Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants

D. J. Benson Department of Mathematics University of Georgia Athens GA 30602, USA djb@sloth.math.uga.edu

1 Introduction

Whatever one may think of a proof that covers over ten thousand pages of journal articles, some of which have still not appeared in print, the classification of finite simple groups is a remarkable theorem. It says that (apart from the cyclic groups of prime order and the alternating groups) most finite simple groups are groups of Lie type. These are the analogues of the compact Lie groups, defined over a finite field. They admit a uniform description in terms of the fixed points of certain automorphisms related to the Frobenius map, on the corresponding algebraic groups in prime characteristic. The classifying space of a finite group of Lie type fibers over that of the corresponding Lie group with fibers which are cohomologically finite away from the defining characteristic. Apart from the groups of Lie type, there are the alternating groups A_{n} (n ≥ 5), and twenty-six other groups called the sporadic simple groups.

The first five sporadic groups were discovered by Mathieu in the late nineteenth century. The remaining twenty-one were discovered in the nineteen sixties and seventies. The largest is the Fischer–Griess Monster, which has order roughly 8 × 10^{53}. For a wealth of information on the sporadic groups and other “small” finite simple groups, the reader is referred to the ATLAS of finite groups [10]. A great deal of effort has gone into trying to understand these sporadic groups. For example, the “Monstrous Moonshine” [8] (see also [9]) is a remarkable series of observations, still not fully understood, connecting various of the sporadic groups, especially those involved in the Monster, with the theory of modular functions. This article is about another attempt to understand some of the sporadic groups from a different point of view, namely that of the topology of their classifying spaces.

The story started with the construction by Bill Dwyer and Clarence Wilkerson [13] of a new finite loop space at the prime two, which they call DI(4). The mod two cohomology of its classifying space is equal to the rank four Dickson invariants, whence the name. Finite loop spaces are sufficiently like compact Lie groups that this prompted the question of what finite simple groups could be associated to them in the way that the finite groups of Lie type are associated to the compact Lie groups.

The first group that was considered in this regard was Conway’s group Co_{3}, which has a 2-local structure closely associated with the 2-local structure of DI(4). It turns out that the classifying space of Co_{3} fibers over the classifying space of DI(4) with fibers cohomologically finite at the prime two. However, this turns out to be more akin to the relation between M_{12} and G_{2} than to the relation between G_{2}(q) and G_{2}.

The finite simple groups which should be regarded as the groups “of Lie type DI(4)” over the field of q elements don’t literally exist as groups. They were first considered by Ron Solomon [27] in an attempt to prove that Co_{3} is determined by a Sylow 2-subgroup. He was led to consider simple groups in which the centralizer of an involution was Spin(q) (q odd), possibly with some decoration of odd order. In order to prove nonexistence, Solomon was obliged to write down what appeared to be a consistent fusion pattern for the Sylow 2-subgroup, and then pass to an odd prime to obtain a contradiction. It turns out that despite the nonexistence of these groups, they have perfectly good 2-completed classifying spaces, which map to the classifying space of DI(4) with fibers which are cohomologically finite at the prime two.

This prompts the speculation that there should exist a theory of “p-local groups”, in which one only gives a Sylow p-subgroup and a fusion pattern. The fusion pattern should obey a set of axioms which are strong enough to be able to build a p-completed classifying space. One possible candidate for such a set of axioms has been written down by Puig [23] in connection with his attempts to formalise the local theory of blocks, but much work remains to be done in this area.

2 Classifying spaces

We begin by describing the theory of classifying spaces, and explain why they are rigid enough that new maps between them should be considered to be interesting.

For any topological group G, there exists a contractible space EG together with a free action of G on EG. A uniform construction was found by Milnor [22]. Roughly speaking, the idea is to form an infinite join G∗G∗⋯ of copies of G as a topological space, and let G act diagonally. The join X ∗ Y of two spaces X and Y consists of the space obtained from X × I × Y by identifying (x, 0, y) with (x′, 0, y) for all x, x′ ∈ X and y ∈ Y, and (x, 1, y) with (x, 1, y′) for all x ∈ X and y, y′ ∈ Y. The difficult point in Milnor’s construction is the exact description of the topology on the infinite join. However, we shall mostly be interested in the case where G is either discrete or a compact Lie group. In this case, we can just give the infinite join the weak topology with respect to the finite joins. Then EG is a CW-complex, so to see that it is contractible, it suffices to see that any map from the n-sphere S^{n} to EG extends to a map from the (n + 1)-disk D^{n+1}. Since S^{n} is compact, any such map has image contained in some finite join of copies of G, and then the next higher join contains the cone on this join, so that the map can be extended to D^{n+1} in this next join.

