-theory in the sense described above, such that:
(a) The complex realization functor (5) takes K to Ktop and KO to KOtop.
39
Chapter II. Comparison Maps
(b) We have an exact triangle in SH(C) of the form
KO ∧ S1,1 η ,2 KO
t|ppp
ppp
K
dl (10)
which corresponds to the usual triangle in SH.
These results are announced in [Mor06]. Independent constructions of spectra repre-
senting hermitian K-theory can be found in [Hor05] and in a recent preprint of Panin and
Walter [PW10].
2c The comparison maps
Given the assumptions above, the existence of comparison maps with good properties is
immediate. In fact, by assumption (a), complex realization induces comparison maps in
arbitrary bidegrees. Thus, we have comparison maps
k˜p,q : K˜p,q(X)→ K˜p(X(C))
k˜p,qh : K˜O
p,q(X)→ K˜Op(X(C))
for any pointed space X, and similarly comparison maps kp,q and kp,qh for any unpointed
space, including in particular the case of a smooth complex scheme X. The maps we
originally intended to study appear as special cases in degrees 0 and −1. They will be
denoted by the same letters as before:
k := k0,0 : K0(X)→ K0(X(C))
gw q := k2q,qh : GW
q(X)→ KO2q(X(C))
w q := k2q−1,q−1h : W
q(X)→ KO2q−1(X(C))
In the following, we will usually writeX forX(C) when this is unambiguous. The following
properties of the comparison 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 that we may identify kp,qh
with kp+8,q+4h (and hence w
q with w q+4 and gw q with gw q+4).
• They are compatible with long exact sequences arising from exact triangles in SH(C).
The crucial advantage of the homotopy-theoretic approach is, of course, the last point.
It applies in particular to the following two kinds of sequences.
40
2 A homotopy-theoretic approach
Localization sequences. As in topology, any morphism of smooth schemes f : X ′ → X
gives rise to long exact sequences relating the cohomology groups of X ′ to those of X. In
particular, if X has a smooth closed subscheme Z with open complement U , we have an
exact triangle of the form
Σ∞(X − Z)+ ,2 Σ∞(X+)
qxjjjj
jj
Σ∞(X/U)
fm
in SH(C), where X/U is obtained as the homotopy quotient of X by U in the category
of spaces1. The induced long exact sequences of cohomology groups may be referred to
as localization sequences. To clarify how the bigrading works, we write out one sequence
explicitly in the example of hermitian K-theory:
· · · → K˜Op,q(X/U)→ KOp,q(X)→ KOp,q(U)→ K˜Op+1,q(X/U)→ . . . (11)
As a further analogy with topology, Morel and Voevodsky show that the space X/U only
depends on the normal bundle N of Z in X. To make this more 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:
ThomZ(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] thatX/U is canonically isomorphic
to ThomZ(N) in the unstable pointed A1-homotopy category.
The comparison maps induce commutative ladder diagrams of the following form:
. . . ,2 K˜Op,q(ThomZ N) ,2

KOp,q(X) ,2

KOp,q(U) ,2

K˜Op+1,q(ThomZ N) ,2

. . .
. . . ,2 K˜Op(ThomZ N) ,2 KOp(X) ,2 KOp(U) ,2 K˜Op+1(ThomZ N) ,2 . . .
Of course, part of the statement here is that the complex realization of the Thom space
of N over Z can be identified with the corresponding Thom space in topology, but since
we are working in the stable category, there is no need to check this explicitly: the two
spaces are isomorphic in SH because they fit into isomorphic exact triangles. To see that
they are already isomorphic in the unstable homotopy category H•, we would need to be
more careful and use a realization functor defined on the level of model categories. Such
a functor is constructed in [Dug01].
1In the model category of spaces used in [MV99], inclusions are cofibrations, so the homotopy quotient
of X by U coincides with the actual quotient.
41
Chapter II. Comparison Maps
Karoubi/Bott sequences. The Karoubi sequence arising from triangle (10) may be
written explicitly as follows:
. . .→ KOp−1,q(X)→ KOp−2,q−1(X)→ K2q−p(X)→
KOp,q(X)→ KOp−1,q−1(X)→ K2q−p−1(X)→ . . .
By Assumption 2.2(b), the comparison maps induce a commutative ladder diagram with
this sequence in the first row and the Bott sequence (I.20) in the second. Near degree
zero, it takes the following familiar form:
. . . ,2 KO2i−1,iX ,2
k2i−1,ih

GWi−1X ,2
gw i−1

K0X ,2
k

GWiX ,2
gw i

WiX ,2
w i

0 ,2

. . .
. . . ,2 KO2i−1X ,2 KO2i−2X ,2 K0X ,2 KO2iX ,2 KO2i−1X ,2 K1X ,2 . . .
(12)
As a consequence, the comparison map w i factors through (KO2i/K)(X)
η→ KO2i−1(X),
so that we have an induced map
w i : Wi(X)→ (KO2i/K)(X)
We will often switch between these two versions of w i without further comment.
Groups with restricted support. Comparing the localization sequences (I.7) and (I.9)
discussed in Chapter I to the localization sequence (11) above, we see that the groups
K˜Op,q(X/U) 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:
2.3 Lemma. Let Z be a smooth closed subvariety of a smooth variety X, with open
complement U . We have the following isomorphisms:
K˜O2q,q(X/U) ∼= GWqZ(X)
K˜Op,q(X/U) ∼= Wp−qZ (X) for 2q − p < 0
Proof. Consider Z = Z×{0} as a subvariety of X×A1. Its open complement (X×A1)−Z
contains X = X × {1} as a retract. Thus, the projection from X × A1 onto X induces a
splitting of the localization sequences associated with (X × A1 − Z) ↪→ X × A1, and we
have
GWi+1Z (X × A1) ∼= coker
(
GWi+11 (X × A1) ↪→ GWi+11 (X × A1 − Z)
)
K˜O2i+2,i+1( X×A
1
X×A1−Z )
∼= coker (KO2i+1,i+1(X × A1) ↪→ KO2i+1,i+1(X × A1 − Z))
By (9), we can identify the groups appearing on the right, so we obtain an induced
42
2 A homotopy-theoretic approach
isomorphism of the cokernels. The quotient X ×A1/(X ×A1 − Z) can be identified with
the suspension of X/U by S2,1, so we have an isomorphism
K˜O2i+2,i+1( X×A
1
X×A1−Z )
∼= K˜O2i,i(X/U)
On the other hand, by Lemma 1.1, we have an isomorphism
GWi+1Z (X × A1) ∼= GWiZ(X)
The proof in lower degrees is analogous.
Twisting by line bundles. In the homotopy-theoretic approach, we can define hermitian
K-groups with twists in a line bundle, or more generally in any vector bundle, in the same
way as we did for KO-theory in Section I.2d.
2.4 Definition. 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
KOp,q(X;E) := K˜Op+2r,q+r(ThomE)
The twisted groups in degrees zero and below agree with the usual twisted Grothendieck-
Witt and Witt groups:
2.5 Lemma. For a vector bundle E over a quasi-projective variety X, we have isomor-
phisms
KO2q,q(X;E) ∼= GWq(X; detE)
KOp,q(X;E) ∼= Wp−q(X; detE) for 2q − p < 0
Proof. This follows from Lemma 2.3 and Nenashev’s Thom isomorphisms for Witt groups:
for any vector bundle E of rank r there is a canonical Thom class in WrX(E) which induces
an isomorphism Wi(X; detE) ∼= Wi+rX (E) by multiplication [Nen07, Theorem 2.5]. This
Thom class actually comes from a class in GWrX(E). Using the Karoubi sequence (I.4), one
sees that multiplication with this class also induces an isomorphism of the corresponding
Grothendieck-Witt groups.
Remark. The isomorphisms of Lemmas 2.3 and 2.5 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 IV.3.
43
Chapter II. Comparison Maps
2d Comparison of the coefficient groups
As before, the comparison maps gw i and w i in degrees 0 and −1 are isomorphisms on
a point. Given that we now also have maps in arbitrary degrees at our disposal, it is
natural to ask whether they, too, are isomorphisms. However, a comparison with the
situation in K-theory shows that this is not what we should expect. Indeed, it is easy to
find counterexamples:
• In degrees −3 or less, the comparison map is necessarily zero. The problem is
that while η : Wp−q(X)→Wp−q(X) is an isomorphism in all negative degrees, the
topological η is nilpotent (η3 = 0).
• In degree −2, we have KO0,−1(point) ∼= W1(point) mapping to KO0(point). The
first group is zero while the second is isomorphic to Z.
• In degree 1, it is known that KO−1,0(point) = Z/2 [Kar05, Example 18], from which
we may deduce via the Karoubi sequence that KO1,1(point) ∼= C∗. In particular,
KO1,1(point) cannot map isomorphically to KO1(point) = 0.
We now show that the map in degree 1 is always surjective, while the map in degree −2
is always injective.
2.6 Proposition. For a complex point, the comparison maps in degrees 1, 0, −1 and −2
have the properties indicated by the following arrows.
KO2q−1,q(point) KO2q−1(point)
KO2q,q(point)
∼=→ KO2q(point)
KO2q+1,q(point)
∼=→ KO2q+1(point)
KO2q+2,q(point) KO2q+2(point)
Proof. The isomorphisms in degree zero can be deduced exactly as in Proposition 1.3. To
deal with the map in degree 1, we note that the odd KO-groups of a complex point p are
all trivial except for KO−1(p), so k2q−1,qh is trivially a surjection unless q ≡ 0 mod 4. In
that case, we claim that the surjectivity of k−1,0h is clear from the following diagram:
. . . ,2 KO−1,0(p) ,2
k−1,0h

GW−1(p) ,2
∼=

K0(p) ,2
∼=

. . .
. . . ,2 KO−1(p) ,2 KO−2(p) ,2 K0(p) ,2 . . .
44
2 A homotopy-theoretic approach
Indeed, if we compute the groups appearing here, we obtain:
. . . ,2 KO−1,0(p) // //
k−1,0h

Z/2 0 ,2
∼=

Z ,2
∼=

. . .
. . . ,2 Z/2
∼= ,2 Z/2 0 ,2 Z ,2 . . .
In degree −2, three out of four cases are again trivial as KO2q+2,q(p) = Wq+2(p) is
zero unless q ≡ 2 mod 4. For the non-trivial case, consider the map η appearing in tri-
angle (10). As the negative algebraic K-groups of p are zero, η yields automorphisms of
Wp−q(p) in negative degrees. In topology, the corresponding maps are given by multipli-
cation with a generator η of KO−1(p), and η2 generates KO−2(p). So the commutative
square
W0(p)
∼= ,2
∼=

W0(p)
k0,−2h

KO−1(p)
η
∼=
,2 KO−2(p)
shows that k0,−2h is an injection (in fact, an isomorphism), as claimed.
2e Comparison with Z/2-coefficients
Another advantage of defining the comparison maps as above is that we can easily pass to
cohomology groups with torsion coefficients. Although these will not lie within our main
focus, it seems worthwhile to point out some simple observations that can be made in the
case of Z/2-coefficients.
In general, suppose E∗,∗ is some cohomology theory associated with a spectrum E in
SH(k). A version of E∗,∗ with Z/2-coefficients can be defined in terms of the spectrum
E/2 that fits into the following exact triangle:
E 2 ,2 E
t}qqq
qqq
E/2
aj
Here, 2 denotes the twofold sum of the identity map on E. The triangle defines E/2 up
to isomorphism. It follows from the definition that the associated cohomology groups
E∗,∗(X;Z/2) fit into the following long exact sequences, which we refer to as Bockstein
sequences:
· · · → Ep,q(X) ·2−→ Ep,q(X)→ Ep,q(X;Z/2)→ Ep+1,q(X) ·2−→ · · ·
In particular, we can apply this definition to obtain algebraic and hermitian K-groups
45
Chapter II. Comparison Maps
with Z/2-coefficients. Moreover, since the same definitions work in topology, we obtain
comparison maps
ki : Ki(X;Z/2)→ K−i(X;Z/2)
kp,qh : KO
p,q(X;Z/2)→ KOp(X;Z/2)
These cohomology theories and the maps between them share all the formal properties
of their integral counterparts. For example, the existence of Karoubi and Bott sequences
with Z/2-coefficients may be deduced from the 3 × 3-Lemma in triangulated categories
[May01, Lemma 2.6], and the comparison maps are again compatible with these.
The behaviour of the comparison maps for K-theory with Z/2-coefficients was pre-
dicted by the Quillen-Lichtenbaum conjectures: they are isomorphisms in all degrees
i ≥ dim(X)− 1 and injective in degree dim(X)−2. Proofs may be found in [Lev99, Corol-
lary 13.5 and Remark 13.2] and [Voe03b, Theorem 7.10]. In particular, on a complex point
we have isomorphisms for all i ≥ 0. This implies the same statement for the hermitian
comparison maps. Namely, it is not difficult to see that the comparison maps on Witt
groups with Z/2-coefficients
w i : Wi(X;Z/2)→ (KO2i/K)(X;Z/2)
are also isomorphisms when X is a point. For example, this can be deduced from Propo-
sition III.4.13 below, where we show more generally that these maps are isomorphisms
whenever they are isomorphisms on the corresponding integral groups and the odd topo-
logical K-groups of X contain no 2-torsion. Thus, the following result may be obtained
via Karoubi-induction.
2.7 Corollary. The comparison maps
kp,qh : KO
p,q(point;Z/2)→ KOp(point;Z/2)
are isomorphisms in all non-negative degrees, i. e. for all (p, q) with 2q − p ≥ 0.
46
Chapter III
Curves and Surfaces
In the previous chapter, we introduced comparison maps
gw i : GWi(X)→ KO2i(X)
w i : Wi(X)→ (KO2i/K)(X)
for any smooth complex variety X and showed that they are isomorphisms when X is a
point. Here, we analyse what happens when X has dimension one or two. Our final result,
proved in Theorems 4.4 and 4.12, is the following:
Theorem. If X is a smooth complex curve, the comparison maps gw i are surjective and
the maps w i are isomorphisms. If X is a smooth complex surface, the same claim holds
if and only if every continuous complex line bundle over X is algebraic, i. e. if and only
if the natural map Pic(X)→ H2(X;Z) is surjective.
Some more detailed statements concerning the situation for surfaces are given in Propo-
sition 4.6. For projective surfaces, the condition on line bundles is equivalent to the con-
dition that the geometric genus h2,0 of X is zero.
Our proof of this result is largely computational. Namely, we explicitly compute both
the Witt groups Wi(X) and the groups (KO2i/K)(X) of arbitrary smooth complex vari-
eties of dimension at most two, from which we see that they are abstractly isomorphic in
the cases claimed. In degree zero, we then deduce that the isomorphism is induced by the
comparison map w0 from the fact that all elements in W0(X) are detected by the first two
Stiefel-Whitney classes. The shifted groups W1(X) and W2(X) require more work. To
obtain the assertions concerning the comparison maps, we decompose an arbitrary surface
into a union of curves and an affine piece whose Picard group vanishes modulo two.
The computation of the classical Witt group W0(X) of a complex curve or surface is
not new. It was accomplished by Ferna´ndez-Carmena more than 20 years ago [FC87], and,
at its core, his calculation is very similar to the one we present. However, we have found it
convenient to rewrite the computation in terms of more general machinery developed over
the past years by Balmer, Walter and Pardon. Not only does this make more transparent
which steps become more difficult in higher dimensions, but also the values of the shifted
Witt groups Wi(X) drop out of the calculation for free. It turns out that we may in fact
work over any algebraically closed field of characteristic not two.
47
Chapter III. Curves and Surfaces
The structure of this chapter is as follows. In the first section, we recall some facts
about Stiefel-Whitney classes in algebraic geometry and topology, and explain why they
coincide for complex varieties. In Section 2, we include a computation of the Grothendieck-
Witt groups of smooth complex projective curves. The result is not needed to prove the
above comparison theorem, but it completes the picture in this dimension. The main
calculations of Witt groups and of the groups (KO2i/K)(X) are presented in Section 3,
leading to a proof of the comparison theorem in Section 4. Finally, we briefly analyse how
our result is related to the Quillen-Lichtenbaum conjecture and its analogue for hermitian
K-theory, which has recently appeared in [Sch10].
1 Stiefel-Whitney classes
As we will see, all elements in the Witt groups W0(X) of a curve or surface can be detected
by the rank homomorphism and the first two Stiefel-Whitney classes. We therefore include
a brief account of the general theory of these characteristic classes in this section. In short,
Stiefel-Whitney classes are invariants of symmetric bundles that take values in the e´tale
cohomology groups ofX with Z/2-coefficients. First, we summarize a construction of these
classes mimicking Grothendieck’s construction of Chern classes via projective bundles, and
we state their main properties. We then specialize to the case of Stiefel-Whitney classes
over fields, as studied by Milnor. The particular appeal of the first two classes stems from
the fact that they can be used to identify certain graded pieces of the Witt group of a field
with e´tale cohomology groups. The existence of similar identifications in higher degrees
was one of the claims of the now famous Milnor conjectures. However, we sketch only the
barest outlines of this story. Finally, we explain the relation of these classes over complex
varieties to the classical Stiefel-Whitney classes used in topology.
Stiefel-Whitney classes over varieties
The following construction of Stiefel-Whitney classes due to Delzant [Del62] and Laborde
[Lab76] is detailed in [EKV93, § 5]. It works for any scheme X over Z[12 ].
The first Stiefel-Whitney class w1 of a symmetric line bundle over X is defined by
the correspondence of isometry classes of such bundles with elements in H1et(X;Z/2)
(see Example I.1.1). For an arbitrary symmetric bundle (E, ε), one considers the scheme
Pnd(E, ε) given by the complement of the quadric in P(E) defined by ε. The restriction of
the universal line bundle O(−1) on P(E) to Pnd(E, ε) carries a canonical symmetric form.
Let w be its first Stiefel-Whitney class in H1et(Pnd(E, ε);Z/2). Then the cohomology of
Pnd(E, ε) can be decomposed as
H∗et(Pnd(E, ε);Z/2) =
r−1⊕
i=0
p∗H∗et(X;Z/2) · wi (1)
48
1 Stiefel-Whitney classes
where p is the projection of Pnd(E, ε) onto X. In particular, wr is a linear combination of
smaller powers of w, so that for certain coefficients
wi(E, ε) ∈ H i(X;Z/2)
we have
wr + p∗w1(E, ε) · wr−1 + p∗w2(E, ε) · wr−2 + · · ·+ p∗wr(E, ε) = 0
These coefficients wi(E, ε) are defined to be the Stiefel-Whitney classes of (E, ε). They are
characterized by the following axiomatic description [EKV93, § 1]:
Normalization. The first Stiefel-Whitney class of a symmetric line bundle w1 is as
defined above.
Boundedness. wi(E, ε) = 0 for all i > rk(E)
Naturality. For any morphism f : Y → X, we have f∗wi(E, ε) = wi(f∗(E, ε)).
Whitney sum
formula.
The total Stiefel-Whitney class wt := 1 +
∑
i≥1wi(E, ε)t
i satisfies
wt((E, ε) ⊥ (F, ϕ)) = wt(E, ε) · wt(F, ϕ)
The Stiefel-Whitney classes of a metabolic bundle only depend on the Chern classes
of its Lagrangian. More precisely, Proposition 5.5 in [EKV93] gives the following formula
for a metabolic bundle (M, µ) with Lagrangian L:
wt(M, µ) =
rk(L)∑
j=0
(1 + (−1)t)n−jcj(L)t2j (2)
For example, for the first two Stiefel-Whitney classes we have
w1(M, µ) = rk(L)(−1) (3)
w2(M, µ) =
(
rk(L)
2
)
(−1,−1) + c1(L) (4)
It follows that wt descends to a well-defined homomorphism from the Grothendieck-Witt
group of X to the multiplicative group of invertible elements in
⊕
iH
i
et(X;Z/2)ti:
wt : GW0(X) −→
(⊕
i
H iet(X;Z/2)ti
)×
In particular, the individual classes descend to well-defined maps
wi : GW0(X)→ H iet(X;Z/2)
49
Chapter III. Curves and Surfaces
In general, none of the individual Stiefel-Whitney classes apart from w1 define homo-
morphisms on GW0(X). It does follow from the Whitney sum formula, however, that
w2 restricts to a homomorphism on the kernel of w1, and in general wi restricts to a
homomorphism on the kernel of (the restriction of) wi−1.
It is not generally true either that the Stiefel-Whitney classes factor through the Witt
group of X: the right-hand side of (2) may be non-zero. However, we may deduce from
(3) that the restriction of the first Stiefel-Whitney class to the reduced group G˜W0(X)
factors through W˜0(X), yielding a map
w1 : W˜0(X) −→ H1et(X;Z/2)
We use a different notation to emphasize that the values of w1 and w1 on a given symmetric
bundle may differ. That is, if (E, ε) is a symmetric bundle of even rank defining an element
Ψ in W˜0(X), then in general w1(Ψ) 6= w1(E, ε). Rather, w1 needs to be computed on a lift
of (E, ε) to G˜W0(X). Explicitly, if H denotes the constant hyperbolic bundle (O⊕O, ( 0 11 0 )),
then
w1(Ψ) = w1
(
(E, ε)− rk(E)2 H
)
= w1(E, ε)− rk(E)2 (−1)
We see similarly from (4) that w2 induces a well-defined map
w2 : W˜0(X) −→ H2et(X;Z/2)
/
Pic(X)
As before, w1 is a surjective homomorphism, while w2 restricts to a homomorphism on
ker(w1).
Stiefel-Whitney classes over fields
When X is a field F (of characteristic not 2), the Stiefel-Whitney classes factor through
Milnor’s K-groups modulo two, which are commonly denoted kMi (F ). We will denote the
classes with values in kMi (F ) by w
M
i :
GW0(F )
wMi  )K
KKK
KKK
KKK
wi ,2 H iet(F ;Z/2)
kMi (F )
α
4=qqqqqqqqqq
Both the groups kMi (F ) and the classes w
M
i were constructed in [Mil69]. In the same paper,
Milnor asked whether the map α appearing in the factorization was an isomorphism, a
question that later became known as one of the Milnor conjectures. For i ≤ 2, which
50
1 Stiefel-Whitney classes
will be the range mainly relevant for us, an affirmative answer was given by Merkurjev
[Mer81]. A general affirmation of the conjecture was found more recently by Voevodsky
[Voe03b].
A second conjecture of Milnor, also to be found in [Mil69], concerned the relation of
W0(F ) to kMi (F ). To state it, we introduce the fundamental filtration. If we view W
0(F )
as a ring, then the reduced Witt group W˜0(F ) becomes an ideal inside W0(F ), which is
traditionally written as I(F ). The powers of this ideal yield a filtration
W0(F ) ⊃ I(F ) ⊃ I2(F ) ⊃ I3(F ) ⊃ . . .
on the Witt ring of F , known as the fundamental filtration. Milnor conjectured that the
associated graded ring was isomorphic to kM∗ (F ) := ⊕ikMi (F ). As a first step towards
a proof, he constructed maps kMi (F ) → Ii(F )/Ii+1(F ) in one direction. Moreover, in
degrees one and two, Milnor could show that these are isomorphisms, with explicit inverses
induced by the Stiefel-Whitney classes wM1 and w
M
2 . In combination with the isomorphisms
α above, one obtains the following identifications:
rk : W
0(F )
/
I(F )
∼=−→ H0et(F ;Z/2)
w1 : I(F )
/
I2(F )
∼=−→ H1et(F ;Z/2)
w2 : I
2(F )
/
I3(F )
∼=−→ H2et(F ;Z/2)
[Mil69, § 4]. Today, these isomorphisms are commonly denoted e0, e1, e2. It was clear
from the outset, however, that the higher isomorphisms
ei : I
i(F )
/
Ii+1(F )
∼=−→ H iet(F ;Z/2) (5)
conjectured by Milnor could not be induced by higher Stiefel-Whitney classes. Their
existence was ultimately proved in [OVV07]. Unlike in the case of Stiefel-Whitney classes,
it does not seem to be clear how these maps may be generalized to varieties.
Stiefel-Whitney classes over complex varieties
Here, we show that over a complex variety the e´tale Stiefel-Whitney classes of symmetric
vector bundles are compatible with the Stiefel-Whitney classes of real vector bundles used
in topology.
Suppose first that Y is an arbitrary CW complex. If we follow the construction of
Stiefel-Whitney classes described above, with singular cohomology in place of e´tale coho-
mology, we obtain classes wi(E, ε) in H i(Y ;Z/2) for every complex symmetric bundle
(E, ε) over Y . On the other hand, given a real vector bundle F over Y , we have the usual
Stiefel-Whitney classes wi(F). We claim that these classes are compatible in the following
51
Chapter III. Curves and Surfaces
sense. Recall from Corollary I.2.4 that we have a one-to-one correspondence
< :
 isometry classes ofcomplex symmetric
bundles over Y
→
 isomorphism classesof real vector bundles
over Y

