2. Transfer and $Q_n$

The Milnor operation (in $H^*({-};\,{\mathbb Z}/p)$ ) is defined by $Q_0=\beta$ and for $n\ge 1$

\begin{equation*} Q_n=P^{\Delta_n} \beta-\beta P^{\Delta_n}, \quad \Delta_n=\left(0,..,0,\stackrel{n}{1},0,...\right) \end{equation*}

(for details see [ Reference Milnor8 ], [ Reference Voevodsky18 , section 3·1]), where $\beta$ is the Bockstein operation and $P^{\alpha}$ for $\alpha=(\alpha_1,\alpha_2,...)$ is the fundamental base of the module of finite sums of products of reduced powers.

The above lemma is known (see the proof of [ Reference Yagita23 , lemma 7·1]). The transfer $f_*$ is expressed as $g^*f^{\prime}_*$ such that

\begin{equation*} f^{\prime}_*(x)=i^*(Th(1)\cdot x), \quad x\in H^*(X;\,{\mathbb Z}/p)\end{equation*}

for some maps g, f ^′, i and the Thom class Th(1). Since $Q_n(Th(1))=0$ and $Q_n$ is a derivation, we get the lemma. However, we give here the another computational proof.

Proof of Lemma 2·1. Recall the following Grothendieck formula (e.g., [Q1])

(1) \begin{equation} P_t(f_*(x))=f_*(c_t\cdot P_t(x)).\end{equation}

Here the total reduced powers $P_t(x)$ are defined

\begin{equation*}P_t(x)=\sum_{\alpha}P^{\alpha}(x)t^{\alpha}\in H^*\left(X;\,{\mathbb Z}/p\right)\left[t_1,t_2,...\right]\quad with\ t^{\alpha}=t_1^{\alpha_1}t_2^{\alpha_2}...,\end{equation*}

where $\alpha=(\alpha_1,\alpha_2,...)$ and $degree(t^{\alpha})=\sum_i 2\alpha_i\left(p^i-1\right)$ (each element in the cohomology $H^*(X;\,{\mathbb Z}/p)$ is represented as a homogeneous part respective to the above degree). 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., $P_t(x)\in H^*(X;\,{\mathbb Z}/p)\otimes \Lambda(t_n)$ . That means

(2) \begin{equation} P_t(f_*(x))=\left(1+P^{\Delta_n}t_n\right)(f_*(x))\end{equation}

(3) \begin{align} f_*\left(c_t\cdot P_t(x)\right) & =f_*\left(\left(1+c_{p^n-1}t_n\right)\left(x+P^{\Delta_n}(x)t_n\right)\right)\nonumber\\& =f_*\left(x+\left(c_{p^n-1}x+P^{\Delta_n}(x)\right)t_n\right).\end{align}

From (1), we see $(2)=(3)$ and we have

(4) \begin{equation} P^{\Delta_n}\left(f_*(x)\right)=f_*\left(c_{p^n-1}x+P^{\Delta_n}(x)\right).\end{equation}

By the definition, $\beta$ commutes with $f_*$ , and we have

(5) \begin{equation}P^{\Delta_n}\beta(f_*(x))=P^{\Delta_n}f_*(\beta x)=f_*\left(c_{p^n-1}\beta x+P^{\Delta_n}(\beta x)\right).\end{equation}

On the other hand

(6) \begin{equation}\beta P^{\Delta_n}f_*(x)\stackrel{(4)}{=}\beta f_*\left(c_{p^n-1}x+P^{\Delta_n}(x)\right)=f_*\left(c_{p^n-1}\beta x+\beta P^{\Delta_n}(x)\right).\end{equation}

Then (5)–(6) gives that $\left(P^{\Delta_n} \beta-\beta P^{\Delta_n}\right)f_*(x)=f_*\left(P^{\Delta_n} \beta-\beta P^{\Delta_n}\right)(x).$ Thus we can prove Lemma 2·1.

By the definition, each cohomology operation h (i.e., an element in the Steenrod algebra) is written $\big($ with $Q^B=Q_0^{b_0}Q_1^{b_1}...\big)$ by

\begin{equation*}h=\sum_{A,B}P^AQ^B\quad {with}\ A=(a_1,...),\ B=(b_0,...)\ b_i=0\ {or}\ 1.\end{equation*}

Hence cohomology operations h (for $H^*({-};\,{\mathbb Z}/p))$ which commute with all transfer $f_*$ are cases $c_t=1$ , i.e. $A=0$ which are only products $Q^B$ of Milnor operations $Q_i$ .

3. Coniveau filtrations

Bloch–Ogus [ Reference Bloch and Ogus2 ] give a spectral sequence such that its $E_2$ -term is given by

\begin{equation*}E(c)_2^{c,*-c}\cong H_{Zar}^c\!\left(X,\mathcal{H}_{A}^{*-c}\right)\Longrightarrow H_{et}^*(X;\,A),\end{equation*}

where $\mathcal{H}_{A}^*$ is the Zariski sheaf induced from the presheaf given by $U\mapsto H_{et}^*(U;\,A)$ for an open $U\subset X$ .

The filtration for this spectral sequence is defined as the coniveau filtration

\begin{equation*}N^cH_{et}^*(X;\,A)= F(c)^{c,*-c},\end{equation*}

where the infinite term $ E(c)^{c,*-c}_{\infty}\cong F(c)^{c,*-c}/F(c)^{c+1,*-c-1}$ and

\begin{equation*} N^cH^*_{et}(X;\,A)=\sum_{Z\subset X;codim_X(Z)\le c} ker\!\left(j^*\,:\,H^*_{et}(X;\,A)\to H^*_{et}(X-Z,A)\right).\end{equation*}

Here we recall the motivic cohomology $H^{*,*^{\prime}}(X;\,{\mathbb Z}/p)$ defined by Voevodsky and Suslin ([ Reference Voevodsky18 , Reference Voevodsky20 , Reference Voevodsky21 ]) so that

\begin{equation*} H^{i,i}(X;\,{\mathbb Z}/p)\cong H^i_{et}(X;\,{\mathbb Z}/p)\cong H^i(X;\,{\mathbb Z}/p).\end{equation*}

Let us write $H^*_{et}(X;\,{\mathbb Z} )$ simply by $H^*_{et}(X)$ as usual. Note that $H^*_{et}(X)\not \cong H^*(X)$ in general, while we have the natural map $H_{et}^*(X)\to H^*(X)$ .

Let $0\not =\tau \in H^{0,1}(Spec({\mathbb C});\,{\mathbb Z}/p)$ . Then by the multiplying $\tau$ , we can define a map $H^{*,*^{\prime}}(X;\,{\mathbb Z}/p)\to H^{*,*^{\prime}+1}(X;\,{\mathbb Z}/p)$ . By Deligne ([ Reference Bloch and Ogus2 , foot note (1) in Remark 6·4]) and Paranjape ([ Reference Paranjape9 , corollary 4·4]), it is proven that there is an isomorphism of the coniveau spectral sequence with the $\tau$ -Bockstein spectral sequence $E(\tau)_r^{*,*^{\prime}}$ (see also [ Reference Tezuka and Yagita16 , Reference Yagita22 ]).

