KERNELS OF RESTRICTION MAPS IN

THE mod-ppp COHOMOLOGY OF ppp-GROUPS

Võ Thanh Tùng

Dedicated to Professor Huỳnh Mùi on his sixtieth birthday

Abstract. Let p be a prime number. The purpose of this paper is to investigate kernels of restriction maps in the mod-p cohomology of p-groups. This result is applied to prove that, if K is a 2-group which is not elementary abelian, and if ξ, ξ_{1}, …, ξ_{r}, … is a sequence of mod-2 cohomology classes of K which restrict trivially to all proper subgroups, then Sq^{n−r}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r} = 0 where n = deg(ξ) and ξ_{0} = 1.

1. Statement of results

Let p be a prime number and let G be a p-group. Denote by H^{∗}(G) = H^{∗}(G, ℤ_{p}) the cohomology ring of G. Suppose that H is a maximal subgroup of G. We can consider a decomposition G = H ∐ Ht ∐ … ∐ Ht^{p−1} into disjoint right cosets of H with t ∈ G. Let σ = (1 p p − 1 … 2) be the p-cycle of the symmetric group S_{p}. The elementary abelian group ℤ_{p}^{p} becomes a G-module under the action defined by

t^{k}(a_{1}, …, a_{p}) = (a_{σ^{k}(1)}, …, a_{σ^{k}(p)}), 1 ≤ k ≤ p − 1,

and H acts trivially on ℤ_{p}^{p}. Consider the short exact sequence of G-modules

0 ⟶ ℤ_{p} ⟶^{i} ℤ_{p}^{p} ⟶^{j} ℤ_{p}^{p−1} ⟶ 0  (1)

with diagonal homomorphism

i(a) = (a, …, a), a ∈ ℤ_{p}

and

j(a_{1}, …, a_{p}) = (a_{1} − a_{p}, …, a_{p−1} − a_{p}), a_{k} ∈ ℤ_{p}, 1 ≤ k ≤ p.

The cohomology long exact sequence associated to (1) is the following (see e.g. [1])

… ⟶ H^{n}(G) ⟶^{Res_{H}^{G}} H^{n}(H) ⟶^{j^{∗}} H^{n}(G, ℤ_{p}^{p−1}) ⟶^{θ̄} H^{n+1}(G) ⟶ ….  (2)

with θ̄ the connecting homomorphism.

We have the following result.

Typeset by AMS-TEX

Theorem A. There exist a G-pairing θ : ℤ_{p}^{p−1} ⊗ ℤ_{p}^{p−1} → ℤ_{p} and an element [u_{H}] ∈ H^{1}(G, ℤ_{p}^{p−1}) such that, for every ξ ∈ Ker Res_{H}^{G},

ξ = θ^{∗}([u_{H}] × [η])

where [η] ∈ H^{∗}(G, ℤ_{p}^{p−1}).

For p = 2, we obtain the following exact sequence (which is known to be the Gysin sequence of the O-sphere bundle BH → BG)

… ⟶ H^{n}(G) ⟶^{Res_{H}^{G}} H^{n}(H) ⟶^{cor_{G}^{H}} H^{n}(G) ⟶^{[u_{H}]∪} H^{n+1}(G) ⟶ ….  (3)

We now assume that K is a 2-group which is not elementary abelian. An approach to study H^{∗}(K) is to investigate the restrictions of its elements to cohomology of all proper subgroups of K. Let Ess(K) be the essential cohomology of K consisting of all mod-p cohomology classes of K which restrict trivially to all proper subgroups of K. The following conjecture, stated by Mui, claims that Ess(K) is small in some meaning. In other words, it predicts that the discussed approach is some how essential.

Conjecture (Mui [3]). Ess(K) is of nilpotency degree ≤ 2, that is,

Ess(K)^{2} = {0}.

Our work is also an attempt to contribute to a solution of Mui’s conjecture. It was shown by Minh in [2] that ξ^{2} = 0 for every ξ ∈ Ess(K). In order to generalize the result of Minh, we prove the following

Theorem B. Let ξ, ξ_{1}, …, ξ_{r}, … be a sequence of elements of Ess(K) with deg(ξ) = n and ξ_{0} = 1 then

Sq^{n−r}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r} = 0.

Especially, for r = 0, we obtain the mentioned result of Minh. The proof of Theorem B, which is given in Section 3, is based on the Gysin exact sequence in mod-2 cohomology of 2-groups.

2. Proof of Theorem A

Let θ : ℤ_{p}^{p−1} ⊗ ℤ_{p}^{p−1} → ℤ_{p} be the G-pairing represented by the following (p − 1) × (p − 1) matrix

( \begin{matrix}0 & 1 & … & (−1)^{p−3} \\ −1 & 0 & … & (−1)^{p−4} \\ ⋮ & ⋮ & & ⋮ \\ −(−1)^{p−3} & −(−1)^{p−4} & … & 0 \end{matrix} )  for p odd

KERNELS OF RESTRICTION MAPS IN THE mod-p COHOMOLOGY OF p-GROUPS 3

and θ(a ⊗ b) = a · b, a, b ∈ ℤ_{2}, for p = 2.

