On The Geometry and Cohomology of the Simple Groups G_{2}(q) and ^{3}D_{4}(q): II

R. James Milgram†

The group GL_{3}(2) is very much involved in the structure of the simple groups. For example, there are precisely 2 extensions of the form

( /2^{n})^{3}⟶E_{n}^{i}⟶GL_{3}(2) i = 1, 2

with GL_{3}(2) acting in the usual way on ( /2)^{3} ⊂ ( /2^{n})^{3}. The first, E_{n}^{1}, is split while the second, E_{n}^{2}, is not. The non-split extension

( /2)^{3}⟶E_{1}^{2}⟶GL_{3}(2)

is a maximal subgroup of G_{2}(q) for all odd q while E_{2}^{1} is a maximal subgroup of the sporadic group HS, and E_{2}^{2} is maximal in O’N.

In [FM] we reduced the determination of the Poincaré series P_{G}(x) for H^{∗}(G; _{2}) where G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4 to the determination of P_{E_{1}^{2}}(x). Conversely, using the Poincaré series P_{G}(x), which, according to [Q2], [K], is given as

\frac{(1 + x^{3})(1 + x^{5})(1 + x^{6})}{(1 − x^{4})(1 − x^{6})(1 − x^{7})},

we determine the Poincaré series of E_{1}^{2}. (See the discussion in the introduction of [FM].) It is, however, desirable to determine the Poincaré series for E_{1}^{2} without using etale cohomology methods. In fact, since we want the cohomology rings of the groups G_{2}(q), 3D_{4}(q) and E_{1}^{2}, we must use other methods than the spectral sequence of [Q2] to study the cohomology of these groups.

From now on we concentrate on the group E_{1}^{2}, and will write it as E. We use detection theorems and the double coset formula to prove Theorem:

(1) P_{E}(x) =

\frac{1 + x^{2} + 3x^{3} + 2x^{4} + 4x^{5} + 5x^{6} + 4x^{7} + 5x^{8} + 4x^{9} + 2x^{10} + 3x^{11} + x^{12} + x^{14}}{(1 − x^{4})(1 − x^{6})(1 − x^{7})} ,

(2) There is a map e : B_{E}⟶B_{O(7)} with e^{∗}(w_{i}) ≠ 0 if and only if i = 4, 6, 7, and e^{∗} is an injection on _{2}[w_{4}, w_{6}, w_{7}] ⊂ H^{∗}(BO(7); _{2}),

A = _{2}[e^{∗}(w_{4}), e^{∗}(w_{6}), e^{∗}(w_{7})] ⊂ H^{∗}(B_{E}; _{2}).

Moreover H^{∗}(B_{E}; _{2}) is Cohen-Macaulay, i.e. free and finitely generated over A.

† Research partially supported by grants from the N.S.F. and the S.E.R.C. at the University of Edinburgh

(This is the content of 6.1 and 9.1 in the text. Here, w_{i} is the i^{th} Stiefel-Whitney class.)

The calculation in §6 constructs an explicit basis for H^{∗}(E; _{2}) over A, (this is done in 6.11, 6.12, 6.16. 6.17. and 6.18) so it is routine to write down a complete set of relations in the cohomology ring.

However, at this point we are more interested in certain other questions. This theorem gives an independent calculation of P_{G}(x) for G equal to ^{3}D_{4}(q) or G_{2}(q). Also, using this result and the description of the poset spaces, A_{2}(G), for these G given in [FM] it is direct to obtain their cohomology rings. This is done in §7. The result is Corollary:

(1) Let G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4 then the Poincaré series P_{G}(x) is independent of q and equals

\frac{(1 + x^{3})(1 + x^{5})(1 + x^{6})}{(1 − x^{4})(1 − x^{6})(1 − x^{7})}.

(2) H^{∗}(G; _{2}) is Cohen-Macaulay over a polynomial subalgebra _{2}[b_{4}, b_{6}, b_{7}] and if q

divides q′ then the G(q) ⊂ G(q′) and (res_{G(q′)}^{G(q)})^{∗} : H^{∗}(G(q′); _{2})⟶H^{∗}(G(q); _{2}) maps the polynomial algebra for G(q′) isomorphically to the polynomial algebra for G(q).

(3) As a ring H^{∗}(G; _{2}) is generated by b_{4}, b_{6} = Sq^{2}(b_{4}), b_{7} = Sq^{3}(b_{4}), a generator β in dimension 3 and f = Sq^{2}(β) in dimension 5. The relations are β^{4} = b_{7}f + b_{6}β^{2}, f^{2} = b_{4}β^{2} + b_{7}β, both of which are implied by the formula Sq^{4}(f) = b_{4}f + b_{6}β.

(This is primarily 7.6 in the text.)

There is an important connection between the prime ideal structure in H^{∗}(G; _{p}) and the structure of the modular representations of G, [Ca], and we have Corollary: H^{∗}(G; _{2}), for G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4 has exactly two minimal prime ideals I_{I} = (β, f), and the principal ideal I_{II} = (b_{7}^{2} +β^{3}f +b_{4}bv_{7}β +b_{6}fβ). (There is a one to one correspondence between minimal prime ideals and conjugate classes of maximal elementary p-subgroups of G. One of the main results of [FM] is that there are exactly two such classes for p = 2 and G as above. The explicit formulae summarize 7.3, 7.4, and 7.5 in the text, where the quotients are also studied.)

One of our original motivations for studying these questions came from the fact that Syl_{2}(G_{2}(q)) ≅ Syl_{2}(M_{12}) for q ≡ 5 mod 8. Consequently, it was natural to ask what the relation between these groups is, even though it does not appear that M_{12} ⊂ G_{2}(q) for such q. The main result of §8 is Theorem 8.3: For q ≡ 5 mod 8 there is an inclusion of rings

H^{∗}(G_{2}(q); _{2}) ⊂ H^{∗}(M_{12}; _{2})

preserving Steenrod operations and Bocksteins. Indeed, this inclusion is stably split, so the stable summand of B_{Syl_{2}(M_{12})} associated to M_{12} is not irreducible. However, I conjecture that the summand associated to G_{2}(q) is irreducible.

Denote the group Syl_{2}(M_{12}) by H. It has order 64, and it was shown in [AMM] that, in M_{12}, H is contained in two subgroups W, W′, each of order 192, and the mod 2 cohomology of M_{12} is the same as that of the amalgamated product W ∐_{H} W′. This configuration is contained in G_{2}(q) as well when q ≡ 5 mod 8, (8.1). However, there is a second configuration H ⊂ W, H ⊂ W′, contained in E and hence G_{2}(q), (1.11), which is not isomorphic to the first. It is this fact which leads to 8.3.

One of the original questions we asked about M_{12} when we proved that H^{∗}(M_{12}; _{2}) was Cohen-Macaulay over the polynomial ring A, was if there was a manifold N^{14}, rationally equivalent to a product of spheres, on which M_{12} acts freely so that H^{∗}(N/M_{12}; _{2}) has Poincaré series P_{M_{12}}(x)(1 − x^{4})(1 − x^{6})(1 − x^{7}).

I do not know the answer to this question, but E ⊂ G_{2} where G_{2} is the 14-dimensional (exceptional) Lie group of automorphisms of the Cayley projective plane, [Cox], [CW]. Consequently, E acts freely on G_{2} and the main result of §9 is Theorem 9.1: The Poincaré series for H^{∗}(G_{2}/E; _{2}) is P_{E}(x) · (1 − x^{4})(1 − x^{6})(1 − x^{7}). Indeed,

H^{∗}(G_{2}/E; _{2}) ≅ H^{∗}(E; _{2})/I(b_{4}, b_{6}, b_{7})

where I(b_{4}, b_{6}, b_{7}) is the ideal in H^{∗}(E; _{2}) spanned by b_{4}, b_{6}, and b_{7}.

Finally, in §10 we recount some remarks of A. Adem, which show that the difference between H^{∗}(E; _{2}) and H^{∗}(M_{12}; _{2}) is measured, in an appropriate sense, by H^{∗}(E; St ⊗ 2) where St is the Steinberg module for GL_{3}(2). In particular, 10.4 determines the groups H^{∗}(E; St ⊗ _{2}). Remark: The technique used here of detection on elementary subgroups is probably not applicable in the case of E_{2}^{i}, the Sylow subgroups of HS and O′N. The calculation of H^{∗}(E_{2}^{i}; _{2}) and H^{∗}(Syl_{2}(E_{2}^{i}); _{2}), i = 1, 2 would seem to be the next step in the study of the cohomology of the simple groups though.

§1. The basic groups

In this section we define the various groups needed. These include groups H_{1}, W′, which are subgroups of M_{12}, [AMM], and groups H^{(n)}, the Sylow 2-subgroups of G_{2}(q), 3D_{4}(q) with q ≡ 1 mod 4. At the end of the section we describe E and reduce the calculation of H^{∗}(E; _{2}) to the study of H^{∗}(W′; _{2}) and H^{∗}(H_{1}; _{2}).

The group (π × π) ×_{T} ( /2), where the twisting action on π × π is (g, h) ↔ (h, g) is the wreath product of π and /2 and is written π ≀ /2. In particular the dihedral group D_{8} of order 8 is /2 ≀ /2. A specific representation is given by

D_{8} = {u, w | u^{2} = w^{2} = (uw)^{2} = 1}

where u generates one of the summands of ( /2)^{2} and wuw generates the other. The generalized dihedral group D_{2^{n+1}} (of order 2^{n+1}), is given as the semi-direct product

D_{2^{n+1}} = /2^{n} ×_{T} /2

with the /2 generated by w and acting on /2^{n} by wuw = u^{−1}. A presentation is given as

D_{2^{n+1}} = {u, w | u^{2^{n}} = w^{2} = (uw)^{2} = 1}.

We also need some groups closely related to the D_{2^{n+1}}, Definition 1.1: Let n ≥ 2, then H_{2}^{(n)} is the semi-direct product of /2^{n} × /2^{n} with /2,

H_{2}^{(n)} = ( /2^{n} × /2^{n}) ×_{T} /2.

Here we let ⟨a, m⟩ = /2^{n} × /2^{n}, and the generator y of the /2 acts on ⟨a, m⟩ by yθy = θ^{−1}.

The first of these groups, H_{2}^{(2)}, is usually written H_{2} from now on.

There are very useful index 2 embeddings ϕ : H_{2}^{(n)} ⊂ D_{2^{n+1}} × D_{2^{n+1}} given by setting a ↦ (u, u), m ↦ (u^{−1}, 1), and y ↦ (w, w). We have an extension of H_{2}^{(n)} by /2, using the inclusion D_{2^{n+1}} × D_{2^{n+1}} ⊂ D_{2^{n+1}} ≀ /2,

Definition 1.2: H^{(n)} ⊂ D_{∗} ≀ /2 is the semi-direct product H_{2}^{(n)} ×_{T} ( /2). In more detail, if we denote the generator of the extending ( /2) by T, we have that H is generated by ((u^{−1}, 1), 1) = m, ((u, u), 1) = a, ((w, w), 1) = y, and ((1, 1), T) = v. Note that a^{−1}m^{−1} = ((1, u^{−1}), 1) = vmv, vav = a, vyv = y describes the action of v. Remark 1.3: H^{(n)} is the Sylow 2-subgroup of G_{2}(q) or ^{3}D_{4}(q) where q − 1 = 2^{n}c with c odd when q ≡ 1 mod 4. H^{(2)} is of special importance so we generally write H for H^{(2)} from now on.

There are obvious containments H^{(n)} ⊂ H^{(n+1)} sending a ↦ a^{2}, m ↦ m^{2}, v ↦ v, y ↦ y, and we have commutative diagrams

1.4  \begin{matrix}H^{(n)} & ⟶ & H^{(n+1)} & ⟶ ⋯ ⟶ & H^{(n+k)} & ⟶ ⋯ \\ ↓ & & ↓ & & ↓ \\ G_{2}(q) & ⟶ & G_{2}(q^{2}) & ⟶ ⋯ ⟶ & G_{2}(q^{2^{k}}) & ⟶ ⋯ \end{matrix}

for q ≡ 1 mod 4 where H^{(n+k)} ∈ Syl_{2}(G_{2}(q^{2^{k}})). (Similarly for ^{3}D_{4}(q).)