Lemma 3·1. (Deligne) Let $A={\mathbb Z}/p$ . Then we have the isomorphism of spectral sequence

\begin{equation*}E(c)_r^{c,*-c}\cong E(\tau)_{r-1}^{*,*-c} \quad for \ r\ge 2.\end{equation*}

Hence the filtrations are the same, i.e. $N^cH_{et}^*(X;\,{\mathbb Z}/p)= F_{\tau}^{*,*-c}$ where

\begin{equation*}F_{\tau}^{*,*-c}=Im({\times}\tau^c\,:\,H^{*,*-c}(X;\,{\mathbb Z}/p)\longrightarrow H^{*,*}(X;\,{\mathbb Z}/p)).\end{equation*}

Proof. Consider the exact sequences

.

By the assumption of this lemma, we can take $x^{\prime}\in H^{*,*-c}(X;\,{\mathbb Z}/p)$ such that $r_2(x)=f_2(x^{\prime})$ . So $\delta_2f_2(x^{\prime})=0$ . Since $f_3$ is injective, we see $\delta_1(x^{\prime})=0$ , Hence there is $x^{\prime\prime}\in H^{*.*-c^{\prime}}(X)$ such that $r_1(x^{\prime\prime})=x^{\prime}$ . Thus we have the lemma.

Let $cl\,:\, CH^*(X)\otimes A\to H^{2*}(X;\,A)$ be the cycle map, and $Im(cl)^+$ be the positive degree parts of its image.

Proof. Recall that $H^{*.*^{\prime}}(X;\,A)\to N^{*-*^{\prime}}H^*(X;\,A)$ . We have $H^{2*,*}(X;\,A)\cong CH^*(X)\otimes A.$ Since $2*>*$ for $*\ge 1$ , we see $cl(y)\in N^{1}H^{2*}(X;\,A)$ .

Each element $y\in CH^*(X)\otimes A$ is represented by closed algebraic set supported Y, while Y may be singular. On the other hand, by Totaro [ Reference Totaro17 ], we have the modified cycle map $\bar cl$

\begin{equation*} cl\,:\,CH^*(X)\otimes A\stackrel{\bar cl}{\longrightarrow} MU^{2*}(X)\otimes_{MU^*}A\stackrel{\rho}{\longrightarrow} H^{2*}(X;\,A)\end{equation*}

for the complex cobordism theory $MU^*(X)$ . It is known [ Reference Quillen11 ] that elements in $MU^{2*}(X)$ can be represented by proper maps to X from stable almost complex manifolds Y. (The manifold Y is not necessarily a complex manifold.)

The following lemma is well known.

Proof. Recall the connective Morava K-theory $k(i)^*(X)$ with $k(i)^*={\mathbb Z}/p[v_i]$ , $|v_i|=-2p^i+2$ , which has natural maps

\begin{equation*} \rho\,:\,MU^*(X)/p \stackrel{\rho_1}{\longrightarrow} k(i)^*(X)\stackrel{\rho_2}{\longrightarrow} H^*(X\,:\,{\mathbb Z}/p).\end{equation*}

It is known that $d_{2p^i-1}=Q_i$ for the first nonzero differential $d_{2p^i-1}$ of the Atiyah-Hirzebruch spectral sequence converging to $k(i)^*(X)$ ,

\begin{equation*} E_2^{*,*^{\prime}}\cong H^*(X;\,{\mathbb Z}/p)\otimes k(i)^*\Longrightarrow k(i)^*(X).\end{equation*}

Hence $Q_i\rho_2(x)=0$ which implies $Q_i\rho(x)=0$ .

Proof. Suppose we have $f\,:\,Y\to X$ with $f_*(a^{\prime})=a$ . Then

\begin{equation*} f_*\left(a^{\prime}f^*(g)\right)=f_*\left(a^{\prime}\right)g=ag\end{equation*}

by Frobenius reciprocity law.

Let G be an algebraic group (over ${\mathbb C}$ ) and r be a complex representation $r\,:\,G\to U_n$ for the unitary group $U_n$ . Then we can define the Chern class $c_i=r^*c_i^U$ . Here the Chern classes $c_i^U$ in $H^*(BU_n)\cong {\mathbb Z}\!\left[c_1^U,...,c_n^U\right]$ are defined by the Gysin map $c_n^U=i_{n*}(1)$ for the inclusion $i_n\,:\,\{0\}\subset {\mathbb C}^{\times n}$ , that is,

\begin{equation*} i_{n*} \,:\, H^*(BU_n)\cong H_{U_n}^*(\{0\})\stackrel{i_{n*}}{\longrightarrow} H_{U_n}^{*+2n}({\mathbb C}^{\times n})\cong H^{*+2i}(BU_n),\end{equation*}

where $H_{U_n}({-})=H^*(EU_n\times_{U_n}-)$ is the $U_n$ -equivariant cohomology. Hence for the approximation $XU_n$ for $U_n$ , we see $c_i^{U}\in \tilde N^1H^*(XU_n)$ . So $c_i=r^*c_i^U\in\tilde N ^{1}H^*(X)$ for the approximation X for BG.

By the reciprocity law (Lemma 3·5) we have

The following lemma is proved by Colliot Thérène and Voisin [ Reference Colliot Thérène and Voisin3 ] by using the affirmative answer of the Bloch–Kato conjecture by Voevodsky ([ Reference Voevodsky20 , Reference Voevodsky21 ]).

4. The main lemmas

The following lemma is the $Q_i$ -version of one of results by Benoist and Ottem.

Lemma 4·1. Let $\alpha\in N^1H^s(X)$ for $s=3$ or 4. If $Q_i(\alpha)\not =0\in H^*(X;\,{\mathbb Z}/p)$ for some $i\ge 1$ , then

\begin{equation*} DH^s(X)\supset {\mathbb Z}/p\{\alpha\},\quad DH^s\left(X;\,{\mathbb Z}/p^t\right)\supset {\mathbb Z}/p\{\alpha\}\ \ for\ t\ge 2.\end{equation*}

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 $f_*\left(\alpha^{\prime}\right)=\alpha$ for $\alpha^{\prime}\in H^*(Y)$ . Then for $s=4$

\begin{align*} Q_i\left(\alpha^{\prime}\right) & =\left(P^{\Delta_i}\beta-\beta P^{\Delta_i}\right)\left(\alpha^{\prime}\right)=\left({-}\beta P^{\Delta_i}\right)\left(\alpha^{\prime}\right) = -\beta \!\left(\alpha^{\prime}\right)^{p^i}\\[4pt]& =-p^i\left(\beta \alpha^{\prime}\right)\left(\alpha^{\prime}\right)^{p^i-1}=0 \quad (by\ Cartan\ formula)\end{align*}

since $\beta\!\left(\alpha^{\prime}\right)=0$ and $P^{\Delta_i}(y)=y^{p^i}$ for $deg(y)=2$ . (For $s=3$ , we get also $Q_i\left(\alpha^{\prime}\right)=0$ since $ P^{\Delta_i}(x)=0$ for $deg(x)=1$ .) This contradicts the commutativity of $Q_i$ and $f_*$ .

