\section{Introduction}


Let $p$ be a prime number and $A=\bQ,\bZ,\bZ_{(p)}$ or $\bZ/p^i$ for $i\ge 1$.
Let $X$ be a smooth algebraic variety over $k=\bC$.  
Let us recall the coniveau filtration
of the (Betti) cohomology with coefficients in  $A$,
\[ N^cH^i(X;A)=\sum_{Z\subset X} ker(j^*:H^i(X;A)\to
H^i(X-Z,A))\]
where $Z\subset X$ runs through the closed subvarieties of codimension at least $c$ of $X$,
and $j:X-Z\subset X$ is the complementary open immersion.

Similarly, we can define the $strong$ coniveau filtration 
by 
\[ \tilde N^cH^i(X;A)=\sum_{f:Y\to X} im(f_*:H^{i-2r}(Y;A)\to
H^i(X,A))\]
where the sum is over all proper morphism $f: Y\to X$ from a smooth complex variety $Y$ of $dim(Y)=dim(X)-r$ with $r\ge c$,
and $f_*$ its transfer (Gysin map).
It is immediate $\tilde N^cH^*(X;A)\subset 
N^cH^*(X;A)$.

It was hoped that 
the strong coniveau filtration was  just the coniveau filtration,
i.e., 
$ \tilde N^cH^i(X;A)=N^cH^i(X;A)$.
In fact Deligne shows they are the same for $A=\bQ$.
However, Benoist and Ottem ([Be-Ot]) show that they are not equal
for $A=\bZ$.

Let $G$ be an algebraic group such that $H^*(BG;\bZ)$ has
$p$-torsion for the classifying space $BG$.
The purpose of this paper is to compute the stable birational
invariant of $X$ (Proposition 2.4 in [Be-Ot])
\[ DH^*(X;A)=N^1H^*(X;A)/(p,\tilde N^1H^*(X;A)) \]
for a projective approximation $X$ for $BG$ ([Ek]).
That is the projective variety $X=X(N)$ such that
there is a map $g:X\to BG\times \bP^{\infty}$ with \[g^*:H^*(BG\times \bP^{\infty};A)\cong H^*(X;A) \quad 
for \ *<N. \]
 (In this paper, we say $X$ is an approximation $for$ $BG$ when it is that $of$ $BG\times \bP^{\infty} $
strictly.)  




 
Here we give an example we can compute a nonzero  $DH^*(X)$.  For  $G=(\bZ/p)^3$ the elementary abelian $p$-group of $rank=3$, we know  (for $p$ odd)
\[ H^*(BG;\bZ/p)\cong \bZ/p[y_1,y_2,y_3 ]\otimes \Lambda(x_1,
x_2,x_3),\quad |x_i|=1,\  Q_0(x_i)=y_i\]
\[H^*(BG;\bZ)/p\cong \bZ/p[y_1,y_2,y_3](1,Q_0(x_ix_j),Q_0(x_1x_2x_3)|1\le i<j\le 3) \]
where $Q_0=\beta$ is the Bockstein operation, and the notation $R(a,...,b)$ (resp. $R\{a,..., b\}$) means the $R$-submodule (resp. free $R$-module) generated by $a,...,b$.
\begin{thm}  Let $G=(\bZ/p)^3$.  Given $N>2p+3$, there are  approximations
$X$ for  $BG$ such that  we have for  $\alpha_{ij}=Q_0(x_ix_j),$   $\alpha=Q_0(x_1x_2x_3)$
\[ DH^*(X;\bZ)\cong \bZ/p\{\alpha_{ij},\alpha|1\le i<j\le 3\}
\quad when\  *<N
.\]
But we have  $DH^*(X;\bZ/p)=0$.
\end{thm}




Benoist and Ottem  give  many examples,  where the above 
invariants are  nonzero  for $A=\bZ_{(2)}$ by using the Steenrod squares.  
We note that some of their arguments can be extended for $A=\bZ_{(p)}$ by using
Milnor operation $Q_1$ for odd primes $p$. However it seems not so easy to give a nontrivial example 
for $A=\bZ/p$ in the case $X$ is an approximation for $BG$
as the above example.

For connected groups. we have
\begin{thm}  Let $G$ be a simply connected group such that $H^*(BG;\bZ)$ has $p$-torsion.
Then there is an approximation $X$ for $BG$ such that 
   $ DH^4(X;\bZ)\not =0.$
\end{thm}
	\begin{thm}
Let $p=2$ and $BG=BSO_n,\ n\ge 3$, or, $p:odd$ and
$G=PGL_p$. 
Then there is an  approximation $X$ for $BG$ such that  
$DH^3(X;\bZ)\not =0$. \end{thm}




\section{transfer and $Q_n$}

The Milnor operation (in $H^*(-;\bZ/p)$)
is defined by $Q_0=\beta$ and for $n\ge 1$
\[ Q_n=P^{\Delta_n} \beta-\beta P^{\Delta_n}, \quad \Delta_n=(0,..,0,\stackrel{n}{1},0,...) \]
where $\beta$ is the Bockstein operation and $P^{\alpha}$
for $\alpha=(\alpha_1,\alpha_2,...)$ is the fundamental base of the the module of sums of products of reduced powers. 
\begin{lemma} Let $f_*$ be the transfer (Gysin) map for $f:Y\to X$.
Then $Q_n$ and $f_*$ are commutative, i.e. $Q_nf_*=f_*Q_n$.
\end{lemma}

The above lemma is known (see the proof Lemma 7.1 in [Ya2]). The transfer $f_*$ is 
expressed as $g^*f_*'$ such that
\[ f_*'(x)=i^*(Th(1)\cdot x) \]
for some maps $g,f', i$ and the Thom class $Th(1)$.  Since 
$Q_n(Th(1))=0$, we can get the lemma.
However, we give here the other computational proof.


\begin{proof}[Proof of Lemma 2.1.]
It is well known as the Grothendieck formula
\[(1)\quad P_t(f_*(x))=f_*(c_t\cdot P_t(x)).\]
Here the total reduced powers $P_t(x)$ are defined 
\[P_t(x)=\sum_{\alpha}P^{\alpha}(x)t^{\alpha}\in H^*(X;\bZ/p)[t_1,t_2,...]\quad with 
\ t^{\alpha}=t_1^{\alpha_1}t_2^{\alpha_2}...\]
where $\alpha=(\alpha_1,\alpha_2,...)$ and $degree(t^{\alpha})=
\sum_i 2\alpha_i(p^i-1)$.
The total Chern class $c_t$ is defined similarly, for the Chern classes of the normal bundle of the map $f$.

