Kernels of the restriction and inflation maps in group cohomology

David J. Rusin

In the mod-2 cohomology of groups, the restriction map from a group to a subgroup of index 2 has an easily described kernel, namely a certain natural principal ideal. This is an extremely useful tool to use when computing cohomology inductively. So it is natural to ask if a corresponding statement holds in more general circumstances. This purpose of this paper is to show that all the obvious candidates for generalization fail. At odd primes there is a natural suggested replacement for the principal ideal; we will show it is in general too small to be kernel of the restriction map. We next show that even at p = 2 one cannot relax the requirement that the subgroup be of index 2. Finally, we provide an example to show that the corresponding natural ideal in the kernel of the inflation map can also be smaller than the complete kernel.

1 Formulation of the conjecture

Suppose F →^{i} E →^{π} B is a fibration. As π ∘ i is null-homotopic, i^{∗} ∘ π^{∗} is the zero map in reduced cohomology, so that Ker(i^{∗}) ⊇ Im(π^{∗}) ∩ H^{+}(E). If we take coefficients in a ring R, then J_{i} = Ker(i^{∗}) will be an ideal in H^{∗}(E), and so will include the ideal

I_{π} = (Im(π^{∗}) ∩ H^{+}(E)) ⌣ H^{∗}(E)

generated by the image of π^{∗} in a positive dimensions. The general conjecture we consider is whether J_{i} = I_{π} for interesting classes of fibrations (and suitable choices of the ring R).

If H is a normal subgroup of a group G and Q = G/H, there is a fibration BH → BG → BQ; the maps i^{∗} and π^{∗} are the restriction and inflation maps, respectively. In this case the general conjecture asks if there is an equality of ideals

Ker(res_{H}^{G}) = (Im(inf_{Q}^{G}))

in H^{+}(G). Write J_{H} and I_{H} for these two ideals J_{i} and I_{π}.

In the special case Q ≅ Z_{2} and R = F_{2}, the cohomology is H^{∗}(Q) = F_{2}[x], a polynomial ring in a single class x ∈ H^{1}(Q), so the conjecture asks if the kernel of the restriction to a subgroup of index 2 is the principal ideal generated by inf(x) ∈ H^{1}(G). Johnson [3] answered this in the affirmative.

If Q is a cyclic p-group not of order 2, and R = F_{p}, then H^{∗}(Q) = F_{p}[x, y]/x^{2} (with x ∈ H^{1}Q, y ∈ H^{2}Q). In this case I_{H} = (inf(x), inf(y)). Quillen and Venkov [4] show that indeed (inf(y)) alone already contains much of J_{H}, including J_{H}^{2} and J_{H} ⌣ inf(x). We will show in section 2, however, that I_{H} properly contains J_{H} in general when p is odd.

Even when p = 2, Johnson’s case is exceptional. If Q = Z_{4}, we have a conjecture as in the previous paragraph, but provide a related counterexample. We also consider in section 4 the case Q = Z_{2} × Z_{2} and show that J_{H} may be strictly larger than I_{H} = (inf(x_{1}), inf(x_{2})).

Similar conjectures may be made for the inflation map. If H is a central subgroup of G, there is a fibration

BG → BQ → K(H, 2);

the general conjecture then asks if the kernel of the inflation map is the ideal generated by the image of H^{+}(K(H, 2)). The simplest case, analogous to the case of the restriction map, would be H = Z_{2} and R = F_{2}. In this case

H^{∗}(K(Z_{2}, 2)) = F_{2}[u, Sq^{1}u, Sq^{2}Sq^{1}u, …].

Since π^{∗}(u) ∈ H^{2}(Q, R) ≅ H^{2}(Q, H) is the extension cocycle q of the extension

1 → Z_{2} → G → Q → 1,

this is equivalent to the conjecture that Ker(inf_{Q}^{G}) is not any larger than the ideal (q, Sq^{1}(q), Sq^{2}Sq^{1}(q), …). We provide a counterexample to this in section 5.

Observe that for a general fibration, H^{∗}(F) may be found from the Eilenberg-Moore spectral sequence. It is not hard to see that the map H^{∗}(E) → E_{2}^{0,∗} has kernel equal to I_{π}, while the map H^{∗}(E) → E_{∞}^{0,∗} ⊆ H^{∗}(F) is i^{∗} and so has kernel J_{i}. Thus the general conjecture for fibrations is equivalent to the conjecture that all differentials d_{r} : E_{r}^{−r,∗} → E_{r}^{0,∗} with r ≥ 2 map to zero on the vertical axis. Early examples of the non-collapse of this spectral sequence [8] thus provide counterexamples to the most general conjecture; conversely, the examples in this paper will imply that the spectral sequence need not collapse even in the very restricted settings considered here. It would be of great interest to describe these higher differentials in a group-theoretic framework.

