ON THE CLASSIFYING SPACE AND COHOMOLOGY OF THOMPSON’S SPORADIC SIMPLE GROUP

D. J. BENSON

Abstract. This paper examines the classifying space of Thompson’s sporadic simple group. Two theorems are proved. The first gives a homotopy colimit decomposition for the 2-completion of the classifying space. The second shows that the mod two cohomology contains a copy of the Dickson invariants of rank five, over which it is finitely generated as a module.

To John Thompson on the occasion of his 70th birthday

1. Introduction

In this paper, we consider the classifying space and the cohomology of John Thompson’s sporadic simple group T h at the prime two. We prove the following theorem.

Theorem 1.1. Let C and N be maximal 2-local subgroups of T h of shape 2^{1+8}A_{9} and 2^{5}L_{5}(2) respectively, intersecting in a group K of shape 2^{1+8}A_{8} = 2^{5}2^{4}L_{4}(2). Then the obvious map of groups from the amalgamated free product C ∗_{K} N to T h is a mod two cohomology isomorphism. The corresponding map on classifying spaces from the homotopy colimit (double mapping cylinder) of the diagram

\xymatrix{BK \ar[r] \ar[d] & BC \\ BN}

to BT h is a homotopy equivalence after 2-completion.

The proof of this theorem is straightforward, and really just involves combining the appropriate results from the literature. One consequence of this theorem is that the Mayer–Vietoris sequence of the amalgamated free product reduces to a short exact sequence

0 → H^{∗}(T h, 𝔽_{2}) → H^{∗}(C, 𝔽_{2}) ⊕ H^{∗}(N, 𝔽_{2}) → H^{∗}(K, 𝔽_{2}) → 0.

In other words, H^{∗}(C, 𝔽_{2}) and H^{∗}(N, 𝔽_{2}) are subrings of H^{∗}(K, 𝔽_{2}) whose linear span is the whole ring, and whose intersection is H^{∗}(T h, 𝔽_{2}). Note that K contains a Sylow 2-subgroup of T h.

We need to work harder to prove the following theorem, which is really the main theorem of the paper.

The author is partly supported by a grant from the NSF.

Theorem 1.2. There is a polynomial subring of H^{∗}(T h, 𝔽_{2}) of the form R = 𝔽_{2}[x_{16}, x_{24}, x_{28}, x_{30}, x_{31}] with the following properties.

(i) If E is any rank five elementary abelian 2-subgroup of T h (note that all maximal elementary abelian 2-subgroups in T h have rank five), then the restriction of R to H^{∗}(E, 𝔽_{2}) is injective, and the image is equal to the rank five Dickson invariants.

(ii) H^{∗}(T h, 𝔽_{2}) is finitely generated as an R-module.

This theorem continues a thread relating the Dickson invariants to sporadic groups, described in the papers [1, 4, 5, 7]. This began when Adem, Maginnis and Milgram computed the cohomology of the Mathieu group M_{12} and noticed that there was a copy of the rank three Dickson invariants, over which H^{∗}(M_{12}, 𝔽_{2}) was a finitely generated free module. There is a maximal 2-local subgroup N of M_{12} which is a nonsplit extension 2^{3}L_{3}(2), and these Dickson invariants restrict to the invariants of N/O_{2}(N) in the cohomology of O_{2}(N). Benson and Wilkerson [7] proved that this inclusion was induced by a map of classifying spaces BM_{12} → BG_{2}, and speculated on higher rank cases.

In the paper [4], a homotopy colimit decomposition of the classifying space BCo_{3} was found. Using this, a map of classifying spaces BCo_{3} → BDI(4) was constructed, realizing a copy of the rank four Dickson invariants as a subring of H^{∗}(Co_{3}, 𝔽_{2}) over which it is finitely generated as a module. Here, DI(4) is the finite loop space at the prime two constructed by Dwyer and Wilkerson [14]. This map corresponds to the existence of a maximal 2-local subgroup which is a nonsplit extension 2^{4}L_{4}(2). In the article [5], some observations were made concerning the relationship between BDI(4) and some nonexistent sporadic simple groups considered by Solomon [31]. These observations and some unpublished work of Puig were among the motivations leading to the notion of a p-local finite group; just enough local information is given to be able to build a p-completed classifying space, and the p-local finite group can be recovered from its p-completed classifying space. For more information, see Broto, Levi and Oliver [10]. In terms of this language, Levi and Oliver [18] made more precise the connections between BDI(4) and Solomon’s groups.