We consider the above equation with the assumption such that 
$t_n^2=0$ and $t_j=0$ for $j\not =n$, i.e., 
$P_t(x)\in H^*(X;\bZ/p)\otimes \Lambda(t_n)$. 
That means
\[(2)\quad  P_t(f_*(x))=(1+P^{\Delta_n}t_n)(f_*(x))\]
\[ (3)\quad f_*(c_t\cdot P_t(x))=f_*((1+c_{p^n-1}t_n)(x+P^{\Delta_n}(x)t_n))
=f_*(x+(c_{p^n-1}x+P^{\Delta_n}(x))t_n).\]
From (1), we see $(2)=(3)$ and we have
\[ (4)\quad P^{\Delta_n}(f_*(x))=f_*(c_{p^n-1}x+P^{\Delta_n}(x)).\]

By the definition, $\beta$ commutes with $f_*$, and we have
\[(5)\quad  P^{\Delta_n}\beta(f_*(x))=P^{\Delta_n}f_*(\beta x)=
f_*(c_{p^n-1}\beta x+P^{\Delta_n}(\beta x)).\]

On the other hand 
\[(6)\quad \beta P^{\Delta_n}f_*(x)\stackrel{(4)}{=}\beta f_*(c_{p^n-1}x+P^{\Delta_n}(x))
=f_*(c_{p^n-1}\beta x+\beta P^{\Delta_n}(x)).\]
Then $(5)-(6)$ gives  that
$(P^{\Delta_n} \beta-\beta P^{\Delta_n})f_*(x)=f_*(P^{\Delta_n} \beta-\beta P^{\Delta_n})(x).$
\end{proof}

By the definition, each cohomology operation $h$ (i.e., an element in the Steenrod algebra) is written (with $Q^B=Q_0^{b_0}Q_1^{b_1}...$)
by
\[h=\sum_{A,B}P^AQ^B\quad with\ A=(a_1,...),\ B=(b_0,...)\
b_i=0\ or\  1.\]

\begin{cor}  We have \ 
$ P_tQ^B(f_*(x))=f_*(c_t\cdot P_tQ^B(x)).$
\end{cor}
Hence cohomology operations $h$ (for $H^*(-;\bZ/p))$  which commute with
all transfer $f_*$ are cases $c_t=1$, i.e. $A=0$ which are only products $Q^B$ of Milnor operations $Q_i$.


\section{coniveau filtrations}

Bloch-Ogus [Bl-Og] give a spectral sequence such that its $E_2$-term is given by
\[E(c)_2^{c,*-c}\cong 
H_{Zar}^c(X,\mathcal{H}_{A}^{*-c})\Longrightarrow H_{et}^*(X;A)\]
where $\mathcal{H}_{A}^*$ is the  Zariski sheaf induced from the  presheaf  given by $U\mapsto H_{et}^*(U;A)$
for an open $U\subset X$. 

{\bf Remark.}  When $X$ is complete, it is known
\[ E(c)_2^{0,*}\cong H_{Zar}^0(X,\mathcal{H}_{\bZ/p}^{*})\cong H^*_{nr}(k(X):\bZ/p)\]
as the unramified cohomology which is a retract rationally invariant for $X$ [Me], [Re-Sc].








The filtration for this spectral sequence is defined as the coniveau filtration, that is 
  \[N^cH_{et}^*(X;A)= F(c)^{c,*-c}\]
where the infinite term $ E(c)^{c,*-c}_{\infty}\cong  F(c)^{c,*-c}/F(c)^{c+1,*-c-1}$ and 
\[ N^cH^*_{et}(X;A)=\sum_{Z\subset X\ ;\ codim_X(Z)\le c} ker(j^*:H^*_{et}(X;A)\to
H^*_{et}(X-Z,A)).\]

Here we recall the motivic cohomology 
$H^{*,*'}(X;\bZ/p)$ defined by Voevodsky and Suslin ([Vo1,Vo3,Vo4])
so that
\[ H^{i,i}(X;\bZ/p)\cong H^i_{et}(X;\bZ/p)\cong H^i(X;\bZ/p).\]

In this paper, let us write the $p$-localized cohomology 
$H^*(X;\bZ_{(p)})$ (resp.  Chow ring $CH^*(X)\otimes \bZ_{(p)}$) simply 
by $H^*(X)$ (resp. $CH^*(X)$).
Note that in general, $H^*_{et}(X)\not \cong H^*(X)$
while we have the natural map $H_{et}^*(X)\to H^*(X)$.

Let $0\not =\tau \in H^{0,1}(Spec(\bC);\bZ/p)$.  Then by the  multiplying by $\tau$, we can define a map
$H^{*,*'}(X;\bZ/p)\to H^{*,*'+1}(X;\bZ/p)$.
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 $\tau$-Bockstein spectral sequence $E(\tau)_r^{*,*'}$
(see also [Te-Ya], [Ya1]).

\begin{thm} (Deligne)
Let $A=\bZ/p$.  Then we have 
the isomorphism of spectral sequence 
\[E(c)_r^{c,*-c}\cong E(\tau)_{r-1}^{*,*-c} \quad 
for \ r\ge 2.\] 
Hence the filtrations are the same, i.e. 
$N^cH_{et}^*(X;\bZ/p)= F_{\tau}^{*,*-c}$
where 
\[F_{\tau}^{*,*-c}=Im(\times \tau^c:H^{*,*-c}(X;\bZ/p)\to H^{*,*}(X;\bZ/p)).\]
\end{thm} 


\begin{lemma}
Let $x\in H^{*,*}(X)$ and $x\in N^cH^*(X;\bZ/p)$. 
Then if  the map $H^{*+1,*-c}(X)\to H^{*+1,*}(X)$
is injective,  then $x\in N^cH^*(X)$.
\end{lemma}
\begin{proof} Consider the exact sequences
\[ \begin{CD} @>{p}>> H^{*,*-c}(X) @>{r_1}>> H^{*,*-c}(X;\bZ/p)
@>{\delta_1}>> H^{*+1,*-c}(X) \\
  @.  @VV{f_1}V  @VV{f_2}V @VV{f_3}V  \\
@>{p}>> H^{*,*}(X) @>{r_2}>> H^{*,*}(X;\bZ/p)
@>{\delta_2}>> H^{*+1,*}(X)
\end{CD} \]
By the assumption of this lemma,  we take $x'\in H^{*,*-c}(X;\bZ/p)$
such that $r_2(x)=f_2(x')$. So $\delta_2f_2(x')=0$.  Since $f_3$ is injective,
we see $\delta_1(x')=0$,  Hence there is $x''\in H^{*.*-c'}(X)$
such that $r_1(x'')=x'$.
\end{proof}

\begin{lemma} If $y\in H^{2*,*}(X;A)$, then
$y\in \tilde N^*H^{2*}(X;A)$.
\end{lemma}
\begin{proof}  Recall that 
\[ y \in H^{2*,*}(X;A)\cong CH^*(X)\otimes A.\]
Hence $y$ is represented by closed algebraic set 
supported $Y$, while $Y$ may be singular.
However by Totaro [To], we have the modified cycle map $\bar cl$
\[ cl:CH^*(X)\otimes A\stackrel{\bar cl}{\to} MU^{2*}(X)/p\otimes_{MU^*}A\to H^{2*}(X;A)\]
for the complex cobordism theory $MU^*(X)$.
It is known [Qu1] that elements in $MU^{2*}(X)$ is written by 
a proper map to X from a smooth complex manifold.
Hence we can take $Y$ smooth.
\end{proof}


\begin{lemma}  Let $cl$ be the cycle map.
If $H^*(X;A)$ is  generated by elements of $deg\le m$, as an $Im(cl)\otimes A$-module, then we have  
\[DH^*(X;A)=N^1H^*(X;A)/(p,\tilde N^1H^*(X;A))
=0\quad  when \  m<*.\]
\end{lemma}
\begin{proof}
Each element $x\in H^*(X;A)$ of $m<*$ is written as 
\[ x=\sum ag\qquad where \ \ a\in Im(cl)^+,\  g: Im(cl)-module \ generator.\]
We have $f:Y\to X$ with $f_*(1)=a$ from the preceding lemma.  Hence 
\[ f_*f^*(g)=f_*(1\cdot f^*(g))=f_*(1)g=ag\]
 by Frobenius reciprocity law.
\end{proof} 