The next group that we need is the central product H_{1} = D_{8} ∗ D_{8}. It is given in terms of generators and relations as

1.5  \begin{aligned}H_{1} = \{r, y, x, v | & r^{2} = y^{2} = x^{2} = v^{2} = (rx)^{2} = (yx)^{2} \\ & = (rv)^{2} = (yv)^{2} = (ry)^{4} = (xv)^{4} = 1\}. \end{aligned}

H_{1} has an embedding H_{1} ⊂ H^{(n)} given by r ↦ a^{2^{n−2}}, y ↦ y, x ↦ x, v ↦ v. It is an extra-special 2-group, [Q], and is also isomorphic to the central product Q_{8} ∗ Q_{8} where Q_{8} is the quaternion group of order 8. H_{1} was studied in [AMM], and it was shown there that H^{∗}(H_{1}; /2) is detected by its abelian subgroups, ( /2)^{3}.

We have two decompositions of H as a semi-direct product,

H_{1} ×_{T} /2 where the extending /2 is generated by my
H_{2} ×_{T} /2 where the extending /2 is generated by v.

Also, H_{2} admits an action of /3 = {t | t^{3} = 1} by

1.6  tat^{−1} = m, tmt^{−1} = a^{−1}m^{−1}, tyt^{−1} = y.

Then tvavt^{−1} = tat^{−1} = m while vt^{2}atv = v(a^{−1}m^{−1})v = m. Similarly, tvmvt^{−1} = ta^{−1}m^{−1}t^{−1} = a while vt^{2}mtv = vav = a. Hence tv and vt^{2} have the same action on H_{2}, and the two elements together generate an action, α, of S_{3} on H_{2}. This action defines the semi-direct product

1.7  W′ = H_{2} ×_{α} S_{3}

together with an inclusion of H as one of the three Sylow 2-subgroups of W′. Similarly, writing H_{1} = ⟨a, x, v, y⟩ ⊂ H, there is an action of /3 on H_{1} defined by

1.8  τaτ^{−1} = xvy, τxvyτ^{−1} = axvy, τxτ^{−1} = y, τyτ^{−1} = xy.

This action can be explicitly constructed as follows. We have, [FM, §2], that H_{1} = Q_{8} ∗ Q_{8} ⊂ SL_{2}(q) ∗ SL_{2}(q) so H_{1} = ⟨a_{1}, a_{2}, b_{1}, b_{2}⟩ where

a_{i} = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , b_{i} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) in the i^{th} copy of SL_{2}(q).

In this representation x = a_{1}a_{2}, a = a_{1}^{−1}, v = xb_{2}, y = b_{1}b_{2}. Moreover, the following matrix has order 3 and normalizes Q_{8} in SL_{2}(q),

A_{i} = \frac{1}{2} ( \begin{matrix}−1 + i & −1 + i \\ 1 + i & −1 − i \end{matrix} )

so AaA^{−1} = b^{−1}, AbA^{−1} = ba, and we set τ = A_{1}A_{2}. Also, we have

1.9  τvτ^{−1} = a^{−1}vy.

As before we check directly that τ^{2}my and myτ have the same action on H_{1}. Consequently they define an action, β, of S_{3} on H_{1} and we define

1.10  W = H_{1} ×_{β} S_{3}.

Again, H ⊂ W as the explicit semi-direct product above.

H_{1} has a normal subgroup ⟨a, xvy⟩, which is a copy of the quaternion group Q_{8}. Moreover, the action of ⟨x, y⟩ = ( /2)^{2} on ⟨a, xvy⟩ is that of Inn(Q_{8}), the group of inner automorphisms. Also, the action β normalizes both Q_{8} and ⟨x, y⟩, so the subgroup

⟨x, y⟩ ×_{β} S_{3} ≅ S_{4} ⊂ W

can be identified with Aut(Q_{8}). Consequently, we see that W is the holomorph of Q_{8}. Note, also, that the Sylow 2-subgroup of both W, W′ is H = ⟨a, m, y, v⟩, and we have constructed a triple

1.11  {W ⊃ H ⊂ W′} = C.

⟨j, x, y⟩ is normal in both W, W′, and, [FM, §4], when G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4, the configuration C is contained in E = N_{G}(⟨j, x, y⟩)/O(N_{G}(⟨j, x, y⟩). In fact, Syl_{2}(E)) ≅ H and in the extension

⟨j, x, y⟩ ↪ E ⟶^{π}GL_{3}(2)

we have

\begin{aligned}imπ|_{W} & = \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & a & b \\ 0 & c & d \end{matrix} ) ∈ GL_{3}(2)\} \\ imπ|_{W′} & = \{( \begin{matrix}a & b & ∗ \\ c & d & ∗ \\ 0 & 0 & 1 \end{matrix} ) ∈ GL_{3}(2)\} \end{aligned}

Also, imπ|_{H} = UT_{3}(2) ⊂ GL_{3}(2), the subgroup of upper triangular matrices, and

v ↦ ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , a ↦ ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , m ↦ ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} )

while

t_{W} = τ ↦ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ) , t_{W′} ↦ ( \begin{matrix}0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

The next result is crucial. It defines the line of attack on H^{∗}(E; _{2}) which will be carried through in the remainder of this note. It is standard that if H ⊂ G is a p-Sylow subgroup, then (res_{G}^{H})^{∗} : H^{∗}(G; _{p})⟶H^{∗}(H; _{p}) is always an injection. However, it is not so easy to identify the image. One method for doing this is the double coset formula of [CE, Prop. 9.4, p. 259].

We assume L ⊂ G, L contains the Sylow p-subgroup of the finite group G, and G = ∐_{1}^{n} Lg_{i}L is a double coset decomposition of G with respect to L. Then (res_{G}^{L})^{∗} is an injection. The double coset formula determines its image. Set

K_{i} = L ∩ g_{i}Lg_{i}^{−1}, K′_{i} = L ∩ g_{i}^{−1}Lg_{i},

and let c_{g_{i}} : K′_{i}⟶K_{i} be given by k ↦ g_{i}kg_{i}^{−1}, then the double coset formula asserts that α ∈ H^{∗}(L; _{p}) is contained in the image of H^{∗}(G; _{p}) if and only if

1.12  c_{g_{i}}^{∗}(res_{L}^{K_{i}})^{∗}(α) = (res_{L}^{K′_{i}})^{∗}(α), 1 ≤ i ≤ n.

In the case of E this condition is both explicit and very simple to verify since there are only two double cosets of W′ in E. We obtain Theorem 1.13: H^{∗}(E; _{2}) ⊂ H^{∗}(W′; _{2}) is the set of elements

α ∈ [(res_{W′}^{H_{1}})^{∗}]^{−1} (H^{∗}(H_{1}; _{2})^{τ}) .

Proof: We have a double coset decompostion of E as W′ ∪ W′t_{W} W′ and

W′ ∩ t_{W} W′t_{W}^{−1} = π^{−1} \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )\} = H_{1}.

Moreover, t_{W} ym = τ normalizes this intersection since we have

τaτ^{−1} = t_{W} a^{−1}t_{W}^{−1} = jxvy
τxτ^{−1} = t_{W} xt_{W}^{−1} = y
τvτ^{−1} = t_{W} a^{−1}xvt_{W}^{−1}
= jxvy · y · a^{−1}vy
= xay
τyτ^{−1} = t_{W} xyt_{W}^{−1} = x.

Now the result is just the double coset formula applied in our particular situation.

§2. The elementary subgroups ( /2)^{3} and their fusions

From 1.13 we are interested in the interaction of H^{∗}(H_{1}; _{2}) and H^{∗}(W′; _{2}). We will determine it by studying the restriction maps to the cohomology of various elementary 2-subgroups ( /2)^{3} ⊂ H_{1}. This process is justified by the next lemma. Motivated by this, in the remainder of this section we determine the conjugacy classes of elementary subgroups of order 8 in H_{j}^{(n)}, H_{1}, W, W′, and E.

Lemma 2.1: The cohomology groups H^{∗}(H^{(n)}; _{2}), H^{∗}(H_{2}^{(n)}; _{2}), and H^{∗}(H_{1}; _{2}) are detected by restriction to elementary subgroups. Proof: For H_{1}, H_{2} this was proved in [AMM, 4.9, and the following remark]. In [FM, 8.8 and 8.9], we show that the H^{∗}(H^{(n)}; _{2}) are all isomorphic as rings. Consequently, by [QV], they are detected by restriction to representatives of their conjugacy classes of maximal elementary 2-groups. In [FM, 7.4], H^{∗}(H_{2}^{(n)}; _{2}) is calculated and shown to have no nilpotent elements. Again, using [QV], it follows that H^{∗}(H_{2}^{(n)}; _{2}) is detected by restriction to representatives for the conjugacy classes of maximal elementary 2-groups.

We introduce the notation

2.2  j = a^{2^{n−1}}, x = m^{2^{n−1}}

in the group H_{2}^{(n)} of §1. We have the following 10 subgroups in H^{(n)}, each isomorphic to ( /2)^{3}.

2.3

Name  Group
I  ⟨j, y, x⟩
II  ⟨j, ay, x⟩
III  ⟨j, y, v⟩
IV  ⟨j, ay, v⟩
V  ⟨j, a^{2^{n−2}}yx, yv⟩
V I  ⟨j, a^{2^{n−2}}yv, yx⟩
V II  ⟨j, x, my⟩
V III  ⟨j, x, amy⟩
IX  ⟨j, a^{2^{n−2}}y, v⟩
X  ⟨j, a^{2^{n−2}}y, x⟩.

In the remainder of this note we will generally refer to these groups by their roman numerals in 2.3.

There are 4 conjugacy classes of subgroups ( /2)^{3} ⊂ H_{2} or H_{2}^{(n)}, [FM, Lemma 4.4]. Representatives can be chosen as I, II, V II, and V III. In particular, X is conjugate to I unless n = 2, in which case X = II. Proposition 2.4: There are 5 conjugacy classes of subgroups ( /2)^{3} ⊂ H^{(n)}. Representatives for these 5 conjugacy classes are I, II, III, IV, and V II. In particular, V III fuses with V II and V, V I, IX fuse with III. Proof: Consider the 4 conjugacy classes of ( /2)^{3}’s in H_{2}^{(n)} , with representatives given above. Conjugation by v interchanges the last two, while the first two are normal in H^{(n)}. Hence, in H^{(n)} there are 3 conjugacy classes of ( /2)^{3}’s with representatives contained in H_{2}^{(n)}. All the other conjugacy classes must contain an element in the coset H_{2}^{(n)}v. We break this coset into two parts ⟨a, m⟩yv and ⟨a, m⟩v. For the first part we have

(a^{i}m^{j}yv)^{2} = a^{i}m^{u}yva^{i}m^{j}yv
= a^{i}m^{j}ya^{i−j}m^{−j}y
= a^{j}m^{2j}

so the elements of order two in this part of the coset have the form a^{i}yv and are all conjugate in H^{(n)}. Next,

(a^{i}m^{j}v)^{2} = a^{i}m^{j}va^{i}m^{j}v
= a^{i}m^{j}a^{i−j}m^{−j}
= a^{2i−j},

so the elements of order two in the second part of the coset are jv, a^{2^{n−2}}xv, a^{2^{n−2}}jxv, and v. The first and fourth are interchanged by conjugation by x, as are the second and third. Also conjugation by m^{2^{j−2}} interchanges the second and fourth elements. It follows that the entire coset contains exactly two conjugacy classes of elements of order 2 with representatives v and yv.

The centralizer of v is

C(v) = ⟨a, y, v⟩ = D_{2^{n+1}} × /2

which has two conjugacy classes of subgroups ( /2)^{3},

⟨j, y, v⟩, ⟨j, ay, v⟩.

The centralizer of yv is ⟨am^{2}, y, yv⟩ = D_{2^{n+1}} × /2. This also has two conjugacy classes of subgroups ( /2)^{3}, with representatives

⟨j, y, v⟩, ⟨j, axy, yv⟩ = ⟨j, axy, axv⟩

but m⟨j, ay, v⟩m^{−1} = ⟨j, axy, axv⟩. Thus there are no more than 5 conjugacy classes and it is clear that the 5 groups listed are not conjugate in H^{(n)}. The result follows.