We complete the picture by displaying a copy of the rank five Dickson invariants as a subring of H^{∗}(T h, 𝔽_{2}), corresponding to the maximal 2-local subgroup 2^{5}L_{5}(2). It is a theorem of Lin and Williams [19] that there can be no space whose cohomology is the rank five Dickson invariants, so there is no obvious way to realize the inclusion of the Dickson invariants topologically. Furthermore, there is no nonsplit extension of the form 2^{6}L_{6}(2), so there is no next case to consider, and the game stops here.

It is also worth observing that if we allow the generators to be raised to a high enough p-power power, there is always a copy of the Dickson invariants over which the cohomology of a finite group with coefficients in 𝔽_{p} is finitely generated as a module. See for example theorem 2.1 of [7], theorem 8.1.8 of Neusel [23], or theorem 8.2 of [6]. The Landweber–Stong conjecture, proved by Bourguiba and Zarati [8] (see also Henn [16]), says in this situation that the depth of the cohomology ring can be computed by seeing how many of the Dickson invariants form a regular sequence, taken in ascending order of degree. A simpler proof of the Landweber– Stong conjecture, but restricted to the context of cohomology of finite groups, can be found in [6].

Since the triality group ^{3}D_{4}(2) : 3 is a maximal subgroup of T h, and has the same 2-rank, the following is an easy corollary of Theorem 1.2.

Corollary 1.3. Both H^{∗}(^{3}D_{4}(2), 𝔽_{2}) and H^{∗}(^{3}D_{4}(2):3, 𝔽_{2}) contain a copy of the rank five Dickson invariants R = 𝔽_{2}[x_{16}, x_{24}, x_{28}, x_{30}, x_{31}] as a subring over which they are finitely generated as a module.

We remark that this can also be proved in a different way. Namely, ^{3}D_{4}(2) : 3 is a finite subgroup of the compact Lie group F_{4}. Since H^{∗}(BF_{4}; 𝔽_{2}) is known from the work of Borel to be a polynomial ring 𝔽_{2}[y_{4}, y_{6}, y_{7}, y_{16}, y_{24}] (see for example Vavpetič and Viruel [33]), it is not hard to find a suitable copy of the Dickson invariants as a subring over which this ring is a finitely generated module, and then restrict. One can show that the restriction maps from H^{∗}(BF_{4}; 𝔽_{2}) to H^{∗}(^{3}D_{4}(2), 𝔽_{2}) and H^{∗}(^{3}D_{4}(2):3, 𝔽_{2}) are injective, by restricting further to a rank five elementary abelian 2-subgroup.

Information on Thompson’s simple group of order 2^{15}.3^{10}.5^{3}.7^{2}.13.19.31 can be found in Thompson [32], Markot [22], Reifart [27], Smith [29], Parrott [24], Lyons [21], Holt [17], Yoshiara [36] and Havas, Soicher and Wilson [15]. The character table can be found in the Atlas [11], and the maximal subgroups were found by Wilson [35] and Linton [20]. Some authors write E or F_{3} or F_{3|3} for this group.

Throughout this paper, the word “space” means “simplicial set.” When we refer to the p-completion of a space X, we mean Bousfield–Kan 𝔽_{p}-completion, denoted R_{∞}X (where R = 𝔽_{p}) in Bousfield and Kan [9]. We note that the classifying spaces we are considering are 𝔽_{p}-good, so that their 𝔽_{p}-completions are 𝔽_{p}-complete. Specifically, it is shown in Ch. VII §§3–5 of [9] that spaces with finite fundamental group and spaces with perfect fundamental group are 𝔽_{p}-good; most of our groups are finite, and C ∗_{K} N is perfect.

Background material on the Dickson invariants can be found in Wilkerson’s primer [34], or in §8.1 of [3].

