Some Homotopy Equivalences for Sporadic Geometries

Stephen D. Smith ^{∗} Department of Mathematics (m/c 249) University of Illinois at Chicago 851 S. Morgan Chicago Illinois 60607-7045 USA

              e-mail: smiths@math.uic.edu

and Satoshi Yoshiara Division of Mathematical Sciences Osaka Kyoiku University Kashiwara, Osaka 582 JAPAN

         e-mail: yoshiara@cc.osaka-kyoiku.ac.jp

(drafted Apr95; revised for J. Algebra Aug 96)

Abstract

A previous work [RSY90] established projectivity of the reduced Lefschetz modules of certain sporadic group geometries, and the present paper continues that work in a wider context:

Recent developments in sporadic-group cohomology include some applications of [RSY90], which in turn suggested treatment of a broader class of geometries. Recurring similarities in the proofs also led to a more unified treatment—establishing the stronger result of homotopy equivalence of the p-local geometry with the usual elementary poset A_{p}(G). One equivalence method proceeds by means of a new “closed set” in a standard technique of Quillen. It was further observed that the larger list of simple groups now treated essentially co-incides with those of characteristic p-type: suggesting another equivalence method via the poset B_{p}(G) of radical (or stubborn) p-subgroups. In particular, one finds that these sporadic groups satisfy an analogue of the Borel-Tits theorm—that normalizers of p-groups lie in simplex stabilizers. Still further intriguing coincidences remain to be explained.

∗Partially supported by NSA grant MDA 904-93-H-3039.

This work is a somewhat unanticipated continuation of earlier joint work of Ryba-Smith-Yoshiara [RSY90]. The paper is organized on the following outline:

Section 0 summarizes the results in the form of a Table extending that of [RSY90]; a number of new geometries are treated, and in most cases the earlier result (namely projectivity of the reduced Lefschetz module) can be strengthened to homotopy equivalence of the standard p-local geometry for G with the Quillen elementary poset A_{p}(G). Section 1 gives details of the new equivalence methods, including also some history of developments since the earlier paper (especially the interaction with research in group cohomology). The subsequent sections are then devoted to verifying the sufficient conditions given in Section 1 in the various individual cases: the greater number occur for for p = 2 in Section 2, with cases for p odd treated in Section 3. Finally Section 4 discusses informally the current state of knowledge about the remaining (typically larger) sporadic geometries; for these, the present methods for projective modules and homotopy equivalences are known to fail—but recent techniques from homotopy theory strongly suggest that the local geometries should still provide the weaker result of a Webb-type alternating-sum decomposition for group cohomology.

0 Summary of projectivity and equivalence results

In the Table below, we will continue certain notational conventions from [RSY90]. For each row, G will denote a finite group, acting on a geometry Δ defined as a certain simplicial complex. (Details of the geometry will be given later, as each case is treated). A particular prime p is also indicated: we take coefficients for homology of Δ in the p-adic integers Z_{p}, and we establish the projectivity of the p-modular representation given by the reduced Lefschetz module L̃(Δ) of Δ (namely, the alternating sum of the chain groups of the complex). The Table indicates only the corresponding dimension of L̃(Δ), given by the reduced Euler characteristic χ̃(Δ); recall it is standard [Isa76, 2.8,2.10] that projectivity forces the p-part |G|_{p} of the group order to divide that dimension. In the fourth column, we indicate by ∼ or = that in most cases we are able to verify the stronger result of homotopy equivalence of Δ with the Quillen elementary complex A_{p}(G). In contrast to the earlier paper [RSY90], here we do not attempt to decompose the new modules into projective covers of individual irreducibles (though sometimes we add relevant remarks as we treat the cases later). Finally, a + in the first column indicates a “new” geometry, namely one not previously considered [RSY90]. For the old geometries, the proofs of stronger equivalence results in this paper are new.

new? G p ∼ A_{p}? χ̃(Δ)
A_{7} 2 No 2^{3}7
A_{7} 2 No 2^{6}
+  L_{3}(3) 2 = 2^{5}11
U_{4}(3) 2 ∼ 2^{7}31
G_{2}(3) 2 No 2^{7}85
G_{2}(3) 2 ∼ 2^{6}181
+  M_{11} 2 = 2^{4}31
+  M_{11} 2 No 2^{6}5
M_{22} 2 ∼ 2^{8}7
+  M_{23} 2 ∼ 2^{7}421
+  M_{23} 2 No 2^{10}21
M_{24} 2 ∼ 2^{10}21
McL 2 No 2^{8}521
McL 2 ∼ 2^{7}7483
+  J_{1} 2 ∼ 2^{3}601
J_{3} 2 ∼ 2^{8}511
+  J_{4} 2 ∼ 2^{21}2520315
+  Co2 2 ∼ 2^{19}77
+  T h 2 ∼ 2^{18}28729
L_{3}(4) 3 No 3^{2}25
U_{5}(2) 3 ∼ 3^{6}7
M_{11} 3 No 3^{2}5
+  Ru 3 = 3^{3}36281363
+  J_{4} 3 = 3^{3}2789248600017427
+  ON 3 ∼ 3^{4}1755889
McL 3 No 3^{6}106
McL 3 ∼ 3^{6}169
+  Ly 3 ∼ 3^{8}80967584
Ly 5 No 5^{6}7065863
Ly 5 ∼ 5^{6}1769293
+  T h 5 = 5^{3}241989183701
ON 7 No 7^{3}162487

Table: Projective modules L̃(Δ), extended from Table I of [RSY90]

We mention the intersection of these results with other work known to us: For the odd-p cases Ru, J_{4}, T h and ON(p = 7), the group cohomology has been described in Tezuka-Yagita [TY96, 4.1]—though those authors did not refer to projectivity. The homotopy equivalence for M_{24} was first established in unpublished work of Ronan (mid-1980s).

The remainder of this introductory Summary is devoted to certain striking features that emerge from further examination of the Table.

The inclusion of the new geometries was motivated (as we will discuss later) by developments in group cohomology. However, there is a purely group-theoretic viewpoint suggesting that this expanded Table is more natural than that of [RSY90], namely that the resulting sublist of groups with Δ ∼ A_{p}(G) essentially coincides with the class of simple groups of characteristic p-type, familiar from the classification of finite simple groups. We discuss this in further detail:

First we recall that the “actual” characteristic-p groups—namely the groups G of Lie type defined over a field of characteristic p—should be regarded as implicitly belonging in the Table: for their natural p-local geometry Δ is given by the Tits building, and Quillen’s result [Qui78, Thm 3.1] establishes the equivalence of Δ with A_{p}(G).

Now the general definition of characteristic p-type is motivated by a particular property of the groups G of Lie type: the Borel-Tits theorem ([BT71], see also the discussion in [Ben91, Thm 6.8.4]) states that for Q a non-trivial p-subgroup of Lie-type G, the normalizer N_{G}(Q) is contained in some parabolic subgroup of G. This parabolic is the stabilizer G_{σ} for some simplex σ of the building Δ, with the property that F^{∗}(G_{σ}) = O_{p}(G_{σ}). ^{1} It follows in a standard way [Asc86a, 31.16] that for the p-local N ≤ G_{σ} we similarly get F^{∗}(N) = O_{p}(N).

So we say (cf. [Asc86a, p.260]) a group G is of characteristic p-type if every p-local subgroup N := N_{G}(Q) for Q ≠ 1 satisfies F^{∗}(N) = O_{p}(N). ^{2} It is also standard (see [Asc86a, 31.16]) that it suffices to require this condition only for N given by the centralizers of elements of order p. From the classification one knows that there are few simple-group examples beyond the Lie-type groups in characteristic p; but a number of sporadic groups do arise in this way—especially for p = 2.

We will now be able to describe the above-mentioned conincidence somewhat more precisely. We will use the expression “∼-subTable” as an abbreviation for the sublist in the Table with ∼ (or =) in the fourth column, together with the Lie-type groups in characteristic p; that is, ∼-subTable gives the pairs (G, p) for which the usual p-local geometry Δ for G is homotopy equivalent to A_{p}(G).

Observation 1: The ∼-subTable consists essentially of the groups of characteristic-p type.

We need to say “essentially ”, since ∼-subTable examples such as J_{1} and McL for p = 2 have non-constrained involution centralizers, and so are not in fact of characteristic 2-type.

We use the term observation in the sense of experimental observation: since the coincidence originally was observed a posteriori, from verification of the homotopy equivalences in the various individual cases. It appears that our recent methods focusing on the poset B_{p}(G) may begin to offer a more uniform explanation; we discuss the method in detail in the following section, but it seems natural to single out one particular aspect in the present discussion.

1Indeed this is given by the unipotent radical of G_{σ}, which in our later geometric notation appears as O_{p}(K_{σ}), where K_{σ} denotes the kernel of the action of G_{σ} on the residue (or link) of σ in Δ.

2In the literature sometimes one also requires a lower bound on the p-rank. The condition F^{∗}(N) = O_{p}(N)

is the special case for O_{p′}(N) = 1 of the more general property that N should be p-constrained, which we discuss in the final section.

We can re-phrase our statement of the Borel-Tits theorem via a condition emphasizing the role of the geometry Δ:

(BT/Δ): Each p-local N_{G}(Q) lies in a stabilizer G_{σ} for some simplex σ ∈ Δ.

A second striking feature of our equivalence results, emerging prominently in the separate proofs, is that the more general groups of characteristic p-type exhibit the same behavior for their geometries Δ:

Observation 2: The ∼-subTable pairs (G, Δ) satisfy the Borel-Tits property (BT/Δ).

This coincidence is perhaps not so surprising when viewed in the light of Observation 1: for in characteristic-p type, maximal locals will tend to appear as stabilizers G_{v} of vertices in Δ. However, cases such as M_{23} below indicate that maximal locals need not always be maximal subgroups, so the connection is not a tautology.

In the following section, we indicate some steps toward a more unified approach to the coincidence in Observation 2, essentially via an N(Q)-equivariant collapsing of the fixed subcomplex Δ^{Q}. It appears that this geometrically-explicit view has not previously been taken in the literature, even for the standard Borel-Tits theorem for Lie-type groups. The first author intends to develop this direction in a future work (namely an equivariant version of Quillen’s ordinary-contractibility proof for Δ^{Q} [Qui78, 3.1(b)] based on the Solomon-Tits argument).

In our final discussion in Section 4, we will indicate how Observations 1 and 2 appear to generalize to other groups which have local geometries, but which are not of characteristic-p type.

1 Group cohomology and homotopy equivalence

The purpose of this section is twofold: we want to view our earlier work [RSY90] in an appropriate historical context—which leads naturally to the development of our new homotopy equivalence methods, to be applied in the later sections.

Subgroup complexes, projectivity, decompositions of cohomology

We continue the notation of the introductory section, for dealing with action of finite G on a complex Δ.

Brown [Bro75] and Quillen [Qui78] in the 1970s established fundamental topological properties, for Δ given by the order complex of the poset S_{p}(G) of all non-trivial p-subgroups, and especially the elementary subposet A_{p}(G)—where G is any finite group. They were partly motivated by the work of Tits [Tit74] where Δ is the building for simple G of Lie type; in another direction, Buekenhout [Bue79] and others (e.g. [RS80]) formulated finite geometries Δ for other simple G.

Various authors studied further connections between these topological and geometric viewpoints. We mention particularly a result of Webb (following Brown-Quillen [Qui78, Cor 4.3] for Δ = A_{p}(G)) showing that projectivity of the reduced Lefschetz module for more general Δ follows from a local-contractibility condition. ^{3} We had stated a version of Webb’s result as [RSY90, Thm 1.4], and for convenience re-state it here:

Theorem 1.1 (Thm A′ in Webb [Web87]) If for all subgroups P of order p in G the fixed subcomplex Δ^{P} is contractible, then L̃(Δ) is a projective (virtual) module.

The main aim of [RSY90] was to examine the known sporadic geometries, and prove or disprove, the contractibility condition—thus deciding in which sporadic cases L̃(Δ) was projective (since the converse essentially holds [Web91, 2.7.4]). Our interest was primarily representation-theoretic, and we did not there follow another direction of Webb’s work, namely his alternating-sum formula for group cohomology; Webb recalls that projective L̃(Δ) is certainly acyclic; that is, application of G-module cohomology functors yields H_{G}^{∗}(L̃(Δ)) = 0. Then the Eckmann-Shapiro lemma translates module cohomology into group cohomology of the simplex stabilizers G_{σ}, resulting in:

Theorem 1.2 (Thm A in Webb [Web87]) For G and Δ satisfying the hypothesis of 1.1,

H^{∗}(G)_{p} = ∑_{σ∈Δ/G} H^{∗}(G_{σ})_{p}.

Indeed, this conclusion holds more generally for L̃(Δ) acyclic.

When [RSY90] was written (around 1988), this formula had already been applied in the cohomologcal literature to describe some fairly small sporadic groups (see e.g. [Web87, Sec 7]). Since around 1990, there has been considerable progress on larger sporadic groups. In particular, the work of Adem and Milgram on M_{22} [AM95a, p 389] [AM94, p 269] and on McL [AM96, Lemma 1.1(2)] quotes the corresponding cases in [RSY90] in order to obtain the decomposition 1.2. Indeed, alternating sums are also prominent in other recent cohomological work, when L̃(Δ) is acyclic but not projective—a topic we will return to in the final section of this paper.

So our work in the present paper is at least partly motivated by possible cohomological applications. Conversely we emphasize that cohomological developments have directly benefited our work—notably by suggesting a broader class of geometries treated beyond those in [RSY90]. In particular, Milgram’s determination of the cohomology of M_{23} [Mil93] [AM94, p 271] demonstrated close analogies with the earlier (geometric) case of M_{22}; and with this clue, it was not difficult for us to find an appropriate geometry for M_{23} satisfying 1.1 and 1.2. Generalizing from this case, it seemed natural to relax the restriction in such sources as [RS80] that simplex stabilizers be p-local subgroups, instead considering also other possible maximal subgroups; or alternatively, using as simplex stabilizer p-local subgroups which are not required to be maximal subgroups. Consequently the first draft of this paper added to [RSY90] a list of further geometries Δ (denoted with a + in our Table) for which the contractibility condition of 1.1 was also verified. We have tried to discuss the wider class of geometries reasonably comprehensively (though less formally than in [RSY90])—in the sense that that there are many well-known geometries which do not appear in our Table, and typically we have verified that those fail to satisfy the properties of projectivity and homotopy equivalence.

3Webb’s condition is visible earlier Brown [Bro74, Sec 6,Lemma (ii)].

Of course, future developments might motivate analysis of yet wider classes of geometries.

Equivalence with A_{p}(G) via closed sets in products

A closer analysis of similarities in our contractibility proofs led to the coincidences we eventually formulated as the homotopy equivalences underlying Observation 1.

One explanation of those similarities had long been suspected—the analogue of Quillen’s building result [Qui78, Thm 3.1], namely that these sporadic Δ might be homotopy equivalent with the corresponding A_{p}(G). So far as we know, the only positive result in this direction had been obtained by Ronan (unpublished, around 1985) for the 2-local geometry of M_{24}. However, we observed that our contractibility proofs via 1.1 could be applied in a method of Quillen, to establish in most cases at least the weaker result of homology equivalence:

We now recall Quillen’s technique of “closed sets in products”, as specialized to our present notation. We consider the Cartesian product A_{p}(G)×Δ of posets, ^{4} and say a subset R is closed ^{5} if: whenever (P, σ) ∈ R with Q ≤ P and τ ⊆ σ, we also have (Q, τ) ∈ R. We define the fibers of the two projections, namely

\begin{aligned}R_{P} & := \{σ ∈ Δ : (P, σ) ∈ R\} \\ R_{σ} & := \{P ∈ A_{p}(G) : (P, σ) ∈ R\} \end{aligned}

Then our special case of Quillen’s result ^{6} takes the form:

Theorem 1.3 (Cor 1.8 in Quillen [Qui78]) Suppose R is closed, and all R_{P} and R_{σ} are contractible. Then A_{p}(G) and Δ are homotopy equivalent.

In fact, it seems that in most applications of this result in the literature, the subset R has the specific structure of “stabilizing pairs”: define S := {(P, σ) : P ≤ G_{σ}}. This definition guarantees the property of closure, since Q ≤ P ≤ G_{σ} ≤ G_{τ}, ^{7} so that also (Q, τ) ∈ S.

4That is, regard Δ as the poset of its simplices under the face relation.

5or an order-ideal, in the combinatorial literature.

6We remark that Thévenaz and Webb in [TW91] do not explicitly state an equivariant version of this result; however it is straightforward to formulate such a version based on their equivariant version of the underlying result [Qui78, Thm 1.6] of which [Qui78, Cor 1.8] is a corollary.

7The final containment assumes that action of G on Δ is admissible, namely that G_{σ} stabilizes all faces of σ. It is standard that we can always obtain this by passing to a barycentric subdivision.

Notice furthermore that for S the fibers have a very natural interpretation: namely, S_{P} is the fixed subcomplex Δ^{P}, and S_{σ} is the poset A_{p}(G_{σ}).

