CONWAY’S GROUP Co_{3} AND THE DICKSON INVARIANTS

D. J. BENSON

In this paper, we construct a map from the classifying space BCo_{3} of Conway’s sporadic simple group Co_{3} to the classifying space BDI(4) of the new finite loop space at the prime two DI(4) of Dwyer and Wilkerson. This map has the property that it injects the mod two cohomology of BDI(4) (which is equal to the Dickson invariants of rank four) as a subring over which the mod two cohomology of BCo_{3} is finitely generated as a module.

1. Introduction

A few years ago, Adem, Maginnis and Milgram [1] calculated the mod two cohomology of the sporadic Mathieu group M_{12}. They found that it contains a copy of the Dickson invariants of rank three, on generators of degrees four, six and seven. It is finitely generated and free as a module over this subring. Since this algebra of Dickson invariants is equal to the cohomology of the classifying space BG_{2} of the compact Lie group G_{2}, one is tempted to speculate that there is a group homomorphism M_{12} → G_{2} inducing this inclusion in mod two cohomology. That this is not the case is easily seen by noticing that G_{2} has a seven dimensional faithful complex representation, while M_{12} does not. Nevertheless, it turns out that there is a map of classifying spaces BM_{12} → BG_{2} which induces the above inclusion in mod two cohomology. This map was constructed by Milgram (unpublished) using étale homotopy, and more directly by the author in joint work with Clarence Wilkerson [2].

Dwyer and Wilkerson [7] recently constructed a new finite loop space at the prime two DI(4) ≃ ΩBDI(4). The mod two cohomology of BDI(4) is the algebra of Dickson invariants of rank four, on generators of degrees 8, 12, 14 and 15. The goal of this paper is to construct a map from the classifying space BCo_{3} of Conway’s sporadic group Co_{3} to the space BDI(4). We do this in such a way that the corresponding map in mod two cohomology embeds the Dickson invariants of rank four into the cohomology of Co_{3}, and so that the latter is finitely generated as a module over the former. Our main theorem is the following.

Theorem 1.1. The mod two cohomology H^{∗}(Co_{3}, 𝔽_{2}) contains a copy of the algebra of Dickson invariants of rank four, on generators of degrees 8, 12, 14 and 15, in such a way that the Steenrod operations act in the normal way on these Dickson invariants. The whole cohomology ring is finitely generated as a module over this subring.

There is a map of classifying spaces BCo_{3} → BDI(4) realizing the inclusion of the Dickson invariants in the mod two cohomology of Co_{3}.

We use Robert Curtis’ MOG (Miracle Octad Generator) notation for displaying vectors in the Leech lattice. In this notation, the 24 basis vectors are arranged in a 4 × 6 array, subdivided into three 4 × 2 blocks. For an explanation of this notation, we refer the reader to Chapter 11 of Conway and Sloane [5]. For a description of the Leech lattice and some of the sporadic groups associated with it, see Chapter 10 of the same reference. Further information about Co_{3} may be found in Conway [4], Finkelstein [10], Solomon [13], and the ATLAS of finite groups [6]. For background on the Dickson invariants, see Wilkerson [14].

Our topological constructions are performed in the category of simplicial sets, so the word “space” means “simplicial set”. If X is a space, we write X̂ for the Bousfield–Kan 𝔽_{p}-completion of X. In particular, B̂G denotes the 𝔽_{p}-completion of the simplicial set of singular simplices on the classifying space of a compact Lie group G. For standard properties of 𝔽_{p}-completion, see Bousfield and Kan [3]. We are mostly interested in the case p = 2 for the purposes of this paper.

The paper is organized as follows. In Section 2 we write down some explicit generators for Co_{3} and describe the 2-local structure in terms of these generators. In Section 3, we show how to express B̂Co_{3} as a homotopy colimit in terms of the 2-local structure. In Sections 4 and 5 we discuss the structure of BDI(4) in more explicit detail than was done in [7], in preparation for the construction of the maps. In Sections 6, 7 and 8 we construct maps from the 2-completions of the classifying spaces of the relevant 2-local subgroups of Co_{3} to BDI(4), and show how to put these together to make a map BCo_{3} → BDI(4) with the relevant properties.

2. Some subgroups of Co_{3}

We take Co_{3} to be the stabilizer of the vector

\begin{matrix}−2 & 2 & 4 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 0 & 0 \end{matrix}

in the Leech lattice. This is a sporadic finite simple group of order

495, 766, 656, 000 = 2^{10}.3^{7}.5^{3}.7.11.23.

We begin by giving names to some elements of Co_{3}.

a_{1} = \begin{matrix}+ & + & + & + & − & − \\ + & + & + & + & − & − \\ + & + & + & + & − & − \\ + & + & + & + & − & − \end{matrix} a_{2} = \begin{matrix}+ & + & + & − & + & − \\ + & + & + & − & + & − \\ + & + & + & − & + & − \\ + & + & + & − & + & − \end{matrix} a_{3} = \begin{matrix}+ & + & + & + & + & + \\ + & + & + & + & + & + \\ + & + & − & − & − & − \\ + & + & − & − & − & − \end{matrix}

a_{4} = \begin{matrix}+ & + & + & + & + & + \\ + & + & − & − & − & − \\ + & + & + & + & + & + \\ + & + & − & − & − & − \end{matrix} b_{1} = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix} b_{2} = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}

c_{1} = \begin{matrix}− & + & + & − & + & + \\ − & + & − & + & + & + \\ + & + & + & + & − & + \\ + & + & − & − & − & + \end{matrix} . \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix} c_{2} = \begin{matrix}− & + & + & − & + & + \\ + & + & − & − & − & + \\ − & + & − & + & + & + \\ + & + & + & + & − & + \end{matrix} . \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}

c_{3} = \begin{matrix}− & − & + & − & − & − \\ + & + & − & + & + & + \\ + & + & − & + & + & + \\ + & + & − & + & + & + \end{matrix} . \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix} d = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}

e = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix} f = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix} g = \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}

h = ε ε ε −ε −ε −ε . \begin{matrix}· & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}

Here, ε denotes the 4 × 4 matrix

( \begin{matrix}−\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & −\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & −\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & −\frac{1}{2} \end{matrix} ) .