1.1 Lemma. For any complex symmetric vector bundle (E, ε) over a CW complex Y , the
classes wi(E, ε) and wi(<(E, ε)) in H i(Y ;Z/2) agree.
Proof. For a complex symmetric line bundle (L, λ) over Y , the first Stiefel-Whitney classes
of (L, λ) and <(L, λ) in H1(Y ;Z/2) agree because both bundles have the same associated
principal Z/2-bundle. In general, the Stiefel-Whitney classes of a real vector bundle F
may be defined in the same way as the classes wi(E, ε), using its real projectivization
RP(F) in place of the space Pnd(E, ε) [Hus94, Theorem 2.5 and Definition 2.6]. Thus, the
lemma follows from the considerations of Section I.2a. Indeed, the spaces Pnd(E, ε) are
instances of the non-degenerate Grassmannians introduced there. In particular, for any
complex symmetric bundle (E, ε), the space Pnd(E, ε) contains RP(<(E, ε)) as a homotopy
equivalent subspace. Let us denote the inclusion and the bundle projections as in the
following diagram:
RP(<(E, ε))
p
%% %%K
KK
KK
KK
KK
K
  j ,2 Pnd(E, ε)
pnd
zzzzvv
vv
vv
vv
v
Y
The universal symmetric bundle O(−1)nd over Pnd(E, ε) restricts to the universal real
bundle O(−1) over RP(<(E, ε)) in the sense that <(j∗O(−1)nd) = O(−1). In particular,
by the naturality of Stiefel-Whitney classes and the above observation concerning line
bundles, we have
j∗w1(O(−1)nd) = w1(O(−1))
Let r := rk(E). Then the Stiefel-Whitney classes of (E, ε) are defined by the following
relations in the cohomology groups Hr(Pnd(E, ε);Z/2) and Hr(RP(<(E, ε);Z/2), respec-
tively:
w1(O(−1)nd)r =
r∑
i=1
w1(O(−1)nd)r−ip∗ndwi(E, ε)
w1(O(−1))r =
r∑
i=1
w1(O(−1))r−ip∗wi(<(E, ε))
The claim follows by applying j∗ to the first line and comparing it with the second.
Now let X be a complex variety. Then the above lemma implies that the e´tale and the
topological Stiefel-Whitney classes are compatible. Moreover, if we specialize equation (2)
for the Stiefel-Whitney classes of a metabolic bundle (M, µ) with Lagrangian L to the case
52
2 Curves
of a complex variety, then since −1 is a square in C we find that
w2i(M, µ) = ci(L) ∈ H2iet (X;Z/2)
and all odd Stiefel-Whitney classes of (M, µ) vanish. This corresponds to the well-known
fact in topology that the even Stiefel-Whitney classes of a complex vector bundle agree
with its Chern classes modulo two, whereas its odd Stiefel-Whitney classes are zero [MS74,
Problem 14.B]. It follows in particular that the odd Stiefel-Whitney classes factor through
the (reduced) Witt group of X, while the even classes induce maps
w2i : W˜0(X) −→ H
2i
et (X;Z/2)
im(ci)
We summarize the situation as follows.
1.2 Proposition. Let X be a complex variety. The Stiefel-Whitney classes factor through
the reduced Grothendieck-Witt and KO-group of X to yield commutative diagrams
G˜W0(X)

wi ,2 H iet(X;Z/2)
∼=

K˜O0(X)
wi ,2 H i(X;Z/2)
Moreover, for all odd i we have induced maps
W˜0(X)

wi ,2 H iet(X;Z/2)
∼=

(K˜O0/K˜)(X)
wi ,2 H i(X;Z/2)
and for even i we have induced maps
W˜0(X)

wi ,2
H iet(X;Z/2)
im(ci)

(K˜O0/K˜)(X)
wi ,2
H i(X;Z/2)
im(ci)
2 Curves
In the next section, we will compute the Witt groups and the groups (KO2i/K)(X) for
smooth varieties of dimension at most two. Here, we briefly summarize the results we
will obtain in the case of curves. Moreover, we give a complete description of their
53
Chapter III. Curves and Surfaces
Grothendieck-Witt groups. This description will not be used in the proofs of the main
comparison theorems of this chapter but is included merely for the purposes of illustration
and completeness.
2a Grothendieck-Witt groups of curves
Let C be a smooth curve over an algebraically closed field k of characteristic not 2, and
let Pic(C) be its Picard group.
If C is projective, say of genus g, we may write Pic(C) as Z ⊕ Jac(C). The free
summand Z is generated by a line bundle O(p) associated with a point p on C, while
Jac(C) denotes (the closed points of) the Jacobian of C, a g-dimensional abelian variety
parametrizing line bundles of degree zero over C. As a group, Jac(C) is two-divisible, and
Jac(C)[2] has rank 2g (e. g. [Mil08, Chapter 14]). In particular, when C is projective, we
have
H1et(C;Z/2) = Pic(C)[2] = (Z/2)2g
H2et(C;Z/2) = Pic(C)/2 = Z/2
If C is not projective, it is affine and may be obtained from a smooth projective curve
by removing a finite number of points. Note that when we remove a single point p from
a projective curve C, the Picard group is reduced to Pic(C − p) = Jac(C). It follows that
the Picard group of any affine curve is two-divisible. In particular, for any affine curve C
we have
H2et(C;Z/2) = Pic(C)/2 = 0
The reduced algebraic K-group of a smooth curve may be identified with its Picard
group via the first Chern class, so that we have an isomorphism
K0(C) ∼= Z⊕ Pic(C) (6)
The following proposition shows that, similarly, the Grothendieck-Witt and Witt groups
of C are completely determined by Pic(C) and the group H1et(C;Z/2) of symmetric line
bundles.
2.1 Proposition. Let C be a smooth curve over an algebraically closed field of character-
istic not 2. The Grothendieck-Witt and Witt groups of C are as follows:
GW0(C) = [Z]⊕H1et(C;Z/2)⊕H2et(C;Z/2) W0(C) = [Z/2]⊕H1et(C;Z/2)
GW1(C) = Pic(C) W1(C) = H2et(C;Z/2)
GW2(C) = [Z] W2(C) = 0
GW3(C) = [Z/2]⊕ Pic(C) W3(C) = 0
54
2 Curves
Here, the summands in square brackets are the trivial ones coming from a point, i. e. those
that disappear when passing to reduced groups. In particular, for a projective curve of genus
g we obtain:
GW0(C) = [Z]⊕ (Z/2)2g+1 W0(C) = [Z/2]⊕ (Z/2)2g
GW1(C) = Z⊕ Jac(C) W1(C) = Z/2
GW2(C) = [Z] W2(C) = 0
GW3(C) = [Z/2]⊕ Z⊕ Jac(C) W3(C) = 0
For affine curves, no non-trivial twists are possible. When C is projective, the groups
twisted by a generator O(p) of Pic(C)/2 are as follows:
GW0(C,O(p)) = Z⊕ (Z/2)2g W0(C,O(p)) = (Z/2)2g
GW1(C,O(p)) = Z⊕ Jac(C) W1(C,O(p)) = 0
GW2(C,O(p)) = Z W2(C,O(p)) = 0
GW3(C,O(p)) = Z⊕ Jac(C) W3(C,O(p)) = 0
2.2 Remark (Explicit generators). The isomorphism between W˜0(C) andH1et(X;Z/2)
is the obvious one, i. e. W˜0(C) is generated by symmetric line bundles (see Example I.1.1).
When C is projective of genus g, the group W1(C) has a single generator Ψg, which for
g = 0 we may take to be the 1-symmetric complex given in Example I.1.6. When g ≥ 1,
a generator may be constructed as follows: Pick two distinct points p and q on C. Let s
and t be sections O→ O(p) and O→ O(q) of the associated line bundles that vanish at p
and q, respectively. Choose a line bundle L ∈ Pic(C) that squares to O(p− q). Then
Ψg :=

