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\begin{document}

\title[Coniveau spectral sequence]{Coniveau spectral sequences of classifying spaces for 
exceptional and Spin groups }

\author{M. Kameko} 

\address{Department of Mathematical Sciences
College of Systems Engineering and Science
Shibaura Institute of Technology,  Saitama, Japan}

\email{kameko@shibaura-it.ac.jp}
 
  
\author{M. Tezuka}
 
\address{ Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan}
 
\email{ tez@sci.u-ryukyu.ac.jp}


\author{N. Yagita}

 \address{Faculty of Education, Ibaraki University,
Mito, Ibaraki, Japan}
\email{yagita@mx.ibaraki.ac.jp}



\keywords{coniveau spectral sequence, motivic cohomology, exceptional groups}
\subjclass{Primary 11E72, 12G05, 55P35; Secondary 55T25,57T05}

\begin{abstract}
Let $k$ be an algebraically closed field of $ch(k)=0$ and $G$ be a simple simply connected 
algebraic group $G$ over $k$.  By using results of cohomological invariants, 
we compute the coniveau spectral sequence for classifying spaces $BG$.
%For each even degree $n\ge 4$, we give an example of  $p$-group $G$ which has a representation
%$W$ over $k$ such that the unramified cohomology $H_{nr}^{n}(k(W)^G;\bZ/p)$ is nonzero.
%Hence $k(W)^G$ is not purely transcendental over $k$.
\end{abstract}


\maketitle



\section{Introduction}

Let $G$ be a simple simply connected algebraic group over
an algebraically closed field $k$ in $\bC$.
The cohomological invariant $Inv^*(G;\bZ/p)$ is (roughly speaking)
the ring of natural maps $H^1(F;G)\to H^*(F;\bZ/p)$ for finitely generated
field $F$ over $k$.
(For detailed definition and properties, see the book [Ga-Me-Se] by Garibaldi, Merkurjev
and Serre.)

Let $BG$ be the classifying space of $G$.  Totaro showed that
\[Inv^*(G;\bZ/p)\cong H^0(BG;H_{\bZ/p}^*)\]
where $H^*(X;H^{*'}_{\bZ/p})$ is the cohomology of the Zarisky sheaf
induced from the presheaf
$H^*_{et}(V;\bZ/p)$ for open subsets $V$ of $X$.
This sheaf cohomology is also the $E_2$-term 
\[E_2^{*,*'}\cong H^*(BG;H_{\bZ/p}^{*'})\Longrightarrow H^*(BG;\bZ/p)\]
of the coniveau spectral sequence by Bloch-Ogus [Bl-Og].

We restrict to consider a group $G$ such that it has only one conjugacy 
class of nontoral maximal elementary abeilan $p$-group $A$.
For exceptional cases, $G=G_2,F_4, E_6$  for $p=2$, $G=F_4,E_6,E_7$ for $p=3$, 
and $G=E_8$ for $p=5$. We also consider groups $Spin_n$, $n\ge 7$.

Let $W_G(A)$ be the Weyl group of $G$ for $A$.
Then by using Rost, Serre and Garibaldi's results [Ga], we easily see that
\[ Res_{Inv} : Inv^*(G;\bZ/p)\cong Inv^*(A;\bZ/p)^{W_G(A)}\]
for cases of the above groups except for $(E_6,p=2)$ and $(Spin_n, p=2),$ $n\ge 10$. 

Let $Q_i$ be the Milnor operation and let 
\[  Q(n)=\Lambda(Q_0,...,Q_n). \]
We easily see that   operations $Q_i$ can extend on $H^*(BG;H_{\bZ/p}^*)$ for these fields
$k$.
In particular, $H^*(BA;H_{\bZ/p}^{*'})^{W_G(A)}$ has also the $Q(\infty)$-module
structure. We can prove that for the above cases except for $(E_7,p=3)$, 
the invariant is generated as $Q(\infty)$-algebras
by elements in $Res(Inv^{*'}(G;\bZ/p))$ and $Res(H^*(BG;H_{\bZ/p}^*))=Res(CH^*(BG)/p)$.
(Moreover  it is a direct sum of free $Q(n)$-modules.)

These facts imply the following theorem. 
\begin{thm}
Let $G=G_2,Spin_n (7\le n\le 9), F_4$ for $p=2$, $G=F_4,E_6$ for $p=3$, or $G=E_8,\ p=5$.
Then the following restriction map 
\[ Res_{E_2}: H^*(BG;H^{*'}_{\bZ/p}) \to  H^*(BA;H^{*'}_{\bZ/p})^{W_G(A)}\]
is an epimorphism.
\end{thm}
For $(E_7,p=3)$, the map $Res_{E_2}$ is not epic, while $Res_{Inv}$ is epic.

We note that the restriction map
\[Res_{H\bZ/p}: H^*(BG;\bZ/p)\to H^*(BA;\bZ/p)^{W_G(A)}\]
is not an epimorphism for $p\ge 3$, 
while $H^*(BA;\bZ/p)\cong H^*(BA;H^{*'}_{\bZ/p})$ as algebras.
(Indeed, $BG$ is $3$-connected but $H^0(BG;H_{\bZ/p}^3)\not =0$.)  
Note also that the right hand side invariant and $Im(Res_{H\bZ/p})$
are computed by Kameko-Mimura [Ka-Mi] for odd primes $p$.

When $p=2$, the maps  $Res_{H\bZ/2}$ are even isomorphic except for the case $E_6$.  
However note ([Or-Vi-Vo]))
\[H^*(BA;H_{\bZ/2}^{*'})\cong grH^*(BA;\bZ/2)\cong \bZ/2[y_1,...,y_n]\otimes
\Lambda(x_1,...,x_n)\]
with $\beta x_i=y_i$ but $y_i\not =x_i^2=0$ as the cases $p=odd$.
So we know 
\[ Res_{H\bZ/2}\cong gr(H^*(BA;\bZ/2)^{W_G(A)})\stackrel{\subset}{\not =} H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}\]
for the above groups.

The arguments seem something subtle and we give here an example, the case
$G=G_2$ and $p=2$.
Then $A\cong (\bZ/2)^3$ and $W_G(A)\cong GL_3(\bZ/2)$,  moreover
\[ H^*(BG_2;\bZ/2)\cong H^*(BA;\bZ/2)^{W_G(A)}\cong \bZ/2[w_4,w_6,w_7]\quad |w_i|=i.\]
The cohomological invariant  is known by Rost and Serre
\[Inv^*(G_2;\bZ/2)\cong \bZ/2\{1,u_3\}\quad |u_3|=3\]
where $u_3=x_1x_2x_3$ in $H^0(BA;H_{\bZ/2}^3)$.
From Mui and Kameko-Mimura results [Ka-Mi], we  can show
\[ H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}\cong \bZ/2[c_4,c_6]\otimes (\bZ/2\{1\}\oplus \bZ/2[c_7]\otimes Q(2)\{u_3\})\]
where $Q_0Q_1Q_2(u_3)=c_7$, $deg(c_i)=(i,i)$ (and $w_i^2=c_i$ in $H^*(BA;\bZ/2)$). 
These $c_i$ are represented by Chern classes, and hence $Res_{E_2}$ is an epimorphism. 

Of course $u_3\not \in Res_{H\bZ/2}$, and moreover we see
\[H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}/Res_{H\bZ/2}\cong \bZ/2[c_4,c_6]\{u_3\}.\]
For example, we have
\[Q_0(u_3)=w_4,\quad Q_1(u_3)=w_6,\quad Q_0Q_1(u_3)=w_7,\quad Q_2(u_3)=w_4w_6.\]
Here $u_3$ does not exist in $H^*(BG_2;\bZ/2)$,  and hence $d_r(u_3)=y\not =0$ for some $r\ge 2$
and $y\in H^*(BG_2;H_{\bZ/2}^*)$ in the coniveau spectral sequence.  We can see this $r=2$.

\begin{thm} 
We have the epimorphism (as bidegree $Q(2)$-modules) from 
 $H^*(BG_2;H_{\bZ/2}^{*'})$ onto 
\[H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}\oplus  (H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}/Res_{H\bZ/2})(-1)[2]\]
\[\cong \bZ/2[c_4,c_6]\otimes
(\bZ/2\{1,y\}\oplus \bZ/2[c_7]\otimes Q(2)\{u_3\})\]
where $(-1)[2]$ is the degree shift operation so that $deg(y)=(2,2)$.
Moreover  $d_2(u_3)=y$ in the coniveau spectral sequence.

 \end{thm}

Moreover if the Gottlieb transfer exists in the motivic cohomology, then
the above epimorphism is indeed isomorphism.
The similar fact also holds for $G=Spin_7$ and $p=2$.

%\begin{cor}  Let $p=2$ and $G=G_2$ or $Spin_7$.  Then as bidegree modules,
%the motivic cohomology $H^{*+*',*'}(BG;\bZ/2)$ is isomorphic to
%\[\bZ/2[\tau]\otimes(Res_{H\bZ/2})\oplus %(H^*(BA;H_{\bZ/2}^{*'})^{W_G(A)}/Res_{H\bZ/2})(-1)[2]\]
%where $\tau\in H^{0,1}(k;\bZ/2)\cong \bZ/2$ is the generator.
%\end{cor}
%The above corollary was given in [Ya3]
%with the assumption of the existence of the motivic Gottlieb transfer.


Note that $y$ in the above theorem, is a ($mod(p)$) Griffith element, namely, 
\[y\in Ker(cycle \ map : CH^*(BG)/p\to H^{2*}(BG;\bZ/p)).\]
Each non zero element in $(H^*(BA;H_{\bZ/p}^{*'})^{W_G(A)}/Res_{H\bZ/p})$
of $deg=(*-2,*+1)$ corresponds to a Griffith element of $deg=(*,*)$.  So we can construct many
Griffith elements in $CH^*(BG)/p$ for the above groups $G$.  

An outline of this paper is following.  In $\S 2$, we recall the relation between the motivic cohomology
$H^{*,*'}(X;\bZ/p)$ and the sheaf cohomology $H^*(X;H_{\bZ/p}^{*'})$.
In $\S 3$, we show that $H^*(X;H_{\bZ/p}^{*'})$ has the $Q_i$-action.
     In $\S 4$,
 we recall the cohomological invariant and give a sufficient condition such that
$Res_{E_2}$ is epic when $Inv^*(G;\bZ/p)$ is known.  
  In $\S 5$, we study the Dickson invariant for $H^*(BA;H_{\bZ/p}^{*'})$
using $Q_i$ actions by Kameko-Mimura.     In $\S 6-\S 8$, we  compute $H^*(BG;H_{\bZ/p}^{*'})$
for concrete cases, e.g., $(G_2,p=2)$ is studied in $\S 6$.
In $\S 9$, we study the relation between $H^*(BG;H_{\bZ/p}^{*'})$ and the Brown-Peterson theory
$BP^*(BG)$.  In the last section, we study the image of Griffith elements to $BP^*(BG)\otimes_{BP^*}\bZ/p$,
in particular, for $(Spin_9,p=2)$. 


\section{motivic cohomology}  

Let $X$ be a smooth (quasi projective) variety
over a field  $k\subset \bC$.
Let $H^{*,*'}(X;\bZ/p)$ be the $mod(p)$ motivic cohomology defined by Voevodsky and Suslin
([Vo1-3]).



Recall that the ($ mod\ p$) $B(n,p)$ condition holds if 
\[H^{m,n}(X;\bZ/p)\cong H_{et}^m(X;\mu_p^{\otimes n})\qquad for\ all\ m \le n.\]
Recently M.Rost and V.Voevodsky ([Vo5],[Su-Jo],[Ro]) proved  
that $B(n,p)$ condition holds for
each $p$ and $n$. Hence the Bloch-Kato conjecture also holds.
Therefore in this paper, we $always$ $assume$ the $B(n,p)$-$condition$ 
and also the Bloch-Kato conjecture for all $n,p$.


 Moreover we always assume that $k$ contains a primitive $p$-th $root$ $of$
 $unity$.
 For these cases, we see the isomorphism
$H_{et}^m(X;\mu _p^{\otimes n})\cong H_{et}^m(X;\bZ/p).$ 
Let $\tau$ be a generator of  $H^{0,1}(Spec(k);\bZ/p)\cong \bZ/p$,
so that 
\[ colim_{i}\tau^iH^{*,*'}(X;\bZ/p)\cong  H_{et}^*(X;\bZ/p).\]

Let $H^*(X;H_{\bZ/p}^{*'})$ be the sheaf cohomology 
 where $H_{\bZ/p}^n$ is the Zarisky sheaf induced from the
  presheaf $H_{et}^n(V;\bZ/p)$ for open subset $V$ of $X$. 
 
Let $X=\bigcup U_{\lambda}$ for Zarisky open sets $U_{\lambda}$.
The sheaf cohomology $H^*(X;H_{\bZ/p}^{*'})$ is defined as the colimit of 
the cohomology of the following $\check{C}$eck complex
\[ (2.1)\qquad \to \prod \Gamma_{(i_1,...,i_n)}\stackrel{\delta}{\to}  
     \prod \Gamma_{(j_1,...,j_{n+1})} \to\]
 \[\qquad     where \quad \Gamma_{(i_1,...,i_n)}=\Gamma(U_{i_1}\cap...\cap U_{i_n};
      H^*(U_{i_1}\cap...\cap U_{i_n};\bZ/p)^a) \]
and  where $H^*(-;\bZ/p)^a$ is a sheaficication of the presheaf $H^*(-;\bZ/p)$. 
Here $\delta$  is induced map from  the inclusions $U_{i}\cap U_j \subset U_i,$
 $U_{i}\cap U_{j}\subset U_j$.
   
