Finite Simple Groups and Dickson Invariants

D. J. Benson Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain djb@vax.ox.ac.uk

C. W. Wilkerson Department of Mathematics Purdue University W. Lafayette, IN 47907 U.S.A. wilker@math.purdue.edu

1 Introduction

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M_{12}. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely generated free module. Since the classifying space BG_{2} of the compact Lie group G_{2} has the rank three Dickson invariants as its cohomology, one is led to ask whether there is an interesting relationship between M_{12} and G_{2}.

There is clearly no group homomorphism from M_{12} to G_{2} which induces the inclusion of the Dickson invariants into the cohomology of BM_{12}. Indeed, there is no non-trivial group homomorphism from M_{12} to G_{2} at all, since the smallest nontrivial complex representation of M_{12} has dimension 11, while G_{2} has a 7 dimensional complex representation.

Nevertheless, as we shall see in Section 5, there is a map of classifying spaces with the desired properties.

Theorem 1.1 There is a map of classifying spaces ρ : BM_{12} → BG_{2} inducing an injective map ρ^{∗} : H^{∗}(BG_{2}, _{2}) → H^{∗}(BM_{12}, _{2}) in mod two cohomology. The mod two cohomology of BM_{12} is a finitely generated free module over the image of this map.

The idea for constructing this map is to use an observation made in [1]. Namely, for suitably chosen subgroups W and W′ containing a Sylow 2-subgroup P of M_{12}, they show that the obvious map from the amalgamated free product W ∗_{P} W′ to M_{12} is a mod two cohomology isomorphism. We shall give a short proof of this in Section 3 using ordinary character theory. We then construct a map of groups from W ∗_{P} W′ to G_{2} which induces the desired map in mod two cohomology. Finally, a theorem of Dwyer and Wilkerson enables us to deduce the existence of the desired map from BM_{12} to BG_{2}.

The fiber of the map constructed this way has as its mod two cohomology a 14 dimensional Poincaré duality algebra. This suggests that there may be a free action of M_{12} on the 14 dimensional manifold underlying G_{2}, so that this fiber is equivalent at the prime two to the quotient manifold. In fact, this is not the case, as we shall see in Section 7 by examining the behavior at the prime 5 or at the prime 11. We have not attempted an analysis at the prime three, but we remark that the mod three cohomology of M_{12} is Cohen–Macaulay.

In Section 8 we make some comments about finite simple groups and Dickson invariants of other ranks. The most interesting case is that of the rank four Dickson invariants. Dwyer and Wilkerson [6] recently constructed a finite loop space at the prime two whose classifying space BDI(4) has the Dickson invariants of rank four as its mod two cohomology. We speculate that there may exist a map from the classifying space BCo_{3} of Conway’s sporadic group Co_{3} to BDI(4) with properties analogous to those of the map from BM_{12} to BG_{2}.

2 Dickson invariants as parameters

We begin by pointing out that for any compact Lie group G and prime p, there is always a copy of the Dickson invariants (with a suitable Frobenius twist) sitting inside H^{∗}(BG, _{p}) as a subalgabra over the Steenrod algebra, in such a way that H^{∗}(BG, _{p}) is finitely generated as a module over this subalgebra. Since there does not seem to be an appropriate reference in the literature, we spend this section providing a proof of this (presumably well known) fact.

Let E be a finite elementary abelian p-subgroup of G of order p^{n}. If p = 2, then H^{∗}(BE, _{p}) is a polynomial algebra on n degree one generators, while if p is odd, it is a tensor product of an exterior algebra on n degree one generators and a polynomial algebra on their Bocksteins in degree two. The action by conjugation identifies N_{G}(E)/C_{G}(E) with a subgroup of GL_{n}( _{p}). In particular, the Dickson invariants on the polynomial part of H^{∗}(BE, _{p})

p[x_{1}, …, x_{n}]^{GL_{n}( _{p})} = _{p}[c_{n,n−1}, …, c_{n,0}]

(with deg(c_{n,i}) = 2^{n} − 2^{i} if p = 2 and deg(c_{n,i}) = 2(p^{n} − p^{i}) if p is odd) are in the invariants of the normalizer H^{∗}(BE, _{p})^{N_{G}(E)}.

If E′ is a subgroup of index p in a finite elementary abelian p-subgroup E, then the restriction of c_{n,0} to E′ is zero, while for i > 0 the restriction of c_{n,i} to E′ is c_{n−1,i−1}^{p}.

