The motivic cohomology of BSO_n

TL;DR

This paper computes the motivic cohomology groups of the classifying space of special orthogonal groups over complex numbers with Z/2 coefficients, advancing understanding in algebraic topology and motivic cohomology.

Contribution

It provides the first explicit calculation of the motivic cohomology of BSO_n over C with Z/2 coefficients, filling a gap in the literature.

Findings

Explicit description of H^{*,*}(BSO_n, Z/2)

New computational techniques for motivic cohomology

Insights into the structure of classifying spaces

Abstract

We will determine the motivic cohomology $H^{*,*}(BSO_n , Z/2)$ with coefficients in $Z/ 2$ of the classifying space of special orthogonal groups $SO_n$ over the complex numbers $C$.

Equations

$$\underset{i,j\ge 0}{\oplus }{H}^{i,j}(X)=\underset{i,j\ge 0}{\oplus }{H}^i(X,\mathbb{Z}/2(j)),$$

$$t_X^{i,j}:H^{i,j}(X,\mathbb{Z}/2)\to H^i(X,\mathbb{Z}/2),$$

$$t_{BS{O}_{2m}}^{\ast ,\ast }(y_{0,m})=0.$$

$$\mathrm{Ker}(t_{BS{O}_{2m}}^{\ast ,\ast })=\mathbb{Z}/2[c_2,c_4,\dots ,c_{2m}]\{y_{0,m},\dots ,y_{m-2,m}\}.$$

$$Q_k:H^{i,j}(X)\to H^{i+2^{k+1}-1,j+2^k-1}(X).$$

$$H_G^{m,\ast }(X)=H^{m,\ast }((E\times X)/G).$$

$$H_G^{\ast ,\ast }((X\times G)/H)=H_H^{\ast ,\ast }(X).$$

$$\rightarrow H_{G}^{\ast-1,\ast}(X-Y)\xrightarrow{\delta}H_{G}^{\ast-2s,\ast-s}(Y)\xrightarrow{i_{*}}H_{G}^{\ast,\ast}(X)\xrightarrow{j^{*}}H_{G}^{\ast,\ast}(X-Y)\xrightarrow{\delta}\$$

$$\to H_{S{O}_n}^{\ast -2n,\ast -n}(\{0\})\stackrel{s_{\ast }}{\to }{H}_{S{O}_n}^{\ast ,\ast }({\mathbb{A}}^n)\stackrel{t^{\ast }}{\to }{H}_{S{O}_n}^{\ast ,\ast }({\mathbb{A}}^n\setminus \{0\})\to$$

$$\to H_{S{O}_n}^{\ast -2,\ast -1}(C)\stackrel{i_{\ast }}{\to }{H}_{S{O}_n}^{\ast ,\ast }({\mathbb{A}}^n\setminus \{0\})\stackrel{j^{\ast }}{\to }{H}_{S{O}_n}^{\ast ,\ast }(B)\stackrel{\delta }{\to }.$$

$$H_{S{O}_n}^{\ast ,\ast }({\mathbb{A}}^n\setminus \{0\})=H_{S{O}_n}^{\ast ,\ast }/c_n.$$

$$g\mapsto \left(\begin{array}{cc}\det (g)& 0\\ 0& g\end{array}\right).$$

$$H_{S{O}_n}^{\ast ,\ast }(B)$$

$$=$$

$$H_{O_{n-1}}^{\ast ,\ast }({\mathbb{G}}_m)=H_{O_{n-1}}^{\ast ,\ast }/c_1$$

$$H_{S{O}_n}^{\ast ,\ast }(C)=H_{S{O}_{n-2}⋉{\mathbb{A}}^{n-1}}^{\ast ,\ast }=H_{S{O}_{n-2}}^{\ast ,\ast }.$$

$$\to H_{S{O}_{n-2}}^{\ast -2,\ast -1}\stackrel{i_{\ast }}{\to }{H}_{S{O}_n}^{\ast ,\ast }/c_n\stackrel{{\kappa }^{\ast }}{\to }{H}_{O_{n-1}}^{\ast ,\ast }/c_1\stackrel{\delta }{\to }.$$

$$H_{O_{n-2}}^{\ast -2,\ast -1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{S{O}_{n-1}}^{\ast -2,\ast -1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^{\ast }(B{O}_n)\to H^{\ast }((B{\mu }_2{)}^n{)}^{{\Sigma }_n}=\mathbb{Z}/2[x_1,\dots ,x_n{]}^{{\Sigma }_n}$$

$$H^{\ast }(B{O}_n)=\underset{k\ge -1}{⨁}Q(k)G_k,$$

$$H^{\ast ,\ast }(B{O}_n)=({\otimes }^n{H}^{\ast ,\ast }(B{\mu }_2){)}^{{\Sigma }_n}=\mathbb{Z}/2[\tau ]\otimes (\oplus Q(k)G_k)$$

$${\kappa }^{-1}{ϵ}^{-1}{\gamma }_n=\det ({\gamma }_{n-1})\oplus {\gamma }_{n-1}$$

$$1+w_1({\kappa }^{-1}{ϵ}^{-1}{\gamma }_n)+w_2({\kappa }^{-1}{ϵ}^{-1}{\gamma }_n)+\cdots +w_n({\kappa }^{-1}{ϵ}^{-1}{\gamma }_n)$$

$$=$$

$${\kappa }^{\ast }{Q}_{j_0}\dots Q_{j_{k-1}}\overline{x}={\tau }^f{Q}_{j_0}\dots Q_{j_{k-1}}\overline{y}.$$

$$\to H_{S{O}_n}^{\ast -2}\stackrel{t(s_{\ast })}{\to }{H}_{S{O}_n}^{\ast }\stackrel{t(t^{\ast })}{\to }{H}_{S{O}_n}^{\ast }({\mathbb{A}}^n\setminus \{0\})\to .$$

$$\to H_{S{O}_{n-2}}^{\ast -2}\stackrel{t(i_{\ast })}{\to }{H}_{S{O}_n}^{\ast }/c_n\stackrel{t({\kappa }^{\ast })}{\to }{H}_{O_{n-1}}/c_1\stackrel{t(\delta )}{\to }{H}_{S{O}_{n-2}}^{\ast -1}\to$$

$$H_{O_{2m-1}}^{\ast ,\ast }/c_1=H_{\mathbb{Z}/2\times S{O}_{2m-1}}^{\ast ,\ast }({\mathbb{G}}_m)=H_{S{O}_{2m-1}}^{\ast ,\ast }\{1,x\}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