Next we consider the conjugacy classes of ( /2)^{3}’s in H_{1}. Since H_{1} is extra special with form

( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} )

the maximal isotropic subspaces of the quotient by the center, ( /2)^{4} are ( /2)^{2}’s. Consequently, any ( /2)^{3} ⊂ H_{1} must contain j, and two ( /2)^{3}’s which are conjugate in H_{1} must actually be equal. Proposition 2.5: H_{1} has exactly 6 subgroups ( /2)^{3}. They are (as subgroups of H^{(n)} using the inclusion described after 1.5), I, III, V, V I, IX and X. Here m^{2^{n−2}} acts to interchange III with V I and IX with V. It normalizes I, X. (Note that when n = 2, IV = IX.) Proof: We write H_{1} = D_{8}×_{γ} < v, x >, where D_{8} =< a^{2^{n−2}}, y >. In particular D_{8} × /2 ⊂ H_{1} is given as < a^{2^{n−2}}, y, v >. This group has exactly 2 ( /2)^{3}’s, ⟨j, y, v⟩, and ⟨j, a^{2^{n−2}}y, v⟩. Next, consider the coset ⟨a^{2^{n−2}}, y, v⟩x. We find (a^{i}yvx)^{2} = j, (a^{i}vx)^{2} = a^{2i}j, (a^{i}x)^{2} = a^{2i}, (a^{i}yx)^{2} = 1, so the elements of order 2 in this coset are a^{(2i+1)2^{n−2}}vx, a^{i2^{n−2}}yx. Any ( /2)^{3} must contain j, so we obtain 4 possiblities x, a^{2^{n−2}}vx, yx, and a^{2^{n−2}}yx. The centralizer of x is D_{8} × /2 = ⟨a^{2^{n−2}}, x, y⟩, so this gives 2 groups ⟨j, y, x⟩ and ⟨ja^{2^{n−2}}y, x⟩. The centralizer of a^{2^{n−2}}vx is D_{8} × /2 = ⟨a^{2^{n−2}}, yv, a^{2^{n−2}}vx⟩, and there are again 2 groups ⟨j, yv, a^{2^{n−2}}vx⟩, and ⟨j, a^{2^{n−2}}yv, a^{2^{n−2}}vx⟩. The centralizer of yx is D_{8} × /2 = ⟨va^{2^{n−2}}, y, xy⟩, which makes no further contribution to the list. Finally, the centralizer of a^{2^{n−2}}yx is again D_{8} × /2 = ⟨a^{2^{n−2}}v, yv, a^{2^{n−2}}yx⟩, and this again gives no further addition to the list.

We still need to check the action of m^{2^{n−2}}. Note that

\begin{aligned}m^{2^{n−2}}vm^{−2^{n−2}} & = a^{2^{n−2}}xv \\ m^{2^{n−2}}ym^{−2^{n−2}} & = xy \\ m^{2^{n−2}}am^{−2^{n−2}} & = a \end{aligned}

so I and II are normalized, while

m^{2^{n−2}}⟨j, y, v⟩m^{−2^{n−2}} = ⟨jxy, a^{2^{n−2}}xv⟩ = V I.

Similarly,

m^{2^{n−2}}⟨j, ay, v⟩m^{−2^{n−2}} = ⟨j, a^{2^{n−2}}xy, a^{2^{n−2}}xv⟩ = V.

Proposition 2.6: W′ contains exactly four conjugacy classes of subgroups ( /2)^{3}, each with a representative in H_{1}. Indeed, suitable representatives for these classes can be taken as I, II, III, and IV. I is normal in W′. Proof: From the definition of t, we have

\begin{aligned}t⟨j, x, y⟩t^{−1} & = ⟨j, x, y⟩ \\ t⟨j, x, ay⟩t^{−1} & = ⟨j, x, my⟩ \end{aligned}

while subgroups ( /2)^{3} represented by ⟨j, y, v⟩, ⟨j, ay, v⟩, which project non-trivially into S_{3} are conjugated to the corresponding subgroups of the other two Sylow 2-subgroups of W′. Thus there is exactly one fusion which takes place among the 5 conjugacy classes of ( /2)^{3}’s contained in H in W′.

We also will need the explicit action of τ in the six subgroups, ( /2)^{3} ⊂ H_{1}. A direct calculation gives Proposition 2.7: τ normalizes ⟨j, y, x⟩ = I, ⟨j, ayv, yx⟩ = V I, and interchanges the groups II, III. It also exchanges IV and V. The exact action is given as follows:

\begin{aligned}τ⟨j, y, x⟩τ^{−1} & = ⟨j, x, y⟩ \\ τ⟨j, ay, x⟩τ^{−1} & = ⟨j, vy, y⟩ \\ τ⟨j, y, v⟩τ^{−1} & = ⟨j, x, axy⟩ \\ τ⟨j, ay, v⟩τ^{−1} & = ⟨j, vy, axy⟩ \\ τ⟨j, ayx, yv⟩τ^{−1} & = ⟨j, v, ay⟩ \\ τ⟨j, ayv, yx⟩τ^{−1} & = ⟨j, axv, yx⟩ \end{aligned}

Consequently, we have Proposition 2.8: W has four conjugacy classes of groups ( /2)^{3}, with representatives given as I, II, IV, V II. Moreover, I is normal in W while the others are not. Proof: Clearly, V II, since it projects non-trivially into S_{3}, is permuted among the 3 Sylow 2-subgroups in W by the action of τ. Also, I is normalized by τ, as are IV, V. However, m interchanges these last two. Finally, τ(II) = III, τ(III) = V I. From this, the result follows. Corollary 2.9: E has exactly three conjugacy classes of subgroups ( /2)^{3}. Representatives can be given as I, which is normal in E, II, and IV. Proof: E is generated by t, τ, H. Consequently, it suffices to note the fusions determined by t, τ on the five conjugacy classes of ( /2)^{3}’s in H. Using the result for W′, and comparing it with that for W, we see that II, III are fused by τ, and t fuses II, V II. On the other hand, IV is not fused with II. Remark 2.10: II projects to a copy of ( /2) ⊂ GL_{3}(2) while IV projects to a copy of ( /2)^{2}. This provides another way of seeing that these three subgroups cannot be conjugate in E.

§3. Cohomology calculations for the groups H^{(n)} and H_{2}^{(n)}

We begin this section by recalling a fundamental construction of N. Steenrod which will allow us to explicitly calculate the restriction maps from the cohomology of H_{2}^{(n)}, H^{(n)}, and W′ to their elementary subgroups ( /2)^{3}. After this we describe the rings H^{∗}(H_{2}^{(n)}; _{2}) and H^{∗}(H^{(n)}; _{2}), paying particular attention to explicit descriptions of their generators. In §4 we will study the restriction maps to representatives for their conjugacy classes of subgroups ( /2)^{3}.

Let X be a CW -complex and set

3.1  E(X) = X × X ×_{T} E_{/2}

where the action of /2 on X^{2} is to switch coordinates, and E_{/2} is any contractible CW - complex with a free, cellular /2-action, for example S^{∞} = lim_{n↦∞} S^{n} with the antipodal action. Let B_{/2} be the quotient space ∗×_{/2} E_{/2} and π : E_{/2} ⟶B_{/2} be the projection, then there is a map

X × B_{/2} ⟶^{Δ×id}E(X)

defined on points by Δ × id(x, b) = {x, x, e} where e is either point of E_{/2} with π(e) = b. Steenrod, [SE], proves that there is a natural class Γ(α) ∈ H^{2i}(E(X); /2) associated to any class α ∈ H^{i}(X; /2), and Theorem 3.2[SE]: There is one and only one functor Γ : H^{∗}(X; _{2})⟶H^{2∗}(E(X); _{2}) for X a CW -complex so that

(1) Γ is natural,

(2) The inclusion J: X × X ⊂ E(X) as π^{−1}(b), satisfies

J^{∗}(Γ(α)) = α ⊗ α ∈ H^{∗}(X) ⊗ H^{∗}(X) = H^{∗}(X × X)

(3) Δ × id^{∗}(Γ(α)) = ∑_{j=0}^{i} Sq^{j}(α) ⊗ e^{i−j},

(4) Γ(α ∪ β) = Γ(α) ∪ Γ(β).

A special case of this construction is when X = B_{π} for some group π. Then E(B_{π}) is also a K(G, 1) for the group (π × π) ×_{T} ( /2), where the twisting action on π × π is (g, h) ↔ (h, g). This group is, of course, the wreath product of π and /2.

As has been pointed out in §1 a key example is the dihedral group D_{8} = /2 ≀ /2. The cohomology ring of D_{8} is determined in many places, see e.g. [MII], [DM]. The result is

H^{∗}(D_{8}; _{2}) = _{2}[x_{1}, y_{1}, f_{2}]/xy = 0

where the subscript on an element denotes its dimension. The action of the mod2 Steenrod algebra, A(2), on H^{∗}(D_{2^{m}}; _{2}) is determined by Sq^{1}(f) = (x+y)f. There is one higher Bockstein determined by β_{4}(xf) = f^{2}. Note that f = Γ(e) where e is the non-zero class in H^{1}( /2; _{2}). More generally, H^{∗}(D_{2^{n+1}}; _{2}) is also equal to _{2}[x_{1}, y_{1}, f_{2}]/(x_{1}y_{1} = 0), and Sq^{1}(f) = (x + y)f. The only difference is that the Bockstein on xf is β_{2^{n}}(xf) = f^{2}.

All these groups have their cohomology detected on elementary 2-groups and D_{2^{n+1}} has exactly 2 conjugacy classes of subgroups of the form ( /2)^{2} represented by P_{1} = ⟨u^{2^{n−1}}, w⟩, and P_{2} = ⟨u^{2^{n−1}}, uw⟩, so, writing D_{2^{n+1}} = D we find that

(res_{D}^{P_{1}})^{∗} : H^{∗}(D; _{2})⟶H^{∗}(P_{1}; _{2})

is given by

\begin{matrix}x & ↦ x \\ y & ↦ 0 \\ f & ↦ y^{2} + xy, \end{matrix}

while (res_{D}^{P_{2}})^{∗} is given as

\begin{matrix}x & ↦ 0 \\ y & ↦ x \\ f & ↦ y^{2} + xy, \end{matrix}

where H^{∗}(( /2)^{2}; _{2}) is written as _{2}[x, y], so the formulae above follows from 3.2. The following result is Proposition 7.4 of [FM],

Proposition 3.3: H^{∗}(H_{2}^{(n)}; _{2}) = _{2}[r_{2}, s_{2}, c_{1}](1, λ_{1}, τ_{1}, λ_{1}τ_{1}), the free sum of 4 copies of the polynomial algebra, each multiplied by the stated generator. The multiplication is determined by λ^{2} = λc, τ^{2} = τc, and

res : H^{∗}(D_{2^{r+1}} × D_{2^{r+1}}; _{2})⟶H^{∗}(H_{2}^{(n)}; _{2})

is a surjection. The action of v and t on H^{∗}(H_{2}^{(n)}; _{2}) is given by

v^{∗}(r) = s, v^{∗}(s) = r, v^{∗}(λ) = c + λ + τ, v^{∗}(τ) = τ, v^{∗}(c) = c,
t^{∗}(r) = s, t^{∗}(s) = r + s, t^{∗}(λ) = τ, t^{∗}(τ) = c + λ + τ, t^{∗}(c) = c,

and is independent of n for n ≥ 2.