Let X ⟶ ℤ be the Bar resolution of G. We define u_{H} ∈ Hom_{G}(X_{1}, ℤ_{p}^{p−1}) by

u_{H}(g) = \{ \begin{matrix}(\underbrace{0, …, 0}_{p−1 times}) & if g ∈ H, \\ (\underbrace{0, 1, …, 1}_{p−1 times}) & if g ∈ Ht, \\ tu_{H}(t^{k−1}) + u_{H}(t) & if g ∈ Ht^{k}, 2 ≤ k ≤ p − 1, \end{matrix}  for p odd

and u_{H}(g) = \{ \begin{matrix}0 & if g ∈ H, \\ 1 & otherwise \end{matrix}  for p = 2,

and extend u_{H} linearly to X_{1}.

Lemma A. u_{H} is a 1-cocycle.

Proof. It is easy to check that the lemma is true for p = 2. We will prove the lemma for the case p > 2. For every element g ∈ G, we write g = ht^{p}, with h = gt^{−p} ∈ H if g ∈ H and g = ht^{k}, k > 0 in otherwise. By the definition above, it may be concluded that

u_{H}(g) = tu_{H}(t^{k−1}) + u_{H}(t) = (t^{k−1} + … + t + 1)u_{H}(t)

by noting that u_{H}(h) = (t^{p−1} + … + t + 1)u_{H}(t) = (0, …, 0) ∈ ℤ_{p}^{p−1}. So, for every g_{1}, g_{2} ∈ G, g_{1} ∈ Ht^{k}, g_{2} ∈ Ht^{ℓ}, 1 ≤ k, ℓ ≤ p,

∂u_{H}(g_{1}, g_{2}) = g_{1}u_{H}(g_{2}) − u_{H}(g_{1}g_{2}) + u_{H}(g_{1}) = (t^{k+ℓ−1} + … + t^{k})u_{H}(t)
− (t^{k+ℓ−1} + … + t + 1)u_{H}(t) + (t^{k−1} + … + t + 1)u_{H}(t)
= (0, …, 0) ∈ ℤ_{p}^{p−1}.

Hence, u_{H} is a 1-cocycle. □

The case p = 2 in the following proposition was due to Rusin [4, Th. 5] (see also Serre [5, p.12, ex.2]).

Proposition A. θ̄[η] = θ^{∗}([u_{H}] × [η]) for every [η] ∈ H^{∗}(G, ℤ_{p}^{p−1}).

Proof. We are now interested in the case p > 2. The exact sequence (2) is the cohomology long exact sequence associated to

0 ⟶ Hom_{G}(X, ℤ_{p}) ⟶^{i^{∗}} Hom_{G}(X, ℤ_{p}^{p}) ⟶^{j^{∗}} Hom_{G}(X, ℤ_{p}^{p−1}) ⟶ 0.

We define a map Ω : Hom_{G}(X, ℤ_{p}^{p−1}) ⟶ Hom_{G}(X, ℤ_{p}^{p}) as follows. For each η ∈ Hom_{G}(X_{n}, ℤ_{p}^{p−1}), write η(x) = (η_{1}(x), …, η_{p−1}(x)), x ∈ X_{n}, with η_{i} a map from X_{n} to ℤ_{p}. Define a homomorphism Ω(η) ∈ Hom_{G}(X_{n}, ℤ_{p}^{p}) by

Ω(η)(x) = (η_{1}(x), …, η_{p−1}(x), 0), ∀x = (g_{1}, …, g_{n}) ∈ X_{n}, g_{i} ∈ G,

4 VÕ THANH TÙNG

and extend Ω(η) linearly to X_{n}. Then

j^{∗}(Ω(η))(x) = j(Ω(η)(x)) = η(x), ∀x ∈ X_{n}.

So

j^{∗}(Ω(η)) = η.

For each [η] ∈ H^{n}(G, ℤ_{p}^{p−1}) with η(x) = (η_{1}(x), …, η_{p−1}(x)), x ∈ X_{n}, we have

0 = ∂(η)(g_{1}, …, g_{n+1})

= g_{1}η(g_{2}, …, g_{n+1}) + ∑_{i=1}^{n}(−1)^{i}η(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}) + (−1)^{n+1}η(g_{1}, …, g_{n})

= g_{1}(η_{1}(g_{2}, …, g_{n+1}), …, η_{p−1}(g_{2}, …, g_{n+1}))

+ ∑_{i=1}^{n}(−1)^{i}(η_{1}(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}), …, η_{p−1}(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}))

+ (−1)^{n+1}(η_{1}(g_{1}, …, g_{n}), …, η_{p−1}(g_{1}, …, g_{n})).

