manuscripta math. 85, 177 - 193 (1994) m a n u s c r i p t a m a t h e m a t i c a ~ Springer-Verlag 1994

Conway's Group Co3 and the Dickson Invariants

Dave Benson

In this paper, we construct a map from the classifying space BCo3 of Conway's sporadic simple group Co3 to the classifying space BDI(4) of the new finite loop space at the prime two DI(4) of Dwyer and Wilkerson. This map has the property that it injects the mod two cohomology of BDI(4) (which is equal to the Dickson invariants of rank four) as a sub- ring over which the rood two cohomology of BCo3 is finitely generated as a module.

1 I n t r o d u c t i o n

A few years ago, Adem, Maginnis and Milgram [1] calculated the mod two coho- mology of the sporadic Mathieu group M12. They found that it contains a copy of the Dickson invariants of rank three, on generators of degrees four, six and seven. It is finitely generated and fi'ee as a module over this subring. Since this algebra of Dickson invariants is equal to the cohomology of the classifying space BG2 of the compact Lie group G2, one is tempted to speculate that there is a group homomor- phism M n ~ G2 inducing this inclusion in mod two cohomology. That this is not the case is easily seen by noticing that G2 has a seven dimensional faithful complex representation, while M12 does not. Nevertheless, it turns out that there is a map of classifying spaces BMn ~ BG2 which induces the above inclusion in mod two coho- mology. This map was constructed by Milgram (unpublished) using 6tale homotopy, and more directly by the author in joint work with Clarence Wilkerson [2].

Dwyer and Wilkerson [7] recently constructed a new finite loop space at the prime two DI(4) ~_ flBDI(4). The rood two cohomology of BDI(4) is the algebra of Dickson invariants of rank four, on generators of degrees 8, 12, 14 and 15. The goal of this paper is to construct a map from the classifying space BCo3 of Conway's sporadic group Co3 to the space BDI(4). We do this in such a way that the corresponding map in mod two cohomology embeds the Dickson invariants of rank four into the cohomology of Co3, and so that the latter is finitely generated as a module over the former. Our main theorem is the following.

~78 BENSON

T h e o r e m 1.1 The rood two cohomology H*(Co3, [:=2) contains a copy of the algebra of Dickson invariants of rank four, on generators of degrees 8, 12, 14 and 15, in such a way that the Steenrod operations act in the normal way on these Dickson invariants. The whole cohomology ring is finitely generated as a module over this subring.

There is a map of classifying spaces BCo3 ---* BDI(4) realizing the inclusion of the Dickson invariants in the rood two cohomology of Co3.

We use Robert Curtis' MOG (Miracle Octad Generator) notation for displaying vectors in the Leech lattice. In this notation, the 24 basis vectors are arranged in a 4 x 6 array, subdivided into three 4 • 2 blocks. For an explanation of this notation, we refer the reader to Chapter 11 of Conway and Sloane [5]. For a description of the Leech lattice and some of the sporadic groups associated with it, see Chapter 10 of the same reference. Further information about Co3 may be found in Conway [4], Finkelstein [10], Solomon [13], and the ATLAS of finite groups [6]. For background on the Dickson invariants, see Wilkerson [14].

Our topological constructions are performed in the category of simplicial sets, so the word "space" means "simplicial set". If X is a space, we write )( for the Bousfield-Kan Fp-completion of X. In particular, /?G denotes the Fp-completion of the simplicial set of singular simplices on the classifying space of a compact Lie group G. For standard properties of Fp-completion, see Bousfield and Kan [3]. We are mostly interested in the case p = 2 for the purposes of this paper.

The paper is organized as follows. In Section 2 we write down some explicit generators for Coa and describe the 2-local structure in terms of these generators. In Section 3, we show how to express [3Co3 as a homotopy colimit in terms of the 2-local structure. In Sections 4 and 5 we discuss the structure of BDI(4) in more explicit detail than was done in [7], in preparation for the construction of the maps. In Sections 6, 7 and 8 we construct maps from the 2-completions of the classifying spaces of the relevant 2-local subgroups of Co3 to BDI(4), and show how to put these together to make a map BCo3 ---, BDI(4) with the relevant properties.

2 Some subgroups of Co3

We take Co3 to be the stabilizer of the vector

- 2 2 4 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0

in the Leech lattice. This is a sporadic finite simple group of order

495,766,656,000 = 21~

We begin by giving names to some elements of Co3.