We write BG for the quotient of EG by the action of G. It turns out that the classifying space BG is determined up to homotopy by G, independently of the choice of contractible space EG. Thus if G is a subgroup of G′, then using EG′ instead of EG, we see that there is an induced fibration BG → BG′ with fiber the coset space G′/G. Similarly if N is a normal subgroup of G, we may use EG × E(G/N) instead of EG, with G acting on both factors, and obtain an induced fibration BG → B(G/N) with fiber BN. In this way, for any homomorphism of topological groups α : G → G′, we have an induced map of classifying spaces Bα : BG → BG′, unique up to homotopy.

If G is discrete, BG is an Eilenberg–Mac Lane space K(G, 1), and EG is its universal cover. In any case, the loop space ΩBG is homotopy equivalent to G as a topological space.

Classifying spaces are homotopically rather rigid. For example, if G and G′ are discrete groups, then the set of homotopy classes of maps [BG; BG′] is in natural bijection with Hom_{grp}(G, G′)/Inn(G′). Here, the group Inn(G′) of inner automorphisms of G′ acts in the obvious way by composition on the set Hom_{grp}(G, G′) of group homomorphisms from G to G′. The bijection sends a map α : G → G′ to Bα : BG → BG′. The inverse of this bijection sends a map BG → BG′ to its effect on π_{1}. This, of course, depends on a choice of basepoints, but the orbit under the action of Inn(G′) does not.

On the other hand, if G is a connected simple compact Lie group, then a theorem of Jackowski, McClure and Oliver (Theorem 3 in [18]) says that there is a natural bijection

Hom_{Liegrp}(G, G)/Inn(G) ∧ {k ≥ 0 : k = 0 or(k, |W |) = 1} → [BG; BG]

sending (α, k) to Ψ^{k} ∘ Bα. Here, the smash product denotes the quotient of the direct product given by identifying all elements in which at least one coordinate is zero. For k > 0, the operations Ψ^{k} are the unstable Adams psi operations, which are known to exist for any connected simple compact Lie group and any integer k coprime to the order of the Weyl group W. They commute up to homotopy with maps coming from Lie group homomorphisms. If k = 0, Ψ^{k} is taken to be the identity map. The map Ψ^{k} is characterised by the property that its effect on degree 2i rational cohomology is multiplication by k^{i} for each i ≥ 0.

These theorems indicate that it is of interest to construct maps of classifying spaces which do not come from group homomorphisms and Adams operations. Some examples for connected Lie groups can be found in [18]. For maps from the classifying space of a finite group to that of a compact Lie group, the situation is rather unclear, and it is possible that one should try to build a branch of representation theory based on such maps.

A theorem of Dwyer and Wilkerson [12] indicates that we may work one prime at a time in this situation. Writing X_{p}^{∧} for the Bousfield–Kan p-completion of a space X, their theorem states the following.

Theorem 2.1 If π is a finite group and G is a connected compact Lie group then

Map(Bπ, BG) ≃ ∏_{p} Map((Bπ)_{p}^{∧}, (BG)_{p}^{∧}).

Here, and from now on in this article, one must work with simplicial sets rather than topological spaces, and “Map” denotes the simplicial mapping space. The homotopy category of topological spaces is equivalent to the homotopy category of simplicial sets (these homotopy categories are formed by inverting weak equivalences), via singular simplices and geometric realization. So we use the term “space” from now on to mean “simplicial set”, and we replace a classifying space by its simplicial set of singular simplices.

The effect of Bousfield–Kan p-completion on the classifying space of a finite group G is as follows. Without changing the mod p cohomology, it gets rid of the mod l cohomology for every prime l ≠ p. Its effect on the fundamental group G is to replace it by G/O^{p}(G). So for example if G is a nonabelian finite simple group then the p-completion of BG is simply connected. The group G may not be recovered from the p-completion of BG, though the Sylow p-subgroup and the strong fusion of its subgroups may be recovered.