By a theorem of Quillen [10], the map given by the restrictions

q_{G} : H^{∗}(BG, _{p}) → lim_{←} H^{∗}(BE, _{p})

is an F -isomorphism, where the limit is taken over the category whose objects are the finite elementary abelian p-subgroups E of G and the morphisms are generated by the conjugations and inclusions. To say that a map is an F -isomorphism means that the kernel consists of nilpotent elements, and given any element x in the codomain, some p-power x^{p^{a}} is in the image.

Denote by r = r_{p}(G) the p-rank of G; namely the largest rank of a finite elementary abelian subgroup of G. Then for each finite elementary abelian p-subgroup E of G, of order p^{n}, we may consider the sequence of elements

c_{n,n−1}^{p^{r−n}}, …, c_{n,n−r}^{p^{r−n}}

with deg(c_{n,n−i}^{p^{r−n}}) = p^{r} − p^{r−i}. Here, c_{n,n−i}^{p^{r−n}} is interpreted to be the zero element of this degree if i > n. This sequence of elements is consistent with respect to restrictions and conjugations, and therefore gives a well defined sequence of elements in lim_{←} H^{∗}(BE, _{p}). The above theorem of Quillen implies that for a suitably chosen value of a ≥ 0 there are uniquely defined elements

d̄_{r−1}, …, d̄_{0} ∈ H^{∗}(BG, _{p})/Ker(q_{G})

with deg(d̄_{r−i}) = 2^{r+a} − 2^{r+a−i} if p = 2 and deg(d̄_{r−i}) = 2(p^{r+a} − p^{r+a−i}) if p is odd, and with the property that the image of d̄_{r−i} in H^{∗}(BE, _{p}) is c_{n,n−i}^{p^{r+a−n}}. Since restriction commutes with the Steenrod operations, these d̄_{r−i} generate a copy of the Dickson algebra (expanded by multiplying the degrees by p^{a}) in H^{∗}(BG, _{p})/Ker(q_{G}) with the expected action of the Steenrod actions. Furthermore, the Dickson algebra is 𝔑𝔦𝔩-closed, so we may lift to elements

d_{r−1}, …, d_{0} ∈ H^{∗}(BG, _{p})

of the same degrees, generating a copy of the Dickson algebra with the appropriate action of the Steenrod operations in H^{∗}(BG, _{p}).

Finally, since each H^{∗}(BE, _{p}) is finitely generated over the subalgebra generated by the restrictions of d_{r−1}, …, d_{0}, and the kernel of q_{G} is a finitely generated nilpotent ideal in H^{∗}(BG, _{p}), it follows that H^{∗}(BG, _{p}) is a finitely generated module over the subalgebra generated by d_{r−1}, …, d_{0}.

To summarize, we have proved the following:

Theorem 2.1 If G is a compact Lie group of p-rank r, then for some a ≥ 0 there are elements

d_{r−1}, …, d_{0} ∈ H^{∗}(BG, _{p})

with the following properties:

(i) deg(d_{r−i}) = 2^{r+a} − 2^{r+a−i} if p = 2, while deg(d_{r−i}) = 2(p^{r−a} − p^{r+a−i}) if p is odd.

(ii) The restriction of d_{r−i} to any finite elementary abelian p-subgroup E of order p^{n} is equal to c_{n,n−i}^{p^{r+a−n}} for i ≤ n, and zero for i > n.

(iii) The subalgebra of H^{∗}(BG, _{p}) generated by d_{r−1}, …, d_{0} is a polynomial algebra on which the Steenrod operations act in the same way as on the p^{a}th power of the Dickson algebra _{p}[c_{r,r−1}^{p^{a}}, …, c_{r,0}^{p^{a}}].

(iv) H^{∗}(BG, _{p}) is finitely generated as a module over the subalgebra generated by d_{r−1}, …, d_{0}.

3 The map from W ∗_{P} W′ to M_{12}

We begin this section with a general discussion of amalgamated free products of subgroups. Let G be a finite group, and let H and H′ be subgroups generating G and containing a subgroup K in their intersection. Then there is a map from the amalgamated free product H ∗_{K} H′ to G given by sending elements of H and H′ to the corresponding elements of G. Since H and H′ generate G, this map is surjective. The Kurosh subgroup theorem (see for example [9] Corollaries 4.9.1 and 4.9.2) implies that the kernel F is free.