These elements satisfy the following relations (∗∗): a_{i}^{2} = 1, [a_{i}, a_{j}] = 1 (1 ≤ i, j ≤ 4), b_{1}^{2} = b_{2}^{2} = [b_{1}, b_{2}] = 1, [a_{1}, b_{1}] = [a_{2}, b_{1}] = [a_{1}, b_{2}] = [a_{2}, b_{2}] = 1, [a_{3}, b_{1}] = a_{1}, [a_{4}, b_{1}] = a_{2}, [a_{3}, b_{2}] = a_{1}a_{2}, [a_{4}, b_{2}] = a_{1}, c_{1}^{2} = c_{2}^{2} = c_{3}^{2} = 1, [c_{1}, c_{2}] = 1, [c_{1}, c_{3}] = [c_{2}, c_{3}] = a_{1}, [a_{1}, c_{1}] = [a_{2}, c_{1}] = [a_{3}, c_{1}] = [a_{1}, c_{2}] = [a_{2}, c_{2}] = [a_{4}, c_{2}] = [a_{1}, c_{3}] = [a_{3}, c_{3}] = [a_{4}, c_{3}] = 1, [a_{4}, c_{1}] = [a_{3}, c_{2}] = [a_{2}, c_{3}] = a_{1}, [b_{1}, c_{1}] = [b_{2}, c_{1}] = [b_{2}, c_{2}] = a_{1}a_{2}, [b_{1}, c_{2}] = 1, [b_{1}, c_{3}] = a_{4}c_{1}, [b_{2}, c_{3}] = a_{3}a_{4}c_{2}, d^{3} = 1, [a_{1}, d] = [a_{2}, d] = 1, da_{3}d^{−1} = a_{4}, da_{4}d^{−1} = a_{3}a_{4}, db_{1}d^{−1} = b_{2}, db_{2}d^{−1} = b_{1}b_{2}, dc_{1}d^{−1} = c_{2}, dc_{2}d^{−1} = a_{1}a_{2}c_{1}c_{2}, [c_{3}, d] = 1, e^{2} = 1, [a_{1}, e] = [a_{3}, e] = 1, [a_{2}, e] = a_{1}, [a_{4}, e] = a_{3}, [b_{1}, e] = 1, [b_{2}, e] = b_{1}, [c_{1}, e] = a_{3}, [c_{2}, e] = a_{2}a_{3}a_{4}c_{1}, [c_{3}, e] = 1, ede = d^{2}, f^{2} = 1, [a_{1}, f] = [a_{3}, f] = 1, [a_{2}, f] = a_{1}a_{3}, [a_{4}, f] = a_{1}, [b_{1}, f] = e, (b_{2}f)^{3} = 1, [c_{1}, f] = 1, [c_{2}, f] = a_{2}a_{4}c_{3}, [c_{3}, f] = a_{1}a_{3}, (df)^{3} = 1, g^{3} = 1, ga_{1}g^{−1} = a_{2}, ga_{2}g^{−1} = a_{1}a_{2}, [a_{3}, g] = [a_{4}, g] = 1, gb_{1}g^{−1} = b_{1}b_{2}, gb_{2}g^{−1} = b_{1}, gc_{1}g^{−1} = a_{1}a_{2}a_{3}b_{1}c_{2}, gc_{2}g^{−1} = a_{4}b_{2}c_{1}c_{2}, (c_{3}g)^{2} = 1, [d, g] = 1, (eg)^{2} = 1, (fg)^{4} = 1, h^{2} = 1, [a_{1}, h] = 1, [a_{2}, h] = a_{1}, ha_{3}h = a_{2}a_{4}b_{1}c_{1}, ha_{4}h = a_{2}a_{3}a_{4}b_{2}c_{2}, [b_{1}, h] = [b_{2}, h] = 1, hc_{1}h = a_{1}a_{4}b_{1}b_{2}c_{2}, hc_{2}h = a_{2}a_{3}a_{4}b_{1}c_{1}c_{2}, (c_{3}h)^{3} = 1, [d, h] = 1, [e, h] = 1, (a_{2}fh)^{4} = a_{1}, (gh)^{2} = d.

The subgroup

P = ⟨a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, c_{1}, c_{2}, c_{3}, e⟩

is a Sylow 2-subgroup of Co_{3}. Let Z = Z(P) = ⟨a_{1}⟩ ≅ ℤ/2, U = ⟨a_{1}, a_{2}⟩ ≅ (ℤ/2)^{2} and V = ⟨a_{1}, a_{2}, a_{3}, a_{4}⟩ ≅ (ℤ/2)^{4}. Then the maximal 2-local subgroups containing P are as follows:

C = C_{Co_{3}}(Z) = ⟨P, d, f, h⟩ ≅ 2S_{6}(2),
N = N_{Co_{3}}(U) = ⟨P, d, g, h⟩

of shape 2^{2+6}3^{1+2}2^{2}, and

X = N_{Co_{3}}(V) = ⟨P, d, f, g⟩.

Note that O_{2}(N) is a special group of shape 2^{2+6} with U as its center, and that the elements a_{3}, a_{4}, b_{1}, b_{2}, c_{1} and c_{2} are coset representatives forming a basis of O_{2}(N)/U ≅ (ℤ/2)^{6}. The group X/V is isomorphic to A_{8} ≅ GL(4, 𝔽_{2}), and the extension does not split. There is a unique isomorphism class of non-split extension, so this information determines the group X uniquely up to isomorphism.

