ON A HOMOTOPY RELATION BETWEEN THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR THE SPORADIC GROUP Co_{3}

JOHN MAGINNIS AND SILVIA ONOFREI

Abstract. We study the homotopy relation between the standard 2-local geometry Δ and the Bouc complex for the sporadic group Co_{3}. We also give a result concerning the relative projectivity of the reduced Lefschetz module L̃(Δ).

1. Introduction

The relationship between the p-local geometry and the Quillen poset of elementary abelian psubgroups or the Bouc poset of p-radical subgroups for certain sporadic groups was first noticed by Smith and Yoshiara [SY]. They studied the projectivity of the reduced Lefschetz module for sporadic group geometries and observed that the sporadic groups of characteristic p-type behave similarly to the finite Lie groups in defining characteristic p.

If G is a finite group of Lie type in characteristic p then the Bouc poset B_{p}(G) is the barycentric subdivision of the Tits building. If G is a sporadic group of characteristic p-type, the p-local geometry Δ(G) is G-homotopy equivalent to B_{p}(G). Moreover, similar to the case of Lie groups, there is a clear correspondence between the barycentric subdivision of Δ(G) and B_{p}(G). For the remaining sporadic groups the relationship is not obvious anymore; in fact, Δ(G) and B_{p}(G) are not always G-homotopy equivalent.

It is the purpose of this paper to give such a relationship for the Conway’s third sporadic group Co_{3}. Section 2 provides some notation and reviews standard results which will be used in proofs. In Section 3 we describe in detail the 2-local maximal “parabolic” geometry Δ for Co_{3}. In Section 4 we describe the relevant elements of the Bouc complex B_{2} of Co_{3} in geometric terms. In Section 5 we prove that the 2-local geometry Δ is G-homotopy equivalent with a subcomplex B̂_{2} of the Bouc complex. In Section 6 we prove that the fixed point set Δ^{z} of a central involution z is contractible. Then, using a result of Thévenaz [Th87], we conclude that the reduced Lefschetz module L̃(Δ) is projective relative to the collection 𝔛 = B_{2} \ B̂_{2} of subgroups of Co_{3}.

2. Notations, terminology and standard results

2.1. Subgroup complexes. Let G be a finite group and p a fixed prime which divides the order of G. A p-subgroup R of G is called p-radical if and only if R = O_{p}(N_{G}(R)). The p-subgroup R

Date: Monday, November 1, 2004.

1991 Mathematics Subject Classification. 20G05, 20C20, 51D20.

2 JOHN MAGINNIS AND SILVIA ONOFREI

is p-centric if Z(R) is a Sylow p-subgroup in C_{G}(R). The following two posets of p-subgroups are standard in the literature:

A_{p}(G) = {E | E nontrivial elementary abelian p-subgroup of G}
B_{p}(G) = {R | R nontrivial p-radical subgroup of G}

By abuse of notation we will use the same symbols for the associated simplicial complexes. The collection A_{p}(G) is known as the Quillen complex and the collection B_{p}(G) is known as the Bouc complex. The two complexes are G-homotopy equivalent.

Let G be a finite group. Let S be a Sylow p-subgroup of G and B_{p}(G) be the Bouc complex of p-radical subgroups. Define the following subcomplex of the Bouc complex:

B̂_{p}(G) = {U ∈ B_{p}(G) | Z(U) ∩ Z(S) ≠ ∅, for some S ∈ Syl_{p}(G)}.

We shall call B̂_{p}(G) the special Bouc complex.

Let E_{p}(G) denote the smallest collection of elementary abelian p-subgroups of G which contains elements of order p in the center of a Sylow p-subgroup, is closed under conjugation in G and is closed under taking products of commuting elements. This complex was introduced by Benson [Ben94] in order to study the mod 2 cohomology of Co_{3}.

The Robinson complex R_{p}(C) is defined over any collection C of subgroups of G and has chains of the form {H_{1} ≤ H_{2} … ≤ H_{n}; H_{i} ⊴ H_{n}; H_{i} ∈ C, i = 1, … n, ∀n}.

2.2. Homotopy techniques. Let us assume that Δ is a simplicial complex. We say that G acts admissibly on Δ if whenever g ∈ G fixes a simplex σ ∈ Δ then g fixes every face of σ. If v is a vertex of Δ, then Star_{Δ}(v) is the collection of simplices containing it and the residue of v, denoted Res_{Δ}(v) is the subcomplex of next lower dimension, obtained by deleting v from each simplex of Star_{Δ}(v). Then Star_{Δ}(v) = v ∗ Res_{Δ}(v), the simplicial join.

For σ ∈ Δ, let G_{σ} be the stabilizer of σ under this action and let U_{σ} denote the kernel of the action of G_{σ} on the residue of σ. The following is inspired from a standard property of the finite groups of Lie type acting on their buildings:

The Borel-Tits property (BT): Let G be a group which acts admissibly on the simplicial complex Δ. For each nontrivial p-subgroup U of G, there exists a simplex σ of Δ such that N_{G}(U) ≤ G_{σ}.

Notation 2.1. For P ≤ G, denote by Δ^{P} the subcomplex of Δ fixed by P.

Lemma 2.2 ([RSY], Lemma 2.1). For a vertex v of a simplicial complex Δ, if the residue Res_{Δ}(v) is contractible, then Δ is homotopy equivalent to Δ \ Star(v).

Lemma 2.3 ([RSY], Section 1). Let σ ∈ Δ be a simplex of maximal dimension with τ as a face. Assume that σ is the only simplex of maximal dimension with τ as a face. Then the process of removing σ from Δ, by collapsing σ down onto its faces other than τ, is a homotopy equivalence.

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR Co_{3} 3

3. The 2-local geometry Δ of Co_{3}

Let Δ denote the 2-local maximal “parabolic” geometry associated with the third Conway sporadic group G = Co_{3}. This geometry was first mentioned in [RStr84]. This is a rank 3 geometry, whose objects will be denoted points P, lines L, and M-spaces. The objects of this geometry correspond to pure central elementary abelian subgroups of G, as follows: the points correspond to subgroups A_{1} = 2, lines to A_{2} = 2^{2} and M-spaces to A_{4} = 2^{4}. Incidence is given by containment. Δ can also be regarded as a simplicial complex of dimension two with three types of vertices. The group G acts (faithfully) flag-transitively on the geometry. The stabilizers of the three types of objects are:

G_{p} = 2^{.}S_{6}(2) = N_{G}(A_{1})  for a point p ∈ P;
G_{L} = 2^{2+6}3.(S_{3} × S_{3}) = N_{G}(A_{2}) for a line L ∈ L;
G_{M} = 2^{4.}L_{4}(2) = N_{G}(A_{4})  for an M-space M.