3 Dickson invariants

Let _{q} be the finite field of q elements, with q = p^{a} a prime power. Let V = ( _{q})^{n} be an n-dimensional vector space over _{q}. Then the finite general linear group G = GL_{n}( _{q}) acts on V in the obvious way, and hence on the coordinate ring

q[V] = _{q} ⊕ V^{∗} ⊕ S^{2}(V^{∗}) ⊕ ⋯

We regard this as a graded ring, with _{q} in degree zero, V^{∗} = Hom_{_{q}} (V, _{q}) in degree one, S^{2}(V^{∗}) in degree two, and so on. It is a polynomial ring over _{q} on generators forming an basis for V^{∗} in degree one.

L. E. Dickson’s theorem [11] states that the invariants of this action also form a polynomial ring:

q[V]^{G} = _{q}[c_{n,0}, …, c_{n,n−1}]

in generators c_{n,i} (the Dickson invariants) of degree q^{n} − q^{i}. In fact, there is an explicit formula for c_{n,i} as follows:

c_{n,i} = ∑_{{W ⊆V \atop dim_{q} (W)=i}} ∏_{{ϕ∈V^{∗} \atop ϕ|_{W} ≠0}} ϕ.

If you are a topologist, you may wish to double all the degrees in the above discussion, because if p is odd, elements of odd degree are supposed to anticommute rather than commuting. Having done that, the polynomial ring on generators in degree two over _{p} acquires an action of the Steenrod operations, since it is the mod p cohomology of a torus T^{n} = ^{n}/ ^{n} = B ^{n} of rank n. The group GL(n, ) acts on T^{n}, and induces an action of the quotient group GL(n, _{p}) on its mod p cohomology. This action commutes with the Steenrod operations, and so the invariants inherit an action of the Steenrod algebra.

Even if you are a topologist, in the case where q = p = 2, you may also wish to keep the polynomial generators in degree one. In this case, the polynomial ring is the cohomology of ( P^{∞})^{n} = B( /2)^{n}, and so the mod two Steenrod algebra acts on this polynomial ring. Again, the invariants inherit an action of the Steenrod algebra. In this case, let us look at the question of when the algebra of Dickson invariants can be the cohomology of a topological space, as an algebra over the Steenrod algebra. When n = 1, the group GL(1, _{2}) is the trivial group, and the invariants form a polynomial ring on a single generator in degree one:

2[c_{1,0}] = H^{∗}( P^{∞}, _{2}) = H^{∗}(B /2, _{2}).

The situation gets slightly more interesting in rank two. GL(2, _{2}) is isomorphic to the symmetric group of degree three, and the invariants form a polynomial ring on generators in degrees two and three:

2[c_{2,0}, c_{2,1}] ≅ H^{∗}(BSO(3), _{2}).

The group SO(3) contains a subgroup E_{2} isomorphic to ( /2)^{2}, and the normalizer quotient is N_{SO(3)}(E_{2})/E_{2} ≅ GL(2, _{2}). The restriction map

H^{∗}(BSO(3), _{2}) → H^{∗}(B( /2)^{2}, _{2})^{GL(2, _{2})}

identifies the cohomology with the Dickson invariants.

In rank three, a similar thing happens, but with the exceptional compact Lie group G_{2}. The Dickson invariants of rank three form a polynomial ring on generators in degrees four, six and seven:

2[c_{3,0}, c_{3,1}, c_{3,2}] ≅ H^{∗}(BG_{2}, _{2}).

The group G_{2} contains a subgroup E_{3} isomorphic to ( /2)^{3}, and the normalizer quotient is N_{G_{2}}(E_{3})/E_{3} ≅ GL(3, _{2}). The restriction map

H^{∗}(BG_{2}, _{2}) → H^{∗}(B( /2)^{3}, _{2})^{GL(3, _{2})}

again identifies the cohomology with the Dickson invariants.

In rank four, there is no longer a compact Lie group with the right cohomology. However, Dwyer and Wilkerson [13] recently constructed a space BDI(4) with the Dickson invariants of rank four as its cohomology. The Dickson invariants in this case form a polynomial ring on generators in degrees eight, twelve, fourteen and fifteen:

2[c_{4,0}, c_{4,1}, c_{4,2}, c_{4,3}] ≅ H^{∗}(BDI(4), _{2}).