2. Local structure and the classifying space

In this section, we give two closely related proofs of Theorem 1.1, and prepare the ground for the cohomological computation in the next section. We begin by recalling from Wilson [35] some details of the 2-local structure of Thompson’s sporadic simple group T h. All the involutions (elements of order two) in T h are conjugate. If t is an involution then the centralizer C = C_{T h}(t) is a nonsplit extension 2^{1+8}A_{9} where the normal subgroup P = O_{2}(C) is an extraspecial group 2^{1+8} of “plus” type (i.e., of 2-rank five) and the quotient is an alternating group A_{9}. The 8-dimensional 2-modular representation of C/P ≅ A_{9} on P/Z(P) is not the deleted permutation representation, but is one of the two representations which may be obtained from this one by applying a triality automorphism of Ω_{8}^{+}(2). So in the permutation action of the quotient C/P ≅ A_{9} on the maximal elementary abelian subgroups of P, there is one orbit of size nine, which may be put in one-one correspondence in a natural way with the nine points permuted by A_{9}. A maximal elementary abelian subgroup of P is said to be nice if it is in this orbit. Of the 135 maximal elementary abelian subgroups of P, nine are nice and 126 are not. The 126 which are not nice also form a single orbit under the action of A_{9}, with stabilizer (A_{5} × A_{4}) : 2. The centralizer of a maximal elementary abelian 2-subgroup which is not nice has shape (2^{4} × Q_{8}). The normalizer modulo the centralizer is GL_{2}(4) extended by a field automorphism, namely ΓL_{2}(4) = 2^{4}(A_{5} × 3) : 2.

Each of the 270 noncentral involutions in P in is in a unique nice subgroup of P. These involutions are all conjugate in C, and if we choose one then the four group V_{4} it generates with t has centralizer in T h of shape 2^{2}[2^{9}]L_{3}(2). An involution in C/P ≅ A_{9} of cycle type (2^{2}1^{5}) does not lift to an involution in C, so the only other conjugacy class of four group in T h has a representative V′_{4} generated by t and an involution in C whose image in A_{9} has cycle type (2^{4}1). The centralizer of V′_{4} has order 2^{10}.3. If we choose a four group V′′_{4} in P which does not contain t and is not contained in a nice elementary abelian subgroup of P (all such four groups are conjugate in C), then it is conjugate in T h to V′_{4}. The involutions in the centralizer in C of V′′_{4} all lie in P. From this, Wilson deduces the following.

Lemma 2.1. Every elementary abelian 2-subgroup of T h is conjugate to a subgroup of P. Every elementary abelian subgroup of P is either contained in a nice elementary abelian subgroup of P, or it is contained in a unique conjugate in T h of P, and its normalizer is contained in C. □

Remark 2.2. Note that an elementary abelian 2-subgroup which is not nice can be contained in more than one conjugate of C.

Let Q be a nice elementary abelian subgroup of P. Then N = N_{T h}(Q) is a nonsplit extension 2^{5}L_{5}(2) where the normal subgroup is the elementary abelian subgroup Q = O_{2}(N) ≅ (ℤ/2)^{5} and the quotient is the group L_{5}(2) of five by five invertible matrices over the field of two elements. The group N is called the Dempwolff group [12]. There are exactly two conjugacy classes of maximal 2-local subgroups of T h, and they are represented by C and N.

Proof of Theorem 1.1. We give two proofs. The first proof goes as follows. Let A_{2}(T h) be the poset of nontrivial elementary abelian 2-subgroups of T h. Denote by P the subposet of A_{2}(T h) consisting of the conjugates of Z(C) = Z(P) ≅ ℤ/2 and of O_{2}(N) ≅ (ℤ/2)^{5}. Ryba, Smith and Yoshiara showed that the reduced Lefschetz module of P is projective.

Denote by F the kernel of the map C ∗_{K} N → T h. By the Kurosh subgroup theorem, this is a free group. The exact sequence coming from the simplicial cochains on |P| is

0 → 𝔽_{2} → 𝔽_{2}[T h/C] ⊕ 𝔽_{2}[T h/N] → 𝔽_{2}[T h/K] → H^{1}(F, 𝔽_{2}) → 0.

This sequence is split exact, because C, N and K all contain a Sylow 2-subgroup of T h. This shows that the reduced Lefschetz module of P is isomorphic to H^{1}(F, 𝔽_{2}).

It follows that H^{1}(F, 𝔽_{2}) is a projective 𝔽_{2}T h-module. Lemma 3.1 of [7] then proves the theorem.