The following lemma is proved by Colliot-Thelene and Voisin
[Co-Vo] by using the Bloch and Ogus spectral sequence and the 
affirmative answer of the Bloch-Kato conjecture
by Voevodsky. ([Vo3]. [Vo4])
\begin{lemma}  ( [Co-Vo])  Let $X$ be a smooth complex variety.
Then any torsion element in $H^*(X)$ is in
$N^1H^*(X)$.
\end{lemma}


{

\section{the main lemma}
 



The following lemma is the $Q_i$-version of  one of results by Benoist and Ottem.
\begin{lemma}
Let $\alpha\in N^1H^s(X)$ for $s=3$ or $4$.
If $Q_i(\alpha)\not =0\in H^*(X;\bZ/p)$ for some $i\ge 1$,  then
\[ DH^s(X)\supset \bZ/p\{\alpha\},\quad 
DH^s(X;\bZ/p^t)\supset \bZ/p\{\alpha\}\ \ for\ t\ge 2.
\]
\end{lemma}
\begin{proof}
Suppose $\alpha\in \tilde N^1H^s(X)$ for $s=3$ or $4$, i.e.
there is  a smooth $Y$ with $f:Y\to X$ such that
the transfer $f_*(\alpha')=\alpha$ for $\alpha'\in H^*(Y)$.
Then for $s=4$
\[ Q_i(\alpha')=(P^{\Delta_i}\beta-\beta P^{\Delta_i})(\alpha')=(-\beta P^{\Delta_i})(\alpha') 
 = -\beta (\alpha')^{p^i}\]
\[ =-p^i(\beta \alpha')(\alpha')^{p^i-1}=0 \quad (by\ Cartan formula)\]
since $\beta(\alpha')=0$ and $P^{\Delta_i}(y)=y^{p^i}$ for $dim(y)=2$.
(For $s=3$, we get also $Q_i(\alpha')=0$ since  $ P^{\Delta_i}(x)=0$ for $dim(x)=1$.)
 This is a contradiction
\[0\not =Q_i(\alpha)=Q_i(f_*\alpha')=f_*(Q_i\alpha')=0.\]

The case $A=\bZ/p^t$, $t\ge 2$ is proved similarly,  since 
for $\alpha'\in H^*(X;A)$ we see $\beta \alpha'=0\in H^*(X;\bZ/p)$.
\end{proof}
\begin{cor}
As an $\Lambda(Q_0,Q_i)$-module, let (for $i\ge 1$)
\[ H^*(X;\bZ/p)\supset \Lambda(Q_0,Q_i)\{a\}\quad |a|=2\ or \ 3.\]
Then $DH^*(X)\supset \bZ/p\{Q_0a\}$. 
\end{cor}
 




\section{classifying spaces for finite groups }

Let $G$ be a finite group or an algebraic group, and $BG$ its classifying space.
For example, when $G=G_m$ is the multiple group, we see
\[ BG_m=BS^1\cong \bP^{\infty}, \quad 
H^*(\bP^{\infty})\cong \bZ[y]\ \  with\  |y|=2,\]
for the infinite (complex) projective space $\bP^{\infty}$.  Note that $BG_m$ is a colimit of complex projective spaces.
Though $BG$ itself is not a colimit of complex projective varieties, we can take a  complex projective variety
$X(N)$ ([Ek]) for $N\ge 3$ such that there is a map $ j:X(N)\to BG\times \bP^{\infty} $  with
\[ H^*(BG\times \bP^{\infty};A)\stackrel{j^*}{\cong} H^*(X(N);A)\quad for \ 
*<N.\]

Note that  $N^nH^*(X;A)/\tilde N^nH^*(X;A)$ is invariant under replacing 
$X$ with $X\times \bP^m$ for all $n$ and all abelian groups $A$.  In fact, from Kunneth formula,
\[ H^*(X\times \bP^m; A)\cong H^*(X;A)\otimes \bZ[y]/(y^{m+1}),\]
here $y\in CH^1(\bP^m)$, and we see $Ideal(y)\subset  \tilde N^*H^*(X\times \bP^{m};A)$ by the Frobenius reciprocity law (Lemma 3.4). 

Moreover Benoist and Ottem  show the above quotient is the stable birational invariant (Proposition 2.4 in [Be-Ot]).

In this paper, we say the above $X(N)$ as an (complex  projective) approximation $for$ $BG$ (which  is an approximation $of$ $BG\times \bP^{\infty}$ strictly), and
let us write the stable rational invariant by 
\[DH^*(X;A)=N^1H^*(X;A)/(p,\tilde N^1H^*(X;A)).\]



Hereafter, we consider $DH^*(X;A)$ when $A=\bZ_{(p)}$.

Let $p$ be an odd prime. (When $p=2$, the different but the similar argument works.)
Let $G=(\bZ/p)^3$ the $rank=3$ elementary abelian $p$-group.
Then the $mod(p)$ cohomology is 
 \[H^*(BG;\bZ/p)\cong \bZ/p[y_1,y_2,y_3]\otimes \Lambda(x_1,x_2,x_3).\]
Here $|y_i|=2, |x_i|=1, \beta(x_i)=y_i$, and $\Lambda(a,...,b)$ is the $\bZ/p$-exterior algebra generated by $a,...,b$. The integral cohomology 
(modulo $p$) 
is isomorphic to
\[ H^*(BG)/p\cong Ker(Q_0)\cong H(H^*(BG;\bZ/p);Q_0)\oplus
Im(Q_0)\]
where $H(-;Q_0)$ is the homology with the differential $Q_0$.
It is immediate \\ $H(H(B\bZ/p;\bZ/p); Q_0)\cong \bZ/p$.
By the Kunneth formula, we have \\
$H(H(BG;\bZ/p);Q_0)\cong \bZ/p$.  Thus we have
\[ H^*(BG)/p\cong \bZ/p\{1\}\oplus Im(Q_0) \]
\[\cong \bZ/p[y_1,y_2,y_3](1,Q_0(x_ix_j),
Q_0(x_1x_2x_3)|i<j)\]
where the notation $R(a,...,b)$ (resp. $R\{a,..., b\}$)
means the $R$-submodule (resp. the free $R$-module)
generated by $a,...,b$.
Here we note $H^+(BG)$ is just $p$-torsion.




