The Cohomology of the McLaughlin Group

and Some Associated Groups

Alejandro Adem† Department of Mathematics University of Wisconsin Madison, WI 53706 R. James Milgram† Department of Math and Statistics University of New Mexico Albuquerque, NM 87131

§0 Introduction

The group of order 2^{7}, Syl_{2}(M_{22}) = H seems quite remarkable – being the Sylow 2-subgroup of three of the sporadics, M_{22}, M_{23}, McL, as well as the simple group of Lie type P SU_{4}(3). Also, it is a particularly nice index two subgroup of Syl_{2}(Ly). We are interested in determining the mod(2) cohomology of these groups, and throughout this note all cohomology groups will be taken with 𝔽_{2} coefficients, so the coefficients will generally be suppressed. We use the standard Atlas (see [Co]) notation.

We have already determined the rings H^{∗}(M_{22}) ⊂ H^{∗}(H), [AM2] and H^{∗}(M_{23}) ⊂ H^{∗}(H), [M]. In this article we determine the rings

H^{∗}(P SU_{4}(3)) ⊂ H^{∗}(H) and H^{∗}(McL) ⊂ H^{∗}(H).

Additionally, the group extension

N = 3 · McL: 2 ⊂ Ly

is a maximal subgroup of odd index in Ly, and we are able to determine H^{∗}(N) which contains H^{∗}(Ly) as a subring. This is very close to determining H^{∗}(Ly) itself. Indeed, what is required is a close study of the double coset decomposition of Ly determined by N, which apparently is already reasonably well understood [Co].

The results are

Theorem A: There is a long exact sequence

\begin{matrix}0⟶𝔽_{2}[d_{8}, d_{12}](a_{2}, & a_{7}, a_{11}, a_{14})⟶H^{∗}(P SU_{4}(3))⟶ \\ H^{∗}(2^{4})^{A_{6}} ⊕ H^{∗}(2^{4})^{A_{6}}⟶𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, w̄_{3}b_{15})⟶0. \end{matrix}

† Both authors were partially supported by grants from the National Science Foundation and the ETH-Zürich. Also, the first author was partially supported by an NSF Young Investigator Award.

Here the subscript on an element denotes its dimension, 𝔽_{2}[x, …] is the polynomial algebra on the stated generators, and the elements in parenthesis following the polynomial algebra are module generators. Also, in [AM4] the algebra of A_{6}-invariants was determined as

𝔽_{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15}).

In the case of McL the result takes the form

Theorem B: There is a long exact sequence

\begin{matrix}0⟶𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}) & ⟶H^{∗}(McL)⟶ \\ H^{∗}(2^{4})^{A_{7}} ⊕ H^{∗}(2^{4})^{A_{7}}⟶𝔽_{2}[d_{8}, d_{12}](1, x_{18})⟶0. \end{matrix}

Again, in [AM4] the algebra of A_{7}-invariants was determined as

𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, x_{18}, x_{20}, x_{21}, x_{24}, x_{25}, x_{27}, x_{45})

where 𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}] is the Dickson algebra of GL_{4}(2) invariants in 𝔽_{2}[x_{1}, y_{1}, z_{1}, w_{1}]. In particular, H^{∗}(McL; 𝔽_{2}) is 6 connected, and this is the largest connectivity of the 2-summand that we have seen yet in dealing with the sporadic groups.

Finally we have

Theorem C: The mod(2) cohomology of the maximal subgroup N ⊂ Ly has the form

𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}) ⊕ H^{∗}(2^{4})^{A_{7}} ⊕ 𝔽_{2}[d_{8}, d_{12}, e_{1}](1, a_{7}, a_{11}, b_{18})e_{1}

where e_{1} is the generator coming from the split ℤ/2-subgroup of N, and the remaining classes restrict to H^{∗}(McL).

We also show in §6 that the first two summands in the display above are in the image of restriction from H^{∗}(Ly), but, as indicated, more work is necessary to complete the determination of H^{∗}(Ly). S. Smith and K. Umland are currently doing the necessary analysis of Ly, the results will appear in a sequel.

The techniques used here are quite a bit different from the methods usually used in the cohomology of finite groups. It is first critical to determine H^{∗}(P SU_{4}(3)). To do this we use our knowledge of H^{∗}(SU_{4}(3)) and a spectral sequence with E_{2}-term Ext_{H^{∗}(ℤ/4)}(H^{∗}(SU_{4}(3)); 𝔽_{2}) which converges to H^{∗}(P SU_{4}(3)). This E_{2}-term is calculated and the d_{2}-differentials are calculated explicitly in §3. There may well be higher differentials in the spectral sequence, but E_{3} = E_{∞} through at least dimension 8. This is enough to determine that there are two distinct classes in dimension 5 in H^{∗}(P SU_{4}(3)). In turn, the existence of these two classes forces the remaining classes to have one and only one form – thus determining H^{∗}(P SU_{4}(3)). Moreover, once this ring is determined, the calculation of H^{∗}(McL) is quite direct, as is the determination of H^{∗}(N).

The description of H^{∗}(P SU_{4}(3)) is given in 3.7, the description of H^{∗}(McL) is given in 5.2 and 5.3, and the description of H^{∗}(N) is given in 6.1. The discussion in §4 gives the structure of the Bockstein spectral sequence for H^{∗}(P SU_{4}(3)) which gives considerable information about the connections between the classes in these cohomology rings as well as determining the structure of the higher two–torsion there.

§1 Preliminaries on the Subgroup Structure of McL

The McLaughlin group McL, first discovered in 1969 [McL], is a sporadic simple group of order 898, 128, 000 = 2^{7} · 3^{6} · 5^{3} · 7 · 11. At the prime p = 2 it has rank four i.e. its largest 2–elementary abelian subgroup is (ℤ/2)^{4}. It shares its 2–Sylow subgroup with the Mathieu groups M_{22}, M_{23} as well as with Ã_{8}, the unique perfect central extension of A_{8} by ℤ/2, and P SU_{4}(3) = P SU_{4}(𝔽_{9}).

In this section we describe some basic facts about the subgroup structure of McL, which will be used in our cohomology calculations. To begin, we describe H = Syl_{2}(M_{22}). We refer to [AM2] for complete details. This group can be described in six distinct but equally useful ways:

• in exactly two ways as a semidirect product (ℤ/2)^{4} ×_{T} D_{8};

• as a central extension 2^{.}UT_{4}(2) (where UT_{4}(2) denotes the group of upper triangular 4 × 4 matrices over 𝔽_{2});

• in exactly two ways as a split extension UT_{4}(2): 2;

• as a split extension Syl_{2}(L_{3}(4)).2, where the ℤ/2 action is induced by the Frobenius

map. From now on we write S for the group S = Syl_{2}(L_{3}(4)) ⊂ H.

Lemma 1.1:

(1) There are exactly two 2^{4}’s contained in H, each normal, so H = 2^{4}: D_{8} in exactly two distinct ways.

(2) The two 2^{4}’s above are not conjugate in any of the simple groups above and their normalizers are given by the following table

( \begin{matrix}& M_{22} & M_{23} & P SU_{4}(3) & McL \\ 2_{I}^{4} & 2^{4}: A_{6} & 2^{4}: A_{7} & 2^{4}: A_{6} & 2^{4}: A_{7} \\ 2_{II}^{4} & 2^{4}: S_{5} & 2^{4}: GL_{2}(4): 2 & 2^{4}: A_{6} & 2^{4}: A_{7} \end{matrix} )

Here the action of the ℤ/2 on GL_{2}(4) is via the Galois automorphism of the field 𝔽_{4}.

(3) Both Aut(P SU_{4}(3)) and Aut(McL) contain elements of order two which exchange 2_{I}^{4} and 2_{II}^{4}.

(The first statement is proved in [J]. The second and third statements are contained in [Co], a more detailed description of the normalizer of 2_{II}^{4} for M_{23} is given in [M].)

Remark 1.2: The group S is given as the span ⟨2_{I}^{4}, 2_{II}^{4}⟩. In particular, from 1.1(1) it follows that there is only one copy of S contained in H, and any copy of L_{3}(4) ⊂ G where G is one of the four simple groups above must intersect some G–conjugate of H in S. Also, the subgroup 2_{II}^{4}: GL_{2}(4) ⊂ 2_{II}^{4}: GL_{2}(4): 2 ⊂ M_{23} has S as its 2-Sylow subgroup.

Consider one of the two copies of UT_{4}(2) ⊂ H, say the one which contains 2_{I}^{4} for definiteness. It can be written in the form 2_{I}^{4}: W, where

W = ⟨α, β⟩ ≅ 2^{2}.