B E N S O N 179

r I _- a 2 a 3

+ + + -[- - - - -

Jl- - - - -

a 4 b I ~-- b 2

C 1 -~-

§ - l l +

C 2 -~ �9 i ii

C 3 d = Iii iili 1 f =

h =

H e r e , e d e n o t e s t h e 4 x 4 m a t r i x

_ ! 1 ! ! 2 2 2 2

1 1 1 ! 2 2 2 2

! 1 _ 1 1 2 2 2 2

1 i 1 1 2 2 2 2

~80 BENSON

These elements satisfy the following relations (**):

s 1, [a,,aj] = 1 (1 < i , j < 4), b~ = b~ = [bl,b2] = 1, [ai,bl] = [a2, bx] = [al, bs] = a i ~ _ _

[as, b2] = 1, [a3, b,] = a l , [a4, b,] = as , [a3, b~] = a i a s , [a4, b~] = a , , c~ = d = c~ = 1, [cl,cs] = 1, [cl,c3] = [c2,c~1 = a,, lax, Cl] = [ a s , e l ] = [ a 3 , c1 ] = [al, c2] =

[as, c21 = [a4, cs] = [al, c3] = [a3, c3] = [a4, c3] = 1, [a4, cl1 = [a3, cs] = [a2, c31 = al, [bl, cl1 = [b2, c , ] = Ibm, csl = a l a 2 , [hi, c,] = 1, [bl, c31 = a~c l , [b~, c~] = a ~ a c s , d ~ = 1, [a~,d] = [a2,d] = 1, da3d -~ = a4, da4d -1 = a3a4, dbld -~ = b2, db2d -1 = bib2, dcld -1 = cs, dc2d -1 = aaa2cac2, [c3, d] = 1, e 2 = 1, [al, e] = [a3, e] = 1, [a2, e] = al, [a4,e ] = a3, [bl,e ] = 1, [b2, e] = bl, [cl,e I = a3, [c2,e] = a2a3a4cx, [c3, e] = 1, ede = d 2, f s = 1, [a l , f ] = [a3, f] = 1, [a2, f] = ala3, [a4,f] = ai, [bl,f] = e, (b2f) 3 = 1, [c~, f ] -- 1, [c2, f ] = a2a~c3, [cs, f ] = a~a3, (dr) 3 = 1, g3 = 1, g a a g -1 = a2,

gasg -~ = ala2, [a3,g] = [a4, g] = 1, gblg -1 = bxb~, gb2g -~ = b~, gclg -1 = a~a2a3b~cs, ~ c ~ -1 = a , ~ s c i c ~ , (c3~) ~ = ~, [d ,~] = ~, ( e~ ) s = ~, ( f ~ ) 4 = ~, h 2 = ~, [ . ~ , h ] = ~, [as, h] = a , , ha3h = a~a4b~c~, ha4h = a2a3a462c2, [bx,h] = Ibm, hi = 1, hclh =

a~a3aabac~c~, (c3h) 3 = 1, [d,h] = 1, [e,h] = 1, (a~fh) 4 = el , aaa4blb2c2, hcsh = (gh ) s = d.

The subgroup P = (al, a2, as, a4, bl, b2, Cl, c2, c3, e)

is a Sylow 2-subgroup of Co3. Let Z = Z ( P ) = (al) '~ l / 2 , U = (al,a2) '~ ( l / 2 ) 2 and V = (al, a2, a3, a4) = (]?/2) 4. Then the maximal 2-10cal subgroups containing P are as follows:

C = CCo3(Z)= { P , d , f , h ) ~- 2Ss(2),

U = Nvo,(U) = ( P , d , g , h )

of shape 22+631+222, and

X = Neo3(V) = ( P , d , f , g ) .

Note that O2(N) is a special group of shape 22+s with U as its center, and that the elements a3, a4, hi, b2, el and c2 are coset representatives forming a basis of 0 2 ( N ) / U ~ ( l / 2 ) s. The group X / V is isomorphic to As -~ GL(4, F2), and the extension does not split. There is a unique isomorphism class of non-split extension, so this information determines the group X uniquely up to isomorphism.

The group C A X = ( P , d , f )

has shape 2~_ +6 : GL(3, F2). The subgroup

02(C 91 X ) = (al,a2, a3,a4, cl,c2,c3)

is an extraspecial group 2~_ +6 with center Z generated by al. The central quotient (C n x ) / z = 2 6 : GL(3, F2) has two conjugacy classes of complements (this is an easy calculation using for example Shapiro's lemma for degree one cohomology). One of them lifts to a conjugacy class of complements for 0 2 ( C N X ) in C 91 X, a representative of which is

H = (bl, b2, d, e, f) ~- GL(3, F2).

The other lifts to a conjugacy class of supplements isomorphic to the non-trivial double cover SL(2 , FT) of GL(3, Fs).