0. Introduction

We denote by $BG_{k}$ the classifying space of a linear algebraic group $G_{k}$ over a field $k$. It is a colimit of smooth schemes defined by B. Totaro [16], and F. Morel and V. Voevodsky in [9, §4], where it is called the geometric model. $SO_{n}$ is the special orthogonal group of dimension $n$, and we will treat cases where $k$ is the field of complex numbers ${\mathbb{C}}$. By $H^{\ast,\ast}(X)$ we mean the bi-graded motivic cohomology group with coefficients in $\mathbb{Z}/2$,

[TABLE]

of a smooth scheme $X$. We use a definition used in [19] and [20]. We call the first index $i$ by degree and $2j-i$ by weight. (Total weight may be appropriate. But we simply call this way.) Since we will only consider cohomology with coefficients in $\mathbb{Z}/2$, the coefficients are usually omitted in the notation. The motivic cohomology of $BSO_{n}$ is defined as a limit of motivic cohomology of smooth schemes.

The motivic cohomology groups are related to the Chow groups and ordinary cohomology groups in the following way. $H^{2i,i}(X)$ is isomorphic to the mod $2$ reduction $CH^{i}(X)\otimes\mathbb{Z}/2$ of the $i$-th Chow group for a smooth scheme $X$. In our convention $\oplus H^{2i,i}(X)$ is the subgroup of elements having weight [math]. There are no elements having negative weights. There exists a realization mapping

[TABLE]

where $t^{2i,i}_{X}$ is identified to the cycle class mapping. It was called the Beilinson-Lichtenbaum conjecture [19, §6] that it becomes an isomorphism when $i\leq j$, which is proved in [19].

$H^{\ast}(BG,\,\mathbb{Z}/2)=H^{\ast}(BG)$ is the singular cohomology group of CW-complex $BG(\mathbb{C})$, which is homotopy equivalent to the classifying space of a maximal compact subgroup of $G(\mathbb{C})$.

When $G=GL_{n},SL_{n},Sp_{n}$, the motivic cohomology of $BG$ is known over arbitrary fields integrally. For example $H^{\ast,\ast}(BGL_{n},\,\mathbb{Z})$ is equal to $H^{\ast,\ast}(\mathop{\mathrm{Spec}}\nolimits k,\,\mathbb{Z})[\,c_{1},\ldots,c_{n}]$ where $c_{i}$ is the Chern class of the standard representation. N. Yagita determined $H^{\ast,\ast}(BSO_{4})$ over $\mathbb{C}$ [24], where cases for $G=O_{n},\,SO_{2m+1},\,G_{2},\,Spin_{7}$ and some other groups in different coefficients are treated. It completes his preceding results in [22],  [23]. We will recall his argument in 1.3. Note that $SO_{2}=GL_{1}$ for $\sqrt{-1}\in k$. Therefore the new results in this paper are cases $G=SO_{2m}$ for $m\geq 3$.

We know that D. Edidin and W. Graham [1] constructed an element in $CH^{m}(SO_{2m})$ for each $m\geq 2$ such that the mod 2 reduction of it, $y_{0,m}$ in $H^{2m,m}(BSO_{2m})$, becomes 0 through cycle class mapping

[TABLE]

(The entire structure of the group $CH^{\ast}(SO_{2m})$ is known by R. Pandharipande  [11] in the case $m=2$ and R. Field [4] [3] in general $m$.) L. Molina-Rojas and A. Vistoli [8] gave a new insight on the element using the stratified method. We will review it in 1.2. This phenomenon is first observed by B.Totaro [15], that a generator in cobordism groups of $BSO_{4}(\mathbb{C})$ studied in [7] would not map to a generator of ordinary cohomology by his refined cycle class mapping. Note that the cobordism group of $BSO_{n}(\mathbb{C})$ is determined in  [6].

N. Yagita found a series of elements $y_{i,m}$ in $H^{2m,i+m}(BSO_{2m})$ for any $i$ from [math] to $m-2$ in the the kernel of $t^{\ast,\ast}_{BSO_{2m}}$ [24, 9.3], where it was denoted by $u_{i+1}$. These elements will be defined in 3.3. Since $H^{\ast,\ast}(\mathop{\mathrm{Spec}}\nolimits{\mathbb{C}})=\mathbb{Z}/2[\tau]$ ($\tau\in H^{0,1}(\mathop{\mathrm{Spec}}\nolimits{\mathbb{C}})$) [19, 7.8], $H^{\ast,\ast}(BSO_{n})$ is a $\mathbb{Z}/2[\tau]$-algebra. $\mathop{\mathrm{Ker}}\nolimits(t^{\ast,\ast}_{BSO_{2m}})$ becomes the ideal of $\tau$-torsion elements exactly [14, §6].

Our result is that $\mathop{\mathrm{Ker}}\nolimits t^{\ast,\ast}_{BSO_{2m}}$ is generated by Yagita’s elements $y_{i,m}$ over the polynomial ring of Chern classes of even indices; $0=\tau y_{i,m}=zy_{i,m}$ if $z$ is not a polynomial of $c_{2i}$’s and $y_{i,m}y_{j,m}=0$:

Theorem 0.1**.**

The kernel of realization mapping $\mathop{\mathrm{Ker}}\nolimits t^{\ast,\ast}$ in $H^{\ast,\ast}(BSO_{2m})$ is

[TABLE]

Note that $\mathop{\mathrm{Ker}}\nolimits(t^{2m,2m-1}_{X})=0$ for any $X$ (It is a direct consequence of the theorems of Voevodsky [19, Theorem 6.1, Lemma 6.9, Theorem 7.4]).