The space BDI(4) is not the classifying space of a group, but rather of a finite loop space DI(4) at the prime two. We shall have more to say about this in Section 6. This finite loop space may be thought of as containing a subgroup E_{4} isomorphic to ( /2)^{4}, with normalizer quotient N_{DI(4)}(E_{4})/E_{4} ≅ GL(4, _{2}). The “restriction map”

H^{∗}(BDI(4), _{2}) → H^{∗}(B( /2)^{4}, _{2})^{GL(4, _{2})}

again identifies the cohomology with the Dickson invariants.

The game stops here. A theorem of Lin and Williams [20] states that there is no topological space whose mod two cohomology is the ring of Dickson invariants of rank five or more (as an algebra over the Steenrod algebra).

4 Finite groups of Lie type

As we pointed out in the introduction, most finite simple groups are of Lie type. Actually, just as in the representation theory, from the point of view of the classifying space, it is better to work with certain “almost simple” groups. For example, instead of working with P SL(n, _{q}), it is usually easier to work with SL(n, _{q}), or even GL(n, _{q}).

With this proviso, the data needed to specify a finite group of Lie type are a compact Lie group G, a finite field _{q}, and a diagram automorphism α of the corresponding Dynkin diagram. For a large portion of the groups of Lie type (the untwisted groups), α is the identity map. For example, both the finite linear groups and the finite unitary groups come from the Dynkin diagram A_{n}, with the identity automorphism in the first case, and a diagram automorphism of order two in the second case. The finite group associated with the above data is denoted ^{α}G(q), or just G(q) if α is the identity.

The following theorem of Quillen and Friedlander (see Quillen [24], and Friedlander [16], Chapter 12) relates the classifying space of the finite group ^{α}G(q) to the classifying space of G.

Theorem 4.1 There is a commutative diagram

B^{α}G(q) ⟶  BG
↓  ↓Δ
BG ⟶^{id×(Ψ^{q}∘Bα)} BG × BG

which becomes a homotopy pullback square after Bousfield–Kan completing at any prime l not dividing q.

In particular, there is a fibration B^{α}G(q) → BG whose fibers have finite homology at primes l not dividing q.

Here, Δ denotes the diagonal map from BG to BG × BG. Construction of this diagram involves replacing the compact Lie group by the corresponding algebraic group defined over , and lifting from characteristic p using étale homotopy theory. The details may be found in Friedlander [16], Chapter 12.

This diagram gives a method of calculating the cohomology of the finite groups of Lie type away from their defining characteristic. First, one uses the Eilenberg– Moore spectral sequence on this homotopy pullback to calculate the cohomology modulo a certain filtration, then one uses the subgroup structure to ungrade. This was carried out by Quillen [25] in the case of GL(n, _{q}), Fiedorowicz and Priddy [15] in the case of the other classical groups, and Kleinerman [19] for the exceptional Lie types.

To give an example, consider the fibration BG_{2}(q) → BG_{2} for q an odd prime power. The homology of the fiber at the prime two is an exterior algebra on generators in degrees three, five and six. So we get an embedding

H^{∗}(BG_{2}, _{2}) = _{2}[c_{3,0}, c_{3,1}, c_{3,2}] ⊆ H^{∗}(BG_{2}(q), _{2}).

The corresponding Eilenberg–Moore spectral sequence calculation gives a filtration of H^{∗}(BG_{2}(q), _{2}) whose associated graded is a tensor product of a polynomial ring on the Dickson generators in degrees four, six and seven, with an exterior algebra on generators in degrees three, five and six. After ungrading, we see that H^{∗}(BG_{2}(q), _{2}) is a free module of rank eight over the Dickson algebra. In particular, it is a Cohen–Macaulay ring, and the Poincaré series for the cohomology is

∑_{i=0}^{∞} t^{i} dim_{_{2}} H^{i}(BG_{2}(q), _{2}) = \frac{(1 + t^{3})(1 + t^{5})(1 + t^{6})}{(1 − t^{4})(1 − t^{6})(1 − t^{7})}.

5 The Mathieu group M_{12}

A few years ago, Adem, Maginnis and Milgram [2] computed H^{∗}(BM_{12}, _{2}). Their answer was rather complicated, but they observed that the answer was a finitely generated free module over a subring isomorphic to the Dickson invariants of rank three:

H^{∗}(BG_{2}, _{2}) = _{2}[c_{3,0}, c_{3,1}, c_{3,2}] ⊆ H^{∗}(BM_{12}, _{2}).