The group

C ∩ X = ⟨P, d, f⟩

has shape 2_{+}^{1+6} : GL(3, 𝔽_{2}). The subgroup

O_{2}(C ∩ X) = ⟨a_{1}, a_{2}, a_{3}, a_{4}, c_{1}, c_{2}, c_{3}⟩

is an extraspecial group 2_{+}^{1+6} with center Z generated by a_{1}. The central quotient (C ∩ X)/Z = 2^{6} : GL(3, 𝔽_{2}) has two conjugacy classes of complements (this is an easy calculation using for example Shapiro’s lemma for degree one cohomology). One of them lifts to a conjugacy class of complements for O_{2}(C ∩ X) in C ∩ X, a representative of which is

H = ⟨b_{1}, b_{2}, d, e, f⟩ ≅ GL(3, 𝔽_{2}).

The other lifts to a conjugacy class of supplements isomorphic to the non-trivial double cover SL(2, 𝔽_{7}) of GL(3, 𝔽_{2}).

The group

C ∩ N ∩ X = ⟨P, d⟩

has order 2^{10}.3. The subgroup

O_{2}(C ∩ N ∩ X) = ⟨a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, c_{1}, c_{2}, c_{3}⟩

has index two in P, and the quotient is

(C ∩ N ∩ X)/O_{2}(C ∩ N ∩ X) ≅ Σ_{3}.

Theorem 2.1. (i) The group defined by generators a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, c_{1}, c_{2}, c_{3}, d, e, f, g and h, and relations (∗∗), is isomorphic to Co_{3}.

(ii) The group defined by the same set of generators with g removed, and those relations among (∗∗) which do not involve g, is isomorphic to C.

(iii) The group defined by the same set of generators with f removed, and those relations among (∗∗) which do not involve f, is isomorphic to N.

(iv) The group defined by the same set of generators with h removed, and those relations among (∗∗) which do not involve h, is isomorphic to X.

Proof. This was proved by performing coset enumerations using John Cannon’s program MAGMA. Furthermore, since the coset enumeration was performed by first copying the relations (∗∗) to a text file readable by MAGMA, this provides comforting evidence that these relations are correct. □

3. B̂Co_{3} as a homotopy colimit

Let G be a compact Lie group and p a prime. Let A_{p}(G) denote the Quillen category, whose objects are the non-trivial finite elementary abelian p-subgroups E ≤ G, and whose morphisms are generated by the conjugations and inclusions in G. In [12], Jackowski and McClure prove the following.

Theorem 3.1. The natural map

hocolim ; ⟶_{E∈Ap(G)}(EG/C_{G}(E)) → BG

is a mod p cohomology equivalence.

We are interested in a variation of this theorem, where under certain conditions on G, we may cut down considerably on the set of centralizers used in this homotopy colimit.

Suppose that E is a collection of elements of order p in G with the following properties:

(i) E is closed under conjugation in G.

(ii) If x ∈ E then every power of x is in E ∪ {1}.

(iii) If x and y are commuting elements of E then xy ∈ E ∪ {1}.

(iv) Every element of order p in the center of a maximal p-toral subgroup of G lies in E. Let E(G) denote the full subcategory of A_{p}(G) whose objects are the non-trivial finite elementary abelian p-subgroups E ≤ G with the property that E ⊆ E ∪ {1}.

Theorem 3.2. The natural map

hocolim ; ⟶_{E∈E(G)}(EG/C_{G}(E)) → BG

is a mod p cohomology equivalence.

Proof. The subcategory E(G) satisfies the conditions (i), (ii) and (iii) of Theorem 7.7 in [12]. □

Now let G = Co_{3} and p = 2, and let E be the set of involutions which are in the center of some Sylow 2-subgroup of Co_{3}. The centralizer of an element of E is isomorphic to 2S_{6}(2), and in this group, every involution outside the center is conjugate to its product with the central involution (this is easy to read off from the ATLAS entry [6] for S_{6}(2)). So E satisfies condition (iii) of the last section. The other conditions are easy to verify, and so we may apply the above theorem to deduce that the map

hocolim ; ⟶_{E∈E(G)} ECo_{3}/C_{Co_{3}}(E) → BCo_{3}

is a mod two cohomology equivalence.

The category E(Co_{3}) contains four conjugacy classes of subgroups, of orders 2, 4, 8 and 16, with representatives Z = ⟨a_{1}⟩, U = ⟨a_{1}, a_{2}⟩, Y = ⟨a_{1}, a_{2}, a_{3}⟩ and V = ⟨a_{1}, a_{2}, a_{3}, a_{4}⟩. Now V = C_{Co_{3}}(V) = C_{Co_{3}}(Y), and so each conjugate of Y is contained in a unique conjugate of V. It follows that if we denote by E′(Co_{3}) the full subcategory of E(Co_{3}) consisting of the conjugates of Z, U and V, then hocolim ; ⟶_{E∈E′(Co_{3})} ECo_{3}/C_{Co_{3}}(E) is a strong deformation retract of hocolim ; ⟶_{E∈E(Co_{3})} ECo_{3}/C_{Co_{3}}(E).

Finally, we replace E′(Co_{3}) by a skeletal subcategory C consisting of just the objects Z, U and V and the morphisms between them. The category C has the following shape:

Z ⇒  U  ⇒  V
|  |  |
{1}  GL(2, 𝔽_{2})  GL(4, 𝔽_{2})

Here, we have used the notation of [7]. Under each object is its monoid of self-maps, and “⇒” stands for an appropriate set of morphisms. To summarize, we have proved the following theorem:

Theorem 3.3. The map

hocolim ; ⟶_{E∈C} ECo_{3}/C_{Co_{3}}(E) → BCo_{3}

is a mod two cohomology equivalence; hence after 𝔽_{2}-completing, it is a homotopy equivalence. □

Another way of writing this 𝔽_{2}-completed homotopy colimit, which is possibly easier to visualize, is as the homotopy colimit of the following diagram:

\xymatrix{B̂C &  & B̂(C ∩ N) \ar[ll] \ar[rr] &  & B̂N \\  &  & B̂(C ∩ N ∩ X) \ar[u] \ar[dl] \ar[dr] \\  & B̂(C ∩ X) \ar[uul] \ar[dr] &  & B̂(N ∩ X) \ar[uur] \ar[dl] \\  &  & B̂X}

This homotopy colimit may be written as a quotient of

B̂C ∐ B̂N ∐ B̂X ∐ B̂(C ∩ N) × Δ^{1} ∐ B̂(C ∩ X) × Δ^{1}
∐ B̂(N ∩ X) × Δ^{1} ∐ B̂(C ∩ N ∩ X) × Δ^{2}

(where Δ^{1} and Δ^{2} are a standard 1-simplex and 2-simplex respectively) by the obvious identifications coming from the arrows.