In fact, somewhat more was proved there. It was shown that the restriction map induced from the index 2 inclusion

H_{2}^{(n)}⟶D_{2^{n+1}} × D_{2^{n+1}}

maps onto H^{∗}(H_{2}^{(n)}; _{2}) for each n. Indeed, the precise definitions of the generators above

for H^{∗}(H_{2}^{(n)}; _{2}) are given as follows. Abbreviate (res_{D×D}^{H_{2}^{(n)}})^{∗} as res^{∗}, then we have

3.4  \begin{aligned}r & = res^{∗}(f(1)) \\ s & = res^{∗}(f(2)) \\ c & = res^{∗}(x(1) + y(1)) \\ λ & = res^{∗}(x(1)) \\ τ & = res^{∗}(x(2)). \end{aligned}

We now turn to the groups H^{(n)}. The Lyndon spectral sequence associated to the extension

H_{2}^{(n)} ◁ H^{(n)}⟶ /2

has E_{2}-term

H^{∗}( /2; H^{∗}(H_{2}^{(n)}; _{2}))

Using the action of v given above we obtain the E_{2}-term explicitly as

3.5  _{2}[c, \overline{rs}, \overline{r + s}]1, λ + τ, \overline{λs + τr}, \overline{λτ}) ⊕ _{2}[c, \overline{rs}, e](e, e\overline{λτ}),

and E_{2} = E_{∞}. (The reference is [FM, 8.9 and the remarks which follow].)

The inclusion H_{2}^{(n)} ⊂ H^{(n)} is associated to a transfer map

(tr_{H_{2}^{(n)}}^{H^{(n)}})^{∗} : H^{∗}(H_{2}^{(n)}; _{2})⟶H^{∗}(H^{(n)}; _{2})

and the inclusion H^{(n)} ⊂ D_{2^{n+1}} ≀ /2 is associated to a restriction map

(res_{D_{2^{n+1}}}^{H^{(n)}} )^{∗} : H^{∗}(D_{2^{n+1}} ≀ /2; _{2})⟶H^{∗}(H^{(n)}; _{2}).

When we use these maps the new classes occuring above can be defined explicitly for H as

3.6  \begin{aligned}\overline{r + s} & ∼ (tr_{H_{2}}^{H})^{∗} (r) \\ \overline{λτ} & ∼ (res_{D_{8}≀ /2}^{H} )^{∗} (Γ(x)) \\ \overline{λs + τr} & ∼ (tr_{H_{2}}^{H})^{∗} (λσ) \\ \overline{rs} & ∼ (res_{D_{8}≀ /2}^{H} )^{∗} Γ(f), \end{aligned}

while (c, λ + τ, e) are in the image of the cohomology map associated to the projection H^{(n)}⟶H^{(n)}/[H^{(n)}, H^{(n)}] = ( /2)^{3},

ϕ^{∗} : H^{∗}(H^{(n)}/[H^{(n)}, H^{(n)}]; _{2})⟶H^{∗}(H; _{2}).

§4. The restriction maps from H to its elementary 2 groups

The object of this section is to evaluate the various restriction maps for the five conjugacy classes of subgroups ( /2)^{3} ⊂ H. At the end of the section, 4.7, 4.8, we indicate the result in the general situation, at least as far as we need it. To start, we consider the generators in H^{1}(H; _{2}). For these we have the table

4.1  \begin{aligned}& v & m & y \\ c & 0 & 0 & 1 \\ λ + τ & 0 & 1 & 0 \\ e & 1 & 0 & 0 \end{aligned}

and on H_{2} we have

4.2  \begin{aligned}& a & m & y \\ e & 0 & 0 & 1 \\ λ & 1 & 1 & 0 \\ τ & 1 & 0 & 0 \end{aligned}

which gives directly, the restriction map for these classes. We next want to evaluate the map on the classes associated to r, s. To do this we use the definitions of these classes in terms of the cohomology of D_{8} × D_{8} or D_{8} ≀ /2, depending on which is more convenient. As an example of how this calculation proceeds we consider the group I of §1, ⟨j, x, y⟩. We use the commutative diagram

⟨j, x, y⟩  ⟶  H_{2}
↓θ  ↓in

( /2)^{2} × ( /2)^{2}  ⟶^{(Δ×id)^{2}} D_{8} × D_{8}

where

\begin{aligned}θ(Δ × id)^{2}(j) & = (j̄, 1), (j̄, 1)) (j̄ = (uw)^{2}) \\ θ(Δ × id)^{2}(x) & = ((j̄, 1), (1, 1)) \\ θ(Δ × id)^{2}(y) & = ((1, w), (1, w)). \end{aligned}

Now, r = in^{∗}(Γ(e) ⊗ 1), s = in^{∗}(1 ⊗ Γ(e)), so it follows that

i_{I}^{∗}(r) = (θ ∘ (Δ × id)^{2})^{∗}(Γ(e) ⊗ 1)
= θ^{∗}((e^{2} + eσ) ⊗ 1)
= (α + β)^{2} + (α + β)σ

while

i_{I}^{∗}(s) = (θ ∘ (Δ × id)(Δ × id))^{∗}(1 ⊗ Γ(e))
= θ^{∗}(1 ⊗ (e^{2} + eσ))
= α^{2} + ασ.

Here α is dual to j, β is dual to x, and σ is dual to y.

To obtain i_{II}^{∗}(r), i_{II}^{∗}(s), note that we have the commutative diagram

⟨j, x, y⟩  ⟶^{λ} ⟨j, x, ay⟩ ⟶^{i_{2}}  H_{2}
↓θ  ↓in

( /2)^{2} × ( /2)^{2}  ⟶^{(Δ×id)^{2}} D_{8} × D_{8} ⟶^{ϕ×ϕ} D_{8} × D_{8}

where ϕ(u) = w, ϕ(w) = u, and λ(j) = j, λ(x) = x, λ(y) = ay. Clearly, ϕ^{∗}(Γ(e)) = Γ(e) while ϕ^{∗}(x) = y, ϕ^{∗}(y) = x. Thus i_{II}^{∗}(r) = i_{I}^{∗}(r), i_{II}^{∗}(s) = i_{I}^{∗}(s).

The other conjugacy class of group in H which is contained in H_{2} has two representatives,

V II = ⟨j, x, my⟩, V III = ⟨j, x, amy⟩,

and the calculation here works as above except that ϕ × ϕ is replaced by ϕ × 1, 1 × ϕ respectively. Thus, we obtain the following table for the subgroups in H_{2},

4.3  \begin{matrix}i_{I}^{∗}(r) = (α + β)^{2} + (α + β)σ ; i_{I}^{∗}(s) = α^{2} + ασ ; i_{I}^{∗}(c) = σ ; i_{I}^{∗}(λ) = 0 ; i_{I}^{∗}(τ) = 0 & i_{II}^{∗}(r) = (α + β)^{2} + (α + β)σ ; i_{II}^{∗}(s) = α^{2} + ασ ; i_{II}^{∗}(c) = σ ; i_{II}^{∗}(λ) = σ ; i_{II}^{∗}(τ) = σ \\ i_{V II}^{∗}(r) = (α + β)^{2} + (α + β)σ ; i_{V II}^{∗}(s) = α^{2} + ασ ; i_{V II}^{∗}(c) = σ ; e_{V II}^{∗}(λ) = σ ; i_{V II}^{∗}(τ) = 0 & i_{V III}^{∗}(r) = (α + β)^{2} + (α + β)σ ; i_{V III}^{∗}(s) = α^{2} + ασ ; i_{V III}^{∗}(c) = σ ; i_{V III}^{∗}(λ) = 0 ; i_{V III}^{∗}(τ) = σ \end{matrix}

Moreover, as is clear, these images detect H^{∗}(H_{2}; _{2}). (Here, as before, α is dual to j, β is dual to x, and σ is dual to y, ay, my, or amy, depending on the group ( /2)^{3}. This becomes clear when we remember that j is represented as (xy, xy) ∈ D_{8} × D_{8}.) Hence

(r + s) ↦ (β^{2} + βσ, β^{2} + βσ, β^{2} + βσ, β^{2} + βσ)
rs ↦ Δ^{4}(α^{4} + α^{2}β^{2} + (αβ^{2} + α^{2}β)σ + (α^{2} + αβ)σ^{2})
= Δ^{4}[(α^{2} + αβ)^{2} + (αβ^{2} + α^{2}β)σ + (α^{2} + αβ)σ^{2}]
= Δ^{4}[(α^{2} + αβ)(α^{2} + αβ + βσ + σ^{2})]

Also,

\begin{matrix}λτ & ↦ (0, σ^{2}, 0, 0), \\ λs + τr & ↦ (0, β^{2}σ + βσ^{2}, α^{2}σ + ασ^{2}, (α + β)^{2}σ + (α + β)σ) \\ λ + τ & ↦ (0, 0, σ, σ). \end{matrix}

The four groups ⟨j, x, y⟩, ⟨j, x, ay⟩, ⟨j, x, my⟩, ⟨j, x, amy⟩ detect all of H^{∗}(H; _{2}) except for the terms in _{2}[c, \overline{r + s}^{2} + \overline{rs}, e](e, e\overline{λτ}). These are detected by the subgroups

III = ⟨j, y, v⟩, IV = ⟨j, ay, v⟩.

Proposition 4.4: The maps

i_{III}^{∗} : H^{∗}(H; _{2})⟶H^{∗}(⟨j, y, v⟩; _{2}) and i_{IV}^{∗} : H^{∗}(H; _{2})⟶H^{∗}(⟨j, ay, v⟩; _{2})

are determined by

\begin{aligned}i_{III}^{∗}(\overline{rs}) & = (α^{2} + ασ)(α^{2} + ασ + σν + ν^{2}), \\ i_{III}^{∗}(\overline{r + s}) & = 0, \\ i_{III}^{∗}(c) & = σ, \\ i_{III}^{∗}(λ + τ) & = 0 \\ i_{III}^{∗}(\overline{λs + τr}) & = 0, \\ i_{III}^{∗}(\overline{λτ}) & = 0, \\ i_{III}^{∗}(e) & = ν, \end{aligned}

\begin{aligned}i_{IV}^{∗} (\overline{rs}) & = (α^{2} + ασ)(α^{2} + ασ + σν + ν^{2}), \\ i_{IV}^{∗} (\overline{r + s}) & = 0, \\ i_{IV}^{∗} (c) & = σ, \\ i_{IV}^{∗} (λ + τ) & = 0 \\ i_{IV}^{∗} (\overline{λs + τr}) & = 0, \\ i_{IV}^{∗} (\overline{λτ}) & = σ^{2} + σν, \\ i_{IV}^{∗} (e) & = ν, \end{aligned}

Proof: Consider the group III. We use the diagram

⟨j, y, v⟩ ⟶^{(Δ×id)×id} D_{8} × /2
↓in  ↓Δ×id
H  ⟶^{in}  D_{8} ≀ /2

and the definitions of the classes c, \overline{r + s}, \overline{rs}, e, \overline{λτ}, given in 3.6. Note first that for D_{8},

\{ \begin{matrix}res^{∗}(x) & = 0 \\ res^{∗}(y) & = σ \\ res^{∗}(f) & = α^{2} + ασ \end{matrix} in⟨a^{2}, y⟩, \{ \begin{matrix}res^{∗}(x) & = σ \\ res^{∗}(y) & = 0 \\ res^{∗}(f) & = α^{2} + ασ \end{matrix} in ⟨a^{2}, ay⟩.

Consequently, since res^{∗}(Γ(λ)) = \overline{λτ}, res^{∗}(Γ(f)) = \overline{rs}, we have

(res_{H}^{III})^{∗} (\overline{rs}) = α^{4} + α^{2}σ^{2} + (α^{2}σ + ασ^{2})ν + (α^{2} + ασ)ν^{2}

which is also equal to (res_{H}^{IV} )^{∗} (\overline{rs}), while

\begin{aligned}(res_{H}^{III})^{∗} (\overline{λτ}) & = 0 \\ (res_{H}^{IV} )^{∗} (\overline{λτ}) & = σ^{2} + σν. \end{aligned}

In both cases

\begin{aligned}(res_{H}^{III})^{∗} (e) & = (res_{H}^{IV} )^{∗} (e) = ν \\ (res_{H}^{III})^{∗} (c) & = σ \\ (res_{H}^{IV} )^{∗} (c) & = σ. \end{aligned}

Now, the other generators of H^{∗}(H; _{2}) are all in the image of the transfer. But then, when we restrict to H^{∗}(III; _{2}) or H^{∗}(IV; _{2}) we obtain, (writing J for III or IV), the image (res_{H}^{J})^{∗} (tr_{H_{2}}^{H})^{∗} (γ). From the double coset formula these images are obtained by first taking restriction and then transfer according to the formula

∑ (tr_{K^{c}}^{J}) c^{∗} (res_{H_{2}}^{K})^{∗} (γ)

where ∐ JcH_{2} = H gives the double coset decomposition and

K = H_{2} ∩ c^{−1}(J)c.