  
  
  
  
  
  
  
  
The
Beilinson and Lichtenbaum conjecture ( hence $B(n,p)$-condition )
(see [Vo2,5]) implies the exact sequences of cohomology theories
 \begin{thm} ([Or-Vi-Vo], [Vo5]) There is a long exact sequence 
  \[  \to H^{m,n-1}(X;\bZ/p)\stackrel{\times \tau}{\to}
  H^{m,n}(X;\bZ/p)\qquad \qquad \qquad \]
  \[\qquad \qquad \to
H^{m-n}(X;H_{\bZ/p}^n)\to H^{m+1,n-1}(X;\bZ/p)\stackrel{\times \tau}{\to} .\]
\end{thm}
 In particular, we have 
\begin{cor} We have the additive isomorphism
\[H^{m-n}(X;H_{\bZ/p}^n)\cong H^{m,n}(X;\bZ/p)/(\tau)\oplus
Ker(\tau)|H^{m+1,n-1}(X;\bZ/p)\]
where $H^{m,n}(X;\bZ/p)/(\tau)=H^{m,n}(X;\bZ/p)/(\tau H^{m,n-1}(X;\bZ/p))$.
\end{cor}



Note that the long exact sequence in Theorem 2.1 
induces the $\tau$-Bockstein spectral sequence
\[E(\tau)_1=H^{m-n}(X;H_{\bZ/p}^n)\Longrightarrow 
colim_{i}\tau^iH^{*,*'}(X;\bZ/p)\cong  H_{et}^*(X;\bZ/p).
\] 


On the other hand, the filtration $coniveau$ is given by
\[N^cH^m_{et}(X;\bZ/p)=\cup_{Z} Ker\{H_{et}^m(X;\bZ/p) \to H^m_{et}(X-Z;\bZ/p)\}\]where $Z$ runs in the set of closed subschemes of $X$ of $codim=c$. The induced spectral sequence is called the coniveau spectral sequence.
Bloch-Ogus [Bl-Og] proved that its $E_2$-term is given by
\[E(c)_2^{c,m-c}\cong 
%\frac{Ker(\Pi _{x\in X^{(c)}}H^{c-m}_{et}(k(x);\bZ/p)
%\to \Pi_{x\in X^{(c+1)}}H^{m-1}_{et}(k(x);\bZ/p))}
%{Im(\Pi _{x\in X^{(c-1)}}H^{c-m+1}_{et}(k(x);\bZ/p)
%\to \Pi_{x\in X^{(c)}}H^{m}_{et}(k(x);\bZ/p))}\]
%\[\cong 
H^c(X,H_{\bZ/p}^{m-c}).\]






By Deligne ( foot note (1) in Remark 6.4 in [Bl-Og]) and
Paranjape (Corollary 4.4 in [Pj]), it is proven that
there is an isomorphism of the coniveau spectral sequence
with the Leray spectral sequence for the natural map
of the sites.
Hence we have ; 
   


\begin{thm} (Deligne, Parajape)
There is the isomorphism $E(c)_r^{c,m-c}\cong E(\tau)_{r-1}^{m,m-c}$
for $r\ge 2$ of spectral sequences.
Hence the filtrations are the same $N^cH_{et}^m(X;\bZ/p)= F_{\tau}^{m,m-c}$
where 
\[F_{\tau}^{m,m-c}=Im(\times \tau^c:H^{m.m-c}(X;\bZ/p)\to H^{m,m}(X;\bZ/p)).\]
\end{thm} 


 
 
 
 
 
 
 
 
 






\section{cohomology operation}

Let $t_{\bC}: H^{*,*'}(X;\bZ/p)\to H^*(X(\bC);\bZ/p)$ be the realization map
([Vo1]) for the inclusion $k\subset \bC$.
The motivic cohomology has (Bockstein, reduced powered) cohomology operations
 ([Vo2,4])
\[\beta:H^{*,*'}(X;\bZ/p)\to H^{*+1,*'}(X;\bZ/p)\]
\[P^i:H^{*,*'}(X;\bZ/p)\to H^{*+2i(p-1),*'+i(p-1)}(X;\bZ/p)\]
which are compatible with the usual (topological) cohomology operations
by the realization map $t_{\bC}$.
Voevodsky  defines the Milnor operation $Q_i$ also in the mod $p$ motivic cohomology
\[Q_i:H^{*,*'}(-;\bZ/p)\to H^{*+2p^i-1,*'+p^i-1}(-;\bZ/p).\]


Here we define the weight degree by
\[ w(x)=2n-m\ (resp.\ =n'-m')\]
for $ 0\not =x\in H^{m,n}(X;\bZ/p)$ (resp. $H^{m'}(X;H_{\bZ/p}^{n'})$).
Similarly, we also define the weight degree for cohomology operations and 
differentials of spectral sequences,e.g.,
\[ w(\tau)=2,\quad  w(P^i)=0,\quad w(Q_j)=-1.\]




Let $\rho_p=(\xi_p)\in  k^*/(k^*)^p=H^{1,1}(Spec(k);\bZ/p)$ where 
$\xi_p$ is the primitive $p$-th
root of unity.  The $Q_i$ operation has
the same property as the topological case only with $mod(\rho_2)$.  For example,
$Q_i$ is a derivative only $mod(\rho_2)$.

Let $A_p$ be the mod $p$ Steenrod algebra generated by all cohomology operations 
on $H^{*,*'}(X;\bZ/p)$. ( Voevodsky proved that $A_p$ is multiplicatively generated
by elements in $H^{*,*'}(Spec(k);\bZ/p)$, $P^j$ and $Q_i$.)
\begin{lemma}
Suppose $\rho_p=0$.  Then the  Steenrod algebra $A_p$ acts on
the etale cohomolgy $H^*(X;\bZ/p)$.
\end{lemma}
\begin{proof} 
In $H^{*,*'}(Spec(k);\bZ/p)$, we know
\[ P^i(\tau)=0\ for \ i>0,\quad and \quad \beta(\tau)=\rho_p=0.\]
When $p\ge 3$, the Cartan formula holds in the motivic cohomology
(Proposition 9.6 in [Vo4]), and we have 
  \[ P^i(\tau x)=\tau P^i(x)\ for \ i>0,\quad and \quad \beta(\tau x)=\tau \beta(x).\]
From the $B(n,p)$ condition, 
$H_{et}^*(X;\bZ/p)=colim _{i}\tau^{i}H^{*,*'}(X;\bZ/p),$
which implies the lemma.

For $p=2$, we also know from Proposition 9.6 in [Vo4],
\[Sq^{2*}(xy)=\sum_iSq^{2i}(x)Sq^{2*-2i}(y)+\tau \sum_iSq^{2i+1}(x)Sq^{2*-2i-1}(y),\]
\[Sq^{2*+1}(xy)=\sum_jSq^{j}(x)Sq^{2*+1-j}(y)+\rho_2\sum_iSq^{2i+1}(x)Sq^{2*-2i-1}(y).\]
Since $\rho_2=0$, we see $Sq^{2i+1}(\tau)=0$, and so $Sq^*(\tau x)=\tau Sq^*(x)$.  
This also induces the lemma.
\end{proof}

\begin{thm}
Suppose $\rho_p=0$.  Then the cohomology operation $Q_i$ and $P^i$ can be extended  on
the $\tau$-Bockstein spectral sequence and so on
the coniveau spectral sequence $E_r$, $r\ge 2$ (e.g., on $H^*(X;H_{\bZ/p}^{*'}$)).
\end{thm}
\begin{proof}In the stable $\bA^{1}$-homotopy category $SHot$,
let $H\bZ/p$ be the Eilenberg-MacLane spectrum representing the mod $p$ motivic cohomology
\[ H^{*,*'}(X;\bZ/p)\cong Hom_{SHot}(X,S^{*,*'}\wedge H\bZ/p)\]
where $S^{*,*'}$ is the sphere of bidegree $(*,*')$.

Let $op.=Q_i$ or $P^i$ of bidegree $(m,n)$.  Consider the diagram
\[ \begin{CD}
  S^{0,1}\wedge H\bZ/p   @>{\times \tau}>> H\bZ/p @>{\rho}>> cone\\
        @V{op.}VV  @V{op.}VV @. \\
       S^{m,n+1}\wedge  H\bZ/p @>{\times \tau}>> S^{m,n}\wedge  H\bZ/p @>{\rho}>> S^{m,n}\wedge cone.
\end{CD} \]
Here $cone$ is the mapping cone of $\tau$ so that
\[ H^{*+*'}(X;H_{\bZ/p}^{*'})\cong Hom_{SHot}(X,S^{*,*'}\wedge cone).\]
In the above diagram, we see  $\rho\cdot op.\cdot \tau=0$.
Hence there is a map $op'.: cone \to S^{m,n}\wedge cone$ such that
$\rho\cdot op.=op'.\cdot \rho$. 
\end{proof}

Here we do not see yet that $A_p$ acts on $E_r$,
e.g., we do not see that $Q_i$ generates the exterior algebra $Q(\infty)$.
However when $r=2$, the following theorem holds.
        
\begin{lemma}
Let $k$ be an algebraically closed field.  Then the  Steenrod algebra $A_p$ acts on
 $H^*(X;H_{\bZ/p)}^{*'})$.
\end{lemma}
\begin{proof} 
Recall that $H^*(X;H_{\bZ/p}^{*'})$ is defined as the cohomology of
the $\check{C}$eck complex.
 Given $op. \in A_p$, by the universality of sheaficication,  the following diagram
from (2.1) is commutative 
\[\begin{CD}  \prod \Gamma_{({i_1},...,{i_n})}
       @>{\delta}>>   
     \prod \Gamma_{({j_1},...,{j_{n+1}})} \\
        @V{op.}VV       @V{op.}VV   \\
        \prod \Gamma_{({i_1},...,{i_n})}
       @>{\delta}>>   
     \prod \Gamma_{({j_1},...,{j_{n+1}})}. 
 \end{CD}\]
Thus we have the desired result.
\end{proof} 





%\begin{lemma}
%Suppose $\rho_2=0$.  Let $i:Z\to X$ be a regular closed embedding of $codim=c$.
%Then the Gysin map
%\[ i_*:H^{*,*'}(Z;\bZ/p)\to H^{*+2c,*'+c}(X;\bZ/p)\]
%commutes $Q_i$ for all $i\ge 0$.
%\end{lemma}
%\begin{proof}
%Let $Th_Z(N)$ be the Thom space of the normal bundle $N$ for the embedding $i$,
%and $Th(1)\in H^{2c,c}(Th_Z(N);\bZ/p)$ the Thom class so that the Thom isomorphism
%\[ Th: H^{*,*'}(Z;\bZ/p)\cong H^{*+2c,*'+c}(Th_Z(N);\bZ/p)\]
%is given by $Th(x)=Th(1)\cdot x$. By Voevodsky ([Vo1,2]), it is known
%\[ H^{*,*'}(Th_N(X);\bZ/p)\cong H^{*,*'}(X/(X-Z);\bZ/p).\]
%
%The Gysin map $i_*$ is given as
%\[ i_*=q^*Th: H^{*,*'}(Z;\bZ/p)\to H^{2*+2c,*'+c}(Th_Z(X);\bZ/p)\cong\]
%\[ H^{*+2c,*'+c}(X/(X-Z);\bZ/p)\to  H^{2*+2c,*'+c}(X;\bZ/p)\]
%where $q:X\to X/(X-Z)$ is the projection.
%Here note $Q_i(Th(1))=0$ since $w(Th(1))=0$.
%Since  $Q_i$ is a derivation, we get
%\[ Q_i(i_*(x))=q^*(Q_i(Th(1)x))=q^*(Th(1)Q_i(x))=i_*(Q_i(x)).\]
%\end{proof}
%
%\begin{thm}
%Suppose $\rho_p=0$.  Then the cohomology operation $Q_i$ acts on
%the coniveau spectral sequence (e.g., on $H^*(X;H_{\bZ/p}^{*'}$)).
%\end{thm}
%\begin{proof} 
%The coniveau filtration is defined as
%\[N^cH^m_{et}(X;\bZ/p)=\cup_{Z} Ker\{H_{et}^m(X;\bZ/p) \to H^m_{et}(X-Z;\bZ/p)\}\]
%where $Z$ runs in the set of closed subschemes of $X$ of $codim=c$. 
%The coniveau spectral sequence is given by taking some limit (see  ? ) of the following arguments.%
%
%Let 
%\[...\subset Z_{d+1}\stackrel{i_{d+1}}{\subset}
% Z_{d}\stackrel{i_{d}}{\subset}...\subset Z_0=X\]
% be a chain of smooth closed embeddings of $Z_d$ with $codim(Z_d)=d$.
% 
% For ease of notations, let us write $H^*_{et}(-;\bZ/p)$ simply by $H^*(-)$.
% Let $j_i: Z_{i-1}-Z_{i} \subset Z_{i-1}$ be the open imersion, and 
% $j^*$ is the induced map in cohomology.  Let 
% $\delta_{i}^*: H^*(Z_{i-1}-Z_{i})\to H^{*+1}(Z_{i-1}/Z_{i-1}-Z_i)$
%be the boundary map. 
%Then we have the diagram
%such that the triangles  $ \ j^* \stackrel{\stackrel{i_*}{\gets}}{ \searrow
%\uparrow}  Th^{-1}\delta^*\ $  are exact ;
%\[
%\begin{CD}
%H^*(X)@<{i_{1*}}<< H^{*-2}(Z_1) @<{i_{2*}}<< H^{*-4}(Z_2)
%@<{i_{3*}}<< H^{*-6}(Z_3) \\
%@.  @A{j_{1}^*\searrow\qquad Th^{-1}\delta_{1}^*}A{\qquad j_{2}^* \searrow}A   
%      @A{Th^{-1}\delta_{2}^*}A{\qquad j_{3}^* \searrow}A 
%@A{Th^{-1}\delta_{3}^*}AA \\
%@. H^{*}(X-Z_1)@>d_{1}>> H^{*-2}(Z_1-Z_2)@>d_{1}>> H^{*-4}(Z_2-Z_3)
%\end{CD}
%\]
%This diagram induces the spectral sequence with the differential
%\[ d_r=j^*(i_*)^{-r+1}Th^{-1}\delta^*.\]
%Hence the differential commutes $Q_i$ for all $i\ge 0$
%from the preceding lemma.
%Since the coniveau spectral sequence is the limit of this spectral sequence,
%we can see the theorem.
%\end{proof}

Let us write $H^{*,*'}=H^{*,*'}(Spec(k);\bZ/p)$ and $H^*=K_M^*(k)/p$
so that $H^{*,*'}\cong H^*[\tau].$  (Note if $k$ is algebraically closed, $H^{*,*'}\cong \bZ/p[\tau]$.)
For an elementary abelian $p$-group $A=A_n\cong (\bZ/p)^n$, the $mod(p)$ motivic cohomology
is given by Voevodsky ([Vo2,4])
\[  H^{*,*'}(BA;\bZ/p)\cong H^{*,*'}[y_1,...,y_n]\otimes \Delta (x_1,..,x_n)
\]
with $x_i^2=y\tau +x\rho_2$ for $p=2$ and $x_i^2=0$ otherwise.

