THE COHOMOLOGIES OF THE SYLOW 2- OF A OF DEGREE SIX AND OF THE THIRD

JOHN MAGINNIS

Abstract. The third Conway group Co3 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2- S is computed, an important step in

calculating the mod 2 cohomology of Co3. The spectral sequence for the central extension of S is described and collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotient S′ = S/2 is the Sylow 2-subgroup of the symplectic group Sp6(F2) of six by six matrices over the field of two elements. The cohomology of S′ is computed, and is detected by restriction to elementary abelian 2-subgroups.

1. Introduction

A presentation for the Sylow 2-subgroup S of the sporadic finite Co3 is given by Benson [1]. We will use a different presentation, with generators which can be described in terms of Benson’s generators {a1, a2, a3, a4, b1, b2, c1, c2, e} as follows.

w = c3, x = b2c1c2, y = ec3, z = a4, a = a3a4c2,

b = a3a4b1c2, c = a3, t = a4c1, u = a2, v = a1.

We have the following set of relations for the generators {w, x, y, z, a, b, c, t, u, v}.

w2 = x2 = y2 = z2 = c2 = u2 = v2 = [w, y] = [w, z] = [w, c] = [w, v] = [x, z] = [x, u] = [x, v] = [y, c] = [y, u] = [y, v] = [a, z] = [c, z] = [u, z] = [v, z] = [c, t] = [c, u] = [c, v] = [a, u] = [a, v] = [b, c] = [b, t] = [b, u] = [b, v] = [t, u] = [t, v] = [u, v] = 1, tu = [a, y], t = [w, b], av = [w, x], b = [x, y], c = [y, z] u = b2 = [x, c] = [b, z] = [b, y] = [b, x], uv = [t, x] = [a, b], v = t2 = a2 = [w, u] = [a, c] = [t, z] = [a, w] = [t, w] = [t, a] = [t, y] = [a, x].

The group S has 1024 and a center Z(S) of order two, generated by v = a1. The subgroup generated by v = a1, u = a2, c = a3 and z = a4 is elementary abelian, and the 4 S/2 is isomorphic to the Sylow 2-subgroup U4 of the

Date: 21 October 2009. 1991 Mathematics Subject Classification. 20. 1 2 JOHN MAGINNIS

4 GL4(F2). One of the maximal subgroups of Co3 is 2 .A8 ; see Finkelstein [5]. Note that we are using the Atlas [3] notation for groups, where pn denotes an elementary abelian p-group of rank n, H.G denotes a group extension with H and quotient group G (with H : G the ), and p1+n = p.pn denotes an extraspecial p-group. The quotient by the center will be denoted S′ = S/⟨v⟩, which is the Sylow 2-subgroup of the symplectic group Sp6(F2); another maximal subgroup of Co3 is 2.Sp6(F2) [5]. We will also denote S′′ = S′/⟨u⟩ = S/⟨v, u⟩,S′′′ = S/⟨v, u, c⟩ and S′′′′ = S/⟨v, u, c, z⟩. We will denote various subgroups of S,S′,S′′ et cetera by subscripts which denote generators not in the subgroup; for example Sx is the subgroup generated by ⟨v, t, u, a, b, c, w, y, z⟩. This index two subgroup is the of a S → ℤ/2ℤ which represents a cohomology class of degree one that we will also denote as x. ′′′′ ′′′ We have S ≃ U4 and S ≃ U4 ×2. We will determine the Lyndon-Hochschild-Serre spectral ′′ ′ ′′ ′ sequences for the central extensions 2 → S → U4 × 2, 2 → S → S and 2 → S → S . We obtain the following.

Theorem 1.1. The cohomology ring of S′ is generated by sixteen classes, with 4 degree one classes, 5 degree two classes, 4 degree three classes and 3 degree four classes. This cohomology ring contains no nilpotent elements.

Theorem 1.2. The cohomology ring of S is generated by seventeen classes, with 4 degree one classes, 4 degree two classes, 2 degree three classes, 1 degree four class, 1 degree five class, 2 degree six classes, 2 degree seven classes, and 1 degree eight class.

′ Although this result for the Sylow 2-subgroup S of the symplectic group Sp6(F2) appears to be completely new, the latter result for the cohomology ring of S has also been obtained by Green and King [6] using a computer program and methods similar to those used by Carlson, et al [2]. The program computes a projective resolution (through some finite degree) and cohomology classes are represented as chain maps. Green and King determine the Poincare series for the cohomology ring of S (also verified in the present paper), show the depth of the ring is three, and compute the restrictions of the generators to representatives of the twenty conjugacy classes of maximal elementary abelian 2-subgroups. Green and King also list 78 relations for these 17 generators. The methods used in the present paper are quite different, and the computations were done mostly by hand. In addition, generators are described using the Evens norm [4] or transfers from subgroups. The webpage of Green and King does not give any details on the definition of their generators, so that we cannot describe our generators in terms of theirs.

2. The cohomology of S′′

The cohomology of the group of unipotent upper triangular matrices U4 ∈ Syl2(GL4(F2)) was originally computed by Tezuka and Yagita [10], although a more complete description appears in the author’s thesis [7] and in [8]. We will for the most part use the notation 2-COHOMOLOGY OF CO3 3 from these latter papers, although one generator, a degree three cohomology class D given as the Steenrod square of a degree two class, D = Sq1(d), will in this paper be replaced by a different generator Δ which can be described as a transfer from an index two subgroup. We will also write  in place of d.

Theorem 2.1. The cohomology ring of U4 is generated by eight classes, with 3 degree one classes w, x and y, 3 degree two classes , and , 1 degree three class Δ, and 1 degree four class T . We have the following additive description.

∗ H (U4; F2) = F2[x, , , T ](1, )(1, Δ)+wF2[w, , T ]+yF2[y, , T ]+wyF2[w, y, T ]+y F2[y, , T ].

The relations are given by wx = xy = 0, w = y = w = y , wΔ = yΔ = 0, 2 = xΔ + x2 + x2 + and Δ2 = x2T + x( + + )Δ + x2  + 2 + 2 + x2 . 4 The cohomology of U4 is detected on five elementary abelian 2-subgroups, 2 = ⟨t, a, b, x⟩, 23 = ⟨t, a, w⟩, 23 = ⟨t, b, y⟩, 23 = ⟨t, w, y⟩, and 23 = ⟨t, ab, wy⟩.

Proof. The above information appears in [10], [7] and [8]. The class Δ replaces the class D from [7] and [8], and we have Δ = D + x + y . The transfer map from the index ′′′′ two subgroup Sy = ⟨t, a, b, w, x⟩ satisfies T r(b) = x, T r() =  + and T r(b) = Δ. ′′′′ The subgroup Sy ⊆ U4 is isomorphic to the central quotient U4/2 and has cohomology ring generated by 3 degree one classes w, x and b and 3 degree two classes ,  and , with relations 2 2 2 ∗ wx = wb = w = 0 and  = bx + x  + b . The classes and T in H (U4, F2) are the ∗ Evens norm of b and , and the classes and  ∈ H (U4/2, F2) are in turn Evens norms from an elementary abelian subgroup of order 16. 2 4 2 2 Let us also remark that U4 is isomorphic to the 2 ≀ 2 = 2 : 2 , with 2 ⊆ S4, 2 4 and the central quotient U4/2 is isomorphic to the wreath product 2 ≀ 2 = 2 : 2. The cohomology of a wreath product can be computed using a theorem of Nakaoka proven in his paper on the cohomology of symmetric groups [9]. Also, a notation such as R(a, b, c) (used in the additive descriptions of cohomology rings) denotes a module, free over the ring R, generated by the elements a, b and c. □ Theorem 2.2. The cohomology ring of the group S′′ is generated by thirteen classes, with 4 degree one classes w, x, y, z, 5 degree two classes , , , , K, 2 degree three classes Δ,L and 2 degree four classes T,M. The relations are given below.

wx = xy = yz = 0, w = y = w = y , wΔ = yΔ = 0 2 = xΔ + x2 + x2 + , Δ2 = x2T + x( + + )Δ + x2  + 2 + 2 + x2 K2 = xzK + z2 + x2 , L2 = z( + )L + z2T + (xΔ + x2 + x2 + 2 + ) M 2 = xz(+ )M+z (+ )L+x (+ )Δ+xzKT +(x2 +z2 )T + (xΔ+x2+x2 + 2+ ) 4 JOHN MAGINNIS

xL = zΔ + ( + )(K + xz),KΔ = xM + z ( + )

KL = zM + x ( + ), ΔL = ( + )M + xzT, KM = xzM + z L + x Δ

ΔM = x( + )M + ( + )L + xKT, LM = z( + )M + ( + )Δ + zKT

yK = yL = yM = wK = wM = 0.