Now turning attention to the particular geometries in our list: we see they are usually “fully” p-local, in the sense that for all simplices σ we have a non-trivial normal p-subgroup: O_{p}(G_{σ}) ≠ 1; so in these cases we get contractibility of S_{σ} = A_{p}(G_{σ}) by Quillen’s standard result [Qui78, Prop 2.4]. On the other hand, our arguments verifying 1.1 only check contractibility of S_{P} = Δ^{P} for those P which have order exactly p. Now any larger-order Q ∈ A_{p}(G) certainly contains such a P, and contractibility has the consequence that Δ^{P} is mod-p acyclic; so a standard application of the P. A. Smith theorem (just as in Webb [Web87, p.148]) guarantees that Δ^{Q} is also mod-p acyclic. So for our fully p-local cases, if we replace “contractible” by “mod-p acyclic” we get the hypotheses of the analogue ^{8} of Quillen’s result: where the conlusion “homotopy equivalent” is replaced by “homology equivalent”. This shows that Δ at least has the same mod-p homology as A_{p}(G), preliminary evidence that homotopy equivalence should hold.

And indeed that stronger result was soon established, via a different modification of Quillen’s technique 1.3: using a (seemingly new) closed set I—which had been essentially implicit in our original contractibility proofs for various Δ^{P}. Those proofs almost always took the form of a series of applications of a standard homological lemma, which we had stated as [RSY90, Lemma 2.1]: if the residue (i.e. link) of a vertex is contractible, then removal of that vertex is a homotopy equivalence. This allowed us to reduce the original Δ^{P} to the (full) subcomplex on vertices which we called “Res-fixed” in [RSY90]: namely those vertices v for which we have not just P ≤ G_{v}, but in fact P ≤ K_{v}—where K_{v} denotes the kernel of the action of G_{v} on the residue of v.

In the present paper, it will be convenient to extend this notation of K_{v} in several ways. For an arbitrary simplex σ, we define K_{σ} to be the kernel of G_{σ} on Res(σ). Clearly K_{σ} contains the product of the kernels K_{v} of the various vertices v ∈ σ; indeed we can usually check that it equals this product, as expected by our analogy of stabilizers with parabolic subgroups of Lie-type groups. Following this same analogy, the kernel K_{σ} is ordinarily an extension of a normal Sylow p-group O_{p}(K_{σ}) (which we will denote by U_{σ}) by a p′-group; again U_{σ} is typically the product of the U_{v} for v ∈ σ. We remark also that for p = 2 (particularly when geometric parameters correspond to the field F_{2}), we usually get U_{v} = K_{v}.

Notice that, under the assumption that U_{v} is a normal Sylow p-group of K_{v}, the above inclusion of the p-group P ≤ K_{v} actually gives a containment P ≤ U_{v} of p-groups. This suggests a way of incorporating our earlier “Res-fixed” observations into a complex of pgroups.

Motivated by this observation, we will define for each simplex σ ∈ Δ a certain “intersection” subgroup I_{σ}. We begin with vertices v, ordinarily taking I_{v} := U_{v}, the normal p-group of the kernel K_{v} of the stabilizer G_{v}; but there will also be some cases when proofs are simplified by taking I_{v} to be some natural proper subgroup of U_{v}. For simplices of larger dimension, we then use the intersection ^{9} of vertex groups:

8A standard alternative to Quillen’s proof (e.g. Walker [Wal81, Sec 3]) proceeds via the Contractible Carrier theorem, so the weaker analogue follows using the Acyclic Carrier Theorem.

I_{σ} := ∩I_{v} over vertices v ∈ σ .

We define a corresponding subset of the Cartesian product by:

I := {(P, σ) : P ≤ I_{σ}} .

Notice that I is automatically closed, but for a different reason than S was: we have Q ≤ P ≤ I_{σ} ≤ I_{τ}—since the latter intersection is over a subset of the vertices of σ; giving (Q, τ) also in I. And again the fibers of the projections have natural interpretations:

I_{σ} = A_{p}(I_{σ})
I_{P} = the subcomplex Res-fixed by P in [RSY90].

(Indeed I_{P} is even smaller if we have chosen some I_{v} < U_{v}.) Now just ss in our earlier discussion of the usual stabilizing-pairs set S, the p-local nature of the geometries will typically give O_{p}(I_{σ}) > 1, hence contractibility follows by [Qui78, Prop 2.4]. So via 1.3 we obtain a sufficient condition for equivalence:

Proposition 1.4 If all O_{p}(I_{σ}) > 1, and all I_{P} are contractible, then A_{p}(G) and Δ are homotopy equivalent.

By comparison with the more ad-hoc techniques of [RSY90], we feel the equivalence method 1.4 usually gives a better insight into common features of the various local geometries Δ. So we will apply it by preference whenever it is practical—that is, whenever the elementary poset A_{p}(G) is reasonably easy to describe. This will cease to be the case as the size of G becomes really large; and then our later, more inductive method for describing B_{p}(G) will be more efficient.

We make a few general remarks about the application of 1.4. We can start our proofs at the Res-fixed subcomplex I_{P}, in contrast to [RSY90]—where we had to reduce from Δ^{P} down to it. This represents a very considerable saving of casework.

Furthermore the proofs in [RSY90] already establish the contractibility of this subcomplex for those P of order exactly p. We do now have to consider I_{Q} for arbitrarily large elementary Q; but Q contains some P of order p, so that I_{Q} is contained in I_{P} —of known, contractible structure. Typically it is straightforward to exhibit the contractibility of the subcomplex I_{Q}. (Unfortunately it need not be exactly the fixed subcomplex (I_{P})^{Q}, so that P. A. Smith-type approaches are not available).

In particular, note that every elementary Q should be contained in some I_{v} (typically in some vertex kernel U_{v})—otherwise we could get I_{Q} empty, hence not contractible. Ordinarily we will check this necessary condition at the start of our proofs; or if it fails, possibly replace A_{p}(G) by some larger equivalent poset for which we can check the needed containment.

We close this subsection with some technical results, which are used for the contractibility proofs in the method of 1.4.

9Readers may note that this condition means that A_{p}(G) is covered by the cones below vertex kernels—so that our intersections I_{σ} are closely related to the nerve of that covering; but note that we are allowing not arbitrary intersections, but only those for incident vertices (hence of different types).

For convenience we recall results on homotopy equivalence among certain standard posets. The Brown poset S_{p}(G) consists of all non-trivial p-subgroups (not necessarily elementary or even abelian); its equivalence with A_{p}(G) was shown by Quillen [Qui78, Prop 2.1]. Later in this section we will emphasize the poset B_{p}(G) of radical p-groups, those P satisfying P = O_{p}(N_{G}(P)); equivalence with the above posets was shown by Bouc [Bou84, Cor,p.50]. Probably a less known subposet of A_{p}(G) is Z_{p}(G)—namely those elementaries P satisfying P = Ω_{1}O_{p}Z(C_{G}(P)); Benson observes equivalence as [Ben91, Exer,p. 211].

We should also remark that Thévenaz and Webb [TW91] showed that most of the standard homotopy equivalences can be extended to G-equivariant maps—even to equivalences of G-homotopy (that is, restrictions to fixed-points of subgroups are also homotopy equivalences). As observed in [Web88, p. 358], this stronger equivalence shows that the reduced Lefschetz modules L̃ of the two geometries agree in the representation (Green) ring, and not just in the Grothendieck group (as in the original treatment of Quillen [Qui78, Cor 4.3]). Our proofs below explicitly address only ordinary homotopy equivalence, but the methods of [TW91] extend these to G-homotopy equivalences—occasionally we add further remarks where that extension may not be immediate.

For convenience, we reproduce from our earlier work [RSY90] a well-known condition for a strong deformation retraction, which may be used in many consecutive steps in our contractibility proofs:

Lemma 1.5 (Lemma 2.1 in [RSY90]) Suppose for a vertex v that Res_{Δ}(v) is contractible. Then collapsing Δ to Δ \ Star(v) is a homotopy equivalence. □

The nature of our subcomplexes I_{P} often leads to a cone configuration, from which we can often conclude that suitable subcomplexes are also cones:

Lemma 1.6 Let P ≤ Q be elementary. Suppose I_{P} is a cone on some particular simplex σ_{P}. If I_{Q} contains σ_{P}, it must be a subcone on σ_{P}.

Proof: Our I-subcomplexes are full subcomplexes on their vertices. So the hypothesis says that I_{P} must have the form of a join σ_{P} ∗ R, where R is some full subcomplex of the residue of σ_{P}. And then if the full subcomplex I_{Q} contains σ_{P}, it can only have form σ_{P} ∗ R′, for R′ some full subcomplex of R. □

Finally we mention that implementing the method of 1.4 ordinarily also includes verification of Observation 2. (In particular this gives Step 1 in the method 1.8 of the next subsection—though in practice we reserve that later method for cases where Observation 2 is not so automatic).

Lemma 1.7 If in 1.4 the contractibilility of each I_{P} is N(P)-equivariant, then G satisfies (BT/Δ).

Proof: Given a p-group Q, set P := Ω_{1}(Z(Q)), so that N_{G}(Q) ≤ N_{G}(P). Now N(P) acts on I_{P}, and by the equivariant hypothesis this contracts to a simplex σ which is necessarily N(P)-fixed. Then N(Q) ≤ N(P) ≤ G_{σ}. □

We mention also that it suffices to verify (BT/Δ) just for those Q lying in B_{p}(G); this is well known, and follows from the construction in the original Borel-Tits proof: Set Q_{1} := O_{p}(N(Q)) so that N(Q) ≤ N(Q_{1}) then Q_{2} := O_{p}(N(Q_{1})), and so on. In finite G this must stabilize at some Q_{n} = Q_{n+1}, so that Q_{n} ∈ B_{p}(G); and we have N(Q) ≤ … ≤ N(Q_{n}). So it is enough to show N(Q_{n}) lies in some G_{σ}.

Equivalence via inductive determination of B_{p}(G)

To deal with equivalence for larger groups G, where the members of A_{p}(G) may be too numerous to classify easily, it is usually more convenient to determine the poset B_{p}(G). We recall this poset is equivalent to S_{p}(G) (hence to A_{p}(G)) by the standard argument of Bouc, indeed equivariantly as noted by Thévenaz and Webb [TW91].

When G is of Lie type in characteristic p, it is a standard consequence of the Borel-Tits theorem (e.g. [Ben91, Thm 6.8.4]) that B_{p}(G) consists precisely of the unipotent radicals of parabolic subgroups—which are the stabilizers of simplices in the standard p-local geometry given by the building. For sporadic G, in contrast, it is not necessarily so apparent that the radical subgroups Q ∈ B_{p}(G) should usually coincide with the U_{σ} (i.e. the normal Sylow p-groups of kernels, as noted earlier) of the stabilizers G_{σ} of simplices in the p-local geometry Δ. However, for G of characteristic p-type (the main objects of study in this paper) we have been able to verify that behavior by following a common (essentially inductive) procedure in the various individual cases:

Algorithm 1.8 To determine possibilities for Q ∈ B_{p}(G) for G of characteristic p-type: 1) Establish (BT/Δ): that N(Q) ≤ G_{σ} for some σ ∈ Δ. (Conclude Q ≥ U_{σ}).

2) Now in known \overline{G_{σ}} := G_{σ}/U_{σ}, determine possible Q̄ ∈ B_{p}(\overline{G_{σ}}).

We summarize how the method will be applied, justifying certain details.

For Step 1, usually fairly crude methods will suffice. Set N := N_{G}(Q). In our simple group G, we may suppose that N lies in some proper maximal subgroup M. Ordinarily the stabilizers G_{v} of vertices v ∈ Δ will be among the maximal subgroups, so if M = G_{v} then we are done; we may as well assume that M is some other maximal subgroup. Now the maximal local subgroups of all G in our Table have been determined, and are listed in the Atlas [C^{+}85], so we are reduced to considering N ≤ M for M in those lists. As our G is normally p-constrained, the case of M a p-local is typically already completed via G_{v} above. If the p-local N lies in a q-local M for q ≠ p, the quotient M/O_{q}(M) is often either essentially in a p-local above, or else a smaller quasisimple group treated as an earlier case in this paper, so that its p-locals are known “inductively” and can be embedded in those of G. So typically we are quickly reduced to the case of M itself an almost-simple group; again there is often inductive leverage, when F^{∗}(M) and its p-locals have been analyzed as an earlier case in the paper.

It should also be possible to verify (BT)/Δ without quoting the full determination of maximal subgroups M of G—by instead collapsing the fixed subcomplex Δ^{Q} in an N-equivariant way down to a suitable N-fixed simplex. Even for G of Lie type, Quillen’s contractibility argument for Δ^{Q} [Qui78, 3.1(b)] is not equivariant. However, his alternative argument for G of type GL_{n} is equivariant. And indeed on fixing any other specific Lie type, it is not hard to describe an explicit collapsing in terms of the action of Q on an appropriate “natural” module V for G; for example if G is symplectic, collapse to the radical of V^{Q}. In the same way, for sporadic G there is usually a small natural module V in terms of which we can describe the local geometry Δ and the fixed subcomplex Δ^{Q}—and then a collapse to a suitable σ fixed by N = N_{G}(Q) is typically apparent. In summary this would give a common approach to (BT)/Δ—but one still requiring considerable separate case analysis (seemingly even in the optimal situation of G of Lie type).

So now assume we have accomplished Step 1, so that we have N(Q) ≤ G_{v} for some vertex v. Passage to the more visibly inductive Step 2 is allowed by an elementary lemma:

Lemma 1.9

(1) If Q ∈ B_{p}(G) with N(Q) ≤ G_{σ}, then U_{σ} ≤ Q.

(2) If further \overline{Q} ≠ 1 in \overline{G_{σ}} = G_{σ}/U_{σ}, then also \overline{Q} ∈ B_{p}(\overline{G_{σ}}).

Proof: Set U := U_{σ}. As N := N_{G}(Q) lies in G_{σ}, it must normalize Q.N_{U}(Q). Thus Q.N_{U}(Q) ≤ O_{p}(N); so as Q ∈ B_{p}(G) we must have Q.N_{U}(Q) = Q. But the Dedekind modular law shows N_{UQ}(Q) = QN_{U}(Q). As Q is self-normalizing in the p-group QU, we must have Q = QU; so that U ≤ Q, establishing (1).

For (2), just observe that the pre-image of O_{p}(N̄) is a p-group normal in N, and since Q ∈ B_{p}(G) that could not exceed Q. So Q̄ = O_{p}(N̄), and N̄ = N\overline{G_{σ}}(Q̄), as required. □

We remark that for G of Lie type, the standard proof of the Borel-Tits theorem also embeds Q in the unipotent radical U_{σ}, so that part (1) shows any Q ∈ B_{p}(G) must be a unipotent radical. That unipotent radicals conversely lie in B_{p}(G) is standard from the Lie theory.

We turn to Step 2, with the goal of showing a member Q ∈ B_{p}(G) must actually equal the kernel of some simplex stabilizer of Δ. By part (1) of the Lemma we have Q ≥ U_{σ}, so that if equality holds we are done. So we may as well assume that U_{σ} < Q; and then part (2) of the Lemma reduces us to determining B_{p}(\overline{G_{σ}}). Typically this is already known “by induction”: for in most cases the quotient group \overline{G_{σ}} is either a Lie-type group in characteristic p, or a smaller group of characteristic p-type described earlier in the paper. In either case we have B_{p} given by kernels for simplices of the geometry Γ = Res_{Δ}(σ) for \overline{G_{σ}}. Thus inductively Q̄ = U_{τ} for some τ ∈ Γ; and then for the preimage we have Q = U_{σ∗τ}, namely the kernel for the simplex σ ∗ τ for Δ.

In some cases (notably those whose diagram contains a square node) the poset B_{p}(G) contains more groups than just the vertex kernels for Δ; but we can use elementary arguments like 1.5 to homotopically collapse B_{p}(G) down to Δ.

Again we conclude the subsection with some technical details that will be required for the inductive procedure of Step 2.

First, we want the analogue for B_{p} of Quillen’s result [Qui78, Prop 2.6] on direct products. the following is a consequence of that result along with Quillen’s standard equivalence of A_{p} with B_{p}, but a direct proof is instructive in comparison with our earlier argument for Lemma 1.9:

Lemma 1.10 B_{p}(G_{1} × G_{2}) is homotopy equivalent with the join B_{p}(G_{1}) ∗ B_{p}(G_{2}).

Proof: The rest of the proof in Quillen goes through, if we can show for Q ∈ B_{p}(G_{1} × G_{2}) that the projection Q_{1} := pr_{G_{1}}(Q) lies in B_{p}(G_{1}). So set R_{1} := O_{p}(N_{G_{1}}(Q_{1})); we must show R_{1} = Q_{1}. Certainly in G := G_{1} × G_{2} we see N_{G}(Q) must normalize Q_{1}, hence also R_{1} and then Q.N_{R_{1}×1}(Q). By the B_{p} assumption on Q we must have Q.N_{R_{1}×1}(Q) ≤ Q. However by the Dedekind modular law, we have N_{Q(R_{1}×1)}(Q) = Q.N_{R_{1}×1}(Q). Since Q is then selfnormalizing in the p-group Q(R_{1} × 1), we must have Q = Q(R_{1} × 1), so that R_{1} × 1 ≤ Q. It follows that R_{1} = Q_{1}, which in turns says that Q_{1} ∈ B_{p}(G_{1}). □

Various of our geometries involve as a residue the rank-2 geometry of points and triples for S_{5} described in [RS80]. Notice this is not the building-geometry of rank 1 for S_{5} regarded as Aut(P GL_{2}(4)), so the usual Borel-Tits theorem does not apply. So for later use we directly verify:

Lemma 1.11 The group S_{5} satisfies (BT/Δ) for its rank-2 geometry Δ.