The action of W on 2_{I}^{4} satisfies the condition that the intersection of the centralizer for each of the three non-trivial elements in W with 2_{I}^{4} is a 2^{2}, and these three 2^{2}’s have a single ℤ/2 intersection which we write ⟨c⟩. Also, c generates the center of UT_{4}(2) and of H. Thus there are three 2^{3} ⊂ UT_{4}(2) each of which intersects W in a single 2 that we call W_{α}, W_{β}, and W_{αβ} for their interesections with W.

In H, following [AM2], we can assume that W_{α} and W_{β} are conjugate, but W_{αβ} ⊂ 2_{II}^{4} is not conjugate to the others. Also, there is another 2^{3} subgroup given as B = ⟨α, β, c⟩. Under the automorphism of 1.1(3) which exchanges the two 2^{4}’s in H, the image of B is W_{α} or W_{β}. In fact we have the following result from [AM2], [M].

Lemma 1.3:

(1) There are 4 conjugacy classes of extremal 2-elementary subgroups contained in H, the two 2^{4}’s, and two classes of 2^{3}’s, each class containing 2 subgroups, the first {W_{α}, W_{β}} and the second {B, B̄}.

(2) In M_{22} there is an element γ ∈ 2_{I}^{4}: A_{6} of order three which conjugates W_{α} to W_{αβ}. Thus, the two extremal 2^{3}’s in the first class fuse with a 2^{3} ⊂ 2_{II}^{4} in M_{22}.

(3) In M_{22} and M_{23} the 2^{3} represented by B remains extremal.

We next recall the notion of weak closure.

Definition 1.4: Given a triple of groups V ⊂ K ⊂ G, we say that V is weakly closed in K if any subgroup V′ ⊂ K which is conjugate to V in G is already conjugate to V in K.

Now we have

Lemma 1.5: In P SU_{4}(3) and McL the two 2^{4}’s are weakly closed in H ⊂ P SU_{4}(3) and the 2^{3} represented by B ⊂ H is conjugate to a subgroup of one of the two 2^{4}’s while W_{α} is conjugate to a subgroup of the other 2^{4}. Consequently, the only extremal 2-elementaries in P SU_{4}(3) are the two 2^{4}’s.

Proof: Since the two 2^{4}’s are not conjugate in P SU_{4}(3) they are weakly closed from the definition. Thus it remains to show that both families of extremal 2^{3}’s are conjugate to subgroups of these 2^{4}’s.

Consider the normalizers of the two 2^{4}’s in P SU_{4}(3). Each has the form of a split extension 2^{4}: A_{6}, hence is completely determined by the action of A_{6} on 2^{4}, but there is only one conjugacy class of subgroups isomorphic to A_{6} contained in A_{8} = L_{4}(2), so these extensions 2^{4}: A_{6} are each isomorphic to the group with the same name in M_{22}. Consequently, the first identifies W_{α} with a subgroup of the other 2^{4} exactly as in M_{22}.

One of the outer automorphisms of P SU_{4}(3) when restricted to H gives the automorphism above exchanging the two 2^{4}’s and the families of extremal 2^{3}. It follows that B must be conjugate to a subgroup of the other 2^{4} in P SU_{4}(3) as asserted.

There is a sporadic geometry discovered by Ronan associated to the group McL (see [RSY] for details). From the topological point of view, this means that there is a finite CW–complex X with an action of McL arising from its internal combinatorial structure. Further, in [RSY] it was shown that for this complex the fixed point set under any element of order two in McL is contractible, and hence its stabilizers can be used to compute H^{∗}(McL). However, as this would involve extensive computations we have chosen instead to make a calculation using more direct methods. Nevertheless it contains essential information about the subgroup structure of McL. The orbit space is a single solid 3–dimensional simplex, described by the following table:

Simplex  Stabilizer
a  P SU_{4}(3)
b, c  2^{4} : A_{7}
d Ã_{8}
ab, ac  2^{4} : A_{6}
ad  2^{1+4} : (S_{3} × S_{3})

Simplex  Stabilizer
bc  2^{4} : (3 × A_{4}) : 2
bd, cd  2^{4} : L_{3}(2)
abd, acd, bcd  2^{4} : S_{4}
abc  2^{4} : S_{4}
abcd  2^{4} : D_{8}

We can best visualize this geometry by considering the following figure, where the larger type identifies the isotropy groups of the vertices, the smaller type gives the isotropy groups of the edges, each face has isotropy group 2^{4}: S_{4}, and the interior has isotropy group H.

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{2^{4}:L_{3}(2)} P SU_{4}(3)  2^{4}: A_{7} 2^{4}: A_{7} Ã_{8}  _{2^{4}:((3×A_{4}):2)}^{2^{4}:A_{6}}  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Actually in [RSY] it is also shown that there is a sub–geometry corresponding to the bottom face of the simplex above from which the cohomology of McL can be determined.

This 2-dimensional geometry is similar to the one associated to M_{22} (see [AM2]). In fact, that group is known to be isomorphic to the generalized amalgam of the stabilizers. Whether or not an analogue of this result is true for McL remains to be verified.

From the above, the group 2^{4}: (3 × A_{4}).2 is the intersection of the two 2^{4}: A_{7}’s which normalize 2_{I}^{4} and 2_{II}^{4} respectively. Since McL contains M_{22} we see that Syl_{2}(2^{4} : 3×A_{4}) = S, and that there are two elements of order three, denoted T_{1} and T_{2} acting on S which gives an embedding 3^{2} ⊂ Out(S).

Lemma 1.7: There is an inclusion S ⊂ 2_{II}^{4}: S_{5} ⊂ M_{22} as the inverse image of the Klein group K ⊂ S_{4} ⊂ S_{5}.

Proof: We know that in M_{22} we have

2_{II}^{4}: S_{5} ∩ 2_{I}^{4}: A_{6} = 2_{II}^{4}: S_{4}

where the first two groups are the normalizers of the two distinct 2^{4}’s in Syl_{2}(M_{22}) = 2^{2+4}: 2. There is only one conjugacy class of subgroups S_{4} in S_{5} and the intersection of the second 2^{4} with it must be a normal subgroup, hence K as asserted.

Corollary 1.8: Aut(S) has the form

V: S_{3} × S_{3}

where V is a finite 2-group.

Proof: Consider the sequence of inclusions

2^{4}: S_{4} ↪ 2^{4}: S_{5} ↪ 2^{4}: A_{7} ↪ 2^{4}: A_{8}

where A_{8} = L_{4}(2) and, as in 1.7, S = 2^{4}: K ⊂ 2^{4}: S_{4}. It is direct to check that if an automorphism, λ, of S is the identity on 2_{II}^{4} then it is given by specifying a homomorphism ϕ: K⟶C ⊂ 2_{II}^{4} where C = 2^{2} ⊂ 2_{II}^{4} is the center of S, and setting λ(k) = k + ϕ(k) for each k ∈ K. Clearly, this subgroup V ⊂ Aut(S) is normal in the subgroup M ⊂ Aut(S) which normalizes 2_{II}^{4}, and, since V = 2^{4}, we have

M = 2^{4} · N_{A_{8}}(K)

where K ⊂ S_{4} is the Klein group. Also, the entirety of the Aut-group is the extension of M by the 2 which switches the two 2^{4}’s. But we have easily that

N_{A_{8}}(K) = A_{4} × A_{4}: 2

and from this the result is a direct calculation.

§2 Preliminaries on the cohomology of P SL_{4}(3) and McL

From the Quillen-Venkov detection theorem [AM1, p.144] we have an exact sequence for any finite group G,

0⟶rad(H^{∗}(G))⟶H^{∗}(G)⟶^{res^{∗}} ∐_{V} H^{∗}(V)^{W_{G}(V)},

where the V run over representatives for the conjugacy classes of maximal 2-elementary subgroups of G and W_{G}(V) = N_{G}(V)/V is the Weyl group of V in G.

In [AM2] we proved that the Lyndon–Hochschild–Serre spectral sequence associated to writing H as the extension S.2 collapses at the E_{2}–stage, yielding a precise description of H^{∗}(H). It is most convenient to describe the result there in the following manner. First, the radical is given as

2.1  rad(H^{∗}(H)) ≅ rad(H^{∗}(S))^{ℤ/2} = 𝔽_{2}[t_{4}, t_{8}](a_{2}, α_{3}, T_{6}, T_{7}).

Here, for the sake of clarity, we emphasize that the radical of H^{∗}(H) injects to rad(H^{∗}(S)) under the restriction map. Next, the image of H^{∗}(H) in each of the four extremal 2^{n}’s given in 1.2 is described. In particular, the image in each 2^{4} is the full invariant subalgebra H^{∗}(2^{4})^{D_{8}}. (The image in each of the two conjugacy classes of extremal 2^{3}’s is a proper subset of its invariant algebra, however in the current situation, as they fuse with subgroups of the two 2^{4}’s, this need not concern us.)