′′′ Proof. The group S is isomorphic to the product U4 ×2, and we shall denote by z the degree one cohomology class corresponding to the direct factor 2 (there is a group homomorphism ′′ U4 × 2 → ℤ/2ℤ with kernel the U4). The spectral sequence of the extension 2 → S → U4 × 2 has d2 differential determined by

d2(c) = yz, obtained by restricting to the various subgroups: 26 = ⟨t, a, b, c, x, z⟩, 25 = ⟨t, a, c, w, z⟩, 2 2 2 D8 × 2 = ⟨t, b, c, y, z⟩,D8 × 2 = ⟨t, c, w, y, z⟩, and D8 × 2 = ⟨t, ab, c, wy, z⟩. We have new generators, in the kernel of the d2 differential, K = cx, L = c( + ) and M = cΔ. 2 1 The d3 differential is given by the Steenrod square of the transgression, d3(c ) = Sq (d2(c)) = 1 2 2 Sq (yz) = y z + yz , which is congruent to zero modulo the d2 differential. Thus the spectral 2 sequence collapses at the E3 page, yielding another generator = c . The index two subgroup ′′ 2 2 Sy is isomorphic to the product (U4/2) × 2 with the factor 2 corresponding to cohomology S′′ classes c and z. Then can be defined as an Evens norm = N ′′ (c). We can define the other Sy ′′ new generators as transfers from Sy with T r(b) = x, T r(c) = z, T r() =  + , T r(bc) = K, T r(b) = Δ, T r(c) = L and T r(bc) = M. G With these definitions, we are able to compute relations. We use the formulas A ⋅ T rH (B) = G G G n n G T rH (ResH (A) ⋅ B) and T rH (Sq (A)) = Sq (T rH (A)), as well as the double coset formula for the restriction of a transfer. For example, KΔ = T r(bc)T r(b) = T r(Res(T r(bc))b) = T r((bz + cx + xz)b) = T r((b2 + bx)z + cxb) = z T r() + xT r(bc) = z ( + ) + xM. Note we are using Res( ) = b2 + bx, which follows from the double coset formula for the restriction of an Evens norm. Also, L2 = Sq3(L) = Sq3(T r(c)) = T r(Sq3(c)) = T r(c2 2) = T r((c2 +cz) 2)+cz( 2 +(+ ))+cz(+ )) = T r( 2)+zT ⋅T r(c)+z(+ )T r(c) = T r(Sq2())+zT ⋅z +z(+ )L = Sq2(T r()) + z2T + z( + )L = ( + )2 + z2T + z( + )L = z( + )L + z2T + (xΔ + 2 2 2 x  + x + + ) . □

Proposition 2.3. The cohomology of S′′ is detected on five elementary abelian 2-subgroups: 26 = ⟨t, a, b, c, x, z⟩, 25 = ⟨t, a, c, w, z⟩, 24 = ⟨t, b, c, y⟩, 24 = ⟨t, c, w, y⟩ and 24 = ⟨t, ab, c, wy⟩.

Proof. The additive description of the mod 2 cohomology of S′′ is given below.

∗ ′′ H (S ; F2) = F2[x, z, , , , T ](1, )(1, Δ,K,M) + F2[w, z, , , T ](w, wL) + yF2[y, , , T ]+

wyF2[w, y, , T ] + y F2[y, , , T ] + F2[z, , , , T ](L, L). 2-COHOMOLOGY OF CO3 5

We then compute the intersection of the kernels of the restrictions to these elementary abelian subgroups. The restriction map to the elementary abelian 26 satisfies Res(w) = Res(y) = 0, Res(x) = x, Res(z) = z, Res( ) = a2 + ax, Res( ) = b2 + bx, Res( ) = c2 + cz, Res() = tx + ab, Res(K) = bz + cx + xz, Res(Δ) = t2x + tx2 + a2b + ab2 + a2x + ax2, Res(L) = z(t2 + tb + tx + ab + a2 + ax) + c(tx + ab + a2 + ax), Res(T ) = t4 + t3x + t2(a2 + ab + b2 + ax + bx)+t(a2b+ab2 +abx), Res(M) = (bz+cx+xz)(t2 +tb+tx+ab+a2 +ax)+bc(tx+ab+a2 +ax). These are all computed using the additive form of the double coset formula for the restriction of a transfer, and the multiplicative form of the double coset formula for the restriction of an S′′ S′′ 2 2 2 Evens norm. For example, Res(Δ) = Res 6 (T r ′′ (b)) = (id + Con )(b(t + tb)) = t b + tb + 2 Sy y 2 2 S′′ S′′ (t+a) (b+x)+(t+a)(b+x) . Also, Res(T ) = Res26 (N26 (t)) = tConw(t)Cony(t)Conwy(t) = t(t + b)(t + a)(t + a + b + x). The notation Cong refers to the map on cohomology induced by conjugation by the group element g. 6 The kernel of the restriction map to the elementary abelian 2 is F2[w, z, , , T ](w, wL) + yF2[y, , , T ] + wyF2[w, y, , T ] + y F2[y, , , T ]. We then restrict the elements in this kernel to 25 = ⟨t, a, c, w, z⟩. The intersection of the two kernels of restriction is yF2[y, , , T ] + wyF2[w, y, , T ] + y F2[y, , , T ].

The intersection of three kernels of restriction is wyF2[w, y, , T ] + y F2[y, , , T ].

The intersection of four kernels is y F2[y, , , T ], and the intersection of all five kernels of restriction is zero. □

′ 3. The cohomology of S ∈ Syl2(Sp6(F2)) Theorem 3.1. The cohomology ring of S′ is generated by sixteen classes, with 4 degree one classes w, x, y and z, 5 degree two classes , , ,  and N, 4 degree three classes Δ, L, P and Q and 3 degree four classes M,T and U.

′′ Proof. The five elementary abelian subgroups of S are sufficient to detect the d2 differential for the spectral sequence of the extension 2 → S′ → S′′, which is

d2(u) = K +  + + xz.

′ 2 1+4 To see this, consider the following subgroups of S . First, ⟨u, t, a, b, c, x, z⟩ = 2 × 2+ , and 2 1+4 6 2 the spectral sequence for 2 → 2 ×2+ → 2 has differential d2(u) = cx+bz+tx+ab+b +bx. The restriction map from the cohomology of S′′ to 26 satisfies Res(K) = cx+bz+xz, Res() = tx + ab, Res( ) = b2 + bx, Res( ) = a2 + ax, and Res( ) = c2 + cz. 2 2 4 Next, ⟨u, t, b, c, y⟩ = 2 × D8, and the spectral sequence for 2 → 2 × D8 → 2 has differential 2 ′′ 4 2 d2(u) = b + by. The restriction map from S to 2 satisfies Res( ) = b + by, Res(K) = Res() = Res( ) = 0, and Res( ) = c2 + cy. 6 Then ⟨u, t, a, c, w, z⟩ = 2 , with d2(u) = 0 and Res(K) = Res() = Res( ) = 0, Res( ) = 2 2 5 a + aw and Res( ) = c + cz. Also, ⟨u, t, c, w, y⟩ = 2 with d2(u) = 0 and Res(K) = 6 JOHN MAGINNIS

Res() = Res( ) = Res( ) = 0 and Res( ) = c2 + cy. Finally, ⟨u, t, ab, c, wy⟩ = 25 with 2 2 d2(u) = 0 and Res(K) = 0, Res() = Res( ) = Res( ) = a + aw and Res( ) = c + cw, where a is the cohomology class dual to the element ab ∈ 24 and w is dual to wy.