Proof: It suffices to consider Q ∈ B_{2}(G). If Q is given by either the 4-group of an A_{4}, or a full dihedral Sylow 2-group, then N(Q) is contained in the stabilizer S_{4} of a point. The only other possibility for Q ∈ B_{2}(G) is generated by a transposition (1, 2); and then N(Q) ≅ Q × S_{3} is the stabilizer of a triple 345. □

2 Cases with p = 2

We recall that for most of these cases, the kernels K_{σ} are already 2-groups, and then we have U_{σ} = K_{σ}. So usually our initial description of the geometries will just give the kernel K_{σ}; but we can use it for U_{σ} where needed in the techniques of Section 1.

2.1 A_{7} revisited

There are no new results for this group; we refer back to the original projectivity proofs for the two geometries in [RSY90], and comment on the failure of the stronger property of equivalence.

We recall that the first geometry Δ is the C_{3}-diagram geometry of Neumaier [Neu84], and the second is Δ_{Π} obtained by truncation, namely removing the planes. Since we can compute that A_{2}(G) has different reduced Euler characteristic χ̃ = 176 = 2^{3}22, it is clear a priori that equivalence with these geometries cannot hold. But we can also see immediately that the method of 1.4 must fail, in that the geometries are not fully 2-local: the plane stabilizer G_{π} = L_{3}(2) and point stabilizer G_{p} = A_{6} are instead simple groups.

We also mention a known relationship of reduced Lefschetz modules, which so far seems to have no corresponding explanation at the level of geometries. First, an outer automorphism g ∈ S_{7} \ A_{7} conjugates Δ to another geometry g(Δ)g^{−1} with planes stabilized by a different conjugacy class of subgroups L_{3}(2). Second, the rank-2 truncation has the same Lefschetz module as the rank-3 restriction of the building for L_{4}(2) ≅ A_{8} ⊃ A_{7}. And there is a direct decomposition of modules:

L̃(A_{2}(A_{7})) = L̃(Δ) ⊕ L̃(g(Δ)g^{−1}) ⊕ L̃(Δ_{Π}).

In later cases we will find further suggestive interrelations of Lefschetz modules.

2.2 The 2-local geometry for L_{3}(3)

The characteristic-3 Lie-type group L_{3}(3) can be regarded as exhibiting sporadic behavior in characteristic 2. Due to its smallness, the only reasonable geometry Δ is given just by its Quillen complex A_{2}(G), so that projectivity of L̃(Δ) follows from Quillen [Qui78, Rk 4.4]. As there is no strictly-smaller local geometry, we did not consider this group in [RSY90]; we include it here for completeness. In particular, the Borel-Tits property (BT/Δ) follows from Δ = A_{2}(G) (since N(Q) ≤ N(P) for elementary P := Ω_{1}(Z(Q))). We note also that the group’s involution geometry (in the sense of many of the examples below) also turns out to coincide exactly with A_{2}(G)—as is the case for M_{11} later.

2.3 U_{4}(3) revisited

This fairly small case will afford a good introductory demonstration of the equivalence method 1.4. Though the arguments are rather elementary, this first time we will give them in fuller detail, with very similar arguments in later cases given in more abbreviated form.

We recall from [RSY90] (where projectivity was proved) that Δ is the near-hexagon B̃_{2}- diagram geometry described in Kantor [Kan81] and Ronan-Smith [RS80]. However, since our equivalence methods focus on elementary 2-subgroups (consisting of commuting involutions), it will be advantageous to replace the near-hexagon view by the involution-geometry viewpoint of large-extraspecial theory. A general reference for the approach is [Asc86b, Sec 17]; for the particular geometry of U_{4}(3), see for example Bardoe [Bar95]—the features we state below can be deduced from the suborbit diagram given as Fig. 1 of the latter paper. We begin by changing terminology for the vertices via the transformation:

{points,lines,quads} of [RSY90] ↦ {left-quads,points,right-quads} The new points can be identified with the involutions p ∈ G. ^{10} There are also new lines, given the 3 points in the intersection of an incident left quad L and right quad R: equivalently they are the involutions p, q, r = pq of an elementary 2^{2}, satisfying the further (symmetric) relation q ∈ O_{2}(C_{G}(p)). Collinearity also defines the edge relation in a graph on the involutions; and for this geometry it happens that commuting involutions have graph distance 1 or 2. The left and right quads can be identified with the sets of 15 points they contain; these involutions have pairwise graph distance 1 or 2, just as in Sp_{4}(2) quadrangles.

10In this section the prime previously denoted by p has been fixed as 2; so now it is convenient to use standard geometric notation of p for a point.

To begin the verification of 1.4, we first observe that the geometry is fully 2-local: The vertex kernels K_{v} are in fact 2-groups: for quads they are elementary K_{L} ≅ 2^{4} ≅ K_{R}, and for points they are extraspecial K_{p} ≅ 2_{+}^{1+4}. In view of this last, it is convenient to vary the usual method slightly: In defining I, we take I_{p} to be Z(K_{p}) or order 2, rather than K_{p} itself. (Compare with the later case of McL, where point kernels are already of order just 2). For larger simplices σ, the corresponding intersections are then also (non-trivial) elementary 2-groups: and I_{LR} ≅ 2^{2} and I_{Lp} = 2^{1} = I_{Rp} = I_{LRp}. So certainly we have all O_{2}(I_{σ}) > 1, hence all I_{σ} contractible, as required for 1.4.

What remains for 1.4 is to show for all elementary P that the complexes I_{P} are contractible. In particular, it is necessary that I_{P} be non-empty. Since these are full subcomplexes, they should then contain at least one vertex; that is, any P should lie in some vertex group I_{v}. For U_{4}(3), up to conjugacy any elementary does lie in a quad kernel I_{L} = K_{L} or I_{R} = K_{R} of order 2^{4}. This is well-known: it can be deduced from the graph above; or see e.g. Adem-Milgram [AM96, Lemma 1.1] and its references to Janko and the Atlas; or argue directly, using complete reducibility of 2-groups in the orthogonal covering group O_{6}^{−}(3) in characteristic 3. So we conclude that any I_{P} contains at least a quad.

Towards contractibility, we first quote the appropriate part of the proof in [RSY90]; but recall that it applies only for P of order exactly 2. In fact that proof begins with the full fixed subcomplex Δ^{P}; but proceeds via a series of reductions: In particular it reaches the subcomplex Res-fixed by P (there called Δ_{PLQ}^{P}) toward the end of the argument. Subsequently the non-central involutions of K_{p} (where the point p corresponds to the involution generating P) are removed, and the result is what we are presently calling I_{P}. Consequently the remaining contractibility proof, from that point onward, establishes the contractibility needed for P of order 2 in 1.4.

So we turn to larger elementaries Q. We recall as in [RSY90] that U_{4}(3) has just one conjugacy class of involutions (such facts are easily checked in the Atlas. So our elementary Q contains some P of order 2 as treated above, and then I_{Q} is a subcomplex of I_{P}. Consequently it will be useful to recall from [RSY90] the explicit structure of I_{P}: it is a cone p∗X, where X is the residue of p, namely the complete bipartite graph on 3 left quads L_{1}, L_{2}, L_{3} and 3 right quads R_{1}, R_{2}, R_{3}.

First we assume Q contains a pair of points p, r at graph distance at least 2—hence exactly 2, since we noted that involutions at distance ≥ 3 do not commute. From our preparatory remarks, we know that Q lies in some quad kernel, so that p and r lie in at least one quad. In C_{G}(p) = G_{p} we may check that kernels of a pair of quads over p of the same type (both left or both right) intersect only at p. Similarly we find that quad kernels of different types intersect in at most 3 involutions at mutual distance 1. So p and r in fact determine a unique common quad—say of left type, L: and since p, r are involutions of the elementary group Q, this says Q lies in a unique quad kernel K_{L}. That is, I_{Q} contains just the single quad L. Also by our choice I_{p} has order just 2, and so no point group I_{p} can contain Q, hence I_{Q} has no points. Thus I_{Q} consists just of the single vertex L, contractible.

So we are reduced to the case that the involutions in Q are pairwise at graph distance at most 1. A feature of the geometry is that it has no proper triangles: so a triple of points at mutual distance 1 determines just a single line {p, q, r} of the geometry, giving Q of order exactly 2^{2}. Such subgroups lie in at least one K_{L} and K_{R}, and inside either kernel all line groups are conjugate under the action of A_{6} in G_{L} or G_{R}. So by conjugacy we may assume Q is given by I_{LR} = K_{L} ∩ K_{R} for a left-right pair as above. As noted above, distinct left quads over the point p intersect in the single involution p; so L is the unique left quad whose kernel contains all three involutions of Q—and similarly R the unique such right quad. Finally Q has order 2^{2}, so cannot fit into any point group I_{p} of order 2. Thus I_{Q} is just the edge L ∗ R, contractible. This completes our proofs of contractibility of the I_{Q}. By 1.4, we conclude that Δ is homotopy equivalent to A_{2}(G).

It would also be elementary to describe B_{2}(G) via 1.8. Instead we will simply note that the Borel-Tits property (BT/Δ) holds via 1.7: For we can check that our above arguments collapsing I_{P} are in fact N(P)-equivariant. Or in a more elementary view, an N(P)-fixed simplex is readily apparent from uniqueness considerations: For |P | ≥ 2^{2} we have P either contained in a unique left or right quad kernel, or equal to the intersection of a unique left-right pair. Otherwise |P | = 2, so that the involution centralizer N(P) is a point-vertex stabilizer.

We conclude by remarking that the mod-2 cohomology of U_{4}(3) has been determined by Adem-Milgram [AM96, Secs 3,4]. In view of the original contractibility proof in [RSY90], one sees that the alternating-sum decomposition of cohomology in 1.2 is available (though not required by those authors). And indeed the present equivalence proof gives a deeper explanation of why Δ can be used in place of A_{2}(G) in that formula.

2.4 G_{2}(3) revisited

We recall from [RSY90], where projectivity was proved, that the first case for G_{2}(3) is the G̃_{2}-diagram geometry discovered by Cooperstein, and described in [AS83]. As in the previous case of U_{4}(3), it is convenient to change to the involution-geometry viewpoint; so we transform terminology for vertices by:

{points,lines,planes} of [RSY90] ↦ {hexagons,points,planes} .

In fact, many local features of this geometry are just as for U_{4}(3), including: Again G has one class of involutions, corresponding to points. There are new-lines of 3 points, now given by the intersection of a plane kernel 2^{3} and a hexagon stabilizer G_{H} ≅ G_{2}(2); again the corresponding involutions satisfy q ∈ O_{2}(C_{G}(p)). Commuting involutions have distance 1 or 2 in the collinearity graph. The planes are subsets of 7 points, whose collinearity subgraph is that of the projective plane over F_{2}. The second geometry is the truncation obtained by removing the hexagons from the first.

The Table shows that the geometries have different Euler characteristics, so certainly homotopy equivalence with A_{2}(G) cannot hold for both. In fact the first geometry is not fully 2-local, since we have almost-simple hexagon stabilizer G_{H} ≅ G_{2}(2), so that the hexagon kernel K_{H} is trivial. This at least prevents application of our method 1.4; so henceforth we let Δ be the second geometry, consisting just of the points and planes of the first. Once we have successfully applied 1.4 to Δ to get equivalence, failure of equivalence for the first geometry follows.

The truncated geometry Δ is now fully 2-local: we have 2-groups for point kernel K_{p} ≅ 2_{+}^{1+4} and plane kernel K_{π} ≅ 2^{3}; with also I_{pπ} = K_{π}, so that all O_{2}(I_{σ}) > 1.

Again it is useful to check the necessary that condition that all I_{P} are non-empty; indeed we will show containment of P in some point kernel: Certainly we may fix some involution p ∈ P, so that P ≤ C_{G}(p) = G_{p} ≅ 2^{1+4}3^{2}2. In this centralizer, we check that elementaries have maximal order 2^{3}. If P already lies inside K_{p} ≅ 2^{1+4}, we are done. If not, P also contains some r at distance 2 from p; by the maximal order 2^{3} above, the involution r centralizes p and at most one other involution pair q, s = pq from K_{p}. By distance 2, such a further pair must occur; and furthermore one of them (say q) must be at distance 1 from both p and r (and indeed only one, since in this geometry triangles only occur inside planes). But now P is generated by p and q and possibly r, and so lies inside K_{q}—since that group is characterized by distance 1 from q.

For contractibility of I_{P}, again the case of P of order 2 is handled by the end of the corresponding proof for Δ^{P} in [RSY90]. Furthermore the structure of I_{P} is isomorphic to the truncation of the structure we saw for U_{4}(3): namely the cone p ∗ {π_{1}, π_{2}, π_{3}} for the 3 planes π_{i} containing p—together with 18 more involutions q at distance 1 from p, partitioned equally by incidence with the planes.

So we turn to larger elementary Q ⊃ P. Much as in the case of U_{4}(3), we first consider Q of order 2^{2}, generated by involutions p and r at distance 2. Then I_{Q} contains no planes, since plane kernels K_{π} have involutions at mutual distance 1. Furthermore since a point kernel K_{q} contains just the involutions at distance ≤ 1 from q, we see that the present Q is contained in K_{q} just for the unique point q at common distance 1 from p and r in our earlier analysis. Thus I_{Q} is just the point q, contractible. And for any Q′ ⊃ Q, our earlier argument that I_{Q′} must contain a point forces that point to be q, so that the standard inclusion I_{Q′} ⊆ I_{Q} must in fact be an equality, giving the desired contractibility.

Consequently we may now assume that involutions in Q lie at pairwise distance 1. Consider first Q of order 2^{2}, with 3 involutions forming a line. We saw in the partition above that these points lie in a unique plane π; as the points of π are the involutions of the kernel K_{π}, we see π is the unique plane of I_{Q}. Furthermore Q lies in K_{q} for all 7 points of π. We conclude I_{Q} is the cone on π with these 7 points. Finally for Q′ ⊃ Q, the distance-1 restriction (together with earlier maximal elementary size 2^{3}) means that the involutions of Q′ are just all 7 points of π. Again we find that the usual inclusion I_{Q′} ⊆ I_{Q} is an equality, giving the desired contractibility. This completes verification of 1.4, so we have homotopy equivalence.

As for the previous case of U_{4}(3), N(P)-equivariance of our contractibility arguments above also establishes (BT/Δ), using 1.7.

The mod-2 cohomology of G_{2}(3) (indeed of more general G_{2}(odd)) is described by Milgram [Mil95]. The group has the same Sylow 2-subgroup as M_{12}, for which the corresponding alternating sum (in a non-projective case) appears in the work of Adem-Maginnis-Milgram [AMM91, after Thm 4.17]. Further connections of M_{12} with G_{2}(odd) and with the compact Lie group G_{2} were considered by Milgram [Mil95] and by Benson-Wilkerson [BW95].

2.5 Some 2-local geometries for M_{11}

We had omitted M_{11} from consideration in [RSY90], since its 2-local geometries seemed too small to be of interest at that time. We include them now partly for completeness, but also because they provide further instances of connections with group cohomology and representation theory.

The first geometry for M_{11} in our Table is its involution geometry, in the sense of the previous subsections. Again the lines consist of 3 commuting involutions p, q, r = pq, but in this case they do not satisfy the relation q ∈ O_{2}(C_{G}(p)). In fact, the points and lines, regarded as groups of order 2 and 2^{2}, represent the only conjugacy classes of elementary abelian 2subgroups; so this geometry actually coincides with the Quillen complex A_{2}(G) (instead of being a proper subcomplex, as happens for most larger sporadic groups). In particular, the Borel-Tits property (BT/Δ) is now immediate from the definition of Δ. Correspondingly, projectivity of the reduced Lefschetz module follows from the work of Quillen [Qui78, Remark 4.4]. Alternatively, one can easily verify 1.1 explicitly in the geometry, using 1.5 as in [RSY90].

