ON THE HOMOTOPY THEORY ASSOCIATED TO CERTAIN FINITE GROUPS OF 2-RANK TWO

F.R. Cohen^{∗}, J.R. Harper, and R. Levi

§1: Introduction.

In this article, the homotopy theory associated to certain finite groups Γ is studied. In particular, assume that Γ is 2-perfect and thus H_{1}(Γ; 𝔽_{2}) = 0. Consider properties of the completion of BΓ at the prime 2, BΓ_{2}^{∧}, and its loop space where the 2-Sylow subgroup of Γ is π = SD_{2^{n}} the semi-dihedral group of order 2^{n}. The main result here is a comparison theorem which implies that Ω(BΓ_{2}^{∧}) can be “resolved by finitely many spheres”. This comparison theorem holds in a mildly more general setting. The homotopy groups of BΓ_{2}^{∧} are then shown to have some possibly unusual properties. The restriction of π to SD_{2^{n}} may appear to be somewhat specialized. The point of this calculation is that in conjunction with [7], the results here give that in many cases if Γ is 2-perfect of 2-rank at most 2, then (ΩBΓ_{2}^{∧})⟨1⟩ can be “resolved by spheres”. As explained after Corollary 1.2, these results handle the cases of finite simple groups of 2-rank at most two with the single possible exception of U_{3}(4). All spaces here are tacitly assumed to have the homotopy type of a CW complex.

In this article a word from popular literature is borrowed in order to convey the main technical idea. Namely, the Pushtu word “jezzailchi” literally means musket bearer [15, p. 48] in the official Afghan language which was adopted in 1936, but in [3, p. 40] this word is used to describe metaphorically a character who has disguised himself by adopting raiments and habits of a jezzailchi. In the current article the idea conveyed by a jezzailchi is that of one space disguising itself as another with enveloping features of the cohomology for the classifying space of a finite p-group.

Definition 1.0. A space X is said to be a (p-local) jezzailchi [3, p. 40] for Y provided

Typeset by AMS-TEX 2 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

*Partially supported by the NSF

(1) there exist maps

j_{X} : A → X and
j_{Y} : A → Y

such that

(2) there are stable maps σ : Σ^{∞}X → Σ^{∞}A with Σ^{∞}(j_{Y}) stably homotopic to Σ^{∞}(j_{Y}) ∘ σ ∘ Σ^{∞}(j_{X}),

(3) j_{X}^{∗} and j_{Y}^{∗} are monomorphisms in mod-p cohomology with image (j_{X}^{∗}) = image (j_{Y}^{∗}), and

(4) X and Y are simply-connected.

If X and Y are CW complexes, then the conditions in Definition 1.0 imply that X and Y are stably homotopy equivalent after completion at p. The point of the jezzailchi condition is given in Section 6; this condition allows a comparison of Postnikov towers one stage further than might be expected. Some examples of X and Y are given by the p-completions of the classifying spaces of certain p-perfect finite groups where A is the classifying space of the p-Sylow subgroup of these finite groups. Some specific examples which are considered here are Γ = SL(3, 𝔽_{q}) or Γ = M_{11} the sporadic finite simple group given by the Mathieu group of order 2^{4} ·3^{2} ·7·11. Thus there is the specific example of Y = BSL(3, 𝔽_{3})_{2}^{∧}, X = (BM_{11})_{2}^{∧}, and A = BSD_{16} [8].

Theorem 1.1. Let X be a simply-connected 2-local space which is a jezzailchi for BΓ, Γ = SL(3, 𝔽_{q}) with q ≡ 3(mod 4) or Γ = M_{11}, then ΩX is homotopy equivalent to Ω(BΓ_{2}^{∧}).

Remark. It follows that if Γ is any 2-perfect finite group with 2-Sylow subgroup SD_{2^{n}} then the same conclusion follows [8]. The cases for which q ≡ 1(mod 4) will be considered elsewhere.

Recall from [1] that a space Y is said to be spherically resolvable of weight ≤ n provided there are fibrations (1) p_{i} : Y_{i} → B_{i}, with fibre Y_{i+1},

(*) \begin{matrix}Y_{0} & ⟵ & Y_{1} & ⟵ & … & ⟵ & Y_{k} & ⟵ & … & ⟵ & Y_{n−1} \\ ^{p_{0}}↓ & ^{p_{1}}↓ & & & ^{p_{k}}↓ & & & ^{p_{n−1}}↓ \\ B_{0} & B_{1} & & & B_{k} & & & B_{n−1}, \end{matrix}

FINITE GROUPS OF 2-RANK TWO 3

(2) p_{n−1} = 1 : Y_{n−1} → B_{n−1}, (3) B_{i} is a sphere or the iterated loop space of a sphere Ω^{m}S^{m+t}, t ≥ 0, and (4) Y_{0} = Y. A diagram (∗) is said to be a resolution for Y.

Corollary 1.2. Let X be a 2-local simply-connected space which is a jezzailchi for BΓ_{2}^{∧}, Γ = SL(3, 𝔽_{q}) with q ≡ 3(mod 4) or Γ = M_{11}. Then ΩX is spherically resolvable of weight 4 and there is a resolution

ΩX = Y_{0} ⟵ Y_{1} ⟵ Y_{2} ⟵ Y_{3}
↓  ↓  ↓  ↓
S^{5}  S^{3}  ΩS^{5}  ΩS^{3}.