BENSON 181

The group C V1N Cl X = (P,d)

has order 21~ The subgroup

02(C VI N fl X) = (al, a2, a3, a4, bl, b2, cl, c2, c3)

has index two in P, and the quotient is

(C n g n X ) /02 (C Cl g fl X) ~ E3.

T h e o r e m 2.1 (i) The group defined by generators al, as, a3, a4, bl, b2, cl, e2, c3, d, e, f , g and h, and relations (**), is isomorphic to C03.

(ii) The group defined by the same set of generators with g removed, and those relations among (**) which do not involve g, is isomorphic to C.

(iii) The group defined by the same set of generators with f removed, and those relations among (**) which do not involve f , is isomorphic to N.

(iv) The group defined by the same set of generators with h removed, and those relations among (**) which do not involve h, is isomorphic to X.

P r o o f This was proved by performing coset enumerations using John Cannon's program MAGMA. Furthermore, since the coset enumeration was performed by first copying the relations (**) to a text file readable by MAGMA, this provides comforting evidence that these relations are correct. []

3 BCo3 a s a h o m o t o p y c o l i m i t

Let G be a compact Lie group and p a prime. Let .Ap(G) denote the Quillen category, whose objects are the non-trivial finite elementary abelian p-subgroups E _< G, and whose morphisms are generated by the conjugations and inclusions in G. In [12], Jackowski and McClure prove the following.

T h e o r e m 3.1 The natural map

hocolim(EG/Ca(E)) --* BG EEAp( G)

is a mod p cohomology equivalence.

We are interested in a variation of this theorem, where under certain conditions on G, we may cut down considerably on the set of centralizers used in this homotopy colimit.

Suppose that s is a collection of elements of order p in G with the following properties:

(i) g is closed under conjugation in G. (ii) If x e g then every power of x is in E U {1}. (iii) If x and y are commuting elements of g then zy E s U {1}. (iv) Every element of order p in the center of a maximal p-torsi subgroup of G

lies in E.

Let •(G) denote the full subcategory of .Ap(G) whose objects are the non-trivial finite elementary abelian p-subgroups E < G with the property that E C E U {1}.

182 BENSON

T h e o r e m 3.2 The natural map

hocolim(EG/Ca(E)) --* BG E e C ( a )

is a mod p cohomology equivalence.

P r o o f The subcategory E(G) satisfies the conditions (i), (ii) and (iii) of Theorem 7.7 in [121. []

Now let G = Co3 and p = 2, and let g be the set of involutions which are in the center of some Sylow 2-subgroup of Co3. The centralizer of an element of g is isomorphic to 25'6(2), and in this group, every involution outside the center is conjugate to its product with the central involution (this is easy to read off from the ATLAS entry [6] for Ss(2)). So g satisfies condition (iii) of the last section. The other conditions are easy to verify, and so we may apply the above theorem to deduce that the map

hocolim EC03/Cco3 (E) ~ BC03 )

E e E ( ~ )

is a mod two cohomology equivalence. The category 8(Co3) contains four conjugacy classes of subgroups, of orders 2,

4, 8 and 16, with representatives Z = (al), U = (al,a2), Y = (al,a2, a3) and V = (a,,a2,a3, a4). Now V = CCo3(V) = CCo3(Y), and so each conjugate of Y is contained in a unique conjugate of V. It follows that if we denote by $~(C'o3) the full subcategory of $(Co3) consisting of the conjugates of Z, U and V, then hocolim ECo3/Cco3 (E) is a strong deformation retract of hocolim ECo3/Cco3 (E).

) ) E E ~ ( C o 3 ) EEE(Co~)

Finally, we replace $'(Co3) by a skeletal subcategory C consisting of just the objects Z, U and V and the morphisms between them. The category C has the following shape:

Z ~ U ~ V

I I I {1} GL(2, F2) GL(4, F2)

Here, we have used the notation of [7]. Under each object is its monoid of self-maps, and "=~" stands for an appropriate set of morphisms. To summarize, we have proved the following theorem:

T h e o r e m 3.3 The map

hocolim EC03/Cco3 (E) ~ BC03 )

EEC

is a rood two cohomology equivalence; hence after F2-completing, it is a homotopy equivalence. []

BENSON 183

Another way of writing this F2-completed homotopy colimit, which is possibly easier to visualize, is as the homotopy colimit of the following diagram:

hc " h(c n N) "[~N

~ ( C n N n x )

h(c n x) B(N n x)

~x

This homotopy colimit may be written as a quotient of

/3C H /3N II .BX II /3(Cr3 N )x A ~ I_I /3(C n x ) x A'

H /~(N n x ) x a 1 lI /~(C n N n x ) x a 2

(where A 1 and A 2 are a standard 1-simplex and 2-simplex respectively) by the obvious identifications coming from the arrows.