It is clear that there can be no homomorphism of groups from M_{12} to G_{2} inducing the above map in mod two cohomology, because G_{2} has a seven dimensional complex representation, while the smallest nontrivial complex representation of M_{12} is eleven dimensional. However, it turns out that there is a fibration BM_{12} → BG_{2} whose fibers have finite mod two cohomology. There are (at least) two possible strategies for constructing such a map. One (Milgram [21]) is to construct maps from BM_{12} to BG_{2}(q) for suitable q, and then use the Quillen–Friedlander theorem. A more direct approach (Benson and Wilkerson [4]) goes as follows. Let P be a Sylow 2subgroup of M_{12}, of order 2^{6}. Let W be the centralizer of the center of P, and W′ be the normalizer of a normal fours group in P. Then |W : P | = |W′ : P | = 3, but W ≅ W̸′. The obvious map from the amalgamated free product W ∗_{P} W′ to M_{12} is a cohomology isomorphism, and so after completing at the prime two, we obtain a homotopy equivalence (B(W ∗_{P} W′))_{2}^{∧} → (BM_{12})_{2}^{∧}. It turns out that there is a group homomorphism from W ∗_{P} W′ to G_{2}, whose image is dense, and having the desired effect in cohomology. Now using Theorem 2.1, we may pass from a map between 2-completions to a map between the original classifying spaces BM_{12} → BG_{2} by specifying any map we want at odd primes.

The classifying space B(W ∗_{P} W′) can be obtained by taking the homotopy colimit (see Chapter XII of Bousfield and Kan [5]) of the diagram

BP → BW
↓
BW′

which may be thought of in this case as a “double mapping cylinder”.

6 Finite loop spaces

From the point of view of a homotopy theorist, the two properties of a compact Lie group G which stand out are:

(i) G is a finite CW-complex, and

(ii) G has the homotopy type of a loop space (namely G ≃ ΩBG). We say that a space X is a finite loop space if it satisfies properties (i) and (ii). It turns out that finite loop spaces enjoy many of the properties of compact Lie groups. There is an algebraically defined maximal torus, and a Weyl group which is a finite reflection group [1].

Since homotopy theorists like to work one prime at a time using Bousfield– Kan p-completion, it is appropriate also to introduce the p-complete version of this definition. We say that X is a finite loop space at the prime p if it satisfies the following two properties:

(i)_{p} X is p-complete and H^{∗}(X, _{p}) is finite, and

(ii)_{p} X has the homotopy type of a loop space ΩBX.

Note that BX is now just the name of the space whose loops are homotopy equivalent to X. To give a finite loop space, we must really give both X and BX, but X is determined by BX so it is only necessary to construct BX.

Again, a finite loop space at the prime p has an algebraically defined maximal torus, and a Weyl group which is a finite p-adic reflection group. The finite reflection groups over the complex numbers were classified by Shephard and Todd [26]. Clark and Ewing [6] determined which of these were defined over the p-adics for p not dividing the group order, and demonstrated how to produce a finite loop space at the prime p in this “non-modular” case. There remains a well defined list of questions, one for each prime dividing the order of each of the groups in the list of Shephard and Todd.

There is essentially only one known example of a connected finite loop space at the prime two, which is not the 2-completion of a compact Lie group. All other known examples are given by taking products. The example in question is the space DI(4) ≃ ΩBDI(4) constructed by Dwyer and Wilkerson [13]. The Weyl group of DI(4) is /2 × GL(3, _{2}), acting as a three dimensional 2-adic reflection group. It is group number 24 in the list of Shephard and Todd. The corresponding “root system” has 21 pairs of opposite roots, so the dimension of DI(4) is 3 + 2 × 21 = 45. The 2-adic integral cohomology of BDI(4) is a polynomial algebra on three variables in degrees 8, 12 and 28, corresponding to (twice) the degrees of the fundamental invariants of the Weyl group. The mod two cohomology is the Dickson algebra on four generators in degrees 8, 12, 14 and 15.