Lemma 2.2:

(a) rad(H^{∗}(P SU_{4}(3))) ⊂ rad(H^{∗}(M_{22})) ≅ 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}).

(b) rad(H^{∗}(McL)) ⊂ rad(H^{∗}(M_{23})) = 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}).

Proof: From [Co] we have that P SU_{4}(3) ⊂ McL contains a subgroup L_{3}(4). Consequently, it follows from 2.1 and 1.2 that

rad(H^{∗}(P SU_{4}(3))) ⊂ rad(H^{∗}(L_{3}(4))) ∩ H^{∗}(S)^{ℤ/2}.

But from [AM2], the intersection above is rad(H^{∗}(M_{22})) = 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}).

In [M] the radical of H^{∗}(M_{23}) was shown to be

rad(M_{22})^{ℤ/3} = 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11})

where, from 1.1, the 3-Sylow subgroup of N_{M_{23}}(S) is equal to (ℤ/3)^{2} with the first ℤ/3 ⊂ L_{3}(4) and the second giving the action above. Consequently, from 1.7, 1.8, we see that rad(H^{∗}(McL)) is also invariant under the same ℤ/3-action and 2.2 follows.

It remains to describe the exact structure of these radicals, and to describe the restriction of H^{∗}(P SU_{4}(3)) and H^{∗}(McL) to the cohomology of the two 2^{4}’s. It turns out that the function of the radicals is to provide elements with higher Bocksteins to make sure that all the elements in H^{∗}(G) are torsion where G is P SU_{4}(3) or McL. Consequently, we will be able to show that a sufficient number of elements are in the radicals in each case once we have some information on the image of restriction in H^{∗}(2^{4}).

Recall the Cardenas-Kuhn theorem [AM1, p.113]:

Theorem 2.3: Let L ⊊ K ⊊ G be weakly closed with L a p-elementary subgroup of G. Suppose that N_{K}(L) contains a p-Sylow subgroup of N_{G}(L), then the image of

res^{∗}: H^{∗}(G; 𝔽_{p})⟶H^{∗}(L; 𝔽_{p})

is exactly equal to the intersection

H^{∗}(L; 𝔽_{p})^{W_{G}(L)} ∩ im(res^{∗}: H^{∗}(K; 𝔽_{p})⟶H^{∗}(L; 𝔽_{p})).

Thus we have, from 1.5 and the remarks before 2.1

Corollary 2.4: The restriction image from H^{∗}(P SU_{4}(3)) ⊂ H^{∗}(2_{J}^{4}) is the full invariant algebra H^{∗}(2^{4})^{A_{6}} for J = I, II. Similarly the restriction image from H^{∗}(McL) ⊂ H^{∗}(2_{J}^{4}) is the full invariant algebra H^{∗}(2^{4})^{A_{7}} for J = I, II.

From [AM4] we have that

2.5  H^{∗}(2^{4})^{A_{6}} = 𝔽_{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15})

and

2.6  H^{∗}(2^{4})^{A_{7}} = 𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, x_{18}, x_{20}, x_{21}, x_{24}, x_{25}, x_{27}, x_{45})

which specifies these invariant subalgebras. Consequently, in order to specify the rings H^{∗}(P SU_{4}(3)) and H^{∗}(McL) we need to finish the determination of the radicals in each case and the image in

H^{∗}(2_{I}^{4})^{W_{G}(2^{4})} ⊕ H^{∗}(2_{II}^{4})^{W_{G}(2^{4})}

which really means specifying which elements (a, 0) are in the restriction image for a in the invariant subalgebra. We know from our analysis of H^{∗}(H) in [AM2] that the element (w̄_{3}, 0) is not in the image of restriction and neither are (d_{8}, 0), (d_{12}, 0). In fact, the quotient algebra of elements not of this form must contain at least

𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}).

In any case, a_{2} ∈ rad and (w̄_{3}, w̄_{3}) are certainly elements in H^{∗}(P SU_{4}(3)). In fact it will turn out that rad(H^{∗}(P SU_{4}(3))) ≅ rad(H^{∗}(L_{3}(4)) ∩ H^{∗}(S)^{ℤ/2} by a fairly easy argument. But the determination of which elements in H^{∗}(E_{1})^{A_{6}} ⊕ H^{∗}(E_{2})^{A_{6}} are not doubled in the image from H^{∗}(P SU_{4}(3)) will necessitate an independent calculation of H^{∗}(P SU_{4}(3)) which we turn to next.

§3 The Structure of H^{∗}(P SU_{4}(3))

The procedure we use in this section to determine the ring H^{∗}(P SU_{4}(3)) is to use our knowledge of the ring H^{∗}(SU_{4}(3)) (see [FP]) and an Eilenberg-Moore spectral sequence with E_{2}-term Ext_{H_{∗}(C(SU_{4}(3)))}(H_{∗}(SU_{4}(3)), 𝔽_{2}) that converges to H^{∗}(P SU_{4}(3)) to make initial calculations (here C(SU_{4}(3)) denotes the center of SU_{4}(3)). Then, coupling these initial results with the discussion in the previous section will enable us to determine H^{∗}(P SU_{4}(3)) completely.

The Eilenberg-Moore spectral sequence for P SU_{4}(3)

We know H^{∗}(SU_{4}(3)) = 𝔽_{2}[b_{4}, b_{6}, b_{8}] ⊗ E(e_{3}, e_{5}, e_{7}) from [FP]. We also have that SU_{4}(3) is a central extension of P SU_{4}(3), C◁SU_{4}(3)⟶P SU_{4}(3) with center C = ℤ/4 ⊂ 𝔽_{9}^{∗}. Each central extension C ⟶^{◁}G⟶G/C is classified by an element α ∈ H^{2}(B_{G/C}; C), which is the same thing as a map α: B_{G/C}⟶K(C, 2) with K(C, 2) an Eilenberg-MacLane space. That is to say K(C, 2) is a CW-complex with homotopy groups given as follows:

π_{i}(K(C, 2)) = \{ \begin{matrix}C & if i = 2, \\ 0 & if i ≠ 2, \end{matrix}

and it is well known that H^{2}(X; C) is naturally identified with the set of homotopy classes of maps [X, K(C, 2)] for X any CW-complex. In fact, in our situation, if the map α: B_{P SU_{4}(3)}⟶K(C, 2) is chosen to represent the K-invariant of the extension, then there is a fibration sequence for the classifying spaces

B_{ℤ/4}⟶B_{SU_{4}(3)}⟶B_{P SU_{4}(3)} ⟶^{α}K(ℤ/4, 2)

and the leftmost fibration is principal.

There are many spectral sequences which can be used to determine the cohomology of the total space of a fibration F ⟶E⟶B. The Serre spectral sequence is the most commonly used, with E_{2}-term H^{∗}(B; H^{∗}(F)). However, when we know the fiber and total space, but want the cohomology of the base then the most efficient tool is the Eilenberg-Moore spectral sequence provided the fibration is principal. Its E_{2}-term is equal to

Ext_{H_{∗}(F;𝔽)}(H_{∗}(E; 𝔽), 𝔽)

and it converges to H^{∗}(B; 𝔽) for 𝔽 any field.

To determine this E_{2}-term we need to study the action of H_{∗}(B_{ℤ/4}) on H_{∗}(B_{SU_{4}(3)}). In fact this action arises from the homomorphism ℤ/4 × SU_{4}(3)⟶SU_{4}(3) obtained by multiplication by the center, on passing to classifying spaces,

(∗)  B(×): B_{ℤ/4} × B_{SU_{4}(3)} = B_{ℤ/4×SU_{4}(3)}⟶B_{SU_{4}(3)}.