The flag stabilizers can be easily determined and they are:

\begin{matrix}G_{pL} = 2^{2+6}(S_{3} × S_{3}) & G_{pM} = 2_{+}^{1+6}L_{3}(2) \\ G_{LM} = 2^{2+6}(S_{3} × S_{3}) & G_{pLM} = [2^{9}] .S_{3} = [2^{10}3]. \end{matrix}

The geometry Δ is pure 2-local, in the sense of [SY], that is, all the simplex stabilizers are 2-local subgroups of G. All vertex stabilizers are maximal 2-local subgroups. Δ has the following diagram:

∘^{P}  ∘^{L} □  ∘^{M}

Notation 3.1. For X ∈ {P, L, M} and F a flag of Δ, we will denote by X_{F} the collection of all objects in X incident with F.

Notation 3.2. For p ∈ P let p^{⊥} = {q ∈ P | q and p are incident with the same line}.

In what follows it will be useful to regard Δ as a point-line geometry also, that is, the lines and the M-spaces are identified with the subsets of points they are incident with. Some of the properties of Δ which can be read from the diagram:

1). For p ∈ P let Res_{Δ}p = (L_{p}, M_{p}), the collection of all lines and M-spaces containing p. Then Res_{Δ}p is isomorphic with the geometry of lines and planes of a polar space of type C_{3} over the field 𝔽_{2}. Note there are 315 lines and 135 M-spaces on a given point p ∈ P.

2). If M ∈ M, then Res_{Δ}M = (P_{M}, L_{M}), the collection of points and lines lying in M is the truncation to points and lines of a projective space P G(3, 2). It is immediate that there are 15 points and 35 lines incident with M.

3). For L ∈ L, then Res_{Δ}L = (P_{L}, M_{L}) is a digon, consists of three points and three M-spaces

incident with L. Every point in P_{L} is incident with every element of M_{L}. 4). Given a point p and a line L, with p ∉ L then p^{⊥} ∩ L can be empty, a point or L. 5). If (p, M) ∈ P × M; p ∈ M or p^{⊥} ∩ M is a line, a point or the empty set. 6). Two distinct M-spaces can intersect at a line, a point or the empty set.

4 JOHN MAGINNIS AND SILVIA ONOFREI

Remark 3.3. Recall that G = Co_{3} has two classes of involutions, which we shall denote 2A (the central involutions) whose centralizer is 2^{.}S_{6}(2), and 2B (the noncentral involutions) whose centralizer is 2 × M_{12}. Given any two commuting involutions of central type, their product is an involution of central type. The product of a central involution and a non-central involution (if an involution) is a non-central involution. The product of two commuting non-central involutions can be either central or noncentral. The complex E_{2} = E_{2}(Co_{3}) contains four conjugacy classes of subgroups, of orders 2, 2^{2}, 2^{3}, 2^{4}; see [Ben94]. Each subgroup 2^{3} is contained in a unique subgroup 2^{4} (the centralizer of the 2^{3}). Let E_{2}^{−} be the subcomplex of E_{2} with the subgroups of the form 2^{3} removed. It is a direct consequence of Lemma 2.3 that E_{2} and E_{2}^{−} are G-homotopy equivalent. Note that E_{2}^{−} = Δ.

Remark 3.4. Let Δ̃ denote the involution geometry of G; that is, the point-line geometry whose points are the involutions of G. Two involutions are collinear if they commute. Then Δ can be identified with a (geometric) subspace Δ̃ whose points correspond to the central involutions, that is every exterior line of Δ̃ intersects Δ at a point or the empty set. Therefore, we can regard Δ as a “central involution geometry”.

4. A geometric description of the special Bouc complex of Co_{3}

In this section G = Co_{3} and Δ represents its 2-local geometry; further let B_{2} = B_{2}(Co_{3}) and B̂_{2} = B̂_{2}(Co_{3}). The Bouc complex of Co_{3} was determined in [An99], from which we reproduce the following table:

Simplex  Name  R  Z(R)  N_{G}(R)
p  R_{p}  2  A^{1}  2^{.}S_{6}(2)
−  R_{2}  2  B^{1}  2 × M_{12}
−  R_{3}  2^{2}  B^{3}  A_{4} × S_{5}
−  R_{4}  2^{3}  B^{7}  (2^{3} × S_{3})F_{7}^{3}
M  R_{M}  2^{4}  A^{15}  2^{4}A_{8}
p□  R_{p□}  2^{1+5}  A^{1}  2^{1+5}S_{6}
pM  R_{pM}  2_{+}^{1+6}  A^{1}  2_{+}^{1+6}L_{3}(2)
M□  R_{M□}  2^{3+4}  A^{7}  2^{3+4}L_{3}(2)
𝕃  R_{𝕃}  2^{2+6}  A^{3}  2^{2+6}3.(S_{3} × S_{3})
pM𝕃  R_{pM𝕃}  2^{4}.2^{2+3}  A^{1}  2^{4}.2^{2+3}.S_{3}
M𝕃□  R_{M𝕃□}  2^{4}.2^{2+3}  A^{1}  2^{4}.2^{2+3}.S_{3}
pM□  R_{pM□}  2^{4}.2_{+}^{1+4}  A^{1}  2^{4}.2_{+}^{1+4}.S_{3}
p𝕃□  R_{p𝕃□}  [2^{9}]  A^{1}  [2^{9}] .S_{3}
pM𝕃□  R_{pM𝕃□}  [2^{10}]  A^{1}  [2^{10}]

The notation □ stands for a typical vertex of the truncated type in Res_{Δ}v, where v ∈ {P, M}. It corresponds to the square node in the diagram of Δ. Note that □ does not represent a class of

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR Co_{3} 5

objects in the geometry Δ. In Res_{Δ}p, □ stands for a structure of 15 lines and 15 M-spaces which form a generalized quadrangle. In Res_{Δ}M, □ can be identified with a plane of the projective space P G(3, 2). Groups involving □, such as R_{p□} or R_{M□} can be constructed as the inverse images under the quotient maps 2^{.}S_{6}(2) ↠ S_{6}(2) or 2^{4.}L_{4}(2) ↠ L_{4}(2) of unipotent radicals of corresponding parabolics. The groups R_{pM} and R_{pM𝕃} can either be constructed in the same manner or as the kernels of actions on the residues of the corresponding flags. Note, R_{𝕃} is the maximal normal 2-subgroup of the kernel 2^{2+6}.3 of the action on the residue of a line.