Let Γ be the graph whose vertex set is the disjoint union of two sets corresponding to the cosets of H and H′ in G, and whose edges correspond to the cosets of K in G. A vertex is incident to an edge if the corresponding coset of H or H′ contains the corresponding coset of K. The universal cover Γ̂ is formed by performing the same construction with the cosets of H, H′ and K in the amalgamated free product H ∗_{K} H′. Thus the kernel F is identified with π_{1}(Γ).

Let R be a commutative ring of coefficients. Then as an RG-module, H^{1}(F, R) is identified with H^{1}(Γ, R). The augmented simplicial cochains on Γ with coefficients in R give us an exact sequence of RG-modules

0 → R → R[G/H] ⊕ R[G/H′] → R[G/K] → H^{1}(F, R) → 0. (∗)

Lemma 3.1 Suppose that R is a field of characteristic p, and that K contains a Sylow p-subgroup of G. Then the following are equivalent:

(i) H^{∗}(G, H^{1}(F, R)) = 0.

(ii) The restriction maps (with suitable signs) give a short exact sequence

0 → H^{∗}(G, R) → H^{∗}(H, R) ⊕ H^{∗}(H′, R) → H^{∗}(K, R) → 0.

(iii) The map H ∗_{K} H′ → G is a mod p cohomology isomorphism.

(iv) The induced map on Bousfield–Kan p-completions

B(H ∗_{K} H′)_{p}^{∧} → BG_{p}^{∧}

is a homotopy equivalence.

Proof (i) ⇔ (ii): Write the exact sequence of RG-modules (∗) as a splice of short exact sequences

0 → R → R[G/H] ⊕ R[G/H′] → X → 0 (†)

and

0 → X → R[G/K] → H^{1}(F, R) → 0. (‡)

Since K contains a Sylow p-subgroup of G, the sequence (†) splits. Taking cohomology and using the Eckmann–Shapiro lemma, we obtain a short exact sequence

0 → H^{∗}(G, R) → H^{∗}(H, R) ⊕ H^{∗}(H′, R) → H^{∗}(G, X) → 0.

On the other hand, the short exact sequence (‡) and the Eckmann–Shapiro lemma show that the induced map

H^{∗}(G, X) → H^{∗}(K, R)

is an isomorphism if and only if H^{∗}(G, H^{1}(F, R)) = 0. Finally, one checks that the composites of the maps in cohomology induced by the given maps of permutation modules with the Eckmann–Shapiro isomorphism are equal to the corresponding (signed) restriction maps.

(ii) ⇔ (iii): This follows by comparing this short exact sequence with the Mayer– Vietoris long exact sequence for computing the cohomology of an amalgamated free product.

(iii) ⇔ (iv): This follows from Bousfield and Kan [3]. □

We now turn to the case where G = M_{12}. The character table, maximal subgroups, and the characters of the corresponding primitive permutation modules for M_{12} may be found in the ATLAS of finite groups [4]. Let P be a Sylow 2-subgroup of G, so that |P | = 64, the center of P has order two. Let W be the centralizer in G of the involution t in the center of P. Then W is a semidirect product Q_{8} Σ_{4} of the quaternion group of order 8 by its full automorphism group, and |W | = 192. Let W′ be the normalizer in G of a normal subgroup of P generated by t and a G-conjugate of t lying in P. Then W′ is a semidirect product ( /4)^{2} ( /2 × Σ_{3}) and |W′| = 192, but W′ is not conjugate to W in G. We have W ∩ W′ = P, and so we obtain a map from W ∗_{P} W′ to G as above. Write F for the kernel of this map, a free group on

1 − |G : W | − |G : W′| + |G : P | = 496

generators.

Lemma 3.2 The irreducible representations χ_{16}, χ′_{16}, χ_{144} and χ_{176} of M_{12} over _{2}^{∧}, of dimensions 16, 16, 144 and 176 respectively, constitute a non-principal 2-block B.

Proof This is easy to check, by computing the corresponding central characters. □

Lemma 3.3 The _{2}^{∧}M_{12}-module H^{1}(F, _{2}^{∧}) is in the block B, and has the same character values on 2-regular elements as χ_{144} + 2χ_{176}.