For the second proof, it is proved in Smith and Yoshiara [30] that the inclusion of P into A_{2}(T h) is a homotopy equivalence. This also follows directly from Lemma 2.1. Since the inclusion is also a T h-equivariant map, the theorem then follows from theorem 1.10 of Dwyer [13]. □

The above description of the 2-local structure also allows us to describe the cohomology variety V_{T h}, namely the maximal ideal spectrum of H^{∗}(T h, k), where k is an algebraically closed field of characteristic two. Namely, there are just two conjugacy classes of maximal elementary abelian 2-subgroups in T h. They both have rank five. One is the nice subgroups, represented by O_{2}(N), and the other is represented by a rank five elementary abelian subgroup of P which is not nice. Using the work of Quillen [25, 26] we see that V_{T h} has two irreducible components, both of dimension five. Their intersection has dimension four.

3. The cohomology

Next, we describe in a little more detail the structure of the subgroup C of T h of shape 2^{1+8}A_{9}, and produce an element w of degree 16 in H^{∗}(BC, 𝔽_{2}) with nontrivial restriction to Z(C) = ⟨t⟩. We shall show that for a suitable choice of such an element w, x_{16} = Tr_{C,T h}(w) ∈ H^{16}(T h, 𝔽_{2}) plays the role of the lowest Dickson invariant. The remaining Dickson invariants are obtained from this one by applying Steenrod operations.

We start by describing a different but closely related group manufactured from the extraspecial group P. Since P is an extraspecial 2-group of order 2^{1+8} and 2rank five, it may be generated by involutions a_{1}, …, a_{4}, b_{1}, …, b_{4} which commute with the exception of the following nontrivial commutators:

[a_{1}, b_{1}] = [a_{2}, b_{2}] = [a_{3}, b_{3}] = [a_{4}, b_{4}] = t,

where t is the central involution. The ordinary irreducible representations of P can all be written over the reals. They consist of 2^{8} one dimensional representations and one irreducible representation on a real vector space V of dimension 16 (note that |P | = 2^{8} + 16^{2}). The representation V may be described as an induced representation ℝP ⊗_{ℝQ} V′ where V′ is the one dimensional representation of the subgroup Q = ⟨t, a_{1}, a_{2}, a_{3}, a_{4}⟩ in which t ↦ (−1) and a_{i} ↦ (1) for 1 ≤ i ≤ 4. In terms of matrices, V is a representation in which t is a diagonal matrix of −1’s, a_{1}, a_{2}, a_{3} and a_{4} act as diagonal matrices with eight 1’s and eight −1’s, and b_{1}, b_{2}, b_{3} and b_{4} act as permutation matrices.

The automorphism group of P has shape 2^{8}SO_{8}^{+}(2), where SO_{8}^{+}(2) is the corresponding special orthogonal group (which has a simple subgroup of index 2 denoted in the Atlas [11] by O_{8}^{+}(2) and elsewhere by Ω_{8}^{+}(2)). Given any automorphism α of P, the conjugate of V by α is isomorphic to V. So there is a 16 by 16 matrix A_{α}, conjugation by which has the same effect as the action of α. In other words, if ρ : P → Aut(V) is the representation of P on V, then for all g ∈ P we have ρ(α(g)) = A_{α}ρ(g)A_{α}^{−1}. The matrix A_{α} is uniquely determined up to multiplication by scalars (by Schur’s lemma), and if we demand that A_{α} have determinant one then it is unique up to sign. It follows that the group formed by taking both possibilities of A_{α} for each automorphism α is a double cover of Aut(P) of shape 2^{1+8}SO_{8}^{+}(2).

There are three conjugacy classes of subgroups of Ω_{8}^{+}(2) isomorphic to A_{9}, permuted transitively by the triality automorphism. For one of them the restriction of the natural eight dimensional representation is the deleted permutation representation. The other two are conjugate in SO^{+}(8). Choosing a representative of one of the latter two conjugacy classes, we obtain a group C_{0} of shape 2^{1+8}A_{9}. This is still not isomorphic to the subgroup C of T h, but they are isomorphic modulo centers. To obtain C from C_{0}, we form the Baer sum of this central extension with the central extension pulled back from the nontrivial double cover of A_{9}. In other words, take C_{0} × C_{1}, where C_{1} is the double cover of A_{9}, pass to the subgroup consisting of pairs with the same image in A_{9}, and then quotient out the diagonal central element of order two. The smallest faithful complex representation of C is 128 dimensional, and is obtained by tensoring V with an irreducible faithful eight dimensional real representation of C_{1}.