The case $A={\mathbb Z}/p^t$ , $t\ge 2$ is proved similarly, since for $\alpha^{\prime}\in H^*(X;\,A)$ we see $\beta \alpha^{\prime}=0\in H^*(X;\,{\mathbb Z}/p)$ . Thus we have this lemma.

We will extend Lemma 4·1 to $s>4$ , by using MU-theory of Eilenberg–MacLane spaces. Recall that $K=K({\mathbb Z},n)$ is the Eilenberg–MacLane space such that the homotopy group $[X,K]\cong H^n(X\,:\,{\mathbb Z})$ , i.e., each element $x\in H^n(X;\,{\mathbb Z})$ is represented by a homotopy map $x\,:\, X\to K$ . Let $\eta_n\in H^n(K;\,{\mathbb Z})$ corresponding the identity map. (For $K^{\prime}=K({\mathbb Z}/p,n)$ define $\eta^{\prime}_n\in H^n(K^{\prime};\,{\mathbb Z}/p)$ by the identity element of K ^′.) We know the image $\rho(MU^*(K))\subset H^*(K;\,{\mathbb Z})/p$ .

Lemma 4·2. ([ Reference Ravenel, Wilson and Yagita13 , Reference Tamanoi15 ]) We have the isomorphism

\begin{equation*} \rho \,:\, MU^*(K)\otimes _{MU^*}{\mathbb Z}/{p}\cong{\mathbb Z}/{p}\left[Q_{i_1}...Q_{i_{n-2}}\eta_n|0<i_1<...<i_{n-2}\right],\end{equation*}

\begin{equation*} \rho \,:\, MU^*(K^{\prime})\otimes _{MU^*}{\mathbb Z}/{p}\cong{\mathbb Z}/{p}\left[Q_{i_1}...Q_{i_{n-1}}Q_0\eta^{\prime}_{n}|0<i_1<...<i_{n-1}\right] , \end{equation*}

where the notation ${\mathbb Z}/p[a,...]$ exactly means ${\mathbb Z}/p[a,...]/\left(a^2|\ |a|=odd\right)$ .

The following lemma is an extension of Lemma 4·1 to $s> 4$ .

Lemma 4·3. Let $\alpha\in N^cH^{n+2c}(X)$ , $n\ge 2$ , $c\ge 1$ . Suppose that there is a sequence $0<i_1< \cdots <i_{n-1}$ with

\begin{equation*} Q_{i_1}...Q_{i_{n-1}}\alpha \not =0 \quad in\ H^*(X;\,{\mathbb Z}/p).\end{equation*}

Then $D^cH^{*}(X)=N^cH^{*}(X)/(p,\tilde N^cH^{*}(X)) \supset {\mathbb Z}/p\{\alpha\}$ .

Proof. Suppose $\alpha\in \tilde N^cH^{n+2c}(X)$ , i.e. there is a smooth Y of $dim(Y)=dim(X)-c$ with $f\,:\,Y\to X$ such that the transfer $f_*\left(\alpha^{\prime}\right)=\alpha$ for $\alpha^{\prime}\in H^n(Y)$ .

Identify the map $\alpha^{\prime}\,:\, Y\to K$ with $\alpha^{\prime}=\left(\alpha^{\prime}\right)^*\eta_n.$ We still see from Lemma 4·2,

\begin{equation*}Q\!\left(\alpha^{\prime}\right)= Q_{i_1}...Q_{i_{n-2}}(\left(\alpha^{\prime}\right)^*\eta_n)\in Im(MU^*(Y)\longrightarrow H^*(Y;\,{\mathbb Z}/p)).\end{equation*}

From Lemma 3·4, we see

\begin{equation*}Q_{i_{n-1}}Q\!\left(\alpha^{\prime}\right)=Q_{i_{n-1}}Q_{i_1}...Q_{i_{n-2}}\left(\alpha^{\prime}\right)=0 \in H^*(Y;\,{\mathbb Z}/p).\end{equation*}

Therefore $Q_{i_{n-1}}Q(\alpha)$ must be zero by the commutativity of $f_*$ and $Q_i$ .

5. Classifying spaces for finite groups

Let G be a finite group or an algebraic group, and BG its (geometric) classifying space. For example, when $G=G_m$ is the multiplicative group, we see

\begin{equation*} BG_m=BS^1\cong {\mathbb P}^{\infty}, \quad H^*({\mathbb P}^{\infty})\cong {\mathbb Z}[y]\ \ with\ degree\ |y=c_1|=2,\end{equation*}

for the infinite (complex) projective space ${\mathbb P}^{\infty}$ . 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) ([ Reference Ekedahl4 ]) for a given $N\ge 3$ such that there is a map $ j\,:\,X(N)\to BG\times {\mathbb P}^{\infty} $ with

\begin{equation*} H^*(BG\times {\mathbb P}^{\infty};\,A)\stackrel{j^*}{\cong} H^*(X(N);\,A)\quad {for \ all} < N.\end{equation*}

In this paper, we call the above X(N) a (degree $\ N$ ) complex projective approximation for BG (which is an approximation of $BG\times {\mathbb P}^{\infty}$ strictly speaking).

Note that the quotient

\begin{equation*} N^nH^*(X;\,A)/(\tilde N^nH^*(X;\,A)) \end{equation*}

is an invariant under replacing X with $X\times {\mathbb P}^m$ for all n and all abelian groups A. In fact, from K $\ddot{{u}}$ nneth formula,

\begin{equation*} H^*\left(X\times {\mathbb P}^m;\, A\right)\cong H^*(X;\,A)\otimes {\mathbb Z}[y]/\left(y^{m+1}\right),\end{equation*}

where $y\in CH^1({\mathbb P}^m)$ is the first Chern class. Let Ideal(y) be the ideal of $H^*(X\times {\mathbb P}^m;\,A)$ generated by y. Then $Ideal(y)\subset \tilde N^*H^*(X\times {\mathbb P}^{m};\,A)$ by the Frobenius reciprocity law (Lemma 3·5). Moreover Benoist and Ottem show that the above quotient when $n=1$ is a stable birational invariant of X ([ Reference Benoist and Ottem1 , proposition 2·4]).

In this paper, we will study the following (mod(p)) stable rational invariant

\begin{equation*}DH^*(X;\,A)=N^1H^*(X;\,A)/\left(p,\tilde N^1H^*(X;\,A)\right).\end{equation*}

Hereafter, we consider $DH^*(X;\,A)$ when $A={\mathbb Z}$ . Let p be an odd prime. (The case $p=2$ is different but a similar argument works.) Let $G=({\mathbb Z}/p)^3$ the $rank=3$ elementary abelian p-group. Then the mod(p) cohomology is

\begin{equation*}H^*(BG;\,{\mathbb Z}/p)\cong H^*(B{\mathbb Z}/p;\,{\mathbb Z}/p)^{3\otimes }\cong {\mathbb Z}/p[y_1,y_2,y_3]\otimes \Lambda(x_1,x_2,x_3).\end{equation*}

Here degree $|y_i|=2, |x_i|=1, \beta(x_i)=y_i$ , and $\Lambda(a,...,b)$ is the ${\mathbb Z}/p$ -exterior algebra generated by $a,...,b$ .