We know $Q_i(y_j)=y_j^{p^i}$ and $Q_j$ is a derivation.  Let us write 
\[ \alpha=Q_0(x_1x_2x_3)=y_1x_2x_3-y_2x_1x_3+y_3x_1x_2.\]
Note $\alpha\in H^4(X)$, $p\alpha=0$,  and $\alpha\in 
N^1H^*(X)$ from Lemma 3.5.  Moreover
 \[Q_1(\alpha)=y_1y_2^px_3-y_1y_3^px_2-...\not =0 \in H^*(X;\bZ/p).\]
Similarly, for $\alpha_{ij}=Q_0(x_ix_{j})$, we see $Q_1(\alpha_{ij})\not =0$.
Hence from Lemma 3.5 and $y_i\in \tilde N^1H^*(X)$, we see 
$y_i\alpha\in \tilde N^1H^*(X)$ by Lemma 3.4.
Thus we have 
\begin{thm}  Let $X=X(N)$ with  $N> 2p+3$ be an approximation for $(B\bZ/p)^3$.  Then
we have 
\[DH^*(X)\cong 
\bZ/p\{\alpha_{ij}, \alpha|1\le i<j\le 3\}\quad *<N.\]
\end{thm}
\begin{proof}
We see $H^*(BG)/(p,y_1,y_2,y_3)\cong \bZ/p\{1,\alpha_{ij},\alpha\}$. Of course $1\not \in N^1H^*(X)$, we have the theorem from Lemma 4.1.
\end{proof}
\begin{cor}  Let $X=X(N)_n$ with  $N>2p+3$ 
be an approximation for $(B\bZ/p)^n$.  Then
we have 
\[DH^*(X)\supset 
\bZ/p\{\alpha_{ij}, \alpha_{i'j'k'}|1\le i<j\le n, 1\le i'<j'<k'\le n\}\]
for $*<N$
where $\alpha_{ij}=Q_0(x_ix_j)$ and $\alpha_{i'j'k'}=Q_0(x_{i'}x_{j'}x_{k'})$.
\end{cor}
\begin{cor}  If $n\not =m\ge 3$, then $X(N)_n$ and $X(N)_m$
are not stable birational equivalent.
\end{cor}

Let $rank_p(G)=r$, i.e., 
there is an injection $j:(\bZ/p)^{r}\subset G$.
Let $j^*:H^*(BG)\to H^*(B(\bZ/p)^{r})$ be the induced map.   \begin{cor}
Let $rank_p(G)\ge s-1$ for $s=4$ (or $s=3$).
If the image $Im(j^*)$ contains $\alpha$ (or $\alpha _{12}$
),  then for the approximation $X=X(N)$ for  $BG$, we have 
$ DH^s(X) \not =0$ for $N>2p+3$.
\end{cor}


Let $G$ be a non abeian group of order $p^3$ (see $\S 8$,
for details).  Then  
$H^{even}(BG)$ is generated by Chern classes, and
$H^{odd}(BG)$ is a $p$-torsion, and the operation $Q_1$ induces the injection
\[ 0\to H^{odd}(BG)\stackrel{Q_1}{\to} H^{even}(BG).\]

Such groups are four types (see $\S 8$ below), and they are called extraspecial $p$ groups $G=p_{\pm}^{1+3}$ order $p^3$.
When $G=Q_8=2_-^{1+2}$ the quaternion group of order $8$, we know  $H^{odd}(X)=0$.
However when $G=D_8=2_+^{1+2}$ the dihedral group of order $8$, the cohomology $H^{odd}(BG)$ is generated as an $H^{even}(BG)$ by an element $e$ of $deg(e)=3$.
When $G=E=p_+^{1+2}$ for $p\ge 3$, $H^{odd}(BG)$ is generated by
$e_1,e_2$ with  $deg(e_i)=3$.
When $G=M=p_-^{1+2}$ for $p\ge 3$, $H^{odd}(BG)$ is generated by $e$ but  $deg(e)=2p+1$.

From Lemma 3.4 (Frobenius reciprocity) and the main lemma
(Lemma 4.1),
we have the following theorem. 

\begin{thm}  Let $X=X(N)$ with  $N>2p+3$ be an approximation for an extraspecial $p$-group $G$ of 
order $p^3$.  Then
we have for $*<N$
\[ DH^*(X)\cong 
\begin{cases} 0\quad for \ G=Q_8\\
           \bZ/2\{e\}\quad for \ G=D_8\\
               0\ or \ \bZ/p\{e\} \quad for\ G=M \\
\bZ/p\{e_1,e_2\} \quad for \ G=E.
\end{cases} \]
\end{thm}

\section{connected groups}


At first, we consider when $G=U_n$, $SU_n$  or $Sp_{2n}$
for all $p$,
the cohomology $H^*(BG)$ has no torsion.
Then $H^*(BG)$ is generated by Chern classes, e.g., 
\[H^*(BU_n)\cong CH^*(BU_n)\cong \bZ_{(p)}[c_1,...,c_n],
\]
\[ H^*(BSp_{2n})\cong CH^*(BSp_{2n})\cong 
\bZ_{(p)}[c_2,c_4,...,c_{2n}].\]
 Hence $DH^*(X)=0$
for the approximations $X$ for these groups.

Next we consider the case $G=SO_3$ and  $p=2$.  Then
\[ H^*(BG;\bZ/2)\cong \bZ/2[w_1,w_2,w_3]/(w_1)\cong \bZ/2[w_2,w_3],\] 
where $w_i$ is the $i$-th Stiefel-Whitney class for $SO_3\subset O_3$ and $w_i^2=c_i$
is the $i$-th Chern class for $SO_3\subset U_3$.
 Here we know 
$Q_0(w_2)=w_3,$ and $Q_1(w_3)=w_3^2$.  Therefore we have 
[Ya1]
\[ H^*(BG;\bZ/2)\cong \bZ/2[c_2,c_3]\{1,w_2,w_3=Q_0(w_2),w_2w_3=Q_1w_2\}\]
\[ \cong \bZ/2[c_2,c_3]\{w_2,Q_0(w_2),Q_1(w_2),Q_0Q_1(w_2)=c_3\}\oplus \bZ/2[c_2] \]
\[\cong \bZ/2[c_2,c_3]\otimes \Lambda(Q_0,Q_1)\{w_2\}
\oplus \bZ/2[c_2].\]
In particular $H^*(BG)/2\cong Ker(Q_0)\cong \bZ/2[c_2,c_3]\{1,w_3\}. $

From Corollary 4.2, we have the following theorem.
\begin{thm} Let $X_n=X_n(N)$ be approximations for $BSO_n$
for $n\ge 3$.
Then
\[  DH^*(X_3)\cong \bZ/2\{w_3\}\subset DH^*(X_n).\]
\end{thm}
\begin{proof} We have seen 
$H^*(BG)/(2, c_2,c_3)\cong \bZ/2\{1,w_3\}$.
Here $1\not \in N^1H^*(X)$ but $w_3=Q_0w_2\in N^1H^*(X)$.
 Therefore from Corollary 4.2,  the approximation $X$ for $BSU_3$ satisfies 
\[DH^*(X)\cong \bZ/2\{Q_0w_2\}\cong \bZ/2\{w_3\}.\]
\end{proof}

We consider simply connected groups. Let us write by $X$ an 
approximation of $BG_2$ for the exceptional simple 
group $G_2$ of $rank=2$.
The $mod(2)$ cohomology is generated by the Stiefel-Whitney classes $w_i$ of
the real representation $G_2\to SO_7$
\[ H^*(BG_2;\bZ/2)\cong \bZ/2[w_4,w_6,w_7],
\quad P^1(w_4)=w_6,\ Q_0(w_6)=w_7, \]
\[H^*(BG_2)\cong (D'\oplus D'/2[w_7]^+)\quad where \ \ D'=\bZ_{(2)}[w_4,c_6].\]
Then we have  $Q_1w_4=w_7, Q_2(w_7)=w_7^2=c_7$ (the Chern class).