Let w_{16} ∈ H^{16}(C_{0}, 𝔽_{2}) be the degree 16 Stiefel–Whitney class of the real representation V. Since the restriction of V to Z(C_{0}) is equal to a direct sum of 16 copies of the sign representation, the formula for the Stiefel–Whitney classes of a direct sum shows that w_{16} restricts to

z_{0}^{16} ∈ H^{∗}(BZ(C_{0}), 𝔽_{2}) = 𝔽_{2}[z_{0}].

It follows that in the spectral sequence

H^{∗}(C_{0}/Z(C_{0}), H^{∗}(Z(C_{0}), 𝔽_{2})) ⇒ H^{∗}(C_{0}, 𝔽_{2})

the element z_{0}^{16} on the fiber is a universal cycle. The element

x = d_{2}(z_{0}) ∈ H^{2}(C_{0}/Z(C_{0}), 𝔽_{2})

classifies the central extension by Z(C_{0}), and Serre’s transgression theorem (§II.9c of [28]) shows that Sq^{8}Sq^{4}Sq^{2}Sq^{1}x = 0.

Next, we examine C_{1} ≅ 2A_{9}. The deleted permutation representation of A_{9} of dimension eight displays it as a subgroup of SO(8). Let

1 → ⟨γ⟩ → Spin(8) → SO(8) → 1

be the double cover map, where γ is a central element of order two. The inverse image of A_{9} in Spin(8) is isomorphic to C_{1}, so we regard C_{1} as a subgroup of Spin(8) in this way. Now if we let Δ be one of the two spin representations of Spin(8) of dimension 8, then the Stiefel–Whitney class w_{8}(Δ) is an element of H^{8}(BSpin(8); 𝔽_{2}) whose restriction to H^{∗}(⟨γ⟩, 𝔽_{2}) is the eighth power of the degree one generator. It follows that the restriction

u = res_{Spin(8),C_{1}}(w_{8}(Δ)) ∈ H^{8}(C_{1}, 𝔽_{2})

is an element whose restriction to Z(C_{1}) is nontrivial. So in the spectral sequence

H^{∗}(C_{1}/Z(C_{1}), H^{∗}(Z(C_{1}), 𝔽_{2})) ⇒ H^{∗}(C_{1}, 𝔽_{2}),

the element z_{1}^{8} ∈ H^{∗}(Z(C_{1}), 𝔽_{2}) = 𝔽_{2}[z_{1}] on the fiber is a universal cycle. The element

y = d_{2}(z_{1}) ∈ H^{2}(C_{1}/Z(C_{1}), 𝔽_{2})

classifies the central extension by Z(C_{1}), and so Sq^{4}Sq^{2}Sq^{1}y = 0.

Now C/Z(C) ≅ C_{0}/Z(C_{0}), of shape 2^{8}A_{9}, and C/O_{2}(C) ≅ C_{1}/Z(C_{1}) ≅ A_{9}. So there is an inflation map

inf : H^{∗}(C_{1}/Z(C_{1}), 𝔽_{2}) → H^{∗}(C/Z(C), 𝔽_{2}),

and the element x + inf(y) ∈ H^{2}(C/Z(C), 𝔽_{2}) classifies the central extension of C/Z(C) by Z(C). Since the Steenrod operations are 𝔽_{2}-linear, and commute with the inflation map, we have

Sq^{8}Sq^{4}Sq^{2}Sq^{1}(x + inf(y)) = Sq^{8}Sq^{4}Sq^{2}Sq^{1}(x) + inf(Sq^{8}Sq^{4}Sq^{2}Sq^{1}(y)) = 0

in H^{∗}(C/Z(C), 𝔽_{2}). Again using Serre’s transgression theorem, it follows that z^{16} is a universal cycle in the spectral sequence

H^{∗}(C/Z(C), H^{∗}(Z(C), 𝔽_{2})) ⇒ H^{∗}(C, 𝔽_{2}).

From this, we deduce that there is an element w ∈ H^{16}(C, 𝔽_{2}) with the property that the restriction to Z(C) is z^{16}.

Lemma 3.1. With this definition of w, the transfer from C to T h of w is an element x_{16} ∈ H^{∗}(T h, 𝔽_{2}) with the property that the restriction to Z(C) of x_{16} is equal to z^{16}.