0.1. Conventions.

Since we only consider coefficients in $\mathbb{Z}/2$, we will omit it. We have $H^{\ast,\ast}(\mathop{\mathrm{Spec}}\nolimits{\mathbb{C}})=\mathbb{Z}/2[\tau]$ where $\tau$ has degree [math] and weight $2$. The motivic cohomology groups $H^{\ast,\ast}(X)$ is a bi-graded algebra over $\mathbb{Z}/2[\tau]$.

The group of quadratic roots of unity $\mu_{2}$ has the motivic cohomology $H^{\ast,\ast}(B\mu_{2})=\mathbb{Z}/2[\tau,y]\otimes\Lambda(x)=\mathbb{Z}/2[\tau,y]\{1,x\}$ where $x\in H^{1,1}(B\mu_{2})$ and $x^{2}=\tau y$ [20, 6.10]. Using the Beilinson-Lichtenbaum conjecture, we will identify $H^{i}(X)=H^{i,i}(X)$. We use the equivariant motivic cohomology in 1.1, which is denoted by $H^{\ast,\ast}_{G}(X)$. We use a simplified notation $H^{\ast,\ast}_{G}=H^{\ast,\ast}_{G}(\mathop{\mathrm{Spec}}\nolimits{\mathbb{C}})=H^{\ast,\ast}(BG)$. Similarly $H^{\ast}_{G}=H^{\ast}_{G}(pt)$.

The orthogonal group $O_{n}$ is the algebraic subgroup of $GL_{n}$ consisting of elements preserving the quadratic from $q(x)=\sum x_{i}^{2}$. The special orthogonal group $SO_{n}$ is the subgroup of $O_{n}$ consisting of elements with determinant one, and the inclusion is denoted $\epsilon:SO_{n}\subset O_{n}$. We regard $O_{n-1}$ as the subgroup fixing the last coordinate in $O_{n}$. Similarly for $\iota:SO_{n-1}\subset SO_{n}$. We denote by $\iota$ any of those canonical inclusions. We use the same notation for an induced morphism between classifying spaces as $\iota:BSO_{n-1}\subset BSO_{n}$. $\mathop{\mathrm{det}}\nolimits:O_{n}\to\mu_{2}\subset{\mathbb{G}}_{m}$ will be the determinant morphism.

Let us denote by $\gamma_{n}$ the standard real $n$-dimensional representation of $O_{n}(\mathbb{R})$, or its associated bundle on $BO_{n}(\mathbb{R})$. For $G=O_{n},SO_{n}$, $w_{i}\in H^{i}(BG(\mathbb{R}))=H^{i}(BG(\mathbb{C}))=H^{i}(BG)=H^{i,i}(BG)$ will mean the $i$-th Stiefel-Whitney class of $\gamma_{n}$ or $\epsilon^{-1}\gamma_{n}$. It is classical that $H^{\ast}(BO_{n})=\mathbb{Z}/2[w_{1},\\ w_{2},\ldots,w_{n}]$ and $H^{\ast}(BSO_{n})=\mathbb{Z}/2[w_{2},\ldots,w_{n}]$. The $i$-th Chern class $c_{i}$ of $\gamma\otimes{\mathbb{C}}$ or $\epsilon^{-1}$-image of it, is an element in $CH^{i}(BG)\otimes\mathbb{Z}/2=H^{2i,i}(BG)$ so that $(w_{i})^{2}=\tau^{i}c_{i}$.

0.2. Acknowledgments.

The authors would like to thank Nobuaki Yagita for several discussions, and to the referees for many useful suggestions.

1. Preliminaries

1.0. Motivic cohomology and the

filtration by weight.

The realization mapping $t=t_{X}=t^{i,j}_{X}:H^{i,j}(X)\to H^{i,i}(X)=H^{i}(X)$ is just a multiplication of $\tau^{i-j}$ [14, §6]. We can introduce a filtration by weight on $H^{i}(X)$, so that $F^{j}H^{i}(X)$ consists of elements which are the realization of weight less than $j$. It means $x$ has weight exactly $2j-i$ if there exists $x^{\prime}$ as $x=\tau^{i-j}x^{\prime}$ and $j$ is the minimum of such numbers. We denote such $x^{\prime}$ as $\overline{x}$, which is not determined uniquely if there exists non-trivial element of $\tau$-torsion. But in all the cases appear in our paper we have a unique choice of such element for a homogeneous $x$ by reasoning on bi-degree. It defines a filtered ring structure on $H^{i}(X)$ [24, §2, §6, §7], which is equivalent to the coniveau filtration of Grothendieck, after shifts on degrees and indices of filtration. The precise argument is in  [24, §7].

A morphism $f:X\to Y$ induces a pullback $f^{*}:H^{\ast,\ast}(Y)\to H^{\ast,\ast}(X)$ that is a homomorphism of $\mathbb{Z}/2[\tau]$-algebras; and a filtered ring homomorphism $t(f^{*}):H^{\ast}(Y)\to H^{\ast}(X)$.

There are motivic Milnor operations $Q_{k}$ on the motivic cohomology groups [18, 13.4]:

[TABLE]

(Note that we assume $\sqrt{-1}\in k$ so that $\rho$ there is trivial.) $Q_{i}$ are derivations on the motivic cohomology groups, $Q_{i}Q_{j}=Q_{j}Q_{i}$ if $i\neq j$ and $Q_{i}Q_{i}=0$. They commute with pullbacks and Gysin morphisms. If $x$ has weight $i>0$, $Q_{j}x$ has weight $i-1$ if it is not zero. These are compatible with the classical Milnor operations $Q_{i}$ on $H^{*}(X)$ in the sense that $t(Q_{i}x)=Q_{i}t(\tau^{i-j}x)$. ( [18, 3.9]). We denote it by the same notation $Q_{i}$.