There is an intriguing relationship with the 2-local geometry of M_{12}. It is not quite an embedding of geometries: For though the points for M_{11} are also points for M_{12}, the lines of M_{12} do satisfy the usual condition q ∈ O_{2}(C_{G}(p)). Correspondingly the chain spaces of the complexes for the two groups are different; but in the respective alternating sums, cancellations leave exactly the same 496-dimensional module—i.e., that for M_{11} is just the restriction of that for M_{12}. (The equality of the characters can be checked easily using the Atlas . We know of no deeper explanation of this coincidence. However, this equality does lead to yet another proof of projectivity for M_{11}, starting at the level of M_{12}: If we let Γ denote the 2-local geometry of M_{12}, then our M_{11}-involution z is 2-central (i.e., central in a Sylow 2-subgroup) in M_{12}. It was known to us and others (e.g. Example 2 in Maginnis [Mag96]) that Γ^{z} is contractible. It then follows from Thévenaz’ extension [Thé87, Thm 2.1] of Webb’s result 1.1 that L̃(Γ) is projective relative to 2-subgroups of M_{12} of non-2-central type. (It cannot be fully projective for M_{12}, since the module dimension is not divisible by the 2-part 2^{6} of the group order). Since z represents the unique involution class for M_{11}, we see M_{11} has no 2-subgroups of non-2-central type. Consequently the restriction to M_{11} of the module for M_{12} must be projective relative to the identity—that is, projective.

It was in fact for the larger Mathieu group M_{23} that cohomological results first suggested consideration of geometries using maximal subgroups larger than 2-locals; but the idea is also of interest for M_{11}. We note that the above line stabilizer ≅ S_{4} is not maximal in M_{11}, since it lies in a larger subgroup M_{10}.

We are led to define a corresponding geometry Δ, the second entry in our Table for M_{11}. Again we start with involutions: since these act with cycle type 1^{3}2^{4} on the 11 letters permuted by M_{11}, we may identify the involutions with their fixed triples—and we will call the corresponding vertices of our geometry triples (rather than points). Since we are taking the other vertex-stabilizers to be subgroups M_{10}, we will call the other vertices letters (rather than lines). which can be identified with the 11 letters themselves, The letter-triple incidence is just containment of a 1-set in a 3-set. Since the Euler characteristic of our new Δ is different from that of A_{2}(G), we cannot have homotopy equivalence; however, the weaker conclusion of projectivity of L̃(Δ) is very easy to establish using 1.1:

Let P of order 2 be generated by involution p. Then the fixed subcomplex Δ^{P} contains just the 3 individual letters α_{1}, α_{2}, α_{3} fixed by the involution p; fixed triples include the above pointwise-fixed triple T_{P} = {α_{1}, α_{2}, α_{3}} as well as triples of the form {α_{i}, β, γ} where the involution p transposes β and γ. However, we see triples of the latter type have residue in Δ^{P} consisting just of the single letter α_{i}—i.e., not just contractible but already contracted to a point. We may by 1.5 homotopically remove such triples from Δ^{P}, reducing to the cone given by the join of the triple T_{P} with its three individual letters α_{i}, contractible. This completes the verification of 1.1, establishing projectivity for the second entry for M_{11} in our Table.

We note that the mod-2 cohomology of M_{11} was described by Webb [Web87, Sec 7.9], using the alternating-sum formula 1.2 for A_{2}(G). Several alternative treatments followed, e.g. by Benson-Carlson and by Adem-Milgram (see [AM94, p.252] for further reference).

2.6 M_{22} revisited

We recall from [RSY90], where projectivity was proved, that Δ is the geometry for M_{22} described in Ronan-Smith [RS80]; its vertices are called octads, hexads, and quintets— determined by intersecting the more usual octads and sextets of M_{24} (again see [RSY90] or [RS80]) with the 2 letters fixed by M_{22} from the 24 letters permuted by M_{24}. In contrast to most cases above, we will maintain this original viewpoint on the geometry; the involutiongeometry is less natural here, since involutions do not correspond directly with vertices.

To establish equivalence via 1.4, we first check that the geometry is fully 2-local: Vertex kernels are elementary 2-groups: K_{H} ≅ 2^{4} ≅ K_{Q} and K_{O} ≅ 2^{3}. And intersections for incident vertices are non-trivial: I_{QH} ≅ 2^{2} ≅ I_{QO} and I_{HO} ≅ 2^{1} ≅ I_{QHO}. In particular we have O_{2}(I_{σ}) > 1 for all σ.

It is also well-known (stated for example in [AM95a, p 389]) that the three vertex kernels above represent the classes of elementary 2-groups maximal under inclusion. For this reason, one could also give an explicit contraction of A_{2}(G) onto Δ, using techniques like 1.5. However, our method 1.4 will be quite efficient: The previous remark gets us started by establishing for any elementary P that I_{P} is non-empty.

We begin the contractibility proofs with the case of P of order 2; generated by an involution z (there is just one conjugacy class in M_{22}). As usual, we can quote ^{11} the end of the proof in [RSY90] to establish contractibility of I_{P}. For later use, we also recall that subcomplex is a cone:

H ∗ {O_{1}, O_{2}} ∗ {Q_{1}, Q_{2}, Q_{3}}

where z generates the intersection of K_{H} and the K_{O_{i}}, while the three quads Q_{j} give the residue of the edges H ∗ O_{i}.

We turn to elementaries Q of order at least 2^{2}, and as usual we may assume Q ⊃ P so that I_{Q} ⊆ I_{P}. We notice above that H is the unique hexad of I_{P}; if Q ≤ K_{H}, we see I_{Q} contains H, and so must be a subcone using 1.6.

By our earlier remark on maximal choices, we must still have Q in some K_{O_{i}} or K_{Q_{j}}. So we consider next the case that Q ≤ K_{O} (where O is one of the O_{i} above); but by the above, may assume that Q ≰ K_{H}, so that I_{Q} contains no hexads. We further restrict to the subcase where Q has order 2^{2} in K_{O}. All such subgroups are conjugate under the action of G_{O}/K_{O} ≅ L_{3}(2), so we may assume in fact that Q = K_{O} ∩ K_{Q_{j}} = I_{O,Q_{j}} for one of the j above. Furthermore the distinct K_{O_{i}} have intersection inside K_{H}, and we remarked above that K_{H}∩K_{O} has order 2, so we see that I_{Q} contains just the one octad O. From containment in I_{P}, we conclude I_{Q} must be a cone on O with possibly some quintets, contractible. And now in the other subcase, for Q < Q′ ≤ K_{O} with Q′ of order larger than 2^{2} (that is, Q′ = K_{O}), we see I_{Q′} contains O, and so must be a subcone by 1.6.

So we have reduced to the case that Q ≤ K_{Q_{j}} for some j, but Q is not contained in any K_{H} or K_{O}, so that I_{Q} consists only of quintets. Furthermore the distinct quintet kernels above intersect only in z: for the 2^{2}-groups given by their intersections with K_{H} must cover the 3-space z^{⊥} of K_{H} regarded as symplectic module for G_{H}/K_{H} ≅ Sp_{4}(2)′. So I_{Q} must consist of a unique quintet Q_{j}, hence is already contracted to a point. This completes the proofs of contractibility, and then by 1.4 of homotopy equivalence of Δ with A_{2}(G).

As usual we can check equivariance in the contractibility proofs above, to establish the Borel-Tits property (BT/Δ) using 1.7. It would also be fairly efficient to verify this property as we do in some larger groups below, by consideration of possible maximal subgroups given in the Atlas.

The mod-2 cohomology of M_{22} was determined by Adem and Milgram, and in particular they mention [AM95a, p 389] that Webb’s alternating-sum formula 1.2 is available in view of the earlier work [RSY90]. The present homotopy equivalence continues the theme of a deeper and more uniform explanation of such phenomena.

2.7 Some local geometries for M_{23}

As mentioned earlier, the cohomology of M_{23} was determined by Milgram [Mil93] [AM94, VIII.5]—using the above-cited work on M_{22}, but without reference at p = 2 to any subcomplex of A_{2}(G) or an alternating sum. However, similarities with M_{22}—such as the description of restriction to local subgroups [Mil93, Sec 3]—led us to to believe there should be a geometry satisfying the conclusions of [RSY90]. In fact, various geometries for M_{23} and p = 2 appear in the literature (e.g. [RS84] [Asc86b]); seemingly the following geometries have not appeared elsewhere, but they appear to be the most appropriate for alternating sums in cohomology.

11We take this opportunity to correct a minor misstatement in [RSY90]: the subgroup there called G_{H,O_{1},O_{2}} is actually of index 2 in C_{G}(z), since the latter can switch the octads O_{1} and O_{2}.

We observe first that the elementary group K_{O} for M_{22} above has the same normalizer of structure 2^{3}L_{3}(2) in M_{23}—but this normalizer is no longer maximal, since it is contained in a larger subgroup of structure A_{8} in M_{23}. The discrepancy is the reason that this group was not considered in the local+maximal viewpoint primarily used in our analysis in [RSY90]. It turns out that using either subgroup to define a vertex stabilizer gives a useful geometry.

So for the first geometry Δ mentioned in our Table, we simply extend that of M_{22} from the 2-local viewpoint: namely as vertices we take the conjugates of the elementary subgroups denoted by K_{H}, K_{O}, and K_{Q} in the previous subsection. The vertex stabilizers are then just the normalizers of these groups; we mentioned that for G = M_{23}, the group G_{O} is no larger than for M_{22}, but the other stabilizers do grow:

G_{H} ≅ 2^{4}A_{7} and G_{Q} ≅ 2^{4}Aut(GL_{2}(4)).

In particular, we notice for a quintet Q that our definition of K_{Q} as the kernel on the residue actually produces a group of order 2^{4}3; that is, the group denoted K_{Q} for M_{22} now appears as the normal Sylow 2-group U_{Q} of the natural kernel denoted by K_{Q} for M_{23}. So since here we have the unusual situation that U_{Q} < K_{Q}, we should specifically note that it is U_{Q} we use as I_{Q} in 1.4 below.

Now M_{22} is of odd index in M_{23}, so that the Sylow 2-subgroup does not grow; and it is standard (as noted in Milgram above) that again we have each elementary contained in some K_{v} (hence in U_{v}). Consequently just as in our remark for M_{22}, one can use 1.5 to give a straightforward contraction of A_{2}(G) onto Δ. And the proof of homotopy equivalence via 1.4 is also similar to the case of M_{22}; we will simply mention the minor differences. For M_{23} and P of order 2, the subcomplex I_{P} has a similar cone structure: namely the join of hexad H now with 7 pairs of octads and 7 quintets. These have the incidence pattern of a projective plane over F_{2}; and since each stabilizer is transitive on the involutions in its kernel, we see C_{G}(P) ≅ 2^{4}L_{3}(2) is transitive on these octads and quintets. Again kernels distinct octads over P = ⟨z⟩—as well as for distinct quintets over P —intersect just in that involution z. Now arguments for larger elementary subgroups Q are the same as for M_{22}. But note this time that the 7 four-groups K_{H} ∩ K_{Q_{j}} (as K_{H} is a 2-group this is K_{H} ∩ U_{Q_{j}}) must cover the full 4-space K_{H}, regarded as natural module for G_{H}/K_{H} ≅ A_{7} ⊂ L_{4}(2).

Equivariance in the above contractibility arguments via 1.7 establishes the Borel-Tits property (BT/Δ).

We want the second geometry Δ in the Table to have stabilizer corresponding to octad O given by a subgroup A_{8}. Now the analogy with M_{22} is from the viewpoint of the objects of the Steiner system: We regard M_{23} as the subgroup of M_{24} fixing just 1 of 24 letters; and in analogy with the hexads, octads, quintets of M_{22} we define heptads, octads, quintets: Heptads are just octads of the Steiner system for M_{24} containing the fixed letter (hence in effect 7subsets of 23 letters). Octads come from the remaining octads, those not containing the fixed letter (hence in effect 8-subsets of 23 letters). And (in contrast to the case of quintets for M_{22}) for the quintets we take all of the original sextets of the Steiner system; of course now we regard the quintet as consisting of a single 3-set (the original tetrad containing the fixed letter) and 5 further tetrads. And incidence among these vertex types is induced from that of M_{24}. The stabilizers G_{H} and G_{Q} are the local subgroups of the previous geometry, but now we have non-local G_{O} ≅ A_{8}.

In the Table we see that the new geometry gives Euler characterstic different from that of A_{2}(G), so that we cannot have homotopy equivalence. However, we can still establish projectivity of L̃(Δ) using 1.1 as in [RSY90]:

There is just one conjugacy class of involutions in M_{23}, so we fix such an involution z, and must check contractibility of Δ^{z}. Our z has cycle type 1^{7}2^{8} on 23 letters, and C_{G}(z) ≅ 2^{4} : L_{3}(2) is a subgroup of the stabilizer G_{H} of the heptad H defined by the 7 fixed letters of z. The argument will be a sequence of elementary collapses of Δ^{z} down to a subcomplex of the cone H ∗ Res(H). This will be a simpler version of the corresponding case for 2-central involutions of M_{24} in [RSY90].

We consider first a quintet Q ∈ Δ^{z}, so that z ∈ G_{Q}. The 5 heptads and 10 octads of R := Res(Q) may be identified (compare with the discussion of quintets for M_{22} in [RS80]) with the 5 letters and 10 triples of letters under action of the quotient group G_{Q}/K_{Q} ≅ S_{5} (recall here the kernel K_{Q} ≅ 2^{4} × 3 is slightly larger than a 2-group). As we noted before 1.11, the rank-2 geometry R is not the building for the Chevalley group S_{5} ≅ GL_{2}(4), so we cannot quote [Qui78, 3.1(b)] for contractibility of R^{i} for an involution i; but it is easy to check directly: up to conjugacy, i acts as (1, 2) or (1, 2)(3, 4) and R^{i} is a tree contracting to triple 345 or letter 5, respectively. Now if z ∈ G_{Q} \ K_{Q}, it must act as some such i, giving collapsible residue of Q in Δ^{z}; so by analogy with [RSY90, 2.2] (which is the special case of 1.5 for buildings), we may homotopically remove from Δ^{z} all quintets Q which are not Res-fixed by z—reducing to the subcomplex denoted by Δ_{Q}^{z} in the notation of [RSY90]. In particular, for any remaining quintet Q we have z ∈ K_{Q} ≅ 2^{4} × 3, so in fact z ∈ U_{Q} = O_{2}(K_{Q}) ≅ 2^{4}. Consequently the elementary group U_{Q} must lie in C(z) ⊂ G_{H}. Thus Q must be among the 35 quintets incident with H; indeed even among the subset Q_{1}, …, Q_{7} of those 35 satisfying U_{Q} in C(z).

Next consider some octad O ∈ Δ_{Q}^{z}, so that z ∈ G_{O}. If O is not incident with our heptad H for z above, then z must be a non-2-central involution in G_{O} ≅ A_{8} ≅ L_{4}(2). Let R now denote the residue of O in the full geometry Δ, viewed as the 2-local geometry for M_{24}. It can be identified with the truncation of the L_{4}(2)-building given by the 3-spaces and 2-spaces of a natural 4-space. The action of non-2-central z on this 4-space determines a certain 2-space X as both the commutator and centralizer; then X corresponds to the only Res-fixed quintet in R, so the intersection of R with Δ_{Q}^{z} is the single 2-space X, joined just to the three 3-spaces above X (the only 3-spaces normalized by z). This is contractible, so by 1.5 we may remove such O. Thus any remaining O lie among the 30 incident with H; and indeed must be among the 14 (in 7 pairs) of them normalized by z. Call this further-reduced subcomplex Δ_{QO}^{z}. (Note since we have G_{O} = A_{8} that K_{O} = 1, so that there are no Res-fixed octads; hence this notation differs slightly from the usual convention in [RSY90]).

We turn to heptads H′ ∈ Δ_{QO}^{z} other than H. From the permutation character of M_{23} on the 253 heptads in the Atlas, we see that z fixes exactly 29 heptads—namely H and the other 4 heptads in the above 7 quintets Q_{i} (since these quintets have H as their only common heptad). Furthermore the residue of H′ in Δ_{QO}^{z} is a cone, namely just one Q_{i}, with 3 of the 14 above octads which are incident with Q_{i}. As usual by 1.5 we may remove these H′ ≠ H.

What finally remains a cone, of form H * (the subcomplex on the 7 Q_{i} and the 14 octads above). So the contractibility proof for Δ^{z} is complete, giving L̃(Δ) projective.

We finish with some remarks similar to those relating M_{11} and M_{12} in the previous subsection. Again we find that our second geometry Δ is not quite embedded as a subgeometry of that for M_{24}, so the chain spaces for the two geometries are certainly different; but in the alternating sum, cancellations leave the same submodule—i.e., that for M_{24} restricts to that for M_{23}. This coincidence provides an even-easier alternative proof of projectivity than in the previous subsection: for the geometry Γ for M_{24} has L̃(Γ) projective directly by the work of [RSY90], so this module must restrict to a projective of the subgroup M_{23}. We showed further in [RSY90] that this M_{24}-module is the projective cover P (1792) of the 1792dimensional irreducible. From this it is easy to check, with the 2-modular character tables originally computed by James [Jam73] and now available in the Modular Atlas [JLPW81], that the restriction to M_{23} breaks up as 3(896 + \overline{896}) + P (220) + P ( ¯220).

2.8 M_{24} revisited

We recall from [RSY90], where projectivity was proved, that the 2-local geometry Δ for M_{24} is defined by the octads, trios and sextets of the Steiner system, as in Ronan-Smith [RS80].