L⊗ O(−p) s ,2
t

L
−t

L⊗ O(q − p) −s ,2 L⊗ O(q)
 (7)
is a 1-symmetric complex over C generating W1(C).
Proof of Proposition 2.1, assuming Proposition 3.1:
The values of the untwisted Witt groups may be read off directly from Proposition 3.1
below: we simply note that c1 : Pic(C) → H2et(C;Z/2) is surjective and induces an iso-
morphism Pic(C)/2 ∼= H2et(C;Z/2). The twisted groups Wi(C,O(p)) of a projective curve
C may then be calculated from the localization sequence (I.5) associated with the inclusion
of the complement of a point p of C into C. Indeed, since the line bundle O(p) is trivial
over C − p, this sequences takes the following form:
· · · →Wi−1(p)→Wi(C,O(p))→Wi(C − p)→Wi(p)→ · · ·
55
Chapter III. Curves and Surfaces
We now deduce the values of the Grothendieck-Witt groups, concentrating on the case
when C is projective. The affine case is very similar, except that all extension problems
disappear.
Untwisted case. To compute the untwisted Grothendieck-Witt groups of a projective
curve C, we use the Karoubi sequences (I.4) and their restrictions to reduced groups:
G˜Wi−1(C) F→ K˜0(C) H
i→ G˜Wi(C) W˜i(C)→ 0
Note first that the identification of the K-group of our projective curve C given in (6) may
be written explicitly as
K0(C)
∼=←→ Z⊕ Z⊕ Jac(C)
[E] 7→ (rkE, d,detE(−d))
[L(d)] + (r − 1)[O]←[ (r, d,L)
Here, in the map from left to right, we have written d for the degree of the determinant
line bundle of E. Given this description, the endomorphisms FH i of K0(C) may easily be
computed directly. For the restrictions to the reduced K-group we obtain:
im(FH i) =
0 ⊂ K˜0(C) when i is even2Z⊕ Jac(C) ⊂ K˜0(C) when i is odd (8)
To compute G˜W0(C), we first observe that W3(C) = 0 implies that K˜0(C) surjects
onto G˜W3(C). Therefore, the image of G˜W3(C) in K˜0(C) coincides with the image of
FH3, and by (8) we obtain a short exact sequence
0→ Z/2 −→ G˜W0(C) −→ W˜0(C)
=Jac(C)[2]
→ 0
This sequence splits. To define a splitting of the surjection, pick line bundles L of order 2
defining additive generators eL := (L, id)−(O, id) of W˜0(C). We claim that the homomor-
phism obtained by lifting these to the corresponding elements in GW0(C) is well-defined,
i. e. that 2eL vanishes in GW0(C). Indeed, by Lemma I.1.9 we have 2eL = H0(L − O),
and H0 vanishes on L− O since by (8) the latter is contained in F (G˜W3(X)).
Next, the image of G˜W0(C) in K˜0(C) is Jac(C)[2]. Since the quotient Jac(C)/ Jac(C)[2]
is isomorphic to Jac(C), we obtain a short exact sequence
0→ Z⊕ Jac(C) H1−→ G˜W1(C) −→ W˜1(C)
= Z/2
→ 0
Let Ψg be the 1-symmetric complex given by (I.3) or (7). As an element of K˜0(C) =
56
2 Curves
Z ⊕ Jac(C), its underlying complex is equivalent to (1,O). Thus, by Lemma I.1.9 again,
we have H1(1,O) = H1(F (Ψg)) = 2Ψg. It follows that the sequence does not split and
that Ψg descends to a generator of W˜1(C). Thus, G˜W1(C) = Z⊕ Jac(C).
Carrying on to compute G˜W2(C), we first note that by (8) the image of G˜W1(C) in
K˜0(C) contains 2Z ⊕ Jac(C). Moreover, F (Ψ) = ψ = (1,O), so in fact G˜W1(C) surjects
onto K˜0(C). It follows that G˜W2(C) = 0. Finally, since W˜3(C) also vanishes, we see that
H3 gives an isomorphism between K˜0(C) and G˜W3(C). This completes the computations
in the untwisted case.
Twisted case. To compute the Grothendieck-Witt groups of C twisted by a generator
O(p) of Pic(C)/2 it again seems easiest to compare them with those of the affine curve
C−p, over which O(p) is trivial. More specifically, we will compare the respective Karoubi
sequences via the following commutative diagram:
GWi−1(C − p) F ,2 K0(C − p) H
i
,2 GWi(C − p) ,2Wi(C − p) ,2 0
GWi−1(C,O(p)) F ,2
LR
K0(C)
Hi
O(p)
,2
LR
GWi(C,O(p)) ,2
LR
Wi(C,O(p)) ,2
LR
0
(9)
The restriction map K0(C)→ K0(C−p) is the projection Z⊕Z⊕Jac(C) Z⊕0⊕Jac(C)
killing the free summand corresponding to line bundles of non-zero degrees. The twisted
version of (8) reads as follows:
im(FH iO(p)) =
Z · (2, 1)⊕ 0 ⊂ K0(C) when i is even0⊕ Z⊕ Jac(C) ⊂ K0(C) when i is odd (10)
Now consider (9) with i = 0. Arguing as in the untwisted case, we may compute the
cokernels of the forgetful maps F to reduce the diagram to a comparison of two short
exact sequences:
0 ,2 Z
( 20 ) ,2 Z⊕ Jac(C)[2] ,2 (Z/2)⊕ Jac(C)[2] ,2 0
0 ,2 Z ,2 GW0(C,O(p)) ,2
LR
Jac(C)[2] ,2
( 01 )
LR
0
Since the outer vertical maps are injective, so is the central vertical map. It follows from a
diagram chase that the image of a generator of K0(C) in GW0(C,O(p)) cannot be divisible
by 2. Thus, the lower sequence must split, yielding GW0(C,O(p)) = Z⊕ Jac(C)[2].
Next, we see by (10) that the free summand in GW0(C,O(p)) has image Z ·(2, 1)⊕0 in
K0(C). On Jac(C)[2], on the other hand, F must be the inclusion Jac(C)[2] ↪→ Jac(C), by
comparison with the case of C−p. This gives GW1(C,O(p)) = Z⊕ (Jac(C)/ Jac(C)[2]) ∼=
Z⊕ Jac(C).
57
Chapter III. Curves and Surfaces
The image of GW1(C,O(p)) in K0(C) may be read off directly from (10), and we may
deduce that GW2(C,O(p)) is a free abelian group generated by the twisted hyperbolic
bundle H2O(p)(O). Equally directly, we see that GW
3(C,O(p)) = Z ⊕ Jac(C). This com-
pletes the computation of all Grothendieck-Witt groups of C.
2b KO-groups of curves
Now suppose that C is a smooth curve defined over C. Its topological KO-groups may be
computed from the Atiyah-Hirzebruch spectral sequence.
2.3 Proposition. Let C be a smooth projective complex curve of genus g as above. Then
its KO-groups are given by
KO0(C) = [Z]⊕ (Z/2)2g+1 KO−1(C) = [Z/2]⊕ (Z/2)2g
KO2(C) = Z KO1(C) = Z2g ⊕ Z/2
KO4(C) = [Z] KO3(C) = 0
KO6(C) = [Z/2]⊕ Z KO5(C) = Z2g
Here, the square brackets again indicate which summands vanish when passing to reduced
groups.
In Section 3b, we will explain in detail how the values of the groups (KO2i/K)(C)
may also be obtained from the spectral sequence. They may be read off as special cases
from Proposition 3.8. In fact, we can also read off the values of the twisted groups
(KO2i/K)(C;O(p)) by applying the same proposition to the Thom space ThomC(O(p))
(see Definition I.2.9). We then find that these groups agree with the corresponding Witt
groups in Proposition 2.1:
(KO0/K)(C) = [Z/2]⊕ (Z/2)2g (KO0/K)(C,O(p)) = (Z/2)2g
(KO2/K)(C) = Z/2 (KO2/K)(C,O(p)) = 0
(KO4/K)(C) = 0 (KO4/K)(C,O(p)) = 0
(KO6/K)(C) = 0 (KO6/K)(C,O(p)) = 0
3 Surfaces
This section contains the main calculations of this chapter: the computation of the Witt
groups of smooth curves and surfaces on the one hand, and the computation of the cor-
58
3 Surfaces
responding topological groups (KO2i/K) on the other. The comparison of the results will
be postponed until the next section.
3a Witt groups of surfaces
Consider the algebraic K-group K0(X) of a smooth variety X, and let ci be the Chern
classes ci : K0(X) → CHi(X) with values in the Chow groups of X. Filter K0(X) by
K0(X) ⊃ K˜0(X) ⊃ ker(c1). If X has dimension at most two, then the map (rk, c1, c2)
induces an isomorphism [Ful98, Example 15.3.6]:
gr∗(K0(X)) ∼= Z⊕ CH1(X)⊕ CH2(X) (11)
A similar statement holds for the Witt group.
3.1 Proposition. Let k be an algebraically closed field of characteristic char(k) 6= 2, and
let X be a smooth variety over k of dimension at most two. Filter W0(X) by W0(X) ⊃
W˜0(X) ⊃ ker(w1). Then the map (rk, w1, w2) gives an isomorphism
gr∗(W0(X)) ∼= Z/2⊕H1et(X;Z/2)⊕
(
H2et(X;Z/2)
/
Pic(X)
)
Moreover, if we write S1 for the squaring map CH1(X)/2→ CH2(X)/2, then the shifted
Witt groups are as follows:
W1(X) ∼= ker(S1)⊕H3et(X;Z/2)
W2(X) ∼= coker(S1)
W3(X) = 0
3.2 Example. Suppose X is a smooth complex projective surface of Picard number ρ.
Write bi for the Betti numbers of X and ν for the rank of H2(X;Z)[2]. Then the above
result shows that the Witt groups of X are as follows:
W0(X) = [Z/2]⊕ (Z/2)b1+b2−ρ+2ν
W1(X) =
(Z/2)b1+ρ+2ν if S1 = 0(Z/2)b1+ρ+2ν−1 if S1 6= 0
W2(X) =
Z/2 if S1 = 00 if S1 6= 0
W3(X) = 0
Our computation will follow a route described in [Tot03]. Namely, there is a sub-
tle relationship between Witt groups and e´tale cohomology groups with Z/2-coefficients,
59
Chapter III. Curves and Surfaces
encoded by three spectral sequences:
Es,t2,BO(X) = H
s(X,Ht) ⇒ Hs+tet (X;Z/2) (BO)
Es,t2,Par(X) = H
s(X,Ht) ⇒ Hs(X,W) (P)
Es,t2,GW(X) = H
s(X,Wt) ⇒ Ws+t(X) (GW)
Here, H is the Zariski sheaf attached to the presheaf that sends an open subset U ⊂ X
to Hjet(U ;Z/2). Similarly, W denotes the Zariski sheaf attached to the presheaf sending
U to W0(U), and we set
Wt =
W for t ≡ 0 mod 40 otherwise
Before specializing to the case of a surface, we give a short discussion of each of these
spectral sequences for an arbitrary dimensional variety X. For simplicity, we assume
that X is a smooth variety over a field of characteristic not 2 throughout. More general
statements may be found in the references.
The Bloch-Ogus spectral sequence (BO). The first spectral sequence is the well-
known Bloch-Ogus spectral sequence [BO74]. This is a first quadrant spectral sequence
with differentials in the usual directions, i. e. of bidegree (r,−r + 1) on the rth page. The
E2-page is concentrated above the main diagonal s = t, and the groups along the diagonal
may be identified with the Chow groups of X modulo two [BO74, Corollary 6.2 and proof
of Theorem 7.7]:
Hs(X,Hs) ∼= CHs(X)/2 (12)
When the ground field is algebraically closed, the sheaves Ht vanish for all t > dim(X),
so that the spectral sequence is concentrated in rows 0 ≤ t ≤ dim(X).
· · · · · · · · · · · ·
H0(X,H3)
/6fffffffff
mmm
H1(X,H3)
/6fffffffff
mmm
H2(X,H3)
/6fffffffff
mmm
CH3(X)/2
gggg
mmm
H0(X,H2)
07gggggggg
3;nnnnnnn
H1(X,H2)
07gggggggg
3;nnnnnnn
CH2(X)/2
07ggggggggg
3;nnnnnnn
·
H0(X,H1)
07gggggggg
3;ooooooo
6@
CH1(X)/2
07ggggggggg
3;ooooooo
6@
· ·
CH0(X)/2
d2ggggg
07ggg
d3ooo
3;ooood4
7A
· · ·
s
,2
t LR
Figure 1: Pardon’s spectral sequence. The entries below the diagonal are zero.
60
3 Surfaces
Pardon’s spectral sequence (P). Pardon’s spectral sequence can be indexed to have
the same E2-page as the Bloch-Ogus spectral sequence. The differentials on the rth page
then have bidegree (1, r − 1), as illustrated in Figure 1. In [Tot03], Totaro shows that,
under the identifications (12), the differential d2 on the main diagonal of the E2-page cor-
responds to the Steenrod operation S1 : CHs(X)/2→ CHs+1(X)/2, as defined by Brosnan
and Voevodsky [Bro03,Voe03a].
The sequence converges to the cohomology of X with respect to W in the usual sense
that the ith column of the E∞-page is isomorphic to the associated graded module of
H i(X,W) with respect to some filtration. In order to describe the filtration onH0(X,W) =
W(X), we briefly summarize how the spectral sequence arises.
Let X be as above, and let K be its function field. We denote the residue field of a
scheme-theoretic point x of X by k(x). In [Par04], Pardon shows that the sheaf W has a
flasque resolution by a Gersten-Witt complex W which on open subsets U ⊂ X takes the
form
W(U) : W0(K)→
∐
x∈U(1)
W0(k(x))→ · · · →
∐
x∈U(n)
W0(k(x))→ 0 (13)
Here, U (i) denotes the set of codimension i points of X contained in U , and W0(k(x)) is
to be viewed as a constant sheaf supported on the closure of x in X. In particular, W is
a subsheaf of the constant sheaf W0(K). Let I(K) ⊂W0(K) be the fundamental ideal of
K (see Section 1). The fundamental filtration on W0(K), given by the powers of I(K),
induces a filtration on W which we denote by
It :=W ∩ It(K) (14)
Pardon shows more generally that the fundamental filtrations of the Witt groups W0(k(x))
give rise to flasque resolutions of the sheaves It of the form
It(U) : It(K)→
∐
x∈U(1)
It−1(k(x))→
∐
x∈U(2)
It−2(k(x))→ . . .
Applying the standard construction of the spectral sequence of a filtered complex of abelian
groups [GM03, Chapter III.7, Section 5] to the filtration of W(X) by It(X), we obtain a
spectral sequence
Hs(X, It/It+1)⇒ Hs(X,W)
To conclude, Pardon uses the affirmation of the Milnor conjectures (see (5)) to identify
the sheaves It/It+1 with the sheaves Ht. The following lemma is a direct consequence of
this construction.
61
Chapter III. Curves and Surfaces
3.3 Lemma. The filtration on W(X) appearing on the zeroth column of the E∞-page of
Pardon’s spectral sequence is given by the global sections of the sheaves It defined in (14):
E0,t∞,Par(X) ∼= It(X)/It+1(X)
The edge homomorphism including this column into the zeroth column of the E2-page is
induced by the sheafification map from the presheaf quotient of It by It+1 to the quotient
sheaf It/It+1.
The group W(X), viewed as a subgroup of W0(K), is commonly referred to as the
unramified Witt group of X.
The Gersten-Witt spectral sequence (GW). The Gersten-Witt spectral sequence
of Balmer and Walter, third on the list above, may be thought of as an analogue of
the Bloch-Ogus spectral sequence for Witt groups. In [BW02], Balmer and Walter, too,
construct a Gersten-Witt complex of the form (13). Although it seems likely that their
construction agrees with Pardon’s, this does not yet seem to have been established. It is
known in any case, however, that their complex also provides a flasque resolution of W
[BGPW02, Lemma 4.2 and Theorem 6.1]. The spectral sequence constructed in [BW02]
may therefore be rewritten in the form above:
3.4 Theorem. [BW02] Let X be a smooth variety over a field of characteristic not 2.
There is a spectral sequence with E2-page
Es,t2 =
Hs(W) in rows t ≡ 0 mod 40 elsewhere
The sequence has differentials dr of bidegree (r,−r + 1) and converges to Ws+t(X).
Proof of Proposition 3.1:
We now specialize to the case when dim(X) ≤ 2 and the ground field is algebraically
closed. Then the Bloch-Ogus spectral sequence (BO) collapses immediately and we obtain
descriptions of the groups Hs(X;Ht) in terms of the e´tale cohomology of X with Z/2-
coefficients.
The E2-page of Pardon’s spectral sequence therefore takes the following form:
H2et(X;Z/2)
/
Pic(X) H
3
et(X;Z/2) H4et(X;Z/2)∼= CH2(X)/2
H1et(X;Z/2)
d2gggg
07gggg
Pic(X)/2
S1hhhh
07hhh
·
Z/2 · ·
s
,2
t LR
62
3 Surfaces
On the other hand, we see that the Gersten-Witt spectral sequence also collapses, so that
Wi(X) ∼= H i(X,W). Thus, the ith column of Pardon’s spectral sequence simply converges
to Wi(X). In particular, if we write It(X) for the filtration on W0(X) corresponding to the
filtration It(X) ofW(X) defined in (14), then by Lemma 3.3 we have E
0,t
∞,Par ∼= grIt W0(X).
3.5 Lemma. Let X be as above. Then, for t = 1 or 2, the edge homomorphisms
grIt W
0(X) ∼= E0,t∞,Par ↪→ E0,t2,Par
can be identified with the Stiefel-Whitney classes wt. More precisely, I1(X) = W˜0(X),
I2(X) = ker(w1), and we have commutative diagrams
E0,1∞,Par
  ,2 E0,12,Par
W˜0(X)
/
ker(w1)
w1 ,2 H1et(X;Z/2)
E0,2∞,Par
  ,2 E0,22,Par
ker(w1)
w2 ,2 H2et(X;Z/2)
/
Pic(X)
Proof. The statement of the lemma is not surprising: the only non-obvious maps that enter
into the construction of Pardon’s spectral sequence are the isomorphisms et displayed in
(5); for t = 1 or 2, these can be identified with Stiefel-Whitney classes, and the latter can
be defined globally.
In more detail, the various identifications arising from the Bloch-Ogus spectral sequence
and Pardon’s spectral sequence fit into the following diagram:
E0,t∞,Par
  ,2 E0,t2,Par
It(X)
/
It+1(X)
,2
?