The proof that this homotopy colimit is equivalent to the one described in Theorem 3.3 is essentially the same as the proof of Proposition 5.1 of Jackowski and McClure [11]. The argument is due to Bill Dwyer, and involves going via an intermediate “twisted arrow” category.

Remark. In contrast with the situation for the map BM_{12} → BG_{2} described in Benson and Wilkerson [2] (see in particular Lemma 3.3 of that paper), it is not true that the character of the alternating sum of the permutation modules

ℚ_{2}^{∧}[Co_{3}/Co_{3}] − ℚ_{2}^{∧}[Co_{3}/C] − ℚ_{2}^{∧}[Co_{3}/N] − ℚ_{2}^{∧}[Co_{3}/X] + ℚ_{2}^{∧}[Co_{3}/(C ∩ N)]
+ℚ_{2}^{∧}[Co_{3}/(C ∩ X)] + ℚ_{2}^{∧}[Co_{3}/(N ∩ X)] − ℚ_{2}^{∧}[Co_{3}/(C ∩ N ∩ X)]

has no constituent in the principal 2-block. Indeed, using John Cannon’s program MAGMA, this alternating sum was calculated to be the following generalized character:

(−50378624, 0, 496, 2080, 784, −125, 0, 0, −24, −19, 0, 0, 0, −8, −5, −2,
0, 0, 0, −8, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, −1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0).

Here, the conjugacy classes are listed in the same order as in the ATLAS [6]. This character decomposes as:

−26χ_{42} − 26χ_{41} − 26χ_{40} − 25χ_{39} − 23χ_{38} − 24χ_{37} − ⋯

The character χ_{37} is in the principal 2-block. It seems that the reason why this character cannot avoid the principal block is that the degree is so large in magnitude in comparison with the other character values.

Of course, the reason for wondering about the above alternating sum of characters is that by Theorem 3.3 it is the character of a finite ℤ-free chain complex of ℤCo_{3}- modules whose mod two hypercohomology vanishes.

4. The Weyl group of DI(4)

We need to be more explicit in some of the constructions made by Dwyer and Wilkerson [7], in order to construct the desired maps. We begin with a discussion of the Weyl group W_{DI(4)} ≅ ℤ/2 × GL(3, 𝔽_{2}) and its 3-dimensional 2-adic reflection representation.

We regard SO(7) as acting on a seven dimensional real space spanned by all except the top-left vector of a 4 × 2 array of basis vectors. For the maximal torus T ⊆ Spin(7), we choose the inverse image of the subgroup SO(2)^{3} ⊆ SO(7) embedded as follows:

∗ S

O

· (2)
S S
O O
(2) (2)

Thus there is a short exact sequence 1 → ℤ/2 → T → SO(2)^{3} → 1. Recall from p. 53 of [7] that T_{n} is the subgroup of T consisting of elements x such that x^{2^{n}} = 1, and T_{∞} = ⋃_{n} T_{n}. The maps

GL(3, ℤ̂_{2}) → Aut(T_{∞}) → π_{0}Aut(B̂T_{∞}) → π_{0}Aut(B̂T)

are isomorphisms. Here, GL(3, ℤ̂_{2}) is the subgroup of GL(3, ℚ̂_{2}) consisting of matrices which preserve the length three column vectors with integer coordinates with even sum.

The group W_{1} = W_{Spin(7)}(T) is the subgroup ℤ/2 × Σ_{4} ⊆ GL(3, ℤ̂_{2}) consisting of the monomial matrices with entries 0 and ±1. According to the ATLAS [6], we may extend this to a 2-adic reflection group ℤ/2×GL(3, 𝔽_{2}) ⊆ GL(3, ℤ̂_{2}) as follows. Let α and β be the 2-adic roots of the equation x^{2} + x + 2 = 0, with α ≡ 5 (8) and β ≡ 2 (8). The orbit of the vector (2, 0, 0) consists of the images under W_{1} of the vectors (2, 0, 0), (0, α, α) and (1, 1, β). Here, for typographical convenience, we write our column vectors as row vectors.

In the notation of [7], the subgroup W_{1} ⊆ ℤ/2 × GL(3, 𝔽_{2}) is ℤ/2 × P (1, 2). The subgroup ℤ/2 × U(2, 1) ≅ (ℤ/2)^{3} of W_{1} is generated by the matrices

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

and ℤ/2 × U is generated by these together with ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ). The normalizer of

ℤ/2 × U(2, 1) in ℤ/2 × GL(3, 𝔽_{2}) is ℤ/2 × P (2, 1), which contains an element ζ of order two swapping the second and third of these matrices. This element ζ therefore swaps the (−1)-eigenvector (0, α, α) of the second matrix with the (−1)-eigenvector (2, 0, 0) of the third matrix, and fixes the common (+1)-eigenvector (0, α, −α). Thus we have

ζ = ( \begin{matrix}0 & β/2 & β/2 \\ α/2 & 1/2 & −1/2 \\ α/2 & −1/2 & 1/2 \end{matrix} ) .