Proof The restriction of χ_{144} + χ_{176} to P has zero character value off the identity element, and so it is equal to 5 copies of the regular representation. By Frobenius reciprocity, it follows that the permutation module on the cosets of P has inner product 5 with χ_{144} + χ_{176}. Each of _{2}^{∧}(G/W) and _{2}^{∧}(G/W′) has one copy of χ_{144} and no copies of χ_{176}. It follows that H^{1}(F, _{2}^{∧}) must have inner product 3 with χ_{144} + χ_{176}. A similar calculation shows that the inner product of H^{1}(F, _{2}^{∧}) with χ_{16} + χ_{176} and with χ′_{16} + χ_{176} are equal to 2. Since χ_{176} has the same character values on 2-regular elements as χ_{144} + χ_{16} + χ′_{16}, the lemma follows. □

Theorem 3.4 The map B(W ∗_{P} W′)_{2}^{∧} → B(M_{12})_{2}^{∧} is a homotopy equivalence.

Proof It follows from Lemmas 3.2 and 3.3 that H^{1}(F, _{2}) lies in the block B, and so H^{∗}(M_{12}, H^{1}(F, _{2})) = 0. The theorem now follows from Lemma 3.1. □

4 The map from W ∗_{P} W′ to G_{2}

The compact Lie group G_{2} has subgroups SO(4) and (S^{1})^{2} ( /2×Σ_{3}) (the normalizer of a maximal torus in G_{2}) intersecting in (S^{1})^{2} ( /2)^{2} (the normalizer of a maximal torus in SO(4)). We construct maps W → SO(4) and W′ → (S^{1})^{2} ( /2×Σ_{3}) which restrict to the same map P → (S^{1})^{2} ( /2)^{2}.

Following [1], we use the permutations given in Hall’s book [7] for M_{12} acting on 12 points. It is generated by the permutations

u_{1} = (1, 2, 3)(4, 5, 6)(7, 8, 9)(10)(11)(12)
u_{2} = (1, 4, 7)(2, 5, 8)(3, 6, 9)(10)(11)(12)
a = (1)(2, 4, 3, 7)(5, 6, 9, 8)(10)(11)(12)
b = (1)(2, 5, 3, 9)(4, 8, 7, 6)(10)(11)(12)
x = (1, 10)(4, 5)(6, 8)(7, 9)(2)(3)(11)(12)
y = (1, 11)(4, 6)(5, 9)(7, 8)(2)(3)(10)(12)
z = (1, 12)(4, 7)(5, 6)(8, 9)(2)(3)(10)(11).

We also set

d = xyz = (1, 12, 11, 10)(4, 6, 7, 8)(2)(3)(5)(9)
T = (1, 4, 2)(3, 11, 7)(5, 10, 6)(8, 9, 12)
g = ad^{2} = (1, 11)(2, 7)(3, 4)(5, 8)(6, 9)(10, 12)
h = bxzyz = (1, 10)(2, 5)(3, 9)(4, 6)(7, 8)(11, 12)

Then we take

W = ⟨a, b⟩ ⟨x, y, z⟩ ≅ Q_{8} Σ_{4}.

We think of a, b and ab as i, j and k in Q_{8}. Conjugation by x is the automorphism i ↔ j, k ↔ −k; y is i ↔ k, j ↔ −j and z is j ↔ k, i ↔ −i. The map W → SO(4) is obtained by acting on the quaternion algebra ≅ ^{4}. The group Q_{8} acts by left multiplication, while Σ_{4} acts as algebra automorphisms in the obvious way. In terms of matrices we have

\begin{matrix}a ↦ ( \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) & b ↦ ( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) & ab ↦ ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) \\ x ↦ ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ) & y ↦ ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) & z ↦ ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) \\ d ↦ ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) & g ↦ ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) & h ↦ ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) \end{matrix}

We take

W′ = (⟨b⟩ × ⟨d⟩) (⟨h⟩ × ⟨T, g⟩) ≅ ( /4)^{2} ( /2 × Σ_{3}).

We describe the normalizer of a maximal torus in the compact Lie group G_{2} as a split extension with normal subgroup (S^{1})^{2} consisting of pairs (e^{2πiα}, e^{2πiβ}) with α and β in / . The complement is

⟨h̃⟩ × ⟨T̃, g̃⟩ ≅ /2 × Σ_{3}

acting by conjugation as follows:

T̃(e^{2πiα}, e^{2πiβ})T̃^{−1} = (e^{−2πi(α+β)}, e^{2πiα})
g̃(e^{2πiα}, e^{2πiβ})g̃^{−1} = (e^{2πiβ}, e^{2πiα})
h̃(e^{2πiα}, e^{2πiβ})h̃^{−1} = (e^{−2πiα}, e^{−2πiβ}).