Since $Ker(\tau)|H^{*,*'}(BA;\bZ/p)=0$, from Corollary 2.2, we have 
\[H^*(BA;H_{\bZ/p}^{*'})\cong H^{*,*}(BA;\bZ/p)/(\tau H^{*,*-1}(BA;\bZ/p))\]
\[\cong H^{*}[y_1,...,y_n]\otimes \Lambda(x_1,...,x_n)\quad (mod(\rho_2))\]
for all primes $p$.
Each $Q_i$ is a derivation $mod(\rho_2)$, and hence
\[Q_0...Q_{s-1}( x_{1}...x_{s})=
\sum sgn(j_1,...,j_s) y_{1}^{p^{j_1}}y_2^{p^{j_2}}...y_{s}^{p^{j_s}}\not =0\quad mod(\rho_2)\]
where $(j_1,...,j_s)$ are permutations of $(0,...,s-1)$.

Let us write
\[Q(n)=\Lambda(Q_0,...,Q_n),\]
\[\bar Q(n)=Q(n)-\bZ/p\{Q_0...Q_{n}\}=\bZ/p\{Q_{i_0}...Q_{i_s}| 0\le i_k\le n,\ s<n\}.\]
Let $u_i=x_1...x_i\in H^0(BA;H_{\bZ/p}^i)$.  For example, we have
\[ H^*(BA;H_{\bZ/p}^{*'})\supset H^*[y_1,...,y_n]\otimes (\oplus_i \bar Q(i-1)\{u_i\})\]
\[      \supset \oplus_iH^*[y_1,...,y_i]\otimes Q(i-1)\{u_i\}
\quad (*)\]
since   $Q_0...Q_{i-1}(u_i)\in H^*[y_1,...,y_i]\{y_1...y_i\}$.
In sections bellow, we show that the last sum $(*)$ of free $Q(i-1)$-modules contains
$H^*(BG;H_{\bZ/p}^{*'})^{W_G(A)}$, as a direct summand , for many cases of $G$.
(See Assumption (1) in $\S 4$.)










\section{cohomological invariant}


Let $G$ be a linear algebraic group over $k$.
Recall that $H^1(k;G)$ is the first non abelian Galois cohomology set of $G$, which represents the set
of $G$-torsors over $k$. 
The cohomology invariant is defined by 
\[ Inv^i(G,\bZ/p)=Func( H^1(F;G)\to H^i(F;\bZ/p))\]
where $Func$ means natural functions for each field $F$ which is finitely generated over $k$.
(For details for the definition or properties, see the book [Ga-Me-Se].)

Totaro proved [Ga-Me-Se] the following theorem in the letter to Serre.
\begin{thm} (Totaro)
$Inv^*(G;\bZ/p)\cong H^0(BG;H_{\bZ/p}^*).$
\end{thm}

Hereafter (throughout this paper),  we assume that $k$ is an $algebraically$ $closed$
field in $\bC$.
Moreover, in this paper, we only consider $simple$ $simply$ $connected$ groups $G$ which have
 the following property.
we assume that the algebraic group $G$ has only $one$ conjugacy class $A$ of non toral
maximal elementary abelian
$p$-subgroups. Exceptional groups are
\[ G=\begin{cases} G_2,F_4,E_6 \quad for \ p=2\\
                   F_4,E_6,E_7 \quad for \ p=3\\
                   E_8\quad for\ p=5.
                   \end{cases}\]
                   For spin groups $Spin_n$, we consider the cases $n\le 9$ only in this paper.
                   
We consider the restriction maps (of cohomology) to $A$ and the maximal torus $T_G$
\[ Res_{H\bZ/p} : H^*(BG;\bZ/p)\stackrel{i^*}{\to}H^*(BT_G;\bZ/p)\times H^*(BA;\bZ/p)\]
\[ \stackrel{pr.}{\to}H^*(BA;\bZ/p)^{W_G(A)}.\]
By the Quillen's theorem the above $i^*$ has nilpotent kernel. More strongly,
Toda, Kono, Tezuka and Kameko show $i^*$ is  really injective, namely,
$H^*(BG;\bZ/p)$ is detected by $A$ and $T_G$.
Moreover when $p=2$,  $Res_{H\bZ/2}$ are isomorphic except the case $E_6$.
However $Res_{H\bZ/p}$ is not epic for $p\ge 3$.

On the other hand, by Serre, Rost and Garibaldi([Ga-Me-Se],[Ga]), 
 $Inv^*(G;\bZ/p)$ are computed for these groups, e.g.,
\[ Inv^*(G;\bZ/p)\cong \begin{cases}
    \bZ/p\{1,u_3\}\quad for\ (G_2,E_6,p=2),\ (F_4,E_7,p=3),\\
        \qquad \qquad (E_8,p=5)\\
    \bZ/p\{1,u_3,u_4\}\quad for\ (Spin_7,p=2),\ (E_6,p=3)\\
    \bZ/p\{1,u_3,u_4,u_5\}\quad for\ (Spin_9,p=2)\\
    \bZ/p\{1,u_3,u_4,u_4',u_5\}\quad for\ (Spin_8,p=2)\\
    \bZ/p\{1,u_3,u_5\}\quad for\ (F_4,p=2).
    \end{cases}\]
  (Moreover Rost and Garibaldi determined $Inv^*(Spin_n;\bZ/2)$ for $n\le 12$).
    
    For these groups, we note (Ga-Me-Se],[Ga]) the the restriction
    \[Res_{Inv}: Inv^*(G;\bZ/p)\to  Inv^*(A;\bZ/p)\cong \Lambda(x_1,...,x_n).\]
    is injective (identifying $u_i=x_1...x_i$ and $u_4'=x_1x_2x_3x_5$).   
    We will show the following theorem in $\S 6-8$ bellow
  (by computations of concrete cases)
    \begin{thm} Let $G$ be an above type except for $G=E_6$ and $p=2$.
      Then 
    \[Res_{Inv}: Inv^*(G;\bZ/p)\cong Inv^*(A;\bZ/2)^{W_G(A)}.\]
    \end{thm}
    {\bf Remark.}  When $G=E_6$ and $p=2$, the above $Res_{Inv}$ is not epic.
    
    We want to extend above isomorphism in the theorem to say that
    \[ Res_{E_2}: H^*(BG;H_{\bZ/p}^{*'}) \to H^*(BA;H_{\bZ/p}^{*'})^{W_G(A)}\]
    is an epimorphism. (Of course for $p\ge 3$ the above map is not injective.)
    We will prove the following assumption (in the sections bellow) for the above groups
     except for $(E_6,p=2)$ and $(E_7,p=3)$.
    (When $G=Spin_8$, some modification of Assumption (1) holds.)
    
    {\bf Assumption }  
      When  $Inv^*(G;\bZ/p)\cong \bZ/p\{1,u_{i_1},...,u_{i_m}\}$, 
      there is a bidegree isomorphism
     \[ (1)\quad H^*(BA;H_{\bZ/p}^{*'})^{W_G(A)}
            \cong  \oplus _{s=1}^{m}\bZ/p[f_{s1},...,f_{sk_{s}}]\otimes Q(i_s-1)\{u_{i_s}\}\] 
     \[(2)\quad f_{st}  \in   Res_{H\bZ/p}(H^{2*,*}(BG;{\bZ/p}))=Res(CH^{*}(BG)/p)\]
     \[\qquad \qquad \qquad         for\ all \ 1\le s\le m,\ 1\le t\le k_s.\]

   If this assumption is satisfied then $H^*(BA;H^{*'})^{W_G(A)}$ is generated as 
  bidegree $Q(\infty)$-algebra by $u_{i_s}$ and $Res(CH^*(BG)/p)$.  Hence the surjectivity
of $Res_{E_2}$ is immediate. 
\begin{lemma}
If Assumption (1),(2) are satisfied, then
 \[ Res_{E_2}: H^*(BG;H_{\bZ/p}^{*'}) \to H^*(BA;H_{\bZ/p}^{*'})^{W_G(A)}\]
    is  an epimorphism. 
\end{lemma}
Thus we can prove Theorem 1.1 in the introduction.
As for the statements of differential and (Griffith elements),
the following lemma is useful.
\begin{lemma}
Let $Res_{Inv}(a)\not =0$ for $a\in Inv^i(G;\bZ/p)=H^0(BG;H_{\bZ/p}^*)$.
(Namely,  the above element is a permanent cycle in the coniveau spectral
sequence.)
Moreover let
\[Q_{j_1}...Q_{j_{i-3}}(a)\not \in  H^*(BG;\bZ/p).\]
Then  $d_2(a)=y\not =0\in H^2(BG;H_{\bZ/p}^{i-1})$ in the coniveau spectral sequence, and 
elements  
\[ Q_{j_1}...Q_{j_{i-3}}(y)\not =0 \in CH^*(BG)/p=H^{2*,*}(BG;\bZ/p)\]
are  Griffith elements (i.e., in the kernel of $CH^{2*}(BG)/p\to H_{et}^{2*}(BG;\bZ/p)$).
\end{lemma}
\begin{proof}
Take $q=Q_{j_1}...Q_{j_{i-3}}(a)$.  
Since $q$ does not exist in $H^*(BG;\bZ/p)$, we see
$d_r(q)\not =0$ in the spectral sequence for some $r$.

This $r=2$  because
the following reason of weight degree. First note
\[w(d_r)=wt(1,1-r)=2(1-r)-1=1-2r.\]
Since $w(q)=w(a)-( i-3)=3$, we have
\[ w(d_r(q))=3+1-2r=4-2r.\]
If $r\ge 3$, then the above weight is negative and $d_r(q)=0$.

This implies that $d_2(a)\not =0$.
Otherwise 
\[ d_2(q)=d_2(Q_{j_1}...Q_{j_{i-3}}a)=Q_{j_1}...Q_{j_{n-3}}(d_2(a))=0,\]
 which is a contradiction.
\end{proof}




  
       
    
\section{Dickson invariant} 

At first we assume $p\ge 3$.
Dickson computed the ring of invariants of $\bZ/p[y_1,...,y_n]$ with respect to 
the action of $GL_n(\bZ/p)$. 
The ring of invariants is a polynomial algebra 
\[D_n=\bZ/p[y_1,...,y_n]^{GL_n(\bZ/p)}\cong \bZ/p[c_{n,0},..., c_{n,n-1}]\]
 where the  generators are given by the equation
\[
\mathcal{O}_n(X)=\prod_{y\in \bZ/p\{y_1, \cdots, y_n\}} (X+y)=X^{p^{n}}+\sum_{j=0}^{n-1}(-1)^{n-j} c_{n,j}X^{p^{j}}.
\]

Let $reg:A\to GL_n(\bC)$ be the regular representation and $c(reg)$ the total Chern class.  Then
it is well known that
\[
c(reg)=\mathcal{O}_n(1)=1-c_{n,n-1}+...+(-1)^nc_{n,0}.\]
We also note the following lemma.
\begin{lemma} (Lemma 2.3,2.4 in [Ka-Ya2])
Let $\rho:A_n\to GL_m(\bC)$ be an representation 
such that $c(\rho)\in H^*(BA;\bZ/p)^{SL_n(\bZ/p)}$.
Then $c(\rho)=c(reg)^a$ for some $a\ge 0$.
\end{lemma}
For the invariant ring $SD_n$ under $SL_n(\bZ/p)$, we have
\[ SD_n=\bZ/p[y_1,...,y_n]^{SL_n(\bZ/p)}\]
\[\cong D_n\{1,e_n,...,e_n^{p-2}\}\quad with \ e_n^{p-1}=c_{n,0}\]
\[\cong D_n'\otimes \bZ/p[e_n]\quad with \ D_n'=\bZ/p[c_{n,1},...,c_{n,n-1}].\]
Mui computed the ring of invariants of 
\[ H^*(BA;H_{\bZ/p}^{*'})\cong \bZ/p[y_1,...,y_n]\otimes \Lambda(x_1,...x_n)\]
with respect to the action of $SL_n(\bZ/p)$.
(In fact, Mui studied $H^*(BA;\bZ/p)$ for odd prime $p$, however we study
$H^*(BA;H_{\bZ/p}^{*'})$ for all primes.)
Of course $u_n=x_1...x_n$ is invariant under $SL_n(\bZ/p)$.
In terms of Milnor's operation, we may state Mui's result in the following form.

\begin{thm}(Mui[Mu], Kameko-Mimura [Ka-Mi])
\[H^*(BA;H_{\bZ/p}^{*'})^{SL_n(\bZ/p)}\cong 
\bZ/p[e_n,c_{n,1},...,c_{n,n-1}]\otimes (\bZ/p\{1\}\oplus \bar Q(n-1)\{u_n\})\]
\[\cong D_n'\oplus SD_n\otimes 
Q(n-1)\{u_n\}\]
where $Q_0...Q_{n-1}u_n=e_n$.
\end{thm}

The $Q_i$-operation acts on $u_n$ as follows.
\begin{lemma} (Kameko-Mimura [Ka-Mi],[Ka-Ya1]) For $x\in H^*(BA;H_{\bZ/p}^{*'})$, it holds
\[(Q_n+\sum_{i=0}^{n-1}(-1)^{n-i}c_{n,i}Q_{i})(x)=0,\quad 
Q_0...\hat{Q}_i...Q_n(u_n)=c_{n,i}e_n.\]
\end{lemma}

%\begin{proof}
%Note $Q_i(x_j)=y_j^{p^i}$. When $x=x_j$, we see 
%\[(Q_n+\sum(-1)^{n-i} c_{n,i}Q_i)(x_j)=y_j^{p^n}+\sum c_{n,i}y_j^{p^i}=
%\mathcal{O}_n(x_j)=0.\]
%Since $Q_i$ is a derivation, we get the the first equation.
%Apply $Q_0...\hat{Q}_i...Q_{n-1}$ to the first equation and we get
%the second equation.
%\end{proof}