(
It
/
It+1
)
(X) 

,2
∼=

It(K)
/
It+1(K)
∼=et

Htet(X;Z/2)
sheafification ,2

Ht(X) 

,2 Htet(K;Z/2)
E0,t∞,BO
∼=
(BO collapses)
,2 E0,t2,BO
The claim is that, for t = 1 or 2, the diagram commutes if we take the dotted map to be
the tth Stiefel-Whitney class. Indeed, et can be identified with wt in these cases, and the
horizontal compositions across the two middle lines in the diagram are given by pullback
along the inclusion of the generic point into X.
63
Chapter III. Curves and Surfaces
There are two potentially non-zero differentials in Pardon’s spectral sequence: the
differential S1 : CH1(X)/2 → CH2(X)/2, which is simply the squaring operation, and
another differential d from H1et(X;Z/2) to H3et(X;Z/2). Since the first Stiefel-Whitney
class always surjects onto H1et(X;Z/2), we see from the previous lemma that H1et(X;Z/2)
survives to the E∞-page. So the differential d must vanish. On the other hand, since
E0,22 = E
0,2∞ , the lemma implies that the restriction of the second Stiefel-Whitney class
to the kernel of the first gives an isomorphism with H2et(X;Z/2)/Pic(X). This proves
the first statement of the proposition. Since S1 is the only non-trivial differential left,
the values of the shifted Witt groups also follow easily from the spectral sequence. This
completes the proof of Proposition 3.1.
In the following corollary, the Chern and Stiefel-Whitney classes refer to the corresponding
maps on reduced groups:
ci : K˜0(X) −→ CHi(X)
wi : G˜W0(X) −→ H iet(X;Z/2)
3.6 Corollary. Let X be as above. Then the second Stiefel-Whitney class is surjective
and restricts to an epimorphism
w2 : ker(w1) − H2et(X;Z/2)
Its kernel is given by the image of ker(c1) under the hyperbolic map H0 : K˜0(X)→ G˜W0(X).
Proof. Note first that H0 : K˜0(X)→ G˜W0(X) factors through the kernel of w1. In fact,
we can restrict the Karoubi sequence to an exact sequence forming the first row of the
following diagram.
K˜0(X)
H0 ,2
c1

ker(w1) // //
w2

ker(w1) ,2
w2∼=

0
Pic(X) ,2 H2et(X;Z/2) // // H
2
et(X;Z/2)
/
Pic(X) ,2 0
The lower row is obtained from the Kummer sequence, hence also exact, and the diagram
commutes. We know from the previous proposition that the map w2 on the right is an
isomorphism. Since c1 is surjective, surjectivity of w2 follows.
It remains to show that the restriction H0 : ker(c1)→ ker(w2) is surjective. By an
instance of the Snake Lemma, this is equivalent to showing that the kernel of H0 on K˜0(X)
maps surjectively to Pic(X)2 under c1. The kernel of H0 is given by the image of G˜W3(C)
under the forgetful map F , which coincides with the image of K˜0(X) under the composition
FH3. Thus, it suffices to show that c1 restricts to a surjection c1 : FH3(K0(X))→ Pic(X)2.
This is done in the next lemma.
64
3 Surfaces
3.7 Lemma. Let X be as above. Then for odd i we have commutative diagrams
K0(X)
c1

Hi ,2 GWi(X) F ,2 K0(X)
c1

Pic(X) ,2 Pic(X)
L 7→ L2
For even i, the diagram commutes if we replace the map on Pic(X) by the constant map.
Proof. This can be checked via a direct calculation. The first Chern class c1 takes a vector
bundle E to its determinant line bundle detE. Under FH i, the class of E is mapped to
the class E⊕ (−1)iE∨ in K0(X), which c1 then takes to (detE)⊗(1−(−1)i).
Remark. More generally, the endomorphism of gr∗(K0(X)) ∼= Z ⊕ Pic(X) ⊕ CH2(X)
induced by FH i is given by (2, 0, 2) if i is even, and by (0, 2, 0) if i is odd.
3b KO/K-groups of surfaces
On a topological space X, let us write Sq2Z for the composition
H2(X;Z)/2 ↪→ H2(X;Z/2) Sq
2
−→ H4(X;Z/2)
where Sq2 is the squaring operation.
3.8 Proposition. Let X be a connected CW complex of dimension at most four. Fil-
ter the group (KO0/K)(X) by (KO0/K)(X) ⊃ (K˜O0/K˜)(X) ⊃ ker(w1). Then the map
(rk, w1, w2) induces an isomorphism
gr
(
(KO0/K)(X)
) ∼= Z/2⊕H1(X;Z/2)⊕ (H2(X;Z/2)/H2(X;Z))
The remaining groups (KO2i/K)(X) have the following values:
(KO2/K)(X) ∼= ker(Sq2Z)⊕H3(X;Z/2)
(KO4/K)(X) ∼= coker(Sq2Z)
(KO6/K)(X) = 0
3.9 Example. Suppose X is a compact four-dimensional manifold. Write bi for its Betti
numbers and ν for the rank of H2(X;Z)[2]. Then the above result shows that
65
Chapter III. Curves and Surfaces
(KO0/K)(X) = [Z/2]⊕ (Z/2)b1+2ν
(KO2/K)(X) =
(Z/2)b1+b2+2ν if Sq2Z = 0(Z/2)b1+b2+2ν−1 if Sq2Z 6= 0
(KO4/K)(X) =
Z/2 if Sq2Z = 00 if Sq2Z 6= 0
(KO6/K)(X) = 0
3.10 Remark. Note that both cases — Sq2Z = 0 and Sq
2
Z onto — can occur even for
complex projective surfaces of geometric genus zero. For example, Sq2Z is onto for P2.
On the other hand, the Wu formula [MS74, Theorem 11.14] shows that Sq2 is given by
multiplication with c1(X) mod 2. So Sq2Z = 0 on any projective surface X of geometric
genus zero whose canonical divisor is numerically trivial. Concretely, we could take X to
be an Enriques surface (a quotient of a K3-surface by a fixed-point free involution): see
[Bea96, page 90].
To calculate the groups (KO2i/K)(X), we use the Atiyah-Hirzebruch spectral sequences
for the K- and KO-theory of X. We begin with some general facts.
3.11 Lemma. The differential d1,−23 : E
1,−2
3 −→ E4,−43 is trivial both in the Atiyah-Hirze-
bruch spectral sequence for K-theory and in the spectral sequence for KO-theory.
Proof. This is immediate from the descriptions of these differentials in terms of the second
Steenrod square given in Section I.2e.
3.12 Lemma. Let X be a connected finite-dimensional CW complex. Denote the filtra-
tions on the groups K0(X) and KOi(X) associated with the Atiyah-Hirzebruch spectral
sequences by
K0(X) ⊃ K1(X) = K2(X) ⊃ K3(X) = K4(X) ⊃ . . .
KOi0(X) ⊃ KOi1(X) ⊃ KOi2(X) ⊃ KOi3(X) ⊃ . . .
The initial layers of these filtrations have more intrinsic descriptions in terms of Chern
and Stiefel-Whitney classes:
• K2(X) = K1(X) = K˜0(X), and KOi1(X) = K˜Oi(X)
• The map K2(X) → H2(X;Z) arising from the spectral sequence may be identified
with the first Chern class, at least up to a sign. That is, we have the following
commutative diagram:
E2,−2∞ 

,2 E2,−22
K˜0(X)
OOOO
±c1 // // H2(X;Z)
In particular, K4(X) = ker(c1). Moreover, since c1 is surjective, E
2,−2∞ = E2,−22 .
66
3 Surfaces
• The map K4(X) → H4(X;Z) can be identified with the restriction of the second
Chern class to ker(c1), again up to a sign:
E4,−4∞ 

,2 E4,−42
ker(c1)
OOOO
±c2 ,2 H4(X;Z)
Thus, K6(X) = ker(c2). (Note that the upper horizontal map in this diagram is
indeed an inclusion, by the previous lemma.)
• Likewise, the map KO01(X) → H1(X;Z/2) can be identified with the first Stiefel-
Whitney class:
E1,−1∞ 

,2 E1,−12
K˜O0(X)
OOOO
w1 // // H1(X;Z/2)
Thus, KO02(X) = ker(w1) and E
1,−1∞ = E1,−12 .
• Lastly, the map KO02(X)→ H2(X;Z/2) can be identified with the restriction of the
second Stiefel-Whitney class, i. e. we have a commutative diagram
E2,−2∞ 

,2 E2,−22
ker(w1)
OOOO
w2 ,2 H2(X;Z/2)
Thus, KO04(X) = ker(w2).
Proof. The first statement is clear. The other assertions follow by viewing the maps in
question as cohomology operations and computing them for a few spaces. For lack of
reference, we include a more detailed proof of the statements concerning KO-theory. The
case of complex K-theory can be dealt with analogously.
First, we analyse the map K˜O0(X)→ H1(X;Z/2) arising from the spectral sequence.
A priori, we have defined this map only for finite-dimensional CW complexes. But we can
extend it to a natural transformation of functors on the homotopy category of all connected
CW complexes, using the fact that the canonical map H1(X;Z/2)→ limiH1(Xi;Z/2) is
an isomorphism for any CW complex X with i-skeletons Xi. On the homotopy category
of connected CW complexes, the functor K˜O0(−) is represented by BO. Natural transfor-
mations K˜O0(−)→ H1(−;Z/2) are therefore in one-to-one correspondence with elements
of H1(BO;Z/2) = Z/2 · w1, where w1 is the first Stiefel-Whitney class of the universal
bundle over BO. Thus, either the map in question is zero, or it is given by w1 as claimed.
Since it is non-zero on S1, the first case may be discarded.
To analyse the map KO02(X) → H2(X;Z/2), we note that the previous conclusion
yields a functorial description of KO02(X) as the kernel of w1 on K˜O
0(X). Moreover, given
67
Chapter III. Curves and Surfaces
this description, we may define a natural set-wise splitting of the inclusion of KO02(X) into
K˜O0(X) for any finite-dimensional CW complex X as follows:
K˜O0(X) KO02(X)
[E]− rk(E)[R] 7→ [E]− rk(E)[R]− [detE] + [R]
The composition K˜O0(−) → H2(−;Z/2) can be extended to a natural transformation of
functors on the homotopy category of connected CW complexes in the same way as before,
and it may thus be identified with an element of H2(BO;Z/2) = Z/2 · w21 ⊕ Z/2 · w2.
Consequently, its restriction to KO02(X) is either 0 and w2. Again, the first possibility
may easily be discarded, for example by considering S2.
Lemma 3.12 has some immediate implications for low-dimensional spaces. In the fol-
lowing corollary, all characteristic classes refer to the corresponding maps on reduced
groups:
ci : K˜0(X) −→ H2i(X;Z)
wi : K˜O0(X) −→ H i(X;Z/2)
3.13 Corollary. Let X be a connected CW complex of dimension at most four. Then the
second Chern class is surjective and restricts to an isomorphism
c2 : ker(c1)
∼=−→ H4(X;Z)
The second Stiefel-Whitney class is also surjective, and restricts to an epimorphism
w2 : ker(w1) − H2(X;Z/2)
Its kernel is given by the image of ker(c1) under realification. Moreover, the induced map
w2 on (K˜O0/K˜)(X) restricts to an isomorphism
w2 : ker(w1)
∼=−→
(
H2(X;Z/2)
/
H2(X;Z)
)
Proof. Consider the Atiyah-Hirzebruch spectral sequences for K- and KO-theory.
Lemma 3.11 shows that the sequence for K-theory collapses. Moreover, in the spectral
sequence for KO-theory, no differentials affect the diagonal computing KO0(X). This
implies the first two claims of the corollary. Next, we compare the two spectral sequences
via realification. The description of the kernel of w2 may be obtained from the following
row-exact commutative diagram:
68
3 Surfaces
0 ,2 H4(X;Z) ,2
∼=

K˜(X)
c1 ,2
r

H2(X;Z) ,2
reduction mod 2

0
0 ,2 H4(X;Z) ,2 ker(w1)
w2 ,2 H2(X;Z/2) ,2 0
The final claim of the corollary also follows from this diagram, by identifying ker(w1) with
ker(w1)/K˜(X).
The situation for a connected CW complex of dimension at most four may now be
summarized as follows. Firstly, by Lemma 3.12, the filtrations on the groups K0(X) and
KO0(X) arising in the Atiyah-Hirzebruch spectral sequences can be written as
K0(X) ⊃ K˜0(X) ⊃ ker(c1)
KO0(X) ⊃ K˜O0(X) ⊃ ker(w1)
Secondly, Corollary 3.13 implies that the maps (rk, c1, c2) and (rk, w1, w2) on the associated
graded groups induce isomorphisms
gr(K0(X)) ∼= Z ⊕H2(X;Z) ⊕ H4(X;Z) (15)
gr
(
(KO0/K)(X)
) ∼= Z/2⊕H1(X;Z/2)⊕ (H2(X;Z/2)/H2(X;Z)) (16)
In particular, we have proved the first part of Proposition 3.8.
Proof of the remaining claims of Proposition 3.8:
To compute (KO2i/K)(X) for general i, we identify this quotient with the image of
η : KO2i(X)→ KO2i−1(X)
This image can be computed at each stage of the filtration
KOj(X) = KOj0(X) ⊃ KOj1(X) ⊃ · · · ⊃ KOj4(X)
To lighten the notation, we simply write KOjk for KO
j
k(X) in the following, and we write
H∗(X) for the singular cohomology of X with integral coefficients.
Since we are assuming that X is at most four-dimensional, there are only three possibly
non-zero differentials in the spectral sequence computing KO∗(X). The first two are the
differentials
Sq2 ◦ pi : H2(X) −→ H4(X;Z/2)
Sq2 : H2(X;Z/2)→ H4(X;Z/2)
on the E2-page, depicted in Figure I.1. If E
4,−2
2 = H
4(X;Z/2) is not killed by Sq2, then
69
Chapter III. Curves and Surfaces
we have a third possibly non-trivial differential on the E3-page:
d1,03 : H
1(X)→ coker(Sq2)
The differential d1,−23 vanishes by Lemma 3.11.
Computation of KO2/K. For the last stage of the filtration, we have a commutative
diagram
0 ,2 KO24 ,2

KO23 ,2
η

H3(X; KO−1(point)) ,2
∼=

0
0 ,2 0 ,2 KO13 H
3(X; KO−2(point)) ,2 0
Thus, KO23 maps surjectively to KO
1
3
∼= H3(X;Z/2). Next, we have
0 ,2 KO23 ,2
η

KO22 ,2
η

ker(Sq2 ◦ pi) ,2
pi

0
0 ,2 KO13 ,2 KO
1
2
,2 ker(Sq2) ,2 0
where ker(Sq2 ◦pi) ⊂ H2(X; KO0(point)) and ker(Sq2) ⊂ H2(X; KO−1(point)). This gives
a short exact sequence of images, which must split since all groups involved are killed by
multiplication with 2. Thus, we have
im
(
KO22
η→ KO12
) ∼= KO13⊕ im( ker(Sq2 ◦ pi)→ ker(Sq2))
∼= H3(X;Z/2)⊕ ker(Sq2Z)
where Sq2Z is the composition defined at the beginning of this section. Finally, the diagram
0 ,2 KO22
η

K˜O2(X) ,2
η

0 ,2

0
0 ,2 KO12 ,2 K˜O
1(X) ,2 E1,04 ,2 0
shows that im(K˜O2(X)
η→ K˜O1(X)) ∼= im(KO22 η→ KO12). So (K˜O2/K˜)(X) is isomorphic
to H3(X;Z/2)⊕ ker(Sq2Z), as claimed.
Computation of KO4/K. Proceeding as in the previous case, we consider the diagram
0 ,2 H4(X)

K˜O4(X) ,2
η

0 ,2

0
0 ,2 KO34 ,2 K˜O
3(X) ,2 H3(X) ,2 0
70
4 Comparison
where the vertical map on the left is the composition
H4(X; KO0(point)) H4(X; KO−1(point)) coker(Sq2 ◦ pi) = KO34
Since Sq2 ◦ pi has the same cokernel as Sq2Z, we may deduce that
(K˜O4/K˜)(X) ∼= im(K˜O4(X) η→ K˜O3(X)) ∼= coker(Sq2Z)
Computation of KO6/K. In this case, the commutative diagram
0 ,2 H2(X)

K˜O6(X) ,2
η

0 ,2

0
0 ,2 0 ,2 K˜O5(X) H1(X) ,2 0
demonstrates that the map K˜O6(X)
η→ K˜O5(X) is zero, so (K˜O6/K)(X) is trivial. This
completes the computations of (K˜O2i/K)(X).
4 Comparison
In this section, we finally compare our two sets of results for complex curves and surfaces
and prove the comparison theorem mentioned in the introduction to this chapter. First,
however, we summarize the situation one finds in K-theory.
K-groups
The algebraic and topological K-groups of a smooth complex curve C are given by
K0(C) = Z⊕ Pic(C)
K0(C) = Z⊕H2(C;Z)
The comparison map K0(C) → K0(C) is always surjective. Indeed, the map is an iso-
morphism on the first summand, and for a projective curve the map on reduced groups
is simply the projection from Pic(C) ∼= Z⊕ Jac(C) onto the free part. It follows that the
map is still surjective if we remove a finite number of points from C.
For a smooth complex surface X, we have seen in (11) and (15) that we have filtrations
on the K-groups such that
gr∗(K0(X)) ∼= Z⊕ Pic(X)⊕ CH2(X)
gr∗(K0(X)) ∼= Z⊕H2(X;Z)⊕H4(X;Z)
71
Chapter III. Curves and Surfaces
Both isomorphisms can be written as (rk, c1, c2), and the comparison map K0(X) →
K0(X) corresponds to the usual comparison maps on the filtration. Of course, on the first
summand we again have the identity. Moreover, the map CH2(X)→ H4(X;Z) is always
surjective:
If X is projective, this follows from the fact that H4(X;Z) is generated by a point. In
general, we can embed any smooth surface X into a projective surface X as an open subset
with complement a divisor with simple normal crossings (c. f. Lemma 4.8). Then CH2(X)
surjects onto CH2(X), and similarly H4(X;Z) surjects onto H4(X;Z), so that the claim
follows. Since we will need this observation in a moment, we record it as a lemma.
4.1 Lemma. For any smooth complex surface X, the natural map CH2(X)→ H4(X;Z)
is surjective.
4.2 Corollary. For any smooth complex surface X, the natural map K0(X)→ K0(X) is
surjective if and only if the natural map Pic(X)→ H2(X;Z) is surjective.
When X is projective, we see from the exponential sequence that we have surjections
if and only if X has geometric genus zero. Equivalently, this happens if and only if X has
full Picard rank, i. e. if and only if its Picard number ρ agrees with its second Betti number
b2. More generally, the Picard group of any smooth complex surface can be written as
Pic(X) = Zρ ⊕H2(X;Z)tors ⊕ Pic0(X) (17)
where ρ ≤ b2 is an integer that generalizes the Picard number, H2(X;Z)tors is the torsion
subgroup of H2(X;Z), and Pic0(X) is a divisible group [PW01, Corollary 6.2.1]. Again,
the natural map Pic(X)→ H2(X;Z) is surjective if and only if ρ = b2.
4a The classical Witt group
For the classical Witt group W0(X), the situation can be analysed in a similar way as in
the case of K0(X), using our description in terms of Stiefel-Whitney classes.
4.3 Proposition. For a smooth complex curve C, the map
gw0 : GW0(C) KO0(C) is surjective, and
w0 : W0(C)
∼=→ (KO0/K)(C) is an isomorphism.
Similarly, for a smooth complex surface X, both gw0 and w0 are surjective. If X is
projective, then w0 is an isomorphism if and only if Pic(X) surjects onto H2(X;Z).
Proof. It suffices to show the corresponding statements for reduced groups. Let X be a
smooth complex variety of dimension at most two. Since the first Stiefel-Whitney classes
72
4 Comparison
are always surjective, we have a row exact commutative diagram of the form
0 ,2 ker(w1) ,2