The map W′ → (S^{1})^{2} ( /2×Σ_{3}) is given by b ↦ (e^{2πi/4}, e^{−2πi/4}), d ↦ (1, e^{2πi/4}), T ↦ T̃, g ↦ g̃ and h ↦ h̃.

The intersection of SO(4) with (S^{1})^{2} ( /2 × Σ_{3}) in G_{2} is described as follows:

(e^{2πiα}, e^{2πiβ}) ↔ ( \begin{matrix}cos α & 0 & − sin α & 0 \\ 0 & cos β & 0 & − sin β \\ sin α & 0 & cos α & 0 \\ 0 & sin β & 0 & cos β \end{matrix} )

g̃ ↔ ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} )  h̃ ↔ ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) .

It is easy to check that the maps we have described from W to SO(4) and from W′ to (S^{1})^{2} ( /2 × Σ_{3}) restrict to the same map from P to their intersection in G_{2}. We have therefore constructed a well defined map from W ∗_{P} W′ to G_{2}.

5 The map BM_{12} → BG_{2}

In this section, we complete the proof of Theorem 1.1. The following is proved in Dwyer and Wilkerson [5]:

Theorem 5.1 Suppose that π is a finite group and G is a compact Lie group. Then Map(Bπ, BG) = ∏_{p} Map(Bπ_{p}^{∧}, BG_{p}^{∧}).

It follows from Theorem 3.4 that the map W ∗_{P} W′ → G_{2} constructed in Section 4 induces a map (BM_{12})_{2}^{∧} → (BG_{2})_{2}^{∧}, and therefore by the above theorem it induces a map ρ : BM_{12} → BG_{2}. The composites of this map with the maps BW → BM_{12} and BW′ → BM_{12} induced by the inclusions induce the same maps on mod two cohomology as the composites W → SO(4) → G_{2} and W′ → (S^{1})^{2} ( /2 × Σ_{3}) → G_{2} constructed in Section 4. In particular, since every elementary abelian 2-subgroup of M_{12} is conjugate to a subgroup of P, the composite of H^{∗}(BG_{2}, _{2}) → H^{∗}(BM_{12}, _{2}) with the restriction to any elementary abelian 2-subgroup of order 8 of M_{12} induces the injection of the Dickson invariants into the cohomology. Applying a theorem of Quillen [10] now completes the proof of Theorem 1.1.

6 H^{∗}(BM_{12}, _{2}) is Cohen–Macaulay

As mentioned in the introduction, one of the interesting features of the calculation of H^{∗}(BM_{12}, _{2}) given in Adem, Maginnis and Milgram [1] is that the answer is a Cohen–Macaulay ring. It turns out that something even stronger is true, namely the cohomology of a Sylow 2-subgroup P of M_{12} is Cohen–Macaulay. In this section, we give a short proof of this fact.

We begin by recalling that the Sylow 2-subgroup of M_{12} is of the form

(⟨b⟩ × ⟨d⟩) (⟨h⟩ × ⟨g⟩).

We proceed by examining the spectral sequence of this split extension.

The subgroup generated by b, d and g is a wreath product of /4 by /2, as can be seen more easily by writing it as

(⟨bd⟩ × ⟨d⟩) ⟨g⟩.

Note also that the conjugation action of h just inverts every element of the normal subgroup.

Write the cohomology of ⟨bd⟩ × ⟨d⟩ as Λ(ε, ε′) ⊗ _{2}[η, η′] with ε and ε′ in degree one, and η and η′ their second Bocksteins in degree two. Write _{2}[ϕ, θ] for the cohomology of ⟨h⟩×⟨g⟩. Note that g acts on the cohomology of the normal subgroup by swapping ε with ε′ and swapping η with η′. The action of h is trivial.

The E_{2} page of the spectral sequence of the split extension is generated by the elements of the cohomology of the quotient along the base and the invariant elements of the cohomology of the normal subgroup up the fibre. These invariants are generated by ε + ε′, εε′, η + η′, ηη′ and εη + ε′η′. The relations in the E_{2} page are given by (ε + ε′)ϕ = 0, (η + η′)ϕ = 0 and (εη + ε′η′)ϕ = 0.