The integral cohomology (modulo p) is isomorphic to

\begin{equation*} H^*(BG)/p\cong Ker(Q_0)\cong H(H^*(BG;\,{\mathbb Z}/p);\,Q_0)\oplus Im(Q_0),\end{equation*}

where $H({-};\,Q_0)=Ker(Q_0)/Im(Q_0)$ is the homology with the differential $Q_0$ . It is immediate that $H(H^*(B{\mathbb Z}/p;\,{\mathbb Z}/p);\, Q_0)\cong {\mathbb Z}/p$ . By the K $\ddot{u}$ nneth formula, we have $H(H^*((BG;\,{\mathbb Z}/p);\,Q_0)\cong ( {\mathbb Z}/p)^{3\otimes}\cong {\mathbb Z}/p$ . Hence we have

\begin{equation*} H^*(BG)/p\cong {\mathbb Z}/p\{1\}\oplus Im(Q_0)\end{equation*}

\begin{equation*} \cong {\mathbb Z}/p[y_1,y_2,y_3]\left(1,Q_0(x_ix_j),Q_0\left(x_1x_2x_3\right)|i<j\right),\end{equation*}

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.

Also note that $y_1,y_2,y_3$ are represented by the Chern classes $c_1$ . From Lemma 3·6, we see

\begin{equation*} Ideal(y_1,y_2,y_3)=0\in DH^*(X).\end{equation*}

We know $Q_i(y_j)=y_j^{p^i}$ and $Q_j$ is a derivation. Let us write

\begin{equation*} \alpha=Q_0(x_1x_2x_3)=y_1x_2x_3-y_2x_1x_3+y_3x_1x_2.\end{equation*}

Note $\alpha\in H^4(X)$ , $p\alpha=0$ , and $\alpha\in N^1H^*(X)$ from Lemma 3·7. Moreover

\begin{equation*}Q_1(\alpha)=Q_1(y_1x_2x_3)-...=y_1y_2^px_3-y_1y_3^px_2-...\not =0 \in H^*(X;\,{\mathbb Z}/p).\end{equation*}

Similarly, for $\alpha_{ij}=Q_0(x_ix_{j})$ , we see $Q_1(\alpha_{ij})\not =0$ . Hence from Lemma 4·1 and Lemma 3·6, we have

Theorem 5·1. Let $X=X(N)$ with $N> 2p+3$ be a (degree N) approximation for $B({\mathbb Z}/p)^3$ . Then we have

\begin{equation*}DH^*(X)\cong{\mathbb Z}/p\{\alpha_{ij}, \alpha|1\le i<j\le 3\}\quad for\ all \ *<N.\end{equation*}

Proof. We see $H^*(BG)/(p,y_1,y_2,y_3)\cong {\mathbb Z}/p\{1,\alpha_{ij},\alpha\}$ . Of course $1\not \in N^1H^*(X)$ , we have the theorem from Lemma 4·1.

Theorem 5·2. Let $X=X(N)_n$ be an approximation for $(B{\mathbb Z}/p)^n$ with $N>|Q_0Q_1...Q_{n-1}(x_1...x_n)|$ . Then we have for $\alpha_{i_1,...,i_s}=Q_0(x_{i_1}...x_{i_s})$ ,

\begin{equation*}DH^*(X)\supset{\mathbb Z}/p\!\left\{\alpha_{i_1,...,i_s }|2\le s,\ 0<i_1<i_2...<i_s\le n\right\}\quad for\ *<N.\end{equation*}

Here the notation $DH^*(X)\supset B^*$ means $DH^t(X)\supset B^t$ for the degree t-homogeneous parts of B for all $t<N$ strictly speaking.

Proof. We have the theorem from Lemma 4·3 and $ Q_{i_1}...Q_{i_{s-2}}(\alpha_{i_1,...,i_s})$ is

\begin{equation*} Q_{i_1}...Q_{i_{s-2}}Q_0(x_{i_1}...x_{i_s})=y_{i_1}^{p^{i_1}}...y_{i_{s-2}}^ {p^{i_{s-2}}}y_{i_{s-1}}x_{i_s}+...\not =0.\end{equation*}

(Note the $n=|\alpha^{\prime}|$ in Lemma 4·3 is written by $s-1$ here.)

Next we study small non-abelian p-groups. Let G be a non-abelian group of order $p^3$ (see Section 8, for details). Then $H^{even}(BG)$ is generated by Chern classes, and $H^{odd}(BG)$ is a (just) p-torsion. We can identify $H^{odd}(BG)\subset H^{odd}(BG;\,{\mathbb Z}/p)$ .The operation $Q_1$ acts on $H^{odd}(X)$ , and induces the injection

\begin{equation*} Q_1\,:\, H^{odd}(BG)\hookrightarrow H^{even}(BG).\end{equation*}

Such groups are four types (see Section 8 below), and they are called extraspecial p-groups $G=p_{\pm}^{1+2}$ of 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 $H^{even}(BG)$ module 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 $deg(e^{\prime})=2p+1$ .

From Lemma 3·5 (Frobenius reciprocity) and the main lemma (Lemma 4·1), we have the following theorem.

Theorem 5·4. 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 all $*<N$ :

\begin{equation*} DH^*(X)\cong\begin{cases} 0\quad for \ G=Q_8\\[3pt] {\mathbb Z}/2\{e\}\quad for \ G=D_8\\[3pt] 0\ or \ {\mathbb Z}/p\{e^{\prime}\} \quad for\ G=M \\[3pt] {\mathbb Z}/p\{e_1,e_2\} \quad for \ G=E.\end{cases} \end{equation*}

In particular, the above theorem implies that when $G=p_+^{1+2}$ , all $X=X(N)$ satisfy $DH^3(X)\not =0$ but $ DH^*(X)=0$ for all $4\le *<N.$

In this paper, we can not decide $DH^*(X)$ when $G=M$ .

6. Connected groups

At first, we consider when $G=U_n$ , $SU_n$ or $Sp_{2n}$ for all p, where the cohomology $H^*(BG)$ has no torsion. Then $H^*(BG)$ is generated by Chern classes, e.g.,

\begin{equation*}H^*(BU_n)\cong CH^*(BU_n)\cong {\mathbb Z}_{(p)}[c_1,...,c_n],\end{equation*}

\begin{equation*} H^*(BSp_{2n})\cong CH^*(BSp_{2n})\cong{\mathbb Z}_{(p)}[c_2,c_4,...,c_{2n}].\end{equation*}

Hence $DH^*(X)=0$ for the approximations X for these groups.

Next we consider the case $G=SO_3$ and $p=2$ . Then

\begin{equation*} H^*(BG;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[w_1,w_2,w_3]/(w_1)\cong {\mathbb Z}/2[w_2,w_3],\end{equation*}

where $w_i$ is the ith Stiefel–Whitney class for $SO_3\subset O_3$ and $w_i^2=c_i$ is the ith Chern class for $SO_3\subset U_3$ . (Also it is the elementary symmetric polynomial in ${\mathbb Z}/2[y_1,...,y_i]$ .)