The group W_{DI(4)} is generated by W_{1} and ζ, while ℤ/2 × P (2, 1) is generated by ℤ/2 × U and ζ.

If we change basis using the matrix

( \begin{matrix}2 & 0 & 0 \\ 0 & α & −α \\ 0 & α & α \end{matrix} ) ∈ GL(2, ℚ̂_{2})

then ℤ/2 × P (2, 1) is the group of monomial matrices, because

( \begin{matrix}1/2 & 0 & 0 \\ 0 & β/4 & β/4 \\ 0 & −β/4 & β/4 \end{matrix} ) ( \begin{matrix}0 & β/2 & β/2 \\ α/2 & 1/2 & −1/2 \\ α/2 & −1/2 & 1/2 \end{matrix} ) ( \begin{matrix}2 & 0 & 0 \\ 0 & α & −α \\ 0 & α & α \end{matrix} ) = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )

( \begin{matrix}1/2 & 0 & 0 \\ 0 & β/4 & β/4 \\ 0 & −β/4 & β/4 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}2 & 0 & 0 \\ 0 & α & −α \\ 0 & α & α \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )

( \begin{matrix}1/2 & 0 & 0 \\ 0 & β/4 & β/4 \\ 0 & −β/4 & β/4 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & −1 & 0 \end{matrix} ) ( \begin{matrix}2 & 0 & 0 \\ 0 & α & −α \\ 0 & α & α \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

In the next section, we discuss how to effect this change of basis.

5. The homotopy equivalence B̂Q_{red}^{3} → B̂(Spin(3) ∘ Spin(4))

In order to make explicit the action of GL(2, 𝔽_{2}) ≅ Σ_{3} on B̂Q_{red}^{3} described in Proposition 6.1 of [7], we describe the homotopy equivalence

B̂Q_{red}^{3} → B̂(Spin(3) ∘ Spin(4))

where Spin(3)∘Spin(4) is the inverse image in Spin(7) of SO(3)×SO(4). By choosing this homotopy equivalence carefully, we effect the change of basis described in the last section, so that the action of Σ_{3} becomes the usual permutation action on the three copies of SU(2).

We begin by discussing the well known isomorphisms Q ≅ Spin(3) and Q^{2} ≅ Spin(4). We regard Q as the quaternions of unit norm, Q ⊆ ℍ, and the conjugation action of Q on the subspace ℝ^{3} of pure imaginary quaternions gives a 2-fold covering map Q → SO(3) which lifts to an isomorphism Q → Spin(3). In particular, using

the labelling \begin{matrix}∗ \\ i \\ j \\ k \end{matrix} , we have

1 ↦ \begin{matrix}∗ \\ + \\ + \\ + \end{matrix}  i ↦ \begin{matrix}∗ \\ + \\ − \\ − \end{matrix}  j ↦ \begin{matrix}∗ \\ − \\ + \\ − \end{matrix}  k ↦ \begin{matrix}∗ \\ − \\ − \\ + \end{matrix}

−\frac{1}{2}(1 + i + j + k) ↦ \begin{matrix}∗ \\ · \\ · \\ · \end{matrix} \frac{1}{\sqrt{2}}(j + k) ↦ \begin{matrix}∗ \\ − \\ · \\ · \end{matrix}

Similarly, Q^{2} acts on ℍ ≅ ℝ^{4} by letting (g_{1}, g_{2}) send x to g_{1}xg_{2}^{−1}. This gives us a 2-fold covering map Q^{2} → SO(4), which lifts to an isomorphism Q^{2} ≅ Spin(4). The automorphism which swaps the two copies of Q corresponds to the automorphism of SO(4) given by quaternionic conjugation, which has determinant −1. Using the

labelling \begin{matrix}1 \\ i \\ j \\ k \end{matrix} , we have

(1, −1) ↦ \begin{matrix}− \\ − \\ − \\ − \end{matrix}  (i, i) ↦ \begin{matrix}+ \\ + \\ − \\ − \end{matrix}  (j, j) ↦ \begin{matrix}+ \\ − \\ + \\ − \end{matrix}  (k, k) ↦ \begin{matrix}+ \\ − \\ − \\ + \end{matrix}

(k, j) ↦ \begin{matrix}· \\ · \\ · \\ · \end{matrix}  (i, k) ↦ \begin{matrix}· \\ · \\ · \\ · \end{matrix}  (j, i) ↦ \begin{matrix}· \\ · \\ · \\ · \end{matrix}  (↔) ↦ \begin{matrix}+ \\ − \\ − \\ − \end{matrix}

(−\frac{1}{2}(1 + i + j + k), −\frac{1}{2}(1 + i + j + k)) ↦ \begin{matrix}· \\ · \\ · \\ · \end{matrix}

(−\frac{1}{2}(1 + i + j + k), −\frac{1}{2}(1 − i − j − k)) ↦ \begin{matrix}+ \\ − \\ − \\ − \end{matrix} · ε

Putting these maps together gives us an isomorphism Q_{red}^{3} → Spin(3) ∘ Spin(4) and hence a homotopy equivalence B̂Q_{red}^{3} → B̂(Spin(3) ∘ Spin(4)). This is not the homotopy equivalence we wish to use, because with respect to the chosen bases for the maximal tori of Q^{3} and SO(7), the matrix for this map is

( \begin{matrix}2 & 0 & 0 \\ 0 & 1 & −1 \\ 0 & 1 & 1 \end{matrix} ) .