Let $U_n\subset SL_n(\bZ/p)$ be the maximal unipotent subgroup generated
by upper triangular matrices with diagonals $1$, so that
$U_n$ is a Sylow $p$-subgroup of $SL_n(\bZ/p)$. The invariant under this group 
is given by Mui, Kameko-Mimura.
\begin{thm} (Kameko-Mimura Theorem 4.2 in [Ka-Mi])
Let $G'\subset GL_n(\bZ/p)$ 
such that
$\bZ/p[y_1,...,y_n]^{G'}\cong \bZ/p[f_1,...,f_n]$ and 
\[ H^*(BA_n;H^{*'}_{\bZ/p})^{G'}\cong \bZ/p[f_1,...,f_n]\{v_1=1,...,v_{2^n}\}.\]
Then the invariant under $G=\la G',U_{n+1}\ra \subset GL_{n+1}(\bZ/p)$
is given by
\[(1)\quad \bZ/p[y_1,...,y_{n+1}]^G
\cong \bZ/p[f_1,...,f_n,
\mathcal{O}_n(y_{n+1})]\]
\[(2)\quad H^*(BA_{n+1};H_{\bZ/p}^{*'})^{G}\cong \]
\[ \quad \bZ/p[f_1,...,f_n,\mathcal{O}_n(y_{n+1})]\otimes (\bZ/p\{v_1,...,v_{2^n}\}\oplus Q(n-1)\{u_{n+1}\}).\]
\end{thm}
\begin{cor}
\[H^*(BA;H_{\bZ/p}^{*'})^{U_n}\cong \oplus_{i=0}^n
 \bZ/p[\mathcal{O}_0(y_1),...,\mathcal{O}_{i-1}(y_i)]\otimes
Q(i-1)\{u_i\}.\]
\end{cor}
\begin{cor} (Lemma 5.8 in [Ka-Ya])
\[ \mathcal{O}(y_{n+1})u_{n}=(Q_n+\sum_{i=0}^{n-1}(-1)^{n-i}c_{n,i}Q_{i})(u_{n+1}).\]
\end{cor}

Hereafter this section, we assume $p=2$.
Of course we have the isomorphism
$H^*(BA;\bZ/2)\cong \bZ/2[x_1,...,x_n]$
and its invariant under $GL_n(\bZ/2)=SL_n(\bZ/2)$  is 
\[\bZ/2[x_1,...,x_n]^{GL_n(\bZ/2)}\cong \bZ/2[d_{n,0},..., d_{n,n-1}]\]
 where the  generators are given by the equation
\[
\mathcal{O}_n'(X)=\prod_{x\in \bZ/p\{x_1, \cdots, x_n\}} (X+x)=X^{2^{n}}+\sum_{j=0}^{n-1} d_{n,j}X^{2^{j}}.
\]
Here $d_{n,i}^2=c_{n,i}$ in $H^*(BA;\bZ/2)$ identifying $y_i=x_i^2$.  The Milnor $Q_i$-operations
(see (2.6) in Schuster-Yagita [Sc-Ya]) are given as the case $p$ odd.
(Hereafter let us write $d_{n,i}$ by $d_i$ simply.)
\[d_0=Q_0...Q_{n-2}(u_n),\qquad d_i=Q_0...\hat Q_{i-1}...Q_{n-1}(u_n)/d_0. \]
                      From Lemma 2.1 in [Sc-Ya], we have
                      \[(*)\quad Q_{n-1}(d_i)=d_0d_i,\quad Q_{i-1}(d_i)=d_0.\]
                      
 In $H^*(BA;H_{\bZ/2}^{*'})$, we can get more strong result.
   Let us write simply
  \[ I(GL_n)=gr(H^*(BA_n;\bZ/2)^{GL_n(\bZ/2)})\subset H^*(BA;H_{\bZ/2}^{*'})\]
\[   Igr(GL_n)=H^*(BA_n;H_{\bZ/2}^{*'})^{GL_n(\bZ/2)}.\]
  By Kameko-Mimura theorem, we have showed 
  \[ Igr(GL_n)\cong D_n'\oplus D_n\otimes Q(n-1)\{u_n\} )   \]
  \[   where \quad D_n=\bZ/2[c_{n,n-1},...,c_{n,0}]\quad and \quad
 D_n'=\bZ/2[c_{n,n-1},...,c_{n,1}] .\]
   \begin{lemma}  In $H^*(BA;H_{\bZ/2}^{*'})$, we have
   \[d_i=Q_0...\hat Q_{i-1}...\hat Q_{n-1}(u_n),\quad d_id_0=Q_0...\hat Q_{i-1}... Q_{n-1}(u_n).\]
      \[ d_id_j=Q_0...\hat Q_{i-1}...\hat Q_{j-1}... Q_{n-1}(u_n)\quad i\not =j.\]
      \end{lemma}
       \begin{proof}
   Consider the element
   \[a_i=Q_0...\hat Q_{i-1}...Q_{n-2}\hat Q_{n-1}(u_n)\in Igr(GL_n).\]
   Then $Q_{i-1}(a_i)=d_0$ and $Q_{n-1}a_i=d_0d_i$.
   
   Using property $(*)$, we see
   $ Q_j(a_i-d_i)=0$ for  all $j$.
   Of course $a_i-d_i\in Igr(GL_n)$. From Kameko-Mimura theorem,  we still know
   \[ Igr(GL_n) \cap \cap_jKer(Q_j)=D_n.\]
   This means $d_i=a_i\in Igr(GL_n)$. (In fact $d_i=a_i$ $mod(D_n)$ in $H^*(BA_n;\bZ/2)$,
   but $d_i=a_i$ exactly in $H^*(BA_n;H_{\bZ/2}^{*'})$.)
   
   Therefore we have
   \[d_i=Q_0...\hat Q_{i-1}...\hat Q_{n-1}(u_n),\quad d_id_0=Q_0...\hat Q_{i-1}... Q_{n-1}(u_n).\]
   By the similar arguments, we have
     \[ d_id_j=Q_0...\hat Q_{i-1}...\hat Q_{j-1}... Q_{n-1}(u_n).\]
         \end{proof}
      
    
   






   Of course $I(GL_n)\subset Igr(GL_n)$, but this injection is not an isomorphism for $n\ge 3$.
   \begin{lemma}   Let $n\ge 3$.  Then we have
\[Igr(GL_n)/I(GL_n)\supset  Q(n-1)/(Q(n-1)^+)^{n-2}\{u_n\}.\]
   \end{lemma}
   \begin{proof}
      Consider the element 
   \[ x=Q_0...\hat Q_{i-1}...\hat Q_{j-1}...\hat Q_{k-1}... Q_{n-1}(u_n).\]
   Its image $Q_{i-1}(x)=d_{j}d_{k}$ or $d_{j}$ (when $k=n$.) Hence $x$ is not 
   in $I(GL_n)$ because $Q_i$ maps $n$-product elements into also $n$-product elements.
   (If $x\in I(GL_n)$, then $x$ must be a sum of $d_i$ or $d_id_j$, but it still appeared
   in $(Q(n-1)^+)^{n-2}(u_n)$.)
   Thus we get the lemma.
      \end{proof}
      
      
      Let $A=A_n$ be a maximal elementary abelian $p$-subgroup of $G$ and
      $W_G(A)$ its Weyl group.
      \begin{lemma}
If $Res_{Inv}: Inv^*(G;\bZ/p)\to Inv^*(A_n;\bZ/p)^{W_G(A)}$ is an epimorphism, then
\[ (Q(n-1)^+)^{n-2}\{u_n\}\subset Res_{H_{\bZ/p}}(H^*(BG;\bZ/p)\to H^*(BA_n;\bZ/p))\]
(e.g., $d_1,...,d_{n-1}$ for $p=2$ are in $Res_{H_{\bZ/2}}$).
\end{lemma}
\begin{proof} 
Let $x$ be an element in $H^*(BG;H_{\bZ/p}^{*'})$ with
 \[Res_{E_2}(x)=Q_{i_1}...Q_{i_{n-2}}(u_n).\]
  Then 
$w(x)=n-(n-2)=2$.  However the weight of the differential $d_r$ of the coniveau spectral sequence
is $w(d_r)=1-2r$ (see the proof of Lemma 4.4). Hence $d_r(x)=0$ for $r\ge 2$, namely, $x$ is
a permanent cycle and is an element in $H^*(BG;\bZ/p)$.
\end{proof}

      
      
      
      
      \quad
      
      
      
      
      
  
\section{$W_G(A)\cong SL_3(\bZ/p)$}

We consider the cases of $A\cong (\bZ/p)^3$ and $W_G(A)\cong SL_3(\bZ/p)$,
namely, $(G_2,2)$, $(F_4,3)$ and $(E_8,5)$.  These cases 
\[Inv^*(G;\bZ/p)\cong \bZ/p\{1,u_3\}.\]
(These $u_3$ are  called the Rost invariants.)

 Let $G=G_2$ and $p=2$.   It is well known that
     \[H^*(BG;\bZ/2)\cong I(GL_3)\cong \bZ/2[w_4,w_6,w_7]\]
     where $w_i$ is the Stiefel-Whitney class of $G_2\subset SO_7$.
     We can identify
     \[w_4=d_{3,2},\quad w_6=d_{3,1},\quad w_7=d_{3,0}.\]
     
     On the other hand, Kameko-Mimura theorem implies
     \[Igr(GL_3)\cong \bZ/2[c_4,c_6]\otimes (\bZ/2\{1\}\oplus \bZ/2[c_7]\otimes Q(2)\{u_3\}).\]
          Also by using Lemma 5.8, we can show 
     \[Igr(GL_3)/I(GL_3)\cong \bZ/2[c_4,c_6]\{u_3\}.\]
     In fact (from Lemma 5.7 or from dimensional reason), we have
     \[ Q_0(u_3)=w_4,\ Q_1(u_3)=w_6,\ Q_0Q_1(u_3)=w_7,\ Q_2(u_3)=w_4w_6.\]
     Moreover  we note
     \[c_7u_3=w_4w_6w_7\]
     because both above elements are same after acting $Q_i$, e.g. $Q_0(c_7u_3)=c_7w_4=Q_0(w_4w_6w_7)$
     (see the proof of Lemma 5.7  or see [Ya2]).
     
     Therefore Assumption (1),(2) are satisfied. 
     Moreover from Lemma 4.4, we see $d_2(u_3)=y\not =0$.
     Therefore we have the following theorem.

    
              \begin{thm}There is a $Q(2)$ bidegree module epimorphism
from $H^*(BG_2;H_{\bZ/2}^{*'})$ to
\[ Igr(GL_3)\oplus \bZ/2[c_4,c_6]\{y\}
 \cong \bZ/2[c_4,c_6]\otimes(\bZ/2\{1,y\}\oplus\bZ/2[c_7]\otimes Q(2)\{u_3\}).\]
          \end{thm}
         {\bf Remark.}  If there is a Gottlib transfer in the motivic theory
$H^{*,*'}(-;\bZ/2)$ or the sheaf theory $H^*(-;H^*_{\bZ/2})$, then the 
above epimorphism  is in fact an isomorphism.

          
Next we consider the odd prime cases i.e., $(G,p)=(F_4,3)$ or $(E_8,5)$.
From Kameko-Mimura theorem, we also have
\[ Igr(SL_3)\cong D_3'\oplus SD_3\otimes Q(2)\{u_3\}.\]
Moreover from Kameko (Lemma 5.2 in [Ka-Ya1]), it is known that
\[Igr(SL_3)/Res_{H\bZ/p}\cong SD_3/(e)\{u_3\}\] 
as the case $(G_2,2)$.  Hence Assumption (1) satisfied and Lemma 4.4 can be applied
so that $d_2(u_3)=y$.

To see Assumption (2), we consider the representations.  
We consider the case $(E_8,5)$. (The case $(F_4,3)$ is similar.)
 It is known that there is a non trivial representation ([Ad], [Ka-Ya2])
\[ \rho:E_8\to SO(248).\]
We consider the total Chern class of the representation $\rho|A$ for $A\cong (\bZ/5)^3$,
\[ c(\rho|A)=(1-c_{3,2}+c_{3,1}-c_{3,0})^a\quad for \ a\ge 0\]
from Lemma 5.1.  Since $\rho|A$ is non trivial, $a\ge 1$.
Moreover
\[|c_{3,0}|=2(5^3-1)=248.\]
So $a=1$.  This means that $c_{3,i}$ are represented by  Chern classes.
(We also note $c_{3,1}=P^1c_{3,2}$ for  the reduced power operation $P^1$.)
Hence $w(c_{3,1})=w(c_{3,2})=0$.  Thus we can see Assumption (2).

\begin{thm}  Let $(G,p)=(F_4,3)$ or $(E_8,5)$.
Then there is an epimorphism of $Q(2)$-bidegree modules from $H^*(BG;H_{\bZ/p}^{*'})$ 
to 
     \[ \bZ/p[c_{3,2},c_{3,1}]\otimes(\bZ/p\{1,y\}\oplus\bZ/p[e_3]\otimes Q(2)\{u_3\}).\]
          \end{thm}
          
          \qquad 
          
          
          
          
          
          
 \section{$W_G(A)\cong \la U_4,SL_3(\bZ/p)\ra$}

We consider the cases of $A\cong (\bZ/p)^4$ and 
\[W_4=W_G(A)\cong \la U_4,SL_3(\bZ/p)\ra ,\]
namely, $(Spin_7,2)$, $(E_6,3)$.  
For these cases, we have the isomorphism
\[Inv^*(G;\bZ/p)\cong \bZ/p\{1,u_3,u_4\}.\]
We also study the case $(E_7,3)$, while the above facts do not satisfied.