2 Cyclic Quotient Groups

One can study J_{H} = Ker(res_{H}^{G}) in this case using the Serre spectral sequence, with

E_{2}^{p,q} = H^{p}(Q, H^{q}(H));

J_{H} is the set of elements of H^{∗}(G) representing terms in E_{∞}^{pq} with p > 0.

We will need to study the E_{2} term in detail. The rows are modules over E_{2}^{∗,0} = H^{∗}(Q) which may be presented (for |Q| > 2) as the ring F_{p}[x, y]/x^{2}. This y ∈ H^{2}(Q) establishes the periodicity of Q:

⌣ y : E_{2}^{p,q} → E_{2}^{p+2,q}

is onto, and an isomorphism if p ≠ 0. It is also true that E_{2}^{p,q} ≅ E_{2}^{p+1,q}, but this isomorphism is not necessarily given by ⌣ x. This observation will provide the counterexamples needed.

Suppose first that |Q| = p, an odd prime. Then there are precisely p nontrivial indecomposable F_{p}[Q]-modules M(i), one of each dimension i = 1, 2, …, p as an F_{p}-vector space. On the one hand, we have M(1) = F_{p}, and H^{n}(Q, M(1)) ≅ F_{p} for all n; here, ⌣ x does induce the isomorphisms H^{2n} → H^{2n+1}.

At the other extreme, M(p) is the module induced from the trivial module of {1} < Q, and so by Shapiro’s lemma,

H^{n}(Q, M(p)) ≅ H^{n}({1}, F_{p}) = 0

for n > 0; there is no question of periodicity.

In the remaining cases, we have

\underline{Lemma 1.} If 1 < i < p, H^{n}(Q, M(i)) ≅ F_{p} for all n. The cup product ⌣ x : H^{0} → H^{1} is zero.

\underline{Proof.} The first part is straightforward. Pick a generator g ∈ Q and write the action of g on M(i) in Jordan canonical form. Then one can compute S = 1 − g and T = 1 + g + … + g^{p−1}; the latter turns out to be zero. Hence H^{2n}(Q, M(i)) = Ker(S)/Im(T) ≅ F_{p} and H^{2n+1}(Q, M(i)) = Ker(T)/Im(S) ≅ F_{p} also.

The exact sequence (for i > 1)

0 → M(1) →^{j} M(i) →^{π} M(i − 1) → 0

then gives a natural long exact sequence which alternates between zero maps and isomorphisms. In particular, π_{∗} is an isomorphism on H^{1}(Q, M(i)).

But the naturality of the sequence then implies a diagram

M(i)^{Q} = H^{0}(Q, M(i)) ⟶^{⌣x} H^{1}(Q, M(i))
↓π=0 ↓π_{∗} ↓π_{∗}=(≅)
M(i − 1)^{Q} = H^{0}(Q, M(i − 1)) ⟶^{⌣x} H^{1}(Q, M(i − 1))

whose commutativity now prohibits the top ⌣ x from being an isomorphism.//

Now return to the Serre spectral sequence. Write each term H^{q}(H) on the vertical axis as a sum of copies of the M(i). If there occurs an M(i) with 1 < i < p, Lemma 1 shows E_{2}^{1,q} contains elements not multiples of x ∈ E_{2}^{1,0}. If these elements survive to E_{∞}, then any z ∈ H^{∗}(G) representing such a class [z] ∈ E_{∞}^{1,q} lies in J_{H}, but not in I_{H}. (Indeed, if z = a · inf(x) + b · inf(y), then [a] ∈ E_{∞}^{0,q} and b = 0, so [z] = x ⌣ [a], a contradiction.) The following proposition then follows easily.

\underline{Proposition 2.} If G is the non-abelian group of order p^{3} and exponent p, and if [G : H] = p, then J_{H} is strictly larger than I_{H}.

\underline{Proof.} We have H ≅ Z_{p}^{2} as groups, and in fact H ≅ M(2) as Q-modules; G is the spilt extension M(2) · Q. Then H^{1}(H) is dual to H, and hence also is an M(2), ensuring the existence of the desired class in E_{2}^{1,1}. The only differential that could affect this group is d_{2} : E_{2}^{1,1} → E_{2}^{3,0}, but as the map G → Q is split, inf_{Q}^{G} is an injection, and so maps onto E_{r}^{∗,0} are zero. Thus, E_{∞}^{1,1} ≠ 0, and we apply the analysis preceding the proposition.//

Here is a similar example when |Q| = 4. Let M(2) be the indecomposable 2-dimensional F_{2}[Q]-module. As with Lemma 1 we may show

\underline{Lemma 3.} H^{n}(Q, M(2)) ≅ F_{2} for all n. However, the map ⌣ x : H^{0}(Q, M(2)) → H^{1}(Q, M(2)) is zero.//

Then the analysis preceding Proposition 2 may again be used to prove

\underline{Proposition 4.} Let G = Z_{2}^{2} · Z_{4} be the non-trivial split extension, and H = Z_{2}^{2} · {1}. Then J_{H} is strictly larger than I_{H}.//

3 The Gysin sequence

We will use the following result in the next section. It seems to be well known, but we take this opportunity to make explicit some very useful consequences.