Here we know $Q_0(w_2)=w_3,$ and $Q_1(w_3)=w_3^2=c_3$ . Therefore we have [ Reference Yagita22 ]

\begin{equation*} H^*(BG;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[c_2,c_3]\{1,w_2,w_3=Q_0(w_2),w_2w_3=Q_1w_2\}\end{equation*}

\begin{equation*} \cong {\mathbb Z}/2[c_2,c_3]\{w_2,Q_0(w_2),Q_1(w_2),Q_0Q_1(w_2)=c_3\}\oplus {\mathbb Z}/2[c_2] \end{equation*}

\begin{equation*}\cong {\mathbb Z}/2[c_2,c_3]\otimes \Lambda(Q_0,Q_1)\{w_2\}\oplus {\mathbb Z}/2[c_2].\end{equation*}

In particular $H^*(BG)/2\cong Ker(Q_0)\cong {\mathbb Z}/2[c_2,c_3]\{1,w_3\}. $ Then from Lemma 4·1, we have

Using Lemma 4·3, we have

Theorem 6·2. Let $X_n=X_n(N)$ be approximations for $BSO_n$ for $n\ge 3$ . Moreover, let $|Q_1...Q_{2m-1}(w_{2m+1})|<N$ . Then we have

\begin{equation*} DH^*(X_{2m+1})\supset {\mathbb Z}/2\{w_3,w_5,...,w_{2m+1}\}\quad for \ all \ 2m+1\le *<N.\end{equation*}

Proof. Since $Q_0w_{2i}=w_{2i+1}$ , we see $w_{2i+1}\in N^1H^{2i+1}(X)$ from Lemma 3·7. We have the theorem, from Lemma 4·3 and the restriction to $H^*(B({\mathbb Z}/2)^{2i};\,{\mathbb Z}/2),$

\begin{equation*} Q_1...Q_{2i-2}(w_{2i+1})=Q_1...Q_{2i-2}Q_0(w_{2i})=y_1y_2^2...y_{2i-1}^{2^{2i-2}}x_{2i}+\cdots\not =0.\end{equation*}

Remark. The same inclusion

\begin{equation*} DH^*(X)\supset {\mathbb Z}/2\{w_3,w_5, ...,w_{2m+1}\}.\end{equation*}

holds for $G=SO_{2m+2}$ . Since $O_{2m+1}\cong SO_{2m+1}\times {\mathbb Z}/2$ , the orthogonal group $O_{2m+1}$ (hence $O_{2m+2}$ ) also has the same property.

We next consider simply connected groups. Let us write by X an approximation for $BG_2$ 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 $G_2\to SO_7$

\begin{equation*} H^*(BG_2;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[w_4,w_6,w_7],\quad P^1(w_4)=w_6,\ Q_0(w_6)=w_7, \end{equation*}

\begin{equation*}H^*(BG_2)\cong \left(D^{\prime}\oplus D^{\prime}/2[w_7]^+\right)\quad where \ \ D^{\prime}={\mathbb Z}[w_4,c_6].\end{equation*}

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

\begin{equation*} CH^*(BG_2)\cong \left(D\{1,2w_4\} \oplus D/2[c_7]^+\right)\quad where \ \ D={\mathbb Z}[c_4,c_6]\ \ c_i=w_i^2.\end{equation*}

In particular the cycle map $cl\,:\, CH^*(BG)\to H^*(BG)$ is injective.

It is known $w_4\in N^1H^*(X;\,{\mathbb Z}/2)$ ([ Reference Yagita22 ]) 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 $DH^4(X)\not =0$ . This fact is known in [ Reference Benoist and Ottem1 ]. Moreover $H^*(BG)/(c_4,c_6,c_7)\cong\Lambda(w_4,w_7)$ implies:

Proposition 6·3. For X an approximation for $BG_2$ , we have the surjection

\begin{equation*} \Lambda(w_4,w_7)^+\twoheadrightarrow DH^*(X)\quad for\ all \ *<N.\end{equation*}

By Voevodsky [ Reference Voevodsky18 , Reference Voevodsky19 ], we have the $Q_i$ operation also in the motivic cohomology $H^{*,*^{\prime}}(X;\,{\mathbb Z}/p)$ with $deg(Q_i)=(2p^i-1,p-1)$ . Then we can take

\begin{equation*} deg(w_4)=(4,3),\ deg(w_6)=(6,4),\ deg(w_7)=(7,4),\ deg(c_7)=(14,7).\end{equation*}

By Theorem 3·1, the above means

\begin{equation*} w_7=Q_1w_4\in N^{7-4}H^*(X;\,{\mathbb Z}/2)=N^{3}H^*(X;\,{\mathbb Z}/2).\end{equation*}

We cannot see here that $0\not =w_7\in DH^*(X)$ , but see the following proposition.

Proposition 6·4. Let $N>|Q_2w_7|=14$ . For an approximation $X=X(N)$ for $BG_2$ , we have

\begin{equation*}{\mathbb Z}/2\{w_7\}\subset D^3H^*(X)=N^3H^*(X)/(2,\tilde N^3H^*(X)).\end{equation*}

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 $H^*(Y;\,{\mathbb Z}/2)$ , and

\begin{equation*} Q_2(x)=\left(P^{\Delta_2}\beta +\beta P^{\Delta_2}\right)(x)=0\end{equation*}

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 $Q_2$ .

Proof. We 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 embedding $j\,:\,G_2\subset G$ such that (see [ Reference Pirutka and Yagita10 , Reference Yagita25 ] for details)

\begin{equation*} H^4(BG)\stackrel{j^*}{\cong} H^4(BG_2)\cong {\mathbb Z}\{w_4\}.\end{equation*}

Let $x=(j^{*})^{-1}w_4\in H^4(BG)$ . From [ Reference Yagita25 , lemma 3·1], 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 $U^{\prime}\subset U$ such that $x=0\in H^4(U^{\prime})$ . 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 px is a Chern class [ Reference Yagita25 ], and $Q_1(x)\not =0\in H^*(BG;\,{\mathbb Z}/p).$ In fact, there is embedding $j\,:\,({\mathbb Z}/p)^3\subset G$ with $j^*(x)=Q_0(x_1x_2x_3)$ . Hence we have the theorem.

Remark. Kordonskii [ Reference Kordonskii6 ], Merkurjev ([ Reference Merkurjev7 , corollary 5·8]), and Reichstein–Scavia show [ Reference Reichstein and Scavia14 ] that the classifying space $BSpin_n$ itself is stably rational when $n\le 14$ . Hence the (Ekedahl) approximation X is not stable rationally equivalent to BG. In fact, these X is constructed from a quasi projective variety BG as taking intersections of subspaces of ${\mathbb P}^{M}$ for a large M. (The author thanks Federico Scavia who pointed out this remark.)

