CONIVEAU SPECTRAL SEQUENCES OF CLASSIFYING SPACES FOR EXCEPTIONAL AND

SPIN GROUPS

M. KAMEKO, M. TEZUKA, AND N. YAGITA

Abstract. Let k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results of cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

1. Introduction

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

Let BG be the classifying space of G. Totaro showed that

Inv^{∗}(G; ℤ/p) ≅ H^{0}(BG; H_{ℤ/p}^{∗})

where H^{∗}(X; H_{ℤ/p}^{∗′}) is the cohomology of the Zarisky sheaf induced from the presheaf H_{et}^{∗}(V; ℤ/p) for open subsets V of X. This sheaf cohomology is also the E_{2}-term

E_{2}^{∗,∗′} ≅ H^{∗}(BG; H_{ℤ/p}^{∗′}) ⟹ H^{∗}(BG; ℤ/p)

of the coniveau spectral sequence by Bloch-Ogus [Bl-Og].

We restrict to consider a group G such that it has only one conjugacy class of nontoral maximal elementary abeilan p-group A. For exceptional cases, G = G_{2}, F_{4}, E_{6} for p = 2, G = F_{4}, E_{6}, E_{7} for p = 3, and G = E_{8} for p = 5. We also consider groups Spin_{n}, n ≥ 7.

Let W_{G}(A) be the Weyl group of G for A. Then by using Rost, Serre and Garibaldi’s results [Ga], we easily see that

Res_{Inv} : Inv^{∗}(G; ℤ/p) ≅ Inv^{∗}(A; ℤ/p)^{W_{G}(A)}

1991 Mathematics Subject Classification. Primary 11E72, 12G05, 55P35; Secondary 55T25,57T05.

Key words and phrases. coniveau spectral sequence, motivic cohomology, exceptional groups.

for cases of the above groups except for (E_{6}, p = 2) and (Spin_{n}, p = 2), n ≥ 10.

Let Q_{i} be the Milnor operation and let

Q(n) = Λ(Q_{0}, ..., Q_{n}).

We easily see that operations Q_{i} can extend on H^{∗}(BG; H_{ℤ/p}^{∗}) for these fields k. In particular, H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)} has also the Q(∞)- module structure. We can prove that for the above cases except for (E_{7}, p = 3), the invariant is generated as Q(∞)-algebras by elements in Res(Inv^{∗′}(G; ℤ/p)) and Res(H^{∗}(BG; H_{ℤ/p}^{∗})) = Res(CH^{∗}(BG)/p). (Moreover it is a direct sum of free Q(n)-modules.)

These facts imply the following theorem.

Theorem 1.1. Let G = G_{2}, Spin_{n}(7 ≤ n ≤ 9), F_{4} for p = 2, G = F_{4}, E_{6} for p = 3, or G = E_{8}, p = 5. Then the following restriction map

Res_{E_{2}} : H^{∗}(BG; H_{ℤ/p}^{∗′}) → H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)}

is an epimorphism.

For (E_{7}, p = 3), the map Res_{E_{2}} is not epic, while Res_{Inv} is epic. We note that the restriction map

Res_{Hℤ/p} : H^{∗}(BG; ℤ/p) → H^{∗}(BA; ℤ/p)^{W_{G}(A)}

is not an epimorphism for p ≥ 3, while H^{∗}(BA; ℤ/p) ≅ H^{∗}(BA; H_{ℤ/p}^{∗′}) as algebras. (Indeed, BG is 3-connected but H^{0}(BG; H_{ℤ/p}^{3}) ≠ 0.) Note also that the right hand side invariant and Im(Res_{Hℤ/p}) are computed by Kameko-Mimura [Ka-Mi] for odd primes p.

When p = 2, the maps Res_{Hℤ/2} are even isomorphic except for the case E_{6}. However note ([Or-Vi-Vo]))

H^{∗}(BA; H_{ℤ/2}^{∗′}) ≅ grH^{∗}(BA; ℤ/2) ≅ ℤ/2[y_{1}, ..., y_{n}] ⊗ Λ(x_{1}, ..., x_{n})

with βx_{i} = y_{i} but y_{i} ≠ x_{i}^{2} = 0 as the cases p = odd. So we know

Res_{Hℤ/2} ≅ gr(H^{∗}(BA; ℤ/2)^{W_{G}(A)}) ≠^{⊂} H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)}

for the above groups.

The arguments seem something subtle and we give here an example, the case G = G_{2} and p = 2. Then A ≅ (ℤ/2)^{3} and W_{G}(A) ≅ GL_{3}(ℤ/2), moreover

H^{∗}(BG_{2}; ℤ/2) ≅ H^{∗}(BA; ℤ/2)^{W_{G}(A)} ≅ ℤ/2[w_{4}, w_{6}, w_{7}] |w_{i}| = i.

The cohomological invariant is known by Rost and Serre

Inv^{∗}(G_{2}; ℤ/2) ≅ ℤ/2{1, u_{3}} |u_{3}| = 3

where u_{3} = x_{1}x_{2}x_{3} in H^{0}(BA; H_{ℤ/2}^{3}). From Mui and Kameko-Mimura results [Ka-Mi], we can show

H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)} ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{1} ⊕ ℤ/2[c_{7}] ⊗ Q(2){u_{3}})

where Q_{0}Q_{1}Q_{2}(u_{3}) = c_{7}, deg(c_{i}) = (i, i) (and w_{i}^{2} = c_{i} in H^{∗}(BA; ℤ/2)). These c_{i} are represented by Chern classes, and hence Res_{E_{2}} is an epimorphism.

Of course u_{3} ∉ Res_{Hℤ/2}, and moreover we see

H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)}/Res_{Hℤ/2} ≅ ℤ/2[c_{4}, c_{6}]{u_{3}}.

For example, we have

Q_{0}(u_{3}) = w_{4}, Q_{1}(u_{3}) = w_{6}, Q_{0}Q_{1}(u_{3}) = w_{7}, Q_{2}(u_{3}) = w_{4}w_{6}.

Here u_{3} does not exist in H^{∗}(BG_{2}; ℤ/2), and hence d_{r}(u_{3}) = y ≠ 0 for some r ≥ 2 and y ∈ H^{∗}(BG_{2}; H_{ℤ/2}^{∗}) in the coniveau spectral sequence. We can see this r = 2.

Theorem 1.2. We have the epimorphism (as bidegree Q(2)-modules) from H^{∗}(BG_{2}; H_{ℤ/2}^{∗′}) onto

H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)} ⊕ (H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)}/Res_{Hℤ/2})(−1)[2]
≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{1, y} ⊕ ℤ/2[c_{7}] ⊗ Q(2){u_{3}})

where (−1)[2] is the degree shift operation so that deg(y) = (2, 2). Moreover d_{2}(u_{3}) = y in the coniveau spectral sequence.

Moreover if the Gottlieb transfer exists in the motivic cohomology, then the above epimorphism is indeed isomorphism. The similar fact also holds for G = Spin_{7} and p = 2.

Note that y in the above theorem, is a (mod(p)) Griffith element, namely,

y ∈ Ker(cycle map : CH^{∗}(BG)/p → H^{2∗}(BG; ℤ/p)).

Each non zero element in (H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)}/Res_{Hℤ/p}) of deg = (∗− 2, ∗ + 1) corresponds to a Griffith element of deg = (∗, ∗). So we can construct many Griffith elements in CH^{∗}(BG)/p for the above groups G.

An outline of this paper is following. In §2, we recall the relation between the motivic cohomology H^{∗,∗′}(X; ℤ/p) and the sheaf cohomology H^{∗}(X; H_{ℤ/p}^{∗′}). In §3, we show that H^{∗}(X; H_{ℤ/p}^{∗′}) has the Q_{i}-action. In §4, we recall the cohomological invariant and give a sufficient condition such that Res_{E_{2}} is epic when Inv^{∗}(G; ℤ/p) is known. In §5, we study the Dickson invariant for H^{∗}(BA; H_{ℤ/p}^{∗′}) using Q_{i} actions by Kameko-Mimura. In §6 − §8, we compute H^{∗}(BG; H_{ℤ/p}^{∗′}) for concrete cases, e.g., (G_{2}, p = 2) is studied in §6. In §9, we study the relation between H^{∗}(BG; H_{ℤ/p}^{∗′}) and the Brown-Peterson theory BP^{∗}(BG). In the last section, we study the image of Griffith elements to BP^{∗}(BG)⊗_{BP^{∗}} ℤ/p, in particular, for (Spin_{9}, p = 2).

2. motivic cohomology

Let X be a smooth (quasi projective) variety over a field k ⊂ ℂ. Let H^{∗,∗′}(X; ℤ/p) be the mod(p) motivic cohomology defined by Voevodsky and Suslin ([Vo1-3]).

Recall that the (mod p) B(n, p) condition holds if

H^{m,n}(X; ℤ/p) ≅ H_{et}^{m}(X; μ_{p}^{⊗n}) for all m ≤ n.

Recently M.Rost and V.Voevodsky ([Vo5],[Su-Jo],[Ro]) proved that B(n, p) condition holds for each p and n. Hence the Bloch-Kato conjecture also holds. Therefore in this paper, we always assume the B(n, p)-condition and also the Bloch-Kato conjecture for all n, p.

Moreover we always assume that k contains a primitive p-th root of unity. For these cases, we see the isomorphism H_{et}^{m}(X; μ_{p}^{⊗n}) ≅ H_{et}^{m}(X; ℤ/p). Let τ be a generator of H^{0,1}(Spec(k); ℤ/p) ≅ ℤ/p, so that

colim_{i}τ^{i}H^{∗,∗′}(X; ℤ/p) ≅ H_{et}^{∗}(X; ℤ/p).

Let H^{∗}(X; H_{ℤ/p}^{∗′}) be the sheaf cohomology where H_{ℤ/p}^{n} is the Zarisky sheaf induced from the presheaf H_{et}^{n}(V; ℤ/p) for open subset V of X.

Let X = ⋃ U_{λ} for Zarisky open sets U_{λ}. The sheaf cohomology H^{∗}(X; H_{ℤ/p}^{∗′}) is defined as the colimit of the cohomology of the following Čeck complex

(2.1) → ∏ Γ_{(i_{1},...,i_{n})} →^{δ} ∏ Γ_{(j_{1},...,j_{n+1})} →

where Γ_{(i_{1},...,i_{n})} = Γ(U_{i_{1}} ∩ ... ∩ U_{i_{n}}; H^{∗}(U_{i_{1}} ∩ ... ∩ U_{i_{n}}; ℤ/p)^{a}) and where H^{∗}(−; ℤ/p)^{a} is a sheaficication of the presheaf H^{∗}(−; ℤ/p). Here δ is induced map from the inclusions U_{i} ∩ U_{j} ⊂ U_{i}, U_{i} ∩ U_{j} ⊂ U_{j}.

The Beilinson and Lichtenbaum conjecture ( hence B(n, p)-condition ) (see [Vo2,5]) implies the exact sequences of cohomology theories

Theorem 2.1. ([Or-Vi-Vo], [Vo5]) There is a long exact sequence

→ H^{m,n−1}(X; ℤ/p) →^{×τ} H^{m,n}(X; ℤ/p)

→ H^{m−n}(X; H_{ℤ/p}^{n}) → H^{m+1,n−1}(X; ℤ/p) →^{×τ} .

In particular, we have Corollary 2.2. We have the additive isomorphism

H^{m−n}(X; H_{ℤ/p}^{n}) ≅ H^{m,n}(X; ℤ/p)/(τ) ⊕ Ker(τ)|H^{m+1,n−1}(X; ℤ/p)

where H^{m,n}(X; ℤ/p)/(τ) = H^{m,n}(X; ℤ/p)/(τH^{m,n−1}(X; ℤ/p)).