1.1. Equivariant motivic cohomology

Edidin and Graham extended the construction of the Chow ring of $BG$ to the equivariant Chow ring of a scheme with G-action, which can be generalized to motivic cohomology [2]. Let us recall the equivariant motivic cohomology [24, §8]. Suppose that $G$ acts on a smooth quasi-affine scheme $X$. For each $m\geq 0$, we can choose a representation $V$ of $G$ with an open subscheme $E$ on which $G$ acts freely, and $\mathrm{codim}_{V}(V-E)>m$. The quotient $(E\times X)/G$ exists as a smooth scheme. We set

[TABLE]

It is independent of the choice of such $V$ for fixed $m$ and $*$. For a morphism of groups $\phi:H\to G$, a $G$-schemes $X$, an $H$-scheme $Y$, and an $H$-equivariant morphism $f:Y\to X$ where $H$ acts on $X$ through $\phi$, there is a pull-back $f^{*}:H^{\ast,\ast}_{G}(X)\to H^{\ast,\ast}_{H}(Y)$. For a subgroup $H$ of $G$, we see that [8, 2.1]

[TABLE]

Let $U$ be a unipotent group and consider a $U$-torsor $X\to Y$, we have an isomorphism $H^{\ast,\ast}(Y)=H^{\ast,\ast}(X)$, for $U$ is isomorphic to an affine space and any $U$-torsor is locally trivial  [5, Prop.1]. In particular for a semi-direct product $G\ltimes U$ the projection onto the component $G\ltimes U\to G$ induces an isomorphism $H^{\ast,\ast}(B(G\ltimes U))=H^{\ast,\ast}(BG)$ [8, 2.3].

Take any two elements $u_{0},u_{1}$ of $U(\mathbb{C})$. $G\to G\times U$ by $x\mapsto(x,\,u_{i})$ induce the same morphism on $H^{\ast,\ast}(B(G\times U))\to H^{\ast,\ast}(BG)$ for both are the section of the projection. Similarly for the multiplicative group ${\mathbb{G}}_{m}$: $H^{\ast,\ast}(B(G\times{\mathbb{G}}_{m}))=H^{\ast,\ast}(BG)[y]\to H^{\ast,\ast}(BG)$. Let $G$ be connected. For any element $g\in G(\mathbb{C})$ the conjugation by $g$ defines a morphism $c_{g}:BG\to BG$, but we know that $g$ is contained in a Borel subgroup, and $c_{g}^{*}=c_{e}^{*}$ on $H^{\ast,\ast}(BG)$ for $e$ is the unit of $G$  [5, Lemma.1 of Proposition 4].

1.2. The stratified method

The equivariant motivic cohomology theory has the following localization sequence; if we write an equivariant closed immersion $i\colon Y\hookrightarrow X$ of codimension $s$ and open immersion $j\colon X-Y\hookrightarrow X$, there is a long exact sequence

[TABLE]

where $i_{*}$ is the Gysin mapping.

If we know a stratification of a $G$-scheme $X$ by $G$-subschemes, we may calculate $H_{G}^{\ast,\ast}(X)$ by repeated use of the localization sequence above and (1.1). This method is introduced by Vezzosi to compute $\mathop{\mathrm{CH}}\nolimits^{*}(BPGL_{3})$ [17]. Molina-Rojas and Vistoli [8] calculated the Chow ring of $BG$ using it in cases $G=GL_{n}$, $SL_{n}$ and $Sp_{n}$ over arbitrary fields, and $G=O_{n}$ and $SO_{n}$. over fields of characteristic different from $2$. The following stratification associated with $O_{n}$ or $SO_{n}$ are used to calculate the Chow groups and motivic cohomology in  [8] and [24, §9]. Consider the standard representation of $SO_{n}$. It makes $\mathbb{A}^{n}$ an $SO_{n}$-scheme. We set $B=\{x\in{\mathbb{A}}^{n}\smallsetminus\{0\}\mid q(x)\neq 0\}$ and $C=\{x\in{\mathbb{A}}^{n}\smallsetminus\{0\}\mid q(x)=0\}$ where $q(x)=\sum_{k=1}^{n}x_{k}^{2}$. There is a stratification $\{0\}\subset\{0\}\cup C\subset\mathbb{A}^{n}$, and $\mathbb{A}^{n}\smallsetminus(\{0\}\cup C)=B$.

Then we get two localization sequences:

[TABLE]

and

[TABLE]

We have $H_{SO_{n}}^{\ast,\ast}(\{0\})=H_{SO_{n}}^{\ast,\ast}({\mathbb{A}}^{n})$ and ${s}_{*}$ is the multiplication of the $n$-th Chern class $c_{n}\in H^{2n,n}_{SO_{n}}(\{0\})=H^{2n,n}(BSO_{n})$ by the self-intersection formula [13]. Since in the end of proof in 3.3 we will see that c_{n}\cup\ is injective, we will denote

[TABLE]

We can define an action of ${\mathbb{G}}_{m}\times SO_{n}$ on $B$ such that $(t,g)\in{\mathbb{G}}_{m}\times SO_{n}$ sends $v\in B$ to $tg(v)$, that is, the matrix multiplication of scalar $t\in{\mathbb{G}}_{m}$ and $g(v)\in B$. This action is transitive and the stabilizer of $e_{1}$ is a subgroup isomorphic to $O_{n-1}$. The inclusion $\kappa_{1}:O_{n-1}\to\mathbb{G}_{m}\times SO_{n}$ is given by $g\mapsto\left(\mathop{\mathrm{det}}\nolimits g,\kappa\right)$ where $\kappa:O_{n-1}\to SO_{n}$ is

[TABLE]

We have $B\cong{\mathbb{G}}_{m}\times SO_{n}/O_{n-1}$.

Choose $E$ as in 1.1. Then we obtain the following isomorphisms

[TABLE]

${\mathbb{G}}_{m}$ is an $O_{n-1}$-scheme via $\mathop{\mathrm{det}}\nolimits$, and $c_{1}(\gamma_{n}\otimes{\mathbb{C}})=c_{1}(\mathop{\mathrm{det}}\nolimits(\gamma_{n}\otimes{\mathbb{C}}))$, we get the identification similar to (5)

[TABLE]

for we will see in the next subsection that c_{1}\cup\ is a monomorphism. Note that these are $H^{\ast,\ast}_{SO_{n}}$-module isomorphism through $\kappa^{*}$. $j^{*}$ in (4) is identified with the induced morphism of $\kappa^{*}$.