\underline{Theorem 5.} If [G : H] = 2 and M is any F_{2}[G]-module, there is a natural exact triangle

H^{∗}(G, M)  ⟶^{⌣α}  H^{∗}(G, M)
cor ↖  ↙ res
H^{∗}(H, M)

where α ∈ H^{1}(G, F_{2}) corresponds to H.

(This correspondence pairs each β ∈ Hom(G, F_{2}) to its kernel. Equivalently, it pairs each subgroup K of index 2 with the unique non-zero element in inf H^{1}(G/K, F_{2}).)

\underline{Proof.} It is particularly valuable to label the one map as the corestriction (transfer). Here is a proof that makes this easy. The exact sequence of G-modules

0 → M →^{i} Ind_{H}^{G}(M) →^{π} M → 0

(which is only exact since [G : H] = 2!) induces a long exact sequence in cohomology. By Shapiro’s lemma, H^{n}(G, Ind_{H}^{G}M) ≅ H^{n}(H, M) and i_{∗} is the restriction. The resulting description of π_{∗} is the corestriction by definition (see, e.g., [2, p. 226]).

To see that the connecting homomorphism Δ is (⌣ α), we use dimension– shifting. Write G = H ⋃ g_{0}H. Then the map

Δ : H^{0}(G, M) = M^{G} → H^{1}(G, M) = Deriv(G, M)

is usually described as follows: given m ∈ H^{0} = M^{G}, pick any element m′ ∈ F_{2}[G] ⊗_{F_{2}[H]} M = (1 ⊗ M) + (g_{0} ⊗ M) with m = π(m′) and then define Δ(m) = f_{m} : G → M by f_{m}(g) = m′ − g(m′). We may take m′ = 1 ⊗ m, and see f_{m}(h) = 1 ⊗ (1 − h)m for h ∈ H and f_{m}(g_{0}h) = 1 ⊗ m − g_{0} ⊗ hm for g_{0}h ∈ g_{0}H; as m ∈ M^{G} and −1 = +1, these are f_{m}(h) = 0 and f_{m}(g_{0}h) = m, respectively, so f_{m}(g) = α(g) · m, and thus Δ(m) = α ⌣ m ∈ H^{1}(G, M).

If Δ is known to equal ⌣ α on H^{n}(G, M) for all modules M, we may prove it on H^{n+1} by constructing sequences 0 → M′ → M → M′′ → 0 on which the connecting maps δ : H^{n}(G, M′′) → H^{n+1}(G, M′) are surjections. By the commutativity of 5 sides of Figure 1, one concludes Δ = (⌣ α) on H^{n+1}(G, M′).//

H^{n}(G, M′′)  ^{⌣α}  H^{n+1}(G, M′′)
|  =  |  =
|  H^{n}(G, M′′)  ^{Δ}  |  H^{n+1}(G, M′′)
δ  δ  δ  δ
|  |
H^{n+1}(G, M′)  ^{⌣α}  H^{n+2}(G, M′)
=  =
H^{n+1}(G, M′)  ^{Δ}  H^{n+2}(G, M′)

Figure 1

Here the commutativity of the top side is assumed, amd that of the left and right ends is immediate. The front commutes because of the commutativity of the vertical and horizontal connecting homomorphisms resulting from

M′  →  M  →  M′′
↓  ↓  ↓
Ind(M′) → Ind(M) → Ind(M′′)
↓  ↓  ↓
M′  →  M  →  M′′

and the commutativity of the rear face follows since δ is a derivation (i.e., δ(α ⌣ x) = α ⌣ δ(x).//

We most frequently take M = F_{2}. This makes easy many applications. For example,

\underline{Corollary 6.} Let g_{n} = dim(H^{n}(G)), a_{n} = dim(H^{n}(G) ∩ Ann(α)) and h_{n} = dim(H^{n}(H)) Then a_{n} + a_{n−1} = h_{n} − (g_{n} − g_{n−1}).

Equivalently, the Poincaré series are related:

(1 + t)P_{A}(t) = P_{H}(t) − (1 − t)P_{G}(t).

