%\begin{comment}
\documentclass[12pt]{amsart}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{verbatim}
\textwidth 16cm
\textheight 22cm
\headheight 0.5cm
\evensidemargin 0.3cm
\oddsidemargin 0.2cm
%\end{comment}

\usepackage{amssymb}

%\usepackage{srcltx}

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{problem}[thm]{Problem}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{con}[thm]{Conjecture}

\theoremstyle{definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{example}[thm]{Example}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{notation}[thm]{Notation}

%\newcommand{\iw}{i_{\mathrm{W}}}


\newcommand{\onto}{\rightarrow\!\!\rightarrow}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\colim}{\operatorname{colim}}

\newcommand{\trdeg}{\operatorname{tr.deg}}


\newcommand{\SB}{\mathop{\mathrm{SB}}}
\newcommand{\Br}{\mathop{\mathrm{Br}}}
\newcommand{\Gal}{\mathop{\mathrm{Gal}}}
\renewcommand{\Im}{\mathop{\mathrm{Im}}}
\newcommand{\Pic}{\mathop{\mathrm{Pic}}}
\newcommand{\ind}{\mathop{\mathrm{ind}}}
\newcommand{\rk}{\mathop{\mathrm{rk}}}
\newcommand{\CH}{\mathop{\mathrm{Ch}}\nolimits}
\newcommand{\SO}{\mathop{\mathrm{SO}}\nolimits}

\newcommand{\ICH}{\mathop{\mathrm{CH}}\nolimits}
\newcommand{\BCH}{\mathop{\bar{\mathrm{Ch}}}\nolimits}
\newcommand{\IBCH}{\mathop{\bar{\mathrm{CH}}}\nolimits}
\newcommand{\BCHE}{\mathop{\mathrm{Che}}\nolimits}

\newcommand{\res}{\mathop{\mathrm{res}}\nolimits}

\newcommand{\cd}{\mathop{\mathrm{cd}}\nolimits}

\newcommand{\pr}{\operatorname{\mathit{pr}}}

\newcommand{\inc}{\operatorname{\mathit{in}}}

\newcommand{\mult}{\operatorname{mult}}


%\newcommand{\esdim}{\mathop{\mathrm{dim}_{es}}\nolimits}

\newcommand{\Char}{\mathop{\mathrm{char}}\nolimits}
\newcommand{\Dim}{\mathop{\mathrm{Dim}}\nolimits}
\newcommand{\id}{\mathrm{id}}
\newcommand{\coker}{\mathrm{coker}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\cF}{\mathcal F}
\newcommand{\cO}{\mathcal O}
\newcommand{\Mor}{\operatorname{Mor}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Tors}{\operatorname{Tors}}
%\newcommand{\an}{\hbox{an}}
\newcommand{\an}{0}

\newcommand{\X}{\mathcal{X}}

\newcommand{\pt}{\mathbf{pt}}
\newcommand{\PS}{\mathbb{P}}

\newcommand{\type}{\mathop{\mathrm{type}}}

\newcommand{\Coprod}{\operatornamewithlimits{\textstyle\coprod}}
\newcommand{\Prod}{\operatornamewithlimits{\textstyle\prod}}
\newcommand{\Sum}{\operatornamewithlimits{\textstyle\sum}}
\newcommand{\Oplus}{\operatornamewithlimits{\textstyle\bigoplus}}

\newcommand{\disc}{\operatorname{disc}}


\newcommand{\<}{\left<}
\renewcommand{\>}{\right>}
\renewcommand{\ll}{\<\!\<}
\newcommand{\rr}{\>\!\>}

\newcommand{\stiso}{\overset{st}\sim}
\newcommand{\miso}{\overset{m}\sim}

\newcommand{\Dfn}{\stackrel{\mathrm{def}}{=}}

%\newcommand{\Label}[1]{\label{#1}\marginpar{\scriptsize #1}}
\marginparwidth 2.5cm
\newcommand{\Label}{\label}

\newcommand{\compose}{\circ}
\newcommand{\ur}{\mathrm{ur}}
\newcommand{\Ker}{\operatorname{Ker}}
%\newcommand{\Tors}{\mathop{\mathrm{Tors}}}
\newcommand{\TCH}{\Tors\CH}