Among the 2-radical subgroups all but the first four conjugacy classes of groups are 2-centric. Also it is easy to see that B̂_{2} = B_{2} \ {R_{2}, R_{3}, R_{4}}.

As mentioned in the previous section, the natural 2-local geometry Δ can be regarded as a pure central involution geometry. In this section we will describe the subcomplex B̂_{2} in more geometric terms, using the properties of the central involutions of G.

We start with a weaker version of the Borel-Tits property mentioned in Section 2:

(BT)^{c}: For each non-trivial 2-subgroup U of G such that Z(U) contains a central involution, there exists a simplex σ of Δ such that N_{G}(U) ≤ G_{σ}.

Proposition 4.1. The pair (G, Δ) satisfies the property (BT)^{c}.

Proof. Let U ≤ G be a non-trivial 2-subgroup such that Z(U) contains a central involution. Let H denote the collection of type A involutions in Z(U). Recall that H is closed under taking products. Thus H is an elementary abelian subgroup of rank at most 4 of U. Then N_{G}(U) ≤ N_{G}(H). If H = 2^{3} then N_{G}(H) ≤ N_{G}(2^{4}). Then N_{G}(H) ∈ {G_{p}, G_{L}, G_{M}}. Thus the conclusion follows. □

Remark 4.2. Note that the pair (G, B̂_{2}) satisfies the property (BT)^{c}. To see this let U be a 2subgroup of G with the property that Z(U) contains an involution of type A. It is a direct consequence of Proposition 4.1 that there exists R ∈ B̂_{2}(G) with N_{G}(U) ≤ N_{G}(R).

In what follows we shall describe each of the subgroups of B̂_{2} in geometric terms; this means that we describe the incidence structure given by the central involutions.

R_{p} = 2

This corresponds to a single a central involution and thus a point of the geometry.

R_{M} = 2^{4}

This is a pure A elementary abelian subgroup of rank 4. Its 15 involutions are in one-to-one correspondence with the points of M, which is a projective space of rank 3.

R_{p□} = 2^{1+5}

It is an easy exercise in GAP to show that there are exactly 31 central involutions in this group. Recall that Z(R_{p□}) = A^{1} and let p denote the point corresponding to this central involution. Then R_{p□} contains a collection of 15 lines and 15 planes on the point p which form a generalized quadrangle GQ(2, 2).

R_{𝕃} = 2^{2+6}

Proposition 4.3. A_{2} = 2^{2} ≤ R_{M} = 2^{4} if and only if R_{M} ≤ R_{𝕃} = O_{2}(N_{G}(A_{2})).

Proof. From the geometry, there exist three subgroups of the form 2^{4} containing the central A_{2} in R_{𝕃}, which are transitively permuted by a N_{G}(R_{𝕃})-subgroup^{1} of order 3. Thus, if A_{2} ≤ R_{M} then R_{M} ≤ R_{𝕃}. Conversely, assume that R_{M} ≤ R_{𝕃}, but some element z ∈ A_{2} does not lie in R_{M}. Note that z is central so ⟨z, R_{M}⟩ = 2^{5}, a contradiction since the 2-rank of Co_{3} is 4. Thus every R_{M} ≤ R_{𝕃} contains A_{2}. Note when A_{2} ≤ R_{M} ≤ R_{𝕃} then R_{M} ⊴ R_{𝕃}, since the quotient group 2^{6} is abelian. □

It follows that R_{𝕃} contains 39 = (15−3)×3+3 involutions of central type. Also Z(R_{𝕃}) contains three involutions, and they form a line L. If z ∈ R_{𝕃} but z ∉ Z(R_{𝕃}), then ⟨z, Z(R_{𝕃})⟩ = 2^{3} is contained in a unique subgroup of the form 2^{4}. In geometric terms, Proposition 4.3 says that the line L lies in three M-spaces in 𝕃. Any two of these M-spaces intersect in precisely L. We refer to 𝕃 as “line structure”.

R_{pM} = 2_{+}^{1+6}

Proposition 4.4. Let z ∈ G be a central involution. Fix a 2-subgroup A_{1} = ⟨a_{1}⟩ ⊆ R_{M}. The involution z is in the kernel of the G-action on Res_{Δ}(pM) if and only if z ∈ R_{M}. Thus R_{pM} contains a unique subgroup R_{M} and 15 central involutions.

Proof. If z ∈ R_{M} then z commutes with a_{1} ∈ R_{M}, so z stabilizes the pM-flag and the 7 lines on p which lie in M. Thus z is in the kernel of the G-action on Res_{Δ}(pM). Let now z be an involution in the kernel of the action of G on Res_{Δ}(pM). Assume by way of contradiction that z ∉ R_{M}. Since [a_{1}, z] = 1 it follows that z^{⊥} ∩ M ≠ ∅. We assumed that z stabilized M, thus the action of z on M is linear (since z stabilizes the lines, z acts trivially on the quotient space M/A_{1}). This implies that at least 7 points are fixed. But z ∉ M and since z^{⊥} ∩ M can be at most one line in this case, we reached a contradiction. Thus we must have z ∈ M. The last statement follows at once now. □

Proposition 4.5. There are no adjacencies betwen R_{pM} and R_{𝕃}.