The proof that this homotopy colimit is equivalent to the one described in The- orem 3.3 is essentially the same as the proof of Proposition 5.1 of Jackowski and McClure [11]. The argument is due to Bill Dwyer, and involves going via an inter- mediate "twisted arrow" category.

R e m a r k In contrast with the situation for the map BMI~ ~ BG2 described in Benson and Wilkerson [2] (see in particular Lemma 3.3 of that paper), it is not true that the character of the alternating sum of the permutation modules

Q~[Co3/Co3] - q~[Co3/C] - Q~[Co3/N] - q~[Co3/X] + q~[Coa/(C n g) ]

^ 0 +Q'~[Co3/(C N X)] + Q'~[Co3/(N n X)] - Q2 [C 3/(C n N n X)]

has no constituent in the principal 2-block. Indeed, using John Cannon's program MAGMA, this alternating sum was calculated to be the following generalized char- acter:

(-50378624, 0,496, 2080, 784, - 125, 0, 0, -24 , -19 , 0, 0, 0, - 8 , - 5 , - 2 ,

0, 0,0, - 8 , 1 , 1,0, 1, 1,0, 0, 0, 0, 0, - 1 , 0 , 0 , 0 , 1, 1,1, 1, 1, 0, 0, 0).

Here, the conjugacy classes are listed in the same order as in the ATLAS [6]. This character decomposes as:

-26X42 - 26);41 - 26X4o - 25Xs9 - 23X~ - 24X37 . . . .

184 BENSON

The character X37 is in the principal 2-block. It seems that the reason why this character cannot avoid the principal block is that the degree is so large in magnitude in comparison with the other character values.

Of course, the reason for wondering about the above alternating sum of characters is that by Theorem 3.3 it is the character of a finite Z-free chain complex of lCo3- modules whose mod two hypercohomology vanishes.

4 The Weyl group of DI(4)

We need to be more explicit in some of the constructions made by Dwyer and Wilkerson [7], in order to construct the desired maps. We begin with a discussion of the Weyl group WDI(4) ~ Z/2 x GL(3, F~) and its 3-dimensional 2-adic reflection representation.

We regard SO(7) as acting on a seven dimensional real space spanned by all except the top-left vector of a 4 • 2 array of basis vectors. For the maximal torus T C Spin(7), we choose the inverse image of the subgroup SO(2) 3 C SO(7) embedded as follows:

�9 S O

(2)

S S O O

(2) (2)

Thus there is a short exact sequence 1 --+ l /2 ---+ T --+ SO(2) 3 -+ 1. Recall from p. 53 of [7] that Tn is the subgroup of T consisting of elements x such that x 2" = 1, and Too = Un T~. The maps

GL(3,12) ~ Aut(T~o) ~ 7r0Aut(/)Tcr ~ roAut(BT)

are isomorphisms. Here, GL(3, l~) is the subgroup of GL(3, Q~) consisting of ma- trices which preserve the length three column vectors with integer coordinates with even sum.

The group W1 = Wspin(z)(T) is the subgroup l /2 x E4 C_ GL(3, ~'2) consisting of the monomial matrices with entries 0 and +1. According to the ATLAS [6], we may extend this to a 2-adic reflection group 1/2 x GL(3, F2) C_ GL(3, 7-2) as follows. Let a and /3 be the 2-adie roots of the equation x 2 + x + 2 = 0, with a -- 5 (8) and fl --- 2 (8). The orbit of the vector (2, 0, 0) consists of the images under W1 of the vectors (2, 0, 0), (0, a, a) and (1, 1,/3). Here, for typographical convenience, we write our column vectors as row vectors.

In the notation of [7], the subgroup W1 C_ l /2 x GL(3, F2) is 1/2 x P(1,2). The subgroup 1/2 x U(2, 1) ~ (] ' /2) 3 of W1 is generated by the matrices (100) (100)(100)

0 - 1 0 , 0 0 - 1 , 0 1 0 ,

0 0 - 1 0 - 1 0 0 0 1

BENSON 185

1 0 0) and 1/2 x U is generated by these together with 0 -1 0 . The normalizer of

0 0 1 / / 2 x U(2, 1) in Z/2 x GL(3, F2) is l / 2 x P(2, 1), which contains an element ( of order two swapping the second and third of these matrices. This element ( therefore swaps the (-1)-eigenvector (0, a, a) of the second matrix with the (-1)-eigenvector (2, 0, 0) of the third matrix, and fixes the common (+l)-eigenvector (0, a, - a ) . Thus we have (0

~= ~/2 1/2 -1 /2 . ~/2 -1/2 1/2

The group WDI(4) is generated by W1 and (, while Z/2 x P(2, 1) is generated by Z/2 x U and (.