Note that the long exact sequence in Theorem 2.1 induces the τ-Bockstein spectral sequence

E(τ)_{1} = H^{m−n}(X; H_{ℤ/p}^{n}) ⟹ colim_{i}τ^{i}H^{∗,∗′}(X; ℤ/p) ≅ H_{et}^{∗}(X; ℤ/p).

On the other hand, the filtration coniveau is given by

N^{c}H_{et}^{m}(X; ℤ/p) = ∪_{Z}Ker{H_{et}^{m}(X; ℤ/p) → H_{et}^{m}(X − Z; ℤ/p)}

where Z runs in the set of closed subschemes of X of codim = c. The induced spectral sequence is called the coniveau spectral sequence. Bloch-Ogus [Bl-Og] proved that its E_{2}-term is given by

E(c)_{2}^{c,m−c} ≅ H^{c}(X, H_{ℤ/p}^{m−c}).

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

Theorem 2.3. (Deligne, Parajape) There is the isomorphism E(c)_{r}^{c,m−c} ≅ E(τ)_{r−1}^{m,m−c} for r ≥ 2 of spectral sequences. Hence the filtrations are the same N^{c}H_{et}^{m}(X; ℤ/p) = F_{τ}^{m,m−c} where

F_{τ}^{m,m−c} = Im(×τ^{c} : H^{m.m−c}(X; ℤ/p) → H^{m,m}(X; ℤ/p)).

3. cohomology operation

Let t_{ℂ} : H^{∗,∗′}(X; ℤ/p) → H^{∗}(X(ℂ); ℤ/p) be the realization map ([Vo1]) for the inclusion k ⊂ ℂ. The motivic cohomology has (Bockstein, reduced powered) cohomology operations ([Vo2,4])

β : H^{∗,∗′}(X; ℤ/p) → H^{∗+1,∗′}(X; ℤ/p)
P^{i} : H^{∗,∗′}(X; ℤ/p) → H^{∗+2i(p−1),∗′+i(p−1)}(X; ℤ/p)

which are compatible with the usual (topological) cohomology operations by the realization map t_{ℂ}. Voevodsky defines the Milnor operation Q_{i} also in the mod p motivic cohomology

Q_{i} : H^{∗,∗′}(−; ℤ/p) → H^{∗+2p^{i}−1,∗′+p^{i}−1}(−; ℤ/p).

Here we define the weight degree by

w(x) = 2n − m (resp. = n′ − m′)

for 0 ≠ x ∈ H^{m,n}(X; ℤ/p) (resp. H^{m′}(X; H_{ℤ/p}^{n′})). Similarly, we also define the weight degree for cohomology operations and differentials of spectral sequences,e.g.,

w(τ) = 2, w(P^{i}) = 0, w(Q_{j}) = −1.

Let ρ_{p} = (ξ_{p}) ∈ k^{∗}/(k^{∗})^{p} = H^{1,1}(Spec(k); ℤ/p) where ξ_{p} is the primitive p-th root of unity. The Q_{i} operation has the same property as the topological case only with mod(ρ_{2}). For example, Q_{i} is a derivative only mod(ρ_{2}).

Let A_{p} be the mod p Steenrod algebra generated by all cohomology operations on H^{∗,∗′}(X; ℤ/p). ( Voevodsky proved that A_{p} is multiplicatively generated by elements in H^{∗,∗′}(Spec(k); ℤ/p), P^{j} and Q_{i}.)

Lemma 3.1. Suppose ρ_{p} = 0. Then the Steenrod algebra A_{p} acts on the etale cohomolgy H^{∗}(X; ℤ/p).

Proof. In H^{∗,∗′}(Spec(k); ℤ/p), we know

P^{i}(τ) = 0 for i > 0, and β(τ) = ρ_{p} = 0.

When p ≥ 3, the Cartan formula holds in the motivic cohomology (Proposition 9.6 in [Vo4]), and we have

P^{i}(τx) = τP^{i}(x) for i > 0, and β(τx) = τβ(x).

From the B(n, p) condition, H_{et}^{∗}(X; ℤ/p) = colim_{i}τ^{i}H^{∗,∗′}(X; ℤ/p), which implies the lemma.

For p = 2, we also know from Proposition 9.6 in [Vo4],

Sq^{2∗}(xy) = ∑_{i} Sq^{2i}(x)Sq^{2∗−2i}(y) + τ ∑_{i} Sq^{2i+1}(x)Sq^{2∗−2i−1}(y),

Sq^{2∗+1}(xy) = ∑_{j} Sq^{j}(x)Sq^{2∗+1−j}(y) + ρ_{2} ∑_{i} Sq^{2i+1}(x)Sq^{2∗−2i−1}(y).

Since ρ_{2} = 0, we see Sq^{2i+1}(τ) = 0, and so Sq^{∗}(τx) = τSq^{∗}(x). This also induces the lemma. □

Theorem 3.2. Suppose ρ_{p} = 0. Then the cohomology operation Q_{i} and P^{i} can be extended on the τ-Bockstein spectral sequence and so on the coniveau spectral sequence E_{r}, r ≥ 2 (e.g., on H^{∗}(X; H_{ℤ/p}^{∗′})).

Proof. In the stable 𝔸^{1}-homotopy category SHot, let Hℤ/p be the Eilenberg-MacLane spectrum representing the mod p motivic cohomology

H^{∗,∗′}(X; ℤ/p) ≅ Hom_{SHot}(X, S^{∗,∗′} ∧ Hℤ/p)

where S^{∗,∗′} is the sphere of bidegree (∗, ∗′).

Let op. = Q_{i} or P^{i} of bidegree (m, n). Consider the diagram

S^{0,1} ∧ Hℤ/p ⟶^{×τ}  Hℤ/p  ⟶^{ρ}  cone
op.↓  ^{op.}↓
S^{m,n+1} ∧ Hℤ/p ⟶^{×τ} S^{m,n} ∧ Hℤ/p ⟶^{ρ} S^{m,n} ∧ cone.

Here cone is the mapping cone of τ so that

H^{∗+∗′}(X; H_{ℤ/p}^{∗′}) ≅ Hom_{SHot}(X, S^{∗,∗′} ∧ cone).

In the above diagram, we see ρ · op. · τ = 0. Hence there is a map op′. : cone → S^{m,n} ∧ cone such that ρ · op. = op′. · ρ. □

Here we do not see yet that A_{p} acts on E_{r}, e.g., we do not see that Q_{i} generates the exterior algebra Q(∞). However when r = 2, the following theorem holds.

Lemma 3.3. Let k be an algebraically closed field. Then the Steenrod algebra A_{p} acts on H^{∗}(X; H_{ℤ/p)}^{∗′}).

Proof. Recall that H^{∗}(X; H_{ℤ/p}^{∗′}) is defined as the cohomology of the Čeck complex. Given op. ∈ A_{p}, by the universality of sheaficication, the following diagram from (2.1) is commutative

∏ Γ_{(i_{1},...,i_{n})} ⟶^{δ} ∏ Γ_{(j_{1},...,j_{n+1})}
op.↓  ^{op.}↓
∏ Γ_{(i_{1},...,i_{n})} ⟶^{δ} ∏ Γ_{(j_{1},...,j_{n+1})}.

Thus we have the desired result. □

Let us write H^{∗,∗′} = H^{∗,∗′}(Spec(k); ℤ/p) and H^{∗} = K_{M}^{∗}(k)/p so that H^{∗,∗′} ≅ H^{∗}[τ]. (Note if k is algebraically closed, H^{∗,∗′} ≅ ℤ/p[τ].) For an elementary abelian p-group A = A_{n} ≅ (ℤ/p)^{n}, the mod(p) motivic cohomology is given by Voevodsky ([Vo2,4])

H^{∗,∗′}(BA; ℤ/p) ≅ H^{∗,∗′}[y_{1}, ..., y_{n}] ⊗ Δ(x_{1}, .., x_{n})

with x_{i}^{2} = yτ + xρ_{2} for p = 2 and x_{i}^{2} = 0 otherwise.

Since Ker(τ)|H^{∗,∗′}(BA; ℤ/p) = 0, from Corollary 2.2, we have

H^{∗}(BA; H_{ℤ/p}^{∗′}) ≅ H^{∗,∗}(BA; ℤ/p)/(τH^{∗,∗−1}(BA; ℤ/p))
≅ H^{∗}[y_{1}, ..., y_{n}] ⊗ Λ(x_{1}, ..., x_{n}) (mod(ρ_{2}))

for all primes p. Each Q_{i} is a derivation mod(ρ_{2}), and hence

Q_{0}...Q_{s−1}(x_{1}...x_{s}) = ∑ sgn(j_{1}, ..., j_{s})y_{1}^{p^{j_{1}}}y_{2}^{p^{j_{2}}}...y_{s}^{p^{js}} ≠ 0 mod(ρ_{2}) where (j_{1}, ..., j_{s}) are permutations of (0, ..., s − 1).

Let us write

Q(n) = Λ(Q_{0}, ..., Q_{n}),

Q̄(n) = Q(n) − ℤ/p{Q_{0}...Q_{n}} = ℤ/p{Q_{i_{0}}...Q_{i_{s}}|0 ≤ i_{k} ≤ n, s < n}. Let u_{i} = x_{1}...x_{i} ∈ H^{0}(BA; H_{ℤ/p}^{i}). For example, we have

H^{∗}(BA; H_{ℤ/p}^{∗′}) ⊃ H^{∗}[y_{1}, ..., y_{n}] ⊗ (⊕_{i}Q̄(i − 1){u_{i}})
⊃ ⊕_{i}H^{∗}[y_{1}, ..., y_{i}] ⊗ Q(i − 1){u_{i}} (∗)

since Q_{0}...Q_{i−1}(u_{i}) ∈ H^{∗}[y_{1}, ..., y_{i}]{y_{1}...y_{i}}. In sections bellow, we show that the last sum (∗) of free Q(i−1)-modules contains H^{∗}(BG; H_{ℤ/p}^{∗′})^{W_{G}(A)}, as a direct summand , for many cases of G. (See Assumption (1) in §4.)

4. cohomological invariant

Let G be a linear algebraic group over k. Recall that H^{1}(k; G) is the first non abelian Galois cohomology set of G, which represents the set of G-torsors over k. The cohomology invariant is defined by

Inv^{i}(G, ℤ/p) = F unc(H^{1}(F; G) → H^{i}(F; ℤ/p))

where F unc means natural functions for each field F which is finitely generated over k. (For details for the definition or properties, see the book [Ga-Me-Se].)

Totaro proved [Ga-Me-Se] the following theorem in the letter to Serre.

Theorem 4.1. (Totaro) Inv^{∗}(G; ℤ/p) ≅ H^{0}(BG; H_{ℤ/p}^{∗}).

Hereafter (throughout this paper), we assume that k is an algebraically closed field in ℂ. Moreover, in this paper, we only consider simple simply connected groups G which have the following property. we assume that the algebraic group G has only one conjugacy class A of non toral maximal elementary abelian p-subgroups. Exceptional groups are

G = \begin{cases}G_{2}, F_{4}, E_{6} for p = 2 \\ F_{4}, E_{6}, E_{7} for p = 3 \\ E_{8} for p = 5. \end{cases}

For spin groups Spin_{n}, we consider the cases n ≤ 9 only in this paper.