Clearly, K = /2 or ( /2)^{2}, and in both cases, (tr_{K^{c}}^{J})^{∗} = 0. It follows that the remaining generators map to 0 in H^{∗}(III; _{2}), H^{∗}(IV; _{2}).

In particular we obtain Corollary 4.5: Let L = tr(λs) = \overline{λs + τr}. Then the action of the Steenrod algebra A(2) on H^{∗}(H; _{2}) is determined by

\begin{aligned}Sq^{1}(\overline{r + s}) & = c\overline{r + s} \\ Sq^{1}(\overline{rs}) & = 0 \\ Sq^{1}(\overline{λτ}) & = e\overline{λτ} \\ Sq^{1}(L) & = 0 \\ Sq^{1}(λ + τ) & = c(λ + τ) = (λ + τ)^{2} \\ Sq^{2}(\overline{rs}) & = (\overline{r + s} + c^{2} + ce + e^{2})\overline{rs} \\ Sq^{2}(L) & = \overline{rs}(λ + τ) + c\overline{r + s}\overline{λτ} + (c^{2} + \overline{r + s})L \\ Sq^{3}(\overline{rs}) & = c(\overline{r + s} + ce + e^{2})\overline{rs} \\ L^{2} & = c\overline{rs}(λ + τ) + c(\overline{r + s})L \\ (\overline{λτ})^{2} & = (c^{2} + ce)\overline{λτ}. \end{aligned}

Proof: We begin with \overline{rs}.

i_{j}^{∗}(\overline{rs}) = \{ \begin{matrix}(β^{2} + αβ)(αβ + β^{2} + ασ + σ^{2}) & j ∈ \{I, II, V II, V III\} \\ (α^{2} + ασ)(α^{2} + ασ + σν + ν^{2}) & j = III or IV. \end{matrix}

Also,

\begin{aligned}Sq^{2}(AB) & = Sq^{1}ASq^{1}B + (Sq^{2}A)B + A(Sq^{2}B) \\ & = Sq^{1}ASq^{1}B + (A + B)AB \end{aligned}

if A, B, are both 2-dimensional. Expanding out,

Sq^{1}A = \{ \begin{matrix}αA & j ∈ \{I, II, V II, V III\} \\ σA & j = III or IV, \end{matrix} Sq^{1}B = \{ \begin{matrix}αB & j ∈ \{I, II, V II, V III\} \\ σB & j = III or IV \end{matrix}

in the two cases so

Sq^{2}(AB) = \{ \begin{matrix}(α^{2} + αβ + β^{2})AB & j ∈ \{I, II, V II, V III\} \\ (σ^{2} + σν + ν^{2})AB & j = III or IV. \end{matrix}

Now, note that

(\overline{r + s} + c^{2})\overline{rs} ↦ \{ \begin{matrix}(α^{2} + ασ + σ^{2})AB & j ∈ \{I, II, V II, V III\} \\ (σ^{2} + σν + ν^{2})AB & j = III or V I, \end{matrix}

and

(ce + e^{2})(\overline{rs} ↦ \{ \begin{matrix}0 & j ∈ \{I, II, V II, V III\} \\ (σν + ν^{2})AB & j = III or IV. \end{matrix}

Thus the stated element maps to Sq^{2}(i_{j}^{∗}(\overline{rs}) for j ∈ \{I, II, III, IV, V II, V III\}. But since the sum of these maps is injective the result follows. The other cases are similar. Corollary 4.6: There is a subalgebra _{2}[k_{4}, k_{6}, k_{7}] ⊂ H^{∗}(H; _{2}) and

(1) i_{j}^{∗}(k_{i}) = i_{s}^{∗}(k_{i}) for all j, s, in the set I, II, III, IV, V II, V III, and i = 4, 6, 7,

(2) i_{I}^{∗}( _{2}[k_{4}, k_{6}, k_{7}]) = _{2}[α, β, σ]^{GL_{3}(2)}. In particular

k_{4} = \overline{r + s}^{2} + \overline{rs} + c^{4} + e^{4} + e^{2}c^{2},
k_{6} = (c^{2} + ce + e^{2})(\overline{r + s}^{2} + \overline{rs}) + (c^{2} + e^{2})e^{2}e^{2},
k_{7} = (c + e)ed(\overline{r + s}^{2} + \overline{rs}).

Proof: All that is needed is to check k_{4}, since k_{6} = Sq^{2}(k_{4}), k_{7} = Sq^{1}(k_{6}), and then use the results of the second corollary. The first invariant in _{2}[α, β, σ]^{GL_{3}(2)} is

α^{4} + β^{4} + σ^{4} + α^{2}β^{2} + α^{2}σ^{2} + β^{2}σ^{2} + αβσ^{2} + αβ^{2}σ + α^{2}βσ.

When we check the image of (\overline{r + s})^{2} + \overline{rs} at i_{I}^{∗}, i_{II}^{∗}, i_{V II}^{∗}, i_{V III}^{∗} we find only σ^{4} missing. This is filled in with c^{4}. when we check at i_{III}^{∗}, i_{IV}^{∗}, we find σ^{4} + σ^{2}ν^{2} + ν^{4} missing. The c^{4} fixes the term σ^{4} here, and c^{2}e^{2} + e^{4} adds the remaining terms at i_{III}^{∗}, i_{IV}^{∗}. The result follows. Remark 4.7: Entirely similar results are true for H^{(n)} with n > 2. However all we need for our main applications is Lemma 4.8: For n ≥ 2 (res_{H^{(n)}}^{I})^{∗} is given by exactly the same formulae as in 4.3. In particular

H^{∗}(I; _{2})^{GL_{3}(2)} ⊂ im (res_{H^{(n)}}^{I})^{∗} .

(Indeed, this is direct from 3.3, 3.4, using the Steenrod theorem 3.2, and proceeds exactly as did the proof of 4.3.)

§5. The cohomology of W′

W′ is studied in [AMM] and its cohomology ring is determined there in 4.14. However, the determination of the elements given there is not quite precise enough for our current needs. Consequently we revise the calculation now. We note, as pointed out in [AMM], that

im (H^{∗}(W′; _{2}) ↪ H^{∗}(H; _{2})) = (res^{∗})^{−1} H^{∗}(H_{2}; _{2})^{S_{3}}.

Here, S_{3} is generated by v and an involution μ = vt^{2} where μ interchanges (a, m), leaving y fixed. Thus, to obtain H^{∗}(H_{2}; _{2})^{S_{3}} it suffices to obtain the μ fixed elements in

H^{∗}(H_{2}; _{2})^{v} = _{2}[c, r + a, rs](1, λ + τ, λτ, λs + τr).

We claim this invariant set is given as Proposition 5.1:

H^{∗}(H_{2}; _{2})^{S_{3}} = _{2}[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})](1, \overline{λτ} + (τ + λ)c, \overline{λs + τr}, \overline{λs^{2} + τr^{2}}).

Proof: This involves a check that the stated elements are μ invariant, and a comparison with the result, 4.14, of [AMM]. Indeed, μ fixes the first and fourth groups ( /2)^{2} in the list above, while interchanging the second and third. It also exchanges α, β, leaving σ fixed. A short calculation gives

\begin{matrix}(τ + λ)c + \overline{λτ} & ↦ (0, σ^{2}, σ^{2}, σ^{2}) \\ \overline{λs^{2} + τr^{2}} & ↦ (0, α^{4}σ + α^{2}σ^{3}, β^{4}σ + β^{2}σ^{3}, (α + β)^{4}σ + (α + β)^{2}σ^{3}), \end{matrix}

and these are indeed invariant under S_{3} as required. Corollary 5.2: Write M_{2} = \overline{λτ} + (λ + τ)c, L_{3} = \overline{σs + τr}, N_{5} = (\overline{λs + τr})(\overline{r + s}) + (λ + τ)(\overline{rs}). Then the image im (H^{∗}(W′; _{2}) ↪ H^{∗}(H; _{2})) is given as

\begin{aligned}_{2}[c, (\overline{r + s})^{2} + \overline{rs}, & (\overline{r + s})(\overline{rs})](1, M_{2}, L_{3}, N_{5}) \\ & ⊕ _{2}[c, \overline{rs}, e](e, eM_{2}). \end{aligned}

§6. The cohomology of E

The following is our main calculational result. Theorem 6.1: H^{∗}(E; _{2}) contains a subalgebra _{2}[b_{4}, b_{6}, b_{7}] which restricts to H^{∗}(I; _{2}) as the Dickson algebra _{2}[x, y, z]^{GL_{3}(2)}. Moreover, H^{∗}(E; _{2}) is Cohen-Macaulay over 2[b_{4}, b_{6}, b_{7}] with Poincaré series

P_{E}(x) = \frac{1 + x^{2} + 3x^{3} + 2x^{4} + 4x^{5} + 5x^{6} + 4x^{7} + 5x^{8} + 4x^{9} + 2x^{10} + 3x^{11} + x^{12} + x^{14}}{(1 − x^{4})(1 − x^{6})(1 − x^{7})} .

The explicit generators for H^{∗}(E; _{2}) ⊂ H^{∗}(W′; _{2}) as a module over _{2}[b_{4}, b_{6}, b_{7}] are given after 6.11 (taking advantage of 6.5) and in 6.16, 6.17, 6.18. Proof: From 1.13 we know that

H^{∗}(E; _{2}) = ((res_{W′}^{H_{1}})^{∗})^{−1}(H^{∗}(H_{1}; _{2})^{τ}).

Both H^{∗}(H_{1}; _{2}) and H^{∗}(W′; _{2}) are detected on elementary 2-groups, so we can consider instead the subring

im {(∐_{I}^{V I} res^{∗}) : H^{∗}(W′; _{2})⟶ ∐_{i=I}^{V I} H^{∗}(J_{i}; _{2})} ∩ {∐_{i=I}^{V I} H^{∗}(J_{i}; _{2})}^{τ} ,

where the J_{i} are the groups, I, ⋯, V I, listed in 2.3. This can be cut down still further. Proposition 6.2: Let w : IV ⟶IV, (w^{3} = 1) be defined by

w : \{ \begin{matrix}j & ↦ j \\ ay & ↦ v \\ v & ↦ ayv, \end{matrix}

let τ_{I} : I⟶I be given as

τ_{I} : \{ \begin{matrix}j & ↦ j \\ x & ↦ y \\ y & ↦ x, \end{matrix}

and, finally, let τ_{II} : II⟶III be given as

τ_{II} : \{ \begin{matrix}j & ↦ j \\ ay & ↦ vy \\ x & ↦ y. \end{matrix}

Then, α ∈ H^{∗}(W′; _{2}) is in the image of restriction from H^{∗}(E; _{2}) if and only if the three conditions below are satisfied:

(1) w^{∗} (rex_{W′}^{IV})^{∗} (α) = (res_{W′}^{IV})^{∗} (α),

(2) τ_{II}^{∗} (res_{W′}^{III})^{∗} (α) = (res_{W′}^{II})^{∗} (α), (3) τ_{I}^{∗} (rest_{W′}^{I})^{∗} (α) = (res_{W′}^{I})^{∗} (α). Proof: Note that w = τm. If w^{∗} (res_{W′}^{IV})^{∗} (α) = (res_{W′}^{IV})^{∗} (α) then τ^{∗} (res_{W′}^{V})^{∗} (α) = τ^{∗}m^{∗} (res_{W′}^{IV})^{∗} (α) = (res_{W′}^{IV})^{∗} (α). Also,

(mτ_{II}m)^{∗}(β) = (τ_{II}mτ_{V})^{∗}(β)

when β is in the image of (res_{W′}^{∼})^{∗}, so (mτ_{II}m)^{∗}(α) = m^{∗}τ^{∗} (res_{W′}^{V I})^{∗} (α) = (res_{W′}^{III})^{∗} (α). Applying m^{∗} again, τ^{∗} (res_{W′}^{IV})^{∗} (α) = (res_{W′}^{V I})^{∗} (α) and ∐_{I}^{V I}(res)^{∗}(α) is indeed invariant under τ^{∗}. Remark 6.3: The group ⟨τ_{I}, N_{W′}(I)/I⟩ = GL_{3}(2) so

im (res_{W′}^{I})^{∗} ∩ H^{∗}(I)^{γ} ⊂ H^{∗}(I; _{2})^{GL_{3}(2)}.