\newcommand{\corr}{\rightsquigarrow}

%\newcommand{\IW}{i_W}
\newcommand{\IW}{\mathfrak{i}_0}
\newcommand{\iw}{\mathfrak{i}}
\newcommand{\jw}{\mathfrak{j}}

\newcommand{\Hight}{\mathfrak{h}}
\newcommand{\Height}{\mathfrak{h}}

\newcommand{\IS}{i_S}

\newcommand{\SP}{\operatorname{sp}}


\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\D}{D}


\newcommand{\BC}{\ast}
\newcommand{\NBC}{\compose}
\newcommand{\ABC}{\star}
\newcommand{\NC}{\compose}
\newcommand{\AC}{\bullet}


\title
{Canonical dimension of orthogonal groups}

%\keywords
%{Quadratic forms, Witt indices,
%Chow groups, Steenrod operations, correspondences.
%{\em 2000 Mathematical Subject Classifications:}
%11E04; 14C25}


\date
%{\today}
{July 9, 2004}

\author
[N. Karpenko]
%{An official preliminary version by\\
{Nikita A. Karpenko}


\address
{Laboratoire de Math{\'e}matiques de Lens\\
Facult{\'e} des Sciences Jean Perrin\\
Universit{\'e} d'Artois\\
rue Jean Souvraz SP 18\\
62307 Lens Cedex\\
FRANCE}

\address
{{\it Web page:}
{\tt www.math.uni-bielefeld.de/\~{ }karpenko}}

\email
{karpenko@euler.univ-artois.fr}


\thanks
{Supported in part by
the European Community's Human Potential
Programme under contract HPRN-CT-2002-00287, KTAGS}

%\dedicatory{To my brother}

\begin{document}

\begin{abstract}
We prove Berhuy-Reichstein's conjecture on the canonical
dimension of orthogonal groups showing that
for any integer $n\geq1$, the canonical dimension of
$\SO_{2n+1}$ and of $\SO_{2n+2}$ is equal to
$n(n+1)/2$.
More precisely, for a given ($2n+1$)-dimensional quadratic form $\phi$
defined over an arbitrary field $F$ of characteristic $\ne2$,
we establish certain property of the correspondences on
the orthogonal grassmanian $X$ of $n$-dimensional totally
isotropic subspaces of $\phi$, provided that the degree
over $F$ of any finite splitting field of $\phi$ is divisible by
$2^n$;
this property allows to prove that the function field of $X$
has the minimal transcendence degree among all generic splitting
fields of $\phi$.
\end{abstract}


\maketitle

%\tableofcontents


\section
{Results}

Let $F$ be an arbitrary field of characteristic $\ne2$,
$\phi$ a non-degenerate ($2n+1$)-dimensional quadratic form over
$F$ (with $n\geq1$), $X$ the orthogonal grassmanian of $n$-dimensional totally
isotropic subspaces of $\phi$.
The variety $X$ is projective, smooth, and geometrically
connected; $\dim X=n(n+1)/2$.
We write $d(X)$ for the greatest common divisor of the degrees of
all closed points on $X$.

In this paper, a field extension $E/F$ is called a {\em splitting field} of
$\phi$, if the Witt index (see \cite{Scharlau} for the definition of the Witt
index of a quadratic form) of the form $\phi_E$ is maximal (i.e.,
equal to $n$).
Note that a field extension $E/F$ is a splitting field of
$\phi$, if and only if the set $X(E)$ is non-empty.
We write $d(\phi)$ for the greatest common divisor of the degrees
of all finite splitting fields of $\phi$.

Clearly, $d(\phi)=d(X)$.
Moreover, this integer is a power of $2$ not exceeding $2^n$.
The equality $d(\phi)=2^n$ holds if, for example, the even Clifford
algebra $C_0(\phi)$ of the quadratic form $\phi$ is a division algebra.
Of course,
%$d(\phi)=2^n$
it is so
for the ($2n+1$)-dimensional generic
quadratic form $\<t_1,\dots,t_{2n+1}\>$ (defined over the field $F(t_1,\dots,t_{2n+1})$
of rational functions in variables $t_1,\dots,t_{2n+1}$).