The Chow ring of $BG_2$ is also known
\[ CH^*(BG_2)\cong (D\{1,2w_4\} \oplus D/2[c_7]^+)\quad where \ \ D=\bZ_{(2)}[c_4,c_6]\ \ c_i=w_i^2.\]
In particular the cycle map $cl: CH^*(BG)\to H^*(BG)$ is injective.

It is known $w_4\in N^1H^*(X;\bZ/2)$ ([Ya1])
and from Lemma 3.2, we see $w_4\in N^1H^*(X)$.
Since $Q_1(w_4)=w_7\not =0$,  from Lemma 4.1,
we have
$DH^4(X)\not =0$ ([Be-Ot]).  Moreover we have
\begin{prop} For  $X$ an approximation for $BG_2$,
we have the surjection
\[ \Lambda(w_4,w_7)^+
\twoheadrightarrow 
DH^*(X)\quad for\ *<N.\]
\end{prop}
\begin{proof} We have seen $H^*(BG)/(c_4,c_6,c_7)\cong 
\Lambda(w_4,w_7)$.
\end{proof}

\begin{thm}  Let $G$ be a simply connected group such that $H^*(BG)$ has $p$-torsion.
Let $X=X(N)$ be an approximation for $BG$ for 
$N\ge 2p+3$. Then
      $ DH^4(X)\not =0$.
\end{thm}
\begin{proof}
It is only need to prove the theorem when  $G$ is a simple group
having $p$ torsion in $H^*(BG)$.
Let $p=2$.  
It is well known that there is an emmbedding  $j:G_2\subset G$ such that
\[ H^4(BG)\stackrel{j^*}{\cong} H^4(BG_2)\cong \bZ_{(2)}\{w_4\}.\] 

Let $x=(j^{*})^{-1}w_4\in H^4(BG)$.  
From Lemma 3.1 in [Ya4], we see that $2x$ is represented by Chern classes.
Hence  $2x$ is the image from $CH^*(X)$,  and so 
$2x\in N^1H^4(X)$.  This means there is an open set $U\subset X$ such that
$ 2x=0\in H^*(U)$ that is $x$ is $2$-torsion in $H^*(U)$. Hence from Lemma 3.5,
we have $x\in N^1H^4(U)$, and so there is $U'\subset U$ such that $x=0\in H^4(U')$.
This implies $x\in N^1H^4(X)$. 


Since $j^*(Q_1x)=Q_1w_4=w_7$, we see $Q_1x\not =0$.
From the main lemma (Lemma 4.1), we see $DH^4(X)\not =0$
for $G$.
  


For the cases $p=3,5$, we consider the exceptional groups $F_4,E_8$ respectively.  Each simply connected simple group
$G$ contains $F_4$ for $p=3$, $E_8$ for $p=5$. There is $x\in H^4(BG)$ such  that $Q_1(x)\not =0\in H^*(BG;\bZ/p).$
In fact,  there is embedding $j:(\bZ/p)^3\subset G$ with $j^*(x)=Q_0(x_1x_2x_3)$. 
Hence we have the theorem.
\end{proof} 
\begin{cor}  Let $X$ be an approximation for $BSpin_n$
with $n\ge 7$ or $BG$ for an exceptional group $G$.  Then $X$ is not stable rational.
\end{cor}
{\bf Remark.}
Kordonskii [Ko], Merkurjev (Corollary 5.8 in [Me]),
and Reinstein-Scavia show [Re-Sc]
that  $BSpin_n$ itself is stably rational when $n\le 14$.
 These facts imply that  
the (Ekedahl) approximation $X$  is not stable rationally equivalent to 
$GL_M/G=BG$ where $M$ is a large number such that $G$ acts freely on $GL_M$.  In fact, $X$ is  constructed  from 
a quasi projective variety $BG$ as taking 
 intersections of subspaces of $\bP^{M'}$ for a large $M'$.
(The author thanks Federico Scavia who pointed out this remark.)

Here we 
note the relations to retract rationally. (Recall the first remark in $\S 3$ about the unramified cohomology.)   By Orlov-Vishik-Voevodsky [Or-Vi-Vo], ([Te-Ya] for $p:odd$,)
we know
\begin{lemma} The comology 
$H^0_{Zar}(X;\mathcal{H}^*_{\bZ/p})$ is isomorphic to
\[H^{*,*}(X;\bZ/p)/(\tau)\oplus Ker(\tau: H^{*+1,*-1}(X;\bZ/p)
\to H^{*+1,*}(X;\bZ/p)).\]
\end{lemma}

Recall (the remark in $\S 3$) that  when $X$ is complete, $H_{nr}^*(X;\bZ/p)
\cong H^0_{Zar}(X;\mathcal{H}^*_{\bZ/p})$, and it 
is an invariant of the retract rationality of $X$
[Me].



\begin{thm} ([Ya4]) Let $G$ be the groups in the preceding theorem.
Then there is an element (Rost invariant) 
$ 0\not =a\in H^{0}_{Zar}(X;\mathcal{H}^3_{\bZ/p}).$
Hence $X$ is not retract rational.
\end{thm}
\begin{proof}
Recall $x\in H^4(X)$ such that there is a Chern class $\alpha\in CH^4(X)
\cong H^{4,2}(X;\bZ_{(2)})$
with $\alpha=px\in H^4(X)$.  Of course $\alpha=0\in H^*(X;\bZ/p)$, i.e., $\tau^2\alpha=0$ in $H^{*,*}(X;\bZ/p)$. But it is known [Or-Vi-Vo] that
\[\tau : H^{*,*-1}(X;\bZ/p)\to H^{*,*}(X;\bZ/p)\ \ 
is\ injective.\]  Hence 
$\alpha\in Ker(\tau: H^{4,2}(X;\bZ/p)
\to H^{3,3}(X;\bZ/p)).$
Take $a\in H^0_{Zar}(X;\mathcal{H}^3_{\bZ/p})$ as the corresponding element to $\alpha$ from the preceding lemma.
\end{proof}

{\bf Remark.}
We do $not$  assume $H^0_{Zar}(X;\mathcal{H}_{\bZ/p}^*)\cong 
H^0_{Zar}(X';\mathcal{H}_{\bZ/p}^*)$ for an other approximation $X'$.