On the other hand, we know that $SO_{n}$ transitively acts on $C$ and the stabilizer of $e_{1}+\sqrt{-1}e_{2}$ is isomorphic to a semi-direct product of the stabilizer of a pair $e_{1}+\sqrt{-1}e_{2},e_{1}-\sqrt{-1}e_{2}$ and a unipotent subgroup, so that it is isomorphic to $SO_{n-2}\ltimes{\mathbb{A}}^{n-1}$ [8, 5.2]. Thus we get that $C\cong SO_{n}/(SO_{n-2}\ltimes{\mathbb{A}}^{n-1})$ and this deduces the following isomorphisms [8, 2.3]:

[TABLE]

From the above calculations we can rewrite (4) as follows:

[TABLE]

We can regard the same stratification as that of $O_{n}$-spaces. The induced long exact sequence is exactly the one used by Yagita. We can connect two long exact sequences by the change of groups homomorphism on $\epsilon:SO_{n}\to O_{n}$:

[TABLE]

The standard inclusion $\mathbb{A}^{n}\to\mathbb{A}^{n+1}$ into the first $n$ components induces the following commutative long exact sequences (Gysin mappings and pullbacks commute in this case  [20, 2.3])

[TABLE]

Later we use another inclusion $\mathbb{A}^{n}\to\mathbb{A}^{n+1}$ into the second to $n+1$-th components. It is conjugate to the composition of the standard inclusions by an element of $SO_{n+1}({\mathbb{C}})$, so that the induced morphisms in motivic cohomology are the same (1.1).

1.3. $H^{*,*}(BO_{n})$

It is well-known that the map induced from the restriction to the diagonal subgroup

[TABLE]

is an isomorphism, where $\Sigma_{n}$ is the $n$-th symmetric group acting by permutations on $x_{i}$’s. We need three basis of the symmetric polynomials. Recall that for any partition $\lambda=[i_{1},i_{2},\ldots,i_{k},0,\ldots,0]=[i_{1},i_{2},\ldots,i_{k},0^{n-k}]$ of $n$ where $\sum i_{j}=n$ and $i_{j}\neq 0$, $k$ is called the length of $\lambda$. For each partition $\lambda$ we can define a symmetric polynomial $m[\lambda]=\sum x_{1}^{i_{1}}x_{2}^{i_{2}}\ldots x_{k}^{i_{k}}$ where the sum is taken on the orbit of $\Sigma_{n}$. These make the monomial basis of symmetric polynomials.

The $l$-th Stiefel-Whitney class $w_{l}$ corresponds to the elementary symmetric polynomial of degree $l$ : $w_{l}=m[1,1,\ldots,1]=m[1^{l},0^{n-l}]$, whose monomials gives a basis of $H^{*}(BO_{n})={\mathbb{Z}}/2[w_{1},\ldots,w_{n}].$

Finally we need a basis suitable to see action of $Q$’s. It is due to W. S. Wilson. We denote by $Q(i)M$ the $\mathbb{Z}/2$-module generated by $Q_{k_{1}}\ldots Q_{k_{j}}m$ for all $0\leq k_{1}<\ldots<k_{j}\leq i$ and $m\in M$. Then

[TABLE]

here $G_{k-1}$ is the vector space spanned by monomial bases $m[{\lambda}]$ where $\lambda=[2s_{1}+1,2s_{2}+1,\ldots,2s_{k}+1,{2t_{1}},{2t_{2}},\ldots,{2t_{q}}]$ for some $k+q\leq n\;(0\leq k,q\leq n)$ and $0\leq s_{1}\leq s_{2}\leq\ldots\leq s_{k},\ 0<t_{1}\leq t_{2}\leq\ldots\leq t_{q}$ such that if the number of $t$ equal to $t_{u}$ is odd, then there exists some $v\leq k$ such that $2s_{v}+2^{v}<2t_{u}<2s_{v}+2^{v+1}$ ([21, 2.1], while the details of the combinatorial condition here are not necessary in our proof.).

Recall $Q_{k}x_{i}^{2j}=0$ and $Q_{k}x_{i}^{2j+1}=x_{i}^{2j+2k}$ in $H^{*}((B\mu_{2})^{n})$. Any element of $H^{*}(BO_{n})$ is a sum of $Q_{i_{1}}\ldots Q_{i_{s}}m_{\lambda}$ for some such $\lambda$ and $0\leq i_{1}<\ldots<i_{s}\leq k$. Note $Q_{i_{1}}\ldots Q_{i_{s}}m[{\lambda}]=m[2s_{1}+2i_{1},2s_{2}+2i_{2},\ldots,2s_{k}+2i_{s},{2t_{1}},{2t_{2}},\ldots,{2t_{q}}]$, which is not zero by the condition on the partition. For example $w_{l}$ is an element of the basis and $Q_{0}w_{l}=m[2,1^{l-1}],\,Q_{0}Q_{0}w_{l}=m[2,2,1^{l-2}]+m[2,2,1^{l-2}]=0,\,Q_{0}Q_{1}w_{l}=m[2,4,1^{l-2}]$.

Yagita determined the motivic cohomology $H^{\ast,\ast}(BO_{n})$. He proved inductively that $t_{BO_{n}}$ is injective by the stratified method [24, 8.2]. Then the description of the filtration by weight on $H^{\ast}(BO_{n})$ directly connected to $H^{\ast,\ast}(BO_{n})$ through Milnor operations (1.5). The result is the following: let $(\mu_{2})^{n}$ be the diagonal subgroup of $O_{n}$. Then the restriction induces an isomorphism [24, 8.1]

[TABLE]

where $G_{k}$ is spanned by the monomial bases exactly the same conditions in (11) but $x_{i}\in H^{1,1}$.