G˜W0(X)
w1 ,2
gw0

H1et(X;Z/2) ,2
∼=

0
0 ,2 ker(wtop1 ) ,2 K˜O
0(X)
wtop1 ,2 H1(X;Z/2) ,2 0
Similarly, by Corollaries 3.6 and 3.13 we have a row exact commutative diagram
0 ,2 H0(ker(c1)) ,2

ker(w1)

w2 ,2 H2et(X;Z/2) ,2
∼=

0
0 ,2 r(ker(ctop1 )) ,2 ker(w
top
1 )
wtop2 ,2 H2(X;Z/2) ,2 0
In both diagrams, we have written wtopi and c
top
i for the topological characteristic classes
to avoid ambiguous notation. The kernels of c1 and c
top
1 can be identified with CH
2(X)
and H4(X;Z), respectively, via the second Chern classes. Thus, Lemma 4.1 implies that
the vertical map on the left of the lower diagram is a surjection. The surjectivity of the
comparison map gw0 on GW0(X) follows.
If we apply the same analysis to W0(X), then the second diagram reduces to a com-
mutative square
ker(w1)

w2
∼=
,2 H2et(X;Z/2)
/
Pic(X)

ker(wtop1 )
wtop2
∼=
,2 H2(X;Z/2)
/
H2(X;Z)
When X is a curve, the vertical arrow on the right is an isomorphism, proving the claim.
In general, we have a row-exact commutative diagram of the following form:
0 ,2 Pic(X)
/
2 ,2


H2et(X;Z/2) ,2
∼=

H2et(X;Z/2)
/
Pic(X) ,2

0
0 ,2 H
2(X;Z)
/
2 ,2 H
2(X;Z/2) ,2 H2(X;Z/2)
/
H2(X;Z) ,2 0
If Pic(X) → H2(X;Z) is surjective, then the two outer maps become isomorphisms and
it follows that the comparison map w0 is also an isomorphism. Using the description of
the Picard group of X given by (17), we see that the converse is also true.
4b Shifted Witt groups
We now generalize Proposition 4.3 to shifted groups. The final result is stated in Theo-
rems 4.4 and 4.12. For curves, there is in fact very little left to be shown. We nevertheless
73
Chapter III. Curves and Surfaces
give a detailed proof in preparation for a similar line of arguments in the case of surfaces.
4.4 Theorem. For any smooth complex curve C, the maps
gw i : GWi(C)→ KOi(C) are surjective, and the maps
w i : Wi(C)→ (KOi/K)(C) are isomorphisms.
Proof. By Lemma II.2.6, the comparison maps appearing here are isomorphisms on a
point, so the claims are equivalent to the corresponding claims involving reduced groups.
It suffices to show that the maps gw i : G˜Wi(C) → K˜Oi(C) are surjective and that the
maps w i : W˜i(C)→ K˜O2i−1(C) are injective.
First, suppose C is affine. Then the cohomology of C is concentrated in degrees 0
and 1 and we see that W1(C), W2(C), W3(C) and K˜O2(X), K˜O4(C), K˜O6(C) all vanish.
Thus, the claims are trivially true.
The case that C is projective can be reduced to the affine case. Indeed, if p is any
point on C, then C˜ := C − p is affine. By comparing the localization sequences arising
from the inclusion of C˜ into C, we see that the comparison maps for C must also have the
desired properties:
. . . ,2 GWi−1(p) ,2
∼=

G˜Wi(C) ,2

G˜Wi(C˜) ,2

Wi(p) ,2
∼=

W˜i+1(C) ,2

W˜i+1(C˜) ,2


. . .
. . . ,2 KO2i−2(p) ,2 K˜O2i(C) ,2 K˜O2i(C˜) ,2 KO2i−1(p) ,2 K˜O2i+1(C) ,2 K˜O2i+1(C˜) ,2 . . .
(18)
4.5 Corollary. Theorem 4.4 also hold for groups with twists in any line bundle.
Proof. Introducing a twist by the line bundle O(p) into the localization sequence (18) only
affects the groups of C, so we can conclude as before. More generally, given a line bundle
O(D) associated with a divisor D =
∑
nipi on C, we can similarly reduce to the case of
a trivial line bundle over C −⋃i pi.
We now want to imitate this proof for surfaces, replacing the role of points on the
curve by curves on the surface. We first prove the following.
4.6 Proposition. For any smooth complex surface X, the comparison maps have the
properties indicated by the following arrows:gw0 : GW0(X) KO0(X)w1 : W1(X) KO1(X)
gw2 : GW2(X) KO4(X)w3 : W3(X) KO5(X)
4.7 Lemma. Proposition 4.6 is true when X is affine and Pic(X)/2 vanishes.
74
4 Comparison
Proof. By the theorem of Andreotti and Frankel, a smooth complex affine variety of
dimension n has the homotopy type of a CW complex of real dimension at most n
[AF59,Laz04, 3.1]. In particular, its cohomology is concentrated in degrees ≤ n . For
a smooth affine surface X, Pardon’s spectral sequence shows immediately that W1(X) =
W2(X) = W3(X) = 0. Similarly, the Atiyah-Hirzebruch spectral sequence shows that
K˜O4(X) vanishes. Thus, three of the four claims are trivially satisfied. The fact that
gw0 : GW0(X) KO0(X) is surjective was already shown in 4.3.
Before proceeding with the proof of Proposition 4.6, we make a note of two general
facts that we will use. First, we will need the following standard consequence of Hironaka’s
resolution of singularities:
4.8 Lemma. In characteristic zero, any smooth variety can be embedded into a smooth
compact variety with complement a divisor with simple normal crossings.
Proof. Let X be a smooth variety over a field of characteristic zero, and let X be some
compactification. Any singularities of X may be resolved without changing the smooth
locus [Kol07, Theorem 3.36], so we may assume that X is smooth. Applying the Princi-
palization Theorem [Kol07, Theorem 3.26] to (the ideal sheaf of) the complement of X in
X yields the variety we are looking for.
Note that in dimension two there is no distinction between compactness and projectiv-
ity: any smooth compact surface is projective [Har70, II.4.2]. It follows that an arbitrary
smooth surface is at least quasi-projective. Secondly, we will need the following lemma
concerning generators of the Picard group.
4.9 Lemma. Let X be a smooth quasi-projective variety over an algebraically closed field
of characteristic zero. Then any element of Pic(X)/2 can be represented by a smooth prime
divisor (i. e. by a smooth irreducible subvariety of codimension 1). If X is projective, we
may moreover take the divisor to be very ample (i. e. to be given by a hyperplane section
of X for some embedding of X into some PN ).
Proof. We consider the case when X is projective first. If X is a projective curve,
Pic(X)/2 ∼= Z/2 is generated by a point and there is nothing to show. So we may assume
dim(X) ≥ 2.
Fix a very ample line bundle L on X. Any element of Pic(X)/2 can be lifted to a line
bundle M on X. Tensoring with any sufficiently high power of L will yield a very ample
line bundleM⊗Lm [Laz04, 1.2.10]. In particular, if we take m large and even we obtain a
very ample line bundle that maps to the class of M in Pic(X)/2. The claim now follows
from Bertini’s theorem on hyperplane sections in characteristic zero [Har77, Corollary 10.9
and Remark 10.9.1].
75
Chapter III. Curves and Surfaces
In general, if X is quasi-projective, we may embed it as an open subset into a smooth
resolution X of its projective closure. Then Pic(X) surjects onto Pic(X), and we obtain
smooth prime divisors on X generating Pic(X)/2 by restriction.
4.10 Example. Consider P˜2, the blow-up of P2 at a point p. Its Picard group is given
by Pic(P˜2) = Z[H]⊕ Z[E], where H is a hyperplane section of P2 that misses p and E is
the exceptional divisor, both isomorphic to P1. A divisor a[H]− b[E] is ample if and only
if a > b > 0. Thus, {[H], 2[H] − [E]} is a basis of Pic(P˜2) consisting of ample divisors.
A smooth curve representing 2[H]− [E] is given by the birational transform of a smooth
conic in P2 through p.
Proof of Proposition 4.6. LetX be a smooth surface. By Lemma 4.8, we can find a projec-
tive surface X and smooth curves D1, . . . , Dk on X whose union
⋃
Di is the complement
of X in X. On the other hand, by Lemma 4.9, we can find smooth ample curves generating
Pic(X)/2. Let C1, . . . , Cρ be a subset of these curves generating Pic(X)/2, and put
Ui := X − C1 − C2 − · · · − Ci
For sufficiently large k, the divisors
∑
j Dj + k(C1+ · · ·+Ci) are ample on X. Thus, each
Ui is affine. Moreover, we see from the exact sequences
Z[Ci+1|Ui ]→ Pic(Ui)→ Pic(Ui+1)→ 0
and the choice of the Ci that rkZ/2(Pic(Ui)/2) = ρ−i. Thus, by Lemma 4.7, Proposition 4.6
holds for Uρ.
We can now proceed as in the proof of Theorem 4.4, by adding the curves back in
to obtain X. Namely, consider the successive open inclusions Ui+1 ↪→ Ui. The closed
complements of these are given by restrictions of the curves Ci, so we obtain a diagram
similar to (18) with Ui playing the role of C, Ui+1 in the role of C˜ and Ci+1 in the role of
a point. If the normal bundle of Ci+1 in Ui is not trivial, the sequences will in fact involve
twisted groups of Ci, but in any case we can conclude using Lemma 4.5.
4.11 Corollary. Proposition 4.6 also holds for groups with twists in a line bundle over
X.
Proof. As we have seen, generators of Pic(X)/2 can be represented by smooth curves on
X. Thus, we can argue as in the proof of Corollary 4.5.
4.12 Theorem. Suppose X is a smooth projective surface for which the natural map
Pic(X)→ H2(X;Z) is surjective. Then the comparison maps
gw i : GWi(X)→ KOi(X) are surjective, and the maps
w i : Wi(X)→ (KO2i/K)(X) are isomorphisms.
76
4 Comparison
By Proposition 4.3, the assumption on the map Pic(X)→ H2(X;Z) is clearly necessary.
Proof. Consider the squaring operations S1 and Sq2Z appearing in the computations of the
Witt and (KO/K)-groups. For any smooth complex variety X, we have a commutative
diagram
Pic(X)/2 S
1
,2
 _

CH2(X)/2

H2(X;Z)/2 ,2 _

H4(X;Z)/2 _

H2(X;Z/2) Sq
2
,2 H4(X;Z/2)
When X is a surface, the two vertical maps on the right are both isomorphisms, and
the horizontal map in the middle is essentially Sq2Z. So whenever Pic(X) surjects onto
H2(X;Z), we may identify S1 and Sq2Z. In particular, we may do so for any a smooth pro-
jective surface of geometric genus zero. It then follows by comparison of Propositions 3.1
and 3.8 that the Witt groups of such a surface agree with the groups (KO2i/K)(X). On the
other hand, we see from Proposition 4.6 that each of the maps w i : Wi(X)→ (KO2i/K)(X)
is either surjective or injective. Given that we are dealing with finite groups, these maps
must be isomorphisms. Moreover, since we know from Corollary 4.2 that we also have
a surjection from the algebraic to the topological K-group of X, we may deduce via the
Karoubi/Bott sequences that the maps gw i : GWi(X)→ KO2i(X) are surjective for all
values of i.
4c Comparison with Z/2-coefficients
As we have seen, the comparison maps on Witt groups are isomorphisms for all surfaces
X for which Pic(X) surjects onto H2(X;Z). To obtain isomorphisms on the level of
Grothendieck-Witt groups, we need to pass to Z/2-coefficients (see Section II.2e) and
introduce one additional topological constraint on X.
4.13 Proposition. Let X be a smooth complex variety, of any dimension. If the odd
topological K-groups of X contain no 2-torsion (i. e. if K1(X)[2] = 0), then the integral
comparison maps
Wi(X)→ (KO2i/K)(X)
are isomorphisms for all i if and only if the comparison maps with Z/2-coefficients
Wi(X;Z/2)→ (KO2i/K)(X;Z/2)
are isomorphisms for all i.
Before giving the proof, we make one preliminary observation.
77
Chapter III. Curves and Surfaces
4.14 Lemma. Let X be a topological space such that K1(X)[2] = 0. Then multiplication
by η ∈ KO−1(point) induces an isomorphism
(KO2i/K)(X)
η−→∼= KO
2i−1(X)[2]
Proof. In general, since 2η = 0, the following exact sequence may be extracted from the
Bott sequence:
0→ (KO2i/K)(X) η−→ KO2i−1(X)[2]→ K1(X)[2]
This proves the claim.
Proof of Proposition 4.13. We claim that we have the following row-exact commutative
diagram:
0 ,2 GW
i(X)
K0(X)
,2

GWi(X;Z/2)
K0(X;Z/2)
,2

Wi+1(X) ,2

0
0 ,2 KO
2i(X)
K0(X)
,2 KO
2i(X;Z/2)
K0(X;Z/2)
,2 KO2i+1(X)[2] ,2 0
(∗)
Indeed, the lower exact row may be obtained by applying the Snake Lemma to the following
diagram of short exact sequences induced by the Bockstein sequences for K- and KO-
theory:
0 ,2 K0(X)/2
∼= ,2

K0(X;Z/2) ,2

0

0 ,2 KO2i(X)/2 ,2 KO2i(X;Z/2) ,2 KO2i+1(X)[2] ,2 0
The upper row of (∗) may be obtained similarly from the Bockstein sequences for algebraic
and hermitian K-theory. The vertical maps are induced by the comparison maps in degrees
0, 0 and −1, respectively. Using the canonical identification of GWi(X)/K0(X) with
Wi(X), we may however identify the first vertical map with the usual comparison map w i
in degree −1. Likewise, the second vertical map may be identified with the comparison
map w i for Witt groups with Z/2-coefficients. Lastly, by the previous lemma, the entry
in the lower right corner may be identified with (KO2i+2/K)(X). Thus, diagram (∗) can
be rewritten in a form from which both implications of the proposition may be deduced:
0 ,2Wi(X) ,2
w i

Wi(X;Z/2) ,2
w i

Wi+1(X) ,2
w i+1

0
0 ,2
KO2i(X)
K(X)
,2
KO2i(X;Z/2)
K(X;Z/2)
,2
KO2i+2(X)
K(X)
,2 0
78
4 Comparison
Recall from II.2e that, by the Quillen-Lichtenbaum conjectures, the comparison maps
on K-groups of curves with Z/2-coefficients are isomorphisms in all non-negative degrees,
while for a smooth complex surface X we have isomorphisms in all positive degrees and
an inclusion in degree zero:
Ki(X;Z/2)
∼=−−→ K−i(X;Z/2) for all i > 0
K0(X;Z/2) ↪−→ K0(X;Z/2)
For these low-dimensional cases, proofs may also be found in [Sus95, 4.7; PW01, Theo-
rem 2.2].
4.15 Lemma. Let X be a smooth complex variety of dimension at most two. The map
K0(X;Z/2) ↪→ K0(X;Z/2) is an isomorphism if and only if the map Pic(X)→ H2(X;Z)
is surjective and K1(X)[2] = 0.
Proof. We see from the description of the Picard group (17) that Pic(X) → H2(X;Z) is
surjective if and only if it is surjective after tensoring with Z/2. Moreover, by a similar
argument as in Corollary 4.2, the surjectivity of the map Pic(X)/2 → H2(X;Z)/2 is
equivalent to the surjectivity of the map K0(X)/2 → K0(X)/2. The claim follows from
these equivalences and a commutative diagram induced by the Bockstein sequences:
0 ,2 K0(X)/2
∼= ,2