Write H^{∗}(ℤ/4) = 𝔽_{2}[d]⊗E(e) where e is the 1-dimensional generator, and d is the non-zero element in dimension 2. Then we have

Lemma 3.1: The cohomology map H^{∗}(SU_{4}(3); 𝔽_{2})⟶H^{∗}(ℤ/4; 𝔽_{2}) ⊗ H^{∗}(SU_{4}(3); 𝔽_{2}) associated to (∗) above is given on generators by b_{i} ↦ 1 ⊗ b_{i}, x_{i} ↦ 1 ⊗ x_{i} for i = 3, 4, 5, 6 but

\begin{matrix}x_{7} & ↦ x_{7} + d^{2} ⊗ x_{3} + d ⊗ x_{5} + e ⊗ b_{6}, \\ b_{8} & ↦ d^{4} ⊗ 1 + d^{2} ⊗ b_{4} + d ⊗ b_{6} + 1 ⊗ b_{8}. \end{matrix}

Proof: It is easiest to work with U_{4}(3) with maximal torus (ℤ/4)^{4}. The restriction of H^{∗}(U_{4}(3); 𝔽_{2}) = 𝔽_{2}[b_{2}, b_{4}, b_{6}, b_{8}] ⊗ E(x_{1}, x_{3}, x_{5}, x_{7}) is injective to

𝔽_{2}[σ_{1}, σ_{2}, σ_{3}, σ_{4}] ⊗ E

where E is an exterior algebra with generators

e_{1} + e_{2} + e_{3} + e_{4},
e_{1}(d_{2} + d_{3} + d_{4}) + e_{2}(d_{1} + d_{3} + d_{4}) + e_{3}(d_{1} + d_{2} + d_{4}) + e_{4}(d_{1} + d_{2} + d_{3}),
e_{1}(σ_{2}(d_{2}, d_{3}, d_{4})) + e_{2}(σ_{2}(d_{1}, d_{3}, d_{4}) + e_{3}(σ_{2}(d_{1}, d_{2}, d_{4})) + e_{4}(σ_{2}(d_{1}, d_{2}, d_{3}))
e_{1}d_{2}d_{3}d_{4} + e_{2}d_{1}d_{3}d_{4} + e_{3}d_{1}d_{2}d_{4} + e_{4}d_{1}d_{2}d_{3}

and σ_{i} is the i^{th} symmetric function in the d_{i}’s ([FP], pg. 288).

The inclusion SU_{4}(3) ⊂ U_{4}(3) induces a surjection in cohomology onto

H^{∗}(SU_{4}(3); 𝔽_{2}) = 𝔽_{2}[b_{4}, b_{6}, b_{8}] ⊗ E(x_{3}, x_{5}, x_{7})

where b_{2i} is the image under restriction of b_{2i}, and x_{2i+1} is the image under restriction of x_{2i+1}. More exactly, the inclusion of the maximal torus for SU_{4}(3) can be given as t_{i} ↦ t_{i}t_{4}^{−1}, 1 ≤ i ≤ 3, so in cohomology d_{i} ↦ d_{i}, e_{i} ↦ e_{i}, 1 ≤ i ≤ 3, while e_{4} ↦ e_{1}+e_{2}+e_{3}, d_{4} ↦ d_{1} + d_{2} + d_{3}. Consequently, the images of the generators for H^{∗}(SU_{4}(3); 𝔽_{2}) in the cohomology of the maximal torus are given as

b_{4} ↦ σ_{2} + σ_{1}^{2}
= d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + d_{1}d_{2} + d_{1}d_{3} + d_{2}d_{3}
b_{6} ↦ σ_{3} + σ_{1}σ_{2}
= d_{1}d_{2}(d_{1} + d_{2}) + d_{1}d_{3}(d_{1} + d_{3}) + d_{2}d_{3}(d_{2} + d_{3})
b_{8} ↦ σ_{3}σ_{1}
x_{3} ↦ e_{1}(d_{2} + d_{3}) + e_{2}(d_{1} + d_{3}) + e_{3}(d_{1} + d_{2})
x_{5} ↦ e_{1}(d_{2}^{2} + d_{3}^{2}) + e_{2}(d_{1}^{2} + d_{3}^{2}) + e_{3}(d_{1}^{2} + d_{2}^{2})
x_{7} ↦ e_{1}d_{2}d_{3}(d_{2} + d_{3}) + e_{2}d_{1}d_{3}(d_{1} + d_{3}) + e_{3}d_{1}d_{2}(d_{1} + d_{2}).

We use this to figure out the cohomology map H^{∗}(SU_{3}(4))⟶H^{∗}(ℤ/4) ⊗ H^{∗}(SU_{3}(4)). Restricted to the maximal torus, the central multiplication

ℤ/4 × (ℤ/4)^{3}⟶(ℤ/4)^{3}

is given by (τ, t_{1}, …, t_{3}) ↦ (τt_{1}, τt_{2}, τt_{3}) and consequently, in cohomology we have

\begin{matrix}e_{i} & ↦ e_{i} + e, 1 ≤ i ≤ 3, \\ d_{i} & ↦ d_{i} + d, 1 ≤ i ≤ 3, \end{matrix}

so b_{4} ↦ 1 ⊗ b_{4}, b_{6} ↦ 1 ⊗ b_{6}, but

b_{8} ↦ (σ_{1} + d)(σ_{3} + dσ_{2} + d^{2}σ_{1} + d^{3})
= σ_{1}σ_{3} + d(σ_{3} + σ_{1}σ_{2}) + d^{2}(σ_{1}^{2} + σ_{2}) + d^{4}
= b_{8} + d ⊗ b_{6} + d^{2} ⊗ b_{4} + d^{4} ⊗ 1.

The calculation for the odd dimensional generators is similar.

As a consequence

Δb_{8}^{2^{n}} = d^{2^{n+2}} + d^{2^{n+1}}b_{4}^{2^{n}} + d^{2^{n}}b_{6}^{2^{n}} + b_{8}^{2^{n}}

and H^{∗}(SU_{4}(3)) is free over the quotient coalgebra 𝔽_{2}[d^{4}]. (Dually, H_{∗}(SU_{4}(3)) is free over the subalgebra E((d^{4})^{∗}, (d^{8})^{∗}, ⋯) where H_{∗}(ℤ/4; 𝔽_{2}) = E(e, d^{∗}, (d^{2})^{∗}, (d^{4})^{∗}, …), with generators the dual elements to the cohomology subalgebra 𝔽_{2}[b_{4}, b_{6}] ⊗ E(x_{3}, x_{5}, x_{7}).)

Now that we have the module structure of H_{∗}(SU_{4}(3)) over H_{∗}(ℤ/4) (or dually, the comodule structure of H^{∗}(SU_{4}(3)) over H^{∗}(ℤ/4)) we are ready to determine the groups Ext_{H_{∗}(ℤ/4)}(H_{∗}(SU_{4}(3)), 𝔽_{2}). The procedure will be to use the freeness over the subalgebra above and the change of rings theorem:

Theorem 3.2 (Change of rings): Let A be an augmented Hopf algebra over a field 𝔽 and M an A-module which is free over the sub-Hopf-algebra B ⊂ A. Let C = A//B be the quotient Hopf algebra and M//B the quotient module, then

Ext_{A}(M, 𝔽) ≅ Ext_{C}(M//B, 𝔽).

In the present case let us denote the quotient Hopf algebra

H_{∗}(ℤ/4)//E((d^{4})^{∗}, (d^{8})^{∗}, …) = E(e, d^{∗}, (d^{2})^{∗}) as Λ,

and the quotient module H_{∗}(SU_{4}(3))/I(E((d^{4})^{∗}, (d^{8})^{∗}, …) as N. In N the action of Λ is given by e(b_{6}^{j})^{∗} = (b_{6}^{j−1}x_{7})^{∗}, d(x_{5}θ)^{∗} = (x_{7}θ)^{∗}, d^{2}(x_{3}τ)^{∗} = (x_{7}τ)^{∗} and otherwise the action is trivial.

Using change of rings we have

Ext_{H_{∗}(ℤ/4)}(H_{∗}(SU_{4}(3)), 𝔽_{2}) ≅ Ext_{Λ}(N , 𝔽_{2})
≅ Ext_{Λ}(𝔽_{2}[b_{4}, b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2})
≅ 𝔽_{2}[b_{4}] ⊗ Ext_{Λ}(𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2}),

since the action of Λ does not involve 𝔽_{2}[b_{4}]. Additionally, since

Ext_{A⊗B}(𝔽, 𝔽) = Ext_{A}(𝔽, 𝔽) ⊗ Ext_{B}(𝔽, 𝔽),

and Ext_{E(y)}(𝔽_{2}, 𝔽_{2}) = 𝔽_{2}[|y|]† we have that Ext_{Λ}(𝔽_{2}, 𝔽_{2}) = 𝔽_{2}[λ_{3}, λ_{5}, λ_{7}], the polynomial ring on the stated generators where λ_{i} ∈ Ext_{Λ}^{i−1,1}(𝔽_{2}, 𝔽_{2}).

Before we determine the Ext-groups Ext_{Λ}(𝔽_{2}[b_{4}, b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2}) we point out that they form an associative, commutative, ring with unit – indeed, a ring over Ext_{Λ}(𝔽_{2}, 𝔽_{2}). To see this, note that the diagonal map gives a ring homomorphism

H_{∗}(ℤ/4)⟶H_{∗}((ℤ/4)^{2}) ≅ H_{∗}(ℤ/4) ⊗ H_{∗}(ℤ/4)

and the diagonal map for SU_{4}(3) gives a module map

H_{∗}(SU_{4}(3))⟶H_{∗}(SU_{4}(3)) ⊗ H_{∗}(SU_{4}(3)).