The multiplicative properties of differentials in a spectral sequence determine all other d2 differentials. There is a new generator N = uw. Note that uy also lies in the kernel of the d2 differential, but can be described by the product N. Other important d2 differentials are listed below.

d2(uy) = y + y 2 2 2 2 2 d2(u(K +  + )) = x + xΔ + (x + xz) + + + x + xz + z 2 2 2 d2(uL) = x( + ) + zM + L + L + (xz + z )Δ + z( + + x ) + x z

d2(uM) = x Δ + M + M + z L

d2(uΔ) = xM + ( + + xz)Δ + z  + z 2 2 d2(u(L + L + xM + xK( + ) + xzΔ + x( + ) + z  + x z( + ) + x + (x + z) )) = x( 2 + ) + ( + )L + (x + z)M + (z + x )M + (x3 + x2z)T + ( + + xz + z2)Δ + (x2 + z2 + xz + z2 )Δ+ (x 2 + z 2 + x 2 + z 2 + x3 + xz2 ) + x 2 + x 2 + x3 2 + xz2 .

2 The E3 page is the tensor product of F2[u ] with the following.

F2[z, , , , T ](1, )(1, Δ, L, M) + F2[w, z, , , T ](w, N)(1,L) + F2[y, , , T ](y , N)+

F2[w, y, , T ](y, yN) + xF2[x, z, , , T ](1, )(1, Δ) + F2[z, , , T ](x , x ).

The d3 differential is determined by computing a Steenrod squaring operation on the trans- 2 1 gression, d3(u ) = Sq (d2(u)), yielding

2 d3(u ) = x + Δ + x + z + x .

There are two new generators, in the kernel of the d3 differential, which we will denote by P = u2w and Q = u2y.

Other d3 differentials are listed below.

2 2 2 d3(u x) = + + xz + z 2 2 2 3 2 3 d3(u ( + )) = L + L + zM + Δ + ( + x + z )Δ + (x + z ) + x + x + z + x 2 2 2 2 2 2 d3(u Δ) = M + M + z L + x T + zΔ + x Δ + x  + + + x 2 2 2 d3(u M) = (Δ + Δ + z  + z ) + z M + (  + + z )L + (x + x + x z)T 2 2 3 d3(u (L + z + z )) = (zΔ +  + ) + ( + + z )M + xzT + (x + x + z )Δ+ 2-COHOMOLOGY OF CO3 7

( 2 + 2 + x2 + x2 + xz + xz ) + x2 2 + x2 + xz 2 + xz .

4 There exists a d4 differential, which can be deduced from the computation of d5(u ) = 2 2 Sq (d3(u )) and the fact that the spectral sequence must collapse at the E5 page. There is a degree four class U, represented by u4, defined by an Evens norm N(u+t) from the index ′ four subgroup Sbx = ⟨t, u, a, c, w, y, z⟩, or equivalently by an Evens norm N() from the index ′ 4 two subgroup Sx. For more details, see section 6. Therefore the class u must survive to the E∞ page, and so the spectral sequence collapses at the E5 page. The expression obtained 2 2 from Sq (d3(u )) must be the of some d4 differential. The only possible d4 differential 3 2 in the correct degree is d4(u w) = d4(u N). We obtain the following.

3 2 2 2 2 2 d4(u w) = (Δ + z) + xT + ( + xz + z )Δ + (x + z + x z + xz ) + x + x z + xz

3 2 2 d4(u wz) = ( + M)( + + z ) + (x + x + z + xz )(Δ + z)+ ( 2 + + x2 + x2 + z2 + x2z2) + x2 2 + xz 2 + x2 + x2z2 + xz3 3 2 2 2 4 2 d4(u wL) = M( + + z ) + (z + z + z )M + (  +  + z  + xz )T + (z  + z + x2z + z3 + z5)Δ + (z2 + x3z + z4 )+ z2 2 + (x3z + x2z2 + xz3 + z4) .

The additive description of the mod 2 cohomology of the Sylow 2-subgroup S′ of the sym- plectic group Sp6(F2) is given below.

∗ ′ H (S ; F2) = F2[T,U] ⊗ {F2[w, z, , ](1,L)(1,N)(1,P ) + F2[w, y, ](y, yN, Q, QN)+

F2[y, , ](y , N, Q, NQ) + F2[z, , ]( , L, M,  )+

MF2[z, ] + F2[x, z, ](x, , , Δ, , Δ, Δ, Δ)}. □

Corollary 3.2. The Poincare´ series for the cohomology ring of S′ is:

1 + t2 − t4 . (t − 1)6(t + 1)2(t2 + 1) Proof. The additive description for the cohomology ring of S′ yields the Poincar´eseries. Note (1+t3)(1+t2)(1+t3) that a term such as F2[w, z, , ](1,L)(1,N)(1,P ) yields the series (1−t)2(1−t2)2 . Collecting together those terms with the same denominator yields:

1 n(1 + t3)(1 + t2)(1 + t3) (t + 2t3 + t5) + (t + 2t2 + t3 + t4 + 2t5 + t7) ⋅ + + (1 − t4)2 (1 − t)2(1 − t2)2 (1 − t)2(1 − t2) 8 JOHN MAGINNIS

(t3 + t4 + t5 + t7) + (3t4 + t5) t6 o + . (1 − t)(1 − t2)2 (1 − t)(1 − t2) This is equal to:

1 + 2t + 3t2 + 4t3 + 2t4 − t6 − 2t7 − t8 = (1 − t)2(1 − t2)2(1 − t4)2

1 + t2 − t4 . (t − 1)6(t + 1)2(t2 + 1) □

The generators N,P and Q can all be defined in terms of the transfer from the index two ′ subgroup Sx. We have T r(a) = w, T r(b) = y, T r() = N, T r(a) = P and T r(b) = Q. 2 ′ The class  ∈ H (Sx; F2) can be defined as an Evens norm of a degree one element,  = N(u), 2 from the subgroup generated by < t, u, a, c, w, by, z >, which is isomorphic to (U4/2) × 2 . The degree one class u corresponds to a group homomorphism with kernel ⟨t, uc, a, by, w, z⟩, isomorphic to (U4/2) × 2. Relations involving N,P and Q are computed using the properties of the transfer map or restrictions to subgroups, and are listed below.

xN = xP = xQ = 0, yP = wQ, zQ = N + N, P = Q P = Q = Q + zN, Q = Q + z N + zN P 2 = w2U + wNP + N 2,Q2 = y2U + yNQ + N 2 PQ = wyU + wNQ + N 2, ΔN = zN, ΔQ = ( + ) N N 2 = wP + (w2 + wy + wz)(N + ) + 2 + + (w + z)L + M + T + xΔ+ ( + + x2 + xz + z2) + (wz + x2 + xz + z2) N = N + ( + ) + M + xΔ + ( + + x2 + xz) + x2 + xz z N = z + L + Δ + x  + x + z MQ = LN = L + z + (x + z) T + (  + x2 + xz )Δ+ (x + x3 + x2z ) + x 2 + z 2 + x3 + z3 LQ = MN = M + ( + ) + ( + + z2)M + ( + )T + (x + x + z3)Δ+ ( 2 + + x2 + x2 + xz + z2 ) + x2 2 + x2 + xz 2 + xz .

We will also need these Steenrod squaring operations in the next section.

Sq1(N) = P +yN +zN, Sq2(P ) = wU +NP + P +w2P +wzP +w2Q+y N +z N +w N. 2-COHOMOLOGY OF CO3 9

Theorem 3.3. The cohomology of S′ is detected on six elementary abelian 2-subgroups: 26 = ⟨u, t, a, c, w, z⟩, 25 = ⟨u, t, c, w, y⟩, 25 = ⟨u, t, c, ab, wy⟩, 25 = ⟨u, t, c, az, bz⟩, 25 = ⟨u, tc, a, x, z⟩ and 25 = ⟨u, tc, az, bz, xz⟩.