K0(X;Z/2) ,2

0

0 ,2 K0(X)/2 ,2 K0(X;Z/2) ,2 K1(X)[2] ,2 0
If we combine the results on K-theory with our results for Witt groups and Proposi-
tion 4.13, we obtain the following corollary via Karoubi induction.
4.16 Corollary. Let X be a smooth complex variety of dimension at most two. Assume
that the natural map Pic(X)→ H2(X;Z) is surjective and that K1(X) has no 2-torsion.
Then the hermitian comparison maps
KOp,q(X;Z/2) −→ KOp(X;Z/2)
are isomorphisms in all non-negative degrees, i. e. for all (p, q) such that 2q − p ≥ 0. In
particular, for all shifts i we have isomorphisms
GWi(X;Z/2)
∼=−→ KO2i(X;Z/2)
For example, the conditions of the corollary are satisfied for any smooth complex curve,
79
Chapter III. Curves and Surfaces
and for any simply-connected projective surface of geometric genus zero. The condition
that K1(X)[2] = 0 can be rephrased in terms of the integral cohomology group H3(X;Z):
4.17 Lemma. Let X be a smooth complex variety of dimension at most two. Then the
torsion of K1(X) agrees with the torsion of H3(X;Z).
Proof. By Lemma 3.11, the Atiyah-Hirzebruch spectral sequence for the K-theory of X
collapses. Since the first integral cohomology group H1(X;Z) of a smooth complex curve
or surface is free, we find that K1(X) ∼= H1(X;Z)⊕H3(X;Z), and the lemma follows.
Remark. Conversely, if we assume the analogue of the Quillen-Lichtenbaum conjecture
for hermitian K-theory, i. e. if we assume that the hermitian comparison maps with Z/2-
coefficients are isomorphisms in high degrees, then we can recover our comparison theorem
for Witt groups for all surfaces X with K1(X)[2] = 0. Said analogue appeared recently in
[Sch10]. However, it does not seem possible to relate our result to the Quillen-Lichtenbaum
conjecture for surfaces with 2-torsion in K1(X). Such surfaces do exist. In particular, if X
is an Enriques surface, then Pic(X) surjects onto H2(X;Z) but K1(X)[2] ∼= pi1(X) ∼= Z/2
[Bea96, page 90].
80
Chapter IV
Cellular Varieties
In this chapter, we turn our attention to cellular varieties. As we will see, the comparison
maps behave particularly well on these, allowing us to identify all Grothendieck-Witt and
Witt groups with the corresponding KO-groups. This will be used to compute the Witt
groups of a series of projective homogeneous varieties.
By definition, a smooth cellular variety is a smooth variety X with a filtration by
closed subvarieties ∅ = Z0 ⊂ Z1 ⊂ Z2 ⊂ · · · ⊂ ZN = X such that the complement of Zk in
Zk+1 is an open “cell” isomorphic to Ank for some nk. For example, projective n-space Pn
contains an open cell isomorphic to An with closed complement Pn−1, so such a filtration
can be obtained inductively. It is well-known that, when X is a smooth cellular variety
over the complex numbers, the comparison map on K-groups is an isomorphism:
K0(X)
∼=−→ K0(X)
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. However, the computation of Witt groups
of cellular varieties does not seem to be as straight-forward. It is true, of course, 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, it is possible to prove by an induction over the number of cells of X that
the comparison maps on Grothendieck-Witt and Witt groups are also isomorphisms:
GWi(X)
∼=−→ KO2i(X)
Wi(X)
∼=−→ (KO2i/K)(X)
The map (KO2i/K)(X)
η→ KO2i−1(X) is an isomorphism in this situation (see Lemma 2.2),
so the second line may equivalently be stated as
Wi(X)
∼=−→ KO2i−1(X)
Thus, the Grothendieck-Witt and Witt groups of smooth complex cellular varieties may
be read off directly from their KO-groups.
As in the previous chapter, we use the homotopy-theoretic approach to the comparison
81
Chapter IV. Cellular Varieties
maps. As a by-product of the proof we give, we also obtain partial information on the
maps in degrees 1 and −2. A slightly different proof using only the maps in degrees 0
and −1 and those properties that can be seen in a more elementary way may be found in
[Zib09, Theorem 3.1]. The argument is sketched in Remark 1.5.
After presenting our proof of the comparison result at the beginning of this chapter, we
turn to concrete topological computations. As our main tool will be the Atiyah-Hirzebruch
spectral sequence, we collect some general observations concerning its behaviour on cellular
varieties in Section 2. We then run through the list of all projective homogeneous varieties
that fall within the class of hermitian symmetric spaces. The untwisted KO-groups of
these are known by several papers of Kono and Hara, and we extend their computations
to twisted KO-groups. Combined with the comparison result, this furnishes us with a
complete additive description of the Witt groups of these varieties.
1 The comparison theorem
1.1 Theorem. For a smooth cellular complex variety X, the following comparison maps
are isomorphisms:
k : K0(X)
∼=−→ K0(X)
gw q : GWq(X)
∼=−→ KO2q(X)
w q : Wq(X)
∼=−→ KO2q−1(X)
More generally, the maps gw q and w q are isomorphisms for the corresponding groups
twisted by an arbitrary vector bundle over X (see Section II.2c).
In the proof, we concentrate on the hermitian case. The case of algebraic/complex
K-theory could be dealt with similarly, or deduced from the hermitian case using trian-
gle (II.10). As indicated in the introduction, it will be helpful to consider not only the
maps gw q = k2q,qh and w
q+1 = k2q+1,qh in degrees 0 and −1, respectively, but also the maps
k2q−1,qh in degree 1 and the maps k
2q+2,q
h in degree −2. In fact, we will prove the following
extended version of the theorem, generalizing Proposition II.2.6:
1.2 Theorem. For X as above, the hermitian comparison maps in degrees 1, 0, −1 and
−2 have the properties indicated:
KO2q−1,q(X) KO2q−1(X)
KO2q,q(X)
∼=→ KO2q(X)
KO2q+1,q(X)
∼=→ KO2q+1(X)
KO2q+2,q(X) KO2q+2(X)
The analogous statements for twisted groups are also true.
82
1 The comparison theorem
Proof. The proof will proceed by induction over the number of cells of X. 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. This case was dealt with in Proposition II.2.6.
Spheres. Since the theorem holds for a point, the compatibility of the comparison
maps with suspensions immediately shows that it is also true for the reduced cohomology
of the spheres (P1)∧d = S2d,d. In other words, the following maps in degrees 1, 0, −1 and
−2 have the properties indicated:
K˜O2q−1,q(S2d,d) K˜O2q−1(S2d)
K˜O2q,q(S2d,d)
∼=→ K˜O2q(S2d)
K˜O2q+1,q(S2d,d)
∼=→ K˜O2q+1(S2d)
K˜O2q+2,q(S2d,d) K˜O2q+2(S2d)
Cellular varieties. Now let X be a smooth cellular variety. By definition, X has
a filtration by closed subvarieties ∅ = Z0 ⊂ Z1 ⊂ Z2 · · · ⊂ ZN = X such that the open
complement of Zk in Zk+1 is isomorphic to Ank for some nk. In general, the subvarieties Zk
will not be smooth. Their complements Uk := X − Zk inX, however, are always smooth as
they are open in X. So we obtain an alternative filtration X = U0 ⊃ U1 ⊃ U2 · · · ⊃ UN = ∅
of X by smooth open subvarieties Uk. Each Uk contains a closed cell Ck ∼= Ank with open
complement Uk+1.
Our inductive hypothesis is that we have already proved the theorem for Uk+1, and we
now want to prove it for Uk. We can use the following exact triangle in SH(C):
Σ∞((Uk+1)+) ,2 Σ∞((Uk)+)
pwggggg
gggg
Σ∞Thom(NCk\Uk)
gn
As Ck is a cell, the Quillen-Suslin theorem tells us that the normal bundle NCk\Uk of
Ck in Uk has to be trivial. Thus, Thom(NCk\Uk) is A
1-weakly equivalent to S2d,d, where d
is the codimension of Ck in Uk. Figure 1 on page 85 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|Uk+1) ,2 Σ
∞Thom(E|Uk)
ovffffff
fffff
Σ∞Thom(E|Ck ⊕NCk\Uk)
ho
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.
83
Chapter IV. Cellular Varieties
1.3 Lemma. 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 ) ,2 Σ∞ThomE
pwggggg
ggggg
gg
Σ∞Thom(E|Z ⊕NZ\X)
ho
Proof. From the Thom isomorphism theorem we know that the Thom space of a vector
bundle over a smooth base is A1-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) ∼= ThomU (E|U )
E/(E−X) ∼= ThomX E
E/(E− Z) ∼= ThomZ(E|Z ⊕NZ\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.
1.4 Remark (Comparison in degree −2). As we saw in Section II.2d, the comparison
map in degrees 1 and −2 are not isomorphisms even when X is a point, and in degrees
below −2 the comparison maps are necessarily zero. For cellular varieties, the map in
degree −2 may be identified with the inclusion of the 2-torsion subgroup of KO2q+2(X)
into KO2q+2(X). This follows from the theorem and the description of the KO-groups of
cellular varieties given in Lemma 2.2.
1.5 Remark. We indicate briefly how Theorem 1.1 can alternatively be obtained by
working only with the maps in degrees 0 and −1 defined by more elementary means.
The basic strategy — comparing the localization sequences arising from the inclusion of a
closed cell Ck into the union of “higher” cells Uk — still works. But we cannot deduce that
the comparison maps are isomorphisms on Uk from the fact that they are isomorphisms on
Uk+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 Uk are surjective, and that the maps in degree −1 with domains the
Witt groups of Uk 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.
84
1 The comparison theorem

···

K˜O2q−1,q(S2d,d)

// // K˜O2q−1(S2d)

KO2q−1,q(Uk)

k2q−1,qh ,2____ KO2q−1(Uk)

KO2q−1,q(Uk+1)

// // KO2q−1(Uk+1)

K˜O2q,q(S2d,d)

∼= ,2 K˜O2q(S2d)

KO2q,q(Uk)

k2q,qh ,2_____ KO2q(Uk)

KO2q,q(Uk+1)

∼= ,2 KO2q(Uk+1)

K˜O2q+1,q(S2d,d)

∼= ,2 K˜O2q+1(S2d)

KO2q+1,q(Uk)

k2q+1,qh ,2____ KO2q+1(Uk)

KO2q+1,q(Uk+1)

∼= ,2 KO2q+1(Uk+1)

K˜O2q+2,q(S2d,d)

,2 ,2 K˜O2q+2(S2d)

KO2q+2,q(Uk)

k2q+2,qh ,2____ KO2q+2(Uk)

KO2q+2,q(Uk+1)

,2 ,2 KO2q+2(Uk+1)
···
Figure 1: The inductive step.
85
Chapter IV. Cellular Varieties
2 The Atiyah-Hirzebruch spectral sequence for cellular vari-
eties
We now prepare the ground for the discussion of the KO-theory of some examples of cellular
spaces in the next section. Our main tool will be the Atiyah-Hirzebruch spectral sequence
described in Section I.2e. For cellular varieties, or more generally for CW complexes with
only even-dimensional cells, this spectral sequence becomes simple enough to make some
general deductions.
So let X be a CW complex with cells only in even dimensions. Then the cohomology
of X is free on generators given by the cells and concentrated in even degrees. The Atiyah-
Hirzebruch spectral sequence for the K-theory of such a space collapses immediately: the
entries of all columns and rows of odd degrees are zero, so there cannot be any non-zero
differentials. We arrive at the following well-known lemma:
2.1 Lemma. Let X be a CW complex with cells only in even dimensions. Then K0(X)
is a free abelian group of rank equal to the number of cells of X. The K-groups of odd
degrees, on the other hand, vanish.
After this preliminary observation, we turn to KO-theory. The Atiyah-Hirzebruch
spectral sequence for the KO-theory of a CW complex with cells only in even dimen-
sions does not necessarily collapse, but one can still make some general deductions. We
summarize some lemmas of Hoggar and Kono and Hara.
2.2 Lemma. [Hog69, 2.1 and 2.2] Let X be a CW complex with only even-dimensional
cells. Then:
• The ranks of the free parts of KO0X and KO4X are equal to the number t0 of cells
of X of dimension a multiple of 4.
• The ranks of the free parts of KO2X and KO6X are equal to the number t1 of cells
of X of dimension 2 modulo 4.
• The groups of odd degrees are two-torsion, i. e. KO2i−1X = (Z/2)si for some si.
• KO2iX is isomorphic to the direct sum of its free part and KO2i+1X.
Moreover, multiplication by η ∈ KO−1(point) induces an isomorphism
η : (KO2i/K)(X)
∼=−→ KO2i−1(X)
The bullet points are summarized by Table 2 in Section 3a.
Proof. The first two statements can be seen directly from the Atiyah-Hirzebruch spectral
sequence for KO-theory (e. g. after tensoring with Q). The remaining statements may then
be deduced from the above description of the K-groups and the Bott sequence (I.20).
86
2 The Atiyah-Hirzebruch spectral sequence for cellular varieties
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 Sq2 Sq2 = Sq3 Sq1 must
vanish when the cohomology of X with Z/2-coefficients is concentrated in even degrees.
So we can view (H∗(X;Z/2),Sq2) as a differential graded algebra over Z/2. To lighten
notation, we will write
H∗(X, Sq2) := H∗(H∗(X;Z/2),Sq2)
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 E3-page is given by H∗(X, Sq2) · 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.
2.3 Lemma. Let X be as above. If the Atiyah-Hirzebruch spectral sequence of KO∗(X)
degenerates on the E3-page then
KO2i−1(X) ∼=
⊕
k
H2i+8k(X, Sq2)
Now suppose that E is a complex vector bundle over X. Then the twisted KO-groups
KO∗(X;E) are computed by the Atiyah-Hirzebruch spectral sequence of the Thom space
ThomE. Recall that the reduced cohomology of this space is isomorphic to the cohomology
of X up to a shift in degrees. By Lemma I.2.11, the Steenrod square on H∗(ThomE;Z/2)
is given by Sq2 + c1(E).
If, as before, X has cells only in even dimensions, then Sq2+c1 may again be viewed as
a differential on H∗(X;Z/2) for any c1 in H2(X;Z/2). Extending our previous notation,
we denote cohomology with respect to this differential by
H∗(X, Sq2 + c1) := H∗(H∗(X;Z/2),Sq2 + c1) (1)
Lemma 2.3 has the following corollary.
2.4 Corollary. Let X and E be as above. If the Atiyah-Hirzebruch spectral sequence of
K˜O∗(ThomE) degenerates on the E3-page, then
KO2i−1(X;E) ∼=
⊕
k
H2i+8k(X, Sq2 + c1E)
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].
2.5 Lemma. Let X be as above. If the differentials d3, d4, . . ., dr−1 are trivial and dr is
87
Chapter IV. Cellular Varieties
non-trivial, then r ≡ 2 mod 8. In other words, the first non-trivial differential after d2
can only appear on a page Er 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 dr is non-zero on x.
Proof. We see from the spectral sequence of a point that drη = 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 E2-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 E3-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:
2.6 Corollary. 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 E3-page, and that p∗ induces an injection in row q ≡ −1:
p∗ : H∗(T,Sq2) ↪→ H∗(X, Sq2)
Then the spectral sequence for KO∗(T ) also collapses at this stage.
Proof. Write dr for the first non-trivial higher differential, so r ≡ 2 mod 8. Then, for any
element x in row q ≡ 0, we have p∗(drx) = drp∗(x) = 0 since the spectral sequence for
X collapses. From our assumption on p∗ we can deduce that drx = 0. By the preceding
lemma, this is all we need to show.
88
3 Examples
3 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 Gc
corresponding to G:
3.1 Proposition. Let P be a parabolic subgroup of a simple complex algebraic group G.
Then we have a diffeomorphism
G
/
P
∼= Gc
/
K
where K is a compact subgroup of maximal rank in a maximal compact subgroup Gc 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 ∼= Gc · P [GOV94, Ch. 6, Prop. 1.7], inducing a diffeomorphism of
quotients as claimed for K = Gc ∩ P . Since Gc ↪→ G is a homotopy equivalence, so
is the inclusion Gc ∩ P ↪→ P . On the other hand, if L is a Levi subgroup of P then
P = U o 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 Lc 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 ⊂ Lc ⊂ L ⊂ P and conclude that K ↪→ Lc is a homotopy equivalence.
Since both groups are compact, it follows that in fact K ∼= Lc.
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 1. For the convenience of the 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 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 as new.
89
Chapter IV. Cellular Varieties
G
/
P G Diagram of P G
c
/
K
Grassmannians (AIII) Grm,n SLm+n ◦ · · · ◦ • ◦ · · · ◦
1 n n+m–1
U(m+ n)
U(m)×U(n)
Maximal symplectic
Grassmannians (CI) Xn Sp2n
◦ ◦ · · · ◦ ◦ < • Sp(n)
/
U(n)
Projective quadrics of
dimension n ≥ 3
(BDI)
Qn SOn+2 • ◦ · · · ◦ > ◦ (n odd)
◦
• ◦ · · · ◦}A (n even)◦
SO(n+ 2)
SO(n)× SO(2)
Spinor varieties (DIII) Sn SO2n
◦
◦ ◦ · · · ◦}A•
SO(2n)
/
U(n)
Exceptional hermitian
symmetric spaces:
EIII E6 ◦
◦ ◦ ◦ ◦ •
Ec6
Spin(10) · S1
(Spin(10)∩S1=Z/4)
EVII E7 ◦
◦ ◦ ◦ ◦ ◦ •
Ec7
Ec6 · S1
(Ec6∩S1=Z/3)
Table 1: List of irreducible compact hermitian symmetric spaces. The symbols AIII, CI,
. . . refer to E. Cartan’s classification. In the description of Grm,n we use U(m+n) instead
of Gc = SU(m+ n).
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 Calme`s and Fasel, all
Witt groups with non-trivial twists vanish for these varieties [CF11].
3a 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 2
below. This was shown in Section 2 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
ti and si. Since the ti are just given by counting cells, and since the numbers of odd- and
even-dimensional cells of a Thom space ThomXE only depend on X and the rank of E,
the ti are in fact independent of L. The si, on the other hand, certainly will depend on
the twist, and we will sometimes acknowledge this by writing si(L).
90
3 Examples
KO6(X;L) = Zt1 ⊕ (Z/2)s0 = GW3(X;L)
KO7(X;L) = (Z/2)s0 = W0(X;L)
KO0(X;L) = Zt0 ⊕ (Z/2)s1 = GW0(X;L)
KO1(X;L) = (Z/2)s1 = W1(X;L)
KO2(X;L) = Zt1 ⊕ (Z/2)s2 = GW1(X;L)
KO3(X;L) = (Z/2)s2 = W2(X;L)
KO4(X;L) = Zt0 ⊕ (Z/2)s3 = GW2(X;L)
KO5(X;L) = (Z/2)s3 = W3(X;L)
Table 2: Notational conventions in the examples. Only the si depend on L.
3b Projective spaces
Complex projective spaces are perhaps the simplest examples for which Theorem 1.1
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 landmark events in the history of the theory. In 1980, Arason was able to
show that the Witt group W0(Pn) of Pn 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 KOi(CPn) were first published in
a 1967 paper by Fujii [Fuj67]. It is not difficult to deduce the values of the twisted groups
KOi(CPn;O(1)) from these: the space Thom(OCPn(1)) can be identified with CPn+1, so
KOi(CPn;O(1)) = K˜Oi+2(Thom(O(1)))
= K˜Oi+2(CPn+1)
Alternatively, we could do all required computations directly following the methods out-
lined in Section 2. The result, in any case, is displayed in Table 3, coinciding with the
known results for the (Grothendieck-)Witt groups.
KO∗(CPn;L) L ≡ O L ≡ O(1)
t0 t1 s0 s1 s2 s3 s0 s1 s2 s3
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 3: KO-groups of projective spaces
91
Chapter IV. Cellular Varieties
3c Grassmannians
We now consider the Grassmannians Grm,n of complex m-planes in Cm+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 Calme`s [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 Calme`s state their result by describing an additive basis of the total Witt
group of Grm,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(Grm,n), say the
dual of the determinant line bundle of the universal m-bundle over Grm,n. The result is
displayed in Table 4.
KO∗(Grm,n;L) L ≡ O L ≡ O(1)
t0 t1 s0 s1 s2 s3 s0 s1 s2 s3
m and n odd s.t.
m ≡ n
a
2
a
2
b b 0 0 0 0 0 0
m and n odd s.t.
m 6≡ n
a
2
a
2
b 0 0 b 0 0 0 0
m ≡ n ≡ 0
m ≡ 0 and n odd
n ≡ 0 and m odd
a+ b
2
a− b
2
b 0 0 0 b 0 0 0