If we change basis using the matrix

2 0 0 ) 0 a - a E GL(2, Q2) 0 a a

then 1/2 x P(2, 1) is the group of monomial matrices, because

(1/2 0 0)(0 j2)(20 0) (010) 0 ~3/4 ~/4 a /2 1/2 -1 /2 0 a - a = 1 0 0 0 - /3 /4 /3/4 a /2 -1 /2 1/2 0 a a 0 0 1

(1/2 0 0)(1 00)(20 0) (100) 0 /3/4 /3/4 0 -1 0 0 ~ - a = 0 0 1 0 - /3 /4 3/4 0 0 1 0 a a 0 1 0

o o o o o) o o ) 0 /3/4 /3/4 0 0 - 1 0 a - a = 0 - 1 0 . 0 - j3 /4 /3/4 0 - 1 0 0 a a 0 0 1

In the next section, we discuss how to effect this change of basis.

5 T h e h o m o t o p y e q u i v a l e n c e BQred ^ 3 ~ B(Spin(3) o Spin(4))

~ ' ~ ^ 3 In order to make explicit the action of GL(2, F2) = E3 on BQ,,a described in Proposition 6.1 of [7], we describe the homotopy equivalence

^ 3 ~ Spin(4)) BQ,.~a B(Spin(3) o

where Spin(3)oSpin(4) is the inverse image in Spin(7) of SO(3) x SO(4). By choosing this homotopy equivalence carefully, we effect the change of basis described in the last section, so that the action of E3 becomes the usual permutation action on the three copies of SU(2).

We begin by discussing the well known isomorphisms Q ~ Spin(3) and Q2 Spin(4). We regard Q as the quaternions of unit norm, Q c_ H, and the conjugation

186 BENSON

action of Q on the subspace R 3 of pure imaginary quaternions gives a 2-fold covering map Q --* S0(3) which lifts to an isomorphism Q ~ Spin(3). In particular, using

V;7

the labelling I ~ l , we have

Similarly, Q2 acts on H ~- N 4 by letting (91,92) send x to glzg~ 1. This gives us a 2-fold covering map Q2 __~ SO(4), which lifts to an isomorphism Q2 ~ Spin(4). The automorphism which swaps the two copies of Q corresponds to the automorphism of S0(4) given by quaternionic conjugation, which has determinant -1 . Using the

Vi7

labelling 1~], we have

N

( - �89 + i + j + k ) , - � 89 + i + j + k)) ~-*

(-�89 + i + j + k),-�89 - i - j - k))

Q~d --* Spin(3) o Spin(4) Putting these maps together gives us an isomorphism a and hence a homotopy equivalence/)Q~ -*/)(Spin(3) o Spin(4)). This is not the

BENSON 187

homotopy equivalence we wish to use, because with respect to the chosen bases for the maximal tori of Qa and S0(7) , the matrix for this map is

( 2 0 0 ) 0 1 - 1 . 0 1 1

[Note that we have carefully avoided choosing a basis for the maximal torus of Spin(7) by redefining GL(3,12) at the beginning of Section 4. We have also avoided choosing a basis for the maximal torus of Q3 a by identifying 7hQ~,d as an appropriate submodule of (1/2)3.] Now a is a 2-adic unit, and hence gives rise to a "scalar" self homotopy equivalence of/SSpin(4). If we compose with this, the new matrix for the map is (:00)

0 c~ c~

^ 3 ~ Spin(4)) be the corresponding map. So according We let p0 : BQrea /3(Spin(3) o to the matrix calculation at the end of the last section, we may now assume that the action of GL(2, F2) = Es on BQre a described in Proposition 6.1 of [7] comes from the permutation action on the three copies of Q. Let Spin(3, 4) be the inverse image in Spin(7) of (0(3) x 0(4)) n S0(7). So Spin(3,4) contains Spin(3) o Spin(4) QZ~a as a subgroup of index two, and is a semidirect product of this with a copy

Qred" So P0 extends to a map of Z j 2 pcrmuting the second and third copies of 3 p: B(Qa~ed>~Z/2) ---*/3Spin(3,4).