 Let $G=Spin_7$ and $p=2$.   It is well known that
     \[H^*(BG;\bZ/2)\cong I(W_4)=\bZ/2[x_1,...,x_4]^{W_4}\]
     \[ \cong \bZ/2[w_4,w_6,w_7,w_8]\]
     where $w_8$ is the Stiefel-Whitney class of some spin representation.
     We can identify $w_8=\mathcal{O}_3'(x_4)$.
     
     
 \begin{lemma}
\[ Igr(W_4)\cong Igr(GL_3)\oplus \bZ/2[c_4,c_6]\otimes(\bZ/2[c_8]\{c_8\}\oplus 
 \bZ/2[c_7,c_8]\otimes Q(3)\{u_4\}).\]
\end{lemma}    
\begin{proof}     
    Recall that using $\bar Q(2)$, we have ($Q_2Q_1Q_0(u_3)=c_7$)
     \[Igr(GL_3)\cong \bZ/2[c_4,c_6,c_7]\otimes (\bZ/2\{1\}\oplus \bar Q(2)\{u_3\}).\]
     By Theorem 5.4, we have
     \[ Igr(W_4)\cong \bZ/2[c_4,c_6,c_7,c_8]\otimes (\bZ/2\{1\}\oplus \bar Q(2)\{u_3\}\oplus Q(2)\{u_4\})\]
     \[  \cong \bZ/2[c_4,c_6]\otimes(\bZ/2[c_8]\{1\}\oplus \bZ/2[c_7]\otimes Q(2)\{u_3\}
     \oplus \bZ/2[c_7,c_8]\otimes Q(3)\{u_4\}).\]
     The last isomorphism is shown by using the following facts.
     From Lemma 5.6, we can see (Lemma 5.8 in [Ka-Ya2])
     \[ \mathcal{O}_3(y_4)u_3=(Q_3+c_{3,2}Q_2+c_{3,1}Q_1+c_{3,0}Q_0)(u_4),\]
     namely, $Q_3(u_4)=c_8u_3+c_4Q_2(u_4)+c_6Q_1(u_4)+c_7Q_0(u_4)$.  Hence 
     \[ Q(2)\{u_4\}\oplus Q(2)\{c_8u_3\}\cong Q(3)\{u_4\}.\]
     Using  $Q_0Q_1Q_2Q_3(u_4)=c_7c_8$, we get the last isomorphism.
 \end{proof}
    
     \begin{lemma}
          \[Igr(W_4)/I(W_4)\cong Igr(GL_3)/I(GL_3)\oplus\]
          \[ \bZ/2[c_4,c_6,c_8]\otimes \{1,Q_0,...,Q_3\}\{u_4\}.\]
          \end{lemma}
          \begin{proof}
       At first, we see
\[ Q_1Q_0(u_4)=Q_1Q_0(u_3x_4)=Q_1(w_4x_4+u_3y_4)\]
\[=w_7x_4+w_4y_4^2+ w_6y_4=w_8\quad in\ H^*(BA;H^{*'}_{\bZ/2}).\]
(This fact also follows from $d_{4,3}=w_8$ and Lemma 5.7.)
Similarly, we can compute the $Q_i$ action on $u_4$, which  is given as
  $Q_0Q_1(u_4)=w_8$,\ $Q_0Q_2(u_4)=w_4w_8$,\ $Q_1Q_2(u_4)=w_6w_8$,   
  $Q_0Q_3(u_4)=c_8w_4$,\ $Q_1Q_3(u_4)=c_8w_6$,\ $Q_2Q_3(u_4)=c_8w_4w_6.$
  
  Moreover we have 
\[c_7u_4=c_7u_3x_4=w_4w_6w_7x_4=w_4w_6w_8.\]
    Therefore $Q_iu_3\not \in I(W_4)$ but $Q_{i}Q_{j}(u_3)\in I(W_4)$. Thus we have
    \[Igr(W_4)/I(W_4)\cong \bZ/2[c_4,c_6]\otimes (\bZ/2\{u_3\}\oplus \bZ/2[c_8]\{1,Q_0,...,Q_3\}\{u_4\}).\]
   \end{proof}
          
          
     Therefore Assumption (1),(2) are satisfied.  
Therefore we can compute ;
     
          \begin{thm}There is a $Q(2)$ bidegree module epimorphism from
$H^*(BSpin_7;H_{\bZ/2}^{*'})$ to 
\[  H^*(BG_2;H_{\bZ/2}^{*'})\oplus 
 \bZ/2[c_4,c_6,c_8]\otimes (\bZ/2\{c_8,y_2',Q_0y_2',...,Q_3y_2'\}
\oplus \bZ/2[c_7]\otimes Q(3)\{u_4\})\]
where $Q_0Q_1Q_2Q_3u_4=c_7c_8 $.  The differentials 
$ d_2(u_3)=y_2,$  $d_2(u_4)=y_2'$ in the coniveau spectral sequence.
\end{thm}

{\bf Remark.} If the epimorphism in Theorem 6.1 is an isomorphism,  then
 that in the above theorem is also an isomorphism.
 
{\bf Remark.} The notations in [Ya3] are  given : $a'=u_4$ as a virtual element
and
\[  \xi_3=Q_0y_2',\  \xi_4=Q_1y_2',\ \xi_6=Q_2y_2', \ c_8y_2=Q_3y_2' .\] 

         
Next we consider the odd prime cases i.e., $(G,p)=(E_6,3)$.
Let us denote by $\mathcal{O}$ simply, $\mathcal{O}_3(x_4)$ so that
$e_4=Q_0Q_1Q_2Q_3(u_4)=e_3\mathcal{O}$.
Then from Kameko-Mimura lemma, we also have ([Ka-Mi], 
\[ Igr(W_4)\cong SD/(e)[\mathcal{O}]\oplus SD_3[\mathcal{O}]\otimes Q(2)\{u_3,u_4\}\]
\[ \cong SD_3/(e)[\mathcal{O}]\oplus SD_3\otimes Q(2)\{u_3\}
\oplus SD_3[\mathcal{O}]\otimes Q(3)\{u_4\}.\]
Moreover from Kameko (Lemma 5.2 in [Ka-Ya1]), it is known that
\[Igr(W_4)/Res_{H\bZ/p}\cong SD_3/(e)\otimes(\bZ/3\{u_3\}\oplus \bZ/3[\mathcal{O}]\{u_4,Q_0u_4,...,Q_3u_4\})\] 
as the case $(Spin_7,2)$. 
 Hence Assumption (1) satisfied and Lemma 4.4 can be applied
so that $d_2(u_3)=y$.

To see Assumption (2), we consider the representations.  It is known that there is a non trivial
representation
$E_6\to SO(26)$.  Hence we know that $c_{3,i}$ is represented by Chern classes by the arguments
similar to the case $(F_4,3)$.  As for the element $\mathcal{O}$, we consider the restriction
\[ \la a_4\ra \subset A\subset E_6\stackrel{\rho}{\to} SO(26).\]
Here $a_4\in A$ is the dual of $x_4\in Hom(A;\bZ/3)$.
We see $\mathcal{O}|\la a_4\ra\not =0$ but $SD_3|\la a_4\ra=0$.  Hence the fact
\[ c_{26}(\rho|\la a_4\ra)=y_4^{3^3-1}\not =0\]
implies $\mathcal{O}$ (modulo elements in $SD_3$) can be represented by a Chern class.
Hence Assumption (2) is also satisfied.
\begin{thm}  There is an epimorphism of $Q(3)$-bimodules 

\noindent from 
$H^*(BE_6;H_{\bZ/3}^{*'})$ 
to 
     \[ H^*(BF_4;H^{*'}_{\bZ/3})\oplus 
\bZ/3[c_{3,2},c_{3,1},\mathcal{O}]\otimes(\bZ/3\{\mathcal{O},y_2',Q_0y_2',...,Q_3y_2'\}\] 
\[
\qquad \qquad \oplus\bZ/2[e_3]\otimes Q(3)\{u_4\}).\]
          \end{thm}
     
     At last of this section, we consider the case $(E_7,3)$. 
This case
\[ W_G(A)=W_4'=\la W_4, diag(1,1,1,-1)\ra \subset GL_4(\bZ/3).\]
 The invariant is also computed 
by Kameko-Mimura
\[ Igr(W_4')\cong SD/(e)[\mathcal{O}^2]\oplus SD_3[\mathcal{O}^2]\otimes Q(2)\{u_3,\mathcal{O}u_4\}\]
Moreover from Kameko (page 2279 in [Ka-Ya1]), it is known that
\[Igr(W_4')/Res_{H\bZ/3}\cong SD_3/(e)[\mathcal{O}^2]\otimes(\bZ/3\{u_3\}\oplus Q(2)\{\mathcal{O}u_4\}).\]
It is also known $Inv^*(E_7;\bZ/3)\cong \bZ/3\{1,u_3\}$ and hence
\[Inv^*(E_7;\bZ/3)\cong Inv^*(A;\bZ/3)^{W_4'}\]
from the above result.

There is the natural representation $\rho:E_7\to SO(52)$.  Hence we see that $\mathcal{O}^2$
can be represented by a Chern class.  So Assumption (2) is satisfied.
However Assumption (1) is not.  
\begin{thm} The following restriction map  
\[ Res_{E_2}: H^*(BE_7,H_{\bZ/3}^{*'}) \to H^*(BA,H_{\bZ/3}^{*'})^{W_4'}\]
is not an epimorphism.
\end{thm}
\begin{proof}
Recall arguments in the proof of Lemma 5.9.
Suppose $\mathcal{O}u_4\in Res_{E_2}$ and $Res(x)=\mathcal{O}u_4$
for $x\in H^*(BE_7;H_{\bZ/3}^{*'})$.

Of course $w(x)=w(\mathcal{O}u_4)=4$.
The weight $Q_0Q_1(x)=2$.  Recall $w(d_r)=1-2r$ and 
\[ d_r(Q_0Q_1(x))=0\quad for\ r\ge 2.\]
Hence $Q_0Q_1(x)\in H^*(BE_3;\bZ/3)$ from the coniveau spectral sequence.
So $Q_0Q_1(\mathcal{O}u_4)\in Res_{H\bZ/3}$, which contradicts to  the result above
\[Igr(W_4')/Res_{H\bZ/3}\supset  Q(2)\{\mathcal{O}u_4\}.\]
\end{proof}




\quad

\section{$Spin_9$ for $p=2$}



In this section, we consider the groups $Spin_8, Spin_9,F_4,E_6$ for $p=2$.
At first we consider  the case $G=Spin_9$.
     Then the maximal elementary abelian $2$-group is $rank_2=5$, and the Weyl group is 
      \[ W_G(A)=W_5\cong \la U_5,SL_3\ra \subset SL_5(\bZ/2).\]
         The cohomology is known that
    \[  H^*(BG;\bZ/2)\cong H^*(BA;\bZ/2)^{W_G(A)}\cong  \bZ/2[w_4,w_6,w_7,w_8,w_{16}]\]
    where $w_{16}$ is the Stiefel-Whitney class of some spin representation.
   We can also identify
\[w_{16}=\mathcal{O}_4'(x_5)=x_5^{16}+d_3x_5^8+d_2x_5^4+d_1x_5^2+d_0x_5\]
where $d_3=w_8,\ d_2=w_8w_4,\ d_1=w_8w_6,\ d_0=w_8w_7$ (see the proof of Lemma 7.2 or p1051 in [Sc-Ya]).
   As the case $(*)$ in $\S 5$, we know ([Sc-Ya])
\[ Q_{3}w_{16}=d_0w_{16},\qquad Q_{4}(d_0w_{16})=d_0^2w_{16}^2.\]

     We can prove from Kameko-Mimura theorem
     \begin{lemma}
     \[Igr(W_5)\cong Igr(W_4)\oplus \bZ/2[c_4,c_6,c_{16}]\otimes (\bZ/2[c_8]\{c_{16}\}
\oplus       \]  \[      \bZ/2[c_7]\otimes Q(2)\{c_{16}u_3\}   \oplus  
       \bZ/2[c_7,c_8]\otimes Q(4)\{u_5\})\]
\end{lemma}
\begin{proof} Recall that  
 \[Igr(W_4)\cong \bZ/2[c_4,c_6,c_7,c_8]\otimes (\bZ/2\{1\}\oplus
 \bar Q(2)\{u_3\}\oplus Q(2)\{u_3\}).\]
From Kameko-Mimura theorem (Theorem 5.4) we see that
\[Igr(W_5)\cong \bZ/2[c_4,c_6,c_7,c_8,c_{16}]\otimes (\bZ/2\{1\}\oplus
 \bar Q(2)\{u_3\}\oplus Q(2)\{u_4\}\oplus Q(3)\{u_5\}).\]
 
 Using $Q_2Q_1Q_0(u_3)=c_7$ and $Q_3(u_4)=c_8u_3+...$ as the case for $Spin_7$, 
we have the isomorphism
\[Q(2)\{u_3\}\cong \bZ/2\{c_7\}\oplus \bar Q(2)\{u_3\},\quad
  Q(3)\{u_4\}\cong Q(2)\{c_8u_3\}\oplus Q(2)\{u_4\}.\]
Hence $Igr(W_5)$ is isomorphic to  
 \[ \bZ/2[c_4,c_6,c_{16}]\otimes (\bZ/2[c_8]\oplus
 \bZ/2[c_7]\otimes Q(2)\{u_3\}\oplus \]
\[\bZ/2[c_7,c_8]\otimes Q(3)\{u_4\}\oplus \bZ/2[c_7,c_8]\otimes Q(3)\{u_5\})\]
 
Using the fact $Q_4(u_5)=c_{16}u_4+...$ from Lemma 5.6, we have the isomorphism
\[ Q(4)\{u_5\}\cong Q(3)\{c_{16}u_4\}\oplus  Q(3)\{u_5\}.\]
This induces the isomorphism
 \[Igr(W_5)\cong \bZ/2[c_4,c_6]\otimes (\bZ/2[c_8,c_{16}]\oplus
 \bZ/2[c_7,c_{16}]\otimes Q(2)\{u_3\}\]
\[\oplus    \bZ/2[c_7,c_{8}]\otimes Q(3)\{u_4\}\oplus \bZ/2[c_7,c_8,c_{16}]\otimes Q(4)\{u_5\}).\]
Hence we have  the desired isomorphism. 
     \end{proof}
     The cohomological invariant is known
     \[Inv^*(Spin_9,\bZ/2)\cong \bZ/2\{1,u_3,u_4,u_5\} .\]
     Hence Assumption (1),(2) are also satisfied for $(Spin_9,2)$
  (from the last isomorphism in the above proof).
     