We consider the restriction maps (of cohomology) to A and the maximal torus T_{G}

Res_{Hℤ/p} : H^{∗}(BG; ℤ/p) →^{i^{∗}} H^{∗}(BT_{G}; ℤ/p) × H^{∗}(BA; ℤ/p)

→^{pr.} H^{∗}(BA; ℤ/p)^{W_{G}(A)}.

By the Quillen’s theorem the above i^{∗} has nilpotent kernel. More strongly, Toda, Kono, Tezuka and Kameko show i^{∗} is really injective, namely, H^{∗}(BG; ℤ/p) is detected by A and T_{G}. Moreover when p = 2, Res_{Hℤ/2} are isomorphic except the case E_{6}. However Res_{Hℤ/p} is not epic for p ≥ 3.

On the other hand, by Serre, Rost and Garibaldi([Ga-Me-Se],[Ga]), Inv^{∗}(G; ℤ/p) are computed for these groups, e.g.,

Inv^{∗}(G; ℤ/p) ≅ \begin{cases}ℤ/p\{1, u_{3}\} for (G_{2}, E_{6}, p = 2), (F_{4}, E_{7}, p = 3), \\ (E_{8}, p = 5) \\ ℤ/p\{1, u_{3}, u_{4}\} for (Spin_{7}, p = 2), (E_{6}, p = 3) \\ ℤ/p\{1, u_{3}, u_{4}, u_{5}\} for (Spin_{9}, p = 2) \\ ℤ/p\{1, u_{3}, u_{4}, u′_{4}, u_{5}\} for (Spin_{8}, p = 2) \\ ℤ/p\{1, u_{3}, u_{5}\} for (F_{4}, p = 2). \end{cases}

(Moreover Rost and Garibaldi determined Inv^{∗}(Spin_{n}; ℤ/2) for n ≤ 12).

For these groups, we note (Ga-Me-Se],[Ga]) the the restriction

Res_{Inv} : Inv^{∗}(G; ℤ/p) → Inv^{∗}(A; ℤ/p) ≅ Λ(x_{1}, ..., x_{n}).

is injective (identifying u_{i} = x_{1}...x_{i} and u′_{4} = x_{1}x_{2}x_{3}x_{5}). We will show the following theorem in §6 − 8 bellow (by computations of concrete cases)

Theorem 4.2. Let G be an above type except for G = E_{6} and p = 2. Then

Res_{Inv} : Inv^{∗}(G; ℤ/p) ≅ Inv^{∗}(A; ℤ/2)^{W_{G}(A)}.

Remark. When G = E_{6} and p = 2, the above Res_{Inv} is not epic. We want to extend above isomorphism in the theorem to say that

Res_{E_{2}} : H^{∗}(BG; H_{ℤ/p}^{∗′}) → H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)}

is an epimorphism. (Of course for p ≥ 3 the above map is not injective.) We will prove the following assumption (in the sections bellow) for the above groups except for (E_{6}, p = 2) and (E_{7}, p = 3). (When G = Spin_{8}, some modification of Assumption (1) holds.)

Assumption When Inv^{∗}(G; ℤ/p) ≅ ℤ/p{1, u_{i_{1}}, ..., u_{i_{m}}}, there is a bidegree isomorphism

(1) H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)} ≅ ⊕_{s=1}^{m}ℤ/p[f_{s1}, ..., f_{sk_{s}}] ⊗ Q(i_{s} − 1){u_{i_{s}}}

(2) f_{st} ∈ Res_{Hℤ/p}(H^{2∗,∗}(BG; ℤ/p)) = Res(CH^{∗}(BG)/p)

for all 1 ≤ s ≤ m, 1 ≤ t ≤ k_{s}.

If this assumption is satisfied then H^{∗}(BA; H^{∗′})^{W_{G}(A)} is generated as bidegree Q(∞)-algebra by u_{i_{s}} and Res(CH^{∗}(BG)/p). Hence the surjectivity of Res_{E_{2}} is immediate.

Lemma 4.3. If Assumption (1),(2) are satisfied, then

Res_{E_{2}} : H^{∗}(BG; H_{ℤ/p}^{∗′}) → H^{∗}(BA; H_{ℤ/p}^{∗′})^{W_{G}(A)}

is an epimorphism.

Thus we can prove Theorem 1.1 in the introduction. As for the statements of differential and (Griffith elements), the following lemma is useful.

Lemma 4.4. Let Res_{Inv}(a) ≠ 0 for a ∈ Inv^{i}(G; ℤ/p) = H^{0}(BG; H_{ℤ/p}^{∗}). (Namely, the above element is a permanent cycle in the coniveau spectral sequence.) Moreover let

Q_{j_{1}}...Q_{j_{i−3}}(a) ∉ H^{∗}(BG; ℤ/p).

Then d_{2}(a) = y ≠ 0 ∈ H^{2}(BG; H_{ℤ/p}^{i−1}) in the coniveau spectral sequence, and elements

Q_{j_{1}}...Q_{j_{i−3}}(y) ≠ 0 ∈ CH^{∗}(BG)/p = H^{2∗,∗}(BG; ℤ/p)

are Griffith elements (i.e., in the kernel of CH^{2∗}(BG)/p → H_{et}^{2∗}(BG; ℤ/p)).

Proof. Take q = Q_{j_{1}}...Q_{j_{i−3}}(a). Since q does not exist in H^{∗}(BG; ℤ/p), we see d_{r}(q) ≠ 0 in the spectral sequence for some r.

This r = 2 because the following reason of weight degree. First note

w(d_{r}) = wt(1, 1 − r) = 2(1 − r) − 1 = 1 − 2r.

Since w(q) = w(a) − (i − 3) = 3, we have

w(d_{r}(q)) = 3 + 1 − 2r = 4 − 2r.

If r ≥ 3, then the above weight is negative and d_{r}(q) = 0.

This implies that d_{2}(a) ≠ 0. Otherwise

d_{2}(q) = d_{2}(Q_{j_{1}}...Q_{j_{i−3}}a) = Q_{j_{1}}...Q_{j_{n−3}}(d_{2}(a)) = 0,

which is a contradiction. □

5. Dickson invariant

At first we assume p ≥ 3. Dickson computed the ring of invariants of ℤ/p[y_{1}, ..., y_{n}] with respect to the action of GL_{n}(ℤ/p). The ring of invariants is a polynomial algebra

D_{n} = ℤ/p[y_{1}, ..., y_{n}]^{GL_{n}(ℤ/p)} ≅ ℤ/p[c_{n,0}, ..., c_{n,n−1}]

where the generators are given by the equation

O_{n}(X) = ∏_{y∈ℤ/p{y_{1},··· ,y_{n}}}(X + y) = X^{p^{n}} + ∑_{j=0}^{n−1}(−1)^{n−j}c_{n,j}X^{p^{j}}.

Let reg : A → GL_{n}(ℂ) be the regular representation and c(reg) the total Chern class. Then it is well known that

c(reg) = O_{n}(1) = 1 − c_{n,n−1} + ... + (−1)^{n}c_{n,0}.

We also note the following lemma.

Lemma 5.1. (Lemma 2.3,2.4 in [Ka-Ya2]) Let ρ : A_{n} → GL_{m}(ℂ) be an representation such that c(ρ) ∈ H^{∗}(BA; ℤ/p)^{SL_{n}(ℤ/p)}. Then c(ρ) = c(reg)^{a} for some a ≥ 0.

For the invariant ring SD_{n} under SL_{n}(ℤ/p), we have

SD_{n} = ℤ/p[y_{1}, ..., y_{n}]^{SL_{n}(ℤ/p)}
≅ D_{n}{1, e_{n}, ..., e_{n}^{p−2}} with e_{n}^{p−1} = c_{n,0}
≅ D′_{n} ⊗ ℤ/p[e_{n}] with D′_{n} = ℤ/p[c_{n,1}, ..., c_{n,n−1}].

Mui computed the ring of invariants of

H^{∗}(BA; H_{ℤ/p}^{∗′}) ≅ ℤ/p[y_{1}, ..., y_{n}] ⊗ Λ(x_{1}, ...x_{n})

with respect to the action of SL_{n}(ℤ/p). (In fact, Mui studied H^{∗}(BA; ℤ/p) for odd prime p, however we study H^{∗}(BA; H_{ℤ/p}^{∗′}) for all primes.) Of course u_{n} = x_{1}...x_{n} is invariant under SL_{n}(ℤ/p). In terms of Milnor’s operation, we may state Mui’s result in the following form.

Theorem 5.2. (Mui[Mu], Kameko-Mimura [Ka-Mi])

H^{∗}(BA; H_{ℤ/p}^{∗′})^{SL_{n}(ℤ/p)} ≅ ℤ/p[e_{n}, c_{n,1}, ..., c_{n,n−1}]⊗(ℤ/p{1}⊕Q̄(n−1){u_{n}})

≅ D′_{n} ⊕ SD_{n} ⊗ Q(n − 1){u_{n}}

where Q_{0}...Q_{n−1}u_{n} = e_{n}.

The Q_{i}-operation acts on u_{n} as follows.

Lemma 5.3. (Kameko-Mimura [Ka-Mi],[Ka-Ya1]) For x ∈ H^{∗}(BA; H_{ℤ/p}^{∗′}), it holds

(Q_{n} + ∑_{i=0}^{n−1}(−1)^{n−i}c_{n,i}Q_{i})(x) = 0, Q_{0}...Q̂_{i}...Q_{n}(u_{n}) = c_{n,i}e_{n}.

Let U_{n} ⊂ SL_{n}(ℤ/p) be the maximal unipotent subgroup generated by upper triangular matrices with diagonals 1, so that U_{n} is a Sylow p-subgroup of SL_{n}(ℤ/p). The invariant under this group is given by Mui, Kameko-Mimura.

Theorem 5.4. (Kameko-Mimura Theorem 4.2 in [Ka-Mi]) Let G′ ⊂ GL_{n}(ℤ/p) such that ℤ/p[y_{1}, ..., y_{n}]^{G′} ≅ ℤ/p[f_{1}, ..., f_{n}] and

H^{∗}(BA_{n}; H_{ℤ/p}^{∗′})^{G′} ≅ ℤ/p[f_{1}, ..., f_{n}]{v_{1} = 1, ..., v_{2^{n}}}.

Then the invariant under G = ⟨G′, U_{n+1}⟩ ⊂ GL_{n+1}(ℤ/p) is given by

(1) ℤ/p[y_{1}, ..., y_{n+1}]^{G} ≅ ℤ/p[f_{1}, ..., f_{n}, O_{n}(y_{n+1})]
(2) H^{∗}(BA_{n+1}; H_{ℤ/p}^{∗′})^{G} ≅

ℤ/p[f_{1}, ..., f_{n}, O_{n}(y_{n+1})] ⊗ (ℤ/p{v_{1}, ..., v_{2^{n}}} ⊕ Q(n − 1){u_{n+1}}).

Corollary 5.5.

H^{∗}(BA; H_{ℤ/p}^{∗′})^{U_{n}} ≅ ⊕_{i=0}^{n}ℤ/p[O_{0}(y_{1}), ..., O_{i−1}(y_{i})] ⊗ Q(i − 1){u_{i}}.

Corollary 5.6. (Lemma 5.8 in [Ka-Ya])

O(y_{n+1})u_{n} = (Q_{n} + ∑_{i=0}^{n−1}(−1)^{n−i}c_{n,i}Q_{i})(u_{n+1}).

Hereafter this section, we assume p = 2. Of course we have the isomorphism H^{∗}(BA; ℤ/2) ≅ ℤ/2[x_{1}, ..., x_{n}] and its invariant under GL_{n}(ℤ/2) = SL_{n}(ℤ/2) is

ℤ/2[x_{1}, ..., x_{n}]^{GL_{n}(ℤ/2)} ≅ ℤ/2[d_{n,0}, ..., d_{n,n−1}]

where the generators are given by the equation

O′_{n}(X) = ∏_{x∈ℤ/p{x_{1},··· ,x_{n}}}(X + x) = X^{2^{n}} + ∑_{j=0}^{n−1} d_{n,j}X^{2^{j}}.