m ≡ n ≡ 2
m ≡ 2 and n odd
n ≡ 2 and m odd
a+ b
2
a− b
2
b 0 0 0 0 0 b 0
m ≡ 0 and n ≡ 2
a+ b
2
a− b
2
b 0 0 0 b1 0 b2 0
m ≡ 2 and n ≡ 0
a+ b
2
a− b
2
b 0 0 0 b2 0 b1 0
All equivalences (≡) are modulo 4. For the values of a and b = b1 + b2, put
k := bm/2c and l := bn/2c. Then
a :=
(
m+n
m
)
b :=
(
k+l
k
)
b1 :=
(
k+l−1
k
)
b2 :=
(
k+l−1
k−1
)
Table 4: KO-groups of Grassmannians
Our computation is based on the following geometric observation. Let Um,n and U⊥m,n
be the universal m-bundle and the orthogonal n-bundle over Grm,n, so that U ⊕ U⊥ =
92
3 Examples
O⊕(m+n). We have various natural inclusions between the Grassmannians of different
dimensions, of which we fix two:
Grm,n−1 ↪→ Grm,n via the inclusion of the first m+ n− 1 coordinates into Cm+n
Grm−1,n ↪→ Grm,n by sending an (m − 1)-plane Λ to the m-plane Λ ⊕ 〈em+n〉, where
e1, e2, . . . , em+n are the canonical basis vectors of Cm+n
3.2 Lemma. The normal bundle of Grm,n−1 in Grm,n is the dual U∨m,n−1 of the universal
m-bundle. Similarly, the normal bundle of Grm−1,n in Grm,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:
Grm−1,n+
i
↪→ Grm,n+
p
 Thom(U∨m,n−1) (2)
Grm,n−1+
i
↪→ Grm,n+
p
 Thom(U⊥m−1,n) (3)
These sequences are the key to relating the untwisted KO-groups to the twisted ones.
Following the notation in [KH91], we write Am,n for the cohomology of Grm,n with Z/2-
coefficients, denoting the Chern classes of U and U⊥ by ai and bi, respectively, and the
total Chern classes 1 + a1 + · · ·+ am and 1 + b1 + · · ·+ bn by a and b:
Am,n =
Z/2 [a1, a2, . . . , am, b1, b2, . . . bn]
a · b = 1
We write d for the differential given by the second Steenrod square Sq2, and d′ for Sq2+a1.
To describe the cohomology of Am,n with respect to these differentials, it is convenient to
introduce the algebra
Bk,l =
Z/2
[
a22, a
2
4, . . . , a
2
2k, b
2
2, b
2
4, . . . , b
2
2l
]
(1 + a22 + · · ·+ a22k)(1 + b22 + · · ·+ b22l) = 1
Note that this subquotient of A2k,2l is isomorphic to Ak,l up to a “dilatation” in grading.
Proposition 2 in [KH91] tells us that
H∗(Am,n, d) =
Bk,l if (m,n) = (2k, 2l), (2k + 1, 2l) or (2k, 2l + 1)Bk,l ⊕Bk,l · ambn−1 if (m,n) = (2k + 1, 2l + 1)
Here, the algebra structure in the case where both m and n are odd is determined by
(ambn−1)2 = 0.
93
Chapter IV. Cellular Varieties
3.3 Lemma. The cohomology of Am,n with respect to the twisted differential d′ is as
follows:
H∗(Am,n, d′) =

Bk,l−1 · am ⊕Bk−1,l · bn if (m,n) = (2k, 2l)
Bk,l · am if (m,n) = (2k, 2l + 1)
Bk,l · bn if (m,n) = (2k + 1, 2l)
0 if (m,n) = (2k + 1, 2l + 1)
Proof. Let us shift the dimensions in the cofibration sequences (2) and (3) 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 (Am,n+1, d)- and (Am+1,n, d)-modules,
respectively:
0→ (Am,n, d′) · θ∨ p
∗
−→(Am,n+1, d) i
∗−→ (Am−1,n+1, d)→ 0 (4)
0→ (Am,n, d′) · θ⊥ p
∗
−→(Am+1,n, d) i
∗−→ (Am+1,n−1, d)→ 0 (5)
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 am. Its kernel, the image of Am,n under
multiplication by am, is generated as an Am,n+1-module by its unique element in degree
2m, and thus we must have p∗(θ∨) = am. Likewise, in the second row we have p∗(θ⊥) = bn.
The lemma can be deduced from here case by case. For example, when both m and
n are even, i∗ maps H∗(Am,n+1, d) = Bk,l to the first summand of H∗(Am−1,n+1, d) =
Bk−1,l ⊕ Bk−1,l · am−1bn by annihilating a2m. We know by comparison with the short
exact sequences for the Am,n that the kernel of this map is Bk,l−1 mapping to Bk,l under
multiplication by a2m. Thus, we obtain a short exact sequence
0→ Bk−1,l · am−1bn ∂−→ H∗(Am,n, d′) · θ∨ p
∗
−→ Bk,l−1 · a2m → 0 (6)
For the Steenrod square Sq2 of the top Chern class am of U, we have Sq2(am) = a1am.
This can be checked, for example, by expressing am as the product of the Chern roots of
U. Consequently, d′(am) = 0. Together with the fact that H∗(Am,n, d′) is a module over
H∗(Am,n+1, d), this shows that we can define a splitting of p∗ by sending a2m to amθ∨.
Thus, H∗(Am,n, d′) contains Bk,l−1 · am as a direct summand. If instead of working with
sequence (4) we work with sequence (5), we see that H∗(Am,n, d′) also contains a direct
summand Bk−1,l · bn. These two summands intersect trivially, and a dimension count
shows that together they encompass all of H∗(Am,n, d′). Alternatively, one may check
explicitly that the boundary map ∂ above sends am−1bn to bnθ. The other cases are
simpler.
94
3 Examples
3.4 Lemma. The Atiyah-Hirzebruch spectral sequence for K˜O∗(ThomU∨m,n) collapses at
the E3-page.
Proof. By Proposition 4 of [KH91] we know that the spectral sequence for KO∗(Grm,n)
collapses as this stage, for any m and n. Now, if both m and n are even, we have
(Bk,l−1 · am ⊕Bk−1,l · bn) · θ
in the (−1)st row of the E3-pages of the spectral sequences for ThomU∨ and ThomU⊥,
where θ = θ∨ or θ⊥, respectively. In the case of U∨ we see from (6) that p∗ maps the
second summand injectively to the E3-page of the spectral sequence for KO∗(Grm,n+1).
Similarly, in the case of U⊥, the first summand is mapped injectively to the E3-page of
KO∗(Grm+1,n). Since the spectral sequences for ThomU∨ and ThomU⊥ can be identified
via Corollary I.2.13, we can argue as in Corollary 2.6 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 2.4. The entries of Table 4 that do not appear in
[KH91], i. e. those of the last four columns, follow from Lemma 3.3 by noting that Bk,l is
concentrated in degrees 8i and of dimension dimBk,l = dimAk,l =
(
k+l
k
)
.
3d Maximal symplectic Grassmannians
The Grassmannian of isotropic n-planes in C2n with respect to a non-degenerate skew-
symmetric bilinear form is given by Xn = Sp(n)/U(n). The universal bundle U over the
usual Grassmannian Gr(n, 2n) restricts to the universal bundle over Xn, and so does the
orthogonal complement bundle U⊥. We will continue to denote these restrictions by the
same letters. Thus, U⊕U⊥ ∼= C2n over Xn, and the fibres of U are orthogonal to those of
U⊥ with respect to the standard hermitian metric on C2n. The determinant line bundles
of U and U⊥ give dual generators O(1) and O(−1) of the Picard group of Xn.
3.5 Theorem. The additive structure of KO∗(Xn;L) is as follows:
t0 t1 si(O) si(O(1))
n even 2n−1 2n−1 ρ(n2 , i) ρ(
n
2 , i− n)
n odd 2n−1 2n−1 ρ(n+12 , 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 ΛZ/2(g1, g2, . . . , gn) on n homogeneous generators g1, g2, . . . , gn
of degree 1, i. e.
ρ(n, i) =
∑
d≡i
mod 4
(
n
d
)
A table of the values of ρ(n, i) can be found in [KH92, Proposition 4.1].
95
Chapter IV. Cellular Varieties
It turns out to be convenient to work with the vector bundle U⊥ ⊕ O for the compu-
tation of the twisted groups KO∗(Xn;O(1)). Namely, we have the following analogue of
Lemma 3.2.
3.6 Lemma. There is an open embedding of the bundle U⊥ ⊕ O over the symplectic
Grassmannian Xn into the symplectic Grassmannian Xn+1 whose closed complement is
again isomorphic to Xn.
Proof. To fix notation, let e1, e2 be the first two canonical basis vectors of C2n+2, and
embed C2n into C2n+2 via the remaining coordinates. Assuming Xn is defined in terms of
a skew-symmetric form Q2n, define Xn+1 with respect to the form
Q2n+2 :=
 0 1 0−1 0 0
0 0 Q2n

Then we have embeddings i1 and i2 of Xn into Xn+1 sending an n-plane Λ ⊂ C2n to e1⊕Λ
or e2 ⊕ Λ in C2n+2, respectively.
We extend i1 to an embedding of U⊥⊕O by sending an n-plane Λ ∈ Xn together with
a vector v in Λ⊥ ⊂ C2n and a complex scalar z to the graph ΓΛ,v,z ⊂ C2n+2 of the linear
map (
z Q2n(−, v)
v 0
)
: 〈e1〉 ⊕ Λ→ 〈e2〉 ⊕ Λ⊥
To avoid confusion, we emphasize that v is orthogonal to Λ with respect to a hermitian
metric on C2n. The value of Q2n(−, v), on the other hand, may well be non-zero on Λ.
Consider the above embedding of U⊥ ⊕ O together with the embedding i2:
U⊥ ⊕ O ↪−−→ Xn+1 Xn←−−↩i2
(Λ, v, z) 7→ ΓΛ,v,z
〈e2〉 ⊕ Λ Λ←[
To see that the two embeddings are complementary, take an arbitrary (n+1)-plane W in
Xn+1. If e2 ∈W then we can consider a basis
e2,
( a1
0
v1
)
, . . . ,
( an
0
vn
)
of W , and the fact that Q2n+2 vanishes on W implies that all ai are zero. Thus W can
be identified with i2(〈v1, . . . , vn〉).
If, on the other hand, e2 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′ ∈ C2n. Extend this vector to a basis of W of the
form (
1
z′
v′
)
,
(
0
b1
v1
)
, . . . ,
(
0
bn
vn
)
96
3 Examples
and let Λ := 〈v1, . . . , vn〉. The condition that Q2n+2 vanishes onW implies that Q vanishes
on Λ and that bi = Q2n(vi, v′) for each i. In particular, Λ is n-dimensional. 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
Q2n(vi, v′) = Q2n(vi, v) and our new basis has the form(
1
z
v
)
,
(
0
Q(v1,v)
v1
)
, . . . ,
(
0
Q(vn,v)
vn
)
This shows that W = ΓΛ,v,z.
3.7 Corollary. We have a cofibration sequence
Xn+
i
↪→ Xn+1+
p
 ThomXn(U⊥ ⊕ O)
The associated long exact cohomology sequence splits into a short exact sequence of
H∗(Xn+1)-modules since all cohomology here is concentrated in even degrees:
0→ H˜∗(ThomXn(U⊥ ⊕ O)) p
∗
→ H∗(Xn+1) i
∗→ H∗(Xn)→ 0 (7)
3.8 Lemma. Let ci denote the ith Chern classes of U on Xn. We have
H∗(Xn,Sq2) =
Λ(a1, a5, a9, . . . , a4m−3) if n = 2mΛ(a1, a5, a9, . . . , a4m−3, a4m+1) if n = 2m+ 1
H∗(Xn,Sq2 + c1) =
Λ(a1, a5, . . . , a4m−3) · c2m if n = 2m0 if n is odd
for certain generators ai of degree 2i.
Proof. Consider the short exact sequence (7). The mod-2 cohomology of Xn is an exterior
algebra on the Chern classes ci of U,
H∗(Xn;Z/2) = Λ(c1, c2, . . . , cn)
and i∗ is given by sending cn+1 to zero. Thus, p∗ is the unique morphism ofH∗(Xn+1;Z/2)-
modules that sends the Thom class θ of U⊥ ⊕ O to cn+1.
This short exact sequence induces a long exact sequence of cohomology groups with
respect to the Steenrod square Sq2. The algebra H∗(Xn,Sq2) was computed in [KH92,
2–2], with the result displayed above, so we already know two thirds of this sequence.
Explicitly, we have a4i+1 = c2ic2i+1,1 so i∗ is the obvious surjection sending ai to ai (or to
zero). Thus, the long exact sequence once again splits.
1In [KH92] the generators are written as c2ic
′
2i+1 with c
′
2i+1 = c2i+1 + c1c2i.
97
Chapter IV. Cellular Varieties
If n = 2m we obtain a short exact sequence
0 → H∗(X2m,Sq2 + c1) · θ p
∗
→ Λ(a1, . . . , a4m−3, a4m+1) i
∗→ Λ(a1, . . . , a4m−3) → 0
We see that H∗(X2m,Sq2+c1)·θ is isomorphic to Λ(a1, . . . , a4m−3)·a4m+1 as a module over
Λ(a1, . . . , a4m+1). It is thus generated by a single element, which is the unique element of
degree 8m + 2. Since p∗(c2mθ) = a4m+1, the class of c2mθ 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∗(Xn,Sq2 + c1) must
be trivial.
We see from the proof that p∗ induces an injection ofH∗(Xn,Sq2 + c1) · θ intoH∗(Xn,Sq2).
Since we already know from [KH92, Theorem 2.1] that the Atiyah-Hirzebruch spectral
sequence for KO∗(Xn) collapses, we can apply Corollary 2.6 to deduce that the spectral
sequence for K˜O∗(ThomXn(U⊥ ⊕ O)) collapses at the E3-page as well. This completes the
proof of Theorem 3.5.
3e Quadrics
We next consider smooth complex quadrics Qn in Pn+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 appli-
cations 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 aris-
ing from the inclusion of a linear subspace of maximal dimension [Nen09]. Calme`s 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.
For n ≥ 3 the Picard group of Qn is free on a single generator given by the restriction
of the universal line bundle O(1) on Pn+1. We will use the same notation O(1) for this
restriction.
3.9 Theorem. The KO-theory of a smooth complex quadric Qn of dimension n ≥ 3 is as
described in Table 5.
Untwisted KO-groups. Before turning to KO∗(Qn;O(1)) we review the initial steps in
the computation of the untwisted KO-groups. The integral cohomology of Qn is well-
known:
If n is even, write n = 2m. We have a class x in H2(Qn) given by a hyperplane section,
and two classes a and b in Hn(Qn) represented by linear subspaces of Q of maximal
98
3 Examples
KO∗(Qn;L) L ≡ O L ≡ O(1)
t0 t1 s0 s1 s2 s3 s0 s1 s2 s3
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 5: KO-groups of projective quadrics (n ≥ 3)
dimension. These three classes generate the cohomology multiplicatively, modulo the
relations
xm = a+ b xm+1 = 2ax
ab =
0 if n ≡ 0axm if n ≡ 2 a2 = b2 =
axm if n ≡ 0 mod 40 if n ≡ 2 mod 4
Additive generators can thus be given as follows:
d 0 2 4 . . . n− 2 n n+ 2 n+ 4 . . . 2n
Hd(Qn) 1 x x2 . . . xm−1 a, b ax ax2 . . . axm
If n is odd, write n = 2m + 1. Then similarly multiplicative generators are given by the
class of a hyperplane section x in H2(Qn) and the class of a linear subspace a in Hn+1(Qn)
modulo the relations xm+1 = 2a and a2 = 0.
d 0 2 4 . . . n− 1 n+ 1 n+ 3 n+ 5 . . . 2n
Hd(Qn) 1 x x2 . . . xm a ax ax2 . . . axm
The action of the Steenrod square on H∗(Qn;Z/2) is also well-known; see for example
[Ish92, Theorem 1.4 and Corollary 1.5] or [EKM08, § 78]:
Sq2(x) = x2
Sq2(a) =
ax if n ≡ 0 or 3 mod 40 if n ≡ 1 or 2
Sq2(b) = Sq2(a) (for even n)
As before, we write H∗(Qn,Sq2) for the cohomology of H∗(Qn;Z/2) with respect to
the differential Sq2.
99
Chapter IV. Cellular Varieties
3.10 Lemma. Write n = 2m or n = 2m + 1 as above. The following table gives a
complete list of the additive generators of H∗(Qn,Sq2).
d 0 . . . n− 1 n n+ 1 . . . 2n
Hd(Qn,Sq2) 1 axm if n ≡ 0 mod 4
1 a if n ≡ 1
1 a, b ab if n ≡ 2
1 xm 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 = xm can be
pulled back from Qn+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 Qn to that of the spinor variety Sn
2
+1 discussed in
Section 3f.
Twisted KO-groups. We now compute KO∗(Qn;O(1)).
Let θ ∈ H2(ThomQnO(1)) be the Thom class of O(1), so that multiplication by θ
maps the cohomology of Qn isomorphically to the reduced cohomology of ThomQnO(1).
The Steenrod square on H˜∗(ThomQnO(1);Z/2) is determined by Lemma I.2.11: for any
y ∈ H∗(Qn;Z/2) we have Sq2(y · θ) = (Sq2 y + xy) · θ. We thus arrive at
3.11 Lemma. A complete list of the additive generators of H˜∗(ThomQnO(1),Sq2) is pro-
vided by the following table:
d . . . n+ 1 n+ 2 n+ 3 . . . 2n+ 2
H˜d(. . . ) aθ, bθ if n ≡ 0 mod 4
xmθ axmθ if n ≡ 1
if n ≡ 2
aθ axmθ if n ≡ 3
We claim that all higher differentials in the Atiyah-Hirzebruch spectral sequence for
K˜O∗(ThomQnO(1)) vanish. For even n this is clear. But for n = 8k + 1 the differential
d8k+2 might a priori take xmθ to axmθ, and for n = 8k + 3 the differential d8k+2 might
take aθ to axmθ.
We therefore need some geometric considerations. Namely, the space ThomQnO(1) can
be identified with the projective cone over Qn embedded in Pn+2. This projective cone
can be realized as the intersection of a smooth quadric Qn+2 ⊂ Pn+3 with its projective
tangent space at the vertex of the cone [Har92, p. 283]. Thus, we can consider the following
inclusions:
Qn
j
↪→ ThomQnO(1) i↪→ Qn+2
100
3 Examples
The composition is the inclusion of the intersection of Qn+2 with two transversal hyper-
planes.
3.12 Lemma. All higher differentials ( dk with k > 2) in the Atiyah-Hirzebruch spectral
sequence for KO∗(ThomQnO(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 xm+1 in Hn+1(Qn+2,Sq2) to xmθ in
Hn+1(ThomQnO(1),Sq2). Indeed, j∗i∗ maps the class of the hyperplane section x in
H2(Qn+2) to the class of the hyperplane section x in H2(Qn). So i∗x in H2(ThomQnO(1))
must be non-zero, hence equal to θ modulo 2. It follows that i∗(xm+1) = θm+1. Since
θ2 = Sq2(θ) = xθ, we have θm+1 = xmθ, proving the claim. As we already know that all
higher differentials vanish on H∗(Qn+2,Sq2), we may now deduce that they also vanish
on H∗(ThomQnO(1),Sq2).
When n ≡ 3 mod 4 we claim that i∗ maps the element a in Hn+3(Qn+2,Sq2) to aθ in
Hn+3(ThomQnO(1),Sq2). Indeed, a represents a linear subspace of codimension m+ 2 in
Qn+2 and is thus mapped to the class of a linear subspace of the same codimension in Qn:
j∗i∗(a) = ax in Hn+3(Qn). Thus, i∗(a) is non-zero in Hn+3(ThomQnO(1)), equal to aθ
modulo 2. Again, this implies that all higher differentials vanish on H∗(ThomQnO(1),Sq2)
since they vanish on H∗(Qn+2,Sq2).
The additive structure of KO∗(Qn;O(1)) thus follows directly from the result for
Hd(Qn,Sq2 + x) = H˜d+2(ThomQnO(1)) displayed in Lemma 3.11 via Corollary 2.4.
3f Spinor varieties
Let GrSO(n,N) be the Grassmannian of n-planes in CN 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 QN−2. For each N > 2n, this is an irreducible
homogeneous variety. In particular, for N = 2n + 1 we obtain the spinor variety Sn+1 =
GrSO(n, 2n+1). The variety GrSO(n, 2n) falls apart into two connected components, both
of which are isomorphic to Sn. This is reflected by the fact that we can equivalently
identify Sn with SO(2n− 1)/U(n− 1) or SO(2n)/U(n).
As for all Grassmannians, the Picard group of Sn is isomorphic to Z; we fix a line
bundle S which generates it. The KO-theory twisted by S vanishes:
3.13 Theorem. For all n ≥ 2 the additive structure of KO∗(Sn;L) is as follows:
t0 t1 si(O) si(S)
n ≡ 2 mod 4 2n−2 2n−2 ρ(n2 , 1− i) 0
otherwise 2n−2 2n−2 ρ(bn2 c,−i) 0
101
Chapter IV. Cellular Varieties
The values ρ(n, i) are defined as in Theorem 3.5.
Proof. The cohomology of Sn with Z/2-coefficients has simple generators e2, e4, . . . , e2n−2,
i. e. it is additively generated by products of distinct elements of this list. Its multiplicative
structure is determined by the rule e22i = e4i, and the second Steenrod square is given by
Sq2(e2i) = ie2i+2 [Ish92, Proposition 1.1]. In both formulae it is of course understood that
e2j = 0 for j ≥ n. What we need to show is that for all n ≥ 2 we have
H∗(Sn,Sq2 + e2) = 0
Let us abbreviate H∗(Sn,Sq2+ e2) to (Hn, d′). We claim that we have the following short
exact sequence of differential Z/2-modules:
0→ (Hn, d′) ·e2n→ (Hn+1, d′)→ (Hn, d′)→ 0 (8)
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 Corol-
lary 3.15 that we have such an exact sequence of Z/2-modules with maps respecting the
differentials given by Sq2 on all three modules. Since they also commute with multiplica-
tion by e2, they likewise respect the differential d′ = Sq2 + e2.
The long exact cohomology sequence associated with (8) allows us to argue by induc-
tion: if H∗(Hn, d′) = 0 then also H∗(Hn+1, d′) = 0. Since we can see by hand that
H∗(H2, d′) = 0, this completes the proof.
We close with a geometric interpretation of the exact sequence (8), via an analogue
of Lemmas 3.2 and 3.6. Let us write U for the universal bundle over Sn, i. e. for the
restriction of the universal bundle over Gr(n− 1, 2n− 1) to Sn, and U⊥ for the restriction
of the orthogonal complement bundle, so that U⊕ U⊥ is the trivial (2n− 1)-bundle over
Sn. As in Section 3d, we emphasize that under these conventions the fibres of U and U⊥
are perpendicular with respect to a hermitian metric on C2n−1 — they are not orthogonal
with respect to the chosen symmetric form.
3.14 Lemma. The spinor variety Sn embeds into the spinor variety Sn+1 with normal
bundle U⊥ such that the embedding extends to an embedding of this bundle. The closed
complement of U⊥ in Sn+1 is again isomorphic to Sn.
3.15 Corollary. We have a cofibration sequence
Sn+
i
↪→ Sn+1+
p
 ThomSn 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 Sn. For example, the embedding
of S2 into Gr(1, 3) can be identified with the embedding of the one-dimensional smooth
102
3 Examples
quadric into the projective plane, of degree 2, and the higher dimensional cases can be
reduced to this example. Thus, c1(U) and c1(U⊥) are trivial in Pic(Sn)/2.
proof of Lemma 3.14. The proof is similar to the proof of Lemma 3.6. Let e1, e2 be the
first two canonical basis vectors of C2n+1, and let C2n−1 be embedded into C2n+1 via the
remaining coordinates. Let Sn be defined in terms of a symmetric form Q on C2n−1, and
define Sn+1 in terms of
Q2n+1 :=
0 1 01 0 0
0 0 Q