The proofs are just calculations.//

So, for example, the Poincaré series of the index-2 subgroups of G are determined by H^{∗}(G). Conversely, if P_{G} and P_{H} are known, we can calculate the size of Ann(α). This is useful in the computation of the ring structure of H^{∗}(G), since a spectral sequence argument will leave only imprecise relations such as “αx ≡ 0 modulo filtration”. However, if we know the dimension of Ann(α) is large enough, we can conclude αx = 0 for an appropriate x. Furthermore, with Theorem 5, we see we can describe these elements of H^{∗}(G) as corestrictions, making it possible to calculate ring structure and Steenrod-algebra action. Examples may be found in [5] and [6]. Another interesting application is

\underline{Corollary 7a.} If res_{H}^{G} is onto then G can only be a split extension, G = H · Z_{2}.

\underline{Proof.} Indeed, if res is onto, the corestriction will be zero, so Ann(α) = 0. Thus, P_{A} = 0 and P_{H} = P_{G} · (1 − t). In particular, the order of the pole at t = 1 of P_{H} is one less than that of P_{G}. Since these orders measure the Krull dimension, and since the Krull dimension measures the 2-rank, it follows G has an elementary-abelian subgroup E of larger rank than any in H. Hence E ≮ H. So, pick x ∈ E − H and observe G = H · ⟨x⟩. //

The referee was kind enough to point out that this proposition is not specific to the prime 2, and does not require Theorem 5. I would like to present this person’s proof (with just a few stylistic changes).

\underline{Corollary 7b.} If G is an extension of H by Z_{p} and res_{H}^{G} is onto then G is a split extension.

\underline{Proof.} The image of res_{H}^{G} is contained in H^{∗}(H)^{Z^{p}}, so H^{∗}(H) is a trivial Z_{p}-module. It follows that the E_{2}^{∗,∗} page of the spectral sequence for G as an extension of H by Z_{p} is isomorphic to the graded tensor product of H^{∗}(H) and H^{∗}(Z_{p}), and that this spectral sequence collapses. This gives the same relation P_{H} = (1 − t)P_{G} between the Poincaré series as in the proof of Corollary 7a, so that the argument there applies here as well.//

4 Elementary-abelian quotients

In view of section 2, we see it is unreasonable to suppose I_{H} = J_{H} unless Q is a 2-group and of exponent 2, that is, Q is an elementary abelian group. We will now show that Q = Z_{2} is the only one of these for which I_{H} must equal J_{H}. We will find a group G and an H[normal]G with G/H ≅ Z_{2} × Z_{2} but with Ker(res_{H}^{G}) larger than Im(inf_{Q}^{G}).

Let G be the unique 2-generated group of rank 3 among the groups of order 16. This is G = 16Γ_{2}c_{1} in the tables [1], and may be described as the central extension

1 → Z_{2}^{2} → G → Z_{2}^{2} → 1

corresponding to the extension cocycle (x_{1}x_{2}, x_{2}^{2}) ∈ F_{2}[x_{1}, x_{2}] ⊕ F_{2}[x_{1}, x_{2}] = H^{∗}(Z_{2}^{2}, Z_{2}^{2}). Observe that the maximal subgroup M = Ker(x_{2}) is elementaryabelian of order 8. We also record for use in section 5 that N = Ker(x_{1}) is isomorphic to Z_{2} × Z_{4}, and that H = M ∩ N is the center Z_{2}^{2}.

\underline{Proposition 8.} There is a class λ ∈ H^{2}(G) in Ker(res_{H}^{G}) but not in the ideal generated by H^{1}(G) (and hence not in Im(inf_{Q}^{G})).

\underline{Proof.} We will construct λ so that its restriction to M is a multiple of x′_{1} = res_{M}^{G}(x_{1}) ≠ 0. It then follows from section 3 that λ restricts to zero on H = Ker(x′_{1}).

We first establish some properties of H^{∗}(G). From the Serre spectral sequence, say, we see that inf_{G/H}^{G} : H^{1}(G/H) → H^{1}(G) is an isomorphism, that is, H^{1}(G) = F_{2}x_{1}⊕F_{2}x_{2}. Moreover, the extension cocycles in H^{2}(G/H) inflate to become relations in H^{2}(G): x_{1}x_{2} = x_{2}^{2} = 0.

We next choose a basis for H^{1}(M). Since G is not abelian and G/M is cyclic, M is a non-trivial G/M ≅ Z_{2}-module; so too is H^{1}(M) ≅ Z_{2}^{3} nontrivial, and thus H^{1}(M) is the sum of a trivial module and a permutation module. Since res_{M}^{G}cor_{M}^{G} is the trace tr(g) = ∑_{G/M} g, it follows the invariant subspace of the permutation module lies in res(H^{1}(G)) ≅ H^{1}(G)/(F_{2} · x_{2}) ≅ F_{2}, and so this subspace is spanned by x′_{1}. Choose x′_{2} to complete a basis of the G-invariant subspace of H^{1}(M), and then choose x′_{3} to complete the basis of all of H^{1}(M). Observe that since tr(x′_{3}) is not zero but is a restriction from G, we must have tr(x′_{3}) = x′_{1}.