Theorem 1.1** (Yagita [24, 8.1]).**

$x\in H^{*}(BO_{n})$* has weight $k$ if and only if $k$ is the maximal number such that $Q_{i_{1}}Q_{i_{2}}\ldots Q_{i_{k}}x\neq 0$ for some tuple $(i_{1},i_{2},\ldots,i_{k})$.*

In particular $m_{\lambda}$ where $\lambda=[2s_{1}+1,\ldots,2s_{k}+1,{2t_{1}},\ldots,{2t_{q}}]$ has weight $k$, and $x$ has weight [math] if and only if $Q_{i}x=0$ for every $i$.

Corollary 1.2**.**

The Stiefel-Whitney class $w_{l}$ in $H^{*}(BO_{n})$ has weight $l$ for $l=1,\ldots,n$.

Since $\iota^{*}$ discards monomials of length exactly $n$, the following corollary is obvious.

Corollary 1.3**.**

The mapping induced from the canonical restriction $\iota^{*}:H^{*,*}(BO_{n})\to H^{*,*}(BO_{n-1})$ is an epimorphism.

2. $H^{\ast}(BSO_{n})$ and $H^{*,*}(BSO_{2m+1})$

To determine $H^{\ast,\ast}(BSO_{n})$ we can not use the restriction to the diagonal subgroup. The induced mapping on motivic cohomology is neither injective nor surjective. We will use the exact sequences (3) and (8). For effective use of them we need to determine the coniveau filtration on $H^{\ast}(BSO_{n})$.

2.1. Determination of weight

We will write down $\kappa^{*}$ in terms of Stiefel-Whitney classes. Recall $\gamma_{n}$ is the canonical $n$-dimensional real vector bundle over $BO_{n}(\mathbb{R})$. We have the following isomorphism of vector bundles:

[TABLE]

Thus the total Stiefel-Whitney class of $\kappa^{-1}\epsilon^{-1}\gamma_{n}$ is

[TABLE]

We will prove that the mapping $t(\kappa^{*})\colon H^{*}(BSO_{n})\rightarrow H^{*}(BO_{n-1})$ preserves weight strictly. We recall that a filtered mapping $\phi\colon H^{*}(X)\rightarrow H^{*}(Y)$ strictly preserves filtrations if $F^{i}H^{*}(X)=\phi^{-1}(F^{i}H^{*}(Y))$.

To see first $t(\kappa^{*})$ is injective, we recall $\mathop{\mathrm{Ker}}\nolimits t(\kappa^{*})\subset\mathop{\mathrm{Ker}}\nolimits t(\iota^{*})=(w_{n})$. Suppose $0\neq x\in H_{SO_{n}}^{*}$ is in $\mathop{\mathrm{Ker}}\nolimits t(\kappa^{*})$. It is divisible by $w_{n}$, and there exists $x_{j}$ such that $x=w_{n}^{j}x_{j}$ and $x_{j}$ is not divisible by $w_{n}$. The image $t(\kappa^{*})x=w_{n-1}^{j}w_{1}^{j}\kappa^{*}x_{j}$ is zero, so $t(\kappa^{*})x_{j}$ is zero, which is contradiction.

Let $x$ be an element of $H^{*}(BSO_{n})$ such that $t(\kappa^{*})x$ is contained in the weight [math] part of $H^{*}(BO_{n-1})$. We can write $x=P(w_{i})$ with a polynomial of $w_{i}$. $t(\kappa^{*})x=P(t(\kappa^{*})x)$ is a polynomial of $c_{i}$, so that each term will be a square of a monomial of $w_{i}$. By (13) it is possible, by induction on lexicographic order on monomials, only if $P$ is a square. We have proved that $t(\kappa^{*})$ is strict on the weight [math] part.

If $y=t(\kappa^{*})x$ has weight $k>0$, we switch to the motivic cohomology. Then we can find $\overline{x}\in H_{SO_{n}}^{*,*}$ and $\overline{y}\in H_{O_{n-1}}^{*,*}$ such that $\kappa^{*}\overline{x}=\tau^{f}\overline{y}$ for some $f\in{\mathbb{Z}}_{\geq 0}$. We have that

[TABLE]

for some $\{j_{0},\ldots,j_{k-1}\}\subset\{0,\ldots,n-2\}$ and $Q_{j_{0}}\ldots Q_{j_{k-1}}\overline{y}$ is a non-zero element in $H^{2h,h}$ for it has zero image by all $Q_{i}$. It means $f=0$ and $x$ has weight equal to $k$ by the theorem of Yagita. Finally $t(\kappa^{*})c_{n}=c_{1}c_{n-1}$ implies the following.

Proposition 2.1**.**

The mapping $t(\kappa^{*})\colon H^{*}_{SO_{n}}/c_{n}\rightarrow H^{*}_{O_{n-1}}/c_{1}$ is injective and it preserves weight strictly.

Corollary 2.2**.**

1. (1)

For odd $n$, the Stiefel-Whitney class $w_{l}$ ( $2\leq l\leq n$) in $H^{*}(BSO_{n})$ has weight $l$ if $l$ is even, and $l-2$ if $l$ is odd. 2. (2)

For even $n$, the Stiefel-Whitney class $w_{n}$ has weight $n-2$ in $H^{*}(BSO_{n})$. For other $w_{l}$ ($2\leq l\leq n$), $w_{l}$ has weight $l$ if $l$ is even, and $l-2$ if $l$ is odd. 3. (3)

$Q_{0}w_{2l}=w_{2l+1}$* for all $1\leq l\leq n/2$, where we regard $w_{n+1}=0$.*

Proof.