But 4.6 shows that the Dickson algebra H^{∗}(I; _{2})^{GL_{3}(2)} ⊂ im (res_{W′}^{I})^{∗} so im (res_{E}^{I})^{∗} = H^{∗}(I; _{2})^{GL_{3}(2)} = _{2}[d_{4}, d_{6}, d_{7}]. Remark 4.4: H^{∗}(IV; _{2}) = _{2}[α, β, ν] with ν dual to v, β dual to ay, and α dual to j. Then, with this notation w^{∗}(α) = α, w^{∗}(β) = ν, w^{∗}(ν) = β + ν. Consequently 2[α, β, ν]^{w∗} = _{2}[α] ⊗ _{2}[β, ν]^{w^{∗}}. The calculation of this latter group is standard:

4 ⊗_{_{2}} _{2}[β, ν]^{w^{∗}} = _{4}[β, ν]^{w^{∗}} = _{4}[μ, φ]^{w^{∗}}

where w^{∗}(μ) = ζ_{3}μ, w^{∗}(φ) = ζ_{3}^{2}φ. But the invariant subring here is generated by μφ, μ^{3}, and φ^{3}. Consequently we obtain

6.5  H^{∗}(IV; _{2})^{w} = _{2}[α, d_{2}, e_{3}](c_{3})

with c^{2} + d^{3} = ec. Here d = σ^{2} + ν^{2} + σν, e = σ^{2}ν + σν^{2}, and c = σ^{3} + ν^{3} + νσ^{2}.

We now construct a table giving the images of the generators for H^{∗}(W′; _{2}) in the cohomology rings of the groups I, II, III, and IV. We begin with the group I = ⟨j, x, y⟩, and set α dual to j, β dual to x, σ dual to y,

6.6 \begin{matrix}Group I \\ c & ↦ σ \\ (\overline{r + s})^{2} + \overline{rs} & ↦ α^{4} + α^{2}σ^{2} + (β^{2} + βσ)(α^{2} + β^{2} + (α + β)σ) \\ (\overline{r + s})(\overline{rs}) & ↦ (α^{2} + ασ)(β^{2} + βσ)(α^{2} + β^{2} + (α + β)σ) \\ \overline{λτ} + (τ + λ)c & ↦ 0 \\ \overline{λs + τr} & ↦ 0 \\ (\overline{λs + τr})(\overline{r + s}) + (λ + τ)(\overline{rs}) & ↦ 0 \\ e & ↦ 0 \end{matrix}

Next we enter the data for II = ⟨j, ay, xa⟩ and III = ⟨j, y, v⟩. In both groups α is dual to j, while σ is dual to y is III and is dual to ay in II, β is dual to x in II, and ν is dual to v in III.

6.7 \begin{matrix}Group II \\ c & ↦ σ \\ (\overline{r + s})^{2} + \overline{rs} & ↦ α^{4} + α^{2}σ^{2} + (α^{2} + β^{2} + (α + β)σ)(β^{2} + βσ) \\ (\overline{r + s})(\overline{rs}) & ↦ (β^{2} + βσ)(α^{2} + ασ)(α^{2} + β^{2} + (α + β)σ) \\ \overline{λτ} + (τ + λ)c & ↦ σ^{2} \\ \overline{λs + τr} & ↦ σ(β^{2} + βσ) \\ (\overline{λs + τr})(\overline{r + s}) + (λ + τ)(\overline{rs}) & ↦ σ(β^{2} + βσ)^{2} \\ e & ↦ 0. \end{matrix}

6.8 \begin{matrix}Group III \\ c & ↦ σ \\ (\overline{r + s})^{2} + \overline{rs} & ↦ (α^{2} + ασ)(α^{2} + ασ + σν + ν^{2}) \\ (\overline{r + s})(\overline{rs}) & ↦ 0 \\ \overline{λτ} + (τ + λ)c & ↦ 0 \\ \overline{λs + τr} & ↦ 0 \\ (\overline{λs + τr})(\overline{r + s}) + (λ + τ)(\overline{rs}) & ↦ 0 \\ e & ↦ ν. \end{matrix}

Finlly we write down the table for the group IV = ⟨j, ay, v⟩. Here, α is dual to j, σ is dual to ay, and ν is dual to v,

6.9 \begin{matrix}Group IV \\ c & ↦ σ \\ (\overline{r + s})^{2} + \overline{rs} & ↦ (α^{2} + ασ)(α^{2} + ασ + σν + ν^{2}) \\ (\overline{r + s})(\overline{rs}) & ↦ 0 \\ \overline{λτ} + (τ + λ)c & ↦ σ^{2} + σν \\ \overline{λs + τr} & ↦ 0 \\ (\overline{λs + τr})(\overline{r + s}) + (λ + τ)(\overline{rs}) & ↦ 0 \\ e & ↦ ν. \end{matrix}

We have already seen that I involves just restrictions on _{2}[c, \overline{rs}+(\overline{r + s})^{2}, (\overline{r + s})(\overline{rs})]. Also, note that the ideal e ∪ H^{∗}(W′; _{2}) has image only in III, IV. In particular, in this ideal, the kernel of the map to III is precisely _{2}[c, e, (\overline{r + s})^{2} + \overline{rs}]e(\overline{λτ} + (τ + λ)c). We have

6.10  \begin{aligned}& & I & II & III & IV \\ e((\overline{λτ} + (τ + λ)c) & ↦ & 0 & 0 & 0 & νσ^{2} + ν^{2}σ \end{aligned}

and this is in H^{∗}(IV; _{2})^{w}. Thus, the overall image of _{2}[c, e, (\overline{r + s})^{2} +\overline{rs}]e(\overline{λτ} +(λ+τ)c) is

6.11  {0, 0, 0, _{2}[α^{4} + α^{2}(σ^{2} + ν^{2} + σν) + α(σν^{2} + σ^{2}ν), σ, ν](σ^{2}ν + σν^{2})}

and an element in this image is invariant under w if and only if the polynomial term is. Clearly, the term involving α is in the invariant image. It follows that the Poincaré series for the invariant image is

6.12  \frac{x^{3}(1 + x^{3})}{(1 − x^{4})(1 − x^{2})(1 − x^{3})}.

The rest of the terms in the ideal e cannot be invariant under τ_{II}^{∗}. We can rewrite the series above as

L = \frac{x^{3} + x^{5} + 2x^{6} + x^{7} + 2x^{8} + 2x^{9} + x^{10} + 2x^{11} + x^{121} + x^{14}}{(1 − x^{4})(1 − x^{6})(1 − x^{7})} .

P_{E}(x) is defined in the statement of 6.1. At this point in the proof we do not yet know that it is the Poincaré series for E but a simple calculation gives

P_{E}(x) − L = \frac{(x^{2} + x^{3} + x^{5})(1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})}{(1 − x^{4})(1 − x^{6})(1 − x^{7})}.

With this in mind we study the subring

2[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})](1, M_{2}, L_{3}, N_{5}) ⊂ H^{∗}(W′; _{2}).

First, the restriction to I is _{2}[d_{4}, d_{6}, d_{7}], and M_{2}, L_{3}, N_{5} go to 0 in I. Thus, we are reduced to studying the images in II, III, IV of _{2}[c, (\overline{r + s})^{2} +\overline{rs}, (\overline{r + s})(\overline{rs})] multiplied by M_{2}, L_{3} and N_{5}. Indeed, we will shortly show that each of these elements can be modified slightly to give an element in the cohomology of E. To begin we need to note that 2[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})] is free over _{2}[d_{4}, d_{6}, d_{7}] with 7 generators, 1, c, c^{2}, …, c^{7}. This follows from Lemma 6.13: _{2}[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})] = _{2}[c, d_{4}, d_{6}, d_{7}], and this latter ring can be written as

2[d_{4}, d_{6}, d_{7}](c) = _{2}[d_{4}, d_{6}, d_{7}][x]/(x^{7} + x^{3}d_{4} + xd_{6} + d_{7})

where x^{7} + x^{3}d_{4} + xd_{6} + d_{7} is irreducible in _{2}(d_{4}, d_{6}d_{7})[x], (the polynomial ring in x over the quotient field _{2}(d_{4}, d_{6}, d_{7})). Consequently, _{2}[d_{4}, d_{6}, d_{7}](c) is Cohen-Macaulay over 2[d_{4}, d_{6}, d_{7}] with basis 1, c, c^{2}, c^{3}, c^{4}, c^{5}, c^{6}. Proof: The restriction of _{2}[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})] to I is an injection and d_{4} = im(c^{4} + (\overline{r + s})^{2} + \overline{rs}). Similarly, d_{6} is the image of c^{2}((\overline{r + s})^{2} + \overline{rs}) + (\overline{r + s})(\overline{rs}) and d_{7} is the image of c(\overline{r + s})(\overline{rs}). Thus, the first statement of 6.13 is verified.

Next we establish the relation above for c^{7}. Let D_{2} = α^{2} + αβ + β^{2}, D_{3} = αβ^{2} + α^{2}β. Then

6.14  \begin{aligned}d_{4} & = c^{4} + c^{2}D_{2} + cD_{3} + D_{2}^{2} \\ d_{6} & = c^{4}D_{2} + c^{2}D_{2}^{2} + cD_{2}D_{3} + D_{3}^{2} \\ d_{7} & = c^{4}D_{3} + c^{2}D_{2}D_{3} + cD_{3}^{2}. \end{aligned}

Using the expressions above a formal expansion gives the relation c^{7} = c^{3}d_{4} + cd_{6} + d_{7}.

Note now that x^{7} + d_{4}x^{3} + d_{6}x + d_{7} = f(x) is irreducible since, first _{2}[d_{4}, d_{6}, d_{7}] is integrally closed in its quotient field K = _{2}(d_{4}, d_{6}, d_{7}), so any factorization of f(x) must already occur over _{2}[d_{4}, d_{6}, d_{7}]. Second, in the quotient domain _{2}[x], obtained by sending d_{4} to 0, but d_{6}, d_{7} to 1, the image of f(x) is x^{7} + x + 1 which is easily seen to be irreducible. It is also separable since the derivative goes to x^{6} + 1 under the same map, and this polynomial has no roots in common with x^{7} + x + 1.

It follows that over K the field extension L = K[x]/(f(x)) has degree 7, and generators can be chosen as 1, x, x^{2}, …, x^{6}. In particular, since there are no relations of the form ∑_{0}^{6} γ_{i}x^{i} = 0 in L with the γ_{i} ∈ K, there are no relations of the form

γ_{0} + γ_{1}c + γ_{2}c^{2} + ⋯ + γ_{6}c^{6}

with the γ_{i} ∈ _{2}[d_{4}, d_{6}, d_{7}] which hold in the ring

2[d_{4}, d_{6}, d_{7}](c) = _{2}[d_{4}, d_{6}, d_{7}] + _{2}[d_{4}, d_{6}, d_{7}]c + ⋯ + _{2}[d_{4}, d_{6}, d_{7}]c^{6}.

The lemma follows. Corollary 6.15: The Poincaré series for the ring _{2}[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})] is

\frac{1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6}}{(1 − x^{4})(1 − x^{6})(1 − x^{7})} .