At last of this section, we consider the case $G=PGL_p$ . We have (for example [ Reference Kameko and Yagita5 , theorems 1·5, 1·7]) additively

\begin{equation*}H^*(BG;\,{\mathbb Z}/p)\cong M\oplus N \quad with \ \ M\stackrel{add.}{\cong}{\mathbb Z}/p\!\left[x_4,x_6,...,x_{2p}\right],\quad \end{equation*}

\begin{equation*} N= SD\otimes \Lambda(Q_0,Q_1)\{u_2\}\quad with\ \ SD={\mathbb Z}/p\!\left[x_{2p+2},x_{2p^2-2p}\right],\end{equation*}

where $x_{2p+2}=Q_1Q_0u_2$ and suffix means its degree. The Chow ring is given as

\begin{equation*} CH^*(BG)/p\cong M\oplus SD\{Q_0Q_1(u_2)\}.\end{equation*}

From Lemma 4·1, we have:

Theorem 6·7. Let p be odd. For an approximation X for $BPGL_p$ , we see $ {\mathbb Z}/p\{Q_0u_2\}\subset DH^*(X)$ , and moreover there is a surjection

\begin{equation*} {\mathbb Z}/p\!\left[x_{2p^2-2p}\right]\{Q_0u_2\}\twoheadrightarrow DH^*(X)/(Im(cl))\quad for\ all \ * <N\end{equation*}

for the cycle map $cl\,:\,CH^*(X)\to H^{2*}(X)$ .

In the above case, we do not see here that $DH^*(X)$ for $*<N$ is invariant of BG. (See the remark in the introduction.)

7. ${\mathbb Z}/p$ -coefficient cohomology for abelian groups

In the preceding sections, we have seen that cases $DH^*(X;\,A)\not =0$ are not so rare for $A={\mathbb Z}_{(p)}$ , ${\mathbb Z}/p^i$ , $i\ge 2$ . However currently it seems difficult to make such example for $A={\mathbb Z}/p$ . (Recall the final remark in Section 4.)

At first, we consider the case $G=({\mathbb Z}/p)^3$ .

Proof. Recall the mod p cohomology

\begin{equation*}H^*(BG;\,{\mathbb Z}/p)\cong {\mathbb Z}/p[y_1,y_2,y_3]\otimes \Lambda(x_1,x_2,x_3).\end{equation*}

Here $y_i$ is a Chern class. Hence $x_jy_i=0 \in DH^*(X;\,{\mathbb Z}/p)$ by reciprocity law. Hence we only need to check it for $z\in \Lambda(x_1,x_2,x_3)$ . But these $z\not \in N^1H^*(X;\,{\mathbb Z}/p)$ (see Lemma 7·4 below). Hence $DH^*(X;\,{\mathbb Z}/p)=0$ .

Example of Gysin maps. We can take a quasi projective approximation $\bar X(N)$ of $B{\mathbb Z}/p$ explicitly by the quotient (the N-dimensional lens space)

\begin{equation*}\bar X(N)={\mathbb C}^{N*}/({\mathbb Z}/p)\quad where \ {\mathbb C}^{N*}=\left({\mathbb C}^N-\{0\}\right).\end{equation*}

Next we consider the projective approximation

\begin{equation*}X(N)\longrightarrow \bar X(N)\times {\mathbb P}^{N}\longrightarrow B{\mathbb Z}/p\times {\mathbb P}^{\infty}.\end{equation*}

Let us write $X_i$ (resp. $X^{\prime}_i$ ) for $i=1,2,3$ the above $\bar X(N)$ (resp. $\bar X(N-1))$ for a sufficient large number N. Let

\begin{equation*} i_1\,:\,Y_1=X^{\prime}_1\times X_2\times X_3\longrightarrow X=X_1\times X_2\times X_3.\end{equation*}

Similarly we define $Y_2,Y_3$ , and the disjoin union $Y=Y_1\sqcup Y_2\sqcup Y_3$ .

Recall that for $p\,:\,odd$

\begin{equation*}H^*(X;\,{\mathbb Z}/p)\cong {\mathbb Z}/p[y_1,y_2,y_3]/\left(y_1^{N+1},y_1^{N+1},y_3^{N+1}\right)\otimes\Lambda(x_1,x_2,x_3),\end{equation*}

and $H^*(Y_i;\,{\mathbb Z}/p)\cong H^*(X;\,{\mathbb Z}/p)/\left(y_i^N\right)$ for $i=1,2,3$ . For $p=2$ , some graded ring $grH^*(X;\,{\mathbb Z}/2)$ is isomorphic to the above ring (in fact $x^2_i=y_i$ ).

For the embedding $f_i\,:\,X^{\prime}_i\to X_i$ , it is known $f_{i*}(1)=c_1(N_i)$ where $N_i$ is the normal bundle for $X^{\prime}_i\subset X_i$ . Hence the Gysin map is given by

\begin{equation*} f_{1*}(1)=y_1,\quad f_{2*}(1)=y_2,\quad f_{3*}(1)=y_3.\end{equation*}

Therefore we have for $x=(x_2x_3+ x_3x_1+x_1x_2)\in H^*(Y_1\sqcup Y_2\sqcup Y_3;\,{\mathbb Z}/p)$ ,

\begin{equation*} f_*(x)=y_1x_2x_3+y_2x_3x_1+y_3x_1x_2=Q_0(x_1x_2x_3)=\alpha.\end{equation*}

(Note that the element $x=(x_1x_2+x_2x_3+x_3x_1)$ is not in the integral cohomology $H^*(Y)$ .) Thus we see $\alpha\in \tilde N^cH^*(X;\,{\mathbb Z}/p).$ More generally, we see

Theorem 7·3. Let $X=X(N)$ be an approximation for $(B{\mathbb Z}/p)^n$ with ${\mathbb Z}/p$ -coefficients. Then we have

\begin{equation*}DH^*(X;\,{\mathbb Z}/p)=0 \quad for\ all\ *<N.\end{equation*}

We recall here the motivic cohomology. By Voevodsky [ Reference Voevodsky18 ], $H^{*,*^{\prime}}(B{\mathbb Z}/p;\,{\mathbb Z}/p)$ satisfies the K $\ddot{{u}}$ nneth formula so that (for p odd)

\begin{equation*}H^{*,*^{\prime}}(B({\mathbb Z}/p)^n;\,{\mathbb Z}/p)\cong {\mathbb Z}/p[\tau, y_1,...,,y_n]/\left(y_1^{N+1},...,y_n^{N+1}\right)\otimes\Lambda(x_1,...,x_n).\end{equation*}

Here $0\not =\tau\in H^{0,1}(Spec({\mathbb C});\,{\mathbb Z}/p)$ , and $deg(y_i)=(2,1)$ , $deg(x_i)=(1,1)$ .