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR Co_{3} 7

Proof. Let us assume by way of contradiction that R_{pM} ≤ R_{𝕃} and let ⟨a⟩ = Z(R_{pM}) and ⟨b, c⟩ = Z(R_{𝕃}). If a ∉ Z(R_{𝕃}) then ⟨a, b, c⟩ ≃ 2^{3} and R_{pM} ≤ C_{G}(2^{3}) ≃ 2^{4}, a contradiction. Thus a ∈ Z(R_{𝕃}). Note that the subgroup Z(R_{𝕃}) ≃ 2^{2} is simultaneously the center and the commutator subgroup of R_{𝕃}. Next, consider the composition of maps: R_{pM} → R_{𝕃} ↠^{φ} 2^{6}. Notice that Z(R_{𝕃}) is not a subgroup of R_{pM} since this would imply Z(R_{𝕃}) ≤ Z(R_{pM}). Therefore the composition R_{pM} ↠ 2^{6} is also surjective. Now, choose two elements x_{1}, y_{1} ∈ R_{𝕃} such that [x_{1}, y_{1}] ∈ Z(R_{𝕃}) but it is not in Z(R_{pM}). Consider the images of these elements under the projection map φ and choose x_{2}, y_{2} ∈ R_{pM} such that φ(x_{1}) = φ(x_{2}) and φ(y_{1}) = φ(y_{2}). This implies that x_{1} = x_{2} · x and y_{1} = y_{2} · y with x, y ∈ Z(R_{𝕃}). It follows then [x_{1}, y_{1}] = [x_{2}, y_{2}], a contradiction. □

1This can be seen using the Benson presentation, with H = ⟨a_{1}, a_{2}⟩, 3 = ⟨c_{3}h⟩, M_{1} = ⟨a_{1}, a_{2}, a_{3}, a_{4}⟩, M_{2} = ⟨a_{1}, a_{2}, b_{1}, b_{2}⟩, M_{3} = ⟨a_{1}, a_{2}, a_{2}a_{4}b_{1}c_{1}, a_{2}a_{3}a_{4}b_{2}c_{2}⟩.

R_{M□} = 2^{3+4}

If z ∈ R_{M□} is a central involution of G then either z ∈ 2^{3}◁R_{M□} or ⟨z, 2^{3}⟩ ≃ 2^{4} is a pure A elementary abelian subgroup. But given a subgroup of the form 2^{3}, there exists a unique 2^{4} = C_{G}(2^{3}) containing it, since every plane is contained in a unique M-space, so z ∈ M. In particular R_{M□} contains exactly 15 central involutions. Since R_{pM} ≄ R_{M□} we see that these groups are not incident in the Bouc complex. There are no incidences between R_{M□} and R_{𝕃}, since the commutator subgroup of the first is 2^{3} and the commutator subgroup of the second is only 2^{2}.

Let H_{σ} be the stabilizer of a 2-simplex σ containing an M-space M = ⟨a_{1}, a_{2}, a_{3}, a_{4}⟩. Let a_{1} ∈ Z(S), where S is a Sylow 2-subgroup of G. Let U_{σ} = O_{2}(H_{σ}) be the kernel of the action of G on the residue of the simplex σ. It is clear that H_{σ} ≤ 2^{4}L_{4}(2). Also notice that U_{σ} contains a normal subgroup of the form 2^{4} ≃ R_{M}. Let z ∈ U_{σ} be a central involution. Then z stabilizes M.

In what follows we will determine the collection of involutions in R_{M} which are fixed by the action of z on M; geometrically these correspond to points in z^{⊥} ∩ M. Now recall that in Δ, z^{⊥} ∩ M can be empty, a point, a line or the whole M. Since M contains a_{1} ∈ Z(S), z fixes at least one element of M, thus z^{⊥} ∩ M ≠ ∅. We will assume that z ∉ M.

R_{pM𝕃} ≃ 2^{4}.2^{2+3}

Take σ = pM𝕃. Then \overline{U}_{σ} = U_{σ}/2^{4} = U_{σ}/R_{M} corresponds to the kernel of the action of the stabilizer of a flag of type p𝕃 in L_{4}(2), that is \overline{U}_{σ} = O_{2}(Stab_{Res_{Δ}M}p𝕃), which is the unipotent radical of the corresponding minimal parabolic subgroup of L_{4}(2) (in standard notation). This is a group of the form:

\overline{U}_{σ} ≃ ( \begin{matrix}1 & * & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) ≃ 2^{2+3}

Let us assume that \overline{z} is the image under the projection map of the central involution z. Then \overline{z} is contained in one of the two maximal elementary abelian subgroups of \overline{U}_{σ}, which have ranks 3 and 4 respectively.

If \overline{z} ∈ 2^{4} where:

2^{4} ≃ ( \begin{matrix}1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

then \overline{z} fixes ⟨a_{1}, a_{2}⟩, a line in M. Furthermore, since z ∉ M it follows that z^{⊥} ∩ M is a line in this case; but, this implies that the upper right 2 × 2 corner must be a rank 2 matrix. There are 6 choices of such matrices and thus 6 × 4 corresponding central involutions.

Let us assume now that \overline{z} ∈ 2^{3} where:

2^{3} ≃ ( \begin{matrix}1 & * & * & * \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

In this case we reach a contradiction with the geometric properties; these matrices act trivially on M/A_{1} so they fix at least 7 points. But we assumed z ∉ M, and the set z^{⊥} ∩M cannot be more than a line. Thus, in this case there are too many fixed points. Therefore, in the maximal elementary abelian subgroup of \overline{U}_{σ} of the form 2^{3}, there is no \overline{z} which satisfies our assumptions. Note this 2^{3} is the image of R_{pM}, so consideration of these matrices yields another proof that R_{pM} contains only the 15 central involutions of R_{M}.

Hence, the 2-simplex pM𝕃 contains a total of 39 central involutions: 15 in M and 6 × 4 more. Geometrically this structure can be thought of as a collection of three M-spaces which have a line L in common; therefore this is same configuration of points as in a line structure 𝕃.

R_{M𝕃□} = 2^{4}.2^{2+3}

Assume now that σ = M𝕃□. Thus \overline{U}_{σ} ≃ 2^{2+3} where now:

\overline{U}_{σ} ≃ ( \begin{matrix}1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \end{matrix} ) ≃ 2^{2+3}

Hence, we can use a similar argument as for the previous group. Thus R_{M𝕃□} contains exactly the 39 central involutions of a line structure 𝕃. Note that the 2^{3} given by the fourth column equals the image of R_{M□}, showing again that R_{M□} contains only 15 central involutions.