Dwyer and Wilkerson constructed the space BDI(4) as a homotopy colimit as follows. Consider the category whose objects are the groups ( /2)^{n} for 1 ≤ n ≤ 4, and whose arrows are the injective homomorphisms between these groups. They constructed a functor to the homotopy category taking the following values:

\begin{matrix}/2 ↦ BSpin(7)_{2}^{∧} & ( /2)^{2} ↦ B(SU(2)^{3}/ ± 1)_{2}^{∧} \\ ( /2)^{3} ↦ B(T^{3} /2)_{2}^{∧} & ( /2)^{4} ↦ B( /2)^{4}. \end{matrix}

They then proved that this could be lifted to a functor to the category of spaces, so that they could then take the homotopy colimit. Some of the maps involved do not exist until the spaces have been 2-completed.

The homotopy colimit of the above functor over a full subcategory consisting of a single object plays the role of the classifying space of the “normalizer” of the corresponding elementary abelian 2-subgroup in DI(4). For example, the normalizer in this sense of the subgroup ( /2)^{4} is a non-split extension of this elementary abelian subgroup by its automorphism group GL(4, _{2}).

It is interesting to speculate on the existence of a suitable algebraic system of dimension 45 which should play the role of a tangent space at the identity for DI(4). It should be almost, but not quite, a Lie algebra over the 2-adics. One tempting candidate is the set of 3 × 3 skew-hermitian matrices over a suitable 2-adic version of the Cayley numbers. This would have the right dimension, but on the face of it, it would seem to contain the wrong 21-dimensional Lie subalgebra (C_{3} instead of B_{3}). But there may be some twisted version of this which works.

7 Conway’s group Co_{3}

Conway’s sporadic simple group Co_{3} was discovered as part of a series of groups arising from Leech’s close packing of spheres in twenty-four dimensions [7, 9]. It is defined as the stabilizer of the origin and a type three vector in the group of automorphisms of the Leech lattice, and has order 2^{10}.3^{7}.5^{3}.7.11.23. The following theorem was proved in [3]:

Theorem 7.1 There is a map from BCo_{3} to BDI(4), which induces an embedding

H^{∗}(BDI(4), _{2}) ⊆ H^{∗}(BCo_{3}, _{2})

in such a way that the latter is a finitely generated module over the former.

Prior to the proof of this theorem, nothing was known about the mod two cohomology ring of BCo_{3} (apart from in degrees one and two). It is still not known whether the mod two cohomology of BCo_{3} is a free module over the rank four Dickson invariants, which amounts to the question of whether it is a Cohen–Macaulay ring. It is known, however, that the cohomology of a Sylow 2-subgroup of Co_{3} is not Cohen–Macaulay, because there are maximal elementary abelian 2-subgroups of different ranks, so that it is not even equidimensional. But all maximal elementary abelian 2-subgroups of Co_{3} have rank four.

The idea of the proof of the above theorem is as follows. Let P be a Sylow 2-subgroup of Co_{3}, of order 2^{10}. Then Z(P) = ⟨t_{1}⟩ has order two, and C = C_{Co_{3}}(t_{1}) is a double cover of the simple group S_{6}(2). Now the Weyl group of type E_{7} is /2 × S_{6}(2), and so there is an obvious map from S_{6}(2) to SO(7), which lifts to a map from C to Spin(7).

There is a normal fours group ⟨t_{1}, t_{2}⟩ in P whose normalizer N is a solvable maximal 2-local subgroup of Co_{3} of order 2^{10}.3^{3}. There is also a normal subgroup V = ⟨t_{1}, t_{2}, t_{3}, t_{4}⟩ ≅ ( /2)^{4} whose normalizer X = N_{Co_{3}}(V) is a non-split extension of ( /2)^{4} by GL(4, _{2}). It turns out that the natural map from the homotopy colimit of the diagram formed by the classifying spaces of C, N, X and their intersections to the classifying space of Co_{3} is a mod two cohomology equivalence. This is proved essentially by the methods of Jackowski and McClure [17].

Now we have already chosen a suitable map from C to Spin(7), and hence from BC_{2}^{∧} to BSpin(7)_{2}^{∧}. So we choose a map from BN_{2}^{∧} to B(SU(2)^{3}/ ± 1 GL(2, _{2}))_{2}^{∧} (the classifying space of the “normalizer” of ( /2)^{2} in DI(4)) and a homotopy equivalence from BX_{2}^{∧} to the classifying space of the “normalizer” of ( /2)^{4} in DI(4), in such a way that the diagrams obtained by restricting to intersections commutes up to homotopy.