Let λ = cor_{M}^{G}(x′_{2}x′_{3}). Then res_{M}^{G}(λ) = tr(x′_{2}x′_{3}) = x′_{2} ⌣ tr(x′_{3}) = x′_{2}x′_{1} is indeed a multiple of x′_{1}. But λ ∉ H^{1}(G) ⌣ H^{1}(G): since H^{1}(G) is spanned by x_{1} and x_{2}, and since x_{1}x_{2} = x_{2}^{2} = 0 in H^{∗}(G), we see H^{1}(G) ⌣ H^{1}(G) is spanned by x_{1}^{2}. Yet λ = c · x_{1}^{2} would force resλ = c · (x′_{1})^{2} ≠ x′_{1}x′_{2}, a contradiction. //

The entire cohomology ring H^{∗}(G) can be now computed with the Serre spectral sequence. We will use this in the following section, so we describe the conclusions here. There are two natural representations ρ_{i} : G → M_{2}(R) corresponding to the quotients G/H_{1} ≅ Z_{2} × Z_{4} and G/H_{2} ≅ D_{8} by the two summands H_{i} of the center H ≅ Z_{2}^{2}. These yield two Stiefel-Whitney classes y_{i} = w_{2}(ρ_{i}) ∈ H^{2}(G) which can be used to represent the only other two classes needed to generate this cohomology ring.

The cohomology ring of 16Γ_{2}c_{1} is then

H^{∗}(G) = F_{2}[x_{1}, x_{2}, y_{1}, y_{2}, λ]/(x_{1}x_{2}, x_{2}^{2}, x_{2}λ, λ^{2} + x_{1}^{2}y_{1})

= F_{2}[x_{1}, y_{1}, y_{2}] · (1) + F_{2}[x_{1}, y_{1}, y_{2}] · (λ) + F_{2}[x_{1}, y_{1}, y_{2}]/x_{1} · (x_{2})

with Sq^{1}(y_{1}) = 0, Sq^{1}(y_{2}) = x_{1}y_{2}, and Sq^{1}(λ) = x_{2}y_{2} + x_{1}y_{1} + x_{1}λ.

Here and in the following section we write R · (a, b, c, …) for the free R-module on a, b, c, . . .. We calculate here the action of Sq^{1} with the formula Sq^{1}(w_{2}) = w_{1}w_{2} + w_{3} for the action of the Steenrod algebra on Stiefel-Whitney classes w_{i} = w_{i}(ρ) and the formula Sq^{1} ∘ cor = cor ∘ Sq^{1} for its action on transfers.

5 Kernels of the inflation map

We have calculated in [7] the mod-2 cohomology rings of all the groups of order 32, viewing each as a central extension of a group G of smaller order, and using spectral sequence calculations. As noted in section 1, such calculations are connected with the descriptions of the kernels of the inflation map envisioned in this paper. We found that in almost every case considered, the kernel was precisely the ideal suggested in section 1. However, in some of the most complicated cases, the kernel is somewhat larger.

\underline{Proposition 9.} There is a central extension

1 → Z_{2} → X → G → 1

of the group G = 16Γ_{2}c_{1} corresponding to a cocycle q ∈ H^{2}(G, Z_{2}), such that the kernel of the inflation map inf : H^{∗}(G) → H^{∗}(X) is strictly larger than

I = (q, Sq^{1}q, Sq^{2}Sq^{1}q, …).

(This I is the smallest ideal of H^{∗}(G) containing q and invariant under the mod-2 Steenrod algebra.)

\underline{Proof.} We continue the notation G for the group 16Γ_{2}c_{1} of the previous section; however, we will write \overline{M} for the maximal subgroup we called M there; M will now denote its preimage in X. In the same way we now have subgroups H and N of X mapping to subgroups \overline{H} and \overline{N} of G.

We take the cocycle q = λ + y_{1} + x_{1}^{2}. Then it may be shown that the extension group X is 32Γ_{7}a_{3} in the tables [1]. Let θ = x_{1}^{2}y_{1}y_{2}^{2} ∈ H^{8}(G). We will show θ ∈ Ker(inf) but θ ∉ I; this is sufficient to prove Proposition 9. The second assertion is easier so we begin with it.

\underline{Proposition 10.} θ does not lie in the ideal I = (q, Sq^{1}(q), Sq^{2}Sq^{1}(q), …).

\underline{Proof.} Since θ is of dimension 8, it suffices to show θ does not lie in the ideal (q, Sq^{1}(q), Sq^{2}Sq^{1}(q)), and thus that θ ≠ 0 in the quotient ring H^{∗}(G)/(q, Sq^{1}q, Sq^{2}Sq^{1}(q)).