Here d_{n,i}^{2} = c_{n,i} in H^{∗}(BA; ℤ/2) identifying y_{i} = x_{i}^{2}. The Milnor Q_{i}- operations (see (2.6) in Schuster-Yagita [Sc-Ya]) are given as the case p odd. (Hereafter let us write d_{n,i} by d_{i} simply.)

d_{0} = Q_{0}...Q_{n−2}(u_{n}), d_{i} = Q_{0}...Q̂_{i−1}...Q_{n−1}(u_{n})/d_{0}.

From Lemma 2.1 in [Sc-Ya], we have

(∗) Q_{n−1}(d_{i}) = d_{0}d_{i}, Q_{i−1}(d_{i}) = d_{0}.

In H^{∗}(BA; H_{ℤ/2}^{∗′}), we can get more strong result. Let us write simply

I(GL_{n}) = gr(H^{∗}(BA_{n}; ℤ/2)^{GL_{n}(ℤ/2)}) ⊂ H^{∗}(BA; H_{ℤ/2}^{∗′})
Igr(GL_{n}) = H^{∗}(BA_{n}; H_{ℤ/2}^{∗′})^{GL_{n}(ℤ/2)}.

By Kameko-Mimura theorem, we have showed

Igr(GL_{n}) ≅ D′_{n} ⊕ D_{n} ⊗ Q(n − 1){u_{n}})

where D_{n} = ℤ/2[c_{n,n−1}, ..., c_{n,0}] and D′_{n} = ℤ/2[c_{n,n−1}, ..., c_{n,1}].

Lemma 5.7. In H^{∗}(BA; H_{ℤ/2}^{∗′}), we have

d_{i} = Q_{0}...Q̂_{i−1}...Q̂_{n−1}(u_{n}), d_{i}d_{0} = Q_{0}...Q̂_{i−1}...Q_{n−1}(u_{n}).
d_{i}d_{j} = Q_{0}...Q̂_{i−1}...Q̂_{j−1}...Q_{n−1}(u_{n}) i ≠ j.

Proof. Consider the element

a_{i} = Q_{0}...Q̂_{i−1}...Q_{n−2}Q̂_{n−1}(u_{n}) ∈ Igr(GL_{n}).

Then Q_{i−1}(a_{i}) = d_{0} and Q_{n−1}a_{i} = d_{0}d_{i}.

Using property (∗), we see Q_{j}(a_{i} − d_{i}) = 0 for all j. Of course a_{i} − d_{i} ∈ Igr(GL_{n}). From Kameko-Mimura theorem, we still know

Igr(GL_{n}) ∩ ∩_{j}Ker(Q_{j}) = D_{n}.

This means d_{i} = a_{i} ∈ Igr(GL_{n}). (In fact d_{i} = a_{i} mod(D_{n}) in H^{∗}(BA_{n}; ℤ/2), but d_{i} = a_{i} exactly in H^{∗}(BA_{n}; H_{ℤ/2}^{∗′}).)

Therefore we have

d_{i} = Q_{0}...Q̂_{i−1}...Q̂_{n−1}(u_{n}), d_{i}d_{0} = Q_{0}...Q̂_{i−1}...Q_{n−1}(u_{n}).

By the similar arguments, we have

d_{i}d_{j} = Q_{0}...Q̂_{i−1}...Q̂_{j−1}...Q_{n−1}(u_{n}).
□

Of course I(GL_{n}) ⊂ Igr(GL_{n}), but this injection is not an isomorphism for n ≥ 3.

Lemma 5.8. Let n ≥ 3. Then we have

Igr(GL_{n})/I(GL_{n}) ⊃ Q(n − 1)/(Q(n − 1)^{+})^{n−2}{u_{n}}.

Proof. Consider the element

x = Q_{0}...Q̂_{i−1}...Q̂_{j−1}...Q̂_{k−1}...Q_{n−1}(u_{n}).

Its image Q_{i−1}(x) = d_{j}d_{k} or d_{j} (when k = n.) Hence x is not in I(GL_{n}) because Q_{i} maps n-product elements into also n-product elements. (If x ∈ I(GL_{n}), then x must be a sum of d_{i} or d_{i}d_{j}, but it still appeared in (Q(n − 1)^{+})^{n−2}(u_{n}).) Thus we get the lemma. □

Let A = A_{n} be a maximal elementary abelian p-subgroup of G and W_{G}(A) its Weyl group.

Lemma 5.9. If Res_{Inv} : Inv^{∗}(G; ℤ/p) → Inv^{∗}(A_{n}; ℤ/p)^{W_{G}(A)} is an epimorphism, then

(Q(n − 1)^{+})^{n−2}{u_{n}} ⊂ Res_{H_{ℤ/p}}(H^{∗}(BG; ℤ/p) → H^{∗}(BA_{n}; ℤ/p))

(e.g., d_{1}, ..., d_{n−1} for p = 2 are in Res_{H_{ℤ/2}}).

Proof. Let x be an element in H^{∗}(BG; H_{ℤ/p}^{∗′}) with

Res_{E_{2}}(x) = Q_{i_{1}}...Q_{i_{n−2}}(u_{n}).

Then w(x) = n − (n − 2) = 2. However the weight of the differential d_{r} of the coniveau spectral sequence is w(d_{r}) = 1 − 2r (see the proof of Lemma 4.4). Hence d_{r}(x) = 0 for r ≥ 2, namely, x is a permanent cycle and is an element in H^{∗}(BG; ℤ/p). □

6. W_{G}(A) ≅ SL_{3}(ℤ/p)

We consider the cases of A ≅ (ℤ/p)^{3} and W_{G}(A) ≅ SL_{3}(ℤ/p), namely, (G_{2}, 2), (F_{4}, 3) and (E_{8}, 5). These cases

Inv^{∗}(G; ℤ/p) ≅ ℤ/p{1, u_{3}}.

(These u_{3} are called the Rost invariants.)

Let G = G_{2} and p = 2. It is well known that

H^{∗}(BG; ℤ/2) ≅ I(GL_{3}) ≅ ℤ/2[w_{4}, w_{6}, w_{7}]

where w_{i} is the Stiefel-Whitney class of G_{2} ⊂ SO_{7}. We can identify

w_{4} = d_{3,2}, w_{6} = d_{3,1}, w_{7} = d_{3,0}.

On the other hand, Kameko-Mimura theorem implies

Igr(GL_{3}) ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{1} ⊕ ℤ/2[c_{7}] ⊗ Q(2){u_{3}}).

Also by using Lemma 5.8, we can show

Igr(GL_{3})/I(GL_{3}) ≅ ℤ/2[c_{4}, c_{6}]{u_{3}}.

In fact (from Lemma 5.7 or from dimensional reason), we have

Q_{0}(u_{3}) = w_{4}, Q_{1}(u_{3}) = w_{6}, Q_{0}Q_{1}(u_{3}) = w_{7}, Q_{2}(u_{3}) = w_{4}w_{6}.

Moreover we note

c_{7}u_{3} = w_{4}w_{6}w_{7}

because both above elements are same after acting Q_{i}, e.g. Q_{0}(c_{7}u_{3}) = c_{7}w_{4} = Q_{0}(w_{4}w_{6}w_{7}) (see the proof of Lemma 5.7 or see [Ya2]).

Therefore Assumption (1),(2) are satisfied. Moreover from Lemma 4.4, we see d_{2}(u_{3}) = y ≠ 0. Therefore we have the following theorem.

Theorem 6.1. There is a Q(2) bidegree module epimorphism from H^{∗}(BG_{2}; H_{ℤ/2}^{∗′}) to

Igr(GL_{3})⊕ℤ/2[c_{4}, c_{6}]{y} ≅ ℤ/2[c_{4}, c_{6}]⊗(ℤ/2{1, y}⊕ℤ/2[c_{7}]⊗Q(2){u_{3}}).

Remark. If there is a Gottlib transfer in the motivic theory H^{∗,∗′}(−; ℤ/2) or the sheaf theory H^{∗}(−; H_{ℤ/2}^{∗}), then the above epimorphism is in fact an isomorphism.

Next we consider the odd prime cases i.e., (G, p) = (F_{4}, 3) or (E_{8}, 5). From Kameko-Mimura theorem, we also have

Igr(SL_{3}) ≅ D′_{3} ⊕ SD_{3} ⊗ Q(2){u_{3}}.

Moreover from Kameko (Lemma 5.2 in [Ka-Ya1]), it is known that

Igr(SL_{3})/Res_{Hℤ/p} ≅ SD_{3}/(e){u_{3}}

as the case (G_{2}, 2). Hence Assumption (1) satisfied and Lemma 4.4 can be applied so that d_{2}(u_{3}) = y.

To see Assumption (2), we consider the representations. We consider the case (E_{8}, 5). (The case (F_{4}, 3) is similar.) It is known that there is a non trivial representation ([Ad], [Ka-Ya2])

ρ : E_{8} → SO(248).

We consider the total Chern class of the representation ρ|A for A ≅ (ℤ/5)^{3},

c(ρ|A) = (1 − c_{3,2} + c_{3,1} − c_{3,0})^{a} for a ≥ 0

from Lemma 5.1. Since ρ|A is non trivial, a ≥ 1. Moreover

|c_{3,0}| = 2(5^{3} − 1) = 248.

So a = 1. This means that c_{3,i} are represented by Chern classes. (We also note c_{3,1} = P^{1}c_{3,2} for the reduced power operation P^{1}.) Hence w(c_{3,1}) = w(c_{3,2}) = 0. Thus we can see Assumption (2).

Theorem 6.2. Let (G, p) = (F_{4}, 3) or (E_{8}, 5). Then there is an epimorphism of Q(2)-bidegree modules from H^{∗}(BG; H_{ℤ/p}^{∗′}) to

ℤ/p[c_{3,2}, c_{3,1}] ⊗ (ℤ/p{1, y} ⊕ ℤ/p[e_{3}] ⊗ Q(2){u_{3}}).

7. W_{G}(A) ≅ ⟨U_{4}, SL_{3}(ℤ/p)⟩

We consider the cases of A ≅ (ℤ/p)^{4} and

W_{4} = W_{G}(A) ≅ ⟨U_{4}, SL_{3}(ℤ/p)⟩,

namely, (Spin_{7}, 2), (E_{6}, 3). For these cases, we have the isomorphism

Inv^{∗}(G; ℤ/p) ≅ ℤ/p{1, u_{3}, u_{4}}.

We also study the case (E_{7}, 3), while the above facts do not satisfied. Let G = Spin_{7} and p = 2. It is well known that

H^{∗}(BG; ℤ/2) ≅ I(W_{4}) = ℤ/2[x_{1}, ..., x_{4}]^{W_{4}}

≅ ℤ/2[w_{4}, w_{6}, w_{7}, w_{8}]

where w_{8} is the Stiefel-Whitney class of some spin representation. We can identify w_{8} = O′_{3}(x_{4}).

Lemma 7.1.

Igr(W_{4}) ≅ Igr(GL_{3})⊕ℤ/2[c_{4}, c_{6}]⊗(ℤ/2[c_{8}]{c_{8}}⊕ℤ/2[c_{7}, c_{8}]⊗Q(3){u_{4}}).