To understand the meaning of the scalar multiplication by ct on BSpin(4), we use the description given in Lemma 5.4 and Proposition 5.5 of [7]. Namely, we regard /3Q as the 2-completion of the homotopy pushout of

/~(~4s ~- BQ16 --~/~ li_m Q2-+,. r *

L e m m a 5.1 Let Q2.+, = <x,y [ x 2" = y4 = 1,y2 = x~"- ' , yxy-1 = x - l ) . Then for n > 3, Aut(Q:.+,) = H)~K, where H = CA.t(Q2.+I)(X ) ~- 1/2" and K = CAut(Q2.§ ) ~- Aut(H) ~ 1/2 x l / 2 ~-~.

P r o o f Since n > 3, (x) is characteristic in Q2-*~. So U = CA.qQ~..~)(x) is normal in Aut(Q2.+~), and is generated by x ~ x, y ~-* yx. It follows that H ~ (x) -~ 1/2". Since H acts transitively on the elements of Q2-§ not in (x), the subgroup K = Ch.t(Q~.+,)(y) is a complement to H in Aut(Q2.+~), and is isomorphic to Aut(H) -~ Aut(x). It is generated by x s-. x -~, y ~ y and x ~-* x 3, y ~ y. [::3

The element of order two in H and the elements in the complement K have the property that they respect the inclusions Q2. '-* Q2-+,. In the limit, we obtain an

^

action of l / 2 • Z~ = Z/2 x li _mAut(//2 ~) on limQ2.+,. The subgroup of index

two consisting of the elements"(a, b) with b ~ = 8"a + 1 (rood 16) is isomorphic to ^

1~ (. These are the elements with the property that the restriction to QI~ extends to an automorphism of O4s. The elements with a = 0 act trivially on O4s, while those with a = 1 act by fixing the elements in the subgroup of index two, and

188 BENSON

multiplying the remaining elements by the central element of order two. This way, we obtain an action of Z~ on the diagram O4s ~ Q~6 --* li_m Q2,+1. The elements

+1 induce inner automorphisms on all the groups involved, so we obtain a map Z~/{+1} ~ r0Aut(/3Q), which is the isomorphism described in Lemma 5.4 of [71.

Now a = 5 (mod 8), and so ~ = 9 (mod 16). So the corresponding auto- morphism of Q16 sends x to x 5 and y to yx 4. The corresponding automorphism of g)4s fixes the elements in the subgroup of index two, and multiplies the remaining elements by the central element of order two. It follows that we may write down a

" 3 Spin(4) such that corresponding injective homomorphism a0 : ((-94S)red --* Spin(3) o the following triangle commutes:

~ 3 B(O48)~.d

^ 3 PO BQ,.r /~(Spin(3) o Spin(4)).

The map ao extends to a ,nap c~: (O4s)3ed>~Z/2 --* Spin(3,4), where Z/2 is acting by permuting the second and third copies of O4s. The following diagram commutes:

/3((04s)~d~ Z/2)

I /)(Q~ed~Z/2) " - /)(Spin(3,4)) .

6 T h e m a p C ---, Spin(7)

There is a unique representation (up to conjugation) $6(2) ~ SO(7), which lifts to a map C ~ 2S6(2) --* Spin(7). Restrict this map to C f3 X to obtain a map from (C n X)/Z to SO(7). Now (C f3 X)/Z is a split extension of shape 26 : GL(3, F2), and so the seven dimensional real orthogonal space decomposes as a direct sum of seven eigenspaces of 02((C n X)/Z), permuted by GL(3, F2).

Now all the non-central involutions in Spin(7) are conjugate, and their image in SO(7) have trace -1 . It follows that any element of order two in C maps to an element of trace - 1 in SO(7). So without loss of generality, we may let 02((C n X)/Z) act in the same way as it does on the seven dimensional space spanned by all except the top left basis vector in the middle block of the MOG diagram. Thus al maps to the identity element, and

BENSON 189

Now the above trace condition implies that the complement H is not conjugate to a complement acting by permutat ions (in fact the obvious permutat ions form a supplementary copy of SL(2, FT) of the type discussed at the end of Section 2), and so we must use signed permutat ions as follows:

The element h conjugates a2 to ala2, and so it must fix the eigenspace decomposition of a2; namely the column decomposition into a 3-space and a 4-space. Since h centralizes bl and d, it fixes the vector with a 1 in each coordinate. Also, h conjugates a3 to a2a4blCl, and so it has to act as :t:1 in the first column, and as -I-e in the second column. To have trace - 1 , we therefore have

We may see the isomorphism between $6(2) x g /2 and the Weyl group of type Er using this representation, as follows. The l / 2 here acts as plus and minus the identity. The element -a2ca is a reflection, and so the 14 signed basis vectors are root vectors. The remaining 2 4 x 7 root vectors have +�89 in four coordinates forming the complement of a "line" with respect to the action of H ~ GL(3, F2) on these seven 1-spaces. This makes a total of 126 root vectors, as required.