Proof. The kernels of the restriction maps are computed and the intersection of these kernels is shown to be zero. For example, the restriction map from the cohomology of S′ to the cohomology of 26 satisfies Res(x) = Res(y) = Res( ) = Res() = Res(Δ) = Res(Q) = Res(M) = 0, Res( ) = a2 + aw, Res( ) = c2 + cz, Res(N) = uw + ac + tz + c2 + cz + t2 + ta + tw, Res(L) = c(a2 + aw) + z(t2 + tw + a2 + aw), Res(P ) = w(u2 + uw + uz + t2 + tw + tz) + a(t2 + ta + tz + c2 + ac + cz), Res(T ) = t4 + t2(a2 + aw + w2) + t(a2w + aw2) and Res(U) = (u + c)(u + t)(u + a + c + z)(u + t + a + w + z). Thus the kernel of the restriction map to 26 is

F2[T,U] ⊗ {F2[w, y, ](y, yN, Q, QN)+

F2[y, , ](y , N, Q, NQ) + F2[z, , ]( , L, M,  )+

MF2[z, ] + F2[x, z, ](x, , , Δ, , Δ, Δ, Δ)}. Then the restriction map from the cohomology of S′ to the cohomology of 25 = ⟨u, t, c, w, y⟩ satisfies Res(x) = Res(z) = Res( ) = Res( ) = Res() = Res(Δ) = Res(M) = 0, Res( ) = c2 +cy, Res(N) = uw +t2 +tw +ty +c2 +cy and Res(Q) = u2y +uwy +uy2 +t2y +twy +ty2. Thus the intersection of the kernels of the two restriction maps is

F2[T,U] ⊗ {F2[y, , ](y , N, Q, NQ) + F2[z, , ]( , L, M,  )+

MF2[z, ] + F2[x, z, ](x, , , Δ, , Δ, Δ, Δ)}. The restriction map to 25 = ⟨u, t, c, ab, wy⟩ satisfies Res(x) = Res(z) = Res(Δ) = Res(L) = Res(M) = 0, Res( ) = Res( ) = Res() = a2 + aw, Res( ) = c2 + cw, Res(N) = uw + t2 + c2 + ta + ac + cw and Res(Q) = u2w + (t2 + c2)a + (t + c)a2 + (t + c)aw + t2w, where a is dual to ab and w is dual to wy. The intersection of the kernels of these three restriction maps is

F2[T,U] ⊗ {F2[z, , ](z , ( + ) , L, M) + MF2[z, ]+

F2[x, z, ](x, x , + , Δ, ( + ), Δ, Δ, Δ)}. The intersection of the kernels of four restriction maps is

2 F2[T,U] ⊗ {( + + z )MF2[z, ]+ 2 2 2 F2[x, z, ](x, x , x, Δ+z, Δ+ Δ+z +z , ( + +z ), ( + +z )Δ, ( + +z )Δ)}. The intersection of the kernels of five restriction maps is

2 3 F2[x, z, , T, U](x , x , (Δ + z), (Δ + z + z )) and the intersection of the kernels of the six restriction maps is zero. □ 10 JOHN MAGINNIS

∗ ′ Note this result implies that H (S ; F2) contains no nilpotent elements, since the cohomology ring of an elementary abelian 2-group is a polynomial algebra.

4. The cohomology of S ∈ Syl2(Co3)

Theorem 4.1. The cohomology ring of S is generated by seventeen classes, with 4 degree one classes w, x, y and z, 4 degree two classes , , and , 2 degree three classes L and Q, 1 degree four class U, 1 degree five class R, 2 degree six classes A and B, 2 degree seven classes I and J, and 1 degree eight class V .

′ Proof. The six elementary abelian subgroups of S are sufficient to detect the d2 differential for the spectral sequence of the extension 2 → S → S′, which is

d2(u) = N + +  + .

1+6 To see this, consider the following subgroups of S. First, ⟨v, u, t, a, c, w, z⟩ = 2+ , and the 1+6 6 2 2 spectral sequence for 2 → 2+ → 2 has differential d2(v) = uw+ac+tz+t +ta+tw+a +aw. The restriction map from the cohomology of S′ to 26 satisfies Res(N) = uw + ac + tz + t2 + ta + tw + c2 + cz, Res( ) = c2 + cz, Res() = 0, Res( ) = a2 + aw and Res( ) = 0. 1+4 1+4 5 Next, ⟨v, u, t, c, w, y⟩ = 2 × 2+ , and the spectral sequence for 2 → 2 × 2+ → 2 has 2 ′ 5 differential d2(v) = uw + t + tw + ty. The restriction map from S to 2 satisfies Res(N) = uw + t2 + tw + ty + c2 + cy, Res( ) = c2 + cy and Res() = Res( ) = Res( ) = 0. 1+4 2 Then ⟨v, u, t, c, ab, wy⟩ equals the central product 4∘2+ , with d2(v) = uw+t +at+ac, where a is dual to ab and w is dual to wy. We have Res(N) = uw + t2 + at + ac + c2 + cw, Res( ) = 2 2 1+4 c + cw and Res() = Res( ) = Res( ) = a + aw. Also, ⟨v, u, t, c, az, bz⟩ = 2 × 2+ with 2 2 2 2 2 d2(v) = t + tb + a + ab + ac and Res(N) = t + tb + c + bc, Res( ) = c + ca + cb, Res() = 2 ab, Res( ) = a2 and Res( ) = b (a is dual to az and b is dual to bz). Next, ⟨v, u, tc, a, x, z⟩ = 1+4 2 2 2 2 2×2+ with d2(v) = t +tx+tz+a +ax and Res( ) = t +tz, Res() = tx, Res( ) = a +ax and Res(N) = Res( ) = 0 (where t is dual to tc). 1+4 2 2 Finally, ⟨v, u, tc, az, bz, xz⟩ = 2 × 2+ with d2(v) = t + ta + tb + a + ab + ax and Res(N) = 2 0, Res( ) = t2 + ta + tb + tx, Res() = tx + ab, Res( ) = a2 + ax and Res( ) = b + bx, where t, a, b and x are dual to tc, az, bz, xz respectively.

The multiplicative properties of differentials in a spectral sequence determine all other d2 differentials. There is no kernel for this differential. Other d2 differentials are described below.

d2(vx) = Δ + z, d2(v( + )) = M + xz( + ), d2(vz ) = L + x ( + ) 2 2 d2(v(N + + + + w + wy + wz)) = wP + + (w + z)L + T + ( + + z )+ 2 + + (w2 + y2 + z2) 2 2 d2(vy(N + + + + w + wy)) = w Q + yT + y

d2(v (N + + xz )) = y Q + T + 2-COHOMOLOGY OF CO3 11

2 d2(vΔ) = xT + zM + L + Δ + ( + z )Δ + z  + z 2 2 3 d2(vM) =  + + z M + z L + ( + + x )T + (z + x + z )Δ+ ( 2 + + x2 + xz + z2 ) + x2 2 + xz 2 + z2 2 2 d2(v L) = z + z M + z L + (x + z) T + Δ + (x + z)  + x 2 2 2 2 2 4 d2(v( + + z )M) = ( + + z )T + x(x + x + z + xz )T + ( + + z + z )M+ z( + + z2)Δ + (xz2 + z3 + z5)Δ + (x2 + z2)z2  + (x + z)xz2 3 2 2 2 2 2 2 2 d2(vNP +⋅ ⋅ ⋅ ) = w U+( +w +wL+zL+T + +w +z )P + Q+w +(z +w ) + z M + ( + w2 + wz )L + w T + ( + xz )Δ + (x + z 2 + z + x2z + xz2 )+ w3 2 + y3 2 + x 2 + z 2 + x2z 2 + wz2 2 + xz2 2 2 d2(vNQ + ⋅ ⋅ ⋅ ) = w yU + Q + TQ + y T 3 2 d2(v NQ + ⋅ ⋅ ⋅ ) = y U + Q + T Q + y T.

2 The E3 page is equal to the tensor product of F2[v ] with the following.

2 F2[z, , , T, U](1,L)(1, P, wU, w U)+wF2[w, z, , , T ](1,L)+F2[y, , T, U](y, Q, wQ, wyU)+ 2 2 wyF2[w, y, , T ] + F2[ , , T, U](y , Q, y U) + y F2[y, , , T ] + F2[x, z, , U](x, )(1, )+

F2[z, , T, U](T, ) + F2[z, , T, U]( , ) + F2[ , , T, U].