Proof. To compute the restriction of the transfer, we use the Mackey double coset formula. Double cosets where the intersection of C with the conjugate of Z(C) is trivial do not contribute to the sum. The fact that Z(C) is equal to the center of a Sylow 2-subgroup of C implies that the remaining double cosets apart from the one corresponding to the identity element correspond to conjugacy classes of involutions in C of even size, which therefore contribute zero in characteristic two. □

Lemma 3.2. The restriction res_{T h,O_{2}(N)}(x_{16}) is equal to the degree 16 Dickson invariant in H^{∗}(O_{2}(N), 𝔽_{2})^{N/O_{2}(N)}.

Proof. The restriction must be nonzero by Lemma 3.1, since Z(C) ≤ O_{2}(N). It must also be invariant under the action of N/O_{2}(N). There is only one nonzero invariant in this degree, namely the degree 16 Dickson invariant. □

We can now create all the Dickson invariants, using the action of the Steenrod algebra. Namely, we set x_{24} = Sq^{8}(x_{16}), x_{28} = Sq^{4}(x_{24}), x_{30} = Sq^{2}(x_{28}) and x_{31} = Sq^{1}(x_{30}). Since the Steenrod operations commute with the restriction map, the formulae given in Wilkerson [34] show that the restrictions of x_{16}, x_{24}, x_{28}, x_{30} and x_{31} are the generators for the rank five Dickson invariants in H^{∗}(O_{2}(N), 𝔽_{2}). In particular, these elements are algebraically independent.

Lemma 3.3. Let E be an elementary abelian subgroup of P of rank five which is not nice. Then H^{16}(E, 𝔽_{2})^{N_{T h}(E)} is two dimensional, and is spanned by the degree 16 Dickson invariant and the element u formed by taking the product of the 16 elements of H^{∗}(E, 𝔽_{2}) whose restrictions to Z(C) are nonzero.

Proof. Set H = N_{T h}(E)/C_{T h}(E) = 2^{4}(A_{5} × 3)2. We work our way up a normal series for H. The invariants of O_{2}(H) is a polynomial ring on four generators of degree one and a generator of degree 16. The invariants of A_{5} on the polynomial ring generated by the four generators of degree one with the given action of A_{5} (namely, a two dimensional representation over 𝔽_{4} with the scalars restricted) were computed by Adem and Milgram [2]. From theorem 3.2 of that paper, it is not hard to read off the degree 16 invariants, together with the action of the extra elements of order three and two. There are five dimensions of A_{5}-invariants, of which only one dimension is (A_{5} × 3)2-invariant, namely the square of the degree eight generator. The remaining four dimensions are given by multiplying elements of degrees 5, 8 and 3. The fixed space of this four dimensional space is zero, even under the extra element of order three. This shows that the degree 16 invariants of 2^{4}(A_{5} × 3)2 on H^{∗}(O_{2}(N), 𝔽_{2}) are two dimensional. Since the degree 16 Dickson invariant and u have different restrictions to subgroups of order two in N other than Z(C), it follows that they span this two dimensional space. □

Lemma 3.4. Let E be any elementary abelian 2-subgroup of T h of rank five. Then the restrictions of x_{16}, x_{24}, x_{28}, x_{30} and x_{31} to E are equal to the Dickson invariants.

Proof. If E is nice, this follows from Lemma 3.2. So suppose that E is not nice. By Lemma 2.1, we can assume that E is contained in P. The restriction of x_{16} is certainly invariant under N_{T h}(E), and so by Lemma 3.3, it is in the two dimensional space spanned by the degree 16 Dickson invariant and u. The element x_{16} has nonzero restriction to every involution in T h. The element u restricts to zero on a cyclic subgroup of E other than Z(C), and u plus the Dickson invariant restricts to zero on Z(C). So the only possibility is that the restriction of x_{16} is equal to the degree 16 Dickson invariant. Applying the Steenrod operations shows that x_{24}, x_{28}, x_{30} and x_{31} restrict to the remaining Dickson invariants. □

Proposition 3.5. The cohomology ring H^{∗}(T h, 𝔽_{2}) is finitely generated as a module over the subring 𝔽_{2}[x_{16}, x_{24}, x_{28}, x_{30}, x_{31}].

Proof. It follows from Quillen’s F-isomorphism theorem [25, 26] that it suffices to check that the cohomology of every elementary abelian 2-subgroup of T h is finitely generated over the restriction of this polynomial subring. There are two conjugacy classes of maximal elementary abelian 2-subgroups, both of rank five, and these are dealt with in Lemma 3.4. □

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

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