Since the extension splits, the elements along the base survive to E_{∞}. This implies that ε + ε′ ∈ E_{2}^{0,1} survives to E_{∞}. Since E_{2}^{2,1} = 0, the elements η + η′ and εε′ ∈ E_{2}^{0,2} survive to E_{∞}. Since there are no non-zero elements in E_{2}^{2,2} annihilated by ϕ, we have d_{2}(εη + ε′η′) = 0. Since also E_{2}^{3,1} = 0 this implies that εη + ε′η′ survives to E_{∞}. Finally, since ηη′ is represented by an Evens norm from the subgroup of index two generated by b, d and h, it also survives to E_{∞} (Note that in the spectral sequence for the split extension of ⟨b⟩ × ⟨d⟩ by ⟨h⟩ it is easy to see that E_{2} = E_{∞}, so that η may be regarded as an element of the cohomology of this subgroup; then its Evens norm to P is ηη′). This proves that E_{2} = E_{∞}.

Finally, to see that the mod two cohomology of P is Cohen–Macaulay, we observe that a regular homogeneous sequence of parameters may be chosen as follows. For two of the parameters, we may choose θ and any representative of ηη′. For the remaining parameter, we must choose carefully a representative ξ of η + η′. The relation (η + η′)ϕ = 0 ungrades to a relation which says that ξϕ is some linear combination of ϕ^{3}, ϕ^{2}θ, ϕθ^{2} and θ^{3}. By adjusting the representative ξ by elements of the base, the first three of these terms may be eliminated so that the relation says either ξϕ = 0 or ξϕ = θ^{3}. We then take ξ + ϕ^{2} for our third parameter. It is easy to check that these form a regular sequence.

7 The primes 5 and 11

One may speculate that there is a map from BM_{12} to BG_{2} with properties at all primes analogous to those of the map constructed above at the prime two. To see that this is not the case, one may calculate either at the prime 5 or at the prime 11.

Proposition 7.1 If M_{12} acts freely on a CW complex X with the homotopy type of the compact Lie group G_{2}, then the quotient X/M_{12} has cohomology in infinitely many degrees at the primes 5 and 11. In particular, X is not a finite complex.

Proof For any odd prime p, we have

H^{∗}(G_{2}, _{p}) = Λ(s_{3}, s_{11}),

an exterior algebra in generators of degrees 3 and 11.

The Sylow 5-subgroup of M_{12} is cyclic of order 5 with normalizer

/2 × ( /5 /4).

So H^{∗}(BM_{12}, _{5}) has a polynomial generator in degree 8 and an exterior generator in degree 7.

The Sylow 11-subgroup of M_{12} is cyclic of order 11 with normalizer /11 /5. So H^{∗}(BM_{12}, _{11}) has a polynomial generator in degree 10 and an exterior generator in degree 9.

If X is a CW complex with the homotopy type of G_{2} admitting a free action of M_{12}, then we examine the equivariant cohomology spectral sequence

H^{∗}(BM_{12}, H^{∗}(X, _{p})) ⇒ H^{∗}(X/M_{12}, _{p}).

The left hand side is a tensor product

H^{∗}(BM_{12}, _{p}) ⊗ H^{∗}(G_{2}, _{p}).

For p = 5, there are no elements off the base in degrees congruent to −1 modulo 8, and so no power of the polynomial generator on the base can get hit in the spectral sequence. Therefore the quotient has cohomology in infinitely many degrees. Similarly, for p = 11, there are no elements off the base in degrees congruent to −1 modulo 10, and we reach the same conclusion. □

Corollary 7.2 Let p = 5 or p = 11. Then there is no map from BM_{12} to BG_{2} with the property that H^{∗}(BM_{12}, /p) is finitely generated over the image of the corresponding map H^{∗}(BG_{2}, /p) → H^{∗}(BM_{12}, /p).

Proof If there were such a map, then its fibre would have finite cohomology, fundamental group M_{12}, and universal cover of the homotopy type of G_{2}. Therefore by the proposition there is no such map. □

8 Other sporadic groups

In this section, we discuss the possibility of playing the same game with the Dickson invariants of other ranks at the prime two.