7 T h e m a p N ~ Qr3edXE3

The image of C n N in Spin(7) lies in Ira(or) C_ Spin(3, 4), and so by inverting ~r we obtain a map C n N --* Q~edxl /2 as follows:

aa ~ ( 1 , - 1 , - 1 ) a2 ~ ( - 1 , 1 , - 1 ) a a ~ ( i , i , i ) a t ~ ( j , j , j )

b~ ~ ( i , j , k ) b2 ~ ( j , k , i )

d~--~ ( - �89 + i + j + k ) , - � 89 + i + j + k ) , - � 89 + i + j + k))

ec3 ,--, ~ ( k - j , k - j , k - j )

hcs ~ (1,- �89 + i + j + k) , - �89 - i - j - k)). a

We extend this to a map N + Qred Ea by sending g to the cyclic permutat ion sending the first coordinate to the second, the second to the third and the third to the first:

g ~ ( ~).

190 BENSON

It is easy, using the relations given in Section 2, to check that this is a well defined homomorphism.

8 The map B C o 3 ~ BDI(4) a In w167 and 6, we constructed maps C -4 Spin(7) and N ---* Qred N3 in such a way

that the following diagram commutes up to homotopy:

/)C " BSpin(7) / I /)(C 91 N) "/?(Q~aea n N2) @ / ) S p i n ( 3 , 4)

--.. l / )N " a - B(Qred:~Ea)

Now there is a unique isomorphism class of non-split extensions (Z/2)4GL(4, F2), and this is isomorphic to the group X. There is a map from BX to BDI(4) obtained by regarding / )X as the F2-completion of the homotopy colimit of the following functor. Restrict the functor used by Dwyer and Wilkerson (p. 40-42 of [7]) to the subcategory of the category A consisting of the object A4 and its automorphisms. [Note that in Spin(7), the normalizer of (7/2) 4 is a non-split extension of (][/2) 4 by (-Z/2)aGL(3, F2). This is a subgroup of the fundamental group of the homotopy colimit, which is therefore a non-split extension of ( l /2) 4 by GL(4, F2).]

Since the image of C Cl X in Spin(7) is the normalizer of the image of V in Spin(7), the composite

/}(C n X) --,/}X ~ BDI(4)

is homotopic to the composite

B(C n X) --*/}C --* BSpin(7) --* BDI(4).

Similarly, since the image of N N X in Q~ed:~E3 is the normatizer of the image of V in 3 Q ~ : ~ 3 , the composite

/)(X Cl X) --*/)X --* BDI(4)

is homotopic to the composite

B(N gl X) --*/)N --*/)(Q~d~Na) ~ BDI(4).

Thus we obtain maps from the spaces in the diagram at the end of Section 3 to BDI(4) which agree under composition, up to homotopy. There is an obstruction to lifting to a map from the homotopy colimit of this diagram to BDI(4). There is no obstruction to getting a map from the homotopy colimit of the full subdiagram consisting of all except the middle objec t / ) (C n N n X) to BDI(4). We are then left with the problem of completing a map

B(C Cl N Cl X) x/~2 --, BDI(4)

BENSON 19~

to a map /}(C n N n X) x A 2 --, BDI(4).

The obstruction to doing this lies in

7hMapo(/}(C n N N X), BDI(4)),

where MaPo denotes the connected component consisting of the homotopy class of maps in question.

L e m m a 8.1 The maps

B Z x B ( C n N N X) ---* BDI(4)

B Z x B ( C O X ) ~ BDI(4)

induce homotopy equivalences

B Z --~ Map0(/}(C n g N X), BDI(4)),

B Z --. Mapo(B(C n X), BDI(4)).

In particular, the obvious map

~'aMapo(/}(C n X), BDI(4)) ~ 7rlMapo(/}(C N g n X), BDI(4))

is an isomorphism, and both sides are isomorphic to Z/2.

Proof We apply Proposition 8.4 of Dwyer and Wilkerson [8] to the homomorphism of loop spaces Spin(7) --+ DI(4) and the sequences

Z ---* P ---+ P / Z ,

Z--, c n g n x ~ (CnN n x ) / z

and z c n x --, ( c n x ) / z .