R_{pM□} = 2^{4}.2_{+}^{1+4}

Let σ = pM□ and U_{σ} ≃ 2^{4}.2_{+}^{1+4} with \overline{U}_{σ} = 2_{+}^{1+4} being the unipotent radical of the minimal parabolic subgroup corresponding to the central node of the L_{4}(2) diagram. This is a group of the form:

2_{+}^{1+4} ≃ ( \begin{matrix}1 & * & * & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \end{matrix} )

Let z ∈ R_{pM□} be a central involution of G such that z ∉ M, so that the image \overline{z} in \overline{U}_{σ} is also an involution. From the matrix above, \overline{z} fixes one of the following lines: either ⟨a_{1}, a_{2}⟩, ⟨a_{1}, a_{3}⟩ or ⟨a_{1}, a_{2} + a_{3}⟩. Since we have assumed that z ∉ R_{M}, it follows that \overline{z} cannot fix anything else in M. If z fixes the line ⟨a_{1}, a_{2}⟩ then \overline{z} is of the form:

( \begin{matrix}1 & 0 & 1 & * \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

and there are 2 such matrices. In this case, the corresponding geometric structure consists of M together with 3 × 2 planes which intersect M at the three lines on p = ⟨a_{1}⟩, lying in a fixed plane ⟨a_{1}, a_{2}, a_{3}⟩ of M. This shows that there are 15 + 6 × 4 = 39 central involutions in R_{pM□}.

R_{p𝕃□} = [2^{9}] and R_{pM𝕃□} = [2^{10}]

Note R_{p□} ≤ R_{p𝕃□} and R_{𝕃} ≤ R_{p𝕃□}, which implies that R_{p𝕃□} contains at least 55 central involutions. A computation using GAP shows that the Sylow 2-subgroup R_{pM𝕃□} contains exactly 55 central involutions. The structure of the central involutions in this group can be described as follows: consider the 15 lines and 15 planes on a fixed p□ flag. Since R_{p𝕃□} stabilizes a line structure 𝕃 there are three M-spaces on a fixed line L on p and contained in □. Thus there are 39 points from the line structure. There are 16 points on the remaining 15 − 7 = 8 lines.

Consider the structure corresponding to pM𝕃□; it contains unique p𝕃□, pM□, 𝕃, p□, three Mspaces and 55 points. In the proof of Theorem 1 we will use the detailed structure of pM𝕃□ so we have to develop some more notation. Let M_{1}, M_{2}, M_{3} be the three M-spaces in 𝕃; they have the property that M_{1} ∩ M_{2} ∩ M_{3} = L, a line. Denote p_{1}, p_{2}, p_{3} the three points on L, with p_{1} the central point in p□. Let M_{1} be the unique M-space in pM□. Let now p□ ∩ M_{i} = □_{i}, with i ∈ {1, 2, 3}, these correspond to the three planes which contain the line L.

5. The homotopy relation

In this Section we maintain all the notations introduced in the previous Section. Let Δ_{1} = {R_{p}, R_{M}, R_{𝕃}} denote the subposet of B_{2} with three types of objects. Next let R_{2} denote the complex obtained by applying the Robinson construction to Δ_{1}.

Proposition 5.1. a). R_{2} is homeomorphic to Δ. b). R_{2} is homotopy equivalent to Δ_{1}.

10 JOHN MAGINNIS AND SILVIA ONOFREI

Proof. a). Note that R_{M} ⊴ R_{𝕃} if and only if Z(R_{𝕃}) ≤ R_{M}, and R_{p} ⊴ R_{𝕃} if and only if R_{p} ≤ Z(R_{𝕃}).

b). Recall that R_{𝕃} = 2^{2+6} contains 39 involutions of central type. Of course Z(R_{𝕃}) contains 3 involutions which correspond to the points of a line L. If z ∈ R_{𝕃} but z ∉ Z(R_{𝕃}), then ⟨z, Z(R_{𝕃})⟩ = 2^{3} is contained in a unique R_{M}. The line L lies in three M-spaces, which all lie in 𝕃. Any two intersect in precisely L. If R_{q} = 2 is not normal in R_{𝕃}, then there exists a unique R_{M} with R_{q} ≤ R_{M} ≤ R_{𝕃}. So the edge R_{q} ≤ R_{𝕃} is contained in a unique simplex of maximal dimension (triangle). So we can apply Lemma 2.3 and collapse this edge. □

In this Section we will prove the main theorem of the paper:

Theorem 1. The two complexes Δ and B̂_{2}(G) are homotopy equivalent.

We will proceed in a number of steps; at each step we will perform a homotopy retraction, which will use one of the results given in Lemmas 2.2 or 2.3.

Proposition 5.2. B̂_{2} is homotopy equivalent to the subcomplex B̂_{2}^{I} obtained by removing from B̂_{2} the vertices R_{pM} and R_{M□} together with all the simplices containing them.

Proof. There exists a unique R_{M} in R_{pM}, and there are exactly 15 involutions of central type, all incident to R_{M} (see Proposition 4.4), so we have chains of the form R_{p′} ≤ R_{M} ≤ R_{pM}. It follows that the residue of R_{pM} is a cone on this unique R_{M} and thus contractible. According to Lemma 2.2 we can remove the vertices of the form R_{pM} together with their stars.

There exists a unique R_{M} ≤ R_{M□} and there are no adjacencies bewteen R_{𝕃} and R_{M□}. There are exactly 15 involutions in R_{M□}, all incident to R_{M}. Thus, the residue of R_{M□} contains chains of the form R_{p′} ≤ R_{M} ≤ R_{M□}, with p′ one of the involutions in R_{M}. Therefore the residue of R_{M□} is a cone on a unique R_{M}. Again, an application of Lemma 2.2 gives the result.

Note R_{p□} is not a subgroup of either R_{pM} or R_{M□} since R_{p□} contains 31 central involutions. □

Proposition 5.3. B̂_{2}^{I} is homotopy equivalent to the subcomplex B̂_{2}^{II} obtained by removing from B̂_{2}^{I} the vertices R_{pM𝕃} and R_{M𝕃□} together with all the simplices containing them.