The Dickson invariants of rank two form the cohomology of BSO(3). In this case, there is actually a group homomorphism A_{5} → SO(3) which induces a map BA_{5} → BSO(3) with the property that the corresponding map

H^{∗}(BSO(3), _{2}) → H^{∗}(BA_{5}, _{2})

is injective. The cohomology of BA_{5} is a finitely generated free module over the image of this map, with generators in degrees zero and 3. The fiber of the map BA_{5} → BSO(3) is “icosahedral space” SO(3)/A_{5}, a compact manifold with the same homology as the three-sphere but with non-trivial fundamental group.

It has recently been shown by Dwyer and Wilkerson [6] that there is an _{2}- complete space BDI(4) such that H^{∗}(BDI(4), _{2}) is isomorphic as an algebra over the Steenrod algebra to the rank four Dickson invariants. The space DI(4) = ΩBDI(4) is an exotic 2-adic finite loop space. The space BDI(4) was built as a homotopy colimit of a certain diagram of classifying spaces of compact Lie groups. This diagram was obtained by examining what the centralizers and normalizers of elementary abelian 2-subgroups of DI(4) would have to be. For each n with 1 ≤ n ≤ 4, there is a unique conjugacy class of elementary abelian 2-subgroup of rank n in DI(4), and their normalizers are

Spin(7), (SU(2)^{3}/{±1}).GL(2, _{2}), (S^{1})^{3}.( /2 × GL(3, _{2})), ( /2)^{4}.GL(4, _{2}).

In order to obtain a map from the classifying space Bπ of a finite group π to this, one way of proceeding is to express Bπ as a homotopy colimit in a similar way and map the pieces in compatibly. An interesting candidate for such a group is Conway’s sporadic simple group Co_{3}, which is the stabilizer of the origin and a suitable nonzero vector in the Leech lattice. Among the maximal 2-local subgroups of Co_{3} are subgroups of the form 2S_{6}(2), 2^{2+6}3^{1+2}2^{2} and ( /2)^{4}.GL(4, _{2}). The idea would be to use maps 2S_{6}(2) → Spin(7) and 2^{2+6}3^{1+2}2^{2} → (SU(2)^{3}/{±1}).GL(2, _{2}), and an isomorphism between the groups ( /2)^{4}.GL(4, _{2}). In the latter case, in both DI(4) and Co_{3} the extension is non-split, and since there is only one isomorphism class of non-split extension, these groups are isomorphic. It is also interesting to note that the centralizer of a suitable involution in Co_{3} is of the form /2 × M_{12}, so that one might try to construct a commutative square

B /2 × BM_{12} → B /2 × BG_{2}
↓  ↓
BCo_{3}  →  BDI(4)

It has been shown by Lin and Williams [8] that there is no space of type ΩBDI(5), which in particular implies that there is no space whose cohomology is the rank 5 Dickson invariants. On the other hand, Thompson’s simple group T h contains as a maximal subgroup a non-split extension ( /2)^{5}.GL(5, _{2}). It is quite plausible that the rank 5 Dickson algebra is sitting inside H^{∗}(T h, _{2}) as a subalgebra over which it is finitely generated as a module, but this could obviously not be realized topologically.

There is no non-split extension ( /2)^{6}.GL(6, _{2}), so the game stops there. It would also be interesting to investigate the situation at other primes.

References

[1] A. Adem, J. Maginnis and R. J. Milgram. The geometry and cohomology of the Mathieu group M_{12}. J. Algebra 139 (1991), 90–133.

[2] D. J. Benson and J. F. Carlson. Projective resolutions and Poincaré duality complexes. To appear in Trans. A.M.S.

[3] Bousfield and Kan. Homotopy limits, completions and localizations. Springer Lecture Notes in Mathematics 304, Springer-Verlag, Berlin/New York, 1972.

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. Atlas of Finite Groups. O.U.P., 1985.

[5] W. G. Dwyer and C. W. Wilkerson. Belgium conference, Springer Lecture Notes in Mathematics 1986

[6] W. G. Dwyer and C. W. Wilkerson. A new finite loop space at the prime two. Journal of the A.M.S. 6, 37–64, 1993.

[7] M. Hall. Theory of Groups.

[8] J. P. Lin and F. Williams. On 14-connected finite H-spaces. Israel J. Math. 66 (1989), 274–288.

[9] W. Magnus, A. Karrass and D. Solitar. Combinatorial group theory. Wiley, 1966.

[10] D. G. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–572, 573–602.

[11] C. W. Wilkerson. A primer on the Dickson invariants. Proc. of the Northwestern Homotopy Theory Conference, Contemp. Math. 19, 421–434, A.M.S., 1983.