Proof. Recall that using Q̄(2), we have (Q_{2}Q_{1}Q_{0}(u_{3}) = c_{7})

Igr(GL_{3}) ≅ ℤ/2[c_{4}, c_{6}, c_{7}] ⊗ (ℤ/2{1} ⊕ Q̄(2){u_{3}}).

By Theorem 5.4, we have

Igr(W_{4}) ≅ ℤ/2[c_{4}, c_{6}, c_{7}, c_{8}] ⊗ (ℤ/2{1} ⊕ Q̄(2){u_{3}} ⊕ Q(2){u_{4}})

≅ ℤ/2[c_{4}, c_{6}]⊗(ℤ/2[c_{8}]{1}⊕ℤ/2[c_{7}]⊗Q(2){u_{3}}⊕ℤ/2[c_{7}, c_{8}]⊗Q(3){u_{4}}). The last isomorphism is shown by using the following facts. From Lemma 5.6, we can see (Lemma 5.8 in [Ka-Ya2])

O_{3}(y_{4})u_{3} = (Q_{3} + c_{3,2}Q_{2} + c_{3,1}Q_{1} + c_{3,0}Q_{0})(u_{4}),

namely, Q_{3}(u_{4}) = c_{8}u_{3} + c_{4}Q_{2}(u_{4}) + c_{6}Q_{1}(u_{4}) + c_{7}Q_{0}(u_{4}). Hence

Q(2){u_{4}} ⊕ Q(2){c_{8}u_{3}} ≅ Q(3){u_{4}}.

Using Q_{0}Q_{1}Q_{2}Q_{3}(u_{4}) = c_{7}c_{8}, we get the last isomorphism. □

Lemma 7.2.

Igr(W_{4})/I(W_{4}) ≅ Igr(GL_{3})/I(GL_{3})⊕
ℤ/2[c_{4}, c_{6}, c_{8}] ⊗ {1, Q_{0}, ..., Q_{3}}{u_{4}}.

Proof. At first, we see

Q_{1}Q_{0}(u_{4}) = Q_{1}Q_{0}(u_{3}x_{4}) = Q_{1}(w_{4}x_{4} + u_{3}y_{4})
= w_{7}x_{4} + w_{4}y_{4}^{2} + w_{6}y_{4} = w_{8} in H^{∗}(BA; H_{ℤ/2}^{∗′}).

(This fact also follows from d_{4,3} = w_{8} and Lemma 5.7.) Similarly, we can compute the Q_{i} action on u_{4}, which is given as Q_{0}Q_{1}(u_{4}) = w_{8}, Q_{0}Q_{2}(u_{4}) = w_{4}w_{8}, Q_{1}Q_{2}(u_{4}) = w_{6}w_{8}, Q_{0}Q_{3}(u_{4}) = c_{8}w_{4}, Q_{1}Q_{3}(u_{4}) = c_{8}w_{6}, Q_{2}Q_{3}(u_{4}) = c_{8}w_{4}w_{6}.

Moreover we have

c_{7}u_{4} = c_{7}u_{3}x_{4} = w_{4}w_{6}w_{7}x_{4} = w_{4}w_{6}w_{8}.

Therefore Q_{i}u_{3} ∉ I(W_{4}) but Q_{i}Q_{j}(u_{3}) ∈ I(W_{4}). Thus we have

Igr(W_{4})/I(W_{4}) ≅ ℤ/2[c_{4}, c_{6}]⊗(ℤ/2{u_{3}}⊕ℤ/2[c_{8}]{1, Q_{0}, ..., Q_{3}}{u_{4}}).

□

Therefore Assumption (1),(2) are satisfied. Therefore we can compute ; Theorem 7.3. There is a Q(2) bidegree module epimorphism from H^{∗}(BSpin_{7}; H_{ℤ/2}^{∗′}) to

H^{∗}(BG_{2}; H_{ℤ/2}^{∗′})⊕ℤ/2[c_{4}, c_{6}, c_{8}]⊗(ℤ/2{c_{8}, y′_{2}, Q_{0}y′_{2}, ..., Q_{3}y′_{2}}⊕ℤ/2[c_{7}]⊗Q(3){u_{4}})

where Q_{0}Q_{1}Q_{2}Q_{3}u_{4} = c_{7}c_{8}. The differentials d_{2}(u_{3}) = y_{2}, d_{2}(u_{4}) = y′_{2} in the coniveau spectral sequence.

Remark. If the epimorphism in Theorem 6.1 is an isomorphism, then that in the above theorem is also an isomorphism.

Remark. The notations in [Ya3] are given : a′ = u_{4} as a virtual element and

ξ_{3} = Q_{0}y′_{2}, ξ_{4} = Q_{1}y′_{2}, ξ_{6} = Q_{2}y′_{2}, c_{8}y_{2} = Q_{3}y′_{2}.

Next we consider the odd prime cases i.e., (G, p) = (E_{6}, 3). Let us denote by O simply, O_{3}(x_{4}) so that e_{4} = Q_{0}Q_{1}Q_{2}Q_{3}(u_{4}) = e_{3}O. Then from Kameko-Mimura lemma, we also have ([Ka-Mi],

Igr(W_{4}) ≅ SD/(e)[O] ⊕ SD_{3}[O] ⊗ Q(2){u_{3}, u_{4}}
≅ SD_{3}/(e)[O] ⊕ SD_{3} ⊗ Q(2){u_{3}} ⊕ SD_{3}[O] ⊗ Q(3){u_{4}}.

Moreover from Kameko (Lemma 5.2 in [Ka-Ya1]), it is known that

Igr(W_{4})/Res_{Hℤ/p} ≅ SD_{3}/(e)⊗(ℤ/3{u_{3}}⊕ℤ/3[O]{u_{4}, Q_{0}u_{4}, ..., Q_{3}u_{4}})

as the case (Spin_{7}, 2). Hence Assumption (1) satisfied and Lemma 4.4 can be applied so that d_{2}(u_{3}) = y.

To see Assumption (2), we consider the representations. It is known that there is a non trivial representation E_{6} → SO(26). Hence we know that c_{3,i} is represented by Chern classes by the arguments similar to the case (F_{4}, 3). As for the element O, we consider the restriction

⟨a_{4}⟩ ⊂ A ⊂ E_{6} →^{ρ} SO(26).

Here a_{4} ∈ A is the dual of x_{4} ∈ Hom(A; ℤ/3). We see O|⟨a_{4}⟩ ≠ 0 but SD_{3}|⟨a_{4}⟩ = 0. Hence the fact

c_{26}(ρ|⟨a_{4}⟩) = y_{4}^{3^{3}−1} ≠ 0

implies O (modulo elements in SD_{3}) can be represented by a Chern class. Hence Assumption (2) is also satisfied.

Theorem 7.4. There is an epimorphism of Q(3)-bimodules from H^{∗}(BE_{6}; H_{ℤ/3}^{∗′}) to

H^{∗}(BF_{4}; H_{ℤ/3}^{∗′}) ⊕ ℤ/3[c_{3,2}, c_{3,1}, O] ⊗ (ℤ/3{O, y′_{2}, Q_{0}y′_{2}, ..., Q_{3}y′_{2}}

⊕ℤ/2[e_{3}] ⊗ Q(3){u_{4}}).

At last of this section, we consider the case (E_{7}, 3). This case

W_{G}(A) = W′_{4} = ⟨W_{4}, diag(1, 1, 1, −1)⟩ ⊂ GL_{4}(ℤ/3).

The invariant is also computed by Kameko-Mimura

Igr(W′_{4}) ≅ SD/(e)[O^{2}] ⊕ SD_{3}[O^{2}] ⊗ Q(2){u_{3}, Ou_{4}}

Moreover from Kameko (page 2279 in [Ka-Ya1]), it is known that

Igr(W′_{4})/Res_{Hℤ/3} ≅ SD_{3}/(e)[O^{2}] ⊗ (ℤ/3{u_{3}} ⊕ Q(2){Ou_{4}}).

It is also known Inv^{∗}(E_{7}; ℤ/3) ≅ ℤ/3{1, u_{3}} and hence

Inv^{∗}(E_{7}; ℤ/3) ≅ Inv^{∗}(A; ℤ/3)^{W′_{4}}

from the above result.

There is the natural representation ρ : E_{7} → SO(52). Hence we see that O^{2} can be represented by a Chern class. So Assumption (2) is satisfied. However Assumption (1) is not.

Theorem 7.5. The following restriction map

Res_{E_{2}} : H^{∗}(BE_{7}, H_{ℤ/3}^{∗′}) → H^{∗}(BA, H_{ℤ/3}^{∗′})^{W′_{4}}

is not an epimorphism.

Proof. Recall arguments in the proof of Lemma 5.9. Suppose Ou_{4} ∈ Res_{E_{2}} and Res(x) = Ou_{4} for x ∈ H^{∗}(BE_{7}; H_{ℤ/3}^{∗′}).

Of course w(x) = w(Ou_{4}) = 4. The weight Q_{0}Q_{1}(x) = 2. Recall w(d_{r}) = 1 − 2r and

d_{r}(Q_{0}Q_{1}(x)) = 0 for r ≥ 2.

Hence Q_{0}Q_{1}(x) ∈ H^{∗}(BE_{3}; ℤ/3) from the coniveau spectral sequence. So Q_{0}Q_{1}(Ou_{4}) ∈ Res_{Hℤ/3}, which contradicts to the result above

Igr(W′_{4})/Res_{Hℤ/3} ⊃ Q(2){Ou_{4}}.
□

8. Spin_{9} for p = 2

In this section, we consider the groups Spin_{8}, Spin_{9}, F_{4}, E_{6} for p = 2. At first we consider the case G = Spin_{9}. Then the maximal elementary abelian 2-group is rank_{2} = 5, and the Weyl group is

W_{G}(A) = W_{5} ≅ ⟨U_{5}, SL_{3}⟩ ⊂ SL_{5}(ℤ/2).

The cohomology is known that

H^{∗}(BG; ℤ/2) ≅ H^{∗}(BA; ℤ/2)^{W_{G}(A)} ≅ ℤ/2[w_{4}, w_{6}, w_{7}, w_{8}, w_{16}]

where w_{16} is the Stiefel-Whitney class of some spin representation. We can also identify

w_{16} = O′_{4}(x_{5}) = x_{5}^{16} + d_{3}x_{5}^{8} + d_{2}x_{5}^{4} + d_{1}x_{5}^{2} + d_{0}x_{5}

where d_{3} = w_{8}, d_{2} = w_{8}w_{4}, d_{1} = w_{8}w_{6}, d_{0} = w_{8}w_{7} (see the proof of Lemma 7.2 or p1051 in [Sc-Ya]). As the case (∗) in §5, we know ([Sc-Ya])

Q_{3}w_{16} = d_{0}w_{16}, Q_{4}(d_{0}w_{16}) = d_{0}^{2}w_{16}^{2}.

We can prove from Kameko-Mimura theorem

Lemma 8.1.

Igr(W_{5}) ≅ Igr(W_{4}) ⊕ ℤ/2[c_{4}, c_{6}, c_{16}] ⊗ (ℤ/2[c_{8}]{c_{16}}⊕
ℤ/2[c_{7}] ⊗ Q(2){c_{16}u_{3}} ⊕ ℤ/2[c_{7}, c_{8}] ⊗ Q(4){u_{5}})

Proof. Recall that

Igr(W_{4}) ≅ ℤ/2[c_{4}, c_{6}, c_{7}, c_{8}] ⊗ (ℤ/2{1} ⊕ Q̄(2){u_{3}} ⊕ Q(2){u_{3}}).