     
     
     
\begin{lemma}
\[Igr(W_5)/I(W_5)\cong Igr(W_4)/(W_4)\oplus \bZ/2[c_4,c_6,c_{16}]\otimes (\bZ/2\{c_{16}u_3\}\oplus 
\]
\[\oplus \bZ/2[c_8]\otimes ((Q(4)/(Q(4)^+)^3\{u_5\}\oplus \bZ/2\{c_7 u_5\})).\]
\end{lemma}
\begin{proof}

Since 
\[Q_0(u_5)=Q_0(u_4x_5)=Q_0(u_4)x_5+u_4y_5,\]
we can compute 
\[ Q_2Q_1Q_0(u_5)=Q_2Q_1Q_0(u_4)x_5+Q_1Q_0(u_4)y_5^4+Q_2Q_0(u_4)y_5^2+Q_2Q_1(u_4)y_5\]
\[ =w_7w_8x_5+w_8y_5^4+w_4w_8y_5^2+w_6w_8y_5=w_{16}.\]
(This fact also follows from $d_{5,4}=w_{16}$.)
Let us write $Q_{i_1,...,i_j}(u_5)=Q_{i_1,...,i_j}$ simply.  
Similarly we  can compute
\[ Q_{012}=w_{16},\quad Q_{013}=w_{16}w_{8},\quad Q_{023}=w_{16}w_{4}w_{8},\quad  Q_{123}=w_{16}w_{6}w_{8},\]
\[ Q_{014}=c_{16}w_{8},\quad Q_{024}=c_{16}w_{8}w_{4},\quad Q_{034}=c_{16}c_8w_{4},\]
\[ Q_{124}=c_{16}w_{6}w_{8},\quad Q_{134}=c_{16}c_8w_6,\quad Q_{234}=c_{16}c_8w_4w_6.\]

 We can compute 
\[ Q_0(c_7u_5)=Q_0(c_7u_4x_5)=Q_0(w_4w_6w_8x_5)\]
\[ =w_4w_7w_8x_5+w_4w_6w_8y_5+...=w_4w_{16}.\]
Let us write $Q_{i_1,...,i_j}(c_7 u_5)=Q_{i_1,...,i_j}'$ simply. 
Then we can compute 
\[Q_0'=w_{16}w_4,\quad Q_1'=w_{16}x_6,\quad Q_2'=w_{16}w_{4}w_{6},\quad Q_3'=w_{16}w_{4}w_{6}w_{8},\]
\[ Q_4'=c_{16}w_4w_6w_8,\quad Q_{01}'=w_{16}w_7,\quad Q_{12}'=w_{16}w_6w_7,\quad Q_{04}'=c_{16}w_7w_4w_8.\]

Moreover
\[c_7^2u_5=c_7w_4w_6w_8x_5=w_4w_6w_7w_{16}.\]

There appear all generators of the $\bZ/2[c_4,c_6,c_8,c_{16}]$-module with 
modulo $Ideal(c_7,w_7)$.
Thus we can see  
\[Igr(W_5)/I(W_5)\cong \bZ/2[c_4,c_6]\otimes (\bZ/2[c_{16}]\{u_3\}\oplus \bZ/2[c_8]
(Q(3)/(Q(3)^+)^2\{u_4\}\]
\[\oplus \bZ/2[c_8,c_{16}]\otimes ((Q(4)/(Q(4)^+)^3\{ u_5\}\oplus \bZ/2\{c_7 u_5\})).\]
\end{proof} 
\begin{thm}  There is an epimorphism of $Q(4)$-modules 

from $H^*(BSpin_9;H_{\bZ/2}^{*'})$
to  
     \[ H^*(BSpin_7;H^{*'}_{\bZ/3})\oplus 
\bZ/p[c_4,c_6,c_{16}]\otimes(\bZ/2[c_7]\otimes Q(2)\{c_{16}u_3\}\oplus\bZ/2 \{c_{16}y\}\]
\[ \oplus\bZ/2[c_8,c_7]\otimes Q(4)\{u_5\}\oplus \bZ/2[c_8]\otimes (Q(4)/(Q(4)^+)^3\{y''\})\]
where $d_2u_3=y$ and $d_2(u_5)=y''.$
          \end{thm}
          {\bf Remark.} However note that we can not see $d_2(c_7u_5)\not =0$ or not.
          
          
From Lemma 4.4, we know $d_2(u_5)=y''\not =0$ and we get many Griffith elements
\[    Q_iQ_j(y'')\quad for \ 0\le i< j \le 4.\]
We study the image of the cycle map $\tilde{cl}$ to $BP^*(BSpin_9)\otimes_{BP^*}\bZ/2$ in the 
last section (indeed $\tilde{cl}(Q_iQ_j(y'')\not =0$).


Next  we consider the case $(Spin_8,2)$.
The Weyl group is
\[W_4\subset W_G(A)=W_5''=\{(w_{ij})\in GL_5(\bZ/2)|w_{5,4}=0\}\subset W_5.\]
We can compute
\[ Igr(W_5'')\cong Igr(W_4)\oplus
\bZ/2[c_4,c_6,c_8',c_8]\otimes \]
\[(\bZ/2\{c_8'\}\oplus \bZ/3[c_7]\otimes Q(3)\{u_4'\}\otimes \bZ/2[c_7]\otimes Q(4)\{u_5\}).\]
Hence Assumption (1) (with some modification for $u_4'$) and (2) are satisfied.


We consider the case $(F_4,2)$ also . This case $A\cong (\bZ/2)^5$ but
\[ W_G(A)=W_5'=\la U_5,GL_3(\bZ/2)\oplus GL_2(\bZ/2)\ra \subset GL_5(\bZ/2).\]
The cohomology is given by
\[H^*(BG;\bZ/2)\cong H^*(BA;\bZ/2)^{W_G(A)}\cong
\bZ/2[w_4,w_6,w_7,x_{16},x_{24}]\]
where $x_{16}=w_8^2+w_{16}$ and $x_{24}=w_8w_{16}$.
We consider the representation
\[ \la a_4,a_5 \ra \subset A \subset F_4 \stackrel{\rho}{\to} SO(26) \]
where $a_4,a_5$ are dual of $x_4,x_5$.  Then the total Chern class is
\[c(\rho|\la a_4,a_5\ra)=(1+c_{2,1}+c_{2,0})^{a}\]
from Lemma 5.1.  By dimensional reason, $a\le 8$.  So we see
$x_{16}^2$ and $x_{24}^2$ are represented by Chern classes.
Thus we can write
\[ Igr(W_5')\cong \bZ/2[c_4,c_6]\otimes (\bZ/2[c_{16},c_{24}]\oplus
 \bZ/2[c_7,c_{24}]\otimes Q(2)\{u_3\}\]
\[\oplus   \bZ/2[c_7,c_{16},c_{24}]\otimes Q(4)\{u_5\}).\]
Hence Assumption (1),(2)  are also satisfied.
  So $Res_{E_2}$ is an
epimorphism for $(F_4,2)$.
  

At last of examples, we consider the case  $(E_6,2)$.
This case $W_G(A)\cong W_5'$ same as the case $F_4$.  But it is known 
([Ga-Me-Se]) that
\[ Inv^*(E_6,\bZ/2)\cong \bZ/2\{1,u_3\}.\]
Therefore we see 
\begin{lemma}
When $G=E_6$ and $p=2$, the restriction map
\[ Res_{E_2} : H^*(BG;H^{*'}_{\bZ/2})\to H^*(BA;H^{*'}_{\bZ/2})^{W_G(A)}\]
is not an epimorphism.
\end{lemma}
The above fact also proved by using the cohomology $H^*(BE_6;\bZ/2)$ 
and Lemma 5.9 as follows.
\begin{thm} The restriction maps
\[Res_{Inv}: Inv^*(G;\bZ/2)\to Inv^*(A_5;\bZ/2)^{W_G(A_5)}\]
 are epimorphisms 
for $G=E_6$ and $G=Spin_n$, $n\ge 10$.
\end{thm}
\begin{proof}
Of course there is the embedding $i: Spin _9\subset Spin_{10}$.  From Kono-Mimura [Ko-Mi], we know
\[ H^*(BSpin_{10};\bZ/2)\cong H^*(BSpin_7;\bZ/2)\otimes \bZ/2[x_{10},x_{32}]/(w_7x_{10}).\]
Hence  $H^*(BE_6;\bZ/2)$ does not contain an element $x$ with $i^*(x)=w_{16}$ for
 $w_{16}=d_{5,4}=Q_0Q_1Q_2(u_5)$.  From Lemma 5.9, we see that $Res_{Inv}$ is not an epimorphism.

There is the embedding $F_4\subset E_6$.  From also Kono-Mimura [Ko-Mi], we know
\[ H^*(BE_6;\bZ/2)\cong H^*(BSpin_7;\bZ/2)\otimes \bZ/2[x_{10},x_{18},x_{32},x_{48}]/(relations).\]
Hence $Res_{Inv}$ is not epic from the lack of element $x_{16}$.
\end{proof} 
 



\section{ $BP$-theory and Griffith elements}


In this section, we recall the results in $\S 5$ in [Ya1] and consider the relation
between $BP$-theory and results in the preceding sections.
We always assume $k=\bC$ in this section. 
 Let  $BP^*(-)$
be the Brown-Peterson theory with the coefficient ring
 $BP^*=\bZ_{(p)}[v_1,...],\ |v_i|=-2(p^i-1)$.
The Thom map induces $\rho:BP^*(X)\otimes _{BP^*}\bZ_{(p)} \to H^*(X;\bZ_{(p)})$.  Totaro constructs [To1]
the map 
\[ \tilde{cl}: CH^*(X)_{(p)}\to BP^*(X)\otimes _{BP^*}\bZ_{(p)}\]
\noindent such that the composition $\rho \cdot \tilde{cl}$ is the usual cycle map $cl=t_{\bC}$
which is also the realization map. 
Totaro conjectured that this map is isomorphic for $X=BG$.


Let us write by
\[P(n)^*=BP^*/(p,v_1,...,v_{n-1}),\]
 e.g., $P(0)^*=BP^*$, $P(1)^*=BP^*/p$ and
$P(\infty)^*=\bZ/p$.  

Many cases of $X$([Te-Ya2], [Ko-Ya]), $BP^*(X)$ are computed by the Atiyah-Hirzebruch spectral sequences
\[E_2^{*,*'}=H^*(X)\otimes BP^{*'} \Longrightarrow BP^*(X).\]
 It is known that
 $d_{2p^i-1}(x)=v_i\otimes Q_i(x)\ mod (M_i)$ where $M_i$ is the ideal of $E_{2p^i-1}^{*,*'}$
 generated by elements in $(p,...,v_{i-1})E_2^{*,*'}$.
 
 We assume that  $H^*(X)$ has no higher $p$-torsion and
 all non zero differentials are of form 
\[(9.1)\quad  d_{2p^i-1}(x)=v_i\otimes Q_i(x)\ mod (M_i).\]
Let us write   
\[(9.2)\quad 
 grBP^*(X)\cong E_{\infty}^{*,*}\cong A\oplus B\]
 where $A$ (resp. $B$)
is a $BP^*$-module generated by elements in $H^*(X)/p$ (resp. $pH^*(X)\oplus E_{\infty}^{*,minus}$) so that $B\subset Ker(\rho _p)$. 
Then we can write  
\[A\cong \oplus _{n=-1}P(n+1)^*\tilde G_n\]
 by the prime
invariant ideal theorem of Landweber ; if $P(n)^*/(a)$ is a $BP^*(BP)$-module, then $a=v_n^s$ for some $s\ge1$.

\begin{lemma} (Lemma 5.1 in [Ya1])  Let $H^*(X)_{(p)}$ has no higher $p$-torsion.
Suppose (9.1) and $A=\oplus _{n=-1}P(n+1)^*\tilde G_ n$ in (9.2).
Then there is a injection of $Q(\infty)$-modules
\[H^*(X;\bZ/p)\hookrightarrow \oplus_{n=-1}
Q(n)G_n\quad  with \ Q_0...Q_nG_n=\tilde G_n.\]
\end{lemma}

It is proved ([Ko-Ya],[Ka-Ya],[Ya1]) that all $X=BG$ in Theorem 1.2 satisfy the assumption in the above
lemma. 
  Hence 
  \[H^*(BG;\bZ/p)\hookrightarrow \oplus_{n=-1}^{\infty} Q(n)G_n.\] 
  
  Moreover when $n\not =-1$, we still know 
  \[  w(G_n)=n.\]
  Indeed if $n\ge 0$, then $G_n=G_n'\{u_n\}$ where $G_n'$ is represented by
  elements in $CH^*(BG)/p$ (see Assumption (1),(2)). 
  From Theorem 1.2, we also know
  \begin{cor}  Let $G$ be a group in Theorem 1.2.   Then
  there is a bidegree $Q(n)$-module injection
  \[ \oplus_{i\ge 0}Q(n)G_n \subset H^*(BG;H_{\bZ/p}^{*'}).\]
  \end{cor}
  
  \begin{lemma}(Lemma 5.2 in [Ya])
Let $H^*(X)_{(p)}$ has no higher $p$-torsion. 
 
(1) \quad If  (9.1) is satisfied  and  in (9.2), 
\[ A\cong \oplus _{n=-1}P(n+1)^*\tilde G_ n \quad and \
B\cong \oplus _{s=0}BP^*\{p,v_1,..,v_{s}\}\tilde K_{s},\]
 then we have the isomorphism
\[ H^*(X;\bZ/p)\cong (\oplus _{n=-1}
Q(n)G_n)-(\oplus_s 
\bar Q(s)K_{s})\]
with  $Q_0...Q_{n}G_n=\tilde G_n$ , 
and $Q_1...Q_{s}K_s=\tilde K_s$.

(2)\quad If \ $Q_0...Q_{n}G_n\in Im(\rho)$ and $|Q_1...Q_sK_s|=even$, 
 \   then the converse also holds.
 \end{lemma}
 \begin{cor}
 Let $G$ be a group in Theorem 1.2 so that Assumption (1),(2) are satisfied.
 Then there are epimorphisms from $grBP^*(BG)\cong E_{\infty}^{*,*'}$ to
 \[(9.3)\quad \oplus_s P(i_s)^*[f_{s1},...,f_{sk_s}]\{\tilde u_{i_s}\}
      \oplus \oplus_tBP^*(p,...,v_t)\{\tilde K_t\}  \]
      and from $BP^*(BG)\otimes_{BP^*}\bZ/p$ to
      \[\oplus_s \bZ/p[f_{s1},...,f_{sk_s}]\{\tilde u_{i_s}\}
      \oplus \oplus_t\bZ/p\{p,...,v_t\}\{\tilde K_t\} \] 
where $\tilde u_{i_s}=Q_{i_s-1}...Q_0(u_{i_s})$  and
$ Igr(W)/Res_{H\bZ/p}\cong \oplus _t\bar Q(t)K_t.$
\end{cor} 

We give examples.
 At first we recall the Atiyah-Hirzebruch spectral sequence
 for $BG_2$ in [Ko-Ya].
Since $H^*(BG)$ has no higher torsin, we have
\[H^*(BG_2)_{(2)}\cong \bZ_{(2)}[w_4,c_6]\otimes(\bZ_{(2)}\{1\}\oplus 
\bZ/2[w_7]\{w_7\}).\]
Let us write $B_{i_1,...,i_j}=\bZ_{(2)}[c_{i_1},...,c_{i_j}]$, e.g.,
$B_{4,6}=\bZ_{(2)}[c_4,c_6]$.  