Let i1 and i2 be the embeddings of Sn into Sn+1 sending an (n − 1)-plane Λ ⊂ C2n−1 to
e1 ⊕ Λ or e2 ⊕ Λ in C2n+1, respectively. Given an (n − 1)-plane Λ ∈ Sn together with a
vector v in Λ⊥ ⊂ C2n−1, consider the linear map(
−12Q(v, v) −Q(−, v)
v 0
)
: 〈e1〉 ⊕ Λ→ 〈e2〉 ⊕ Λ⊥
Sending (Λ, v) to the graph of this function defines an open embedding of U⊥ whose closed
complement is the image of i2.
3g 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.
3.16 Theorem. The KO-groups of the exceptional hermitian symmetric spaces EIII and
EVII are as follows:
L ≡ O L ≡ O(1)
t0 t1 s0 s1 s2 s3 s0 s1 s2 s3
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
]/
(u2t, u3 + t12)
H∗(EVII;Z/2) = Z/2
[
t, v, w
]/
(t14, v2, w2)
103
Chapter IV. Cellular Varieties
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 Sq2 u = ut and Sq2 v = Sq2w = 0. Thus, we find
H∗(EIII,Sq2 + t) = Z/2 · u⊕ Z/2 · u2 ⊕ Z/2 · u3
H∗(EVII,Sq2 + t) = 0
By Lemma 2.5, the Atiyah-Hirzebruch spectral sequence for EIII must collapse. This gives
the result displayed above.
104
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.
[AF59] Aldo Andreotti and Theodore Frankel, The Lefschetz theorem on hyperplane sections, Ann. of
Math. (2) 69 (1959), 713–717.
[Ara80] Jo´n Kristinn Arason, Der Wittring projektiver Ra¨ume, 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.
[AR76] Michael Atiyah and Elmer Rees, Vector bundles on projective 3-space, Invent. Math. 35 (1976),
131–153.
[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.
[Bal05] , Witt groups, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 539–576.
[BC09] Paul Balmer and Baptiste Calme`s, 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.
[BGPW02] Paul Balmer, Stefan Gille, Ivan Panin, and Charles Walter, The Gersten conjecture for Witt
groups in the equicharacteristic case, Doc. Math. 7 (2002), 203–217.
[BW02] Paul Balmer and Charles Walter, A Gersten-Witt spectral sequence for regular schemes, Ann.
Sci. E´cole Norm. Sup. (4) 35 (2002), no. 1, 127–152.
[Bea96] Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student
Texts, vol. 34, Cambridge University Press, Cambridge, 1996.
[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.
[BO74] Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci.
E´cole Norm. Sup. (4) 7 (1974), 181–201 (1975).
[Bot69] Raoul Bott, Lectures on K(X), Mathematics Lecture Note Series, W. A. Benjamin, Inc., New
York-Amsterdam, 1969.
[Bro03] Patrick Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003),
no. 5, 1869–1903.
[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.
105
Chapter IV. Cellular Varieties
[CF11] Baptiste Calme`s and Jean Fasel, Trivial Witt groups of flag varieties (2011), available at
arxiv.org/abs/1103.4412. Preprint.
[Del62] Antoine Delzant, De´finition des classes de Stiefel-Whitney d’un module quadratique sur un
corps de caracte´ristique diffe´rente de 2, C. R. Acad. Sci. Paris 255 (1962).
[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 Ro¨ndigs, and Vladimir Voevod-
sky, 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.
[EKV93] He´le`ne Esnault, Bruno Kahn, and Eckart Viehweg, Coverings with odd ramification and Stiefel-
Whitney classes, J. Reine Angew. Math. 441 (1993), 145–188.
[FC87] Fernando Ferna´ndez-Carmena, The Witt group of a smooth complex surface, Math. Ann. 277
(1987), no. 3, 469–481.
[Fuj67] Michikazu Fujii, K0-groups of projective spaces, Osaka J. Math. 4 (1967), 141–149.
[Ful98] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzge-
biete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin,
1998.
[FH91] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics,
vol. 129, Springer-Verlag, New York, 1991.
[GM03] Sergei Gelfand and Yuri Manin, Methods of homological algebra, 2nd ed., Springer Monographs
in Mathematics, Springer-Verlag, Berlin, 2003.
[Gil03] Stefan Gille, Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), no. 2, 211–
233.
[GN03] Stefan Gille and Alexander Nenashev, Pairings in triangular Witt theory, J. Algebra 261
(2003), no. 2, 292–309.
[GOV94] Vladimir Gorbatsevich, Arkadi˘ı Onishchik, and E`rnest Vinberg, Lie Groups and Lie Algebras
III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, vol. 41,
Springer-Verlag, Berlin, 1994.
[Gra75] Brayton Gray, Homotopy theory, Academic Press [Harcourt Brace Jovanovich Publishers], New
York, 1975. An introduction to algebraic topology; Pure and Applied Mathematics, Vol. 64.
[Har92] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag,
New York, 1992. A first course.
[Har70] Robin Hartshorne, Ample subvarieties of algebraic varieties, Notes written in collaboration
with C. Musili. Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin, 1970.
[Har77] , Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics,
No. 52.
[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, A1-representability of Hermitian K-theory and Witt groups, Topology 44
(2005), no. 3, 661–687.
106
3 Examples
[HS04] Jens Hornbostel and Marco Schlichting, Localization in Hermitian K-theory of rings, J. London
Math. Soc. (2) 70 (2004), no. 1, 77–124.
[Hus94] Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-
Verlag, New York, 1994.
[HJJS08] Dale Husemo¨ller, Michael Joachim, Branislav Jurcˇo, and Martin Schottenloher, Basic bundle
theory and K-cohomology invariants, Lecture Notes in Physics, vol. 726, Springer, Berlin, 2008.
[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-the´orie alge´brique, 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.
[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.
[Kne77] Manfred Knebusch, Symmetric bilinear forms over algebraic varieties, Queen’s Univ., Kingston,
Ont., 1977, pp. 103–283. Queen’s Papers in Pure and Appl. Math., No. 46. Also available from
epub.uni-regensburg.de/12783/.
[Koc96] Stanley Kochman, Bordism, stable homotopy and Adams spectral sequences, Fields Institute
Monographs, vol. 7, American Mathematical Society, Providence, RI, 1996.
[Kol07] Ja´nos Kolla´r, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166,
Princeton University Press, Princeton, NJ, 2007.
[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.
[Lab76] Olivier Laborde, Classes de Stiefel-Whitney en cohomologie e´tale, Bull. Soc. Math. France
Suppl. Mem. (1976), no. 48, 47–51. Colloque sur les Formes Quadratiques (Montpellier, 1975).
[Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 48, Springer-Verlag,
Berlin, 2004. Classical setting: line bundles and linear series.
[Lev99] Marc Levine, K-theory and motivic cohomology of schemes (1999), available at www.math.
uiuc.edu/K-theory/0336/. Preprint.
[May01] J. Peter May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1,
34–73.
[Mer81] Alexander Merkurjev, On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR 261
(1981), no. 3, 542–547.
[Mil08] James Milne, Lectures on Etale Cohomology (v2.10) (2008), available at www.jmilne.org/
math/.
[Mil69] John Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.
[MH73] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973.
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73.
107
Chapter IV. Cellular Varieties
[MS74] John Milnor and James Stasheff, Characteristic classes, Princeton University Press, Princeton,
N. J., 1974. Annals of Mathematics Studies, No. 76.
[Mor99] Fabien Morel, The´orie homotopique des sche´mas, Aste´risque (1999), no. 256.
[Mor04] , An introduction to A1-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.
[MV99] Fabien Morel and Vladimir Voevodsky, A1-homotopy theory of schemes, Inst. Hautes E´tudes
Sci. Publ. Math. (1999), no. 90, 45–143 (2001).
[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.
[OVV07] Dmitri Orlov, Alexander Vishik, and Vladimir Voevodsky, An exact sequence for KM∗ /2 with
applications to quadratic forms, Ann. of Math. (2) 165 (2007), no. 1, 1–13.
[PW10] Ivan Panin and Charles Walter, On the motivic commutative ring spectrum BO (2010), avail-
able at www.math.uiuc.edu/K-theory/0978/. Preprint.
[Par04] William Pardon, The filtered Gersten-Witt resolution for regular schemes (2004), available at
www.math.uiuc.edu/K-theory/0419/. Preprint.
[PW01] Claudio Pedrini and Charles Weibel, The higher K-theory of a complex surface, Compositio
Math. 129 (2001), no. 3, 239–271.
[Qui73] Daniel Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc.
Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lec-
ture Notes in Math., Vol. 341.
[Rio06] Joe¨l Riou, Ope´rations sur la K-the´orie alge´brique et re´gulateurs via la the´orie homotopique
des sche´mas, Ph.D. Thesis, l’Universite´ Paris 7 – Denis Diderot, 2006, www.math.u-psud.fr/
∼riou/these/these.pdf.
[Rio07a] , Cate´gorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math.
France 135 (2007), no. 4, 495–547.
[Rio07b] , Ope´rations sur la K-the´orie alge´brique et re´gulateurs via la the´orie homotopique des
sche´mas, 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.
[Sch10] , The homotopy limit problem and (e´tale) hermitian K-theory (2010), available at
arxiv.org/abs/1011.4977. Preprint.
[Seg68] Graeme Segal, Equivariant K-theory, Inst. Hautes E´tudes Sci. Publ. Math. (1968), no. 34,
129–151.
[Sus95] Andrei Suslin, Algebraic K-theory and motivic cohomology, 2 (Zu¨rich, 1994), Birkha¨user,
Basel, 1995.
108
3 Examples
[TT90] Robert Wayne Thomason and Thomas Trobaugh, Higher algebraic K-theory of schemes and
of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkha¨user
Boston, Boston, MA, 1990, pp. 247–435.
[Tot03] Burt Totaro, Non-injectivity of the map from the Witt group of a variety to the Witt group of
its function field, J. Inst. Math. Jussieu 2 (2003), no. 3, 483–493.
[Voe98] Vladimir Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Math-
ematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604.
[Voe03a] , Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes E´tudes
Sci. (2003), no. 98, 1–57.
[Voe03b] ,Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes E´tudes Sci. (2003),
no. 98, 59–104.
[Wal03a] Charles Walter, Grothendieck-Witt groups of triangulated categories (2003), available 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.
[Wei94] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced
Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.
[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 topo-
logical K-groups (2009), available at www.dspace.cam.ac.uk/handle/1810/236624. Smith-
Knight/Rayleigh-Knight Essay.
109