[Note that we have carefully avoided choosing a basis for the maximal torus of Spin(7) by redefining GL(3, ℤ̂_{2}) at the beginning of Section 4. We have also avoided choosing a basis for the maximal torus of Q_{red}^{3} by identifying π_{1}Q_{red}^{3} as an appropriate submodule of (ℤ/2)^{3}.] Now α is a 2-adic unit, and hence gives rise to a “scalar” self homotopy equivalence of B̂Spin(4). If we compose with this, the new matrix for the map is

( \begin{matrix}2 & 0 & 0 \\ 0 & α & −α \\ 0 & α & α \end{matrix} )

We let ρ_{0} : B̂Q_{red}^{3} → B̂(Spin(3) ∘ Spin(4)) be the corresponding map. So according to the matrix calculation at the end of the last section, we may now assume that the action of GL(2, 𝔽_{2}) ≅ Σ_{3} on B̂Q_{red}^{3} described in Proposition 6.1 of [7] comes from the permutation action on the three copies of Q. Let Spin(3, 4) be the inverse image in Spin(7) of (O(3) × O(4)) ∩ SO(7). So Spin(3, 4) contains Spin(3) ∘ Spin(4) ≅ Q_{red}^{3} as a subgroup of index two, and is a semidirect product of this with a copy of ℤ/2 permuting the second and third copies of Q_{red}^{3}. So ρ_{0} extends to a map ρ : B̂(Q_{red}^{3}⋊ℤ/2) → B̂Spin(3, 4).

To understand the meaning of the scalar multiplication by α on B̂Spin(4), we use the description given in Lemma 5.4 and Proposition 5.5 of [7]. Namely, we regard B̂Q as the 2-completion of the homotopy pushout of

B̂Õ_{48} ← B̂Q_{16} → B̂ lim_{\overrightarrow{n}} Q_{2^{n+1}}.

Lemma 5.1. Let Q_{2^{n+1}} = ⟨x, y | x^{2^{n}} = y^{4} = 1, y^{2} = x^{2^{n−1}}, yxy^{−1} = x^{−1}⟩. Then for n ≥ 3, Aut(Q_{2^{n+1}}) = H⋊K, where H = C_{Aut(Q_{2^{n+1}})}(x) ≅ ℤ/2^{n} and K = C_{Aut(Q_{2^{n+1}})}(y) ≅ Aut(H) ≅ ℤ/2 × ℤ/2^{n−2}.

Proof. Since n ≥ 3, ⟨x⟩ is characteristic in Q_{2^{n+1}}. So H = C_{Aut(Q_{2^{n+1}})}(x) is normal in Aut(Q_{2^{n+1}}), and is generated by x ↦ x, y ↦ yx. It follows that H ≅ ⟨x⟩ ≅ ℤ/2^{n}. Since H acts transitively on the elements of Q_{2^{n+1}} not in ⟨x⟩, the subgroup K = C_{Aut(Q_{2^{n+1}})}(y) is a complement to H in Aut(Q_{2^{n+1}}), and is isomorphic to Aut(H) ≅ Aut⟨x⟩. It is generated by x ↦ x^{−1}, y ↦ y and x ↦ x^{3}, y ↦ y. □

The element of order two in H and the elements in the complement K have the property that they respect the inclusions Q_{2^{n}} ↪ Q_{2^{n+1}}. In the limit, we obtain an action of ℤ/2 × ℤ̂_{2}^{×} = ℤ/2 × lim_{\overleftarrow{n}} Aut(ℤ/2^{n}) on lim_{\overrightarrow{n}} Q_{2^{n+1}}. The subgroup of index

two consisting of the elements (a, b) with b^{2} ≡ 8a + 1 (mod 16) is isomorphic to ℤ̂_{2}^{×}. These are the elements with the property that the restriction to Q_{16} extends to an automorphism of Õ_{48}. The elements with a = 0 act trivially on Õ_{48}, while those with a = 1 act by fixing the elements in the subgroup of index two, and multiplying the remaining elements by the central element of order two. This way, we obtain an action of ℤ̂_{2}^{×} on the diagram Õ_{48} ← Q_{16} → lim_{\overrightarrow{n}} Q_{2^{n+1}}. The elements ±1 induce inner automorphisms on all the groups involved, so we obtain a map ℤ̂_{2}^{×}/{±1} → π_{0}Aut(B̂Q), which is the isomorphism described in Lemma 5.4 of [7].

Now α ≡ 5 (mod 8), and so α^{2} ≡ 9 (mod 16). So the corresponding automorphism of Q_{16} sends x to x^{5} and y to yx^{4}. The corresponding automorphism of Õ_{48} fixes the elements in the subgroup of index two, and multiplies the remaining elements by the central element of order two. It follows that we may write down a corresponding injective homomorphism σ_{0} : (Õ_{48})_{red}^{3} → Spin(3) ∘ Spin(4) such that the following triangle commutes:

[figure: B̂(Õ_{48})_{red}^{3}; B̂σ_{0}; B̂Q_{red}^{3}; ρ_{0}; B̂(Spin(3) ∘ Spin(4)).]

The map σ_{0} extends to a map σ : (Õ_{48})_{red}^{3}⋊ℤ/2 → Spin(3, 4), where ℤ/2 is acting by permuting the second and third copies of Õ_{48}. The following diagram commutes:

[figure: B̂((Õ_{48})_{red}^{3}⋊ℤ/2); B̂σ; B̂(Q_{red}^{3}⋊ℤ/2); ρ; B̂(Spin(3, 4)).]

6. The map C → Spin(7)

There is a unique representation (up to conjugation) S_{6}(2) → SO(7), which lifts to a map C ≅ 2S_{6}(2) → Spin(7). Restrict this map to C ∩ X to obtain a map from (C ∩ X)/Z to SO(7). Now (C ∩ X)/Z is a split extension of shape 2^{6} : GL(3, 𝔽_{2}), and so the seven dimensional real orthogonal space decomposes as a direct sum of seven eigenspaces of O_{2}((C ∩ X)/Z), permuted by GL(3, 𝔽_{2}).