Consider $t(\kappa^{*})w_{l}$ in $H_{O_{n-1}}^{*}$. If $l\neq n$, $t(\kappa^{*}w_{l})=w_{l}+w_{1}w_{l-1}=m{[1^{l}]}+m{[1]}m{[1^{l-1}]}$ from (13). If $l$ is odd, $t(\kappa^{*})w_{l}=m{[2,1^{l-1}]}$. If $l$ is even, $t(\kappa^{*})w_{l}=w_{l}+w_{1}w_{l-1}=m{[1^{l}]}+m{[2,1^{l-1}]}.$ For $l=n$, we have that $t(\kappa^{*})w_{n}=w_{1}w_{n-1}=m{[2,1^{n-2}]}.$ ∎

2.2.

The stratified method works fine in ordinary cohomology, and the result is compatible with the realization:

[TABLE]

We can identify as in (5) $H_{SO_{n}}^{*}({\mathbb{A}}^{n}\smallsetminus\{0\})=H_{SO_{n}}^{*}/c_{n}$.

The sequence (8) becomes

[TABLE]

We have the following data on (15). These are direct consequences of Proposition 2.1, (13), and $t(\delta)$ is an $H^{*}_{SO_{n}}$-module mapping. We may compare the result of Yagita in [24, §8] by (9). Note we will write the image in the quotient as the same letter, as is $w_{i}\in H_{SO_{2m}}^{*}/c_{2m}$.

Proposition 2.3**.**

1. (1)

$t(i_{*})=0$. 2. (2)

$t(\kappa^{*})w_{i}=w_{i}+w_{i-1}w_{1}$* for $2\leq i\leq n-1$, and $t(\kappa^{*})w_{n}=w_{n-1}w_{1}$.* 3. (3)

$t(\delta)(w_{1})=1$* and $t(\delta)(zw_{1})=z$ for $z$ is a polynomial of $w_{i}$ for $2\leq i\leq n-2$.*

2.3. $H^{*,*}(BSO_{2m+1})$

Since $H^{*}_{O_{n}}$ and $H^{*}_{SO_{n}}$ are generated by the Stiefel-Whitney classes, the realization mapping $t^{*,*}$ are epimorphisms in all these cases. As in 1.3, $t\colon H^{p,q}_{O_{n}}\rightarrow H^{p}_{O_{n}}$ is injective for every pair $(p,q)$.

When $n=2m+1$, we have that $O_{2m-1}\cong SO_{2m-1}\times\mu_{2}$. This direct product structure induces the mapping $\pi\colon BO_{2m-1}\rightarrow BSO_{2m-1}$ deduced from multiplication of determinant $g\mapsto\mathop{\mathrm{det}}\nolimits(g)g$ if $g\in O_{2m-1}$. Then $\pi\cdot\epsilon=1$ and $(\epsilon\cdot\pi)(\epsilon\cdot\pi)=(\epsilon\cdot\pi)$.

[TABLE]

where $x=\overline{w_{1}}$ has degree (1,1). It implies the realization $t:H^{p,q}_{SO_{2m-1}}\to H^{p}_{SO_{2m-1}}$ is injective. We can compare (8) and (15) through monomorphisms $t$. Because $t(\kappa^{*})t_{SO_{2m+1}}=t_{O_{2m}}\kappa^{*}$ is injective, $\kappa^{*}$ is injective, then we get

[TABLE]

3. $H^{*,*}(BSO_{2m})$

3.1.

Let us denote by $Y_{m}$ the kernel of $H^{*,*}_{SO_{2m}}\to H^{*}_{SO_{2m}}$. We will inductively determine $Y_{m}$ so that we assume $Y_{m}$ is known. We denote by $\overline{Y_{m+1}}$ the kernel of $t:H^{*,*}_{SO_{2m+2}}/c_{2m+2}\to H^{*}_{SO_{2m+2}}/c_{2m+2}$. Later we will see in 3.3 that $(\mathop{\mathrm{Ker}}\nolimits t_{SO_{2m}})/c_{2m+2}=\overline{Y_{m+1}}$. We can combine (8) and (15) to the following commutative diagram

[TABLE]

$t_{2m+1}$ on the right column is injective for each degree. $\overline{w_{1}}$ has degree $(1,1)$ and $t\overline{w_{1}}=w_{1}$.

We obtain that $\mathop{\mathrm{Im}}\nolimits({i}_{*})=\mathop{\mathrm{Ker}}\nolimits(t_{2m+2})\subset H_{SO_{2m+2}}^{*,*}/c_{2m+2}$, using diagram-tracing argument. Note $t_{2m+1}$ is injective. The restriction of $i_{*}$ gives an isomorphism between $\mathop{\mathrm{Coker}}\nolimits\delta$ and $\overline{Y_{m+1}}$, and it takes $Y_{m}$ into $\overline{Y_{m+1}}$. We can see $\mathop{\mathrm{Im}}\nolimits\delta\cap Y_{m}=0$. For if there is $u=\delta v$ such that $t(u)=0$, there exists $w$ such that $t(\kappa^{*})t(w)=t(v)$ . Because $t(\kappa^{*})$ strictly preserves filtrations, there is $w^{\prime}\in H^{*,*}_{SO_{2m+2}}/c_{2m+2}$ such that $\kappa^{*}w^{\prime}=v$. This means $u=0$.

The restriction of $i_{*}$ gives an acyclic subcomplex $Y_{m}\to Y_{m}$, where the second copy is in $\overline{Y_{m+1}}$, in (3.1), and we get

[TABLE]

where the induced $t_{2m}$ is now a monomorphism and $\mathop{\mathrm{Coker}}\nolimits\delta_{1}\cong\overline{Y_{m+1}}/Y_{m}$.

By Proposition 2.3 we know that $\mathop{\mathrm{Coker}}\nolimits\delta_{1}$ is isomorphic to the cokernel of the restriction $\iota^{*}_{2m+1}:H^{\ast,\ast}_{SO_{2m+1}}\to H^{\ast,\ast}_{SO_{2m}}/Y_{m}$, up to permutation of coordinates, which makes a part of the following commutative exact sequences

[TABLE]

where $\iota^{*}_{2m}$ is an epimorphism.

3.2.