Proof. In B̂_{2}^{I} the residues of both vertices R_{pM𝕃} and R_{M𝕃□} are cones on R_{𝕃}, therefore two applications of Lemma 2.2 prove the Proposition. □

5.1. Flags in B̂_{2}^{II}. Let F : x_{1} ⊆ x_{2} … ⊆ x_{n} be a flag in B̂_{2}^{II}, where the object x_{i} is assumed to have type i, and i ∈ {1, … 5}. Then the stabilizer of the object x_{n} acts on the collection of objects of type 1 incident with x_{n}. In most of the cases this action is not transitive. In the first column of the table we give the type of the flag F, in the second column the order of the stabilizer of the object x_{n} in F and in the third column we give the number of elements in the orbits of Stab_{G}(x_{n}) on the collection of objects of type 1 incident with x_{n}.

Rank 1 flags  Rank 3 flags
p□  2^{10} · 3^{2} · 5 1  p ⊆ M ⊆ pM□  2^{10} · 3 1, 6, 8
pM□  2^{10} · 3  1  p ⊆ M ⊆ p𝕃□  2^{10} · 3 1, 2, 12, 24
p𝕃□  2^{10} · 3  1  p ⊆ M ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
pM𝕃□  2^{10}  1  p ⊆ 𝕃 ⊆ p𝕃□  2^{10} · 3 1, 2, 12, 24
p ⊆ 𝕃 ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
Rank 2 flags  p ⊆ p□ ⊆ p𝕃□  2^{10} · 3 1, 2, 12, 16
p ⊆ p□  2^{10} · 3^{2} · 5 1, 30  p ⊆ p□ ⊆ pM□  2^{10} · 3 1, 6, 24
p ⊆ pM□  2^{10} · 3  1, 6, 8, 24  p ⊆ p□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 16
p ⊆ p𝕃□  2^{10} · 3  1, 2, 12, 16, 24  p ⊆ pM□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
p ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16, 16 p ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16, 16
M ⊆ pM□  2^{10} · 3  1  M ⊆ 𝕃 ⊆ p𝕃□  2^{10} · 3 3
M ⊆ p𝕃□  2^{10} · 3  3  M ⊆ 𝕃 ⊆ pM𝕃□  2^{10}  1, 2
M ⊆ pM𝕃□  2^{10}  1, 2  M ⊆ pM□ ⊆ pM𝕃□  2^{10}  1
𝕃 ⊆ p𝕃□  2^{10} · 3  1  M ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2
𝕃 ⊆ pM𝕃□  2^{10}  1  𝕃 ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1
p□ ⊆ pM□  2^{10} · 3  1  p□ ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1
p□ ⊆ p𝕃□  2^{10} · 3  1  p□ ⊆ pM□ ⊆ pM𝕃□  2^{10}  1
p□ ⊆ pM𝕃□  2^{10}  1
pM□ ⊆ pM𝕃□ 2^{10}  1  Rank 4 flags
p𝕃□ ⊆ pM𝕃□ 2^{10}  1  p ⊆ M ⊆ 𝕃 ⊆ p𝕃□  2^{10} · 3 1, 2, 12, 24
p ⊆ M ⊆ 𝕃 ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
p ⊆ M ⊆ pM□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8
p ⊆ M ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
p ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 8, 16
p ⊆ p□ ⊆ pM□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 16
p ⊆ p□ ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2, 4, 8, 16
M ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□  2^{10}  1, 2

Rank 5 flags

p ⊆ M ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ 2^{10}  1, 2, 4, 8, 8, 16

Table I

Proof of the Theorem 1. The rest of the proof of Theorem 1 is straightforward, involving a sequence of 35 steps of homotopy retractions; a few of them use the fact that the residue is contractible, Lemma 2.2, but most of them use the collapsibility property of certain flags, Lemma 2.3.

The list of simplices in B̂_{2}^{II} was given in Table 1. In what follows we will homotopically retract B̂_{2}^{II} down to Δ_{1}. Recall that a maximal simplex Σ is free over some maximal face σ, if Σ is the only maximal simplex with σ as a face. In this case we can use Lemma 2.3 to collapse Σ down onto its faces other than σ. This means that we can remove Σ and σ from the complex to obtain a homotopy equivalent subcomplex. Note that Σ is not necessarily a simplex of maximal dimension; it has to be a maximal simplex, in the sense that it is not a face of a larger simplex in the complex.

12 JOHN MAGINNIS AND SILVIA ONOFREI

Note that since this method depends on the maximality of Σ and since the removal of some simplices might make other simplices maximal, the sequence of homotopy retractions we are performing has to be done in the order indicated here.

We begin by homotopically retracting the following pairs of simplices:

\begin{matrix}Step & Σ & σ \\ 1. & p ⊆ M ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ & p ⊆ M ⊆ 𝕃 ⊆ pM𝕃□ \\ 2. & M ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ & M ⊆ 𝕃 ⊆ pM𝕃□ \\ 3. & p ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ & p ⊆ 𝕃 ⊆ pM𝕃□ \\ 4. & 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ & 𝕃 ⊆ pM𝕃□ \end{matrix}

We will use the notation: (k_{1}) x_{1} ⊆ … (k_{n−1}) x_{n−1} ⊆ x_{n} to denote k_{1} · k_{2} . . . · k_{n−1} simplices of the form x_{1} ⊆ … x_{n}, for which there are k_{i} choices for the vertices of type i ∈ {1, … n − 1}.

Step 5: We will discuss this step in detail since the approach we use here will be repeated. Consider a simplex Σ : p ⊆ M ⊆ p𝕃□ ⊆ pM𝕃□. We consider here the points p ∈ p𝕃□ \ p□. There are 8 × 3 such points, 8 points in each of the three M-spaces of pM𝕃□. This means that for a fixed p𝕃□ ⊆ pM𝕃□, there are 24 simplexes Σ of the type chosen above. We write this as (8) p ⊆ (3) M ⊆ p𝕃□ ⊆ pM𝕃□. Fix one such simplex Σ. This is a simplex of maximal dimension and free over the face σ : p ⊆ p𝕃□ ⊆ pM𝕃□. To see this note that p ∉ L. Thus ⟨p, L⟩ is a plane on L and thus lies in a unique M-space. Note the simplex p ⊆ 𝕃 ⊆ p𝕃□ ⊆ pM𝕃□ was removed in Step 3. Thus Σ is free over σ. Finally we can apply Lemma 2.3 and remove these two simplices.

The remaining steps are listed in the following Table. In the first column we number the steps. In the second column we give the maximal simplex Σ and in the third column we give a face σ of Σ such that Σ is free over σ. In the fourth column we specify the collection of points in σ. We use the notation described above.