From Kameko-Mimura theorem (Theorem 5.4) we see that

Igr(W_{5}) ≅ ℤ/2[c_{4}, c_{6}, c_{7}, c_{8}, c_{16}]⊗(ℤ/2{1}⊕Q̄(2){u_{3}}⊕Q(2){u_{4}}⊕Q(3){u_{5}}).

Using Q_{2}Q_{1}Q_{0}(u_{3}) = c_{7} and Q_{3}(u_{4}) = c_{8}u_{3}+... as the case for Spin_{7}, we have the isomorphism

Q(2){u_{3}} ≅ ℤ/2{c_{7}}⊕Q̄(2){u_{3}}, Q(3){u_{4}} ≅ Q(2){c_{8}u_{3}}⊕Q(2){u_{4}}.

Hence Igr(W_{5}) is isomorphic to

ℤ/2[c_{4}, c_{6}, c_{16}] ⊗ (ℤ/2[c_{8}] ⊕ ℤ/2[c_{7}] ⊗ Q(2){u_{3}}⊕
ℤ/2[c_{7}, c_{8}] ⊗ Q(3){u_{4}} ⊕ ℤ/2[c_{7}, c_{8}] ⊗ Q(3){u_{5}})

Using the fact Q_{4}(u_{5}) = c_{16}u_{4} + ... from Lemma 5.6, we have the isomorphism

Q(4){u_{5}} ≅ Q(3){c_{16}u_{4}} ⊕ Q(3){u_{5}}.

This induces the isomorphism

Igr(W_{5}) ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2[c_{8}, c_{16}] ⊕ ℤ/2[c_{7}, c_{16}] ⊗ Q(2){u_{3}}
⊕ℤ/2[c_{7}, c_{8}] ⊗ Q(3){u_{4}} ⊕ ℤ/2[c_{7}, c_{8}, c_{16}] ⊗ Q(4){u_{5}}).

Hence we have the desired isomorphism. □

The cohomological invariant is known

Inv^{∗}(Spin_{9}, ℤ/2) ≅ ℤ/2{1, u_{3}, u_{4}, u_{5}}.

Hence Assumption (1),(2) are also satisfied for (Spin_{9}, 2) (from the last isomorphism in the above proof).

20 M. KAMEKO, M. TEZUKA, AND N. YAGITA

Lemma 8.2. Igr(W_{5})/I(W_{5}) ≅ Igr(W_{4})/(W_{4}) ⊕ ℤ/2[c_{4}, c_{6}, c_{16}] ⊗ (ℤ/2{c_{16}u_{3}}⊕

⊕ℤ/2[c_{8}] ⊗ ((Q(4)/(Q(4)^{+})^{3}{u_{5}} ⊕ ℤ/2{c_{7}u_{5}})).

Proof. Since

Q_{0}(u_{5}) = Q_{0}(u_{4}x_{5}) = Q_{0}(u_{4})x_{5} + u_{4}y_{5},

we can compute Q_{2}Q_{1}Q_{0}(u_{5}) = Q_{2}Q_{1}Q_{0}(u_{4})x_{5}+Q_{1}Q_{0}(u_{4})y_{5}^{4}+Q_{2}Q_{0}(u_{4})y_{5}^{2}+Q_{2}Q_{1}(u_{4})y_{5}

= w_{7}w_{8}x_{5} + w_{8}y_{5}^{4} + w_{4}w_{8}y_{5}^{2} + w_{6}w_{8}y_{5} = w_{16}.

(This fact also follows from d_{5,4} = w_{16}.) Let us write Q_{i_{1},...,i_{j}}(u_{5}) = Q_{i_{1},...,i_{j}} simply. Similarly we can compute Q_{012} = w_{16}, Q_{013} = w_{16}w_{8}, Q_{023} = w_{16}w_{4}w_{8}, Q_{123} = w_{16}w_{6}w_{8},

Q_{014} = c_{16}w_{8}, Q_{024} = c_{16}w_{8}w_{4}, Q_{034} = c_{16}c_{8}w_{4},

Q_{124} = c_{16}w_{6}w_{8}, Q_{134} = c_{16}c_{8}w_{6}, Q_{234} = c_{16}c_{8}w_{4}w_{6}.

We can compute Q_{0}(c_{7}u_{5}) = Q_{0}(c_{7}u_{4}x_{5}) = Q_{0}(w_{4}w_{6}w_{8}x_{5})

= w_{4}w_{7}w_{8}x_{5} + w_{4}w_{6}w_{8}y_{5} + ... = w_{4}w_{16}.

Let us write Q_{i_{1},...,i_{j}}(c_{7}u_{5}) = Q′_{i_{1},...,i_{j}} simply. Then we can compute

Q′_{0} = w_{16}w_{4}, Q′_{1} = w_{16}x_{6}, Q′_{2} = w_{16}w_{4}w_{6}, Q′_{3} = w_{16}w_{4}w_{6}w_{8}, Q′_{4} = c_{16}w_{4}w_{6}w_{8}, Q′_{01} = w_{16}w_{7}, Q′_{12} = w_{16}w_{6}w_{7}, Q′_{04} = c_{16}w_{7}w_{4}w_{8}.

Moreover

c_{7}^{2}u_{5} = c_{7}w_{4}w_{6}w_{8}x_{5} = w_{4}w_{6}w_{7}w_{16}.

There appear all generators of the ℤ/2[c_{4}, c_{6}, c_{8}, c_{16}]-module with modulo Ideal(c_{7}, w_{7}). Thus we can see Igr(W_{5})/I(W_{5}) ≅ ℤ/2[c_{4}, c_{6}]⊗(ℤ/2[c_{16}]{u_{3}}⊕ℤ/2[c_{8}](Q(3)/(Q(3)^{+})^{2}{u_{4}}

⊕ℤ/2[c_{8}, c_{16}] ⊗ ((Q(4)/(Q(4)^{+})^{3}{u_{5}} ⊕ ℤ/2{c_{7}u_{5}})).

□

Theorem 8.3. There is an epimorphism of Q(4)-modules

from H^{∗}(BSpin_{9}; H_{ℤ/2}^{∗′}) to H^{∗}(BSpin_{7}; H_{ℤ/3}^{∗′})⊕ℤ/p[c_{4}, c_{6}, c_{16}]⊗(ℤ/2[c_{7}]⊗Q(2){c_{16}u_{3}}⊕ℤ/2{c_{16}y}

⊕ℤ/2[c_{8}, c_{7}] ⊗ Q(4){u_{5}} ⊕ ℤ/2[c_{8}] ⊗ (Q(4)/(Q(4)^{+})^{3}{y′′}) where d_{2}u_{3} = y and d_{2}(u_{5}) = y′′.

Remark. However note that we can not see d_{2}(c_{7}u_{5}) ≠ 0 or not. From Lemma 4.4, we know d_{2}(u_{5}) = y′′ ≠ 0 and we get many Griffith

elements

Q_{i}Q_{j}(y′′) for 0 ≤ i < j ≤ 4.

We study the image of the cycle map c̃l to BP^{∗}(BSpin_{9}) ⊗_{BP^{∗}} ℤ/2 in the last section (indeed c̃l(Q_{i}Q_{j}(y′′) ≠ 0).

Next we consider the case (Spin_{8}, 2). The Weyl group is

W_{4} ⊂ W_{G}(A) = W′′_{5} = {(w_{ij}) ∈ GL_{5}(ℤ/2)|w_{5,4} = 0} ⊂ W_{5}.

We can compute

Igr(W′′_{5}) ≅ Igr(W_{4}) ⊕ ℤ/2[c_{4}, c_{6}, c′_{8}, c_{8}]⊗
(ℤ/2{c′_{8}} ⊕ ℤ/3[c_{7}] ⊗ Q(3){u′_{4}} ⊗ ℤ/2[c_{7}] ⊗ Q(4){u_{5}}).

Hence Assumption (1) (with some modification for u′_{4}) and (2) are satisfied.

We consider the case (F_{4}, 2) also . This case A ≅ (ℤ/2)^{5} but

W_{G}(A) = W′_{5} = ⟨U_{5}, GL_{3}(ℤ/2) ⊕ GL_{2}(ℤ/2)⟩ ⊂ GL_{5}(ℤ/2).

The cohomology is given by

H^{∗}(BG; ℤ/2) ≅ H^{∗}(BA; ℤ/2)^{W_{G}(A)} ≅ ℤ/2[w_{4}, w_{6}, w_{7}, x_{16}, x_{24}]

where x_{16} = w_{8}^{2}+w_{16} and x_{24} = w_{8}w_{16}. We consider the representation

⟨a_{4}, a_{5}⟩ ⊂ A ⊂ F_{4} →^{ρ} SO(26)

where a_{4}, a_{5} are dual of x_{4}, x_{5}. Then the total Chern class is

c(ρ|⟨a_{4}, a_{5}⟩) = (1 + c_{2,1} + c_{2,0})^{a}

from Lemma 5.1. By dimensional reason, a ≤ 8. So we see x_{16}^{2} and x_{24}^{2} are represented by Chern classes. Thus we can write

Igr(W′_{5}) ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2[c_{16}, c_{24}] ⊕ ℤ/2[c_{7}, c_{24}] ⊗ Q(2){u_{3}}

⊕ℤ/2[c_{7}, c_{16}, c_{24}] ⊗ Q(4){u_{5}}).

Hence Assumption (1),(2) are also satisfied. So Res_{E_{2}} is an epimorphism for (F_{4}, 2).

At last of examples, we consider the case (E_{6}, 2). This case W_{G}(A) ≅ W′_{5} same as the case F_{4}. But it is known ([Ga-Me-Se]) that

Inv^{∗}(E_{6}, ℤ/2) ≅ ℤ/2{1, u_{3}}.

Therefore we see

Lemma 8.4. When G = E_{6} and p = 2, the restriction map

Res_{E_{2}} : H^{∗}(BG; H_{ℤ/2}^{∗′}) → H^{∗}(BA; H_{ℤ/2}^{∗′})^{W_{G}(A)}

is not an epimorphism.

The above fact also proved by using the cohomology H^{∗}(BE_{6}; ℤ/2) and Lemma 5.9 as follows.

Theorem 8.5. The restriction maps

Res_{Inv} : Inv^{∗}(G; ℤ/2) → Inv^{∗}(A_{5}; ℤ/2)^{W_{G}(A_{5})}

are epimorphisms for G = E_{6} and G = Spin_{n}, n ≥ 10.

Proof. Of course there is the embedding i : Spin_{9} ⊂ Spin_{10}. From Kono-Mimura [Ko-Mi], we know

H^{∗}(BSpin_{10}; ℤ/2) ≅ H^{∗}(BSpin_{7}; ℤ/2) ⊗ ℤ/2[x_{10}, x_{32}]/(w_{7}x_{10}).

Hence H^{∗}(BE_{6}; ℤ/2) does not contain an element x with i^{∗}(x) = w_{16} for w_{16} = d_{5,4} = Q_{0}Q_{1}Q_{2}(u_{5}). From Lemma 5.9, we see that Res_{Inv} is not an epimorphism.

There is the embedding F_{4} ⊂ E_{6}. From also Kono-Mimura [Ko-Mi], we know

H^{∗}(BE_{6}; ℤ/2) ≅ H^{∗}(BSpin_{7}; ℤ/2)⊗ℤ/2[x_{10}, x_{18}, x_{32}, x_{48}]/(relations).