As previously mentioned, Ronan (unpublished) first gave a proof of equivalence—in fact of Δ with the radical poset B_{2}(G); we do not know if that (rather large) poset has ever been explicitly described in the literature, but will include some remarks on the application of our method 1.8 related to B_{2}(G). However, our main aim will be to give a proof based on the method of 1.4 which is complete in the sense of specifiying all subgroups and containments; it is straightforward but tedious to check these structures, so for brevity we may omit some of the details of those verifications.

We also comment at the outset that the elementary 2-subgroups of M_{24} are sufficiently numerous that we must make further preliminary arguments before our standard method in 1.4 will apply; indeed the complications suggest that the size of M_{24} may represent the practical limits of that method.

As usual, we begin by checking locality. Here we have elementary 2-groups for octad and trio kernels: K_{O} ≅ 2^{4} and K_{T} ≅ 2^{6}; but the sextet kernel is slightly larger: K_{S} ≅ 2^{6}3, so as I_{S} we take elementary U_{S} = O_{2}(K_{S}) ≅ 2^{6}; for other vertices we have K_{v} = U_{v}, so either name describes I_{v}. With this change, all further intersections are elementary:

I_{OT} ≅ 2^{3}, I_{T S} ≅ 2^{4}, I_{OS} ≅ 2^{2} ≅ I_{OT S}.

In particular, we have all O_{2}(I_{σ}) > 1, hence all I_{σ} contractible.

But next we face an obstruction to our usual necessary condition, in that not every elementary P lies in some vertex kernel K_{v} (hence in U_{v}); this arises essentially because of the “square node” in the diagram, representing 2-locals not corresponding to vertices. For this reason, we will require an initial argument passing from A_{2}(G) to a homotopy-equivalent subposet whose members do lie in vertex kernels.

In view of [Ben91, Exer,p.211], we may initially replace A_{2}(G) with the subposet Z_{2}(G) of elementaries which contain all the order-p elements of the centers of their centralizers. The conjugacy classes of these can be explicitly determined. Because of length considerations, we will not attempt to provide details; we simply summarize the results in the following table. Here A, B denote the two classes of elements of order 2; with C, D, E, F (appropriately subscripted) denoting classes of orders 2^{2}, 2^{3}, 2^{4}, 2^{6}. (Note that classes of order 2^{5} do not satisfy the condition for Z_{2}). Finally for each representative, the remainder of its row indicates numbers of subgroups from the smaller classes. This information will serve to distinguish classes, and also we will use some of it later in contractibility proofs.

X  m(X) AB  C  D  E
A  1
B  1
C_{1}  2  3, 0
C_{2}  2  3, 0
C_{3}  2  1, 2
C_{4}  2  0, 3
C_{5}  2  0, 3
D_{1}  3  7, 0  3, 0, 0, 0, 0
D_{2}  3  3, 4  1, 0, 6, 0, 0
D_{3}  3  7, 0  0, 7, 0, 0, 0
D_{4}  3  3, 4  0, 1, 2, 0, 0
D_{5}  3  1, 6  0, 0, 3, 0, 4
E_{1}  4  9, 6  3, 3, 9, 2, 0  0, 0, 0, 0, 0
E_{2}  4  3, 12 1, 0, 18, 0, 16 0, 3, 0, 0, 12
E_{3}  4  15, 0  7, 0, 0, 0, 0  7, 0, 0, 0, 0
E_{4}  4  3, 12 0, 1, 6, 0, 16  0, 0, 0, 3, 4
E_{5} = K_{O}  4  15, 0 0, 35, 0, 0, 0 0, 0, 15, 0, 0
F_{1} = K_{T}  6  21, 42 7, 21, 63, 14, 0 0, 0, 3, 63, 0 7, 0, 0, 0, 0
F_{2} = U_{S}  6  45, 18 45, 15, 135, 6, 0 45, 45, 0, 0, 0 15, 0, 0, 0, 0

Among these, we see that E_{2}, E_{3}, E_{4} are not contained in any of the three vertex kernels; and then D_{5} and subsequently C_{5} fail to lie in vertex kernels. So our goal will be to homotopically remove these 5 classes from Z_{2}(G). For this, we will also need to know the types of subgroups above each group, and that information we summarize in the rows of the following table.

X  D  E  E_{5} = K_{O} F_{1} = K_{T} F_{2} = U_{S}
D_{1} D_{2} D_{3} D_{4} D_{5} E_{1} E_{2} E_{3} E_{4}
C_{1} 9 3  3 1 3  1  3
C_{2}  3 9  3  3  1  3  1
C_{3}  2  2 4 1 2  2  1  1
C_{4}  5  5  1
C_{5}  3  1  3
D_{1}  1  1
D_{2}  1  1
D_{3}  1  1
D_{4}  1  1
D_{5}  1  1

Note we have omitted E_{5} since it is itself a vertex kernel K_{O}, and E_{1} since it is contained in (a unique) K_{T} and U_{S}; while we do not list E_{2}, E_{3}, E_{4} since they are maximal in Z_{2}(G).

We now set Z = Z_{2}(G), and proceed to remove the undesirable classes in several steps, using Quillen’s Fiber Theorem [Qui78, Prop 1.6]. So let Z^{1} denote Z with the classes E_{3}, E_{4} removed. For any elementary X ∈ Z^{1}, the fiber i^{−1}(Z_{≤X}) is has unique maximal element X, and so is contractible. So it remains to consider X ∉ Z^{1}, that is, X of type E_{3} or E_{4}; in which case, the corresponding fiber is just Z_{<X}^{1}. In both cases, there is a unique involution z ∈ X with N_{G}(X) ≤ C_{G}(z). Furthermore if Y is a complement to Z = ⟨z⟩ in X, then Y does not satisfy the condition for Z_{2}(G). Consequently if Y ∈ Z_{<X}^{1} then Y Z < X so that also Y Z ∈ Z_{<X}^{1}. And then the inclusions

Y ≤ Y Z ≥ Z

give conical contractibility of Z_{<X}^{1} as in [Qui78, 1.5]. Thus we have the hypotheses of [Qui78, Prop 1.6], and we conclude Z^{1} is homotopy equivalent to Z.

Next let Z^{2} denote Z^{1} with the classes C_{5} and D_{5} removed. This time we apply the Fiber Theorem to preimages of posets above X, and again reduce immediately to X of type C_{5} or D_{5}. Notice in the second table, for either case, that X is contained in a unique subgroup of type E_{2}; since the other overgroups of type E_{4} were removed in the previous step, we see the fiber Z_{>X}^{2}has a unique maximal element, and so is contractible. So again the hypotheses of [Qui78, Prop 1.6] are satisfied, and we conclude Z^{2} is homotopy equivalent to Z^{1}.

Finally we let Z^{3} denote Z^{2} with the class E_{2} removed. The argument is similar to that for Z^{1}: We return to considering fibers of posets below X, and may again assume X has type E_{2}. In this case there is a unique 4-group Z in X of 2-central type with N_{G}(X) ≤ N_{G}(Z), and in particular C_{G}(X) ≤ C_{G}(Z). This time if Y is a complement to Z in X, then either Y fails the condition for Z_{2}(G) or Y has type C_{5} or D_{5} and has already been removed. Thus again we have for Y ∈ Z_{<X}^{2} that also Y Z ∈ Z_{<X}^{2}; so that

Y ≤ Y Z ≥ Z

gives conical contractibility of Z_{<X}^{2} by [Qui78, 1.5]. So we have the hypotheses of [Qui78, Prop 1.5], and conclude that Z^{3} is homotopy equivalent to Z^{2}.

As a result of the above sequence, we have that A_{2}(G) is homotopy equivalent to Z^{3}, where all elementaries of the latter lie in some vertex kernel.

So when we set up our standard apparatus of 1.4, but now for equivalence of Z^{3} rather that A_{2}(G) with Δ, we have the desired necessary condition that all I_{P} are non-empty.

We begin with P of order 2, and can as usual refer back to the end of the proofs in [RSY90]. If P is generated by an involution of class 2B, then I_{P} is a cone on a unique sextets with 7 of its incident lines (this corrects an inessential misstatement in [RSY90], that all 15 lines of the sextet should occur); certainly this is contractible. For P of type 2A, the complex I_{P} consists primarily of a cone: the join of a unique octad with a projective plane structure on 7 trios and 7 sextets; but for each trio we also have its other two octads (so 15 octads in all). Contractibility follows in [RSY90] by using 1.5 to remove the sextets; but we could equally remove the 14 other octads using 1.5 (since their residues are given by a cone on a unique line with 3 sextets). This latter argument leaves us with a cone more in the spirit of the arguments of the present paper.

For P of larger order, we see in Z^{3} it remains to consider the types C_{1} through C_{4}, D_{1} through D_{4}, and E_{1}.

We first consider P containing some involution of class 2B, in view of the smaller configuration just discussed. For P of type C_{4}, we see in the second table that I_{P} will contain a sextet and five trios, so this must be the full subcomplex for type 2B, contractible. Next for P of type C_{3} or E_{1}, we see in the table that I_{P} consists of just one sextet and one trio; by uniqueness of the sextet for type 2B, we see these I_{P} must be subcones as in 1.6. Finally for P of types D_{2} and D_{4}, we see in the table that I_{P} consists of a unique vertex, certainly contractible.

So we turn to cases where P consists entirely of involutions of class 2A. For P of type C_{1} we see in the the table that I_{P} consists of a trio and 3 sextets; and it is easily checked that these do form a cone on that trio. Next P of type D_{1} contains type C_{1} and we see in the table that I_{P} consists of a single sextet, contractible.

Then consider P of type C_{2}: from the table I_{P} consists of an octad, a sextet, and 3 trios; we check easily that the octad and sextet form an edge, this edge is incident with each trio—contractible to that edge and then to either vertex. Finally P of type D_{3} contains type C_{2}, and from the table I_{P} consists of a single octad and trio; by containment in the previous, these form an edge, contractible.

This completes checking of contractibility of I_{P} for all P ∈ Z^{3}. Consequently by 1.4 we may conclude that the 2-local geometry Δ is homotopy equivalent to Z^{3}—and hence to A_{2}(G), as desired.

It may be useful, for comparison, to sketch an equivalence proof based on equivalence with B_{2}(G) following the method of 1.8 (which presumably could not differ greatly from Ronan’s original proof). Arguably the method of 1.8 begins to be preferable for groups of the size of M_{24}, and it seems clearly preferable for our yet-larger cases of Co2 and J_{4} later.

For Step 1 of 1.8 we need to establish the Borel-Tits property (BT/Δ), that for Q ∈ B_{2}(G) we have the normalizer N := N_{G}(Q) contained in some vertex stabilizer G_{v}. For smaller groups it was easy to modify our discussion of contractibility of I_{P} as indicated in 1.7; for M_{24} it seems more efficient to work instead directly with the list of maximal subgroups of M_{24} in the Atlas. (So below we will simply state various subgroup containments; these can be deduced using the Atlas and discussions such as [Con71], but the details though routine would be lengthy). If N already lies in one of the maximal subgroups G_{v} we are done, so we may as well assume that N ≤ M for M one of the other maximal subgroups. If M is one of M_{23} or M_{22}2, our previous discussion of those groups shows that N lies in the stabilizer of a vertex for their geometries, and those in turn may be regarded as vertices of the geometry Δ for M_{24} in the obvious way. Similarly if M = M_{21}S_{3} we can recall that M_{21} ≅ L_{3}(4). If Q ∩ L_{3}(4) = 1 then Q has order 2, so N(Q) is an involution centralizer lying in G_{O} or G_{S}. Otherwise Q_{1} := Q ∩ L_{3}(4) ≠ 1 so that N(Q) ≤ N_{M}(Q_{1}); we can apply the usual Borel-Tits theorem to N_{M}(Q_{1}), to conclude that it lies in a parabolic of structure 2^{4}L_{2}(4) of L_{3}(4).2, which in turn lies in G_{O} or G_{S}. If M = M_{12}2, we see our 2-local N must lie in one of the maximal subgroups X of M_{12}2, which from the discussion in [Con71] are contained in one of the above maximal subgroups of M_{24}—except possibly X = L_{2}(11). But the 2-locals of L_{2}(11) in turn lie in previously-treated subgroups of M_{12} and hence of M_{24}. The same observation treats the remaining maximal subgroups M of M_{24}, of structures L_{2}(23) and L_{2}(7).

Then Step 2 of 1.8 allows the determination of B_{2}(G). For by Lemma 1.9(1) we now have Q containing the kernel U_{v} for some vertex v. If Q = U_{v} then Q already corresponds to a simplex of Δ, so we may as well assume that U_{v} < Q, and hence in \overline{G_{v}} := G_{v}/U_{v} we have Q̄ ≠ 1. So by Lemma 1.9(2) we have Q̄ ∈ B_{2}(\overline{G_{v}}). Now the quotients \overline{G_{v}} have structure L_{4}(2), L_{2}(2) × L_{3}(2), 3Sp_{4}(2). These are Lie-type groups in characteristic 2, so we may apply the usual Borel-Tits theorem to describe their B_{2} (in particular for the direct product use 1.10). That is, Q̄ must be the unipotent radical of a parabolic subgroup of the indicated quotient.

Finally we sketch the reduction of B_{2}(G) as just described to the smaller complex Δ given by the 2-local geometry. For vertex v of type O or T, our Q̄ may be the unipotent radical for a simplex σ involving the square node in the diagram for the residue of v, i.e. not corresponding to a vertex of the geometry Δ; we can correspondingly write σ = τ ∗ □ where τ is some simplex (possibly empty) of Res_{Δ}(v). Then it is standard to check that our group Q contains a unique kernel U_{v∗τ} of the type of v ∗ τ; using 1.5 we may homotopically remove such Q from our poset. In this way we can collapse B_{2}(G) down to Δ.

We conclude our discussion by mentioning that the mod-2 cohomology of M_{24} is being determined by John Maginnis, and it is our understanding that his work is essentially complete. Of course by the earlier paper [RSY90, 3.2] one has available the alternating-sum decomposition for cohomology. We do not know if the present homotopy equivalence would be of any further use for calculation of cohomology.

2.9 McL revisited

We recall from [RSY90], where projectivity was proved, that the first geometry for McL is the rank-4 geometry discovered by Ronan and described in Ronan-Stroth [RS84]. In the Table, we see that the two geometries have different Euler characteristics, so that we cannot have both equivalent to A_{2}(G). This first geometry is non-local in that it has simple line stabilizer G_{ℓ} ≅ U_{4}(3); and in fact I_{ℓ} is the geometry above for U_{4}(3), which in full is certainly not contractible, so that our method would not apply.

Consequently we let Δ denote the second geometry for McL, obtained as a truncation of the first by deleting the lines ℓ; and we will show that Δ is homotopy equivalent to A_{2}(G). In this case, it again happens that involutions (just one G-class) correspond with certain vertices. So as before it is convenient to transform terminology by:

{ spaces, points, planes } of [RSY90] ↦ { points, left spaces, right spaces } . Here the points p correspond to involutions, with stabilizer given by involution centralizer G_{p} ≅ 2.A_{8}. The left and right spaces correspond to elementary subgroups K_{L} ≅ 2^{4} ≅ K_{R}, each with stabilizer of structure 2^{4}A_{7}.

We see that Δ is fully local: the vertex kernels above are elementary 2-groups, as are their intersections I_{LR} ≅ 2^{2} and I_{pL} ≅ I_{pR} ≅ I_{pLR} ≅ 2^{1}. Consequently in 1.4 we have all O_{2}(I_{σ}) > 1 so that all I_{σ} are contractible.

For the necessary condition, it is standard (see e.g. Adem-Milgram [AM96, 1.5,1.6]) that K_{L} and K_{R} represent the conjugacy classes of maximal elementary 2-subgroups. So for any elementary P it follows that I_{P} contains at least some space-vertex L or R.

In fact, these considerations can be slightly extended to an explicit contraction of A_{2}(G) onto Δ: since any elementary of order 2^{3} (or 2^{2}) is contained in a unique L or R (respectively, unique edge L ∗ R), these can be removed by 1.5 from A_{2}(G) to yield Δ. However, it is also quite efficient to proceed by our standard method:

So as usual, for the case of P of order exactly 2, we obtain contractibility of I_{P} by quoting the final part of the proof for Δ^{P} in [RSY90]. Furthermore we recall that I_{P} has the structure of a cone p ∗ R, where the point p is the involution generating P, and R is its residue (with 15 each of left spaces L and right spaces R—the truncation of the projective 3-space over F_{2} given by removing projective lines).

Elementaries Q of order larger than 2 can be quickly treated. If Q has order 2^{2}, we see from flag-transitivity of A_{7} on the spaces K_{L} or K_{R} that by conjugacy we may take Q = I_{LR} = K_{L} ∩ K_{R} for a unique left-right incident pair L ∗ R. Since K_{p} has order 2, we see that I_{Q} has no points—so consists just of this edge, contractible. Finally, any Q′ of order 2^{3} or 2^{4} can be assumed to contain such a Q of order 2^{2}. By our earlier observation on maximality, I_{Q′} must contain at least one of L, R; but not both, since Q cannot lie in I_{LR} of order 2^{2}. So I_{Q′} is already contracted to a single vertex. This completes the proofs of contractibility, and hence by 1.4 of equivalence of Δ with A_{2}(G).