Since Mapo(/}Z , BSpin(7)) ~ Mapo(/}Z , B D I ( 4 ) )

is a homotopy equivalence (both sides are equivalent to/}Spin(7)), it follows that the maps

Mapo(/}P ,/}Spin(7)) ~ Mapo(/}P , BDI(4 ) ) ,

Mapo(/}(C N g O X),/}Spin(7)) ~ Mapo(/}(C N g n X), B D I (4 ) )

and Mapo(/}(C N X),/}Spin(7)) ~ Mapo(/}(C fl X ) , B D I ( 4 ) )

are all equivalences. The centralizer of P in Spin(7) is Z, and so by the main theorem of Dwyer-Zabrodsky [9], the composite map

B Z ~ Mapo(/}(C n X),/}Spin(7)) ---* Mapo(/}(C n g n X),/}Spin(7))

Mapo(BP , BSpin(7))

192 BENSON

is a homotopy equivalence. Now

Map0(B(C N X),/)Spin(7)) = Map0(/)(C N X),/)Spin(7))

is a connected component of the homotopy fixed point set of (C n X)/02(C n X) '~ GL(3, F2) acting on

Mapo( B02( C N X), /)Spin(7) ).

The centralizer of 02(C n X) in Spin(7) is Z, so the latter mapping space is BZ, and hence the homotopy fixed point set is also BZ.

Similarly,

Map0(B(C N N N X),/)Spin(7)) = Map0(/)(C N N O X),/)Spin(7))

is a connected component of the homotopy fixed point set of (C fl N N X)/02(C fl N fl X) ~ E3 acting on

Mapo(BO~(C n N n X),/)Spin(7)).

The centralizer of 02(C n N n X) in Spin(7) is Z, so the latter mapping space is BZ, and hence the homotopy fixed point set is also BZ.

It follows that the maps

BZ ---* Map0(/?(C n X),/?Spin(7)) ~ Map0(/)(C N g N X),/)Spin(7))

--+ Map0(/)P ,/~Spin(7))

are all homotopy equivalences. 0

It follows from the lemma that if necessary, we may adjust the map

/~(C N X) x A ~ --* BDI(4)

so that we can complete to a map from the homotopy colimit to BDI(4). This gives us a map from BOo 3 to BDI(4), and hence a map BCo3 ---* BDI(4).

To check that this map has the desired properties to complete the proof of Theorem 1.1, we note that the composite BP -+ BCoa ---* BDI(4) factors through BSpin(7) ~ BDI(4) (recall that P is a Sylow 2-subgroup of Coa). The mod two cohomology of BP is a finitely generated module over the mod two cohomology of BSpin(7), which in turn is a finitely generated module over the mod two cohomology of BDI(4). Since the mod two cohomology of BP contains the mod two cohomology of BCo3, this proves that the mod two cohomology of BCo3 is also finitely generated as a module over the mod two cohomology of BDI(4).

Acknowledgemen t s I would like to thank Jon Carlson, Bill Dwyer, Richard Lyons, Ron Solomon and Clarence Wilkerson for conversations without which this work could not have been carried out. I would also like to thank the referee for doing a thorough job.

BENSON 193

R e f e r e n c e s

[1] A. Adem, J. Maginnis and R. J. Milgram. The geometry and cohomology of the Mathieu group Mn. J. Algebra 139 (1991), 90-133

[2] D. J. Benson and C. W. Wilkerson. Finite simple groups and Dickson invariants. Preprint, 1993

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

[4] J. H. Conway. A group of order 8,315,553,613,086,720,000. Bull. L.M.S. 1 (1969), 79-88.

[5] J. H. Conway and N. Sloane. Sphere packings, lattices and groups. Grundlehren der mathematischen Wissenschaften 290, Springer-Verlag, Berlin/New York, 1988

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

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

[8] W. G. Dwyer and C. W. Wilkerson. Homotopy fixed point methods for Lie groups and finite loop spaces. Preprint

[9] W. G. Dwyer and A. Zabrodsky. Maps between classifying spaces. Algebraic Topol- ogy, Barcelona 1986. Springer Lecture Notes in Mathematics 1298, Springer-Verlag, Berlin/New York 1987, 106-119

[10] L. Finkelstein. The maximal subgTvups of Conway's group C3 and McLaughlin's group. J. Algebra 25 (1973), 58-89

[11] S. Jackowski and J. McClure. Homotopy approximations for classifying spaces of compact Lie groups. Proceedings of the 1986 Arcata conference on algebraic topology. Springer Lecture Notes in Mathematics 1370, Springer-Verlag, Berhn/New York 1989

[12] S. Jackowski and J. McClure. IIomotopy decomposition of classifying spaces via ele- mentary abelian subgroups. Topology 31 (1992), 113-132

[13] R. Solomon. Finite groups with Sylow 2-subgroups of type .3. J. Algebra 28 (1974), 182-198

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

D. J. BENSON, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF GEORGIA, ATHENS G A 30602, USA. DJBQSLOTH.MATII.UGA.EDU

(Received January 17, 1994; in revised form July 4, 1994)