2 1 The d3 differential is determined by d3(v ) = Sq (d2(v)), yielding

2 d3(v ) = P + (w + y + z) .

There is no kernel for this differential. Other d3 differentials are listed below.

2 2 2 2 d3(v w) = T + + (w + z)L + ( + + z ) + + + wz + z 2 2 2 d3(v x) = xz , d3(v y) = wQ, d3(v wy) = yT + y 2 2 2 2 2 d3(v (w y + wy )) = (wy + y )T, d3(v y ) = T + 2 2 d3(v ) = Q + z , d3(v wQ) = (T + )Q 2 2 2 2 d3(v ( + )) = z + x z + xz  + x z 2 2 2 2 3 2 2 3 4 2 2 3 3 d3(v z) = (+ )T +(x +z +z +x z+x z +xz +z )+xz +x +xz +x z +xz 2 2 2 3 2 d3(v (  +  + z )) = z  + z  + z , d3(v ( + )) = z T 2 2 3 2 2 3 2 2 2 d3(v P ) = w U +(w+z) L+ T +(w +z +w +w z+wz +z )L+( + +w +y +z )T + (x2 + xz + z2 + z2 + x2z2 + xz3 + z4) + w2 2 + y2 2 + x2 + xz + z2 2 + z2 + w3z + w2z2 + wz3 + x2z2 + z4 2 2 2 2 2 2 4 3 3 4 d3(v Q) = wyU +  + ( + )T + ( + + x + x + z + x + x z + xz + z )+ xz 2 + (xz3 + x4) . 12 JOHN MAGINNIS

4 The E4 page is equal to the tensor product of F2[v ] with the following.

F2[z, , , U](1,L)(1, wU) + wF2[w, z, , ](1,L) + w F2[w, z, , T ](1,L)+ 2 F2[y, , U](y, Q) + wyF2[w, y, ] + y F2[ , , U] + y F2[y, , ]+

F2[x, , U](x, )(1, ) + zF2[x, z, U](x, )(1, ) + z F2[z, , U] + F2[ , , U]+

z F2[z, , U] + z F2[z, , U] +  F2[ , U].

4 2 2 The d5 differential is determined by d5(v ) = Sq (d3(v )), which yields

4 d5(v ) = wU. There are several new generators in the kernel of this differential: R = v4x, A = v4(wy + y2),B = v4( + ),I = v4z and J = v4Q. We will show in section 6 that the class A can be defined as a transfer from the index two subgroup Sx, and the classes R,B,I and J can be defined as transfers from the subgroup Sw.

Other d5 differentials are given below.

4 2 2 2 2 2 2 2 2 3 d5(v w) = + (w + wz + y ) + ( + + z + x + xz + x z + xz )+ 3 + 2 + wz 2 + z2 2 + x2 4 2 2 2 2 4 2 2 2 4 d5(v y) =  + ( + + x + z + x + x z ) + x + x 4 2 2 3 2 2 d5(v y ) = + (x + x z + x z )+ x2 2 + xz 2 + x3z + x2z2 4 4 2 2 5 4 2 2 2 5 4 d5(v wy) = d5(v y ) = y , d (v y ) = y , d (v ) = y U.

The spectral sequence now collapses, yielding a new generator V = v8. The additive descrip- tion of the mod 2 cohomology of the Sylow 2-subgroup S of the sporadic finite simple group

Co3 is given below.

2 F2[z, , U, V ](1, )(1,L) + F2[z, , U, V ](1,L) + F2[z, , U, V ] + F2[z, , U, V ]+ 2 F2[ , U, V ] + wF2[w, z, , V ](1, )(1,L) + w F2[w, z, , V ](1,L) + F2[y, , U, V ](y, Q)+ wyF2[w, y, , V ] + y F2[y, , V ](1, ) + F2[x, , U, V ](x, )(1, ) + zF2[x, z, U, V ](x, )(1, )+

z F2[z, , U, V ] +  F2[U, V ] + JF2[y, , U, V ]+

R{F2[x, , U, V ](1, )(1, ) + zF2[x, z, U, V ](1, )(1, )}+

B{F2[z, , U, V ] + F2[z, , U, V ]} + BF2[ , U, V ]+

I{F2[z, , U, V ] + F2[z, U, V ]} + A{F2[w, y, , V ] + UF2[y, , U, V ]}. □ 2-COHOMOLOGY OF CO3 13

Corollary 4.2. [6, Green, King] The Poincare´ series for the cohomology ring of S is:

1 + t + 3t2 + 2t3 + 3t4 + t5 + 3t6 + 2t7 + t8 + t9 . (t + 1)(t − 1)4(t2 + 1)2(t4 + 1) Proof. The additive description for the cohomology ring of S yields the Poincar´eseries. Note (1+t2)(1+t3) that a term such as F2[z, , U, V ](1, )(1,L) yields the series (1−t)(1−t2)(1−t4)(1−t8) . Collecting together those terms with the same denominator yields

(1 + t2)(1 + t3) + t4(1 + t3) + t2 + t4 + (t + t3) + (t + t2)(1 + t2) + t5 + t7 + (1 − t)(1 − t2)(1 − t4)(1 − t8) t5(1 + t2)(1 + t2) + t6(1 + t2) + t7 + t6(t4) t(1 + t2)(1 + t3) + t5(1 + t3) + t2 + t6 + + (1 − t)(1 − t2)(1 − t4)(1 − t8) (1 − t)2(1 − t2)(1 − t8) t(t + t2)(1 + t2) + t5(t)(1 + t2)(1 + t2) t6 + t8 + + (1 − t)2(1 − t4)(1 − t8) (1 − t2)(1 − t4)(1 − t8) t3(1 + t2) t7(t2) t4 + + . (1 − t)(1 − t2)(1 − t8) (1 − t)(1 − t4)(1 − t8) (1 − t4)(1 − t8) After using the common denominator (1 − t)(1 − t2)(1 − t4)(1 − t8), this is equal to

1 + 3t + 5t2 + 8t3 + 9t4 + 8t5 + 9t6 + 10t7 + 9t8 + 6t9 + 3t10 + t11 . (1 − t)(1 − t2)(1 − t4)(1 − t8) The numerator and denominator of this fraction have a common factor of (1+t)2. Cancelling this (and factoring the denominator) yields the expression given by Green and King. □

5. Cohomologies of Subgroups of S

We describe the cohomology rings of several subgroups of S, in particular the subgroups

Sw = ⟨v, t, u, a, b, c, x, y, z⟩ and Sx = ⟨v, t, u, a, b, c, w, y, z⟩, with the goal of defining the generators for the cohomology of S as either Evens norms or transfers.

Theorem 5.1. The cohomology ring of the subgroup Sw = ⟨v, t, u, a, b, c, x, y, z⟩ is generated by twelve classes, with 4 degree one classes a, x, y and z, 4 degree two classes , ,  and , 3 degree three classes L, Q and , and 1 degree four class Λ.

′′′ Proof. The group Sw is isomorphic to 2 × (U4/2) with cohomology equal to

F2[a, x, y, z, , ](1, ) + yF2[y, z, , ].

′′ ′′′ The differential for the spectral sequence 2 → Sw → Sw is given by d2(c) = yz. There are three new generators K = cx, N = ca and L = c. We can define these classes using 6 ′′ the transfer map from the elementary abelian subgroup 2 = ⟨t, a, b, c, x, z⟩ ⊆ Sw where T r(t) = a, T r(b) = x, T r(c) = z, T r(tb) = , T r(tc) = N, T r(bc) = K and T r(tbc) = L. These 14 JOHN MAGINNIS do not quite match up with the classes given in sections 2 and 3; we have restriction maps Res() =  + ax, Res(N) = N + +  + az, Res(K) = K and Res(L) = L + aN + z. The spectral sequence collapses with the new generator = c2, and we have the cohomology ′′ of Sw given below.

∗ ′′ H (Sw; F2) = F2[a, x, z, , , ](1, , K, L) + NF2[a, z, , , ] + yF2[y, , , ]. ′ ′′ The differential for the spectral sequence of the extension 2 → Sw → Sw is d2(u) = K +  + + ax + xz. There is no kernel for this differential. Other differentials are given below.