Finally, in order to obtain a map of classifying spaces, it is necessary to lift to strictly commuting maps. There is an obstruction theory for doing this, and it turns out that the obstruction lies in a group isomorphic to /2. Fortunately, there is just enough room for maneuver to change this obstruction to zero, and so a map of classifying spaces is obtained.

8 Some nonexistent finite simple groups

In his 1974 paper [27], Ron Solomon considered the problem of classifying finite simple groups in which the Sylow 2-subgroups are isomorphic to those of Conway’s group Co_{3}. In this paper, he proved that Co_{3} is the only such group. In the process of proving this, he was forced to examine a configuration in which the centralizer of an involution is isomorphic to Spin(7, _{q}) (q an odd prime power), possibly with some decoration of odd order at the top and bottom. In fact, only the case where q is congruent to 3 or 5 (mod 8) was relevant to the problem he was trying to solve, because for other values of q the Sylow 2-subgroup is larger than that of Co_{3}. However, we shall be interested in all odd prime powers q.

The final result of Solomon’s analysis was that if G is a finite group with an involution z satisfying

O^{2′}C_{G}(z)/(O_{2′}C_{G}(z) ∩ O^{2′}C_{G}(z)) ≅ Spin(7, _{q})

then z ∈ Z^{∗}(G) (that is, zO_{2′}(G) ∈ Z(G/O_{2′}(G))). In particular, G is not simple. In order to reach this conclusion, he was obliged to examine a configuration which appeared to be consistent at the prime two, but which gave rise to a contradiction upon examining a Sylow p-subgroup, where p is the prime of which q is a power.

Despite the nonexistence of Solomon’s groups, they have 2-completed classifying spaces. Namely, Dwyer and Wilkerson [14] show that for each 2-adic unit q, there is a (unique up to homotopy) self homotopy equivalence Ψ^{q} : BDI(4) → BDI(4) whose effect on degree 2i integral cohomology is multiplication by q^{i}. In fact, they prove that the group of homotopy classes of self equivalences inducing the identity on mod two cohomology is precisely the group of such operations Ψ^{q}; since minus the identity is in the Weyl group, Ψ^{q} is indistinguishable from Ψ^{−q}, so the group is isomorphic to the 2-adic units modulo ±1. For an odd integer q, write BSol(q) (Sol for Solomon) for the space defined by the following homotopy pullback diagram:

BSol(q) ⟶  BDI(4)
↓  ↓Δ
BDI(4) ⟶^{id×Ψ^{q}} BDI(4) × BDI(4).

The following diagram of maps of classifying spaces commutes up to homotopy:

B( /2 × M_{12}) ⟶ BCo_{3}
↓  ↓
B( /2 × G_{2}(q)) → BSpin(7, _{q}) → BSol(q)
↓  ↓  ↓
B( /2 × G_{2}) → BSpin(7) → BDI(4).

Using the Eilenberg–Moore spectral sequence on the pullback diagram defining BSol(q), one can easily deduce that the Poincaré series for the cohomology is given by

∑_{i=0}^{∞} dim_{_{2}} H^{i}(BSol(q), _{2}) = \frac{(1 + t^{7})(1 + t^{11})(1 + t^{13})(1 + t^{14})}{(1 − t^{8})(1 − t^{12})(1 − t^{14})(1 − t^{15})}.

We have maps H^{∗}(BDI(4), _{2}) → H^{∗}(BSol(q), _{2}) → H^{∗}(B( /2)^{4}, _{2})^{GL(4, _{2})} whose composite is an isomorphism. Using this, one can show that H^{∗}(BSol(q), _{2}) splits, as an algebra over the Steenrod algebra, as a tensor product of H^{∗}(BDI(4), _{2}) with the cohomology of the fiber of BSol(q) → BDI(4); namely with H^{∗}(DI(4), _{2}). The latter is generated by an element λ in degree 7 whose fourth power is zero, and two elements μ and ν of degrees 11 and 13 which square to zero. The action of the Steenrod algebra is determined by Sq^{4}(λ) = μ, Sq^{2}(μ) = ν, and Sq^{1}(ν) = λ^{2}.

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