 Since $Q_1(w_4)=w_7$, 
we have $d_3(w_4)=v_1\otimes w_7$.
Hence the $E_4$-term of the spectral sequence is
\[E(G_2)_4^{*,*}\cong B_{4,6}\otimes
(BP^*\{1,2w_4\}\oplus P(2)^*[c_7]\{c_7,w_7\}).\]
Next differential is $d_7(w_7)=v_2\otimes Q_2(w_7)=v_2c_7$ and 
\[ E(G_2)_8^{*,*}\cong  B_{4,6}\otimes
(BP^*\{1,2w_4\}\oplus P(3)^*[c_7]\{c_7\}).\]
which is isomorphic to $E(G_2)^{*,*}_{\infty}$.
In particular 
\[ BP^*(BG)\otimes_{BP^*}\bZ/2\cong
   \bZ/2[c_4,c_6]\otimes (\bZ/2\{1,2w_4\}\oplus \bZ/2[c_7]\{c_7\}).\]
   
   This result is also immediate  from Corollary 9.4 and Theorem 6.1,
in fact, we have the epimorphism
\[ H^*(BG_2;H_{\bZ/2}^{*'})\to \bZ/2[c_4,c_6]\otimes (\bZ/2\{1,y\}\oplus \bZ/2[c_7]\otimes Q(2)\{u_3\}).\]
Here $\tilde  u_3=Q_0Q_1Q_2(u_3)=c_7$ and   $d_2(u_3)=y$ in the coniveau spectral sequence, and we have
\[ \bar Q(0)K_0=K_0=\bZ/2[c_4,c_6]\{u\}.\]
Hence the cycle map $\tilde{cl}$ is epimorphism and
$\tilde{cl}(y)=\{2w_4\}$ (which is represented by a Chern class $c_2$).
Moreover we know [Ya2] that this $\tilde{cl}$ is a really isomorphism.


Next consider the case $Spin_7$.
From [Ko-Ya], we can compute
%\[H^*(BSpin_7;\bZ/2)\cong H^*(BG_2;\bZ/2)\otimes \bZ/2[w_8].\]
%Since $Q_0(w_8)=Q_1(w_8)=0$, we see that 
%\[E(Spin_7)_{4}^{*,*}
%\cong E(G_2)_4^{*,*}[w_8]\]
%\[\cong B_{4,6,8}\otimes 
%(BP^*\{1,2w_4\}\oplus P(2)^*[c_7]\{c_7,w_7\})\{1,w_8\}\]
%\[\cong B_{4,6,8}\otimes(BP^*\{1,2w_4,w_8,2w_4w_8\}
%\oplus P(2)^*[c_7]\{c_7,w_7,c_7w_8,w_7w_8\}).\]
%The next differential is $d_7(w_7)=v_2c_7$ and $d_7(w_8)=v_2(w_7w_8).$  
%Hence
%\[E(Spin_7)_8^{*,*}\cong \]
%\[B_{4,6,8}\otimes(BP^*\{1,2w_4,2w_4w_8,2w_8,v_1w_8\}
%\oplus P(3)^*[c_7]\{c_7,w_7w_8\}).\]
%Since $Q_3(w_7w_8)=c_7c_8$, we get 
\[E(Spin_7)_{16}^{*,*}\cong  B_{4,6}\otimes P(3)^*[c_7]\{c_7\}\oplus \]
\[B_{4,6,8}\otimes (BP^*\{1,2w_4,2w_4w_8,2w_8,v_1w_8\}
\oplus P(4)^*[c_7]\{c_7c_8\}).\]
This term is also the infinity term.
Hence we have
\[ BP^*(BG)\otimes_{BP^*}\bZ/2
 \cong  BP^*(BG_2)\otimes_{BP^*}\bZ/2\oplus \]
\[ \bZ/2[c_4,c_6,c_8]\otimes(\bZ/2\{c_8,2c_8w_4,2w_4w_8,2w_8,v_1w_8\}
\oplus \bZ/2[c_7]\{c_7c_8\})\]
This result is also get from Corollary 9.4 and Theorem 7.3, indeed,
we have the epimorphsm
      \[H^*(BSpin_7;H_{\bZ/2}^{*'})\to  H^*(BG_2;H_{\bZ/2}^{*'})\oplus \]
\[ \bZ/2[c_4,c_6,c_8]\otimes (\bZ/2\{c_8,y_2',Q_0y_2',...,Q_3y_2'\}
\oplus \bZ/2[c_7]\otimes Q(3)\{u_4\})\]

This case $d_2(u_3)=y, d_2(u_4)=y'$ in the coniveau spectral sequence.

Recall 
\[ \oplus _s\bar Q(s)K_s \cong Igr(W_4)/I(W_4)\cong I(GL_3)/I(GL_3)\oplus \]
 \[   \bZ/2[c_4,c_6, c_8]\{1,Q_0,...,Q_3\}\{u_4\}.\]
    We can take
    \[ \bar Q(1)K_1\cong \bZ/2[c_4,c_6,c_8]\{1,Q_0,Q_1\}\{u_4\}\]
    \[ \bar Q(0)K_0\cong \bZ/2[c_4,c_6]\otimes(
    \bZ/2\{u_3\}\oplus \bZ/2\{ Q_2u_4,Q_3u_4\}).\]
  The cycle map $\tilde{cl}$ is given by
  \[ Q_0(y')\mapsto v_1w_8\quad,  Q_1y'\mapsto 2w_8\quad,  Q_2y'\mapsto 2w_4w_8\quad,  Q_3y'\mapsto 2c_8w_4.\]
  Of course the cycle map $\tilde{cl}$ is isomorphic. This fact is still proved
by P.Guillot [Gu1], and the case $G=Spin_8$ is computed by Molina [Mo].  





















                    
 
 
 
 
\section{$BP^*(BSpin_9)\otimes _{BP^*}\bZ/2$}
At last of this paper, we consider the case $Spin_9$.
In [Ko-Ya], we can compute (which is quite complicated)
%The mod $2$ cohomology is
%\[H^*(BSpin(9);\bZ/2)\cong H^*(BSpin(7);Z/2)\otimes \bZ/2[w_{16}].\]
%Since $Q_0(w_{16})=Q_1(w_{16})=Q_2(w_{16})=0$, we see that 
%\[E(Spin(9))_{8}^{*,*}
%\cong E(Spin(7))_8^{*,*}\otimes \bZ_{(2)}[w_{16}]\cong \]
%\[B_{4,6,8,16}\otimes(BP^*\{1,2w_4,2w_4w_8,2w_8,v_1w_8,\]
% \[ w_{16}, 2w_4w_{16}, 2w_4w_8w_{16},  2w_8w_{16},v_1w_8w_{16}\}\]
%\[\oplus P(3)^*[c_7]\{c_7,w_7w_8,w_7w_{16},,c_7w_8w_{16}\}).\]
%Since $Q_3(w_7w_8)=c_7c_8, Q_3(w_{16})=w_{16}w_{7}w_{8}$,
% we get 
%\[E(Spin(9))_{16}^{*,*}\cong 
%B_{4,6,8,16}\otimes (BP^*
%\{1,2w_4,2w_4w_8,2w_8,v_1w_8,
%2w'_8,v_1w'_8,\]
%\[ 2w_4w_{16},2w_4w_8w_{16}, 2w_8w_{16},v_1w_8w_{16},
%2w_{16},v_1w_{16}.v_2w_{16}\}\]
%\[\oplus P(4)^*[c_7]
%\{c_7c_8,w_7w_8w_{16}\})
%\oplus  B_{4,6,16}\otimes P(4)^*[c_7]\{c_7\}.\]
%The next differential is
%$d_{31}(w_7w_8w_{16})=v_4\otimes c_7c_8c_{16}$.
%Therefore we get
\[E(Spin_9)_{32}^{*,*}\cong 
B_{4,6,8,16}\otimes (BP^*
\{1,2w_4,2w_4w_8,2w_8,v_1w_8,\]
\[ 2w_4w_{16},2w_4w_8w_{16}, 2w_8w_{16},v_1w_8w_{16},
2w_{16},v_1w_{16}.v_2w_{16}\}\]
\[\oplus P(5)^*[c_7]
\{c_7c_8c_{16}\})
\oplus B_{4,6,7,8}\otimes P(4)^*[c_7]\{c_7c_8\}
 \oplus B_{4,6,16}\otimes P(3)^*[c_7]\{c_7\}.\]
This term is the infinite term.
(See Theorem 8.3 also.)
%\begin{lemma}
%\[BP^*(BSpin(9))\otimes_{BP^*}\bZ_{(2)}\cong\]
%\[B_{4,6,8,16}\otimes(\bZ_{(2)}\{1,2w_4,2w_4w_8,2w_8,
%2w_4w_{16},2w_4w_8w_{16},
% 2w_8w_{16},2w_{16}\}\]
%\[ \oplus \bZ/2\{v_1w_8,v_1w_8w_{16},
%v_1w_{16},v_2w_{16}\}\oplus \bZ/2[c_7]\{c_7\})\]
%\[\cong (CH^*(BSpin(7))\oplus B_{4,6,8}
%(\bZ_{(2)}\{ 2w_4w_{16},2w_4w_8w_{16}, 2w_8w_{16},2w_{16}\}\]
%\[\oplus \bZ/2\{v_1w_8w_{16},v_1w_{16}.v_2w_{16}\})\otimes \bZ_{(2)}[c_{18}].\]
%\end{lemma}
\begin{lemma}([Ko-Ya])
\[BP^*(BSpin_9)\otimes_{BP^*}\bZ/2\cong BP^*(BSpin_7)\otimes_{BP^*}\bZ/2\oplus\]
\[\bZ/2[c_{4},c_{6},c_{8},c_{16}]\otimes( \bZ/2\{c_{16},2w_4c_{16},2w_4c_8c_{16},(2,v_1)w_8c_{16},
2w_4w_8c_{16},\]
\[ (2,v_1,v_2)w_{16},2w_4w_{16},(2,v_1)w_8w_{16},2w_4w_8w_{16}\}\oplus 
 \bZ/2[c_7]\{c_{16}c_7\}).\]
\end{lemma}
We still know there is an epimorphism 
from $H^*(BSpin_9;H_{\bZ/2}^{*'})$
to  
     \[ H^*(BSpin_7;H^{*'}_{\bZ/2})\oplus 
\bZ/2[c_4,c_6,c_{16}]\otimes(\bZ/2[c_7]\otimes Q(2)\{c_{16}u_3\}\oplus\bZ/2 \{c_{16}y\}\]
\[ \oplus\bZ/2[c_8,c_7]\otimes Q(4)\{u_5\}\oplus \bZ/2[c_8]\otimes (Q(4)/(Q(4)^+)^3\{y''\})\]
where $d_2u_3=y$ and $d_2(u_5)=y''.$

We can see the following lemma;

\begin{prop} When $(G,p)=(Spin_9,2)$, the cycle map
\[ \tilde{cl}: CH^*(BG)/2\to (BP^*(BG)\otimes_{BP^*}\bZ/2)\]
is an epimorphism  with $mod(\bZ/2[c_4,c_6,c_8,c_{16}]\{2w_4w_{16}\})$.
\end{prop}



The image of the cycle map $\tilde{cl}$ is given as follows.
By arguments for $G_2$,  we see
\[ d_2(c_{16}u_3)\mapsto  2w_4c_{16}\]
If we can see
\[ d_3(c_7u_5)\mapsto  2w_4w_{16},\]
then the Totaro's map $\tilde{cl}$ is an epimorphism. Unfortunately, we do not see
even $d_2(c_7u_5)=c_7y''=0$ yet.

Let us write $Q_{ij}=Q_iQ_j(d_2u_5)=Q_iQ_j(y'')$.  Then the cycle map $\tilde {cl}$ is written as
\[Q_{01}\mapsto v_2w_{16},\quad Q_{02}\mapsto v_1w_{16},\quad Q_{12}\mapsto 2w_{16},\quad \]
\[Q_{03}\mapsto v_1w_8w_{16},\quad Q_{13}\mapsto 2w_8w_{16},\quad Q_{23}\mapsto 2w_4w_8w_{16},\quad \]
\[Q_{04}\mapsto v_1w_8c_{16},\quad Q_{14}\mapsto 2w_8c_{16},\quad Q_{24}\mapsto 2w_4w_8c_{16},\quad \]
\[Q_{34}\mapsto 2w_4c_8c_{16}.\]
%Hence  (9.4)also holds for this case (e.g.,$Q_0Q_1Q_2(u_5)=w_{16}$).

  By the arguments for $Spin_7$, we still know,
for example $Q_1(d_2u_4)\mapsto 2w_8$, so we get the $Q_4Q_1(d_2u_5)\mapsto 2w_8c_{16}$
using $Q_4(u_5)=c_{16}u_4+...$. Similarly we get the maps for $Q_{*4}$
from that for $Spin_7$.

For other maps, we use  the Quillen operation in $BP^*(-)$ theory.
For a sequence $\alpha=(\alpha_1,...,\alpha_m)$, $\alpha_i\ge 0$,  we have 
 the Quillen cohomology operation
in $BP^*(X)$ (and also in $ABP^*(X)$) (see [Ra], [Ha], [Ya2])
\[r_{\alpha}: BP^*(X)\to BP^{*+|\alpha|}(X)\quad |\alpha|=\sum 2p^i\alpha_i\]
such that $\rho(r_{\alpha})=P^{\alpha}$ the fundamental basis of the reduced power operations
(see [Ha]) and $r_{\alpha}(v_i)\in Ideal(p,...,v_i).$
%\[r_{p^j\Delta_{i-j}}(v_i)=v_j\quad mod(p,v_1,..,v_{j-1}).\] 
Hence  $r_{\alpha}$ acts also on $BP^*(X)\otimes _{BP^*}\bZ/p$. 