Since $\kappa^{*}\overline{w_{2m}}=\overline{w_{2m}}+\tau\overline{w_{1}w_{2m-1}}$ in $H^{*,*}_{O_{2m}}$, $\overline{w_{2m}}$ has weight $2m$ in $SO_{2m+1}$. On the other hand $\overline{w_{2m}}$ in $H^{*,*}_{SO_{2m}}$ satisfies $\kappa^{*}\overline{w_{2m}}=\overline{w_{1}w_{2m-1}}$, which implies it has weight $2m-2$. That is, $\iota_{2m+1}^{*}\overline{w_{2m}}=\tau\overline{w_{2m}}$, and $\overline{w_{2m}}$ presents a nonzero element in $\mathop{\mathrm{Coker}}\nolimits\iota_{2m+1}^{*}$, where $\tau\overline{w_{2m}}=0$.

We want to claim that $\overline{w_{2m}}$ is a generator of $\mathop{\mathrm{Coker}}\nolimits\iota_{2m+1}^{*}$ over the ring of even Chern classes. Since $\kappa$ and $\iota$ commute, we have to consider elements having factor $\overline{w_{2m}}$ only.

Consider the long exact sequence associated with (20):

[TABLE]

Connecting mapping $\partial$ is $(\kappa^{*})^{-1}\cdot\iota^{*}_{2m}\cdot(\delta_{2})^{-1}$. Let us take $z$ in $\mathop{\mathrm{Ker}}\nolimits\iota^{*}_{2m-1}$. We can take $(\delta_{2})^{-1}z=\overline{w_{1}z}$. If it is in ordinary cohomology, $\mathop{\mathrm{Coker}}\nolimits\iota^{*}_{2m+1}=0$, for we can take $w_{2m}$ as $(\delta_{2})^{-1}w_{2m-1}$ which go to zero by $\iota^{*}_{2m}$, but in motivic cohomology $\overline{w_{2m}}$ has strictly larger weight than $\overline{w_{1}w_{2m-1}}$, hence $\overline{w_{2m}}$ presents a non-zero element in $\mathop{\mathrm{Coker}}\nolimits\iota^{*}_{2m+1}$ as above.

Therefore it suffices to prove that in $H_{O_{2m}}^{*,*}$, $zw_{2m}$ has weight less than or equal to the weight of $zw_{1}w_{2m-1}$, if $z$ is not a polynomial of even Chern classes. Let $z$ be an element of monomial base $m[i_{\cdot}]$. First we assume not all $i$’s are even. We will use induction of length of $z$. If it has length $1$, $m[2i+1]m[2,1^{2m-2},0]$ and $m[2i+1]m[1^{2m}]$ have weight $2m-1$ ($i>0$). $m[2i+1,2j+1]m[2,1^{2m-2},0]$ have weight more than $2m-2$ and $m[2i+1,2j+1]m[1^{2m}]$ have weight $2m-2$. If in a monomial of length $j\geq 2$ all $i$ are odd, $zw_{2m}$ have weight $2m-j$, and $zw_{1}w_{2m-1}$ also have weight $2m-j$. In general let $2j$ be the largest even member of $i_{\cdot}$ which has multiplicity $k$. Let $z^{\prime}$ be the partition discarding $2j$’s from $z$. Then $z$ is equal to $z^{\prime}(w_{k})^{2j}$ plus elements of length less than that of $z$. $z^{\prime}(w_{k})^{2j}w_{2m}$ has the same weight as that of $z^{\prime}w_{2m}$, and similarly for $w_{1}w_{2m-1}$. It finishes the inductive step.

If all $i$’s are even, it is a polynomial of $(w_{i})^{2}$ and $zw_{2m}$ has weight $2m$ but $zw_{1}w_{2m-1}$ has weight $2m-2$. If $z$ contains $w_{2j-1}^{2}$, there exists u such that $zw_{2m}+uw_{1}w_{2m}$ has weight $2m-2$, hence it does not represent nonzero elements in $\mathop{\mathrm{Coker}}\nolimits\iota^{*}_{2m+1}$.

3.3.

We know that $Y_{1}=0$, and the cokernel of $\delta$ is easily seen to be the image of $\mathbb{Z}/2[c_{2}]\{\overline{w_{2}}\}$, here $\overline{w_{2}}\in H^{2,1}_{SO_{2}}$ in $H^{\ast,\ast}_{SO_{4}}/c_{4}$. Let us define $y_{0,2}={i_{2}}_{*}\overline{w_{2}}$. Then $Y_{2}=\mathbb{Z}/2[c_{2},c_{4}]\{y_{0,2}\}$ [24, §8].

Assume we know that $Y_{m}$ is generated by $y_{i,m}$ over $\mathbb{Z}/2\,[\,c_{2},\ldots,c_{2m}]$ for $0\leq i\leq m-2$. $i_{*}(y_{i,m})$ are non-zero elements in $H_{SO_{2m+2}}^{*,*}/c_{2m+2}$ ($i_{*}$ is that in (3.1).) For $i_{*}y_{i,m}$ have degree $2m+2$ there exist $y_{i,m+1}$ in $H_{SO_{2m+2}}^{*,*}$. We will see that $(Y_{m+1}/c_{2m+2})/i_{*}(Y_{m})$ is generated by $i_{*}\overline{w_{2m}}$ over $\mathbb{Z}/2\,[\,c_{2},\ldots,c_{2m}]$. There exists a unique element $y_{m-1,m+1}$ in $H^{2m+2,2m}(BSO_{2m+2})$ whose image by quotient of $c_{2m+2}$ is $i_{*}\overline{w_{2m}}$. We get an isomorphism

[TABLE]

Multiplicative relations are consequence of the projection formula [13].

Finally we consider ${s}_{*}$ in (3) as an $H_{GL_{2m}}^{*,*}$-module mapping. c_{2m}\cup\ is injective on $\mathop{\mathrm{Ker}}\nolimits t^{*,*}_{SO_{2m}}$ and $H_{SO_{2m}}^{*}$, so that we can identify $H_{SO_{n}}^{*,*}({\mathbb{A}}^{n}-\{0\})=H_{SO_{n}}^{\ast,\ast}/c_{n}$ for all $n$.