From Lemma 3·1, we can identify $N^cH_{et}^*(X;\,{\mathbb Z}/p)= F_{\tau}^{*,*-c}$ where

$F_{\tau}^{*,*-c}=Im(\times \tau^c\,:\,H^{*,*-c}(X;\,{\mathbb Z}/p)\to H^{*,*}(X;\,{\mathbb Z}/p)).$

Lemma 7·4. ([ Reference Tezuka and Yagita16 , theorem 5·1]) Let $X=X(N)$ be an approximation for $(B{\mathbb Z}/p)^n$ for a sufficient large N. Then we have

\begin{equation*} H^*(X;\,{\mathbb Z}/p)/N^1H^*(X;\,{\mathbb Z}/p)\cong \Lambda(x_1,....,x_n).\end{equation*}

Proof. Let $x\in Ideal(y_1,...,y_n)\subset H^{*,*^{\prime}}(X;\,{\mathbb Z}/p)$ . Then $deg(x)=(*,*^{\prime})$ with $*>*^{\prime}$ , and x is a multiplying of $\tau$ . Hence $x\in N^1H^*(X;\,{\mathbb Z}/p)$ .

Proof of Theorem 7·3. Let $x\in N^1H^*(X;\,{\mathbb Z}/p)$ . From the above lemma, $x\in Ideal(y_1,...,y_n)$ which is in the image of the Gysin map. That is $x\in \tilde N^1H^*(X;\,{\mathbb Z}/p)$ .

We can extend Theorem 7·3, by using the following lemma. Let us write by XG an approximation for BG. Let $j\,:\, BS\to BG$ and $i;\, Y\to XS$ . We consider maps:

Proof. Let $j\,:\, BS\to BG$ so that $j_*=cor_S^G$ is the transfer (with the codimension $c=0$ ) for finite groups. Note that $j^*N^1H^*(XG;\,{\mathbb Z}/p)\subset N^1H^*(XS;\,{\mathbb Z}/p)$ by the naturality of $j^*$ . Hence given $x\in N^1H^*(XG;\,{\mathbb Z}/p)$ , the element $y=j^*(x)$ is in $N^1H^*(XS;\,{\mathbb Z}/p)$ .

By the assumption in this lemma, there are $i\,:\,Y\to XS$ and y ^′ such that $y^{\prime}\in H^*(Y;\,{\mathbb Z}/p)$ with $i_*(y^{\prime})=y$ . We consider maps:

\begin{equation*} H^*(Y;\,{\mathbb Z}/p)\stackrel{i_*}{\to} H^*(XS;\,{\mathbb Z}/p)\stackrel{j_*}{\to}H^*(XG;\,{\mathbb Z}/p).\end{equation*}

Then we have $\ j_*i_*(y^{\prime})=j_*y=j_*j^*(x)=[G;\,S]x.$

Similarly, we can prove:

8. The groups $Q_8$ and $D_8$

When $|G|=p^3$ , we have the short exact sequence

\begin{equation*} 0\longrightarrow C\longrightarrow G\longrightarrow V \longrightarrow 0,\end{equation*}

where $C\cong {\mathbb Z}/p$ is in the center and $ V\cong {\mathbb Z}/p\times {\mathbb Z}/p$ . Let us take generators such that $C={\langle} c{\rangle}, V={\langle} a,b {\rangle}.$ Moreover we can take $[a,b]=c$ when G is non-abelian.

There are two cases, when $p=2$ , the quaternion group $Q_8$ and the dihedral group $D_8$ . We will show here

8·1. The case $G=Q_8$ . Then $a^2=b^2=c$ . Its cohomologies are well known (see [ Reference Quillen12 ]):

\begin{equation*}H^*(BG)/2\cong {\mathbb Z}/2[y_1,y_2,c_2]/\left(y_i^2,y_1y_2\right)\ \|y_i|=2,\end{equation*}

\begin{equation*}H^*(BG;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[x_1,x_2,c_2]/(x_1x_2+y_1+y_2,x_1y_2+x_2y_1)\end{equation*}

\begin{equation*}\cong {\mathbb Z}/2\{1,x_1,y_1,x_2,y_2,w\}\otimes {\mathbb Z}/2[c_2],\end{equation*}

where $ x_i^2=y_i$ $|x_i|=1$ , and $w=y_1x_2=y_2x_1$ , $|w|=3$ .

Therefore, we see

\begin{equation*}H^*(BG;\,{\mathbb Z}/2)/(y_1,y_2,c_2)\cong {\mathbb Z}/2\{1,x_1,x_2\}.\end{equation*}

Of course $deg(x_i)=(1,1)$ in $H^{*,*^{\prime}}(BG;\,{\mathbb Z}/2)$ and they are not in $N^1H^*(BG;\,{\mathbb Z}/2)$ . Thus we have Theorem 8·1 for $G=Q_8$ .

8·2. The case $G=D_8$ . Then $a^2=c, b^2=1$ . It is well known

\begin{equation*} H^*(BG)/2\cong {\mathbb Z}/2[y_1,y_2,c_2]/(y_1y_2)\{1,e\}\quad with \ |e|=3.\end{equation*}

The mod 2 cohomlogy is written [ Reference Quillen12 ]

\begin{equation*} H^*(BG;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[x_1,x_2,u]/(x_1x_2)\quad (with\ |u|=2)\end{equation*}

\begin{equation*} \cong \left(\!\oplus_{j=1}^2 {\mathbb Z}/2[y_j]\{y_j, x_j,y_ju,x_ju\}\oplus{\mathbb Z}/2\{1,u\}\right)\otimes {\mathbb Z}/2[c_2].\end{equation*}

Here $y_j=x_j^2, u^2=c_2$ 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

\begin{equation*} H^*(BG;\,{\mathbb Z}/2)/(y_1,y_2,c_2)\cong\left(\oplus _{j=1}^2{\mathbb Z}/2\{x_j,x_ju\}\right)\oplus {\mathbb Z}/2\{1,u\}.\end{equation*}

Moreover, $deg(x_j)=(1,1)$ , $deg(u)=(2,2)$ in the motivic cohomology

$H^{*,*^{\prime}}(BG;\,{\mathbb Z}/2)$ and they are not in $N^1H^*(BG;\,{\mathbb Z}/2)$ . Here we note $deg(x_ju)=(3,3)$ , but there is $u^{\prime}_j\in H^{3,2}(BG;\,{\mathbb Z}/2)$ with $x_ju=\tau u^{\prime}_j$ from [ Reference Yagita24 , lemma 6·2] (i.e., $x_ju\in N^1H^*(X;\,{\mathbb Z}/2)$ ).