Together these two maps thus give a pairing map

Ext_{H_{∗}(ℤ/4)}(H_{∗}(SU_{4}(3)), 𝔽_{2}) ⊗ Ext_{H_{∗}(ℤ/4)}(H_{∗}(SU_{4}(3)), 𝔽_{2})⟶Ext_{H_{∗}(ℤ/4)}(H_{∗}(SU_{4}(3), 𝔽_{2})

which has the requisite properties. In particular, the Ext groups have the form of the tensor product

𝔽_{2}[b_{4}] ⊗ Ext_{Λ}(𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2})

as a ring.

Theorem 3.3: We have

Ext_{Λ}(𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2}) ≅ 𝔽_{2}[b_{6}, x_{3}, x_{5}, λ_{2}, λ_{3}, λ_{5}]/R

where R is the ideal with generators

R = (x_{3}^{2}, x_{5}^{2}, λ_{2}b_{6} + λ_{3}x_{5} + λ_{5}x_{3}).

† Here E(y) is a regarded as a primitively generated Hopf algebra so that Ext_{E(y)}(𝔽, 𝔽) becomes a commutative ring.

Remark 3.4: The following figure gives the groups Ext_{Λ}(𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2}) through dimension 8, the range which is most important to calculate H^{∗}(P SU_{4}(3)).

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Proof: Since e(b_{6}^{j})^{∗} = (b_{6}^{j−1}x_{7})^{∗} for j ≥ 1 it follows that

F_{i} = E(x_{3}, x_{5}, x_{7})(1, b_{6}, b_{6}^{2}, b_{6}^{3}, …, b_{6}^{i})^{∗}

is a sub-Λ module of 𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗} for each i = 1, 2, . . .. This defines a filtration of 𝔽_{2}[b_{6}] ⊗ E(x_{3}, x_{5}, x_{7})^{∗}, and the resulting spectral sequence has E_{1}^{i}-term equal to Ext_{Λ}(E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2})b_{6}^{i}.

We now determine this Ext-group.

Lemma 3.5:

Ext_{Λ}(E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2}) ≅ 𝔽_{2}[λ_{2}, λ_{3}, λ_{5}](1, x_{3}, x_{5}, x_{3}x_{5}, x_{15}, y_{15})/R

where y_{15} ∈ Ext_{Λ}^{14,1}, x_{15} ∈ Ext^{15,0} and R is the ideal

(x_{3}^{2}, x_{5}^{2}, x_{15}^{2}, y_{15}^{2}, λ_{5}x_{3} + λ_{3}x_{5}).

The multiplication is given by the relations in R above and x_{3}y_{15} = λ_{3}x_{15}, x_{5}y_{15} = λ_{5}x_{15}.

Proof: We write out the module as follows

1, x_{3}, x_{5}, x_{7}, x_{3}x_{5}, x_{3}x_{7}, x_{5}x_{7}, x_{3}x_{5}x_{7},

where d^{∗}x_{5} = (d^{2})^{∗}x_{3} = x_{7} while d^{∗}x_{3}x_{5} = x_{3}x_{7}, (d^{2})^{∗}x_{3}x_{5} = x_{5}x_{7} give the complete action. It follows that V = {x_{7}, x_{3}x_{7}, x_{5}x_{7}, x_{3}x_{5}x_{7}} is a trivial Λ-submodule with quotient W = {x_{3}, x_{5}, x_{3}x_{5}} also a trivial module. From this short exact sequence of modules we get a long exact sequence of Ext-groups:

⋯ ⟶Ext_{Λ}^{∗}(W, 𝔽_{2})⟶Ext_{Λ}^{∗}(E(x_{3}, x_{5}, x_{7})^{∗}, 𝔽_{2})⟶Ext_{Λ}^{∗}(V, 𝔽_{2})⟶^{δ}Ext_{Λ}^{∗+1}(W, 𝔽_{2}) ⋯

and δ(x_{7}) = λ_{3}x_{5} + λ_{5}x_{3}, δ(x_{3}x_{7}) = λ_{3}x_{3}x_{5}, δ(x_{5}x_{7}) = λ_{5}x_{3}x_{5}, while δ(θx) = θδ(x) for θ ∈ Ext_{Λ}(𝔽_{2}, 𝔽_{2}) = 𝔽_{2}[λ_{2}, λ_{3}, λ_{5}]. Also, note that

Ext_{Λ}(W, 𝔽_{2}) ≅ 𝔽_{2}[λ_{2}, λ_{3}, λ_{5}] ⊗ W
Ext_{Λ}(V, 𝔽_{2}) ≅ 𝔽_{2}[λ_{2}, λ_{3}, λ_{5}] ⊗ V,

as Ext_{Λ}(𝔽_{2}, 𝔽_{2})-modules, since both V and W are trivial over Λ. From this the calculation follows directly.

Now, to complete the proof of the theorem we need to determine the differentials in the spectral sequence. Since (d)^{∗}(b_{6}^{n})^{∗} = (x_{7}b_{6}^{n−1})^{∗}, as has been already pointed out, we see that d_{1}(x_{15}b_{6}^{i−1}) = λ_{2}x_{3}x_{5}b_{6}^{i}, and similarly, d_{1}(y_{15}b_{6}^{i−2}) = λ_{2}^{2}b_{6}^{i}. Note that both of these differentials are injective on 𝔽_{2}[λ_{2}, λ_{3}, λ_{5}] tensored with the respective generators.

From this the E_{2}-term is the algebra specified, and it is direct to check that each element is an infinite cycle.

This completes the first stage of the calculation of H^{∗}(P SU_{4}(3)).

The d_{2}-differentials in the Eilenberg-Moore spectral sequence

Now that we have determined the E_{2} term of the Eilenberg-Moore spectral sequence converging to H^{∗}(P SU_{4}(3)) we need to calculate the d^{2}-differentials. We have

Lemma 3.6: The d^{2} differentials in the Eilenberg-Moore spectral sequence converging to H^{∗}(P SU_{4}(3)) are completely determined by d^{2}(x_{3}) = λ_{2}^{2}, d^{2}(b_{4}) = λ_{2}λ_{3}, d^{2}(x_{5}) = 0, d^{2}(b_{6}) = λ_{2}λ_{5}.

Proof: Since the spectral sequence is multiplicative and these are the generators it remains only to check that the stated differentials are correct. The first differential is forced since we know from our results on H^{∗}(M_{22}) that a_{2}^{2} = 0 and λ_{2} must represent a_{2} in the spectral sequence. Since w_{3} ≠ 0 it must be represented by λ_{3} which appears in filtration Ext^{2,1} and so cannot have a d^{2} differential to Ext^{2,2} (it’s image under d^{2} lies in Ext^{1,3} = 0). Thus, the only element available to truncate λ_{2} is x_{3}. Also, this forces d^{2}(b_{6}) = λ_{2}λ_{5}, since x_{5}, if it had a d^{2} differential, could only hit λ_{3}^{2}, a class which is non-zero in H^{∗}(2^{4})^{A_{6}}, and consequently, in H^{∗}(P SU_{4}(3)). It follows, since we have the relation λ_{5}x_{3} + λ_{3}x_{5} + λ_{2}b_{6} = 0, that, on passing to differentials λ_{5}λ_{2}^{2} + λ_{2}d^{2}(b_{6}) = 0, which, as asserted, forces d^{2}(b_{6}) = λ_{2}λ_{5}.

Now we consider the class λ_{2}λ_{3}. Since λ_{2} represents a_{2} ∈ rad(H^{∗}(Syl_{2}(P SU_{4}(3)))), and since L_{3}(4) ⊂ P SU_{4}(3) from [Co], it follows that

rad(H^{∗}(P SU_{4}(3))) ⊂ rad(H^{∗}(L_{3}(4))) ∩ H^{∗}(S)^{ℤ/2} = 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}),

so λ_{3}λ_{2} = 0 in H^{∗}(P SU_{4}(3)). But the only class which can hit this class is b_{4}.

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Notes: λ_{3} represents w̄_{3} and λ_{5} represents Sq^{2}(w̄_{3}) so x_{5} represents an element which restricts to (γ_{5}, 0) in (H^{∗}(2^{4})^{A_{6}})^{2}. Also, γ = {λ_{2}b_{4} + x_{3}λ_{3}} must represent an element which restricts to (w̄_{3}^{2}, 0) in (H^{∗}(2^{4})^{A_{6}})^{2}. We have that a_{7} is represented by λ_{2}x_{5} which implies the existence of the cup product x_{5} ∪ a_{2} = a_{7}. Finally, b_{4}^{2} represents d_{8} and d_{6}^{2} represents d_{12}.