2 2 d2(uy) = y , d2(u) = xL + x  + xz +  + a + az 2 2 2 2 2 2 d2(u(K + + ax)) = x = xL +  + ax + + (xz + z + az) + ax z + a x 2 2 2 2 3 2 d2(u(N + a + az)) = ax + (a + z)L + N + xz + (a + z ) + (a + az) + a x + a xz 2 2 d2(u(L + x)) = x + L + (a + z )N + x + (x + x z + ax ).

2 2 Then we have d3(u ) = x + x + (a + z) + a + a x + axz. The kernel of this differential is generated by Q = u2y. Other differentials are:

2 2 d3(u a) = N + (a + z)L + (ax + xz) + (az + z ) + az 2 2 2 2 d3(u x) = + (a + ax + xz + z ) 2 2 2 2 2 2 3 3 2 d3(u ( + a + az)) = ( + a + az)L + (x + a x + xz ) + (a + a z = z ) + (a + az ) 2 2 2 d3(u N) = a + (a + z) + (a + z) + (a + a x + xz ) 2 3 2 2 2 2 d3(u (L + (a + z) + a + az )) =  + (a + az) +  + ( + (az + z ) )+ (a2 + az)  + (a4 + a2z2) .

′ The cohomology ring of Sw has the following additive description.

∗ ′ H (Sw; F2) = F2[a, z, , ](1, , N, L) + xF2[a, x, z, ](1, )(1, ) + F2[a, z, ](1, )+

F2[z, , ] + yF2[y, , ]. ′ The d2 differential for the spectral sequence of the extension 2 → Sw → Sw is determined by 2 d2(v) = N +  +  + a + az. There is no kernel for this differential. Other differentials are:

d2(vy) = y, d2(vx) = x + (a + x + z) + a + axz 2 2 2 2 2 4 3 d2(v(N +  + a )) = a +  + (az + x + az) + (ax + az) + a + a + a z 2 3 d2(v(L + z)) = a + L + x + (x + z + a x) + a  + a 2 2 2 d2(v) = aL +  + (x + xz) + (a + ax + az) + a 2 2 d2(v( + )) = zL +  + ( + ax + x ) + ( + a + ax) 2 2 2 2 3 2 2 2 d2(v(a +a+z)) = a+z+(a +a x+ax +x z+xz )+(a +a +a x+a z+axz)+a z 2-COHOMOLOGY OF CO3 15

2 2 2 2 4 3 3 2 d2(v( +a +az )) =  +(a +ax+az+x +xz)  +(a +ax) +(a +a x+a z+a xz) 2 2 2 d2(vQ) = Q + L + z + a + L + a + (a + z + a x + axz) + (a z + az ).

2 The E3 page of the spectral sequence is the tensor product of F2[v ] with

F2[a, z, ](1, ) + F2[z, , ]( , a , , ) + F2[a, z](a, ) + LF2[, ]+

xF2[a, x, z](1, )(1, ) + [y, ](y, Q). 2 3 2 Then we have d3(v ) = a + a + a + a z + axz. The kernel of the d3 differential is generated by  = v2y. Other differentials are:

2 2 2 2 2 2 2 2 2 2 d3(v x) = xz(a +ax), d3(v a) =  +(az+z ) +(a +az+x +xz)+(a +ax) +a xz+ax z

2 3 2 3 2 2 3 3 2 2 d3(v ) = L+z  +(a+x+z) +(x +xz )+(a +ax +axz +az ) +a xz +ax z +ax z 2 3 3 2 2 2 2 2 3 2 2 4 3 d3(v L) =  + az  + (a x + a z + a x + a xz + a z + ax + ax z + axz + x + xz )+ (a3x + a2xz + a2z2 + ax3 + axz2) + a4xz + a3x2z + a2x3z + a2x2z2 + ax4z + ax2z3 2 2 2 2 2 3 d3(v ( + )) = z + (a + z) + a  + (a + a x + ax + az + xz + z )+ (a3 + a2x + a2z + axz + az2) + a3xz + a2x2z + axz3 2 2 2 2 3 2 3 2 4 3 2 2 3 4 d3(v z) = ( + ) +z  +(ax+x +xz)  +a x z +a xz +ax z +ax z +ax z +axz + (a3z + a2x2 + a2z2 + ax2z + axz2 + az3 + x4 + z4) + (a3x + a3z + ax3 + ax2z + az3) 2 3 2 2 2 2 3 2 d3(v (a + z + a )) = (a z + a xz + a z + axz ) + (a z + a xz) 2 2 2 4 3 3 2 2 2 d3(v (  + a + az )) = (a z + axz)  + (a z + a xz + a z + a xz ) 2 2 2 2 2 2 3 2 2 2 2 2 3 4 d3(v Q) =  + ( + ) + z  + (a + x + xz)  + a xz + a x z + ax z + axz + (a4 + a3z + a2x2 + az3 + x2z2 + z4) + (a3x + a3z + a2x2 + a2xz + a2z2 + az3) 2 2 2 3 2 2 2 2 2 4 2 2 d3(v (z + L)) =  + z  + (a z + a x + a z + axz + x + x z ) + (a4xz + a3x2z + a3xz2 + a2x3z + x5z + x4z2 + x3z3 + x2z4)+ (a5z + a4x2 + a4xz + a3z3 + a2x4 + a2x3z + a2x2z2 + a2xz3 + ax4z + ax3z2) + a5xz2 + a4x2z2 + a3x3z2 + a2x3z3 + ax5z2 + ax3z4.

The spectral sequence collapses with the new generator Λ = v4 surviving. The additive description of the cohomology of Sw is:

F2[a, z, Λ](1, )(1, ) + F2[z, , Λ](1, )(1, ) + xF2[a, x, Λ](1, )(1, )+

xzF2[x, z, Λ](1, a)(1, )(1, ) + F2[y, , Λ](y, Q, , Q) + LF2[ , Λ]+

F2[z, Λ](1, , z , az ) + [a, Λ](a, ) + azF2[z, Λ](1, a) +  [Λ]. □ 16 JOHN MAGINNIS

Theorem 5.2. The cohomology ring of the group Saw = ⟨v, t, u, b, c, x, y, z⟩ is generated by nine classes, with 4 degree one classes t, x, y and z, 3 degree two classes , and , 1 degree three class Q and 1 degree four class U.

′′ Proof. The group Saw is isomorphic to 2×(U4/2) with cohomology generated by t, x, y, z, , 2 2 and K. We have d2(u) = K + + xz + tx, d2(uy) = y and d2(u(K + + tx)) = x + + 2 2 2 2 2 2 2 2 2 xz + z + t x + tx z. Then d3(u ) = x + t x + txz and d3(u x) = + xz + z . The 2 kernel of the d3 differential is generated by Q = u y. The spectral sequence collapses, and 4 ′ U = u survives. The cohomology of Saw equals

F2[t, z, , U](1, ) + xF2[t, x, z, U](1, ) + F2[t, y, , U](y, Q).

2 2 Then d2(v) = t + tx + ty + tz, and this spectral sequence collapses with  = v surviving.

The cohomology of Saw equals

F2[z, , , U](1, t)(1, ) + xF2[x, z, , U](1, t)(1, ) + F2[y, , , U](1, t)(1,Q).

□

′ Lemma 5.3. The cohomology of the group Sx = ⟨t, u, a, b, c, w, y, z⟩ is generated by nine classes, with 5 degree one classes a, b, w, y and z, 3 degree two classes ,  and , and 1 degree three class L.

3 ′ Proof. There is an elementary abelian subgroup 2 = ⟨t, u, c⟩ ∈ Sx such that the quotient ′ 3 6 group Sx/2 is elementary abelian 2 = ⟨a, b, w, y, z⟩. We first adjoin c to get a group 3 2 isomorphic to 2 × D8, with d2(c) = yz and = c . Then we adjoin u, with differential 2 2 d2(u) = b + ab + by + bz + ay. This collapses at the E3 page, with  = u surviving. Finally we adjoin t, with differential d2(t) = wb + ay. We also have differentials d2(t(a + b + y + z)) = 2 2 2 2 2 2 2 aby + a y + ay + awy and d2(t(wb + aw + wy + wz + ay)) = a y + a wy + aw y + awy . The kernel of this differential is generated by L = tz(a + b + z). The spectral sequence collapses 2 ′ at the E3 page, with  = t surviving. The cohomology of Sx equals:

2 F2[a, w, z, , , ](1,L) + F2[w, y, , , ](y, ay) + a yF2[a, w, , , ]+

bF2[a, z, , , ] + byF2[y, , , ].