Let us write by $\bar Sq^{even}$ the Quillen operation corresponding $Sq^{even}$.
By the definition of $Q_i$, se see the equation
$Sq^{16}Sq^{8}(Q_2Q_1(u_5))=Q_4Q_1(u_5)$ in $H^*(BG;\bZ/2)$.  
We still know the image of the cycle map  
\[   Sq^{16} Sq^{8}(Q_{12})=Q_{14} \mapsto 2w_8c_{16} \in BP^*(BG)\otimes_{BP^*}\bZ/2.\]
Let $\tilde{cl}(Q_{12})=x$.  Then 
\[\bar Sq^{16}\bar Sq^{8}(x)=2w_8c_{16}\quad in \ BP^*(BG)\otimes_{BP^*}\bZ/2.\]
So $x$ is non zero.   
The equation 
 $ Sq^{16}Sq^{8}(w_{16})=w_8c_{16}$ in $H^*(BG;\bZ/2)$ implies
 \[ \bar Sq^{16}\bar Sq^{8}(2w_{16})=2w_8c_{16}\quad mod(v_1,...)\ \ in\ BP^*(BG).\]
Hence  we can take $x=2w_{16}$.

Using $\bar Sq^4\bar Sq^2$ and dimensional reason, we have the first map
$Q_{01}\mapsto v_2w_{16}$. The other cases are also proved similarly.
 

The above $Q_{ij}$ are all Griffith elements.  From Corollary 9.4, we can write 
\[ Q(4)\{u_5\}/(Q(4)^+)^3\{u_5\}\cong \oplus _{t=1}^2 \bar Q(t)K_t'.\]
In fact, we can take
\[\bar Q(2)K_2'=\bZ/2\{u_5,Q_0,Q_1,Q_2,Q_{01},Q_{02},Q_{12}\},\]
\[\bar Q(1)K_1'=\bZ/2\{Q_3,Q_{03},Q_{13},\ \ Q_4,Q_{04},Q_{14}\},\]
\[\bar Q(0)K_0'=\bZ/2\{Q_{23},Q_{24},Q_{34}\}.\]

 Recall Corollary 9.4 and 
$ Igr(W)/Res_{H\bZ/p}\cong \oplus _t\bar Q(t)K_t$.
Let  $k\in K_t$.  Then  we can identify $k\in H^*(BG;H_{\bZ/p}^{*'})$ and 
  \[  k(i)=Q_0...\hat Q_i...Q_t(k)\in \bar Q(t)K_t\subset Igr(W)/Res_{H\bZ/p}.\]
  Moreover suppose $w(k)=t+3$.
Since $w(k(i))=3$ and $w(d_r)=1-2r$, we see 
       \[ d_2(k(i))\not =0\quad (hence\ d_2(k)\not =0).\]
       Let us consider the projection map
       \[ pr.: BP^*(X)\otimes _{BP^*}\bZ/p\to (9.3)\otimes_{BP^*}\bZ/p\to (p,v_1,...,v_t)\{\tilde k\}\]
       where $\tilde k= Q_0...Q_t(k)$.
For $G=G_2,Spin_7$ and $Spin_9$, it holds that
 $ pr.\tilde{cl}(d_2(k(i)))=v_i\tilde k$, that is,
\[  pr.\tilde{cl}(d_2(Q_0...\hat Q_i...Q_t(k)))=v_iQ_0...Q_t(k),\]
while we do not show it for general cases.

%In $pr.(BP^*(X)\otimes_{BP^*}\bZ/p)$, we see
 %       \[r_{p^j\Delta_{i-j}}(v_i\tilde k)=v_j \tilde k.\]
%On the other hand in $CH^*(X)/p$ ( in $ABP^{2*,*}(X)\otimes_{BP^*}\bZ/p$ )
%\[ r_{p^j\Delta_{i-j}}(d_2k(i))=d_2k(j) \quad mod(Q_j)\]
%because it is known $Q_i=[Q_j,P^{p^{j}\Delta_{i-j}} ]$ and
%\[k(j)=P^{p^j\Delta_{i-j}}(k(i))\quad mod(Q_j)\]
%\begin{lemma}
%If (10.1) holds for $i=t$, then
%so does for all $0\le i\le t$.
%\end{lemma}   



   
  


 
 
  \begin{thebibliography}{Ra-Wi-Ya}
  
  
  
  
  \bibitem[Ad]{Ad}
J. F. Adams. {\it Lectures on exceptional Lie groups}, Univ. Chicago Press, Chicago, IL, 1996. 

%\bibitem [A-G-M-V]{A-G-M-V} 
%K. K. S. Andersen\ et al.. The classification of $p$-compact groups for $p$ odd, Ann. of Math. (2) {\bf 167} (2008), %no.~1, 95--210. 




\bibitem[Bl-Og]{Bl-Og}
S.Bloch and A.Ogus.
\newblock  Gersten's conjecture  and the homology of schemes.
\newblock \emph{Ann.Scient.\'Ec.Norm.Sup.} 
\textbf{7} (1974) 181-202. 



\bibitem[Br-Re-Vi]{Bo-Re-Vi}
P.Brosman.  Z.Reichstein and A.Vistoli.
\newblock{Essential dimension, spinor groups, and quadratic forms.}
\newblock \emph{to appear in Ann. Math.}
\textbf{} (2003).



\bibitem[Co]{Co}
J.L.Colliot-Thelene.
\newblock  Cycles algebraiques de torsion et $K$-theorie algebrique.
In: Arithmetic algebraic geometry (Tronto, 1991).
\newblock \emph{Lect. Note Math. Springer, Berlin} 
\textbf{1553} (1993) 1-49. 



\bibitem[Ga]{Ga} 
S.Garibaldi.
\newblock  Cohomological invariants : exceptional groups and Spin groups
 (wth an appendix by D.Hoffmann).
\newblock \emph{Mem. Amer. Math. Soc. 200.}
\textbf{937} (2009)





\bibitem[Ga-Me-Se]{Ga-Me-Se} 
S.Garibaldi, A.Merkurjev and J.P.Serre.
\newblock  Cohomological invariants in Galois cohomology.
\newblock \emph{University lect. series vol(28) AMS}  (2003).




%\bibitem[Gr]{G}
%A.~Grothendieck.
%\newblock Torsion homologique et sections rationnelles.
%\newblock \emph{Sem. C.Chevalley, ENS 1958,expose 5,
%Secreatariat Math., IHP, Paris} (1958). 



\bibitem[Gu1]{Gu1}
P.Guillot.
\newblock  The Chow ring of $G_2$ and $Spin(7)$.
\newblock \emph{J. Reine angew. Math.} 
\textbf{604} (2007) 137-158. 






\bibitem[Gu2]{Gu2}
P.Guillot. 
\newblock Geometric methods for cohomological invariants.
\newblock \emph{Document Math.}  \textbf{12} (2007), 521-545.



%\bibitem[Fr-Mi]{Fr-Mi}
%E.Friedlander and G.Mislin. 
%\newblock Cohomology of classifying spaces of complex Lie groups and related discrete %groups.
%\newblock \emph{Comment. Math. Helvet.}  \textbf{59} (1984), 347-361.






%\bibitem [Kah]{Kah}
%B. Kahn.
%\newblock{Classes de Stiefel-Whitney de formes quadratiques et 
%representations galoisiennes reelles.}
%\newblock \emph{Invent. Math.}
%\textbf{78} (1984), 223-256.



%\bibitem [Kah]{Kah}
%B.Kahn.
%\newblock{Cohomologie non ramifiee des quadriques.
%In Geometric methods in the algebraic theory of quadratic forms.}
%\newblock \emph{Lect. Notes in Math. Springer,Berlin}
%\textbf{1835} (2004), 1-23.





\bibitem [Ka]{Ka}
M.Kameko.
\newblock{Poincar\'e series of cotorsion products.}
\emph{Preprint} (2005).





\bibitem [Ka-Mi]{Ka-Mi}
M. Kameko\ and\ M. Mimura.
M\`ui invariants and Milnor operations,
Geometry and Topology Monographs {\bf 11},  (2007), 107-140.





\bibitem [Ka-Ya1]{Ka-Ya1}
M.Kameko and N.Yagita.
\newblock{The Brown-Peterson cohomology of the classifying spaces of 
the projective unitary groups $PU(p)$
and exceptional Lie group.}
\newblock \emph{Trans. Amer. Math. Soc.}
\textbf{360} (2008), 2265-2284.

\bibitem [Ka-Ya2]{Ka-Ya1}
M.Kameko and N.Yagita.
\newblock{Chern subrings.}
\newblock \emph{ Proc. Amer. Math. Soc.}
\textbf{138} (2010) 367-373.


\bibitem[Ko-Ya]{Ko-Ya}  
A. Kono\ and\ N. Yagita. Brown-Peterson and ordinary cohomology theories 
of classifying spaces for compact Lie groups, Trans. Amer. Math. Soc. {\bf 339} (1993), no.~2, 781--798. 



\bibitem[Mo]{Mo}
L.A. Molina. 
\newblock The Chow ring of classifying space of $Spin_8$.
\newblock \emph{preprint}  (2007).  





\bibitem[Mo-Vi]{Mo-Vi}
L.Molina and A.Vistoli. 
\newblock On the Chow rings of classifying spaces for classical groups. 
\newblock \emph{Rend. Sem. Mat. Univ. Padova}  \textbf{116}
(2006), 271-298.  





 \bibitem[Or-Vi-Vo]{Or-Vi-Vo} 
D.Orlov,A.Vishik and V.Voevodsky.
\newblock An exact sequence for Milnor's K-theory with applications to quadric forms.
\newblock \emph{Ann. of Math.} \textbf{165} (2007) 1-13.





\bibitem[Pa]{Pa}
W.Paranjape  
\newblock Some spectral sequences for filtered complexes and applications.
\newblock \emph{J. Algebra}
\textbf{186} (1996) 793-806.



\bibitem[Qu]{Qu} 
D.Quillen.
\newblock The mod 2 cohomology rings of extra-special 2-groups and the spinor groups.
\newblock \emph{Math. Ann.} \textbf{194} (1971), 197-212.


\bibitem[Re]{Re} 
Z. Reichstein.
\newblock On the notion of essential dimension for algebraic groups.
\newblock \emph{Transform. groups} \textbf{5} (2000), 265-304.



\bibitem[Ro]{Ro}  
M.Rost. 
\newblock On the basic correspondence of a splitting variety.
\newblock \emph{preprint} (2006)


%\bibitem[Sa]{Sa} 
%D. Saltman
%\newblock Noether's problem over an algebraically closed field.
%\newblock \emph{Invent.  Math. } \textbf{77} (1984), 71-84.




\bibitem[Sc-Ya]{Sc-Ya}
B. Schuster\ and\ N. Yagita, Transfers of Chern classes in BP-cohomology and Chow rings, 
Trans. Amer. Math. Soc. {\bf 353} (2001), no.~3, 1039--1054 (electronic).



\bibitem[Su-Jo]{Su-Jo} 
A.Suslin and S.Joukhovistski. 
\newblock Norm Variety.
\newblock \emph{J.Pure and Appl. Algebra} 
\emph{206} (2006) 245-276.



 
%\bibitem[Te-Ya]{Te-Ya}
%M.~Tezuka and N.Yagita.
%\newblock{The varieties of the mod $p$ cohomology rings of extraspecial
%$p$-groups for an odd prime $p$.}
%\newblock \emph{Math. Proc. Cambridge Phil. Soc. } \textbf{94} (1983) 449-459.



\bibitem[To]{To} 
B.~Totaro.  
\newblock The Chow ring of classifying spaces. 
\newblock \emph{Proc.of Symposia in Pure Math. "Algebraic K-theory" 
(1997:University of Washington,Seattle)} \textbf{67} (1999), 248-281.




\bibitem[Vi]{Vi}
A.Vistoli. 
\newblock{On the cohomology and the Chow ring of the 
classifying space of $PGL_p$.} 
\newblock \emph{J. Reine Angew. Math.} \textbf{610} (2007)
181-227.

  



\bibitem[Vo1]{Vo1} 
V.~Voevodsky. 
\newblock The Milnor conjecture.
\newblock \emph{www.math.uiuc.edu/K-theory/0170}    (1996).





\bibitem[Vo2]{Vo2} 
V.~Voevodsky. 
\newblock Motivic cohomology with $\bZ/2$ coefficient.
\newblock \emph{Publ. Math. IHES}
\textbf{98} (2003), 59-104.

\bibitem[Vo3]{Vo3}
V.~Voevodsky (Noted by Weibel).
\newblock Voevodsky's Seattle lectures : $K$-theory and motivic cohomology
\newblock \emph{Proc.of Symposia in Pure Math. "Algebraic K-theory" 
(1997:University of Washington,Seattle)} \textbf{67} (1999), 283-303.

\bibitem[Vo4]{Vo4} 
V.Voevodsky. 
\newblock Reduced power operations in motivic cohomology.
\newblock \emph{Publ.Math. IHES}
\textbf{98} (2003),1-57.


\bibitem[Vo5]{Vo5} 
V.Voevodsky. 
\newblock On motivic cohomology with $\bZ/l$-coefficients.
\newblock \emph{www.math.uiuc.edu/K-theory/0631}    (2003).





%\bibitem[32]{Ya1}
%N.~Yagita.
%\newblock  Chow ring of classifying spaces of extraspecial $p$-groups.
%\newblock \emph{Contem. Math.} \textbf{293}, (2002) 397-409.
 
\bibitem[Ya1]{Ya1}
N.~Yagita.
\newblock  Examples for the mod p motivic cohomology of classifying spaces.
\newblock \emph{Trans. Amer. Math. Soc.}
\textbf{355} (2003),4427-4450. 
 


\bibitem[Ya2]{Ya2}
N.~Yagita.
\newblock  Applications of Atiyah-Hirzebruch spectral 
sequence for motivic cobordism.
\newblock \emph{Proc. London Math. Soc.}
\textbf{90} (2005) 783-816.

%\bibitem[Ya3]{Ya3}
%N.~Yagita.
%\newblock  Coniveau spectral sequence and motivic cohomology of quadrics.
%\newblock \emph{to appear in J. K-theory} 

\bibitem[Ya3]{Ya3}
N.~Yagita.
\newblock  Coniveau filtration of cohomology of groups.
\newblock \emph{to appear in Proc. London Math. Soc.} (2008).




































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