We compute Sq^{1}(q) = x_{1}y_{1}+x_{2}y_{2}+x_{1}λ and Sq^{2}Sq^{1}(q) = x_{1}y_{1}^{2}+x_{2}y_{2}^{2}+x_{1}^{3}y_{1}. Then it is easy to compute quotient rings:

H^{∗}(G)/q = F_{2}[x_{1}, x_{2}, y_{1}, y_{2}]/(x_{1}x_{2}, x_{2}^{2}, x_{2}y_{1},
y_{1}^{2} + y_{1}x_{1}^{2} + x_{1}^{4})
= F_{2}[x_{1}, y_{2}] · (1, y_{1}) + F_{2}[y_{2}] · (x_{2})
H^{∗}(G)/(q, Sq^{1}(q)) = F_{2}[x_{1}, x_{2}, y_{1}, y_{2}]/(x_{1}x_{2}, x_{2}^{2}, x_{1}^{3} + y_{2}x_{2},
x_{2}y_{1}, y_{1}^{2} + y_{1}x_{1}^{2})
= F_{2}[y_{2}] · (1, x_{1}, x_{1}^{2}, x_{2}, y_{1}, y_{1}x_{1}, y_{1}x_{1}^{2})
H^{∗}(G)/(q, Sq^{1}q, Sq^{2}Sq^{1}q) = F_{2}[x_{1}, x_{2}, y_{1}, y_{2}]/(x_{1}x_{2}, x_{2}^{2}, x_{1}^{3} + y_{2}x_{2},
x_{2}y_{1}, y_{1}^{2} + y_{1}x_{1}^{2}, x_{2}y_{2}^{2})
= F_{2}[y_{2}] · (1, x_{1}, x_{1}^{2}, y_{1}, x_{1}y_{1}, x_{1}^{2}y_{1})
⊕F_{2} · (x_{2}, y_{2}x_{2})

In particular, x_{1}^{2}y_{1}y_{2}^{2} ≠ 0, as desired.//

It is a little trickier to show that θ lies in Ker(inf_{G}^{X}). We will need some preparatory work first. Recall that \overline{M} = Ker(x_{2}) < G so that M = Ker(inf_{G}^{X}(x_{2})) < X.

\underline{Lemma 11.} res_{M}^{X} is an injection on H^{7}(X).

\underline{Proof.} Lengthly but straightforward calculations in the Serre or Eilenberg-Moore spectral sequences show H^{∗}(X) has a Poincaré series beginning

P_{X}(t) = 1 + 2t + 3t^{2} + 4t^{3} + 4t^{4} + 5t^{5} + 6t^{6} + …

It is not hard to identify M from the tables (or see step (1) below): M ≅ Q_{8} × Z_{2} and so its series is

P_{M}(t) = 1 + 3t + 5t^{2} + 6t^{3} + 7t^{4} + 9t^{5} + 11t^{6} + …

From Corollary 6 we get the series for Ann(x_{2}):

P_{A}(t) = P_{X}(t) − (1 + t^{2} + t^{3} + t^{5} + 0t^{6} + …)

So the series for Ker(res_{M}^{X}) = (x_{2}) is

t · (P_{X} − P_{A}) = t + t^{3} + t^{4} + t^{6} + 0 · t^{7} + …

and thus res_{M}^{X} is an injection in dimension 7. //

Now let θ_{1} = x_{1}y_{1}y_{2}^{2} ∈ H^{7}(G) (so that θ = x_{1}θ_{1}). Our goal is to show that after inflation to X, inf_{G}^{X}(θ_{1}) = cor_{N}^{X}(φ) for some φ, where N = Ker(x_{1}). For then

inf_{G}^{X}(θ) = cor_{N}^{X}(φ ⌣ res_{N}^{X}(x_{1})) = 0,

i.e., θ lies in the kernel of inf_{G}^{X}. The construction of φ is not straightforward.

We take several steps to clarify the cohomological structure of some groups K related to G. Figure 2 may be helpful. It associates to each K the elements we discuss in each of the cohomology rings H^{∗}(K). The inflation map goes left, the restriction down, and transfer (and norm) up.

(1) resG ; \overline{M}: Recall that \overline{M} = Ker(x_{2}) < G. We discussed this subgroup in the previous section: \overline{M} ≅ Z_{2}^{3}, and its cohomology ring is generated by x′_{1} = res_{M}^{G}(x_{1}), x′_{2} (which is G-invariant) and x′_{3}. With the description of the classes y_{i} ∈ H^{2}(G) as Stiefel-Whitney classes, it is possible to calculate res(y_{1}) = (x′_{2})^{2} and res(y_{2}) = x′_{3}(x′_{3} + x′_{1}), so that res(θ_{1}) = res(x_{1}y_{1}y_{2}^{2}) = (x′_{1})(x′_{2})^{2}(x′_{3})^{2}(x′_{1}+x′_{3})^{2}. Similarly we calculate that the cocycle q = λ+y_{1}+x_{1}^{2} restricts to (x′_{1})^{2} + x′_{1}x′_{2} + (x′_{2})^{2}. (This implies M ≅ Q_{8} × Z_{2}).