Hence for the proof of Lemma 8·1 (for $G=D_8$ ), it is only needed to show

To prove the above lemma, for a G-variety H, we consider the equivariant cohomology (recall the arguments just before Lemma 3·6)

\begin{equation*} H_G^*(H;\,{\mathbb Z}/p)=H^*(E(N)\times _{G}H;\,{\mathbb Z}/p),\end{equation*}

where E(N) is an (approximation of) contractible free G-variety. Let us write

\begin{equation*} X_GH=approx. \ of\ E(N)\times_GH \ so\ that\ H_G^*(H;\,{\mathbb Z}/p)\cong H^*(X_GH;\,{\mathbb Z}/p).\end{equation*}

For a closed embedding $i\,:\,H\subset K$ of G-varieties, we can define the Gysin map

\begin{equation*} i_*\,:\, H_G^*(H;\,{\mathbb Z}/p)\longrightarrow H^*_G(K;\,{\mathbb Z}/p)\quad by \ i\,:\,X_GH\stackrel{id\times_Gi}{\longrightarrow} X_GK. \end{equation*}

Hereafter in this section, let $G=D_8$ . We recall arguments in [ Reference Yagita24 ]. 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 $W={\mathbb C}^{2*}={\mathbb C}^2-\{0\}$ is an G-variety. Note G acts freely on $W\times {\mathbb C}^*$ but it does not act freely on $W={\mathbb C}^{2*}.$

The fixed points set on W under b is

\begin{equation*} W^{{\langle} b{\rangle}}=\{(x,x)|x\in {\mathbb C}^*\}={\mathbb C}^*\{e^{\prime}\} ,\quad e^{\prime}=diag(1,1)\in GL_2({\mathbb C}).\end{equation*}

Similarly $W^{{\langle} bc{\rangle}}={\mathbb C}^*\{a^{-1}e^{\prime}\}.$ Take

\begin{equation*} H_0={\mathbb C}^{*}\{e^{\prime},ae^{\prime}\},\quad H_1={\mathbb C}^{*}\left\{g^{-1}e^{\prime},q^{-1}ae^{\prime}\right\},\end{equation*}

where $g\in GL_2({\mathbb C})$ 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 ${\mathbb C}^{2*}-H$ . In fact it does not contain fixed points of non-trivial stabiliser groups. We consider the transfer for some G-variety H in ${\mathbb C}^{2*}$ , and induced equivariant cohomology

\begin{equation*} i_*\,:\, H^*_G(H;\,{\mathbb Z}/2)\longrightarrow H^*_G\!\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right).\end{equation*}

Lemma 8·3. We have

\begin{equation*}H^*_G(H_0;\,{\mathbb Z}/2)\cong {\mathbb Z}/2[y] \otimes \Lambda(x,z)\quad with \ y=x^2,\ |x|= |z|=1.\end{equation*}

Proof. We consider the group extension $0\to {\langle} a{\rangle} \to G\to {\langle} b {\rangle} \to 0$ and the induced spectral sequence

\begin{equation*}E_2^{*,*^{\prime}}=H^*(B{\langle} b{\rangle};\, H^{*^{\prime}}_{{\langle} a{\rangle} }(H_0;\,{\mathbb Z}/2))\Longrightarrow H_G^*(H_0;\,{\mathbb Z}/2).\end{equation*}

Since ${\langle} a{\rangle}\cong {\mathbb Z}/4$ acts freely on $H_0$ , we see $H_0/{\langle} a{\rangle} \cong {\mathbb C}^{*}\{e^{\prime},ae^{\prime}\}/{\langle} a{\rangle}\cong {\mathbb C}^{*}.$ Therefore we have

\begin{equation*}H^*_{{\langle} a{\rangle}}(H_0;\,{\mathbb Z}/2)\cong H^*({\mathbb C}^*/{\langle} a{\rangle};\,{\mathbb Z}/2)\cong H^*({\mathbb C}^*;\,{\mathbb Z}/2)\cong \Lambda (z),\quad |z|=1.\end{equation*}

Since ${\langle} b{\rangle}$ acts trivially on $\Lambda(z)$ we have this lemma

\begin{equation*} H^*_G(H_0;\,{\mathbb Z}/2)\cong H^*(B{\langle} b{\rangle};\,{\mathbb Z}/2)\otimes \Lambda(z)\cong {\mathbb Z}/2[y]\otimes \Lambda(x,z).\end{equation*}

Note $H_G^*(H_0;\,{\mathbb Z}/2)\cong H_G^*(H_1;\,{\mathbb Z}/2)$ and hence we see

\begin{equation*}H^*_G(H;\,{\mathbb Z}/2)\cong \oplus_{j=1}^2 {\mathbb Z}/2[y_j]\left\{1_j,y_j, x_j,x_jz_j,z_j\right\}.\end{equation*}

We consider the long exact sequence

\begin{equation*} \quad\quad\cdots\longrightarrow H^{*}_G(\{0\};\,{\mathbb Z}/2)\stackrel{i_*=c_2}{\longrightarrow}H^{*+4}_G\left({\mathbb C}^2;\,{\mathbb Z}/2\right) \longrightarrow H^{*+4}_G\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right)\longrightarrow \cdots \qquad({\ast})\end{equation*}

and we have $H^*_G\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right)\cong H^*(BG;\,{\mathbb Z}/2)/(c_2)$ . Hence, we get

\begin{equation*}H^*_G\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right)\cong \left( \oplus_{j=1}^2 {\mathbb Z}/2[y_j]\left\{y_j,x_j,x_ju^{\prime}_j,u^{\prime}_j\right\}\right)\oplus{\mathbb Z}/2\{1,u\}.\end{equation*}

Now we consider the transfer $H^*_G(H;\,{\mathbb Z}/2) \stackrel{i_*}{\to} H^{*+2}_G\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right)$ . We have explicitly ([ Reference Yagita24 , p. 527])

\begin{equation*} i_*(1_j)=y_j,\ i_*(x_j)=y_jx_j,\ i_*(x_jz_j)=x_ju^{\prime}_j,\ i_*(z_j)=u^{\prime}_j.\end{equation*}

Therefore we have Lemma 8·2 and hence Theorem 8·1 for $G=D_8$ .

To see the above $i_*$ , we recall the long exact sequence for $i\,:\,H \subset {\mathbb C}^{2*}$

\begin{equation*} \quad\qquad\cdots\longrightarrow H^{*+1}_G\left({\mathbb C}^{2*}-H;\,{\mathbb Z}/2\right)\stackrel{\delta}{\longrightarrow}H^*_G(H;\,{\mathbb Z}/2) \stackrel{i_*}{\longrightarrow} H^{*+2}_G\left({\mathbb C}^{2*};\,{\mathbb Z}/2\right)\quad\quad({\ast\ast})\end{equation*}

\begin{equation*}\stackrel{j^*}{\longrightarrow} H^{*+2}_G\left({\mathbb C}^{2*}-H;\,{\mathbb Z}/2\right)\longrightarrow\cdots\end{equation*}

The transfer $i_*$ is determined by the following lemma.

Proof. Since G acts freely on ${\mathbb C}^{2*}-H$ , we have

\begin{equation*} H^*_G\left({\mathbb C}^{2*}-H\,:\,{\mathbb Z}/2\right)\cong H^*\left(\left({\mathbb C}^{2*}-H\right)/G;\,{\mathbb Z}/2\right),\end{equation*}

which is zero when $*>4=2dim(({\mathbb C}^{2*}-H)/G)$ . Hence $\delta$ must be zero for $*>4$ , and $i_*$ is injective for $*>4$ . In particular, $i_*\left(y^2_jz_j\right)=y_j^2u^{\prime}_j$ . Since $H^*_G(H;\,{\mathbb Z}/2)$ is ${\mathbb Z}/2[y_1]$ -free (or $Z/2[y_2]$ -free,) we see $i_*(z_j)=u^{\prime}_j$ .