□

Theorem 5.4. The cohomology of the group Sx = ⟨v, t, u, a, b, c, w, y, z⟩ is generated by thir- teen classes, with 5 degree one classes a, b, w, y and z, 2 degree two classes and , 1 degree three class L, 3 degree six classes A, Γ and B, 1 degree seven class J and 1 degree eight class V . 2-COHOMOLOGY OF CO3 17

′ Proof. The differential for the spectral sequence of the extension 2 → Sx → Sx is determined 2 2 3 2 2 2 2 by d2(v) = +a +ab+aw+ay. Then we have d3(v ) = w+a +a +a b+a w+a y+a z+awz, 2 2 as well as d2(v b) = ay+ab and d2(v bz) = abz . There are no kernels for these differentials. 4 The E4 page equals the tensor product of F2[v ] with the following.

2 F2[a, z, , ](1,L) + wF2[a, w, z, ](1,L) + yF2[w, y, ](1, a) + a yF2[a, w, ]+

bF2[a, z, ] + F2[y, , ](by, y) + b F2[a, , ] + bz F2[z, , ].

The d5 differential is determined by

4 3 2 2 2 3 2 2 2 2 d5(v ) = a  + (a + a b + a z) + a + (a + a b + a w + a y + aw + awy + awz) + a4(w + y + z) + a3(bz + w2 + wy + z2) + a2(w2z + wz2).

4 2 4 2 4 2 2 We also have d5(v b) = ab  + ab , d5(v y) = (ab + ay) , d5(v wy) = (a y + awy) and

4 2 2 2 2 2 2 3 2 2 d5(v w) = (a + aw) + (a w + a wy + a wz + aw + aw y + aw z) + a6 + a5(b + w + y) + a4(bz + w2 + wy + wz + z2) + a3(w3 + w2y) + a2(w3z + w2z2).

4 2 4 4 The kernel of the d5 differential is generated by A = v (wy + y ),B = v bz, Γ = v (by + ay) and J = v4(y + b ). Then the spectral sequence collapses, with V = v8 surviving. The cohomology of Sx is equal to:

F2[a, z, , V ](1,L) + wF2[w, a, z, V ](1,L) + F2[z, , , V ](1,L) + w F2[w, z, , V ](1,L)+ 2 2 a F2[a, z, V ](1,L) + aw F2[w, a, z, V ](1,L) + a F2[z, , V ](1,L) + aw F2[w, z, , V ](1,L)+ 2 yF2[w, y, , V ](1, a) + a yF2[a, w, V ](1, ) + bF2[a, z, , V ] + F2[y, , , V ](by, y)+

bz F2[z, , , V ] + b F2[ , , V ] + ab F2[a, V ] + JF2[y, , , V ]+

B{F2[a, z, , V ] + F2[a, , , V ]} + AF2[w, y, , V ](1, a) + ΓF2[y, , , V ]. □

Theorem 5.5. The cohomology ring of Sax = ⟨v, t, u, b, c, w, y, z⟩ is generated by nine classes, with 4 degree one classes b, w, y and z, 3 degree two classes N, and , 1 degree three class , and 1 degree four class Λ.

′′ Proof. The group Sax is isomorphic to D8 ×D8, with cohomology generated by b, w, y, z,  and ′ ′′ . The differential for the spectral sequence of the extension 2 → Sax → Sax is determined 2 by d2(u) = b + by + bz. The kernel of this differential is generated by N = uw. The spectral sequence collapses, and  = u2 survives. ′ The differential for the spectral sequence of the extension 2 → Sax → Sax is determined by 2 d2(v) =  + N. There is no kernel for this differential. We then have d3(v ) = w, with kernel generated by  = v2b. The spectral sequence collapses with Λ = v4 surviving. The cohomology of Sax is equal to: 18 JOHN MAGINNIS

F2[w, y, , Λ](1,N) + F2[y, , , Λ](1,N) + zF2[w, z, , Λ](1,N) + zF2[z, , , Λ](1,N)+

F2[y, , , Λ](b, ) + zF2[z, , , Λ](b, ).

□

Theorem 5.6. The cohomology ring of Sbx = ⟨v, t, u, a, c, w, y, z⟩ is generated by nine classes, with 5 degree one classes u, a, w, y and z, 2 degree two classes and N, 1 degree five class Ω and 1 degree eight class V .

′ 2 Proof. The quotient group Sbx is isomorphic to 2 × (U4/2), with the elementary abelian 2 factor 2 = ⟨u, w⟩, and U4/2 = ⟨tu, a, c, y, z⟩. The degree one class u corresponds to a group ′ homomorphism Sbx → ℤ/2ℤ with kernel ⟨tu, a, c, w, y, z⟩. ′ 2 For the spectral sequence 2 → Sbx → Sbx, we have differentials d2(v) =  + N + uw + a + aw 2 2 2 3 2 2 and d3(v ) = u w + u(w + aw + wy + wz) + a + a + a z + aw + awz. There are no kernels 4 2 2 2 2 2 3 for these differentials. Then d5(v ) = a + (u a + u(a + aw + az) + aw + awz) + u (a + a2z) + u(a4 + a3w + a2wz + a2z2) + a5 + a3(w2 + wz + z2) + a2(w2z + wz2). The kernel of this differential is generated by Ω = v4y. Then the spectral sequence collapses with V = v8 surviving. The cohomology ring of Sbx is equal to:

F2[u, z, , V ](1,N) + aF2[u, a, z, V ](1, )(1,N) + wF2[w, z, , V ](1, u)(1,N)+

awF2[a, w, z, V ](1, u)(1, )(1,N) + F2[u, , V ](y, Ω) + wF2[w, , V ](y, Ω).

□ Theorem 5.7. The cohomology ring of the group H = ⟨v, t, u, a, c, w, by, z⟩ is generated by nine classes, with 5 degree one classes u, a, w, y and z, 2 degree two classes and N, 1 degree five class Ω and 1 degree eight class V .

Proof. The group H is isomorphic to the group Sbx, and an isomorphism is given by conjuga- tion by the group element x ∈ S. The degree one cohomology class u corresponds to a group homomorphism H → ℤ/2ℤ with kernel ⟨v, t, uc, a, w, by, z⟩. The degree one cohomology class y corresponds to a group homomorphism H → ℤ/2ℤ with kernel ⟨v, t, u, a, c, w, z⟩. □

6. The generators

Recall that degree one cohomology classes correspond to group homomorphisms via the Uni- versal Coefficient Theorem and the Hurewicz Theorem:

1 H (G; F2) ≃ Hom(H1(G, ℤ), F2) ≃ Hom(G/[G, G], F2) ≃ Hom(G, ℤ/2ℤ). 2-COHOMOLOGY OF CO3 19

Thus the cohomology class w corresponds to a group homomorphism S → ℤ/2ℤ with kernel the subgroup Sw generated by ⟨v, u, t, a, b, c, x, y, z⟩. Similarly, the cohomology classes x, y and z correspond to homomorphisms with kernels Sx,Sy and Sz respectively. The degree two classes , and , the degree four class U, and the degree eight class V can S S S S be defined using the Evens norm [4]. We have = NSw (a) = NSx (a), = NSx (b) = NSy (b) and = N S (c) = N S (c). Also U = N S (u) = N S () and V = N S (v) = N S (Λ). Sy Sz Sbx Sx Staw Sw The remaining classes can be defined using the transfer map. As briefly mentioned in section S S 2, the degree two class  satisfies  + = T rSw () (or alternatively  + = T rSy ()) where the class  is the Evens norm  = N Sw (t) (or alternatively  = N Sy (t)). Also, the degree Swy Swy S three class L = T rSy (c). As discussed in section 3, the degree class Q can be defined using a transfer from the subgroup S Sx as Q = T rSx (b) with  equal to an Evens norm. However, the relation wQ = 0 following 2 from the differential d3(v y) = wQ is section 4 implies that Q can also be defined as a transfer from the subgroup Sw. The cohomology classes R,B,I and J (represented in the spectral sequence by the filtered or graded expressions v4x, v4( + ), v4z and v4Q) can also be given as transfers from the 4 2 subgroup Sw. The class A (represented in the spectral sequence by v (wy + y )) can be defined using the transfer from the subgroup Sx.