(2) infM ; \overline{M} : Write x′′_{i} for inf(x′_{i}) ∈ H^{∗}(M). Since q′ = resG ; \overline{M}(q) and Sq^{1}(q′) must inflate to zero, we may inflate their definitions to obtain relations

(x′′_{2})^{2} = (x′′_{1})^{2} + x′_{1}x′′_{2}
(x′_{1})^{3} = 0

Hence res_{M}^{G}(θ_{1}) inflates to (x′′_{1})^{2}(x′′_{2})(x′′_{3})^{4}.

(3) N and M ∩ N: We defined \overline{N} = Ker(x_{1}) < G which in section 4 we noted to be isomorphic to Z_{2} × Z_{4}. As was done for \overline{M} we may calculate resG ; \overline{N}(q) and discover N to be the extension

1 → Z_{2} → N → Z_{2} × Z_{4} → 1

usually called the “quasidihedral group”, N ≅ ⟨a, b : a^{8} = b^{2} = 1, a^{b} = a^{5}⟩. Then as M = Ker(inf_{G}^{X}(x_{2})) < X, M ∩ N = Ker(res_{N}^{X}inf_{G}^{X}x_{2}) < N, i.e., Ker(infN ; \overline{N}resG ; \overline{N}x_{2}); then \overline{M ∩ N} = Ker(resG ; \overline{N}(x_{2})) < \overline{N} is the subgroup Z_{2}^{2} < Z_{2} × Z_{4}. It is then easy to check that M ∩ N = ⟨a^{2}, b⟩ < N, so M ∩ N ≅ Z_{4} × Z_{2}.

X  G_{x_{i} y_{i} λ θ θ_{1}}
M_{x′′_{1} x′′_{2} x′′_{3} w}  N_{φ}  \overline{M}_{x′_{1} x′_{2} x′_{3}}  \overline{N}
(Q_{8} × Z_{2})  ⟨a, b⟩ ≅ QD (≅ Z_{2}^{3})  (Z_{4} × Z_{2})
(M ∩ N)_{α β γ}  \overline{M ∩ N} ≅ Z_{2}^{2}

Figure 2

(4) H^{∗}(M ∩ N): We may write H^{∗}(M ∩ N) as F_{2}[α, β]/α^{2} ⊗ F_{2}[γ], since we know the cohomology of Z_{4} and Z_{2}. The elements α, γ ∈ H^{1}(M ∩ N) = Hom(M ∩ N, F_{2}) are characterized by their actions:

α(a^{2}) = 1 α(b) = 0
γ(a^{2}) = 0 γ(b) = 1.

Similarly we describe β as (the restriction mod 2 of) the Chern class of a representation ρ : M ∩ N → GL(1, C):

ρ(a^{2}) = i ρ(b) = 1

(5) Action of N: We calculate the effect of N/(M ∩ N) ≅ Z_{2} on the ring H^{∗}(M ∩ N) by using the formulas in (4): since N = (M ∩ N) ⋃ a · (M ∩ N), the action of N on H^{∗}(M ∩ N) is described by the map c_{a}^{∗} induced by conjugacy by a. But

c_{a}^{∗}(α) = α ∘ c_{a} = α
c_{a}^{∗}(γ) = γ ∘ c_{a} = γ

and c_{a}^{∗}(β) is the Chern class of c_{a}^{∗}(ρ), which sends

c_{a}^{∗}(ρ)(a^{2}) = (ρ ∘ c_{a})(a^{2}) = ρ(a^{2}) = i
c_{a}^{∗}(ρ)(b) = (ρ ∘ c_{a})(b) = ρ(b · a^{4}) = −1

so that c_{a}^{∗}(ρ) = ρ ⊗ γ, and the Chern class is c_{a}^{∗}(β) = c_{1}(c_{a}^{∗}ρ) = c_{1}(ρ ⊗ γ) = c_{1}(ρ) + c_{1}(γ) = β + γ^{2}.

In particular, β(β + γ^{2}) is invariant under the action of N.

(6) H^{∗}(M) and H^{∗}(M ∩ N): We have seen M ≅ Q_{8} × Z_{2} so that one may write

H^{∗}(M) = F_{2}[x′′_{1}, x′′_{2}, x′′_{3}, w]/((x′_{2})^{2} + (x′′_{1})^{2} + (x′′_{1})(x′′_{2}), (x′′_{1})^{3})

The class w ∈ H^{∗}(M) may be chosen arbitrarily among the elements restricting non-trivially on [M, M] ≅ Z_{2}. We will describe such a class momentarily.