%Following Knebusch \cite[def. 3.2]{Knebusch},
%we call
A splitting field $L/F$ of $\phi$ is called {\em generic}, if
it is finitely generated and for any splitting field $E/F$ and any non-zero element
$a\in L$ there exists an $F$-place
$f\!:L\to E$ such that $f(a)$ is neither $0$ nor $\infty$.
The function field $F(X)$ is a generic splitting field of $\phi$.
In fact,
%it is also generic in the stronger sense of \cite[def. 9.3]{Berhuy-Reichstein};
%moreover,
it is even {\em very generic} in the sense of
\cite
%[def. 9.3]
{Berhuy-Reichstein} (where it is also explained how the ``very generic''
property implies the ``generic'' one):
indeed, if $E/F$ is a splitting field, the variety $X_E$ is
rational (as any projective homogeneous variety with a rational point is)
and therefore $F(X)$ is contained in a purely
transcendental extension of $E$ (in $E(X)$ namely).

%Let $\psi$ be a non-degenerate quadratic form over $F$.
Following \cite{Berhuy-Reichstein}, we define the {\em canonical dimension}
$\cd(\phi)$ of $\phi$ as the minimum of the transcendence degrees of all
generic splitting fields of $\phi$
(the canonical dimension of $\SO_{2n+1}$ is then the maximum of $\cd(\phi)$
when $\phi$ runs over all ($2n+1$)-dimensional quadratic forms over (finitely
generated) extensions of $F$; the canonical dimension of $\SO_{2n+2}$ coincides with
the canonical dimension of $\SO_{2n+1}$, see \cite{Berhuy-Reichstein}).

Our main result here reads as follows:

\begin{thm}
\label{thm}
If $d(\phi)=2^n$, then $\cd(\phi)=n(n+1)/2$.
%More precisely, the transcendence degree of any Knebusch's sense
%generic splitting field of $\phi$ is at least $n(n+1)/2$.
In particular,
$$
\cd(\SO_{2n+1})=\cd(\SO_{2n+2})=n(n+1)/2\;.
$$
\end{thm}

The proof is given in section \ref{Proofs}.
%It is essentially the same as
%\cite[proof of th. 4.3]{KM}, but
%is based on
It immediately follows from
Theorem \ref{prop}
(proved too in section \ref{Proofs}),
dealing with correspondences on
%the maximal orthogonal grassmanian of
%$\phi$.
$X$.
% replacing \cite[th. 6.4]{aoi}
A similar situation occurs in the proof of \cite[th. 11.3]{Berhuy-Reichstein}
based on \cite[th. 2.1]{aoi} dealing with correspondences on a Severi-Brauer
variety.
An alternative proof of \cite[th. 2.1]{aoi}, making use of a
degree formula, is given in \cite[\S7.2]{Merkurjev-SODF}.
However, for the similar statement  \cite[th. 6.4]{aoi}, concerning
correspondences on quadrics (producing a similar to Theorem \ref{thm} result
\cite[th. 4.3]{KM} on the minimum of transcendence degree of
generic {\em isotropy} fields of a quadratic form),
there is no proof making use of a degree
formula.
In the present article as well, we use neither degree formulas nor Steenrod
operations.

\begin{thm}
\label{prop}
If $d(X)=2^n$, then the multiplicity of any correspondence
$\alpha\!:X\corr X$ is congruent modulo $2$ to the multiplicity of the
transpose of $\alpha$.
In particular, any rational map $X\to X$ is necessarily dominant.
\end{thm}

Here by a correspondence $X\corr X$ we mean an algebraic cycle on
$X\times X$ of dimension $\dim X$.
The multiplicity $\mult(\alpha)$ of such a correspondence $\alpha$
is defined by the formula
$(\pr_1)_*(\alpha)=\mult(\alpha)\cdot[X]$, where $\pr_1\!:X\times X\to
X$ is the projection onto the first factor,  while $(\pr_1)_*$ is the
push-forward homomorphism of the group of algebraic cycles, see
\cite{Fulton} (we do not use
any equivalence relation on algebraic cycles yet).
For the transpose $\alpha^t$ of $\alpha$ we clearly have:
$\mult(\alpha^t)\cdot[X]=(\pr_2)_*(\alpha)$.
The statement on rational maps is obtained by consideration of
the correspondence given by
the closure in $X\times X$ of the graph of a given rational map
$X\to X$.