We now construct the 21 generators with leading terms c^{i}M_{2}, c^{i}L_{3}, and c^{i}N_{5} for 0 ≤ i ≤ 6. It suffices to construct elements of the proper form which go under restriction to elements invariant under the maps τ_{II}^{∗} and w^{∗} and which restrict to 0 in I. For M_{2} we obtain elements and their images in H^{∗}(II; _{2}), H^{∗}(III; _{2}), H^{∗}(IV; _{2}) satisfying the required invariance properties as follows:

6.16  \begin{matrix}Element & II & III & IV \\ M_{2} + e^{2} & σ^{2} & v^{2} & σ^{2} + σv + v^{2} \\ cM_{2} + e^{3} & σ^{3} & v^{3} & σ^{3} + σ^{2}v + v^{3} \\ (c^{2} + ce)M_{2} + e^{4} & σ^{4} & v^{4} & σ^{4} + σ^{2}v^{2} + v^{4} \\ c^{3}M_{2} + e^{5} & σ^{5} & v^{5} & σ^{5} + σ^{4}v + v^{5} \\ (c^{4} + c^{3}e)M_{2} + e^{6} & σ^{6} & v^{6} & σ^{6} + σ^{4}v^{2} + v^{6} \\ (c^{5} + c^{3}e^{2} + c^{2}e^{3})M_{2} + e^{7} & σ^{7} & v^{7} & σ^{7} + σ^{6}v + σ^{5}v^{2} + σ^{3}v^{4} + v^{7} \\ (c^{6} + c^{5}e + c^{4}e^{2} + c^{3}e^{3})M_{2} + e^{8} & σ^{8} & v^{8} & σ^{8} + σ^{4}v^{4} + v^{8}. \end{matrix}

The elements

6.17  c^{i}L_{3} + (c(c + e) + M_{2})e^{i+1}

give the next set of generators, and, finally, the remaining generators are given by

6.18  c^{i}N_{5} + e^{i+1}c(c + e)[M_{2} + c^{2} + ce].

Thus, the part _{2}[c, (\overline{r + s})^{2} + \overline{rs}, (\overline{r + s})(\overline{rs})](1, M_{2}, L_{3}, N_{5}] in H^{∗}(W′; _{2}) contributes a free module over _{2}[d_{4}, d_{6}, d_{7}] on 22 generators, 1, and the 21 generators above. There can clearly be no further generators, and the theorem follows.

§7. The cohomology rings of G_{2}(q), ^{3}D_{4}(q)

We begin by proving that there is a polynomial subalgebra _{2}[b_{4}, b_{6}, b_{7}] ⊂ H^{∗}(G; _{2}) so that under the restriction maps

(res_{G}^{M_{i}})^{∗} : H^{∗}(G; _{2})⟶H^{∗}(M_{i}; _{2})

im( _{2}[b_{4}, b_{6}, b_{7}] = H^{∗}(M_{i}; _{2})^{GL_{3}(2)}. Here i = 1, 2 and, as in [FM, introduction], M_{1}, M_{2} denote the two conjugacy classes of subgroups ( /2)^{3} ⊂ G where G is G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4. Proposition 7.1: Let G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4, then there is an n so that H^{(n)} ∈ Syl_{2}(G), and the sequence of subgroups ⟨j, x, y⟩ ◁ H^{(n)} ⊂ G has the property that ⟨j, x, y⟩ is weakly closed in G. (This means that if g ∈ G and g⟨j, x, y⟩g^{−1} ⊂ H^{(n)}, then g⟨j, x, y⟩g^{−1} = ⟨j, x, y⟩.) Proof: Note that⟨j, x, y⟩ is normal in H^{(n)}, so it is the only subgroup in its conjugacy class in H^{(n)}. We have also seen that II, III, IV, V II, are fused in G = G_{2}(q) or ^{3}D_{4}(q) [FM, Theorem 4.6] but I is not fused to these other groups in G. Consequently it is the only group in its conjugacy class in G which is contained in H^{(n)} and the result follows. Corollary 7.2: Let G(q) be either G_{2}(q) or ^{3}D_{4}(q), then there is a polynomial algebra A_{G(q)} ≅ _{2}[b_{4}, b_{6}, b_{7}] ⊂ H^{∗}(G(q); _{2}) which satisfies the properties

(1) (res_{G(q)}^{I})^{∗} A_{G(q)} = H^{∗}(I; _{2})^{GL_{3}(2)},

(2) Let II represent the second conjugacy class of groups ( /2)^{3} in G(q), then

(res_{G(q)}^{II})^{∗} A_{G(q)} = H^{∗}(II; _{2})^{GL_{3}(2)},

(3) If q divides q′, then

(res_{G(q′)}^{G(q)})^{∗} A_{G(q′)} = A_{G(q)}.

Proof: We have seen in 4.8 that the image of (res_{H^{(n)}}^{I})^{∗} contains H^{∗}(I; _{2})^{GL_{3}(2)}. Consequently, using [MP, Prop. 6.2, 6.3], we have that

im (res_{G(q)}^{I})^{∗} = H^{∗}(I; _{2})^{GL_{3}(2)} = _{2}[d_{4}, d_{6}, d_{7}].

Since this image is a polynomial algebra, if we pick b_{4}, Sq^{2}(b_{4}) = b_{6}, and Sq^{3}(b_{4}) = b_{7}

so that (res_{G(q)}^{I})^{∗} (b_{4}) = d_{4}, then b_{6} ↦ d_{6}, b_{7} ↦ d_{7}, and the subalgebra of H^{∗}(G(q); _{2}) generated by b_{4}, b_{6}, and b_{7} is a polynomial algebra.

Now, if l^{2} divides q, then, in the inclusion G(l) ⊂ G(q), both the type I subgroup and the type II subgroup have image a type I subgroup. Consequently, the map (res_{G(q)}^{G(l)})^{∗} takes the algebra above to an algebra which restricts isomorphically to both

H^{∗}(II; _{2})^{GL_{3}(2)}, and H^{∗}(I; _{2})^{GL_{3}(2)} for G(l).

On the other hand, the Poincaré series for H^{∗}(G(q); _{2}) is seen to be

P_{E}(x) − \frac{x^{2}(1 + x)}{(1 − x^{2})(1 − x^{3})(1 − x^{4})} = \frac{(1 + x^{3})(1 + x^{5})(1 + x^{6})}{(1 − x^{4})(1 − x^{6})(1 − x^{7})} = 1 + x^{3} + x^{4} + ⋯

when we apply the main result of [FM], and the determination of P_{E}(x) given in §6. It follows that there is a unique non-zero element in H^{4}(G(q); _{2}). From the argument above it must be b_{4}, and there is a unique choice of b_{4} for each G(q). Clearly they must fit together under restriction. This completes the proof. Corollary 7.3: H^{∗}(G(q); _{2}) is Cohen-Macaulay over the subalgebra A_{G(q)} with generators 1, L_{3}, F_{5}, and relations Sq^{2}(L_{3}) = F_{5}, L_{3}^{4} = b_{7}F_{5} + L_{3}^{2}b_{6}, F_{5}^{2} = L_{3}(L_{3}b_{4} + b_{7}). Moreover, H^{∗}(G(q); _{2}) has exactly 2 minimal ideals, I_{I} and I_{II}. The quotient H^{∗}(G(q); _{2})/I_{I} = H^{∗}(I; _{2})^{GL_{3}(2)}, while the second quotient H^{∗}(G(q); _{2})/I_{II} is contained in H^{∗}(II; _{2}) and is generated by the Dickson elements d_{4}, d_{6}, d_{7} together with the two additional elements β^{2}σ + βσ^{2} and β^{4}σ + βσ^{4}.

Proof: We know from [FM, 4.6] that im (res_{G(q)}^{II})^{∗} ⊂ H^{∗}(II; _{2})^{S_{4}}. Another way of

thinking of this is as the subring of H^{∗}(II; _{2})^{UT_{3}(2)} which is also invariant under GL_{2}(2) acting on the subpolynomial algebra _{2}[β, σ]. In particular we see that the generators in the first 4 dimensions are β^{2} + βσ + σ^{2} in dimension 2, β^{2}σ + βσ^{2} in dimension 3, and α^{4} + α^{2}(β^{2} + βσ + σ^{2}) + α(β^{2}σ + βσ^{2}) in dimension 4.

We also know that since H^{∗}(H^{(n)}; _{2}) is detected by restriction to subgroups ( /2)^{3} the same is true of H^{∗}(G(q); _{2}). Consequently, the element in H^{3}(G(q); _{2}) must restrict non-trivially to H^{3}(II; _{2}). The only candidate is β^{2}σ + βσ^{2}, so it must be in the image. Next Sq^{2}(β^{2}σ + βσ^{2}) = β^{4}σ + βσ^{4}, so this class must also be in the image. It remains to show that these two classes generate the entire image.

Let D_{3} = β^{2}σ + βσ^{2}, D_{2} = β^{2} + βσ + σ^{2}, and for convenience write f = β^{4}σ + βσ^{4} = D_{3}D_{2}. Then as in 6.14 we have

d_{4} = a^{4} + a^{2}D_{2} + aD_{3} + D_{2}^{2},
d_{6} = a^{4}D_{2} + a^{2}D_{2}^{2} + aD_{2}D_{3} + D_{3}^{2},
d_{7} = a^{4}D_{3} + a^{2}D_{2}D_{3} + aD_{3}^{2}
= D_{3}(a^{4} + a^{2}D_{2} + aD_{3}),

where a is the remaining generator (dual to j). Consequently, we have D_{3}^{4} = D_{3}^{2}d_{6} + fd_{7}, and, at the same time, f^{2} = D_{3}d_{7} + D_{3}^{2}d_{4}. Lemma 7.4: D_{3} satisfies the irreducible polynomial equation

x^{7} + d_{6}^{2}x^{3} + d_{4}d_{7}^{2}x + d_{7}^{3} = 0

over _{2}[d_{4}, d_{6}, d_{7}]. Consequently the ring, R, generated by D_{3}, f, and _{2}[d_{4}, d_{6}, d_{7}] in H^{∗}(II; _{2}) contains a free submodule

2[d_{4}, d_{6}, d_{7}](1, D_{3}, D_{3}^{2}, D_{3}^{3}, D_{3}^{4}, D_{3}^{5}, D_{3}^{6})

with Poincaré series

\frac{1 − x^{21}}{(1 − x^{3})(1 − x^{4})(1 − x^{6})(1 − x^{7}).}

Proof: We have D_{3}^{8} = D_{3}^{4}d_{6}^{2} + f^{2}d_{7}^{2}. Substituting for f^{2} from above we have

D_{3}^{8} = D_{3}^{4}d_{6}^{2} + D_{3}^{2}d_{4}d_{7}^{2} + D_{3}d_{7}^{3},

but since this is taking place in an integral domain this means

D_{3}^{7} + D_{3}^{3}d_{6}^{2} + D_{3}d_{4}d_{7}^{2} + d_{7}^{3} = 0.

The first statement is verified. We check that the polynomial is irreducible by replacing d_{6} by 0, and setting d_{4} = d_{7} = 1. The resulting polynomial x^{7} + x + 1 is irreducible over 2, and the lemma follows.

Now, f is not integral over R since f = \frac{1}{d_{7}} (D_{3}^{4} + D_{3}^{2}d_{6}), but using this formula, we see that we can use f to replace D_{3}^{4}, D_{3}f to replace D_{3}^{5}, D_{3}^{2}f to replace D_{3}^{6}, and, finally, note the relation

d_{7}^{3} = d_{7}D_{3}^{3}f + d_{4}d_{7}^{2}D_{3} + d_{6}d_{7}fD_{3}

so

7.5  d_{7}^{2} = D_{3}^{3}f + d_{4}d_{7}D_{3} + d_{6}fD_{3}.