As usual, equivariance in contractibility arguments above establishes (BT/Δ) in view of 1.7.

We already noted that in their determination of the mod-2 cohomology of McL, Adem-Milgram used the contractibility in [RSY90] to obtain the alternating-sum formula 1.2 over Δ. Once again, the equivalence of Δ with A_{2}(G) was only observed after that application of the earlier projectivity result.

2.10 The 2-local geometry for J_{1}

For this small sporadic group, the mod-2 cohomology has been treated in several elementary ways. In particular, the Sylow 2-subgroup T is abelian (indeed elementary of order 2^{3})—so by Burnside’s argument [Gor68, Thm 7.1.1], N_{G}(T) controls 2-fusion; and then by Swan’s standard result [Swa60, Lemma 1], restriction from G to N_{G}(T) gives an isomorphism of cohomology. Notice, however, that if we regard this one class of elementaries as defining a complex of dimension 0, the corresponding reduced Lefschetz module is not projective—just acyclic.

The cohomology of J_{1} has also already been treated using the viewpoint of alternating sums, see e.g. Webb [Web87, 7.10] and Adem-Milgram [AM94, V.2.11,V.3.4,VIII.2]. But for completeness and comparison we indicate the application of our standard equivalence method.

The complex A_{2}(G) of all elementaries in fact contains just one conjugacy class of subgroups of each of the orders 2, 2^{2}, 2^{3}. The standard 2-local geometry Δ is the truncation obtained by removing the subgroups of order 2^{2}; we note that this truncation in fact coincides with the radical poset B_{2}(G). In particular, the Borel-Tits property (BT/Δ) holds just because of coinciding with B_{2}(G).

2.11 J_{3} revisited

We recall from [RSY90], where projectivity was proved, that Δ is the rank-3 geometry described in Yoshiara [Yos89]. Again it will be useful to change to the viewpoint of the involution geometry, with new notation

{ vectors, points, lines } of [RSY90] ↦ { points, lines, spaces } ,

where new points correspond to the members of the single G-conjugacy class of involutions. We have the standard large-extraspecial situation: lines are four-groups whose three involutions p, q, r = pq satisfy the (symmetric) relation q ∈ O_{2}(C_{G}(p)).

We remark that one route to an equivalence proof is to check that Δ coincides with Z_{2}(G). But as usual we follow the method of 1.4:

First we note that the geometry is fully local; indeed all kernels are 2-groups: K_{S} ≅ 2^{4}, K_{L} ≅ 2^{2+4}, and K_{p} ≅ 2^{1+4}. As in the case of U_{4}(3) earlier, it is convenient to take I_{p} to be the center Z(K_{p}) or order 2, rather than K_{p} itself: and this time we also take I_{L} to be the line itself, namely the 4-group Z(K_{L}), rather than all of K_{L}. Then further intersections are also 2-groups: I_{LS} ≅ 2^{2} and I_{pL} = I_{pS} = I_{pLS} ≅ 2^{1}. So all O_{2}(I_{σ}) > 1, hence all I_{σ} are contractible, as required in 1.4.

It is not hard to verify the usual necessary condition. Note that any elementary P contains some involution p and hence lies in C_{G}(p) = G_{p}. If P is not contained within extraspecial K_{p}, we find P lies in a unique Sylow 2-subgroup of G_{p}/K_{p} ≅ A_{5}; and check then that P lies in a corresponding kernel K_{S}. Otherwise P lies in extraspecial K_{p} of minus type; the elementaries inside K_{p} (other than the center itself) are of order 4, all conjugate under the previous A_{5}, and hence coincide with our line 4-groups I_{L} = Z(K_{L}). We conclude that any I_{P} contains at least some L or S.

So we turn to proving contractibility of I_{P}, beginning with the case where P has order 2. The proof in [RSY90] reduced first to the subcomplex Res-fixed by P, and then to the further subcomplex which is I_{P} in our present notation: namely the cone p∗R, where p is the involution generating P; and R is its residue—consisting of 5 “claws” of form L ∗ {S_{1}, S_{2}}. In particular, this cone is contractible.

Next we consider Q of order larger than 2, with all points of at mutual distance 1 in the usual graph. In this geometry (as in several previous cases, there are no proper triangles, so that the points of Q form just a single line, with Q of order 4. Our choices of I_{p} and I_{L} of small order now guarantee that I_{Q} contains no points, and contains just one line L as above—the distinct I_{L} above I_{p} intersect only at I_{p}. Furthermore we check that Q is just K_{S_{1}} ∩ K_{S_{2}} for the above pair of spaces incident with L—and that spaces for L′ other than intersect Q only in p. It follows that I_{Q} is just the indicated claw, contractible.

Finally we may assume that Q contains a pair of points p, r at graph distance greater than 1. Then Q is not contained in K_{p}, and our earlier argument shows Q lies in a unique space kernel K_{S}. Furthermore such a pair p, r is not contained in any point group I_{p} or line group I_{L}. Consequently I_{Q} consists of the unique vertex S, contractible. This completes the verification of the hypotheses of 1.4, and hence of equivalence of Δ with A_{2}(G).

As usual, we note that the Borel-Tits property (BT/Δ) follows from equivariance in the above contractibility arguments, using 1.7.

We are not aware of any treatment of mod-2 cohomology of J_{3} in the literature (in particular, of any application of the alternating-sum formula 1.2).

2.12 The 2-local geometry for J_{4}

The group J_{4} is large, and we know of no classification of its vast collection of elementaries in the literature. Additionally, its diagram has a “square node” as in the case of M_{24}, with the attendant complication for the method of 1.4. Consequently we will follow instead the method of 1.8.

The geometry Δ is described ^{12} in Ronan-Smith [RS80]. It has rank 4; one stabilizer has structure 2^{11}M_{24}, so it is convenient to use names octads, trios, sextets for the other vertex types, and refer the vertices of this fourth type as codes. So typical stabilizers are given by

G_{O} ≅ 2^{10}L_{5}(2), G_{T} ≅ 2^{3+12}(S_{5} × L_{3}(2)), G_{S} ≅ 2^{1+12}3M_{22}2, G_{C} ≅ 2^{11}M_{24}.

We remark that that K_{S} = 2^{1+12}3 has normal Sylow 2-group U_{S} = 2^{1+12}; but other vertices have K_{v} = U_{v} a 2-group. It is also useful to view vertices in terms of the elementary subgroups appearing as Z(U_{v}) in the kernel-subgroups of the indicated stabilizers.

We set out to determine possible Q ∈ B_{2}(G) via Algorithm 1.8. Set N := N_{G}(Q).

12Cf. footnote for Co2: it is not clear why we failed to treat this geometry in [RSY90].

Step 1: Show N ≤ G_{v} for some vertex v ∈ Δ.

(a) Case |Q| = 2: Here Q is generated by a single involution. In the Atlas [C^{+}85] we see there are 2 conjugacy classes, with C(2A) = G_{S} and C(2B) ≅ 2^{11}M_{22}2 ≤ G_{C}. In particular these centralizers are 2-constrained, so that G = J_{4} is indeed of characteristic 2-type.

Now N must lie in some maximal subgroup of G, and we are done if it lies in one of the four vertex stabilizers just listed. So we may as well assume that N lies in one of the other maximal subgroups M listed in the Atlas, and consider these various possiblities for M in turn. We will indicate various subgroup containments without details; these can be deduced using the Atlas and more detailed references such as Kleidman-Wilson [KW88].

(b) Case M an odd p-local: We consider M = N_{G}(P) where P has one of the orders 11,29,37,43; with structure 11^{1+2}(5 × 2S_{4}), 29.28, 37.12, 43.14. Visibly any 2-local such as N lies iN A subgroup of structure either 11 × 5 × 2S_{4}, or cyclic of order 28, 12, 14. But in all cases N lies in an involution centralizer, and these were treated in (a).

(c) Case M = L_{2}(32).5: Here M is Lie-type of characteristic 2; indeed it has rank 1 and its unique 2-local has structure 2^{5}.31.5. But we consider an element of order 31 in the L_{5}(2) in G_{O} in its action on K_{0} ≅ 2^{10} and conclude this configuration 2^{5}.31.5 already lies in G_{O}.

(d) Case M = L_{2}(23).2: Notice first if Q has trivial intersection with L_{2}(23) = [M, M], then [Q, N] ≤ Q ∩ [M, M] = 1 so that N lies in an involution centralizer, treated in (a). So we may assume Q_{1} := Q ∩ L_{2}(23) ≠ 1 and N ≤ N_{M}(Q_{1}). Now the subgroup M′ = L_{2}(23) embeds in the M_{24} in G_{C}; so N_{M′}(Q_{1}) has index at most 2 in N_{M}(Q_{1}), and is described by our earlier treatment of 2-locals in M_{24}. In particular it is contained in the stabilizer of an octad, trio, or sextet for M_{24}, and then N is contained in the stabilizer in J_{4} of a simplex of corresponding type OC, T C, or SC.

(e) Case M = U_{3}(11).2: This final maximal subgroup is not of characteristic-2 type, and so its 2-locals have not been treated earlier in this paper. So we can consider how N might lie in one of its maximal subgroups listed in the Atlas. Most of these in fact lie in other maximals of J_{4} just treated: Indeed the odd locals 11^{1+2}40 and 37.13 appear in the groups in (b), and the 3-locals (4^{2} × 3)D_{12} and 3^{2}Q_{8} appear in (non-maximal) 3-locals in G_{S} and G_{C}. The extension 2.(L_{2}(11) × 2).2.2 lies in an involution centralizer, and so is handled by (a). In A_{6} the 2-locals have structure D_{8} (in an involution centralizer again) or S_{4}, which can be found in G_{C}. Finally L_{2}(11).2 lies in a subgroup M_{12}2 in the M_{24} of G_{C}.

This concludes treatment of the maximal subgroups M of J_{4}, and the demonstration of the Borel-Tits property (BT/Δ).

Step 2: Analyze possible Q̄ ∈ B_{2}(\overline{G_{v}}).

Since we have N(Q) ≤ G_{v} from Step 1, we know U_{v} ≤ Q using Lemma 1.9(1). If Q = U_{v} then Q already corresponds to a vertex of the local geometry Δ, as desired. So we may as well assume U_{v} < Q; and then in \overline{G_{v}}, we see from Lemma 1.9(2) that Q̄ ∈ B_{2}(\overline{G_{v}}). Now for the 4 vertex types, the quotient \overline{G_{v}} has structure L_{5}(2), S_{5} × L_{3}(2), M_{24}, or 3M_{22}2: either a Lie-type group in characteristic 2, or a characteristic 2-type group whose B_{2} was analyzed earlier in the paper. (In particular, recall lemmas 1.11 and 1.10 about S_{5} and direct products). In either case, the appropriate result of Borel-Tits type holds, namely that Q̄ must be a kernel U_{σ} for a simplex σ of the usual 2-local geometry for \overline{G_{v}}. This completes, on taking preimages, the determination of B_{2}(G).

However, we note for vertex types other than v = S that the simplices σ occurring above may include the square node, not corresponding to a vertex of the local geometry Δ; so that B_{2}(G) is in fact larger than Δ. Just as for M_{24}, we can homotopically collapse B_{2}(G) down onto Δ: For suppose we get above that Q̄ = U_{σ} for σ of form τ ∗ □. Then the preimage Q contains U_{v∗τ} and contains no other vertex kernel of that type; a standard application of Lemma 1.5 shows we may homotopically remove Q from the poset. Continuing this process removes all such Q involving the square node, and so reduces B_{2}(G) down to our local geometry Δ.

2.13 The 2-local geometry for Co2

As for J_{4}, this group is large and its numerous elementaries do not seem to be determined elsewhere; it seems unlikely that the method of 1.4 will be practical here, so we follow the approach of 1.8.

The geometry is described ^{13} in Ronan-Smith [RS80]. It has rank 4; as it is a one-node extension of the standard geometry for M_{22}, it may be most natural to simply extend that earlier terminology as used in [RSY90]. Thus three vertex types will be called octads, hexads, quintets with corresponding stabilizers of structure

G_{O} ≅ (2^{1+6} × 2^{4})L_{4}(2), G_{H} ≅ 2^{1+8}Sp_{6}(2), G_{Q} ≅ 2^{4+10}(S_{5} × S_{3}).

The fourth vertex type will be the full M_{22} geometry; we will refer to it as a code, since it arises by restriction from the Golay-code module of M_{24}. It has stabilizer

G_{C} ≅ 2^{10}M_{22}2.

Another useful viewpoint on the geometry is via arrangment of involutions in elementary 2-subgroups; the octad and hexad stabilizers are just the centralizers of the involutions in their centers.

Again we set out to determine possible Q ∈ B_{2}(G) via Algorithm 1.8. Set N := N_{G}(Q). Step 1: Show N ≤ G_{v} for some vertex v ∈ Δ.

(a) Case |Q| = 2: Here Q is generated by a single involution. In the Atlas we see there are three conjugacy classes, with C(2A) = G_{H}, C(2B) = G_{O}, and C(2C) ≅ 2^{10}Aut(A_{6}) ≤ G_{C}. In particular these centralizers are all 2-constrained, so that G is of characteristic 2-type.

Now N must lie in some maximal subgroup of G, and we are done if it is one of the four vertex stabilizers G_{v} given above. So we may as well assume that N lies in one of the other maximal subgroups M of G listed in the Atlas; and we consider these possibilities in turn. Again we indicate certain subgroup containments without details; these can be deduced using the Atlas and more detailed references such as Wilson [Wil83].

13And so was among the “catalogued” geometries of [RSY90]; implicitly it was eliminated from consideration there for projectivity, evidently mistakenly. Possibly we had computed the Euler characteristic incorrectly?

(b) Case N ≤ M for M an odd p-locals: We consider for M the groups 3^{1+4}2^{1+4}S_{5} and 5^{1+2}4S_{4}. We regard them as N_{G}(P), for P the indicated extra-special p-group for p = 3, 5. Visibly the only part of P lying in any 2-local is Z(P), so we would have N ≤ Z(P) : N_{1} with N_{1} ≤ 2^{1+4}S_{5} or 4S_{4}. In either case we see N_{1} (hence N) centralizes an involution, so we conclude using case (a) that N lies in some G_{σ}.

(c) Case M = U_{6}(2).2: Notice first if Q has trivial intersection with M′ ≅ U_{6}(2), then [N, Q] ≤ [M, M] ∩ Q = 1—that is, N ≤ C(Q) so that we are done in view of case (a). Thus we may assume that Q_{1} := Q ∩ M′ ≠ 1. Then N ≤ N_{M}(Q_{1}), which by the Borel-Tits theorem must lie in some maximal parabolic subgroup of the Lie-type group M. But for these 3 possibilities we see that 2^{1+8}GU_{4}(2) ≤ G_{H} and 2^{4+8}(A_{5} × S_{3}) ≤ G_{Q} and 2^{9}(L_{3}(4) ≅ M_{21}) ≤ G_{C}.

(d) Now the 3 possible cases M = U_{4}(3)D_{8}, McL, M_{23} will very similar to case (c)—but we replace the Borel-Tits theorem by the property (BT/Γ) for this previously-treated group M of characteristic-2 type and its 2-local geometry Γ. We sketch the details:

For M = U_{4}(3)D_{8}, arguing with M′′ = [M′, M′] ≅ U_{4}(3) instead of M′ as above, via (a) we may assume that Q_{1} := Q ∩ [M, M, M] ≠ 1. Then N ≤ N_{M}(Q_{1}), and by our earlier consideration of M′′ ≅ U_{4}(3) we know N_{M′′}(Q_{1}) must lie in a subgroup of structure 2^{1+4}(S_{3} × S_{3}) or 2^{4}A_{6}. Now the involutions in M′′ are of type 2A in G; so the first subgroup lies in C(2A) = G_{H}. All the involutions in the normal 2^{4} of the second are conjugate under S_{6}; we see this subgroup is the 2^{4} factor of the kernel K_{O} of G_{O}. So in any case N lies in some G_{v}.

For M = McL we see N ≤ N_{M}(Ω_{1}(Z(Q)), and by our earlier consideration of McL (with the possibilities for elementary Ω_{1}(Z(Q)) described by Adem-Milgram [AM96]) the latter must lie in a subgroup of structure 2A_{8} or 2^{4}A_{7}. Considerations like those for U_{4}(3) above show these in turn lie in G_{H} or G_{O}.

For M = M_{23}, our earlier analysis of that group shows that N must lie in the stabilizer of a hexad, octad, or quintet for the geometry of M_{23}. However we see that those lie in the correspondingly-named stabilizers for the full geometry Δ of G = Co2.