{\bf Remark.}
Since stable rationality implies retract rationality, the preceding corollary is the just corollary of the above result. However
note  in general,
\[H_{nr}^*(k(GL_N/G);\bZ/p)\not \cong H^0(BG;\mathcal{H}_{Z/p}^*),\]
\[\quad k(X)\not \cong k(W)^G\quad W: \bC-vector\ space.\]
























By Voevodsky { [Vo1], [Vo2], we have  the $Q_i$ operation also in the 
motivic cohomology $H^{*,*'}(X;\bZ/p)$ with
$deg(Q_i)=(2p^i-1,p-1)$.  Then we can take 
\[ deg(w_4)=(4,3),\ deg(w_6)=(6,4),\ deg(w_7)=(7,4),\ deg(c_7)=(14,7).\]
By Theorem 3.1, the above means
\[ w_7\in N^{7-4}H^*(X;\bZ/2)=N^{3}H^*(X;\bZ/2).\]
We can not see here that $0\not =w_7\in DH^*(X)$,
but see the following proposition.

\begin{prop}  For $X$ the approximation for  $BG_2$, we have
\[\bZ/2\{w_7\}\subset D^3H^*(X)=N^3H^*(X)/(2,\tilde N^3H^*(X)).\]
\end{prop}
\begin{proof}
Suppose $w_7\in \tilde N^3H^*(X)$.  That is, there is
$x\in H^1(Y)$ with $f_*(x)=w_7$ for $f:Y\to X$.
Act $Q_2$ on $H^*(Y)$, and
\[ Q_2(x)=(P^{\Delta_2}\beta +\beta P^{\Delta_2})(x)=0\]
since $\beta(x)=0$ and $P^{i}(x)=Sq^{2i}(x)=0$ for $i>0$. 
But $Q_2w_7=c_7\not =0$.
This contradicts to the commutativity of $f_*$ and $Q_2$. 
\end{proof}

At last of this section, we consider the case $G=PGL_p$.
We have (for example Theorem 1.5,1.7 in [Ka-Ya])  
additively \[H^*(BG;\bZ/p)\cong 
M\oplus N \quad with \ \ 
 M\stackrel{add.}{\cong}\bZ/p[x_4,x_6,...,x_{2p}],\quad \]
\[ N= SD\otimes \Lambda(Q_0,Q_1)\{u_2\}\quad with\ \ 
SD=\bZ/p[x_{2p+2},x_{2p^2-2p}]\]
where $x_{2p+2}=Q_1Q_0u_2$ and suffix means its degree.  The Chow ring is given as 
\[ CH^*(BG)/p\cong M\oplus SD\{Q_0Q_1(u_2)\}.\]
From Corollary 4.2, we have ;
\begin{thm}  Let $p$ be odd.
For an approximation $X$ for $BPGL_p$,
we see $ \bZ/p\{Q_0u_2\}\subset DH^*(X)$, and moreover there is a surjection \[  \bZ/p[x_{2p^2-2p}]\{Q_0u_2\}\twoheadrightarrow
DH^*(X)\quad for\ *<N.\]
\end{thm}
For $G=PGL_p$,  we easily see [Te-Ya]  
\[H^2(X;\bZ/p)/N^1H^2(X;\bZ/p)
\cong H^2(BG;\bZ/p)/N^1H^2(BG;\bZ/p),\]
\[ H_{Zar}^0(X;\mathcal{H}^*_{\bZ/p})\supset 
H^2(BG;\bZ/p)/N^1H^2(BG;\bZ/p)
\supset  \bZ/p\{u_2\}.\]
(When $G=SO_3$, we see $H^*(X;\bZ/2)/N^1H^*(X;\bZ/2)\supset \bZ/2\{1,w_2\}.$)  Hence in both cases $X$ are  not retract rational.


{\bf Remark.}  All examples $X$ such that $H^*(BG)$ have $p$-torsion in this section are not retract rational.



\bf Remark.}
There are primes $\ell\not =p$ such that almost all results in $\S 5,6$ hold
for $X$ varieties over $\bar k$ with $ch(k)=\ell$
letting 
\[ H^*(X)=H_{et}^*(X;\bZ_{p})=Lim_{i}H^*_{et}(X;\bZ/p^i),\]
\[DH^*(X)=N^1H^*(X)/(p,\tilde N^1H^*(X), Im(cl)).\]
The above facts are proved by the similar arguments
in [Pi-Ya] (exchanging $p$ and $\ell$). However, we do not
prove that $DH^*(X)$ is stable birational invariant
of varieties over the field $\bar k$.






\section{$\bZ/p$-coefficient cohomology for abelian groups }

In the preceding sections, we have seen that cases
$DH^*(X;A)\not =0$ are not so rare 
for $A=\bZ_{(p)}$, $\bZ/p^i$, $i\ge 2$. However currently it seems  no 
such example for $A=\bZ/p$.   
\begin{que} We have $DH^*(X;\bZ/p)\cong 0$
for an approximation $X$ for  $BG$ ? 
\end{que}




At first, we consider the case $G=(\bZ/p)^3$.
We can take an (quasi projective) approximation $\bar X(N)$ of $B\bZ/p$ explicitly 
by the quotient (the $N$-dimensional lens space)
\[\bar X(N)=\bC^{N*}/(\bZ/p)\quad where \ \bC^{N*}=(\bC^N-\{0\}).
\]
We consider the approximations 
\[X(N)\to \bar X(N)\times \bP^{N}\to B\bZ/p\times \bP^{\infty}.\]
Let us write $X_i$ (resp. $X_i'$) the above $\bar X(N)$ (resp. $\bar X(N-1))$ for a sufficient large number $N$.
Let \[ i_1:Y_1=X_1'\times X_2\times X_3\to
 X=X_1\times X_2\times X_3
.\] 
Similarly we define $Y_2,Y_3$, and the disjoin union $Y=Y_1\sqcup Y_2\sqcup Y_3$.

Recall that for $p:odd$
\[H^*(X;\bZ/p)\cong \bZ/p[y_1,y_2,y_3]/(y_1^{N+1},y_1^{N+1},y_3^{N+1})\otimes
\Lambda(x_1,x_2,x_3),\]
and $H^*(Y_i;\bZ/p)\cong H^*(X;\bZ/p)/(y_i^N)$ for $i=1,2,3$.
For $p=2$, some graded ring $grH^*(X;\bZ/2)$ is isomorphic to the above ring
(in fact $x^2_i=y_i$).