We have not checked to see if there are higher differentials in the Eilenberg-Moore spectral sequence that we are using, but by inspection of the table above it is clear that through dimension 8 all the classes remaining at E_{3} are infinite cycles. However, we will see now that the results to this point suffice to completely determine H^{∗}(P SU_{4}(3)).

The first thing to note is that the two classes in dimension 5 have Sq^{4}’s on them which construct classes which restrict to (γ_{9}, 0) and (γ_{9}, γ_{9}) respectively. a_{7} and a_{11} must exist in the radical because d_{8} and d_{12} must be in the images of some higher order Bocksteins. The ring structure now implies that 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}) ⊂ rad.

Next we write down all the elements in dimensions 15 and 16. In 16 there are 5 elements, d_{8}^{2}, d_{8}w̄_{3}γ_{5}, d_{8}w̄_{3}τ_{5}, w̄_{3}^{2}γ_{5}^{2}, and w̄_{3}^{3}γ_{5}τ_{5}, the second and third with non-trivial Sq^{1}’s and the fourth and fifth in the image of Sq^{1} (γ_{5}^{3} and γ_{5}^{2}τ_{5} respectively). The first image is in the Bockstein image from d_{8}a_{7}. In 15 there are either two elements corresponding to b_{15} (restricting respectively to (b_{15}, 0) and (0, b_{15}) or there is just one which restricts to (b_{15}, b_{15}). The remaining elements which we are sure of are

{d_{12}w̄_{3}, w̄_{3}^{5}, w̄_{3}^{3}τ_{6}, γ_{5}^{3}, γ_{5}^{2}τ_{5}, γ_{9}w̄_{3}^{2}, τ_{9}w̄_{3}^{2}}.

In this list the third and fourth have Sq^{1} images in 16, while the second is Sq^{1}(w̄_{3}^{3}γ_{5}). Also, the fifth is Sq^{1}(γ_{9}γ_{5}) while the last is Sq^{1}(γ_{9}τ_{5}). Finally, the first is in the Bockstein image from d_{12}a_{2}.

Similarly, in 14 there are the four elements listed above, d_{8}w̄_{3}^{2}, d_{8}τ_{6} and (at most) a_{14}. But the two elements involving d_{8} are in the image of Sq^{1} it follows that a_{14} must be present to have a Bockstein to b_{15}, and there can only be the one element b_{15} in dimension 15.

At this point it is probably convenient to write down the Poincaré series for the quotient by rad, at least as it is determined so far. So let us assume that the quotient ring of doubled elements is 𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, w̄_{3}b_{15}). Thus, H^{∗}(P SU_{4}(3))/rad will have Poincaré series

\begin{aligned}\frac{2(1 + x^{9})(1 + x^{15})}{(1 − x^{3})(1 − x^{5})(1 − x^{8})(1 − x^{(}12))} & − \frac{(1 + x^{3})(1 + x^{15})}{(1 − x^{8})(1 − x^{12})} \\ & = \frac{(1 + x^{15})(1 − x^{5} − x^{6} + 2x^{9} + x^{11})}{(1 − x^{3})(1 − x^{5})(1 − x^{8})(1 − x^{11})} \end{aligned}

which expands into a Taylor series of the form

\begin{aligned}1 + x^{3} & + 2 x^{5} + 2 x^{6} + 3 x^{8} + 4 x^{9} + 2 x^{10} + 3 x^{11} + 5 x^{12} + 4 x^{13} + 6 x^{14} + 8 x^{15} + 5 x^{16} \\ & + 10 x^{17} + 11 x^{18} + 7 x^{19} + 15 x^{20} + 16 x^{21} + 12 x^{22} + 18 x^{23} + 22 x^{24} + O(x^{25}). \end{aligned}

Actually, at this point there is one more possible doubled element, w̄_{3}b_{15} which we assumed was doubled in writing down the Poincaré series above, but which we haven’t yet actually proved is doubled. We do that now.

From the Poincaré series, in dimension 19 a total list of the elements in the quotient by the radical is

{d_{8}^{2}w̄_{3}, d_{8}w̄_{3}^{2}γ_{5}, d_{8}w̄_{3}^{2}τ_{5}, w̄_{3}^{3}γ_{5}^{2}, w̄_{3}^{3}γ_{5}τ_{5}, γ_{9}γ_{5}^{2}, γ′_{9}γ_{5}^{2}}

and we see that two of them are in the image of Sq^{1}, γ_{9}w̄_{3}^{3} ↦ w̄_{3}^{3}γ_{5}^{2}, γ′_{9}w̄_{3}^{3} ↦ w̄_{3}^{3}γ_{5}τ_{5}, d_{8}^{2}w̄_{3} is in the image of a higher Bockstein from d_{8}^{2}a_{2}, and the remaining elments have non-trivial Sq^{1}’s.

In dimension 18 the 11 elements are

{d_{12}w̄_{3}^{2}, d_{12}γ_{6}, b_{15}w̄_{3}, γ_{9}w̄_{3}^{3}, γ_{9}w̄_{3}γ_{6}, γ_{5}^{3}w̄_{3}, γ_{5}^{2}w̄_{3}τ_{5}, d_{8}γ_{5}^{2}, d_{8}γ_{5}τ_{5}, w̄_{3}^{6}, w̄_{3}^{4}γ_{6}},

and two elements are needed for Sq^{1} to dimension 19. This leaves nine elments (actually, Sq^{1}(γ_{5}^{3}w̄_{3}) = Sq^{1}(w̄_{3}^{3}γ_{9}), and Sq^{1}(w̄_{3}γ_{9}γ_{6}) = Sq^{1}(w̄_{3}γ_{5}γ′_{9}), so the 9 elements include 7 monomials and the two sums of the elements above with equal Sq^{1}’s which together form a basis).

The 10 elements in dimension 17 are

{d_{12}γ_{5}, d_{12}τ_{5}, d_{8}γ_{9}, d_{8}γ′_{9}, w̄_{3}^{4}γ_{5}, w̄_{3}^{4}τ_{5}, γ_{9}w̄_{3}τ_{5}, γ_{9}w̄_{3}τ_{5}, d_{8}w̄_{3}^{3}, d_{8}w̄_{3}γ_{6}}

and the last two elements are in the image of Sq^{1} while the first eight map under Sq^{1} to each of the elements in the basis for Ker(Sq^{1}) except b_{15}w̄_{3}. It follows that only b_{15}w̄_{3} (and possibly an element which restricts to (b_{15}w̄_{3}, 0)) are left unaccounted for in the Bockstein spectral sequence.

On the other hand there is one and only one element in the radical, d_{8}a_{11} + d_{12}a_{7}, which is in the kernel of the higher Bocksteins taking a_{7} to d_{8} and a_{11} to d_{12}. It follows that b_{15}w̄_{3} must be doubled, must have a higher Bockstein to d_{8}a_{11} + d_{12}a_{7} and there can be no element in H^{∗}(P SU_{4}(3)) which restricts to (b_{15}w̄_{3}, 0).

The arguments above complete the determination of H^{∗}(P SU_{4}(3)) and we have

Theorem 3.7: rad(H^{∗}(P SU_{4}(3))) ≅ rad(H^{∗}(L_{3}(4))) ∩ H^{∗}(S)^{ℤ/2} which is the module 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}) and

H^{∗}(P SU_{4}(3))/rad⟶^{res^{∗}}H^{∗}(2_{I}^{4})^{A_{6}} ⊕ H^{∗}(2_{II}^{4})^{A_{6}}

is injective where the double image classes form the quotient algebra

𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, b_{15}w̄_{3}).

In particular, there is a class τ_{5} ∈ H^{5}(P SU_{4}(3)) with res^{∗}(τ_{5}) = (γ_{5}, 0). Consequently, the same is true for Sq^{1}(τ_{5}) = γ_{6} which restricts to (Sq^{1}(γ_{5}), 0) = (w_{3}^{2}, 0), and Sq^{4}(τ_{5}) = γ′_{9} which restricts to (γ_{9}, 0).

§4 The Bockstein spectral sequence for P SU_{4}(3)

In order to clarify the relations between the various elements in H^{∗}(P SU_{4}(3)) as described in 3.7 we now analyze the Bockstein spectral sequence for H^{∗}(P SU_{4}(3)).