(e) The final maximal subgroup HS.2 is not of characteristic-2 type, and so its 2-local subgroups have not been described ^{14} earlier in this paper. However, we can examine its maximal subgroups from the Atlas in the usual inductive way. First, the local subgroups appearing in the usual 2-local geometry for HS have structure 2^{4}S_{6}, 4^{3}L_{3}(2), and 4 ∗ 2^{1+4}S_{5}; we check these lie in G_{O}, G_{O}, G_{H}. The involution centralizers 2 × Aut(A_{6}) and S_{8} × 2 centralizing the outer automorphism in HS.2 are handled by (a). The subgroups M_{11}, L_{3}(4).2, and M_{22} are contained in a subgroup M_{23} of Co2 and so have been handled in (d) above. The subgroup 5.4 × A_{5} lies in the normalizer in Co2 of a group of order 5, and so has been handled in (b) above.

14Indeed for HS, the 2-local geometry is not equivalent to B_{2}(G). The mod-2 cohomology of this group is being determined by Milgram et al.; from the viewpoint of decompositions, it can be treated by the methods of Benson or Dwyer, mentioned in the final section of this paper.

So the only maximal subgroup of HS not yet considered is that of structure U_{3}(5).2, and we can in turn consider its maximal subgroups from the Atlas. The involution centralizer 2S_{5} is handled by (a). The subgroup M_{10} lies in an M_{11} in the previous paragraph. The 5^{1+2}8 lies in the 5-normalizer considered in (b). Finally the 2-local subgroups; in fact these either lie in an A_{6} in the M_{10} just considered, or else in the normalizer of a 3-element handled in (b) above.

This concludes the verification of the Borel-Tits property (BT/Δ).

Step 2: Analyze possible Q̄ ∈ B_{2}(\overline{G_{v}}).

The remainder of the proof is routine, using earlier cases inductively. For with N(Q) ≤ G_{v} from Step 1, we have K_{v} ≤ Q using Lemma 1.9(1). If Q = K_{v} then Q itself corresponds to a vertex in the local geometry Δ, as desired. So we may as well assume K_{v} < Q; then in \overline{G_{v}} := G_{v}/K_{v} we see from Lemma 1.9(2) that Q̄ ∈ B_{2}(\overline{G_{v}}). But notice that for each of the 4 vertex types, the quotient L_{4}(2), Sp_{6}(2), M_{22}2, or S_{5} × L_{2}(2) is either a Lie-type group of characteristic 2, or a characteristic 2-type group whose B_{2} was analyzed earlier. (In particular, recall lemmas 1.11 and 1.10 on S_{5} and direct products). In either case the Borel-Tits property (BT/Γ) holds for the appropriate residue Γ, namely that Q̄ must be the kernel K_{σ} for some simplex σ of the usual 2-local geometry for \overline{G_{v}}, namely Res_{Δ}(v). Then the preimage Q is precisely the kernel K_{v∗σ} corresponding to the simplex v ∗ σ of Δ.

This completes the identification of the poset B_{2}(G) with the simplices of complex Δ; hence the desired homotopy equivalence for G = Co2.

In earlier cases, we had related M_{11} to M_{12}, and M_{23} to M_{24}; we can make similar but rather weaker remarks relating Co2 to Co1. At the time of [RSY90] we had considered the group Co1 and verified (unpublished) contractibility of fixed-point subcomplexes for 2 of its 3 classes of involutions—the condition failing for the class with centralizer containing G_{2}(4). In particular this showed via [Thé87, Thm 2.1] that the Lefschetz module for the Co1 geometry is projective relative to 2-subgroups whose involutions are in the bad class. Since the 3 classes in Co2 come from the 2 good classes of Co1, it follows that the restriction of the module to Co2 must also be projective. However, we have not checked that the module for Co2 above is a direct summand of the restriction of that for Co1.

2.14 The 2-local geometry for T h

The 2-local geometry for the Thompson group T h = F_{3} is described in Aschbacher [Asc86b] (and can also be obtained from the minimal-parabolic description in Ronan-Stroth [RS84]). It is a bipartite graph, with points given by involutions with stabilizer G_{p} = C_{G}(p) ≅ 2^{1+8}.A_{9}; and spaces given by certain elementary groups S of order 2^{5}, with stabilizer the non-split Dempwolff extension 2^{5}.L_{5}(2). The incidence is just containment. (Along with most other rank-2 geometries, it was not considered in [RSY90]).

We note first that the geometry is fully local: we have elementary space kernel K_{S} ≅ 2^{5} and extraspecial point kernel K_{p} ≅ 2^{1+8}; with intersection I_{pS} ≅ 2^{5} also elementary. Thus all O_{2}(I_{σ}) > 1, so all I_{σ} are contractible.

Next we indicate a crucial feature of the geometry: A point p is incident with exactly 9 spaces S_{i}. These spaces, as elementary 2-groups K_{S_{i}}, intersect pairwise just in p; and the resulting 1 + 9 · 30 elements exactly cover all the involutions of extraspecial K_{p}.

This allows us to quickly compute the subcomplex I_{P}, for P of order 2 generated by point p: it contains just the 9 spaces S_{i}, as well as the point p, and then all the 270 non-central involutions q of K_{p} in the various S_{i}. By the feature above, the residue of each q in this subcomplex is a single space S_{i}; so we may remove these q using 1.5. What remains is the cone on p with the 9 S_{i} in its residue, contractible.

So we turn to elementary Q of order larger than 2, and since there is just one class of involutions we can assume that Q ⊃ P above, giving I_{Q} ⊆ I_{P}. Now Q must lie in some Sylow group; it must then fix some space S and so lie in G_{S} ≅ 2^{5}L_{5}(2)—and furthermore lie in the subgroup centralizing P, which is covered by a parabolic of L_{5}(2) having structure 2^{4}L_{4}(2). From the structure of the non-split extension in G_{S}, we conclude that Q must be covered by the unipotent radical—that is, Q must lie in the extraspecial kernel for some point of S. Consequently I_{Q} contains some point, and we may as well assume that point is the p above. Now if Q contains further points q, r lying in distinct spaces S_{i} of I_{P}, we recall that those spaces intersect only at p—from which it follows that I_{P} consists exactly of the vertex p, contractible. Otherwise Q lies entirely within some S_{i}, and then we see that I_{Q} is just the cone on S_{i} with some or all of its 31 points—again contractible. This completes the verificatioin of the hypotheses of 1.4, and hence of the equivalence of Δ with A_{2}(G).

As usual, equivariance in contractibility proofs also establishes the Borel-Tits property (BT/Δ) in view of 1.7.

Seemingly the mod-2 cohomology of T h has not been studied. Nonetheless, it should be interesting—it appears likely ^{15} that it contains the rank-5 Dickson invariants. (It is known that the rank-5 invariants cannot as the exact cohomology of a finite group). The invariants in smaller ranks have already arisen from sporadic analysis: in rank 3 in connection with M_{12} [AMM91] [BW95], and in rank-4 for Co3 [Ben95] [Mil95].

3 Cases with odd p

3.1 Cases with extraspecial Sylow of order p^{3}

The work of Tezuka-Yagita [TY96] considers G with Sylow p-group extraspecial of order p^{3} and exponent p. By the p-local geometry Δ in this situation we mean the subcomplex of the elementary p-complex A_{p}(G) consisting of subgroups all of whose elements are from the p-central class. Notice in the cases listed in our Table

Ru, J_{4}, T h (for p = 3, 3, 5)

that G has just one class of elements of order p; so that this definition of Δ must coincide with A_{p}(G). So in these cases, the necessary contractibility is a consequence of Quillen’s standard result [Qui78, Remark 4.4].

15Discussed with Benson and Milgram at AMS Orlando March 1995, this question might now be settled.

3.2 L_{3}(4) and M_{11} at p = 3, and ON at p = 7: nothing new

In [RSY90] we proved projectivity for a certain characteristic-3 geometry Δ for L_{3}(4). However, we see in the Table that the reduced Euler characteristic of Δ is 3^{2}25, while that of A_{3}(G) can be computed as 3^{2}21; so that homotopy equivalence cannot hold. In any case our method would fail, since point stabilizers are non-local (G_{p} ≅ A_{6} is simple).

Similar remarks hold for M_{11} at p = 3: the geometry Δ gives in [RSY90] a projective L̃(Δ); but its reduced Euler characteristic 3^{2}5 differs from that of A_{3}(G), namely 3^{3}2. So homotopy equivalence cannot hold; and certainly our method fails, in view of non-local point stabilizer G_{p} ≅ M_{10}.

And also similarly for ON at p = 7: Here the geometry Δ with projective L̃(Δ) in [RSY90] has reduced Euler characteristic 7^{3}162487, while that of A_{7}(G) is 7^{3}408007, so that equivalence cannot hold. Of course this geometry is not fully local, as both vertex stabilizers have structure L_{3}(7) : 2.

3.3 U_{5}(2) at p = 3 revisited

We recall from [RSY90], where projectivity was proved, that the geometry Δ is described e.g. in Buekenhout [Bue79, Sec 7,no 20,n = 5]. We maintain the terminology use there of points, lines, and quintets—since points now correspond to 3-central elements in the Atlas classes 3A, 3B; and lines correspond to suitable elementary groups 3^{2} generated by pairs of points.

To begin the proof of equivalence with A_{2}(G), we first note that the geometry is fully local: we have 3-groups as vertex kernels

K_{p} ≅ 3^{1} and K_{L} ≅ (3^{1} × 3^{1+2})2 and K_{Q} ≅ 3^{4}.

where just in case L we have a strictly-smaller U_{L} ≅ 3 × 3^{1+2}. For line group in 1.4 it is convenient to take I_{L} = Z(U_{L}) of order 3^{2}, rather than the full U_{L} itself. We then have 3-groups for all incident intersections:

I_{pL} = I_{pQ} = I_{pLQ} = 3^{1} and I_{LQ} = 3^{2}.

Thus we have all O_{3}(I_{σ}) > 1, so that all I_{σ} are contractible. For the necessary condition, one checks that any elementary lies in some K_{Q} ≅ 3^{4}.

So we turn to the subcomplexes I_{P}, beginning with P of order exactly 3. As usual we quote the latter parts of the proofs in [RSY90]; there we saw there are 6 conjugacy classes of order 3, determining 3 different cases for the appropriate Res-fixed subcomplex: For P generated by an element in class 3F, we get I_{P} consisting just of a single quintet Q. For P generated by an element in classes 3C, 3D, 3E, we get I_{P} the cone on a single line L with the 4 quintets Q_{i} in its residue. And for P generated by an element of class 3A, 3B (a point p), we find I_{P} is the cone on p with its residue, namely the 40 lines and 40 quintets of the generalized quadrangle for G_{p}/K_{p} ≅ Sp_{4}(3). Of course, all three of these structures are contractible.

So we turn to elementary R of order at least 3^{2}. Now I_{R} can contain no points, since K_{p} has order just 3. But we also recall by our earlier remark that I_{R} must contain at least some quintet Q. If R contains an element of class 3F, that quintet must be all of I_{R} (an easy case of 1.6). Every elementary of order 3^{4} is conjugate to K_{Q} and so contains some such 3F, so we may assume R has order ≤ 3^{3}. If R has order exactly 3^{3}, we see Q cannot lie in any I_{L} of order 3^{2}, so I_{R} has no lines, hence consists only of quintets. But in the above case 3CDE, the distinct pairs of Q_{i} above some L intersect only in I_{L} of order 3^{2}—so here I_{R} can only consist of a single Q. So we are reduced to the case of R of order exactly 3^{2}. If R is not conjugate to I_{L}, the same argument above still shows that I_{R} is a single Q. The only remaining case is R = I_{L}, in which case I_{R} is the same as the contractible I_{P} above for the case of type 3CDE. This completes the proofs of contractibility for 1.4, hence shows equivalence of Δ with A_{3}(G).

As usual, checking equivariance in the above contractbility proofs leads via 1.7 to the Borel-Tits property (BT/Δ). We mention that an alternative equivalence can be given using the method of 1.5; see Ly below for an outline of an approach of this type.

We do not know of any treatment of the mod-3 cohomology of U_{5}(2).

3.4 The 3-local geometry of ON

For G = ON the O’Nan simple group, the Sylow 3-subgroup is abelian, so (as in our remark for J_{1} at p = 2) the cohomology can be determined by restriction to the Sylow 3-normalizer. But there is also an interesting 3-local geometry, which we can show is equivalent to A_{3}(G).

Since the Sylow group is elementary of order 3^{4}, A_{3}(G) has dimension 3. By contrast, the standard 3-local geometry Δ, described in Aschbacher [Asc86b], has dimension 1; it can be shown to coincide with B_{3}(G). As usual, it is also easy to apply our method 1.4:

We will call the vertices of this geometry by the usual names of points and lines. The points are T I-subgroups of order 3^{2} with stabilizer G_{p} ≅ (3^{2} : 4 × A_{6})2. The lines are just the Sylow groups of order 3^{4}, with stabilizer G_{L} ≅ 3^{4}(2_{−}^{1+4}D_{10}). Here the kernels are K_{p} ≅ 3^{2}4 and K_{L} ≅ 3^{4}2, with normal Sylow 3-groups U_{p} ≅ 3^{2} and U_{L} ≅ 3^{4}. (These groups are described more explicitly in [Yos85, 2.6] as K_{p} = R_{1}⟨v_{1}v_{2}v_{3}^{2}⟩ and K_{L} = (R_{1} × R′_{1})⟨v_{1}^{2}v_{3}^{2}⟩).

Clearly the geometry is fully local: we take I_{p} = U_{p} and I_{L} = U_{L} as the elementary 3-groups above; and then the intersection I_{pL} = 3^{2} is also a 3-group. Thus we have all O_{3}(I_{σ}) > 1, hence all I_{σ} contractible. Furthermore every elementary P lies in some U_{L} just by Sylow’s theorem, so that all I_{P} are non-empty.

So we verify contractibility of I_{P}, beginning with P of order 3—there is just one class of elements with order 3. Since the point groups are T I-sets, we see that P determines a unique point p with P ≤ U_{p}, and thus I_{P} contains p as its unique point. Furthermore if P lies in a line kernel U_{L}, we can consider I_{L} ≤ C_{G}(P) ≤ G_{p}; since the quotient G_{p}/I_{p} ≅ (4 × A_{6})2 has Sylow 3-groups which are T I-sets in A_{6} ≅ P SL_{2}(9), we see the possible L correspond exactly with the 10 lines above p. Consequently I_{P} is the cone on p with these 10 lines, contractible.

So we turn to Q of order at least 3^{2}, and we may assume that Q ⊃ P above. If Q is of order ≥ 3^{3}, or of order 3^{2} other than I_{p}, our remark about T I-Sylows in A_{6} shows that I_{Q} consists just of a single line (that is, the distinct I_{L} above p intersect exactly in I_{p}), so contractible. Finally if Q = I_{p} we have I_{Q} equal to the contractible I_{P} above. This completes the proofs of contractibility for 1.4, and hence establishes equivalence of Δ with A_{3}(G).

Again, checking equivariance in the contractibility proofs leads via 1.7 to the property (BT/Δ).

We are not aware of any treatment of the mod-3 cohomology of ON.

We remark that the occurrence of the 3-elements inside T I-subgroups of order 3^{2} seems analogous to the behavior of root subgroups in a Lie-type group. However N(A) has simple normal subgroup A_{6}, so we see that ON is not even of characteristic-3 type.

3.5 McL at p = 3 revisited

We recall from [RSY90], where projectivity was proved, that the first p = 3 geometry for McL in our Table is that described in Buekenhout [Bue79, Sec 7,No 22], while the second is obtained by removing the quads of the first. Since we see in the Table that the Euler characteristics differ, certainly we could not have both equivalent to A_{3}(G). In fact the first geometry is not fully local, since it has simple quad stabilizers G_{Q} ≅ Ω_{6}^{−}(3). So henceforth we let Δ mean the second geometry defined using only points and lines, and establish its equivalence with A_{3}(G).

This truncated geometry is fully local: it has elementary line kernel K_{L} ≅ 3^{4}, and point kernel K_{p} ≅ 3^{1+4}2 with extraspecial U_{p} ≅ 3^{1+4}. As before taking I_{p} = U_{p} and I_{L} = U_{L}, we get elementary intersection I_{pL} ≅ 3^{3}. Thus all O_{3}(I_{σ}) > 1, so that all I_{σ} are contractible.

Furthermore any elementary P centralizes some element of order 3, corresponding to a point p—so that we may consider P inside C_{G}(p) = G_{p}. If P does not lie inside U_{p}, it determines a unique Sylow 3-group in the quotient G_{p}/K_{p} ≅ S_{5}, covered by the image of a line kernel U_{L}. We see that P lies either in some U_{L} or some U_{p}, so that I_{P} must contain at least a point or a line.

So we turn to the contractibility of the I_{P}, beginning with P of order exactly 3. There are two conjugacy classes of elements of order 3 in G. The Atlas class 3A gives those lying in the center of a Sylow 3-group, which correspond to points p of our geometry. In the final part of the proof in [RSY90], it is shown that the subcomplex I_{P} consists of the point p generating P, together with the 10 lines on p, and all the other points q of those lines (other 3A elements in U_{p}). The following step in that argument shows via 1.5 that the q can be removed—leaving the cone on p with its lines, contractible. In the case of P generated by an element from class 3B, the arguments of [RSY90] shows that I_{P} is the cone on a particular line L with all its points, also contractible.