Hence Res_{Inv} is not epic from the lack of element x_{16}. □

9. BP -theory and Griffith elements

In this section, we recall the results in §5 in [Ya1] and consider the relation between BP -theory and results in the preceding sections. We always assume k = ℂ in this section. Let BP^{∗}(−) be the Brown-Peterson theory with the coefficient ring BP^{∗} = ℤ_{(p)}[v_{1}, ...], |v_{i}| = −2(p^{i} − 1). The Thom map induces ρ : BP^{∗}(X) ⊗_{BP^{∗}} ℤ_{(p)} → H^{∗}(X; ℤ_{(p)}). Totaro constructs [To1] the map

c̃l : CH^{∗}(X)_{(p)} → BP^{∗}(X) ⊗_{BP^{∗}} ℤ_{(p)}

such that the composition ρ · c̃l is the usual cycle map cl = t_{ℂ} which is also the realization map. Totaro conjectured that this map is isomorphic for X = BG.

Let us write by

P (n)^{∗} = BP^{∗}/(p, v_{1}, ..., v_{n−1}),

e.g., P (0)^{∗} = BP^{∗}, P (1)^{∗} = BP^{∗}/p and P (∞)^{∗} = ℤ/p.

Many cases of X([Te-Ya2], [Ko-Ya]), BP^{∗}(X) are computed by the Atiyah-Hirzebruch spectral sequences

E_{2}^{∗,∗′} = H^{∗}(X) ⊗ BP^{∗′} ⟹ BP^{∗}(X).

It is known that d_{2p^{i}−1}(x) = v_{i} ⊗ Q_{i}(x) mod(M_{i}) where M_{i} is the ideal of E_{2p^{i}−1}^{∗,∗′} generated by elements in (p, ..., v_{i−1})E_{2}^{∗,∗′}.

We assume that H^{∗}(X) has no higher p-torsion and all non zero differentials are of form

(9.1) d_{2p^{i}−1}(x) = v_{i} ⊗ Q_{i}(x) mod(M_{i}).

Let us write

(9.2) grBP^{∗}(X) ≅ E_{∞}^{∗,∗} ≅ A ⊕ B

where A (resp. B) is a BP^{∗}-module generated by elements in H^{∗}(X)/p (resp. pH^{∗}(X) ⊕ E_{∞}^{∗,minus}) so that B ⊂ Ker(ρ_{p}). Then we can write

A ≅ ⊕_{n=−1}P (n + 1)^{∗}G̃_{n}

by the prime invariant ideal theorem of Landweber ; if P (n)^{∗}/(a) is a BP^{∗}(BP)-module, then a = v_{n}^{s} for some s ≥ 1.

Lemma 9.1. (Lemma 5.1 in [Ya1]) Let H^{∗}(X)_{(p)} has no higher ptorsion. Suppose (9.1) and A = ⊕_{n=−1}P (n + 1)^{∗}G̃_{n} in (9.2). Then there is a injection of Q(∞)-modules

H^{∗}(X; ℤ/p) ↪ ⊕_{n=−1}Q(n)G_{n} with Q_{0}...Q_{n}G_{n} = G̃_{n}.

It is proved ([Ko-Ya],[Ka-Ya],[Ya1]) that all X = BG in Theorem 1.2 satisfy the assumption in the above lemma. Hence

H^{∗}(BG; ℤ/p) ↪ ⊕_{n=−1}^{∞}Q(n)G_{n}.

Moreover when n ≠ −1, we still know

w(G_{n}) = n.

Indeed if n ≥ 0, then G_{n} = G′_{n}{u_{n}} where G′_{n} is represented by elements in CH^{∗}(BG)/p (see Assumption (1),(2)). From Theorem 1.2, we also know

Corollary 9.2. Let G be a group in Theorem 1.2. Then there is a bidegree Q(n)-module injection

⊕_{i≥0}Q(n)G_{n} ⊂ H^{∗}(BG; H_{ℤ/p}^{∗′}).

Lemma 9.3. (Lemma 5.2 in [Ya]) Let H^{∗}(X)_{(p)} has no higher ptorsion.

(1) If (9.1) is satisfied and in (9.2),

A ≅ ⊕_{n=−1}P (n + 1)^{∗}G̃_{n} and B ≅ ⊕_{s=0}BP^{∗}{p, v_{1}, .., v_{s}}K̃_{s},

then we have the isomorphism

H^{∗}(X; ℤ/p) ≅ (⊕_{n=−1}Q(n)G_{n}) − (⊕_{s}Q̄(s)K_{s})

with Q_{0}...Q_{n}G_{n} = G̃_{n} , and Q_{1}...Q_{s}K_{s} = K̃_{s}.

(2) If Q_{0}...Q_{n}G_{n} ∈ Im(ρ) and |Q_{1}...Q_{s}K_{s}| = even, then the converse also holds.

Corollary 9.4. Let G be a group in Theorem 1.2 so that Assumption (1),(2) are satisfied. Then there are epimorphisms from grBP^{∗}(BG) ≅ E_{∞}^{∗,∗′} to

(9.3) ⊕_{s} P (i_{s})^{∗}[f_{s1}, ..., f_{sk_{s}}]{ũ_{i_{s}}} ⊕ ⊕_{t}BP^{∗}(p, ..., v_{t}){K̃_{t}}

and from BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/p to

⊕_{s}ℤ/p[f_{s1}, ..., f_{sk_{s}}]{ũ_{i_{s}}} ⊕ ⊕_{t}ℤ/p{p, ..., v_{t}}{K̃_{t}}

where ũ_{i_{s}} = Q_{i_{s}−1}...Q_{0}(u_{i_{s}}) and Igr(W)/Res_{Hℤ/p} ≅ ⊕_{t}Q̄(t)K_{t}.

We give examples. At first we recall the Atiyah-Hirzebruch spectral sequence for BG_{2} in [Ko-Ya]. Since H^{∗}(BG) has no higher torsin, we have

H^{∗}(BG_{2})_{(2)} ≅ ℤ_{(2)}[w_{4}, c_{6}] ⊗ (ℤ_{(2)}{1} ⊕ ℤ/2[w_{7}]{w_{7}}).

Let us write B_{i_{1},...,i_{j}} = ℤ_{(2)}[c_{i_{1}}, ..., c_{i_{j}}], e.g., B_{4,6} = ℤ_{(2)}[c_{4}, c_{6}].

Since Q_{1}(w_{4}) = w_{7}, we have d_{3}(w_{4}) = v_{1} ⊗ w_{7}. Hence the E_{4}-term of the spectral sequence is

E(G_{2})_{4}^{∗,∗} ≅ B_{4,6} ⊗ (BP^{∗}{1, 2w_{4}} ⊕ P (2)^{∗}[c_{7}]{c_{7}, w_{7}}).

Next differential is d_{7}(w_{7}) = v_{2} ⊗ Q_{2}(w_{7}) = v_{2}c_{7} and

E(G_{2})_{8}^{∗,∗} ≅ B_{4,6} ⊗ (BP^{∗}{1, 2w_{4}} ⊕ P (3)^{∗}[c_{7}]{c_{7}}).

which is isomorphic to E(G_{2})_{∞}^{∗,∗}. In particular

BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/2 ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{1, 2w_{4}} ⊕ ℤ/2[c_{7}]{c_{7}}).

This result is also immediate from Corollary 9.4 and Theorem 6.1, in fact, we have the epimorphism

H^{∗}(BG_{2}; H_{ℤ/2}^{∗′}) → ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{1, y} ⊕ ℤ/2[c_{7}] ⊗ Q(2){u_{3}}).

Here ũ_{3} = Q_{0}Q_{1}Q_{2}(u_{3}) = c_{7} and d_{2}(u_{3}) = y in the coniveau spectral sequence, and we have

Q̄(0)K_{0} = K_{0} = ℤ/2[c_{4}, c_{6}]{u}.

Hence the cycle map c̃l is epimorphism and c̃l(y) = {2w_{4}} (which is represented by a Chern class c_{2}). Moreover we know [Ya2] that this c̃l is a really isomorphism.

Next consider the case Spin_{7}. From [Ko-Ya], we can compute

E(Spin_{7})_{16}^{∗,∗} ≅ B_{4,6} ⊗ P (3)^{∗}[c_{7}]{c_{7}}⊕

B_{4,6,8} ⊗ (BP^{∗}{1, 2w_{4}, 2w_{4}w_{8}, 2w_{8}, v_{1}w_{8}} ⊕ P (4)^{∗}[c_{7}]{c_{7}c_{8}}). This term is also the infinity term. Hence we have

BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/2 ≅ BP^{∗}(BG_{2}) ⊗_{BP^{∗}} ℤ/2⊕

ℤ/2[c_{4}, c_{6}, c_{8}] ⊗ (ℤ/2{c_{8}, 2c_{8}w_{4}, 2w_{4}w_{8}, 2w_{8}, v_{1}w_{8}} ⊕ ℤ/2[c_{7}]{c_{7}c_{8}}) This result is also get from Corollary 9.4 and Theorem 7.3, indeed, we have the epimorphsm

H^{∗}(BSpin_{7}; H_{ℤ/2}^{∗′}) → H^{∗}(BG_{2}; H_{ℤ/2}^{∗′})⊕

ℤ/2[c_{4}, c_{6}, c_{8}] ⊗ (ℤ/2{c_{8}, y′_{2}, Q_{0}y′_{2}, ..., Q_{3}y′_{2}} ⊕ ℤ/2[c_{7}] ⊗ Q(3){u_{4}}) This case d_{2}(u_{3}) = y, d_{2}(u_{4}) = y′ in the coniveau spectral sequence. Recall

⊕_{s}Q̄(s)K_{s} ≅ Igr(W_{4})/I(W_{4}) ≅ I(GL_{3})/I(GL_{3})⊕
ℤ/2[c_{4}, c_{6}, c_{8}]{1, Q_{0}, ..., Q_{3}}{u_{4}}.

We can take

Q̄(1)K_{1} ≅ ℤ/2[c_{4}, c_{6}, c_{8}]{1, Q_{0}, Q_{1}}{u_{4}}
Q̄(0)K_{0} ≅ ℤ/2[c_{4}, c_{6}] ⊗ (ℤ/2{u_{3}} ⊕ ℤ/2{Q_{2}u_{4}, Q_{3}u_{4}}).

The cycle map c̃l is given by

Q_{0}(y′) ↦ v_{1}w_{8} , Q_{1}y′ ↦ 2w_{8} , Q_{2}y′ ↦ 2w_{4}w_{8} , Q_{3}y′ ↦ 2c_{8}w_{4}.

Of course the cycle map c̃l is isomorphic. This fact is still proved by P.Guillot [Gu1], and the case G = Spin_{8} is computed by Molina [Mo].

10. BP^{∗}(BSpin_{9}) ⊗_{BP^{∗}} ℤ/2

At last of this paper, we consider the case Spin_{9}. In [Ko-Ya], we can compute (which is quite complicated)

E(Spin_{9})_{32}^{∗,∗} ≅ B_{4,6,8,16} ⊗ (BP^{∗}{1, 2w_{4}, 2w_{4}w_{8}, 2w_{8}, v_{1}w_{8},
2w_{4}w_{16}, 2w_{4}w_{8}w_{16}, 2w_{8}w_{16}, v_{1}w_{8}w_{16}, 2w_{16}, v_{1}w_{16}.v_{2}w_{16}}

⊕P (5)^{∗}[c_{7}]{c_{7}c_{8}c_{16}})⊕B_{4,6,7,8}⊗P (4)^{∗}[c_{7}]{c_{7}c_{8}}⊕B_{4,6,16}⊗P (3)^{∗}[c_{7}]{c_{7}}. This term is the infinite term. (See Theorem 8.3 also.)