To begin we write out the ideal, Ker(res^{∗}) where res^{∗} is restriction to H^{∗}(2_{I}^{4}) as follows:

I = 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}) ⊕ x_{5}𝔽_{2}[w̄_{3}, γ_{5}](1, γ_{9}, b_{15}, γ_{9}b_{15})
⊕ γ_{6}𝔽_{2}[w̄_{3}](1, γ_{9}, b_{15}, γ_{9}b_{15})
⊕ γ′_{9}(1, w̄_{3}, b_{15}, b_{15}w̄_{3}))

The relations are x_{5}γ′_{9} = x_{5}γ_{9} = γ_{5}γ′_{9}, and γ_{6}γ_{5} = w̄_{3}^{2}x_{5}, γ_{6}γ′_{9} = w̄_{3}^{2}γ′_{9}, γ_{6}x_{5} = γ_{6}γ_{5} = w̄_{3}^{2}x_{5}. The first differential in the Bockstein spectral sequence is Sq^{1}, and here, Sq^{1}(γ_{5}) = w̄_{3}^{2}, Sq^{1}(x_{5}) = γ_{6}, Sq^{1}(γ^{9}) = γ_{5}^{2}, and Sq^{1}(γ′_{9}) = x_{5}γ_{5}. Consequently we have

Lemma 4.1: In the ideal above, the E_{2}-term of the Bockstein spectral sequence has the form 𝔽_{2}[d_{8}, d_{12}](a_{2}, a_{7}, a_{11}, a_{14}).

Proof: We have that the kernel of Sq^{1} on x_{5}𝔽_{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15}) is exactly x_{5}γ_{5}𝔽_{2}[w̄_{3}, γ_{5}^{2}, d_{8}, d_{12}](1, b_{15}), while x_{5}𝔽_{2}[w̄_{3}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15}) has image

γ_{6}𝔽_{2}[w̄_{3}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15})

(summed with x_{5}𝔽_{2}[w̄_{3}, γ_{5}^{2}, d_{8}, d_{12}](1, b_{15}) from the differential on γ_{9}, but we ignore that here, and simply regard the differential as removing the terms involving γ_{6} from the list). Now, in the full submodule involving multiples of x_{5}γ_{5},

x_{5}γ_{5}𝔽_{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15})

we have that x_{5}γ_{5}γ_{9} ↦ x_{5}γ_{5}(γ_{5}^{2}), x_{5}γ_{5}(γ_{5}) ↦ x_{5}γ_{5}w̄_{3}^{2}, in the kernel of Sq^{1}. Thus, after factoring the kernel of Sq^{1} by these images, we have x_{5}γ_{5}𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, w̄_{3}b_{15}) left. On the other hand, the image under Sq^{1} of the part involving γ′_{9} is exactly this, so we get cancellation of all these terms, leaving just the terms asserted in the lemma.

We also have, as a consequence of the calculations above

Lemma 4.2: In the quotient by the ideal above, 𝔽_{2}[w̄_{3}, γ_{5}, d_{8}, d_{12}](1, γ_{9}, b_{15}, γ_{9}b_{15}) the contribution to the E_{2}-term of the Bockstein spectral sequence is 𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, w̄_{3}b_{15}).

Thus, we conclude that the entire E_{2}-term of the Bockstein spectral sequence has the form

4.3  E_{2} = 𝔽_{2}[d_{8}, d_{12}](1, a_{2}, w̄_{3}, a_{7}, a_{11}, a_{14}, b_{15}, w̄_{3}b_{15})

We know that d_{2}(a_{2}) = w̄_{3}. Also, an early differential on a_{14} must be b_{15}. At this point, the spectral sequence will have the form

𝔽_{2}[d_{8}, d_{12}](1, a_{7}, a_{11}, w̄_{3}b_{15}).

Then, the remaining differentials must have the form d(a_{7}) = d_{8}, d(a_{11}) = d_{12}, and d(w̄_{3}b_{15}) = d_{8}a_{11} + d_{12}a_{7}, though we have no idea of the order in which they occur.

Remark 4.4: As a consequence of the last calculation we have the the higher 2-torsion in H^{∗}(P SU_{4}(3); ℤ) when tensored with 𝔽_{2} has the form 𝔽_{2}[d_{8}, d_{12}](1, w̄_{3}, b_{15}, d_{8}a_{11} + d_{12}a_{7}).

§5 The Determination of H^{∗}(McL)

The calculation of H^{∗}(McL)) will be divided into 2 steps. Namely, we will determine its non–nilpotent part and its radical separately. To start, we have the exact sequence

0 → rad(H^{∗}(McL))⟶H^{∗}(McL) → H^{∗}(2_{I}^{4})^{A_{7}} ⊕ H^{∗}(2_{II}^{4})^{A_{7}}.  (5.1)

As pointed out in 2.4, the image of res_{E_{i}}^{McL} will be the full ring of invariants in each case. From 2.6,

H^{∗}(2^{4})^{A_{7}} = 𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, x_{18}, x_{20}, x_{21}, x_{24}, x_{25}, x_{45})

so, in particular, H^{∗}(McL) ≅ 𝔽_{2}[d_{8}, d_{12}] ⊕ rad through dimension 13, and contains 𝔽_{2}[d_{8}, d_{12}] as a sub-polynomial algebra. It follows for Bockstein reasons that a_{7} and a_{11} must both be contained in the radical, so 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}) ⊂ rad. Hence, from 2.2 we have

Lemma 5.1: The radical of H^{∗}(McL) is exactly 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}).

We now turn to the double image classes. From the inclusion P SU_{4}(3) ⊂ McL we have the commutative diagram of restriction maps,

H^{∗}(McL) \underrightarrow{∐ res^{∗}} H^{∗}(2_{I}^{4})^{A_{7}} ⊕ H^{∗}(2_{II}^{4})^{A_{7}}
↓res^{∗}  ↓in⊕in

H^{∗}(P SU_{4}(3)) \underrightarrow{∐ res^{∗}}  H^{∗}(2_{I}^{4}) ⊕ H^{∗}(2_{II}^{4})

where the bottom restriction map was discussed in §3. As before, to make this meaningful, we must calculate the double image classes i.e. pairs (res_{2_{I}^{4}}^{McL}(x), res_{2_{II}^{4}}^{McL}(x)) where both components are non–trivial. To do this we use the double coset decomposition of [JW]:

5.2  McL = P SU_{4}(3) ⊔ P SU_{4}(3)xP SU_{4}(3) ⊔ P SU_{4}(3)v_{1}P SU_{4}(3)

where x has order 2 and the order of v_{1} is four. The two intersections are L_{3}(4) = xP SU_{4}(3)x ∩ P SU_{4}(3) and 3^{4}: A_{6} where the action of v_{1} leaves invariant the D_{8}-Sylow subgroup, but acts on it as the non-trivial outer automorphism. This only produces a stability condition on the restrictions to 2^{2}’s in McL. The double coset involving x produces the copy of L_{3}(4): 2_{2} ⊂ McL, and additionally gives the extra automorphisms to build the A_{7} action on the two 2^{4}’s from the A_{6}-action in P SU_{4}(3). But these are then the only stability requirements which are placed on the elements of H^{∗}(P SU_{4}(3)) in order that they be in the image from H^{∗}(McL).

Corollary 5.3: Let α ∈ H^{∗}(P SU_{4}(3)), then α is contained in the image of res^{∗} from H^{∗}(McL) if α restricts to (λ_{1}, λ_{2}) ∈ H^{∗}(2_{I}^{4})^{A_{7}} ⊕ H^{∗}(2_{II}^{4})^{A_{7}} and λ_{i}, i = 1, 2 restricts to zero in H^{∗}(2^{2}) for some 2^{2} ⊂ 2^{4} provided the λ_{i} are in a dimension where the radical of H^{∗}(P SU_{4}(3)) is equal to the radical of H^{∗}(McL).

Proof: The action of A_{7} on 2^{4} is transitive on the 35 distinct subgroups of the form 2^{2} ⊂ 2^{4}. Thus, for any α ∈ H^{∗}(2^{4})^{A_{7}}, in order to verify that α restricts to zero in H^{∗}(2^{2}) for each 2^{2} ⊂ 2^{4} it suffices to check on a single subgroup. Moreover, since the radical of H^{∗}(P SU_{4}(3)) is equal to the radical of H^{∗}(McL) in this dimension it follows that α is completely determined by its restrictions to H^{∗}(2_{I}^{4}) and H^{∗}(2_{II}^{4}). Consequently, it is stable for the double coset decomposition in 5.2.

Remark: The only dimensions where there are elements in rad(H^{∗}(P SU_{4}(3))) which are not in rad(H^{∗}(McL)) are of the form 10, 14, 18, 22, 26, 30 in the range we will need to study.

We are now ready to determine H^{∗}(McL).

Theorem 5.4: Let M be the algebra obtained by identifying the two copies of the subalgebra 𝔽_{2}[d_{8}, d_{12}](1, x_{18}) in the direct sum of two copies of the algebra of A_{7}-invariants

𝔽_{2}[d_{8}, d_{12}, d_{14}, d_{15}](1, x_{18}, x_{20}, x_{21}, x_{24}, x_{25}, x_{27}, x_{45}).