For the embedding $f_i:X_i'\to X_i$, it is known 
$f_{i*}(1)=c_1(N)$ where $N$ is the normal   bundle for $X_i'\subset X_i$.
Hence the transfer is given by
\[ f_{1*}(1)=y_1,\quad f_{2*}(1)=y_2\quad f_{3*}(1)=y_3.\]
(Since $y_1\in CH^*(X)$, there is $\tilde i_*: \tilde Y\to X$ such that
$\tilde i_*(1)=y_1$.  We take $\tilde Y=Y_1$.)
 Therefore we have for $x=(x_2x_3+ x_3x_1+x_1x_2)\in H^*(Y_1
\sqcup Y_2\sqcup Y_3;\bZ/p)$,
\[ f_*(x)=y_1x_2x_3+y_2x_3x_1+y_3x_1x_2=Q_0(x_1x_2x_3)=\alpha.\]
Note that the element $\alpha'=(x_1x_2+x_2x_3+x_3x_1)$ is not 
in the integral cohomology $H^*(Y)$.  Similarly we have 
\[ i_*(Q_1\alpha')=i_*( y_1^px_2-y_2^px_1)+....=(y_3y_1^px_2+y_3y_2^px_1)+...=Q_1(\alpha).\]
Thus we see $\alpha\in \tilde N^cH^*(X;\bZ/p).$
More generally, we see 
\begin{thm}
We have the  approximation $X$ for 
$(B\bZ/p)^n$ such that  
\[DH^*(X;\bZ/p)=0 \quad *<N.\]
\end{thm}


We recall here  the motivic cohomology.
By Voevodsky [Vo 1], $H^{*,*'}(B\bZ/p;\bZ/p)$ satisfies the Kunneth formula so that for 
\[H^{*,*'}(B(\bZ/p)^n;\bZ/p)\cong \bZ/p[\tau, y_1,...,,y_n]/(y_1^{N+1},...,y_n^{N+1})\otimes
\Lambda(x_1,...,x_n).\]
Here $0\not =\tau\in H^{0,1}(Spec(\bC);\bZ/p)$, and
$deg(y_i)=(2,1)$, $deg(x_i)=(1,1)$.

From Theorem 3.1, we can identify
$N^cH_{et}^*(X;\bZ/p)= F_{\tau}^{*,*-c}$
where \\ 
$F_{\tau}^{*,*-c}=Im(\times \tau^c:H^{*,*-c}(X;\bZ/p)\to H^{*,*}(X;\bZ/p)).$

\begin{cor} (Theorem 5.1 in [Te-Ya])  Let  $X$ be  the approximation for  $(B\bZ/p)^n$.  Then we have
\[ H^*(X:\bZ/p)/N^1H^*(X;\bZ/p)\cong \Lambda(x_1,....,x_n).\]
\end{cor}
\begin{proof}
Let $x\in Ideal(y_1,...,y_n)\subset H^{*,*'}(X;\bZ/p)$.  Then $deg(x)=(*,*')$ with
$*>*'$, and $x$ is a multiplying  of $\tau$. Hence $x\in N^1H^*(X;\bZ/p)$.
\end{proof}
\begin{proof}[Proof of Theorem 7.2.]
Let $x\in N^1H^*(X;\bZ/p)$.  From the above corollary.
$x\in Ideal(y_1,...,y_n)$ which is in the image of the transfer
(such as $Y_1=X_{1}'\times X_2\times ...\times X_n$).
That is $x\in \tilde N^1H^*(X;\bZ/p)$. 
\end{proof}

We can extend Lemma 7.3, by using the following lemma.
Let us write by $XG$ an approximation for $BG$.
\begin{lemma}
Let $G$ have a Sylow $p$-subgroup $S$.
If $ DH^*(XS;\bZ/p)=0$, then 
so for $BG$.
\end{lemma}
\begin{proof}
Let $j: BS\to BG$ and $i; Y\to XS$.  We consider maps
\[ H^*(Y;\bZ/p)\stackrel{i_*}{\to}
  H^*(XS;\bZ/p)\stackrel{j_*}{\to}
H^*(XG;\bZ/p).\]
Here $j_*=cor_S^G$ is the transfer for finite groups.

Note $j^*N^1H^*(XG;\bZ/p)\subset N^1H^*(XS;\bZ/p)$ by the naturality
of $j^*$.
Hence given $x\in N^1H^*(XG;\bZ/p)$ the element 
$y=j^*(x)$ is in $N^1H^*(XS;\bZ/p)$.

 By the assumption
in this lemma, there is $y'\in H^*(Y;\bZ/p)$ with $i_*(y')=y$.
Thus  we have
$\ j_*i_*(y')=j_*y=j_*j^*(x)=[G;S]x.$ 
\end{proof}


\begin{cor}
Let $G$ have  an abelian $p$ Sylow subgroup, and $X=X(N)$
be an approximation for $BG$.  Then
$ DH^*(X;\bZ/p)=0$ for $  *<N$.\
\end{cor}






\section{the groups $Q_8$ and $D_8$}


When $|G|=p^3$, we have the short exact sequence
\[ 0\to C\to G\to V \to 0\]
where $C\cong \bZ/p$ is in the center and $ V\cong \bZ/p\times \bZ/p$. 
Let us take generators such that $C=\la c\ra, V=\la a,b \ra.$
Moreover  $[a,b]=c$ (when $G$ is non-abelian).

There are four cases, when $p=2$, the quaternion group $Q_8$
and the dihedral group $D_8$. When $p\ge 3$, the extraspecial groups
$p_-^{1+2}=M$ and $p^{1+2}_+=E$.

We will show 
\begin{thm} Let $X=X(N)$ be an approximation for $Q_8$
or $D_8$.  Then $DH^*(X;\bZ/2)=0$ for $*<N$.
\end{thm}




{\bf 8.1.}  The case $G=Q_8$. Then $p=2$ and $a^2=b^2=c$.
 Its cohomologies are well known (see [Qu2])
\[H^*(BG)/2\cong \bZ/2[y_1,y_2,c_2]/(y_i^2,y_1y_2)\ \ 
|y_i|=2,\]
\[H^*(BG;\bZ/2)\cong \bZ/2[x_1,x_2,c_2]/(x_1x_2+y_1+y_2,
x_1y_2+x_2y_1)\]
\[\cong \bZ/2\{1,x_1,y_1,x_2,y_2,w\}\otimes \bZ/2[c_2]\]
where $ x_i^2=y_i$ $|x_i|=1$, and  $w=y_1x_2=y_2x_1$,
$|w|=3$.

Therefore, we see 
\[H^*(BG;\bZ/2)/(y_1,y_2,c_2)\cong \bZ/2\{1,x_1,x_2\}.\]
Of course $deg(x_i)=(1,1)$ in $H^{*,*'}(BG;\bZ/2)$ and they are
not in $N^1H^*(BG;\bZ/2)$.  Thus we have Theorem 8.1 for 
$G=Q_8$.


{\bf 8.2.}  The case $G=D_8$.  That is $p=2$ and $a^2=c, b^2=1$.
 It is well known
\[ H^*(BG)/2\cong \bZ/2[y_1,y_2,c_2]/(y_1y_2)\{1,e\}\quad with \ |e|=3.\]
  The mod $2$ cohomlogy is written [Qu2]
\[ H^*(BG;\bZ/2)\cong \bZ/2[x_1,x_2,u]/(x_1x_2)\quad (with\ |u|=2)\]
\[ \cong (\oplus_{i=1}^2 \bZ/2[y_i]\{y_i, x_i,y_iu,x_iu\}\oplus
\bZ/2\{1,u\})\otimes \bZ/2[c_2].\]
Here $y_i=x_i^2, u^2=c_2$ and $Q_0(u)=(x_1+x_2)u=e,
Q_1Q_0(u)=(y_1+y_2)c_2$.