It remains to consider elementary Q of order at least 3^{2}. Suppose first Q lies in some U_{L} ≅ 3^{4}. Since there are no subgroups 3^{2} consisting only of 3A-elements, Q must contain some 3B element. Now from containment in I_{P} of type 3B above, we see using 1.6 that I_{Q} must be a subcone, contractible. So we may assume that Q lies in no U_{L}; by our earlier remark, we can take Q in some U_{p}, and then from extraspecial structure we have Q of order at most 3^{3}. Now kernels for distinct collinear points can intersect in an elementary group of order 3^{3}, but that intersection is contained in the corresponding U_{L}—the case just considered. Kernels for non-collinear points intersect in a group of order at most 3; so in our remaining case we see that I_{Q} must consist just of the single point p, contractible. This completes the verifications of contractibility, and hence by 1.4 shows Δ is equivalent to A_{3}(G).

Checking equivariance in contractibility proofs leads via 1.7 to the Borel-Tits property (BT/Δ). Again we note that 1.5 provides an alternate approach to the equivalence proof; see Ly below for an outline of such an approach.

We are not aware of any treatment of the mod-3 cohomology of McL.

3.6 The 3-local geometry of Ly

The geometry Δ is described in Aschbacher [Asc86b, p.19] (a truncation is described in Ronan-Stroth [RS84, p.84]. We will refer to the three vertex types as vertices, edges, codes, with corresponding stabilizers:

G_{v} ≅ 3McL2 and G_{E} ≅ 3^{2+4}2A_{5}D_{8} and G_{C} ≅ 3^{5}(2 × M_{11}) .

The geometry is a rather natural extension of that just considered for McL. In particular, the residue of a code is the geometry for M_{11} considered in [RSY90], namely the 11 letters permuted by M_{11} and the 55 pairs (ie, edges in the complete graph on 11 letters); and the residue of a vertex is the rank-2 truncation considered in the previous case for McL.

The geometry is fully local: we have as vertex kernels

K_{v} ≅ 3^{1} and K_{E} ≅ 3^{2+4}2 and K_{C} ≅ 3^{5}2.

with smaller normal Sylow 3-groups U_{E} ≅ 3^{2+4} and U_{C} ≅ 3^{5}. Taking these 3-groups as the various vertex terms in 1.4, we obtain 3-groups as intersections

I_{vE} = I_{vC} = I_{vEC} = 3^{1} and I_{CE} ≅ 3^{4}.

Thus all O_{3}(I_{σ}) > 1, giving all I_{σ} contractible.

For the necessary condition, any elementary P must lie in a Sylow group, and so centralize some element of class 3A in the center of that Sylow; these correspond (much as in McL) with our “vertices”, so we can consider P inside some (G_{v})′ ≅ 3McL, and in particular its image in the quotient McL of this group modulo its center. In McL we had elementaries in a point or line kernel, so here we get elementaries inside an edge or code kernel. In particular, each I_{P} contains at least an edge or a code.

So we turn to verifying contractibility of I_{P}, starting with P of order exactly 3. As in McL there are two classes, in the Atlas denoted 3A and 3B. And the corresponding subcomplexes are rather similar to those for McL earlier: For P of type 3A, we find I_{P} is the cone on the single vertex v (the 3-element generating P) with all the edges and codes in its residue—these are just all the points and lines of the geometry for the McL quotient of G_{v}. For P of type 3B, since I_{v} has order 3 with type 3A, we see I_{P} has no points. We find that P is contained in Z(I_{E}) for a unique edge E. And then I_{P} contains the 10 codes C over E; plus all the remaining edges E′ of those codes, corresponding to cases with P ≤ I_{E′} \ Z(I_{E′}). In this subcomplex, we see that distinct codes C have kernels I_{C} intersecting just in Z(I_{E}); consequently each E′ has residue consisting of a unique code C; so we may remove these E′ using 1.5, and what remains is the cone on E with its 10 codes, contractible.

So we can now consider elementary Q of order at least 3^{2}. As usual, Q contains some P of order 3 as above, so that I_{Q} is contained in the corresponding I_{P}. Furthermore since I_{v} has order 3, we see that I_{Q} can contain no vertices—only edges and codes. Indeed working in G_{v} ≅ 3McL2, we find there are no subgroups of order 3^{2} consisting entirely of 3A elements, so we may assume P has type 3B. The cases proceed much as for McL previously: If Q lies in a code kernel I_{C}, our remark about distinct intersections at Z(I_{E}) shows that I_{Q} has a unique code C—and in fact must be the cone on that C with some or all of its edges. Otherwise, Q must be contained in the kernel for some edge; we can take this to be E above, so that Q ≤ I_{E}. Now if Q has order at least 3^{3}, or has order at 3^{2} but is not equal to Z(I_{E}), the same non-intersection argument gives I_{Q} a unique code on which it has cone structure, contractible. In the remaining case, we have Q = I_{E}, and then I_{Q} is the full subcomplex I_{P} of type 3B above, which we saw was contractible. This completes the proofs of contractibility for 1.4, establishing equivalence of Δ with A_{3}(G).

Checking equivariance in contractibility proofs leads via 1.7 to the property (BT/Δ). We also outline an alternative proof using the method of 1.5. It can be checked that any

element g of class 3B can be uniquely written as the product ab of commuting 3A-elements. Then any elementary E containing g must centralize a and b; and it follows that E can be uniquely embedded in an elementary F generated by 3A-elements. Using this uniqueness, if follows using 1.5 that A_{3}(G) is homotopy equivalent to the subposet of elementaries generated by 3A-elements. Now such subgroups are contained in code kernels of order 3^{5}; and since distinct code kernels intersect in at most 3^{2}, we can also use 1.5 to remove elementaries of orders 3^{3}, 3^{4}. So we have reduced A_{3}(G) to the 3-local geometry Δ defined by the indicated subgroups of order 3^{1}, 3^{2}, 3^{5}.

We are not aware of any treatment of the mod-3 cohomology of Ly.

3.7 Ly at p = 5 revisited

We recall from [RSY90], where projectivity was proved, that the first p = 5 geometry for Ly in our Table is the G̃_{2}-diagram geometry discoverd by Kantor and Tits, and described in [Kan81]. The second geometry is the truncation obtained by removing the points of the first. Since the geometries have different Euler characteristics, we cannot have both equivalent to A_{5}(G). And indeed the first is not fully local; it has point stabilizer G_{p} ≅ G_{2}(5).

So we now let Δ denote the second geometry, and show its equivalence with A_{5}(G). Since the vertices called lines in [RSY90] in fact correspond with elements in the Atlas class 5A, according to our usual philosophy we will rename these as the new points.

The geometry is fully 5-local: We have elementary plane kernel K_{π} ≅ 5^{3} and point kernel K_{p} ≅ 5^{1+4}4, with extraspecial U_{p} ≅ 5^{1+4}. We take I_{v} = U_{v} for vertices, and get get elementary intersection I_{pπ} ≅ 5^{2}. Thus we have all O_{5}(I_{σ}) > 1, hence all I_{σ} contractible.

For the necessary condition, we note that a Sylow 5-group is visible in the non-split extension in the plane group G_{π} ≅ 5^{3}SL_{3}(5); maximal elementaries have order 5^{3}, and those not conjugate to I_{π} are visible in extraspecial I_{p}. In particular, for elementary P the subcomplex I_{P} always contains at least a point or a plane.

For the case of P of order exactly 5, as usual we can quote the latter part of the proofs in [RSY90]; there are two classes of 5-elements, denoted 5A and 5B in the Atlas. For P of type 5A (the 5-central case in [RSY90]), we see I_{P} consists of the point p generating P, along with its 6 planes—and also with all further points q on those planes (corresponding to the remaining 5A-elements of I_{p} outside its center). As we have seen often before, the residue in I_{P} of a point q other than p consists just of a single plane π—since the distinct planes above p have kernels intersecting only at p. These q are then removable by 1.5, leaving the cone on p with its 6 planes, contractible. The case of P of type 5B is similar, but with no q occurring: such a P also determines a unique p of type 5A, and I_{P} consists only of the cone on p with its 6 planes.

So it remains to consider elementary Q of order at least 5^{2}. Suppose first Q contains an element of type 5B, determining P and p as above. Since distinct planes over p have kernels intersecting exactly at p, we see that I_{Q} contains a unique plane π—and so from structure of I_{P} must consist of π alone, or the edge π∗p—both contractible. So we may assume Q consists entirely of elements of type 5A. Again our remark on plane kernels intersecting exactly at p shows that I_{Q} contains at most one plane π; in fact this does occur, as there are just two cases: If such a Q has maximal order 5^{3}, it must be conjugate to I_{π}; and then I_{Q} is the cone on π with its 7 points. Otherwise Q is of order 5^{2}, and is conjugate to I_{pπ} = I_{p} ∩ I_{π}, and the same structure for I_{Q} as for order 5^{3} results. This completes the proofs of contractibility for 1.4, and establishes equivalence of Δ with A_{5}(G).

As usual, checking equivariance in the contractibility proofs leads via 1.7 to the Borel-Tits property (BT/Δ).

4 Toward further alternating-sum decompositions

Many interesting sporadic geometries G, Δ do not appear in our Table, since it is known that they do not satisfy the properties of projective L̃(Δ) and equivalence of Δ with A_{p}(G). However, there has long been at least collateral evidence (notably relative projectivity results using Thévenaz [Thé87, Thm 2.1]) supporting the general conjecture that the mod-p cohomology of G should be described ^{16} by a Webb-type alternating sum over Δ as in 1.2. As we have suggested by our way of stating 1.2, it seems that the reduced Lefschetz module of Δ should in fact be acyclic (though only sometimes projective). Recently there have been exciting breakthroughs in this direction by a number of authors; we breifly survey such results with an overview of the current situation, and indicate how we expect in a future work to give a methodical treatment of the remaining sporadic geometries.

The majority of the local geometries occur for p = 2; so as a yardstick of progress, we note that the conjecture should apply to geometries (in the broader sense of this paper) for 24 of the 26 sporadic groups—all those other than ON and Ly. ^{17} For the 10 sporadics with p = 2 in our Table, the result follows from the stronger assertion of homotopy equivalence with A_{2}(G).

16But we emphasize that this description falls well short of the actual determination of cohomology—as in the case of G of Lie type, with coefficients in the natural characteristic p.

17And we mention that for these two, the mod-2 cohomology is treated by methods not referring to any

Seemingly the first alternating-sum result beyond these (that is, for a local geometry Δ not in the G-homotopy type of A_{2}(G)) arose in the M_{12} work of Adem-Maginnis-Milgram [AMM91]: In this case the authors actually began with the alternating sum for the larger poset A_{2}(G), and demonstrated by direct computation that various terms cancel out—so that the final sum is over the usual 2-local geometry Δ (whose orbit complex is just the single edge on a pair of vertices).

Recent work of Benson [Ben95] establishes more directly the acyclicity result for the group Co3. Here the underlying reduction is from A_{2}(G) down to those elementaries containing only 2-central involutions; and the condition enabling this reduction is that the conjugacy class of such involutions should be closed under commuting products. In fact one can check that this condition makes it possible to treat a total of 6 further sporadic groups for p = 2, namely:

M_{12}, J_{2}, Ru, HS, Co3, Suz.

We mention also that the first 3 of these can be shown to have decompositions ^{18} by a different recent method of Maginnis [Mag96]. That is, the overall conjecture should now hold for 16 of the expected sporadics at p = 2. For p odd one can make a similar analysis of the applicability of the new methods of Benson and Maginnis; here we will just mention that the methods should usually provide an alternative approach to those cases of Tezuka-Yagita [TY96, 4.1] of extra-special Sylow groups of order p^{3} not also covered in the Table of this paper.

The previous paragraph summarizes the state of affairs up to roughly late 1995, when it seemed there might be significant obstructions to further progress. In particular, the new methods eventually produced a decomposition over a simplicial complex, but required considerations of much greater technical depth: Maginnis considered detailed structure of the associated spectral sequence for G-equivariant cohomology of Δ in a suitable relaxation of the hypothesis of 1.1; and Benson made use of results of Jackowski-McClure [JM92, Sec 7] in the context of homotopy colimits and Mackey functors on conjugation categories. There seemed to be no suitable extension of these methods to the next-larger sized sporadic groups such as He and Co1 at p = 2.

But very recent work of Dwyer [Dwya] [Dwyb] seems to open up an avenue that may allow a uniform treatment of the remaining sporadic geometries. We mention first (somewhat digressively) that an important goal of Dwyer’s work is in fact to unify various “homology approximation” results in the topological literature (including those of Quillen/Webb and Jackowski-McClure) by means of an elementary category-theoretic viewpoint.

deomposition: ON by Adem-Milgram [AM95b], and Ly in the 1996 UIC Ph.D. thesis of K. Umland (which in turn is based on the work on McL of Adem-Milgram [AM96]).

18Concerning full determination of cohomology of these groups: At the time of writing, Adem and Milgram are treating HS, and we understand Maginnis is treating J_{2}; possibly other cases have been completed.

A particularly useful technical innovation in Dwyer’s work is the introduction of the collection of subgroups he terms p-centric: namely those p-groups Q such that all p-elements of C_{G}(Q) in fact lie in Z(Q); Dwyer shows that this collection also determines approximations, much like those for the standard posets A_{p}(G) and B_{p}(G). The concept turns out to be closely related to the familiar group-theoretic notion of the normalizer N_{G}(Q) being pconstrained; this was implicitly involved in the definition of characteristic p-type used earlier in this paper (and also prominent in the original discovery of p-local geometries in [RS80]). We recall that a group N is p-constrained if for P a Sylow p-group of the pre-image of O_{p}(N/O_{p′}(N)), we have C_{N}(P) ≤ O_{p′}(N).P. (Most often we have O_{p′}(N) = 1, and then the condition takes the more standard form F^{∗}(N) = O_{p}(N)). The basic observation for our present purposes is that;

Q p-centric ⟹ N(Q) p-constrained.

For set N := N_{G}(Q), with P a Sylow p-group of the preimage of O_{p}(N̄), where N̄ := N/O_{p′}(N). Since Q is normal in N, we have Q̄ normal in N̄—hence Q̄ ≤ O_{p}(N̄) = P̄. Then Q lies in the preimage O_{p′}(N)P and is even normal there, so certainly Q ≤ P for any Sylow group P of this preimage. Thus we have C_{N}(P) ≤ C_{G}(Q). But the p-centric hypothesis on Q in fact shows that C_{G}(Q) = Z(Q) × O_{p′}(C_{G}(Q)) = Z(Q) × O_{p′}(N_{G}(Q)). So we have

C_{N}(P) ≤ C_{G}(Q) ≤ O_{p′}(N) × Z(Q) ≤ O_{p′}(N)P.

This shows that N is p-constrained, as asserted.

Dwyer further shows in [Dwyb] there is a “sharp” homology approximation (implying a corresponding alternating sum decomposition) for the intersection of the p-centric collection with B_{p}(G). ^{19} We will denote this intersection-subposet by B_{p}^{cen}(G); the above argument shows in fact that

B_{p}^{cen}(G) ⊆ B_{p}^{con}(G),

where we use the latter expression to denote the subposet of those Q with constrained normalizer. Indeed for our geometric examples G it usually seems to be the case that equality holds. Furthermore we remark that for G of characteristic p-type we have B_{p}(G) = B_{p}^{con}(G). So it would appear that a natural generalization of our Observation 1 would be

Conjecture 1: For sporadic G having a p-local geometry Δ, B_{p}^{cen}(G) = B_{p}^{con}(G) ∼ Δ.

This would provide an underlying explanation for the conjecture about decompositions over Δ suggested earlier in this section. Correspondingly, the natural extension of Observation 2 would be the Borel-Tits property stated only for constrained normalizers:

Conjecture 2: For (G, Δ) as above, and Q ∈ B_{p}^{con}(G), we have N(Q) in some G_{σ}.

Future work will aim to verify this property (BT^{con}/Δ) for the larger sporadic cases.

19Not that Dwyer follows that topological literature in terming the member of B_{p}(G) p-stubborn rather than p-radical.

We conclude with a brief indication of an important technical advantage of B_{p}^{cen}(G), namely that it ordinarily avoids the smallest and most troublesome p-groups. For B_{p}^{con}(G) the corresponding remark is that when N(Q) is not constrained, neither is N(P) for each 1 ≠ P ≤ Q. This seems to circumvent the obstructions we had encountered in analyzing the larger sporadic groups. For example, for the above-mentioned cases of He and Co1 at p = 2 we have worked out these posets, and shown them to be equivalent to the usual 2-local geometry, as conjectured. Nor does there seem to be any theoretical obstruction to a similar treatment of even the largest sporadic groups and their geometries. So in future work we expect to be able to treat them systematically with the new methods.

However, we should indicate one technical difference: Dwyer’s sharp decomposition for B_{p}^{cen}(G) is a “subgroup decomposition”, namely the alternating sum of the cohomology of the p-groups themselves; not the “normalizer decomposition” involving the cohomology of the full normalizer of the p-groups, which appear as the simplex stabilizers in our phrasing of 1.2. We suspect that the latter decomposition does hold, but it seems that proving it might be rather more subtle.

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