Step Σ  σ  Points in σ
5.  (8) p ⊆ (3) M ⊆ p𝕃□ ⊆ pM𝕃□ (24) p ⊆ p𝕃□ ⊆ pM𝕃□ p𝕃□ \ p□
6.  (16) p ⊆ p□ ⊆ p𝕃□ ⊆ pM𝕃□  (16) p ⊆ p𝕃□ ⊆ pM𝕃□ p□ \ 𝕃
7.  (24) p ⊆ p□ ⊆ pM□ ⊆ pM𝕃□ (24) p ⊆ pM□ ⊆ pM𝕃□ p□ \ M_{1}
8.  (8) p ⊆ M ⊆ pM□ ⊆ pM𝕃□  (8) p ⊆ pM□ ⊆ pM𝕃□ M_{1} \ p□
9.  (12) p ⊆ (3) M ⊆ 𝕃 ⊆ p𝕃□  (36) p ⊆ 𝕃 ⊆ p𝕃□  𝕃 \ L
10. (8) p ⊆ (3) M ⊆ p𝕃□  (24) p ⊆ p𝕃□  p𝕃□ \ p□
11. (16) p ⊆ p□ ⊆ p𝕃□  (16) p ⊆ p𝕃□  p□ \ 𝕃
12. (24) p ⊆ p□ ⊆ pM□  (24) p ⊆ pM□  p□ \ M_{1}
13. (8) p ⊆ M ⊆ pM□  (8) p ⊆ pM□  M_{1} \ p□
14. (8) p ⊆ p□ ⊆ p𝕃□ ⊆ pM𝕃□  (8) p ⊆ p□ ⊆ pM𝕃□  (□_{2} ∪ □_{3}) \ L
15. (4) p ⊆ (2) M ⊆ p𝕃□ ⊆ pM𝕃□ (8) p ⊆ p𝕃□ ⊆ pM𝕃□ (□_{2} ∪ □_{3}) \ L
16. (16) p ⊆ p□ ⊆ pM𝕃□  (16) p ⊆ pM𝕃□  p□ \ 𝕃
17. (8) p ⊆ (3) M ⊆ pM𝕃□  (24) p ⊆ pM𝕃□  𝕃 \ p□
18. (4) p ⊆ (2) M ⊆ pM𝕃□  (8) p ⊆ pM𝕃□  (□_{2} ∪ □_{3}) \ L
19. (3) p ⊆ (2) M ⊆ p𝕃□ ⊆ pM_{1}𝕃□ (3) p ⊆ (2) M ⊆ pM𝕃□ L

M ∈ {M_{2}, M_{3}}

20. (4) p ⊆ p□ ⊆ p𝕃□ ⊆ pM_{i}𝕃□  (12) p ⊆ p□ ⊆ p𝕃□  (□_{1} ∪ □_{2} ∪ □_{3}) \ L

i ∈ {1, 2, 3}

21. (4) p ⊆ M ⊆ p𝕃□ ⊆ pM𝕃□  (4) p ⊆ p𝕃□ ⊆ pM𝕃□ □_{1} \ L
22. (4) p ⊆ p□ ⊆ pM□ ⊆ pM𝕃□  (4) p ⊆ p□ ⊆ pM𝕃□  □_{1} \ L
23. (4) p ⊆ M ⊆ pM□ ⊆ pM𝕃□  (4) p ⊆ pM□ ⊆ pM𝕃□ □_{1} \ L
24. (2) p ⊆ M ⊆ pM□ ⊆ pM𝕃_{i}□  (6) p ⊆ M ⊆ pM□  □_{1} \ p_{1}

i ∈ {1, 2, 3}

25. (2) p ⊆ p□ ⊆ pM□ ⊆ pM𝕃_{i}□ (6) p ⊆ p□ ⊆ pM□  □_{1} \ p_{1}

i ∈ {1, 2, 3}

26. (2) p ⊆ M_{1} ⊆ p𝕃□ ⊆ pM_{1}𝕃□  (2) p ⊆ M_{1} ⊆ pM_{1}𝕃□  L \ p_{1}
27. (2) p ⊆ p□ ⊆ p𝕃□ ⊆ pM𝕃□  (2) p ⊆ p□ ⊆ pM𝕃□  L \ p_{1}
28. (2) p ⊆ (3) M ⊆ 𝕃 ⊆ p𝕃□  (2) p ⊆ (3) M ⊆ p𝕃□  L \ p_{1}
29. (4) p ⊆ (3) M ⊆ p𝕃□  (12) p ⊆ p𝕃□  (□_{1} ∪ □_{2} ∪ □_{3}) \ L
30. (2) p ⊆ pM□ ⊆ pM𝕃_{i}□  (6) p ⊆ pM□  □_{1} \ p_{1}

i ∈ {1, 2, 3}

31. (4) p ⊆ M ⊆ pM𝕃□  (4) p ⊆ pM𝕃□  □_{1} \ L
32. (2) p ⊆ p𝕃□ ⊆ pM𝕃□  (2) p ⊆ pM𝕃□  L \ p_{1}
33. (2)M ⊆ p𝕃□ ⊆ pM_{1}𝕃□  (2) M ⊆ pM𝕃□
M ∈ {M_{2}, M_{3}}  M ∈ {M_{2}, M_{3}}
34. (2) p ⊆ p□ ⊆ p𝕃_{i}□  (30) p ⊆ p□  p□ \ p_{1}

i ∈ {1, … 15}

35. (2) p ⊆ 𝕃 ⊆ p𝕃□  (2) p ⊆ p𝕃□  L \ p_{1}

14 JOHN MAGINNIS AND SILVIA ONOFREI

After these steps we are left with the complex Δ_{1}, with a cone attached at each point p. This is a cone on a complex consisting of (63) p□, (945) p𝕃□, (945) pM□, and (2835) pML□. This is of course homotopy equivalent to Δ_{1}. Recall that by Proposition 5.1, Δ_{1} is homotopy equivalent to R_{2} which, in its turn, is homeomeorphic to Δ. This end the proof of the Theorem. □

6. On the Lefschtez module

In this Section G = Co_{3} and Δ is the 2-local geometry of G described in Section 3.

Proposition 6.1. Let z be a central involution in G. The set Δ^{z} is contractible.