Lemma 10.1. ([Ko-Ya])

BP^{∗}(BSpin_{9}) ⊗_{BP^{∗}} ℤ/2 ≅ BP^{∗}(BSpin_{7}) ⊗_{BP^{∗}} ℤ/2⊕

ℤ/2[c_{4}, c_{6}, c_{8}, c_{16}] ⊗ (ℤ/2{c_{16}, 2w_{4}c_{16}, 2w_{4}c_{8}c_{16}, (2, v_{1})w_{8}c_{16}, 2w_{4}w_{8}c_{16}, (2, v_{1}, v_{2})w_{16}, 2w_{4}w_{16}, (2, v_{1})w_{8}w_{16}, 2w_{4}w_{8}w_{16}} ⊕ ℤ/2[c_{7}]{c_{16}c_{7}}).

We still know there is an epimorphism from H^{∗}(BSpin_{9}; H_{ℤ/2}^{∗′}) to

H^{∗}(BSpin_{7}; H_{ℤ/2}^{∗′})⊕ℤ/2[c_{4}, c_{6}, c_{16}]⊗(ℤ/2[c_{7}]⊗Q(2){c_{16}u_{3}}⊕ℤ/2{c_{16}y}

⊕ℤ/2[c_{8}, c_{7}] ⊗ Q(4){u_{5}} ⊕ ℤ/2[c_{8}] ⊗ (Q(4)/(Q(4)^{+})^{3}{y′′})

where d_{2}u_{3} = y and d_{2}(u_{5}) = y′′.

We can see the following lemma; Proposition 10.2. When (G, p) = (Spin_{9}, 2), the cycle map

c̃l : CH^{∗}(BG)/2 → (BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/2)

is an epimorphism with mod(ℤ/2[c_{4}, c_{6}, c_{8}, c_{16}]{2w_{4}w_{16}}).

The image of the cycle map c̃l is given as follows. By arguments for G_{2}, we see

d_{2}(c_{16}u_{3}) ↦ 2w_{4}c_{16}

If we can see

d_{3}(c_{7}u_{5}) ↦ 2w_{4}w_{16},

then the Totaro’s map c̃l is an epimorphism. Unfortunately, we do not see even d_{2}(c_{7}u_{5}) = c_{7}y′′ = 0 yet.

Let us write Q_{ij} = Q_{i}Q_{j}(d_{2}u_{5}) = Q_{i}Q_{j}(y′′). Then the cycle map c̃l is written as

Q_{01} ↦ v_{2}w_{16}, Q_{02} ↦ v_{1}w_{16}, Q_{12} ↦ 2w_{16},
Q_{03} ↦ v_{1}w_{8}w_{16}, Q_{13} ↦ 2w_{8}w_{16}, Q_{23} ↦ 2w_{4}w_{8}w_{16},
Q_{04} ↦ v_{1}w_{8}c_{16}, Q_{14} ↦ 2w_{8}c_{16}, Q_{24} ↦ 2w_{4}w_{8}c_{16},
Q_{34} ↦ 2w_{4}c_{8}c_{16}.

By the arguments for Spin_{7}, we still know, for example Q_{1}(d_{2}u_{4}) ↦ 2w_{8}, so we get the Q_{4}Q_{1}(d_{2}u_{5}) ↦ 2w_{8}c_{16} using Q_{4}(u_{5}) = c_{16}u_{4} + .... Similarly we get the maps for Q_{∗4} from that for Spin_{7}.

For other maps, we use the Quillen operation in BP^{∗}(−) theory. For a sequence α = (α_{1}, ..., α_{m}), α_{i} ≥ 0, we have the Quillen cohomology operation in BP^{∗}(X) (and also in ABP^{∗}(X)) (see [Ra], [Ha], [Ya2])

r_{α} : BP^{∗}(X) → BP^{∗+|α|}(X) |α| = ∑ 2p^{i}α_{i}

such that ρ(r_{α}) = P^{α} the fundamental basis of the reduced power operations (see [Ha]) and r_{α}(v_{i}) ∈ Ideal(p, ..., v_{i}). Hence r_{α} acts also on BP^{∗}(X) ⊗_{BP^{∗}} ℤ/p.

Let us write by S̄q^{even} the Quillen operation corresponding Sq^{even}. By the definition of Q_{i}, se see the equation Sq^{16}Sq^{8}(Q_{2}Q_{1}(u_{5})) = Q_{4}Q_{1}(u_{5}) in H^{∗}(BG; ℤ/2). We still know the image of the cycle map

Sq^{16}Sq^{8}(Q_{12}) = Q_{14} ↦ 2w_{8}c_{16} ∈ BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/2.

Let c̃l(Q_{12}) = x. Then

S̄q^{16}S̄q^{8}(x) = 2w_{8}c_{16} in BP^{∗}(BG) ⊗_{BP^{∗}} ℤ/2.

So x is non zero. The equation Sq^{16}Sq^{8}(w_{16}) = w_{8}c_{16} in H^{∗}(BG; ℤ/2) implies

S̄q^{16}S̄q^{8}(2w_{16}) = 2w_{8}c_{16} mod(v_{1}, ...) in BP^{∗}(BG).

Hence we can take x = 2w_{16}.

Using S̄q^{4}S̄q^{2} and dimensional reason, we have the first map Q_{01} ↦ v_{2}w_{16}. The other cases are also proved similarly.

The above Q_{ij} are all Griffith elements. From Corollary 9.4, we can write

Q(4){u_{5}}/(Q(4)^{+})^{3}{u_{5}} ≅ ⊕_{t=1}^{2}Q̄(t)K′_{t}.

In fact, we can take

Q̄(2)K′_{2} = ℤ/2{u_{5}, Q_{0}, Q_{1}, Q_{2}, Q_{01}, Q_{02}, Q_{12}},
Q̄(1)K′_{1} = ℤ/2{Q_{3}, Q_{03}, Q_{13}, Q_{4}, Q_{04}, Q_{14}},
Q̄(0)K′_{0} = ℤ/2{Q_{23}, Q_{24}, Q_{34}}.

Recall Corollary 9.4 and Igr(W)/Res_{Hℤ/p} ≅ ⊕_{t}Q̄(t)K_{t}. Let k ∈ K_{t}. Then we can identify k ∈ H^{∗}(BG; H_{ℤ/p}^{∗′}) and

k(i) = Q_{0}...Q̂_{i}...Q_{t}(k) ∈ Q̄(t)K_{t} ⊂ Igr(W)/Res_{Hℤ/p}.

Moreover suppose w(k) = t + 3. Since w(k(i)) = 3 and w(d_{r}) = 1 − 2r, we see

d_{2}(k(i)) ≠ 0 (hence d_{2}(k) ≠ 0).

Let us consider the projection map

pr. : BP^{∗}(X) ⊗_{BP^{∗}} ℤ/p → (9.3) ⊗_{BP^{∗}} ℤ/p → (p, v_{1}, ..., v_{t}){k̃}

where k̃ = Q_{0}...Q_{t}(k). For G = G_{2}, Spin_{7} and Spin_{9}, it holds that pr.c̃l(d_{2}(k(i))) = v_{i}k̃, that is,

pr.c̃l(d_{2}(Q_{0}...Q̂_{i}...Q_{t}(k))) = v_{i}Q_{0}...Q_{t}(k),

while we do not show it for general cases.

References

[Ad] J. F. Adams. Lectures on exceptional Lie groups, Univ. Chicago Press, Chicago, IL, 1996.

[Bl-Og] S.Bloch and A.Ogus. Gersten’s conjecture and the homology of

schemes. Ann.Scient.Éc.Norm.Sup. 7 (1974) 181-202.

[Br-Re-Vi] P.Brosman. Z.Reichstein and A.Vistoli. Essential dimension, spinor

groups, and quadratic forms. to appear in Ann. Math. (2003).

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

[Ga] S.Garibaldi. Cohomological invariants : exceptional groups and Spin groups (wth an appendix by D.Hoffmann). Mem. Amer. Math. Soc.

200. 937 (2009)

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

[Gu1] P.Guillot. The Chow ring of G_{2} and Spin(7). J. Reine angew. Math. 604 (2007) 137-158.

[Gu2] P.Guillot. Geometric methods for cohomological invariants. Document Math. 12 (2007), 521-545.

[Ka] M.Kameko. Poincaré series of cotorsion products. Preprint (2005). [Ka-Mi] M. Kameko and M. Mimura. Mùi invariants and Milnor operations,

Geometry and Topology Monographs 11, (2007), 107-140.

[Ka-Ya1] M.Kameko and N.Yagita. The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups P U(p) and exceptional Lie group. Trans. Amer. Math. Soc. 360 (2008), 2265-2284.

[Ka-Ya2] M.Kameko and N.Yagita. Chern subrings. Proc. Amer. Math. Soc.

138 (2010) 367-373.

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

[Mo] L.A. Molina. The Chow ring of classifying space of Spin_{8}. preprint (2007).

[Mo-Vi] L.Molina and A.Vistoli. On the Chow rings of classifying spaces for

classical groups. Rend. Sem. Mat. Univ. Padova 116 (2006), 271-298.

[Or-Vi-Vo] D.Orlov,A.Vishik and V.Voevodsky. An exact sequence for Milnor’s Ktheory with applications to quadric forms. Ann. of Math. 165 (2007) 1-13.

[Pa] W.Paranjape Some spectral sequences for filtered complexes and applications. J. Algebra 186 (1996) 793-806.

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

[Re] Z. Reichstein. On the notion of essential dimension for algebraic groups. Transform. groups 5 (2000), 265-304.

[Ro] M.Rost. On the basic correspondence of a splitting variety. preprint (2006)

[Sc-Ya] B. Schuster and N. Yagita, Transfers of Chern classes in BPcohomology and Chow rings, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1039–1054 (electronic).

[Su-Jo] A.Suslin and S.Joukhovistski. Norm Variety. J.Pure and Appl. Algebra

206 (2006) 245-276.

[To] B. Totaro. The Chow ring of classifying spaces. Proc.of Symposia in Pure Math. ”Algebraic K-theory” (1997:University of Washington,Seattle) 67 (1999), 248-281.

[Vi] A.Vistoli. On the cohomology and the Chow ring of the classifying space of P GL_{p}. J. Reine Angew. Math. 610 (2007) 181-227.

[Vo1] V. Voevodsky. The Milnor conjecture. www.math.uiuc.edu/Ktheory/0170 (1996).

[Vo2] V. Voevodsky. Motivic cohomology with ℤ/2 coefficient. Publ. Math. IHES 98 (2003), 59-104.

[Vo3] V. Voevodsky (Noted by Weibel). Voevodsky’s Seattle lectures : Ktheory and motivic cohomology Proc.of Symposia in Pure Math. ”Algebraic K-theory” (1997:University of Washington,Seattle) 67 (1999), 283-303.

[Vo4] V.Voevodsky. Reduced power operations in motivic cohomology. Publ.Math. IHES 98 (2003),1-57.

[Vo5] V.Voevodsky. On motivic cohomology with ℤ/l-coefficients. www.math.uiuc.edu/K-theory/0631 (2003).

[Ya1] N. Yagita. Examples for the mod p motivic cohomology of classifying spaces. Trans. Amer. Math. Soc. 355 (2003),4427-4450.

[Ya2] N. Yagita. Applications of Atiyah-Hirzebruch spectral sequence for motivic cobordism. Proc. London Math. Soc. 90 (2005) 783-816.

[Ya3] N. Yagita. Coniveau filtration of cohomology of groups. to appear in Proc. London Math. Soc. (2008).

Department of Mathematical Sciences College of Systems Engineering and Science Shibaura Institute of Technology, Saitama, Japan

  E-mail address: kameko@shibaura-it.ac.jp

Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan

  E-mail address: tez@sci.u-ryukyu.ac.jp

Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

  E-mail address: yagita@mx.ibaraki.ac.jp