\begin{rem}
Assume that $d(X)=2^n$.
Although we have Theorem \ref{prop}, we do not know whether
the variety $X$ is {\em $2$-incompressible} in the sense of
\cite[\S7]{Merkurjev-SODF}.
Note that the only known proof of $p$-incompressibility of
Severi-Brauer varieties of $p$-primary division algebras ($p$ is an
arbitrary prime), given in \cite[\S7.2]{Merkurjev-SODF},
makes use of a degree formula while the incompressibility of
quadrics with first Witt index $1$ \cite[cor. 3.4]{KM} can not be proved by a degree
formula.
%Proposition \ref{???} gives an alternative version of incompressibility
%for $X$.
\end{rem}

On its turn, Theorem \ref{prop} follows
(in a way very similar to the way \cite[th. 2.1]{aoi} follows from
\cite[cor. 2.3]{aoi})
from the following computation of the reduced modulo $2$ Chow group
$\BCH(X)$, defined as the image of the restriction homomorphism
$\CH(X)\to\CH(\bar{X})$ of the usual modulo $2$ Chow groups,
where $\bar{X}$ is $X$ over an algebraic closure $\bar{F}$ of $F$
(a general reference for Chow groups is \cite{Fulton}):

\begin{prop}
\label{lemma}
If $d(X)=2^n$, then
$\BCH^{>0}(X)=0$.
\end{prop}

The next section starts with the proof of Proposition \ref{lemma}.

\section
{Proofs}
\label{Proofs}

In the proof of Proposition \ref{lemma},
we are going to use the description of the integral Chow ring $\ICH(\bar{X})$
given in \cite{Mimura-Toda}
(we borrowed this reference from beautiful Totaro's paper \cite{Totaro}).
The graded ring $\ICH^*(\bar{X})$ is isomorphic to the quotient of
the polynomial ring $\Z[e_1,\dots,e_n]$ by the ideal generated
by the polynomials
$$
e_i^2-2e_{i-1}e_{i+1}+2e_{i-2}e_{i+2}-\dots+(-1)^{i-1}2e_1e_{2i-1}+(-1)^ie_{2i}
$$
with $i=1,\dots,n$ ($e_i$ should be understood to mean $0$ for
$i>n$ in this formula),
where the degree of $e_i$ is $i$.
The element of $\ICH(\bar{X})$ corresponding to the class of $e_i$
is a special Schubert class;
we still write $e_i$ for it.
For any $i$, the element $2e_i$ is the $i$-th
%Segr{\'e}
Chern
class
of the tautological vector bundle on the grassmanian, therefore is
rational, that is, lies in the integral reduced Chow group
$\IBCH(X)\subset\ICH(\bar{X})$.

%We notice that, as the relations show, the ring $\ICH(\bar{X})$ is generated by
%$e_i$ with odd $i$-s.
%The same is of course true for the modulo $2$ Chow group, where,
%moreover, we have
%$$
%\CH(\bar{X})=\bigotimes_{\text{odd
%}i\in[1,\;n]}(\Z/2\Z)[e_i]/(e_i^{n_i})
%$$
%with $n_i=2^{[log_2\frac{n}{i}]+1}$.
%In particular, the class of rational point is equal to the product
%of all $e_i^{n_i-1}$ with odd $i$.


For any subset $I$ of
the set $\{1,2,\dots,n\}$, let us define an element
$e_I\in\ICH(\bar{X})$ as the product $\prod_{i\in I}e_i$.
Defining $|I|$ as $\Sum_{i\in I}i$, we have
$\codim e_I=|I|$.
The element $e_{\{1,2,\dots,n\}}$ of the maximal codimension
$1+2+\dots+n=\dim X$
is equal to the class of a rational point.