Besides, θ̄[η] can be represented by the (n + 1) - cocycle θ̄η given by

(θ̄η)(g_{1}, …, g_{n+1}) = i^{−1} (∂(Ω(η))(g_{1}, …, g_{n+1}))

where ∂(Ω(η))(g_{1}, …, g_{n+1}) = g_{1}.Ω(η)(g_{2}, …, g_{n+1})

+ ∑_{i=1}^{n}(−1)^{i}Ω(η)(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}) + (−1)^{n+1}Ω(η)(g_{1}, …, g_{n})

= g_{1}(η_{1}(g_{2}, …, g_{n+1}), …, η_{p−1}(g_{2}, …, g_{n+1}), 0)

+ ∑_{i=1}^{n}(−1)^{i}(η_{1}(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}), …, η_{p−1}(g_{1}, …, g_{i}.g_{i+1}, …, g_{n+1}), 0)

+ (−1)^{n+1} (η_{1}(g_{1}, …, g_{n}), …, η_{p−1}(g_{1}, …, g_{n}), 0)

= \{ \begin{matrix}(\underbrace{0, …, 0}_{p times}) & if g_{1} ∈ H, \\ (\underbrace{η_{p−k}(g_{2}, …, g_{n+1}), …, η_{p−k}(g_{2}, …, g_{n+1})}_{p times}) & if g_{1} ∈ Ht^{k}, 1 ≤ k ≤ p − 1. \end{matrix}

So

(θ̄η)(g_{1}, …, g_{n+1}) = \{ \begin{matrix}0 & if g_{1} ∈ H, \\ η_{p−k}(g_{2}, …, g_{n+1}) & if g_{1} ∈ Ht^{k}, 1 ≤ k ≤ p − 1. \end{matrix}

KERNELS OF RESTRICTION MAPS IN THE mod-p COHOMOLOGY OF p-GROUPS 5

Furthermore, we also have

θ^{∗}(u_{H} × η)(g_{1}, …, g_{n+1}) = θ(u_{H}(g_{1}) ⊗ g_{1}η(g_{2}, …, g_{n+1}))

= θ(u_{H}(g_{1}) ⊗ g_{1}(η_{1}(g_{2}, …, g_{n+1}), …, η_{p−1}(g_{2}, …, g_{n+1})))

= \{ \begin{matrix}0 & if g_{1} ∈ H, \\ η_{p−k}(g_{2}, …, g_{n+1}) & if g_{1} ∈ Ht^{k}, 1 ≤ k ≤ p − 1. \end{matrix}

So θ̄η = θ^{∗}(u_{H} × η). Therefore, θ̄[η] = θ^{∗}([u_{H}] × [η]). The proposition follows. □ Lemma B. 0 ≠ [u_{H}] ∈ H^{1}(G, ℤ_{p}^{p−1}). Proof. Consider the connecting homomorphism θ̄: H^{0}(G, ℤ_{p}^{p−1}) ⟶ H^{1}(G). Proposition 1 shows that θ̄(a) = θ^{∗}([u_{H}] × a) for every a ∈ H^{0}(G, ℤ_{p}^{p−1}). Choose 0 ≠ a = (1, 2, …, p − 1) ∈ (ℤ_{p}^{p−1})^{G} = H^{0}(G, ℤ_{p}^{p−1}). As in the proof of Proposition 1, we have

θ̄(a)(g) = \{ \begin{matrix}0 & if g ∈ H, \\ p − k & if g ∈ Ht^{k}, 1 ≤ k ≤ p − 1. \end{matrix}

So θ^{∗}([u_{H}] × a) = θ̄(a) ≠ 0. Therefore, [u_{H}] ≠ 0. The lemma follows. □

Proof of Theorem A. The theorem is proved by combining Proposition 1, Lemma A, Lemma B and the exactness of the sequence (2). □

3. Proof of Theorem B

We need some lemmas. For i > 0, set S_{i} = ∑_{0≠u∈H^{1}(K)} u^{i}. The following was given in [2]. Lemma C. S_{i} = 0. □ Remark that Lemma C is not true if K is an elementary abelian group.