We note $y_1,y_2,c_2\in CH^*(BG)/2$ and
\[ H^*(BG;\bZ/2)/(y_1,y_2,c_2)\cong
(\oplus _{j=1}^2\bZ/2\{x_j,x_ju\})\oplus \bZ/2\{1,u\}.\]
Moreover, $deg(x_j)=(1,1)$, $deg(u)=(2,2)$ in the motivic cohomology \\ 
$H^{*,*'}(BG;\bZ/2)$ and they are not in $N^1
H^*(BG;\bZ/2)$.
Here we note $deg(x_ju)=(3,3)$, but there is $u_j'\in 
H^{3,2}(BG;\bZ/2)$ with $x_ju=\tau u_j'$ from Lemma 6.2 in [Ya3]
( i.e., $x_iu\in N^1H^*(X;\bZ/2)$).

Hence for the proof of Lemma 8.1 (for $G=D_8$),
 it is only need to show 
\begin{lemma} We have $x_iu\in \tilde N^1H^*(BG;\bZ/2)$.
\end{lemma}

To prove above lemma, for a $G$ variety
$H$, we consider the equivariant cohomology 
\[ H_G^*(H;\bZ/p)=H^*(U(N)\times _{G}H;\bZ/p)\]
where $U(N)$ is an (approximation of) contractible free $G$-variety.  Let us write
\[ X_GH=appro. \ of\ U(N)\times_GH \ so\ that\ H_G^*(H;\bZ/p)\cong H^*(X_GH;\bZ/p).\]
For the injection $i:H\subset K$ of $G$-varieties, we can
define the transfer
\[ i_*: H_G^*(H;\bZ/p)\to H^*_G(K;\bZ/p)
\quad by \  i:X_GH\stackrel{id\times_Gi}{\to} X_GK. \]



Hereafter in this section, let $G=D_8$.
We recall arguments in [Ya3].
We define the $2$-dimensional representation 
$\tilde c: G\to U_2$ such that 
$\tilde c(a)=diag(i,-i)$
and $\bar c(b)$ is the permutation matrix $(1,2)$. 
By this representation, we identify that
$W=\bC^{2*}=\bC^2-\{0\}$
 is an $G$-variety. Note $G$ acts freely on $W\times \bC^*$
but it does not so on $W=\bC^{2*}.$

 The fixed points set on $W$ under $b$ is
\[  W^{\la b\ra}=\{(x,x)|x\in \bC^*\}=\bC^*\{e'\} ,\quad e'=diag(1,1)\in GL_2(\bC).\]
Similarly $W^{\la bc\ra}=\bC^*\{a^{-1}e'\}.$  
  Take
\[ H_0=\bC^{*}\{e',ae'\},\quad H_1=\bC^{*}\{g^{-1}e',q^{-1}ae'\}\]
where $g\in GL_2(\bC)$ with $g^{-1}bg=ab$ (note $(ab)^2=1$).
Let us write
\[ H=H_0\sqcup H_1.\]
Then $G$ acts on $H_i$ and acts freely on $\bC^{2*}-H$.
In fact it does not contain
fixed points of non-trivial stabilizer groups.



 
We consider the transfer for some $G$-set $H$ in 
$\bC^{2*}$, and induced equivariant cohomology
\[ i_*: H^*_G(H;\bZ/2)\to H^*_G(\bC^{2*};\bZ/2).\]
\begin{lemma} We have 
\[H^*_G(H_0;\bZ/2)\cong \bZ/2[y] \otimes \Lambda(x,z)\quad
 with \ y=x^2,\ |x|= |z|=1.\]
\end{lemma}
\begin{proof}
We consider the group extension $0\to \la a
\ra \to G\to \la b \ra \to 0$ and the induced spectral sequence
\[E_2^{*,*'}=H^*(B\la b\ra; H^{*'}_{\la a\ra }(H_0;\bZ/2))\Longrightarrow
H_G^*(H_0;\bZ/2).\]

Since $\la a\ra\cong \bZ/4$ acts freely on $H_0$, we see
\[H_0/\la a\ra \cong \bC^{*}\{e',ae'\}/\la a\ra
 \cong \bC^{*}/\la a\ra \cong \bC^{*}.\]
Therefore we have 
\[H^*_{\la a\ra}(H_0;\bZ/2)\cong H^*(\bC^*/\la a\ra;\bZ/2)
\cong H^*(\bC^*;\bZ/2)\cong \Lambda (z)\quad |z|=1.\]
Since $\la b\ra$ acts trivially on $\Lambda(z)$ we have
$ H^*_G(H;\bZ/2)\cong H^*(B\la b\ra;\bZ/2)\otimes \Lambda(z).$
\end{proof}

Note $H_G^*(H_0;\bZ/2)\cong H_G^*(H_1;\bZ/2)$ and hence we 
see
\[H^*_G(H;\bZ/2)\cong  \oplus_{j=1}^2 \bZ/2[y_j]\{1_j,y_j, x_j,x_jz_j,z_j\}.\]

We consider the long exact sequence
\[ (*)\quad ...\to H^{*}_G(\{0\};\bZ/2)\stackrel{i_*=c_2}{\to}
H^{*+4}_G(\bC^2;\bZ/2) \to H^{*+4}_G(\bC^{2*};\bZ/2)
\to ...\]
and we have  $H^*(\bC^{2*};\bZ/2)\cong H^*(BG;\bZ/2)/(c_2)$. 
Hence,   we get 
\[H^*_G(\bC^{2*};\bZ/2)\cong ( \oplus_{j=1}^2 \bZ/2[y_j]\{y_j,x_j,x_ju_j',u_j'\})\oplus
\bZ/2\{1,u\}\]

Now we consider the transfer 
$H^*_G(H;\bZ/2) \stackrel{i_*}{\to} H^{*+2}_G(\bC^{2*};\bZ/2)$. 
We have explicitly (page 527 in [Ya3])
\[ i_*(1_j)=y_j,\ i_*(x_j)=y_jx_j,\ i_*(x_jz_j)=x_ju_j',\ i_*(z_j)=u_j'.\]  
Therefore we have Theorem 8.1 for $G=D_8$.

To see the above fact we recall 
 the long exact
sequence for $i;H \subset \bC^{2*}$
\[ (**)\quad ...\to H^{*+1}_G(\bC^{2*}-H;\bZ/2)\stackrel{\delta}{\to}
H^*_G(H;\bZ/2) \stackrel{i_*}{\to} H^{*+2}_G(\bC^{2*};\bZ/2)\] \[\stackrel{j^*}{\to} H^{*+2}_G(\bC^{2*}-H;\bZ/2)\to...\]
The transfer $i_*$ is determined by the following lemma.
\begin{lemma}  In the above $(**)$, we see 
$\delta=0$, and hence $i_*$ is injective.
\end{lemma}
\begin{proof}
Since $G$ acts freely on $\bC^{2*}-H$, we have
\[ H^*_G(\bC^{2*}-H:\bZ/2)\cong H^*((\bC^{2*}-H)/G;\bZ/2),\]
which is zero when $*>4$.   Hence $\delta$ must be zero 
for $*>4$, and $i_*$ is injective for $*>4$.  In particular, $i_*(y^2_jz_j)=y_j^2u_j'$.  Since  $H^*_G(H;\bZ/2)$ is 
$\bZ/2[y_1]$-free (or $Z/2[y_2]$-free,) we see
$i_*(z_j)=u_j'$.
\end{proof}