A basis of the modulo $2$ Chow
group $\CH(\bar{X})$ is given by the classes of the elements
$e_I$, where $I$ runs over all subsets of
the set $\{1,2,\dots,n\}$ (in particular, the dimension of $\CH(\bar{X})$
(as a vector space over $\Z/2\Z$) is
equal to $2^n$).


\begin{proof}[Proof of Proposition \ref{lemma}]
Assume the contrary: there exists a homogeneous element $\alpha\in\IBCH(X)$
of a positive codimension such that $\alpha\pmod{2}$ is a
non-zero element of $\BCH(X)$.
% different from $[X]$.
Decomposing $\alpha$ in a sum of some $e_I$ (without repetitions) plus
$2\beta$ with some $\beta\in\ICH(\bar{X})$,
let us fix a
%non-empty
set $I$ such that the element $e_I$ occurs in the decomposition.
%polynomial of $e_i$ with odd $i$,
%let us take a $\ne1$ monomial $e_1^{r_1}e_3^{r_3}\dots$.
Let $J$ be the complement of $I$.
Let $m$ be the number of elements in $J$
(note that $m<n$).
Then the product $2^me_J$ is rational.
We claim that the degree of the rational $0$-cycle $\alpha\cdot(2^m e_J)$
is an odd multiple of $2^m$:
indeed, the product
$e_I\cdot(2^m e_J)=2^me_{\{1,2,\dots,n\}}$
has the degree $2^m$,
while the product $e_{I'}\cdot(2^m e_J)$ for any $I'\ne I$ with $|I'|=|I|$
as well as the product $(2\beta)\cdot(2^m e_J)$ are $0$ modulo $2^{m+1}$.
We have got a contradiction with the assumption on $d(X)$.
\end{proof}

In the proof of Theorem \ref{prop}, which follows, we
use a motivic decomposition of $X\times X$ (in the category of the integral Chow
motives), produced in \cite{Chernousov-Merkurjev}.
This motivic decomposition arises from the relative cellular structure
on $X\times X$, where the cells are the orbits of the diagonal
$G$-action for $G=\SO(\phi)$.
Every summand of this decomposition is a Tate twist of the motive
of $X$.
%some projective $G$-homogeneous variety having a canonical
%morphism to $X$
%in the sense of \cite[rem. 4.2]{Chernousov-Merkurjev}.
%Since the only projective $G$-homogeneous variety having a canonical
%morphism to our $X$ is $X$ itself,
%every summand of this decomposition is a Tate twist of the motive
%of $X$.
More precisely, there is one copy of the motive of $X$ (without
twist, that is, with the zero twist),
while the remaining summands have some positive
twists (although we do not need the completely precise information,
here it is:
for any $i$, the number of summands
twisted $i$ times is equal to the rank of the group $\ICH_i(\bar{X})$).

To be  absolutely precise, we have to say that the motivic decomposition
of $X$ given in \cite{Chernousov-Merkurjev} is not yet the decomposition
described above:
it also contains motives of certain flag varieties of the tautological
vector bundle on $X$.
However the motive of each such flag variety decomposes in the sum of some
twists of the motive of $X$ by \cite{Koeck}.

\begin{proof}[Proof of Theorem \ref{prop}]
First of all, since $X$ is projective, the multiplicity
homomorphism factors through the Chow group, so that we have
$\mult\!:\ICH_N(X\times X)\to \Z$, where $N=\dim X= n(n+1)/2$.
Since the multiplicity of a cycle is not changed under extensions
of the base field, the multiplicity homomorphism factors even through the reduced
Chow group, so that we may replace $\ICH(X\times X)$ by $\IBCH(X\times
X)$.
Since we are interested in multiplicities modulo $2$, we consider
the induced homomorphism of the modulo $2$ Chow group
(still denoted by $\mult$):
$\mult\!:\BCH_N(X\times X)\to \Z/2\Z$.

Theorem under proof claims that the image
of the homomorphism
$$
f\!:\BCH(X\times X)\to \Z/2\Z\oplus\Z/2\Z\;,\;\;
f\!:\alpha\mapsto\big(\mult(\alpha),\mult(\alpha^t)\big)
$$
is contained in the diagonal subgroup of $\Z/2\Z\oplus\Z/2\Z$.
Using the described above motivic decomposition of $X\times X$, we
get a decomposition of $\BCH_N(X\times X)$ in the direct sum,
where the summands are:
one copy of $\BCH_N(X)$ and several copies of $\BCH_i(X)$
with various $i<N$.
Since $\BCH_i(X)=0$ for any $i<N$ by Proposition \ref{lemma},
the image of the homomorphism $f$
%$f\!:\BCH(X\times X)\to \Z/2\Z\oplus\Z/2\Z$,
%$f\!:\alpha\mapsto\big(\mult(\alpha),\mult(\alpha^t)\big)$
is cyclic.
Since on the other hand, $f([\Delta_X])=(1,1)$, the image of $f$
is generated by $(1,1)$, that is, coincides with
the diagonal subgroup of $\Z/2\Z\oplus\Z/2\Z$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm}]
We repeat the proof of \cite[th. 11.3]{Berhuy-Reichstein} using
Theorem \ref{prop} instead of \cite[\S7.2]{Merkurjev-SODF}
(and meaning by $X$ our orthogonal grassmanian instead of a Severi-Brauer
variety).