Proof. Note that C_{G}(z) = G_{p} for some point p. It follows that p ∗ Res(p) = Star(p) ⊆ Δ^{z}. In [RSY], the point p is called res-fixed. It is easy to see that a point q ∈ Δ^{z} if and only if q ∈ p^{⊥}.

Next we consider the lines of Δ^{z}. Recall that if L ∈ L then p is collinear with none, one or all the points of L. This implies that there are three types of lines we have to consider: (a). lines on p, already included in Res(p); (b). lines with only one point fixed, the other 2 points being interchanged; (c). lines with all three points fixed.

Let us consider case b). The line L does not contain p and has only one point q fixed by the action of z. Then the collection Res_{Δ^{z}}(L) = {q} ∗ {some M − spaces on L} is a cone on q, and thus contractible. It follows that we can remove all such lines from Δ^{z}, yielding a homotopy equivalent subcomplex Δ_{1}^{z} ⊆ Δ^{z}.

Recall that, given a point, M-space pair (p, M), the set p^{⊥} ∩ M can be empty, a point, a line or p ∈ M. This implies that a M-space in Δ^{z} contains one fixed point, three fixed points or is entirely fixed by z. This is easily seen if we recall that M contains 35 points, thus under the action of z, at least one point is fixed.

Let M ∈ Δ_{1}^{z} be such that p^{⊥} ∩ M = L, a line: this means all three points of L are in Δ_{1}^{z}. Then Res_{Δ_{1}^{z}}(M) consists of L and the three points on L, but this is a cone and thus contractible. Note that the other lines of M are either not in Δ_{1}^{z} or were deleted at the previous step. We can remove all such M-spaces from Δ^{z}, yielding a homotopy equivalent complex Δ_{2}^{z} ⊆ Δ_{1}^{z} ⊆ Δ^{z}.

Let us return to lines and consider a line L whose all three points are fixed by the action of z. Then ⟨p, L⟩ generate a plane which lies in a unique M ∈ Res_{Δ}(p). But the other M-spaces from Δ^{z} through L were deleted at the previous step. Thus Res_{Δ_{2}^{z}}L = {M} ∗ {3 points on L}, again a cone and thus contractible. Remove these lines, yielding Δ_{3}^{z} ⊆ Δ_{2}^{z}.

Let M ∈ M be such that p^{⊥} ∩ M = {q}, a single point. Then Res_{Δ_{3}^{z}}M is the point q itself. Remove these M-spaces, yielding Δ_{4}^{z} ⊆ Δ_{3}^{z}.

Next, consider the collection of points collinear with p and look at their residues in Δ_{4}^{z}. Let q ∈ p^{⊥}. Since we have already contracted all the lines on q which do not contain p, all it is left at this stage in Res_{Δ^{z}}(q) is the line pq which is contained in Res_{Δ_{4}^{z}}(p). So we can retract Δ_{4}^{z} to Star(z), which is contractible. □

THE 2-LOCAL GEOMETRY AND THE BOUC COMPLEX FOR Co_{3} 15

In what follows we will use the following results due to Thévenaz; see [Th87], Theorem 2.1 and Corollary 2.3:

Theorem 6.2. (Thévenaz) a). Let 𝔛 be a class of subgroups of G which is closed under subconjugacy. Let Δ be an admissible G-complex such that the reduced Euler characteristic χ̃(Δ^{S}) = 0 for every subgroup S which is cyclic mod p and satisfies O_{p}(S) ∉ 𝔛. Then the reduced Lefschetz module L̃(Δ) is a 𝔽_{2}G-virtual module projective relative to the collection 𝔛. b). Let p^{n} = |G|_{p} be the p-part of G, and let p^{k} be the highest power of p dividing the order of some subgroup belonging to 𝔛. Then χ̃(Δ) is a multiple of p^{n−k}.

We let 𝔛 = B_{2} \ B̂_{2} = {R_{2}, R_{3}, R_{4}}, the family of 2-radical subgroups which are pure non-central. It is clear that 𝔛 is closed under subconjugacy.

Remark 6.3. The subgroups S in G which are mod p cyclic and such that Z(O_{p}(S)) contains no central involutions have O_{p}(S) in 𝔛. To see this, let S be a mod p cyclic subgroup of G; set P = O_{2}(S) and H = Ω_{1}Z(P). Thus P ≤ Syl_{2}(N_{G}(H)). Let us assume H contains no central involutions. Clearly, H is of the form 2, 2^{2} or 2^{3}. The normalizers of R_{2}, R_{3} and R_{4} are given at the beginning of Section 4. It is easy to see that Syl_{2}(N_{G}(H)) = H × K with K some 2-subgroup. If H ≤ P ≤ H × K then P = H × K_{1} for some subgroup K_{1} ≤ K. But Z(P) = H × Z(K_{1}), where Z(K_{1}) is nontrivial if K_{1} is nontrivial. Thus Ω_{1}Z(P) = H × Ω_{1}Z(K_{1}) = H. It follows that Ω_{1}Z(K_{1}) = 1 and also Z(K_{1}) = 1 which, in its turn implies K_{1} = 1. Finally P = H.

Furthermore, G acts admissibly on Δ and χ̃(Δ^{z}) = 0, for every central involution z ∈ G, according to Proposition 6.1. Now, by P.A. Smith theory, it follows that χ̃(Δ^{S}) = 0 also, for every subgroup S which is cyclic mod p and such that O_{p}(S) ∉ 𝔛. Thus, all the necessary conditions required by the Theorem 6.2 are in place and we can formulate the following:

Theorem 2. Let G = Co_{3} and let 𝔛 = B_{2}\B̂_{2}. Let Δ denote the 2-local geometry of G. The reduced Lefschetz module L̃(Δ) is a virtual 𝔽_{2}G-module projective relative to the collection 𝔛. Furthermore χ̃(Δ) is a multiple of 2^{7}.

Remark 6.4. Note that |G|_{2} = 2^{10} and that the 2-part of χ̃(Δ) is 2^{7}. Also, the largest subgroup in 𝔛 has order 2^{3}. Note also that Thévenaz’ result tells us that the reduced Euler characteristic is a multiple of 2^{7}, without giving an upper bound.

Acknowledgements. We would like to express our gratitude to S.D. Smith for suggesting the problem and for helpful comments.

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