Then H^{∗}(McL) is described as an extension

0 → 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}) → H^{∗}(McL) → M → 0.

In other words, we have a short exact sequence

0 → M → H^{∗}(2_{I}^{4})^{A_{7}} ⊕ H^{∗}(2_{II}^{4})^{A_{7}} → 𝔽_{2}[d_{8}, d_{12}](1, x_{18}) → 0

where M is isomorphic to the algebra H^{∗}(McL)/rad.

Proof: Note that in dimensions 20, 21, 24 and 25 there are no elements in the radical of H^{∗}(P SU_{4}(3)). Also, in [AM4] we write down representatives for the four elements x_{20}, x_{21}, x_{24} and x_{25} in 2.4 in the form of symmetric sums. Note that each symmetric sum appearing in the expressions for these elements has the form Sx_{1}^{i}x_{2}^{j}x_{3}^{k} or Sx_{1}^{i}x_{2}^{j}x_{3}^{k}x_{4}^{l} with i, j, k, l all positive. Consequently, restricting each of these elements to the 2^{2} spanned by elements dual to x_{1} and x_{2} does give zero, and (x_{20}, 0), (x_{21}, 0), (x_{24}, 0) and (x_{25}, 0) are all in the image from H^{∗}(McL) if they were not represented by doubled elements in H^{∗}(P SU_{4}(3)). But from 3.7 there are no double image elements in these dimensions except for d_{8}d_{12} which has already been accounted for. It follows that (x_{20}, 0), (x_{21}, 0), (x_{24}, 0) and (x_{25}, 0) are all in the restriction image from classes in H^{∗}(McL).

We now show that the classes (d_{14}, 0) and (d_{15}, 0) are in the restriction image from classes in H^{∗}(McL). From [AM4, 3.14] we have d_{14} = γ_{5}γ_{9} + w̄_{3}^{2}d_{8} + w̄_{3}^{3}γ_{5} from which we see that there is also the element d̄_{14} = γ_{5}γ′_{9} + w̄(γ_{6}d_{8} + γ̄_{5}) ∈ H^{∗}(P SU_{4}(3)) which restricts to (d_{14}, 0) in the two invariant subrings as well as restricting trivially to H^{∗}(2^{2}). There is, however, the element d_{12}a_{2} in the radical in this dimension so we do have to be a bit careful about asserting that d̄_{14} actually lies in H^{∗}(McL). However, in dimension 15, the only element in the radical is d_{8}a_{7} which, as we have seen, is in H^{∗}(McL), and a similar argument constructs d̄_{15} which restricts to (d_{15}, 0), and which therefore does represent an element in H^{15}(McL). Consequently, d̄_{14} must also be present in order to have the Bocksteins work out correctly. In particular, Sq^{1}(d_{14}) = d_{15} and Sq^{1}(d̄_{14}) = d̄_{15}.

From the explicit descriptions of the classes x_{i} in [AM4] we have Sq^{1}(x_{20}) = x_{21}, Sq^{1}(x_{24}) = x_{25} in H^{∗}(2^{4})^{A_{7}}. This gives us sufficient information to understand the first differential Sq^{1} in the Bockstein spectral sequence for H^{∗}(McL), at least through dimension 27. We see that, including the class x_{27} and the (at this stage only possible) class x̄_{27} which restricts to (x_{27}, 0), the E_{2}-term has the form

𝔽_{2}[d_{8}, d_{12}, d_{14}^{2}](1, x_{18}, x_{27}, x̄_{27}, d_{14}d̄_{14}, x_{18}d_{14}d̄_{14}) ⊕ 𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11})

at least assuming that Sq^{1}(x_{27}) = 0. But from this it is clear that x̄_{27} must be present to kill d_{14}d̄_{14}, and that x_{18} must be doubled – there cannot be a class x̄_{18} restricting to (x_{18}, 0) in H^{∗}(McL) since there are no classes in either dimensions 17 or 19 to account for it in the Bockstein spectral sequence. Finally, the role of the class x_{45} must be to hit x_{18}d_{14}^{2} but since the class x_{18}d_{14}d̄_{14} is also present in dimension 46 it follows that there is a class x̄_{45} ∈ H^{∗}(McL) which restricts to (x_{45}, 0).

This completes the construction of the required classes and complete the proof.

Corollary 5.5: The Poincaré series for H^{∗}(McL) is given by

p(t) = \frac{2(1 + t^{18} + t^{20} + t^{21} + t^{24} + t^{25} + t^{27} + t^{45})}{(1 − t^{8})(1 − t^{12})(1 − t^{14})(1 − t^{15})} + \frac{(t^{7} − 1)(1 − t^{11})}{(1 − t^{8})(1 − t^{12})}.

Note in particular that we have that H^{i}(McL) = 0 for 1 ≤ i ≤ 6 and that H^{7}(McL) ≅ H^{8}(McL) = 𝔽_{2}.

§6 The Extension 3^{.}McL.2

According to the Atlas [Co], there is a subgroup N of the form 3^{.}McL.2 which is maximal and of odd index in the sporadic group Ly. The new element of order 2 exchanges 2_{I}^{4} and 2_{II}^{4}, and acts trivially on rad(H^{∗}(McL)). From this we deduce first that 2_{I}^{4} ⊂ Syl_{2}(Ly) ⊂ Ly is weakly closed and that the E_{2} term of the Lyndon–Hochschild–Serre spectral sequence associated to the extension has the form

𝔽_{2}[d_{8}, d_{12}](a_{7}, a_{11}) ⊕ H^{∗}((ℤ/2)^{4})^{A_{7}} ⊕ 𝔽_{2}[d_{8}, d_{12}, e](1, a_{7}, a_{11}, x_{18})e.

Indeed, only the doubled elements can multiply non–trivially with the one dimensional generator e.

Proposition 6.1: The spectral sequence above collapses at the E_{2}–stage, i.e. all differentials are trivial.

Proof: For Bockstein reasons (connecting a_{7} to d_{8}, a_{11} to d_{12}) and the fact, due to weak closure, that the invariants H^{∗}(2_{i}^{4})^{A_{7}} are in the image of the restriction map, it suffices to show that d_{8}, d_{12} are permanent cocycles. From the Atlas [Co], page 175, there exists a real representation χ_{4} for Ly and hence for 3^{.}McL.2 which restricted to (ℤ/2)^{4} has the form 110ϵ ⊕ 2849χ_{reg}, where ϵ is the trivial representation, and χ_{reg} is the regular representation. Hence res(w_{8}(χ_{4})) = d_{8}, and res(w_{12}(χ_{4})) = d_{12}, showing that they are permanent cocycles in the spectral sequence.

In the Atlas [Co], the double coset decomposition of G = Ly with respect to N = 3^{.}McL.2 is explicitly described. In fact we have

G = N ∪ Nx_{1}N ∪ Nx_{2}N ∪ Nx_{3}N ∪ Nx_{4}N

with known intersections: 3^{2+4}.4S_{5}, 2S_{7}, 4S_{6} and 5^{1+2} : S_{3} respectively. This will be used to calculate H^{∗}(Ly) in a sequel.

References

[AM1] Adem, A. and Milgram, R. J. Cohomology of Finite Groups, Springer–Verlag Grundlehren 309 (1994).

[AM2] Adem, A. and Milgram, R. J., “The Cohomology of the Mathieu Group M_{22},” Topology, to appear.

[AM3] Adem, A. and Milgram, R. J., “A_{5}–Invariants, the Cohomology of L_{3}(4) and Related Extensions,” Proc. London Math. Society (3) 66 (1993), 187–224.

[AM4] Adem, A. and Milgram, R. J., “Invariants and Cohomology of Groups,” Bol. Soc. Mat. Mex., to appear.

[Co] Conway, J. et al, Atlas of Finite Groups, Oxford University Press 1985.

[FP] Fiedorowicz, S. and Priddy, S., Homology of Classical Groups over Finite Fields and their Associated Infinite Loop Spaces Springer–Verlag Lecture Notes in Mathematics 674 (1978).

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[JW] Janko, Z. and Wong, W., “A Characterization of McLaughlin’s Simple Group,” Journal of Algebra 20 (1972) 203–225.

[McL] McLaughlin, J., “A simple group of order 898,128,000,” The Theorey of Finite Groups, (Brauer and Sah, eds.), Benjamin, 1969, pps. 109-111.

[M] Milgram, R.J., “The cohomology of the Mathieu group M_{23},” (Preprint, Stanford, 1993).

[RSY] A. Ryba, S. Smith and S. Yoshiara, “Some projective modules determined by sporadic geometries,” J. Algebra 129 (1990), 279–311.