Since the field $F(X)$ is a generic splitting field of $\phi$ and has the
transcendence degree $n(n+1)/2$, the inequality $\cd(\phi)\leq n(n+1)/2$
holds (the assumption on $d(\phi)$ is not needed for this bound).

If now $L$ is another generic splitting field of $\phi$,
%(we may even allow it
%to be non finitely generated),
then we show that $\trdeg(L/F)\geq
n(n+1)/2$ as follows.
%Let $L'$ be a finitely generated subfield of $L/F$ such that the Witt
%index of $\phi_{L'}$ is already maximal (such a subfield always exists).
Let $Y$ be a projective model of $L/F$.
Since both $F(X)$ and $F(Y)$ are generic splitting fields of $\phi$,
there exist rational morphisms
$f\!:X\to Y$ and $g\!:Y\to X$.
Moreover, for any non-empty open subset $U\subset Y$, there exists a
rational morphism $X\to Y$ with an image meeting $U$, so that we
may assume that $f$ and $g$ are composable.
Since the rational map $X\to X$ given by the composition $g\compose f$ is dominant
by Theorem \ref{prop}, the dimension of $Y$ is at least equal to
$\dim X=n(n+1)/2$.
%\cite[th. 4.7]{Hartshorne}
\end{proof}







\begin{thebibliography}{99}
\bibitem{Berhuy-Reichstein}
G. Berhuy and Z. Reichstein.
{\em On the notion of canonical dimension for algebraic groups.}
Linear Algebraic Groups and Related Structures (Preprint Server)
{\bf 140} (2004, Jun 18), 36 p.

\bibitem{Chernousov-Merkurjev}
V. Chernousov, A. Merkurjev.
{\em Motivic decomposition of projective homogeneous varieties
and the Krull-Schmidt theorem.}
Preprint (2004), 16 p.
Available on:
{\tt www.math.ucla.edu/\~{ }merkurev}

\bibitem{Fulton}
W. Fulton.
{\em Intersection Theory.}
Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.


%\bibitem{Hartshorne}
%R. Hartshorne.
%{\em Algebraic Geometry.}
%Graduate Texts in Mathematics {\bf 52}.
%Springer-Verlag; New York, Heidelberg, Berlin, 1977.

\bibitem{aoi}
N. A. Karpenko.
{\em On anisotropy of orthogonal involutions.}
J. Ramanujan Math. Soc. {\bf 15} (2000), no. 1, 1--22.

\bibitem{KM}
N. A. Karpenko, A. S. Merkurjev.
{\em Essential dimension of quadrics.}
Invent. Math. {\bf 153} (2003), 361--372.

%\bibitem{Knebusch}
%M. Knebusch.
%{\em Generic splitting of quadratic forms, I.}
%Proc. London Math. Soc. (3) {\bf 33} (1976), 65--93.

\bibitem{Koeck}
B. K\"{o}ck.
{\em Chow motif and higher Chow theory of $G/P$.}
Manuscripta Math. {\bf 70} (1991), no. 4, 363--372.

\bibitem{Merkurjev-SODF}
A. Merkurjev.
{\em Steenrod operations and degree formulas.}
J. Reine Angew. Math. {\bf 565} (2003), 13--26.


\bibitem{Mimura-Toda}
M. Mimura and H. Toda.
{\em Topology of Lie Groups.}
Providence: ASM (1991).

\bibitem{Scharlau}
W. Scharlau.
{\em Quadratic and Hermitian Forms.}
Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.


\bibitem{Totaro}
B. Totaro.
{\em The torsion index of the spin group.}
Preprint (2004), 35 p.
\end{thebibliography}


\end{document}