Now all the non-central involutions in Spin(7) are conjugate, and their image in SO(7) have trace −1. It follows that any element of order two in C maps to an element of trace −1 in SO(7). So without loss of generality, we may let O_{2}((C ∩ X)/Z) act in the same way as it does on the seven dimensional space spanned by all except the top left basis vector in the middle block of the MOG diagram. Thus a_{1} maps to the identity element, and

a_{2} ↦ \begin{matrix}∗ & − \\ + & − \\ + & − \\ + & − \end{matrix}  a_{3} ↦ \begin{matrix}∗ & + \\ + & + \\ − & − \\ − & − \end{matrix}  a_{4} ↦ \begin{matrix}∗ & + \\ − & − \\ + & + \\ − & − \end{matrix}

c_{1} ↦ \begin{matrix}∗ & − \\ − & + \\ + & + \\ − & − \end{matrix}  c_{2} ↦ \begin{matrix}∗ & − \\ − & − \\ − & + \\ + & + \end{matrix}  c_{3} ↦ \begin{matrix}∗ & − \\ − & + \\ − & + \\ − & + \end{matrix}

Now the above trace condition implies that the complement H is not conjugate to a complement acting by permutations (in fact the obvious permutations form a supplementary copy of SL(2, 𝔽_{7}) of the type discussed at the end of Section 2), and so we must use signed permutations as follows:

b_{1} ↦ \begin{matrix}∗ & − \\ + & − \\ − & + \\ − & + \end{matrix} . \begin{matrix}∗ & · \\ · & · \\ · & · \\ · & · \end{matrix} b_{2} ↦ \begin{matrix}∗ & − \\ − & + \\ + & − \\ − & + \end{matrix} . \begin{matrix}∗ & · \\ · & · \\ · & · \\ · & · \end{matrix}

d ↦ \begin{matrix}∗ & · \\ · & · \\ · & · \\ · & · \end{matrix}  e ↦ \begin{matrix}∗ & − \\ + & − \\ + & − \\ + & − \end{matrix} . \begin{matrix}∗ & · \\ · & · \\ · & · \\ · & · \end{matrix}  f ↦ \begin{matrix}∗ & + \\ − & − \\ + & + \\ − & − \end{matrix} . \begin{matrix}∗ & · \\ · & · \\ · & · \\ · & · \end{matrix}

The element h conjugates a_{2} to a_{1}a_{2}, and so it must fix the eigenspace decomposition of a_{2}; namely the column decomposition into a 3-space and a 4-space. Since h centralizes b_{1} and d, it fixes the vector with a 1 in each coordinate. Also, h conjugates a_{3} to a_{2}a_{4}b_{1}c_{1}, and so it has to act as ±1 in the first column, and as ±ε in the second column. To have trace −1, we therefore have

h ↦ \begin{matrix}∗ \\ − & −ε \\ − \\ − \end{matrix} .

We may see the isomorphism between S_{6}(2) × ℤ/2 and the Weyl group of type E_{7} using this representation, as follows. The ℤ/2 here acts as plus and minus the identity. The element −a_{2}c_{3} is a reflection, and so the 14 signed basis vectors are root vectors. The remaining 2^{4} ×7 root vectors have ±\frac{1}{2} in four coordinates forming the complement of a “line” with respect to the action of H ≅ GL(3, 𝔽_{2}) on these seven 1-spaces. This makes a total of 126 root vectors, as required.

7. The map N → Q_{red}^{3}⋊Σ_{3}

The image of C ∩ N in Spin(7) lies in Im(σ) ⊆ Spin(3, 4), and so by inverting σ we obtain a map C ∩ N → Q_{red}^{3}⋊ℤ/2 as follows:

a_{1} ↦ (1, −1, −1)  a_{2} ↦ (−1, 1, −1)  a_{3} ↦ (i, i, i)  a_{4} ↦ (j, j, j)
b_{1} ↦ (i, j, k) b_{2} ↦ (j, k, i)
c_{1} ↦ (j, k, −k)  c_{2} ↦ (k, i, −i)  c_{3} ↦ (· ↔)
d ↦ (−\frac{1}{2}(1 + i + j + k), −\frac{1}{2}(1 + i + j + k), −\frac{1}{2}(1 + i + j + k))
ec_{3} ↦ \frac{1}{\sqrt{2}}(k − j, k − j, k − j)
hc_{3} ↦ (1, −\frac{1}{2}(1 + i + j + k), −\frac{1}{2}(1 − i − j − k)).

We extend this to a map N → Q_{red}^{3}⋊Σ_{3} by sending g to the cyclic permutation sending the first coordinate to the second, the second to the third and the third to the first:

g ↦ ( ⋯).

It is easy, using the relations given in Section 2, to check that this is a well defined homomorphism.

8. The map BCo_{3} → BDI(4)

In §§3 and 6, we constructed maps C → Spin(7) and N → Q_{red}^{3}⋊Σ_{3} in such a way that the following diagram commutes up to homotopy:

[figure: B̂C; B̂Spin(7); B̂(C ∩ N); B̂(Q_{red}^{3}⋊Σ_{2}); ρ; ≃; B̂Spin(3, 4); B̂N; B̂(Q_{red}^{3}⋊Σ_{3})]

Now there is a unique isomorphism class of non-split extensions (ℤ/2)^{4}GL(4, 𝔽_{2}), and this is isomorphic to the group X. There is a map from B̂X to BDI(4) obtained by regarding B̂X as the 𝔽_{2}-completion of the homotopy colimit of the following functor. Restrict the functor used by Dwyer and Wilkerson (p. 40–42 of [7]) to the subcategory of the category A consisting of the object A_{4} and its automorphisms. [Note that in Spin(7), the normalizer of (ℤ/2)^{4} is a non-split extension of (ℤ/2)^{4} by (ℤ/2)^{3}GL(3, 𝔽_{2}). This is a subgroup of the fundamental group of the homotopy colimit, which is therefore a non-split extension of (ℤ/2)^{4} by GL(4, 𝔽_{2}).]

Since the image of C ∩X in Spin(7) is the normalizer of the image of V in Spin(7), the composite

B̂(C ∩ X) → B̂X → BDI(4)

is homotopic to the composite

B̂(C ∩ X) → B̂C → B̂Spin(7) → BDI(4).