Thus, D_{3}, D_{3}^{2}, D_{3}^{3}, f, D_{3}f, D_{3}^{2}f, and D_{3}^{3}f are free generators for a free _{2}[d_{4}, d_{6}, d_{7}] submodule of R. From this it follows that H^{∗}(G; _{2}) is Cohen-Macaulay over _{2}[b_{4}, b_{6}, b_{7}]. Moreover, the relation 7.5 generates the second minimal prime ideal, while the elements D_{3}, f generate for first minimal prime ideal. Corollary 7.6: Let G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 1 mod 4, then H^{∗}(G; _{2}), as a ring is

2[b_{4}, b_{6}, b_{7}, D_{3}, f_{5}]/ {D_{3}^{4} = b_{7}f + D_{3}^{2}b_{6} \atop f_{5}^{2} = D_{3}(D_{3}b_{4} + b_{7}).}

The Steenrod algebra acts on H^{∗}(G; _{2}) by

\begin{aligned}Sq^{2}b_{4} & = b_{6} \\ Sq^{3}b_{4} & = b_{7} \\ Sq^{4}b_{7} & = b_{4}b_{7} \\ Sq^{2}D_{3} & = f \\ Sq^{1}f & = D_{3}^{2} \\ Sq^{4}f & = fb_{4} + D_{3}b_{6}. \end{aligned}

Proof: The only thing to check at this point is Sq^{4}f = Sq^{4}(D_{2}D_{3}) = D_{3}^{3} + D_{2}^{3}D_{3}. We have

fb_{4} = D_{3}D_{2}(D_{2}^{2} + a^{4} + a^{2}D_{2} + aD_{3}) = D_{3}D_{2}^{3} + D_{3}b_{6} + D_{3}^{3},

so the asserted formula follows. (This also provides a check of our relations since Sq^{5}f = f^{2} = Sq^{1}Sq^{4}f = D_{3}^{2}b_{4} + D_{3}b_{7}, while Sq^{6}Sq^{1}f = D_{3}^{4} = Sq^{2}Sq^{5}f = D_{3}^{2}b_{6} + fb_{7}.)

§8. Relation with the cohomology of M_{12} There is an outer isomorphism ϕ : H⟶H defined by

\begin{matrix}a & ↦ a \\ m & ↦ m \\ y & ↦ ay \\ v & ↦ v, \end{matrix}

so using ϕ we obtain a new configuration

8.1  {W ⊃ H ⟶^{ϕ}H ⊂ W′} = D.

This second configuration is also contained in G, after factoring out by maximal odd order subgroups when q ≡ 5 mod 8, as the triple W = N_{G}(⟨j, x, ay⟩) while W′ = N_{G}(⟨j, x, y⟩). However, it is not isomorphic to C, the configuration of 1.11, since, as we have seen ⟨j, x, y⟩ is normal in both W, W′ for C, while W and W′ do not have any jointly normal subgroup ( /2)^{3} in common. D is also contained in M_{12}, and was studied in [AMM] where the following result was proved: Theorem 8.2: Let Π be the amalgamated product W ∗_{H} W′ for the configuration D, then the surjective homomorphism Π⟶M_{12} induced by the inclusion of the configuration induces an isomorphism in mod 2 cohomology

H^{∗}(Π; _{2})⟶^{≅}H^{∗}(M_{12}; _{2}).

We can now put the results above together to obtain Corollary 8.3: Let q ≡ 5 mod 8 then there is an injection of rings H^{∗}(G; _{2}) ⊂ H^{∗}(M_{12}; _{2}) preserving Bocksteins and Steenrod operations where G = G_{2}(q) or ^{3}D_{4}(q) with q ≡ 5 mod 8. Proof: First we explain what this containment means. Since q ≡ 5 mod 8 it follows that Syl_{2}(G) = Syl_{2}(M_{12}), so, choosing an isomorphism Syl_{2}(G) = Syl_{2}(M_{12}) and also representative Sylow subgroups H ⊂ G, H ⊂ M_{12}, we have the restriction maps

(rest_{G}^{H})^{∗} : H^{∗}(G; _{2})⟶H^{∗}(H; _{2}),

(res_{M_{12}}^{H})^{∗} : H^{∗}(M_{12}; _{2})⟶H^{∗}(H; _{2}).

But both these maps are injections. Consequently, having made the choice above, we can ask if one of the images is contained in the other.

In particular, as remarked above, we can choose the embeddings H ⊂ M_{12}, H ⊂ G so that they extend to give embeddings r, s, r′, s′, so the following diagrams commute

\begin{matrix}H & ⟶^{ϕ^{−1}} & W \\ ↓i & & ↓r \\ W′ & ⟶^{s} & M_{12} \end{matrix}  \begin{matrix}H & ⟶^{ϕ^{−1}} & W \\ ↓i & & ↓r′ \\ W′ & ⟶^{s′} & G. \end{matrix}

It follows that im (res_{G}^{H})^{∗} ⊂ i^{∗}(H^{∗}(W′; _{2})) ∩ (ϕ^{−1})^{∗}H^{∗}(W; _{2}) = H^{∗}(M_{12}; _{2}). Moreover, since (res_{G}^{H})^{∗} commutes with cup products, Steenrod squares, and Bocksteins, being induced by a map of topological spaces, the result follows.

§9. A free action of E on G_{2}

The continuous Lie group G_{2} of automorphisms of the Cayley projective plane is a 14 dimensional compact manifold given as a subspace of the Orthogonal group O(7). H.S.M. Coxeter [Cox] constructed an embedding of E in G_{2}, and a particular representation is given in [CW, pg. 448] as the span of the permutation matrices (1, 2, 3, 4, 5, 6, 7), (1, 2, 4)(3, 6, 5), and the matrix δ_{{1,2,4,7}}(1, 2)(3, 6). Here δ_{{a,b,c,d}} is the diagonal matrix in O(7) which has +1’s in all the diagonal positions but (a, a), (b, b), (c, c), and (d, d). Using this embedding we get a free action of E on G_{2}. Theorem 9.1: The Poincaré series for H^{∗}(G_{2}/E; _{2}) is P_{E}(x) · (1 − x^{4})(1 − x^{6})(1 − x^{7}). Indeed,

H^{∗}(G_{2}/E; _{2}) ≅ H^{∗}(E; _{2})/I(b_{4}, b_{6}, b_{7})

where I(b_{4}, b_{6}, b_{7}) is the ideal in H^{∗}(E; _{2}) spanned by these elements. Proof: There is a fibration G_{2}/E⟶B_{E}⟶B_{G_{2}}. We know that

H^{∗}(B_{G_{2}}; _{2}) = _{2}[b_{4}, b_{6}, b_{7}]

and we claim that the induced map H^{∗}(B_{G_{2}}; _{2})⟶H^{∗}(E; _{2}) is injective in cohomology. Assuming this, if we take the cotor spectral sequence converging to H^{∗}(G_{2}/E; _{2}), it has E_{2} term

Ext_{H^{∗}(B_{G_{2}} ; _{2})}^{∗} (H^{∗}(E; _{2}), _{2}) = Ext_{H^{∗}(B_{G_{2}} ; _{2})}^{0} (H^{∗}(E; _{2}), _{2})

since H^{∗}(E; _{2}) is free as an H^{∗}(B_{G_{2}}; _{2}) module. Consequently, the spectral sequence collapses and the result follows.

It remains to verify the claim of injectivity. To do this we analyze the composite map

( /2)^{3}⟶E⟶G_{2}⟶O(7).

We can write ( /2)^{3} ≅ _{8} so that an additive basis is given by 1, ζ_{7}, ζ_{7}^{2}, with ζ_{7}^{3} = 1 + ζ_{7}. Then the seven non-zero elements can be enumerated as powers of ζ_{7}, and are given explicitly by the table

9.2  \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ ζ_{7} & ζ_{7}^{2} & 1 + ζ_{7} & ζ_{7} + ζ_{7}^{2} & 1 + ζ_{7} + ζ_{7}^{2} & 1 + ζ_{7}^{2} & 1 \end{matrix}

Choose a basis for ^{7} and write A = (1, 2, 3, 4, 5, 6, 7), B = (1, 2, 4)(3, 6, 5), and C = δ_{{1,2,4,7}}(1, 2)(3, 6) where the permutations are of the basis elements. Then A, B, C are elements of O(7). They map into GL_{3}(2) by forgetting the δ_{{a,b,c,d}} part, and regarding the permutation as a permutation of the elements in 9.2 corresponding to 1, …, 7. In fact we find

A ↦ ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) B ↦ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ) C ↦ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) .

As a consequence

CB ↦ ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , (ABCA^{−1})(CB) ↦ ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

C(ABCA^{−1})(CB)C ↦ ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

and each of these elements has order two. Hence

α = [(CB)C(ABCA^{−1})(CB)C]^{2}(ABCA^{−1})(CB) ∈ ( /2)^{3}

and cannot be zero there since otherwise the extension would be split. Expanding out we have that CB = δ_{{1,2,4,7}}(2, 4)(5, 6), ABCA^{−1} = δ_{{1,2,3,5}}(2, 5)(4, 6), so

(ABCA^{−1})(CB) = δ_{{2,3,6,7}}(2, 6)(4, 5).

Finally, conjugating this last expression by C we have

C(ABCA^{−1})(CB)C = δ_{{4,5,6,7}}(1, 3)(4, 5),

and (CB)(C(ABCA^{−1})(CB)C) = δ_{{2,3,4,6}}(1, 3)(2, 5, 6, 4). Now we can calculate α. We have

α = (δ_{{2,3,4,6}}(1, 3)(2, 5, 6, 4))^{2}δ_{{2,3,6,7}}(2, 6)(4, 5)
= δ_{{1,3,5,6}}(2, 6)(4, 5)δ_{{2,3,6,7}}(2, 6)(4, 5)
= δ_{{1,2,5,7}}.

Translating by A we find that ( /2)^{3} has generators α, β = δ_{{1,2,3,6}} and δ = δ_{{2,3,4,7}}. Now, let x_{i} : (Diag(O(7))⟶ /2 be given by x_{i}(D) = D_{ii}. Then, in cohomology, we find

\begin{matrix}x_{1}^{∗} & ↦ α^{∗} + β^{∗} \\ x_{2}^{∗} & ↦ α^{∗} + β^{∗} + ϵ^{∗} \\ x_{3}^{∗} & ↦ α^{∗} + ϵ^{∗} \\ x_{4}^{∗} & ↦ ϵ^{∗} \\ x_{5}^{∗} & ↦ β^{∗} \\ x_{6}^{∗} & ↦ α^{∗} \\ x_{7}^{∗} & ↦ α + β. \end{matrix}

But this then determines the image of the Stiefel-Whitney classes in H^{∗}(B_{O(7)}; _{2}) in H^{∗}(( /2)^{3}; _{2}) as

∏_{ν∈Hom(( /2)^{3}, /2)} (1 + ν) = 1 + d_{4} + d_{6} + d_{7},

as is shown for example in [MM]. Since this map factors through H^{∗}(B_{G_{2}}; _{2}), the assertion follows, and the proof is complete.

§10. Connection with the Steinberg module

The remarks in this section are due to Alex Adem. From [W, §5 and Theorem D’] we have a decomposition into direct sums

10.1  St ⊕ [E/W] ⊕ [E/W′] ≅_{(2)} [E/H] ⊕

where [E/L] is the free -module with basis the left cosets of L in E, E acts on [G/L] via the usual left action on left cosets, and St is the Steinberg module for GL_{3}(2). More exactly, the Tits building, B, for GL_{3}(2) is a graph, with one conjugacy class of edges and two conjugacy classes of vertices, isomorphic to A_{2}(GL_{3}(2)), and H_{1}(B; ) = St while H^{0}(B; ) = . The isotropy group of any edge is conjugate to H, while W is the isotropy group of one vertex and W′ is the isotropy group of the other. Consequently C_{0} = [E/W] ⊕ [E/W′], C_{1} = [E/H], so 10.1 follows from the splitting theorems for permutation modules in [W].

When we tensor 10.1 with _{2} we obtain an isomorphism

10.2  St ⊗ _{2} ⊕ _{2}[E/W] ⊕ _{2}[E/W′] = _{2}[E/H] ⊕ _{2}.

Now, by change of rings [CE], H^{∗}(E; _{2}[E/L]) ≅ H^{∗}(L; _{2}), so we obtain

10.3  H^{∗}(E; St ⊗ _{2}) ⊕ H^{∗}(W; _{2}) ⊕ H^{∗}(W′; _{2}) ≅ H^{∗}(H; _{2}) ⊕ H^{∗}(E; _{2}).

In particular, we have Corollary 10.4: The Poincaré series for H^{∗}(E; St ⊗ _{2}) is given as the difference

P_{E}(x) − P_{M_{12}}(x)

of the Poincaré series for M_{12} and that for E. (Since, from [AMM], the Poincaré series for M_{12} is P_{W} (x) + P_{W′}(x) − P_{H}(x).)

References

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Department of Mathematics Stanford University Stanford, CA 94305