As in step (3) we may observe M ∩ N is the subgroup corresponding to res_{M}^{X}(inf_{G}^{X}(x_{1})) = x′′_{1}. Then x′′_{2} restricts to a non-zero nilpotent class in H^{1}(M ∩ N), necessarily α, and x′′_{3} restricts to a non-nilpotent class, γ or γ + α. In particular, res_{M∩N}^{M}((x′′_{3})^{2}) = γ^{2}.

Here is a choice for w. Recall β = c_{1}(ρ) for a ρ : M ∩ N → GL(1, C). Since (Q_{8} × Z_{2})/(Z_{4} × Z_{2}) acts as inversion, M/(M ∩ N) interchanges ρ and ρ̄; but complex conjugate representations have equal Chern classes (mod 2). Hence β ∈ H^{2}(M ∩ N) is M-invariant, and then res_{M∩N}^{M}norm_{M∩N}^{M}(β) = β^{2}. Since this norm(β) restricts non-trivially on the squares of M, we may take w = norm(β). Thus, β^{2} = res(w).

Finally, we note cor(β) ∈ H^{2}(M) is annihilated by x′′_{1} and so is (x′′_{1})^{2}.

In the next steps we can now show inf(θ_{1}) is a transfer from N of some element φ ∈ H^{7}(N).

(7) Description of φ: We take φ = cor_{M∩N}^{N}(αβ^{2}(β+γ^{2})). Then res_{M∩N}^{N}(φ) is tr(αβ^{2}(β + γ^{2})).

But in (5) we saw that α and β(β+γ^{2}) were N-invariant, so the restriction of φ may be written

res_{M∩N}^{N}(φ) = αβ(β + γ^{2}) · tr(β) = αβ(β + γ^{2})γ^{2}.

We transfer this back to H^{∗}(M), using the formulas in (6):

cor_{M∩N}^{M} (αβ (β + γ^{2})γ^{2})

= cor(αβ^{2}γ^{2}) + cor(αβγ^{4})
= cor(res(x′′_{2})res(w)res((x′′_{3})^{2})) + cor(res(x′′_{2})res((x′′_{3})^{4})β)
= (x′′_{2})(x′′_{3})^{4}cor(β)
= (x′′_{2})(x′′_{3})^{4}(x′′_{1})^{2}.

From the double coset formula we then also have

res_{M}^{X}cor_{N}^{X}(φ) = (x′′_{1})^{2}(x′′_{2})(x′′_{3})^{4}

.

(8) Relation to θ: Steps (1) and (2) showed that infM ; \overline{M}resG ; \overline{M}(θ_{1}) was equal to this same (x′′_{1})^{2}(x′′_{2})(x′′_{3})^{4}. Another double coset formula then shows that res_{M}^{X}inf_{G}^{X}(θ_{1}) = (x′′_{1})^{2}(x′′_{2})(x′′_{3})^{4} also. Comparing to step (7) we deduce by Lemma 11 that inf_{G}^{X}(θ_{1}) = cor_{N}^{X}(φ), i.e., inf_{G}^{X}(θ_{1}) lies in the ideal Im(cor_{N}^{X}) = Ann(inf_{G}^{X}(x_{1})). This means

0 = inf_{G}^{X}(x_{1}) ⌣ inf_{G}^{X}(θ_{1}) = inf_{G}^{X}(θ),

completing the proof of Proposition 9.//

This result implies a non-zero differential d_{2} : E_{2}^{−2,9} → E_{2}^{0,8} in the Eilenberg-Moore spectral sequence, and a non-zero d_{7} : E_{7}^{1,6} → E_{7}^{0,8} in the Serre spectral sequence which is not a result of the transgression. The existence of these examples means it will be necessary to predict non-zero differentials in a group-cohomological setting when doing future calculations. We can offer no interpretation at this time, although it is possible that the corestriction provides the necessary data in general, as it does in this case.

We also observe this ‘extra’ class in H^{n}(G) ∩ Ker(inf) implies an obstruction to the extension problem

BX  ⟶  BG
↘  _{↙}^{↙}
K(Z_{2}, n − 1)

not implied by the ring structure or the primary cohomology operations. It seems plausible that some higher cohomology operations might predict this obstruction, but we have been unable to prove this.

References

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[3] D.L. Johnson, On the Cohomology of finite 2-groups, Invent. Math. \underline{7} (1969) 159-173.

[4] D. Quillen and B. Venkov, Cohomology of finite groups and elementary abelian subgroups, Topology \underline{11} (1972) 317-318.

[5] D. Rusin, The mod-2 cohomology of metacyclic 2-groups. J. Pure Appl. Alg. \underline{44} (1987) 315-327.

[6] D. Rusin, The cohomology of groups generated by involutions, Thesis, Chicago, 1984.

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[8] C. Schochet, A two-stage Postnikov system where E_{2} ≠ E_{∞} in the Eilenberg-Moore spectral sequence, Trans. AMS \underline{157} (1971) 113-118.