Similarly, since the image of N ∩ X in Q_{red}^{3}⋊Σ_{3} is the normalizer of the image of V in Q_{red}^{3}⋊Σ_{3}, the composite

B̂(N ∩ X) → B̂X → BDI(4)

is homotopic to the composite

B̂(N ∩ X) → B̂N → B̂(Q_{red}^{3}⋊Σ_{3}) → BDI(4).

Thus we obtain maps from the spaces in the diagram at the end of Section 3 to BDI(4) which agree under composition, up to homotopy. There is an obstruction to lifting to a map from the homotopy colimit of this diagram to BDI(4). There is no obstruction to getting a map from the homotopy colimit of the full subdiagram consisting of all except the middle object B̂(C ∩ N ∩ X) to BDI(4). We are then left with the problem of completing a map

B̂(C ∩ N ∩ X) × Δ̇^{2} → BDI(4)

to a map

B̂(C ∩ N ∩ X) × Δ^{2} → BDI(4).

The obstruction to doing this lies in

π_{1}Map_{0}(B̂(C ∩ N ∩ X), BDI(4)),

where Map_{0} denotes the connected component consisting of the homotopy class of maps in question.

Lemma 8.1. The maps

BZ × B(C ∩ N ∩ X) → BDI(4)
BZ × B(C ∩ X) → BDI(4)

induce homotopy equivalences

BZ → Map_{0}(B̂(C ∩ N ∩ X), BDI(4)),
BZ → Map_{0}(B̂(C ∩ X), BDI(4)).

In particular, the obvious map

π_{1}Map_{0}(B̂(C ∩ X), BDI(4)) → π_{1}Map_{0}(B̂(C ∩ N ∩ X), BDI(4))

is an isomorphism, and both sides are isomorphic to ℤ/2.

Proof. We apply Proposition 8.4 of Dwyer and Wilkerson [8] to the homomorphism of loop spaces Spin(7) → DI(4) and the sequences

Z → P → P/Z,
Z → C ∩ N ∩ X → (C ∩ N ∩ X)/Z

and

Z → C ∩ X → (C ∩ X)/Z.

Since

Map_{0}(B̂Z, B̂Spin(7)) → Map_{0}(B̂Z, BDI(4))

is a homotopy equivalence (both sides are equivalent to B̂Spin(7)), it follows that the maps

Map_{0}(B̂P, B̂Spin(7)) → Map_{0}(B̂P, BDI(4)),
Map_{0}(B̂(C ∩ N ∩ X), B̂Spin(7)) → Map_{0}(B̂(C ∩ N ∩ X), BDI(4))

and

Map_{0}(B̂(C ∩ X), B̂Spin(7)) → Map_{0}(B̂(C ∩ X), BDI(4))

are all equivalences. The centralizer of P in Spin(7) is Z, and so by the main theorem of Dwyer–Zabrodsky [9], the composite map

BZ → Map_{0}(B̂(C ∩ X), B̂Spin(7)) → Map_{0}(B̂(C ∩ N ∩ X), B̂Spin(7))
→ Map_{0}(B̂P, B̂Spin(7))

is a homotopy equivalence.

Now

Map_{0}(B(C ∩ X), B̂Spin(7)) = Map_{0}(B̂(C ∩ X), B̂Spin(7))

is a connected component of the homotopy fixed point set of (C ∩ X)/O_{2}(C ∩ X) ≅ GL(3, 𝔽_{2}) acting on

Map_{0}(BO_{2}(C ∩ X), B̂Spin(7)).

The centralizer of O_{2}(C ∩ X) in Spin(7) is Z, so the latter mapping space is BZ, and hence the homotopy fixed point set is also BZ.

Similarly,

Map_{0}(B(C ∩ N ∩ X), B̂Spin(7)) = Map_{0}(B̂(C ∩ N ∩ X), B̂Spin(7))

is a connected component of the homotopy fixed point set of (C ∩ N ∩ X)/O_{2}(C ∩ N ∩ X) ≅ Σ_{3} acting on

Map_{0}(BO_{2}(C ∩ N ∩ X), B̂Spin(7)).

The centralizer of O_{2}(C ∩ N ∩ X) in Spin(7) is Z, so the latter mapping space is BZ, and hence the homotopy fixed point set is also BZ.

It follows that the maps

BZ → Map_{0}(B̂(C ∩ X), B̂Spin(7)) → Map_{0}(B̂(C ∩ N ∩ X), B̂Spin(7))

→ Map_{0}(B̂P, B̂Spin(7))

are all homotopy equivalences. □

It follows from the lemma that if necessary, we may adjust the map

B̂(C ∩ X) × Δ^{1} → BDI(4)

so that we can complete to a map from the homotopy colimit to BDI(4). This gives us a map from B̂Co_{3} to BDI(4), and hence a map BCo_{3} → BDI(4).

To check that this map has the desired properties to complete the proof of Theorem 1.1, we note that the composite BP → BCo_{3} → BDI(4) factors through BSpin(7) → BDI(4) (recall that P is a Sylow 2-subgroup of Co_{3}). The mod two cohomology of BP is a finitely generated module over the mod two cohomology of BSpin(7), which in turn is a finitely generated module over the mod two cohomology of BDI(4). Since the mod two cohomology of BP contains the mod two cohomology of BCo_{3}, this proves that the mod two cohomology of BCo_{3} is also finitely generated as a module over the mod two cohomology of BDI(4). Acknowledgements. I would like to thank Jon Carlson, Bill Dwyer, Richard Lyons, Ron Solomon and Clarence Wilkerson for conversations without which this work could not have been carried out. I would also like to thank the referee for doing a thorough job.

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Department of Mathematics, University of Georgia, Athens GA 30602, USA

  E-mail address: djb@byrd.math.uga.edu