The following lemma was proved in [2] by using the Evens norm map. We will provide here a simple proof of the first part of it by means of the Gysin sequence. Lemma D (Minh [2]). If 0 ≠ u ∈ H^{1}(K) and Res_{Ker u}^{K}(ξ) = 0 with deg(ξ) = n, then

ξ^{2} = ∑_{j>0} Sq^{n−j}(ξ) · u^{j}.  (4)

Hence, if ξ ∈ Ess(K), then ξ^{2} = 0. Proof. From the exactness of Gysin sequence (3), we have ξ = x · u where x ∈ H^{n−1}(K). For all j, 1 ≤ j < n, we have

\begin{aligned}Sq^{n−j}(ξ) · u^{j} & = ( ∑_{i≥0} Sq^{(n−j)−i}(x) · Sq^{i}(u)) · u^{j} \\ & = Sq^{n−j}(x) · u^{j+1} + Sq^{n−j−1}(x) · u^{j+2}. \end{aligned}

So

\begin{aligned}∑_{j>0} Sq^{n−j}(ξ) · u^{j} & = ∑_{j=1}^{n−1} (Sq^{n−j}(x) · u^{j+1} + Sq^{n−j−1}(x) · u^{j+2}) + Sq^{0}(ξ) · u^{n} \\ & = Sq^{n−1}(x) · u^{2} = ξ^{2}. \end{aligned}

Finally, if ξ ∈ Ess(K), it follows from what we just proved that

ξ^{2} = ∑_{0≠u∈H^{1}(K)} ξ^{2} = ∑_{j>0} Sq^{n−j}(ξ) · S_{j}.

So, by Lemma C, ξ^{2} = 0. The lemma follows. □

Let T be the ideal of H^{∗}(K) generated by the images of corestrictions from proper subgroups of K. We have the following

Proposition B. Let ξ, ξ_{1}, …, ξ_{r}, … be a sequence of elements of Ess(K) with deg(ξ) = n and ξ_{0} = 1. For r ≥ 1, we have

Sq^{n−r}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r−1} ∈ T .

Proof. By induction on r. For any element 0 ≠ u ∈ H^{1}(K), set

ξ_{u}^{(r)} = ∑_{i=r}^{n} Sq^{n−i}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r−1} · u^{i−r}.

For r = 1, by Lemma D, we have ξ_{u}^{(1)} · u = ξ^{2} = 0. Following the exactness of the Gysin sequence (3), we have ξ_{u}^{(1)} ∈ T . Hence, we have

∑_{0≠u∈H^{1}(K)} ξ_{u}^{(1)} = ∑_{0≠u∈H^{1}(K)} Sq^{n−1}(ξ)u^{0}+∑_{i=2} Sq^{n−i}(ξ)( ∑_{0≠u∈H^{1}(K)} u^{i−1}) = Sq^{n−1}(ξ) ∈ T .

Suppose that r > 1 and the result holds for all k, 1 ≤ k ≤ r − 1, that is

Sq^{n−k}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{k−1} ∈ T .

By the Gysin exact sequence (3), we can decompose ξ_{k} = u·x_{k} with x_{k} ∈ H^{deg(ξ_{k})−1}(K), 1 ≤ k ≤ r − 1. Note that, if α ∈ T and β ∈ Ess(K) then α · β = 0 by the Frobenius formula (see [3, Remark]). Therefore, by multiplying (4) with x_{1} · . . . · x_{r−1}, we have ξ_{u}^{(r)} · u = 0. It follows from the Gysin exact sequence that there exist η_{u} ∈ H^{∗}(Ker u), ξ_{u}^{(r)} = cor_{K}^{Ker u}(η_{u}). Hence,

Sq^{n−r}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r−1} = ∑_{0≠u∈H^{1}(K)} ξ_{u}^{(r)} (by Lemma C)

= ∑_{0≠u∈H^{1}(K)} cor_{K}^{Ker u}(η_{u}) ∈ T .

(5)

The proposition follows. □

Proof of Theorem B. By Lemma D, the theorem holds for r = 0. For r > 1, from (5), we have

Sq^{n−r}(ξ) · ξ_{0} · ξ_{1} · . . . · ξ_{r−1} · ξ_{r} = ∑_{0≠u∈H^{1}(K)} cor_{K}^{Ker u}(η_{u}) · ξ_{r}

= ∑_{0≠u∈H^{1}(K)} cor_{K}^{Ker u}(η_{u} · Res_{Ker u}^{K}(ξ_{r}))

= 0 since Res_{Ker u}^{K}(ξ_{r}) = 0.

This completes the proof of the theorem. □

Acknowledgements

The author is greatly indebted to Pha.m Anh Minh for suggesting the problem and for many stimulating conversations.

References

1. G. Lewis, The integral cohomology rings of groups of order p^{3}, Trans. Amer. Math. Soc. 32 (1968), 501-529.

2. P. A. Minh, Essential mod-p cohomology classes of p-groups: an upper bound for nilpotency degrees, Bull. London Math. Soc. 32 (2000), 285-291.

3. H. Mui, The mod p cohomology algebra of the group E(p^{3}), Unpublished essay.

4. D. J. Rusin, Kernels of the restriction and inflation maps in group cohomology, J. Pure Appl. Alg. 79 (1992), 191-204.

5. J. P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Springer - Verlag, 1994.

Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

E-mail address: tung sp@yahoo.com