Lemma 6.1. Define Q = T rS () = T rS (t), where  = N Saw (v). Also define Q = Sw Saw Staw S Sx T rSx (b), where  = NH (u) with H = ⟨v, t, u, a, by, c, w, z⟩. Then Q = Q + z.

1+6 Proof. We restrict these classes to the six subgroups H1 = 2 = ⟨v, t, u, a, c, w, z⟩,H2 = 2 × 1+4 1+4 1+4 2+ = ⟨v, t, u, c, w, y⟩,H3 = 2×2+ = ⟨v, tc, u, a, x, z⟩,H4 = 4∘2+ = ⟨v, t, u, c, ab, wy⟩,H5 = 1+4 1+4 2 × 2+ = ⟨v, t, u, c, az, bz⟩ and H6 = 2 × 2+ = ⟨v, tc, u, az, bz, xz⟩. See the computation of d2(u) in Theorem 4.1. Compute restrictions using the double coset formula. We can compute Res(Q) to the elementary abelian subgroups in the group S′ and then induce to 2 2 S. We have ResH1 (Q) = Res(Q) = Res() = 0. Then ResH2 (Q) = u y + uy , ResH2 (Q) = 2 2 2 2 u y + uwy + uy + t y + twy + ty and ResH2 (z) = 0. But also in the spectral sequence for 5 2 2 → H2 → 2 we have d2(v) = uw + t + tw + ty, so that ResH2 (Q) = ResH2 (Q) + ResH2 (z).

Next, ResH3 (Q) = xzt, ResH3 (Q) = 0 and ResH3 (z) = xzt. Then ResH4 (Q) = ResH4 (z) = 0 2 2 2 2 2 2 2 and ResH4 (Q) = u w+t a+t w+ta +taw+a c+ac +awc. But also d2(v) = uw+t +ta+ac 2 2 2 2 2 2 2 2 2 and d3(v ) = u w +uw +t a+ta +a c+ac . Next, ResH5 (Q) = abc+b c+bc , ResH5 (Q) = 2 2 2 2 2 2 2 2 2 t b + tb + b c + bc , ResH5 (z) = a b + ab and d2(v) = t + tb + a + ab + abc. Finally, 2 2 2 ResH6 (Q) = ResH6 (z) = tx + tax + tbx + a b + ab + abx and ResH6 (Q) = 0. □

Theorem 6.2. Using the transfer map for Sw ⊆ S, we can define the following generators ∗ for H (S; F2): R = T r(aΛ),B = T r(Λ), I = T r(aΛ) and J = T r(Λ). 20 JOHN MAGINNIS

Proof. The expressions I and J are not equal to I and J, in the sense of the representing expressions in the spectral sequence, but are linear combinations involving I and J and so can be substituted for them in the set of generators.

In the cohomology ring of Sw, the classes  and Λ can be defined as the Evens norms of degree one classes  = N Sw (t) and Λ = N Sw (v). The degree three class  can be defined as Swy Staw a transfer from the subgroup S = ⟨v, t, u, b, c, x, y, z⟩, with  = T rSw (t). This class  is aw Saw the Evens norm N Saw (v). The transfer map from the subgroup S to the group S satisfies Staw w T r(a) = x, T r() =  + and T r() = Q, where the expression Q = Q + z. To verify these claims we compute various restrictions. Also note that the image of the transfer S map T rSw equals the kernel of multiplication by w, and the kernel of the restriction map S S S ResSw equals the multiples of w. We have ResSw T rSw (a) = (id+Conw)(a) = a+(a+x) = x. In the context of group elements, the corresponding computation involves the commutator Sw 4 [w, x] = av. We also have Res⟨v⟩(Λ) = v , using the multiplicative version of the double coset formula for the restriction of an Evens norm. Thus Λ is represented by the expression v4 in the spectral sequences. Similarly,  is represented by v2. S S Now ResSw T rSw (aΛ) involves the term xΛ, and so is represented in the spectral sequence by 4 S v x. Therefore T rSw (aΛ) is a choice of a definition of the class R. Next, ResS T rS () = (id + Con )(ResSw () = (id + Con )(t(t + a)) = t2 + ta + (t + b)2 + Swy Sw w Swy w 2 S S S (t+b)(a+x) = (tx+ab)+(b +bx) = ResSwy ( + ). Then ResSw T rSw (Λ) involves ( + )Λ, 4 S and is represented in the spectral sequence as v ( + ). Thus T rSw (Λ) is a possible choice of a definition of the class B. S S We have ResSw T rSw (a) = x + (a + x)( + ax + ). Using the differential d2(v) = x + (a + S S x + z) + a + axz described in Theorem 5.1, we obtain ResSw T rSw (a) = z( + ax) + x + 2 S x(a + ax) = ResSw (z + x( + )). Thus I = I + ( + )R could be used to define the class S I in terms of I = T rSw (aΛ). S S S Since Q = T rSw () is equal to Q + z, we have that ResSw T rSw (Λ) is represented in the spectral sequence by a term involving v4(Q + z). Therefore J = T r(Λ) can be expressed as the linear combination I + J, allowing us to define I. □

Theorem 6.3. Using the transfer map for Sax ⊆ S, we can define the generator A = S ∗ T rSax (NΛ) for H (S; F2).

Proof. In the cohomology ring of S , we have N =  = N Sax (t). Alternatively, N is the sum ax Sabx T rSax (tu) + w(y + z). The class Λ is the Evens norm Λ = N Sax (v). Sabx Stawx The restriction of A to the elementary abelian subgroup ⟨v, w, y⟩ is, by the double coset formula,

S S Sax Sax Res⟨v,w,y⟩T rSax (NΛ) = Res⟨v,w,y⟩(NΛ) + ConaRes⟨v,vw,tuy⟩(NΛ)+

Sax Sax ConxRes⟨v,vaw,by⟩(NΛ) + ConaxRes⟨v,aw,vtby⟩(NΛ). 2-COHOMOLOGY OF CO3 21

The restriction of N is zero in three of these terms, leaving only the term involving ⟨v, vw, tuy⟩. We obtain the restriction of A as y(w+y)(v)(v+w)(v+y)(v+w+y). Note the term v4(wy+y2) appears in the expansion, representing A in the spectral sequence. Next, restrict to the group ⟨v, x⟩ to show that the class R is not involved. We have

S ⟨v,x⟩ Sax Res⟨v,x⟩T rSax (NΛ) = (id + Cona)T r⟨v⟩ Res⟨v⟩ (NΛ). ⟨v,x⟩ But the transfer map T r⟨v⟩ from a proper subgroup of an elementary is the zero map. Finally, restrict to the ⟨v, acz, bcz⟩, which would detect the class B. S ⟨v,acz,bcz⟩ Sax We have Res⟨v,acz,bcz⟩T rSax (NΛ) = (id + Conx)T r⟨v,bcz⟩ Res⟨v,bcz⟩(NΛ). Again, the transfer map is the zero map. S 4 2 Therefore A = T rSax (NΛ) is represented in the spectral sequence by v (wy + y ), and so is a possible choice of a definition of the class A. □

References

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[5] L. Finkelstein, The maximal subgroups of Conway’s group C3 and McLaughlin’s group, J. Algebra 25 (1973) 58–89. [6] D. J. Green, S. King, http://users.minet.uni-jena.de/ ˜king/cohomology/ [7] J. Maginnis, Braids and mapping class groups, Ph.D. Thesis, Stanford University 1987.

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Department of Mathematics, Kansas State University, 137 Cardwell Hall, Manhattan, Kansas 66506, Email: [email protected]