The finite simple groups Γ of 2-rank two are listed in the Second Main Theorem of [14]; the terminology in this last paper disagrees with that in [2, 8] and that which is adopted here. Namely the term quasi-dihedral group is used in [14] for the groups which are called semi-dihedral in [2, 8]. With the terminology from [2, 8] and the results in [14] one of the following holds for Γ:

(i) Γ has dihedral Sylow 2-subgroups and Γ is isomorphic to either A_{7} or P SL(2, 𝔽_{q}) for q odd,

(ii) Γ has semi-dihedral Sylow 2-subgroups (in the terminology of [2, 8]) and Γ is isomorphic to either P SL(3, 𝔽_{q}) for q ≡ 3(mod 4), U_{3}(q) for q ≡ 1(mod 4), or M_{11},

(iii) Γ has wreathed Sylow 2-subgroup and Γ is isomorphic to either P SL(3, 𝔽_{q})

for q ≡ 1(mod 4), or U_{3}(q) for q ≡ 3(mod 4); or (iv) Γ is isomorphic to U_{3}(4). By inspecting (1) the classification listed above which was given in [14], (2) the

results in Corollary 1.2 here, and (3) the results in [7, chapter 5], it follows that Ω(BΓ)_{2}^{∧} is spherically resolvable of weight ≤ 4 if Γ is a finite simple group of 2-rank at most 2 with the single possible exception of Γ = U_{3}(4).

The homotopy groups of spaces X in Corollary 1.2 appear to be mildly unusual and thus a few of them are listed below.

Corollary 1.3. Let X = BΓ_{2}^{∧} or Γ = M_{11} with Γ = SL(3, 𝔽_{q}) and q ≡ 3(mod 4). Then X is 2-connected,

π_{3}X = ℤ/2^{n−1}ℤ

4 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

where the 2-Sylow subgroup of M_{11} or SL(3, 𝔽_{q}) is SD_{2^{n}},

π_{4}X = {0}, and
π_{i}X = ℤ/2ℤ if 5 ≤ i ≤ 10.

In [12], Oka described certain periodic elements in the homotopy groups of SU(3). These periodic elements contribute copies of ℤ/2ℤ to the homotopy groups of X in Corollary 1.3. One might wonder whether the homotopy groups of the 3-connected cover of X have exponent 2.

The homology of ΩX for X as in 1.1 also appears to be unusual. The following result was proven in [7] for X = BΓ_{2}^{∧}.

Proposition 1.4. If X is a jezzailchi for BΓ_{2}^{∧}, Γ = SL(3, 𝔽_{q}), q ≡ 3(mod 4), or Γ = M_{11}, then H_{∗}(ΩX; 𝔽_{2}) is isomorphic as an algebra to

E[x_{2}, x_{3}] ⊗ 𝔽_{2}[x_{4}]

where degree (x_{i}) = i, β_{n−1}(x_{3}) = x_{2},β_{n−1}(x_{2}x_{3}) = x_{4}, and the 2-Sylow subgroup of Γ is SD_{2^{n}}. Furthermore, H_{∗}(ΩX; 𝔽_{2}) is completely determined as a Hopf algebra by requiring the coproduct for x_{4} to be given by Δ(x_{4}) = x_{4} ⊗ 1 + x_{2} ⊗ x_{2} + 1 ⊗ x_{4} and that both x_{2} and x_{3} be primitive. The homology ring H_{∗}(ΩX; ℤ_{(2)}) is abelian and periodic in the sense that there are isomorphisms

H̄_{i}(ΩX; ℤ_{(2)}) = \{ \begin{matrix}ℤ/2^{n}ℤ if i = 4k > 0 \\ ℤ/2^{n−1}ℤ if i = 4k + 2, and \\ 0 otherwise . \end{matrix}

The main technical tool used here is a 4-stage Moore-Postnikov system which is described briefly below and in more detail in section 5:

E_{3}

↓
E_{2} ⟶^{e_{9}} K(ℤ/2, 9)
P_{2}↓
E_{1}  ⟶^{ẽ_{7},e_{8}} K(ℤ/2, 7) × K(ℤ/2, 8)
P_{1}↓
K(ℤ/2^{n−1}, 3) = E_{0} ⟶^{k_{0}}  K(ℤ/2, 6)

FINITE GROUPS OF 2-RANK TWO 5

The map k_{0} is determined, up to homotopy, by the class i_{3}^{2} + Sq^{2}ν where ν is the generator of H^{4}(K(ℤ/2^{n−1}, 3); 𝔽_{2}). The class ẽ_{7} will be uniquely determined by the cohomology ring structure of the spaces X given in Lemma 1.6 below; however the class e_{8} will be chosen from a 2 dimensional subspace of H^{8}(E_{1}; 𝔽_{2}). Information for ẽ_{7} and e_{8} provides enough information about e_{9} to prove the next theorem.

Theorem 1.5. If X is simply-connected, 2-local, and a jezzailchi for BΓ with Γ 2-perfect and Sylow 2-subgroup given by SD_{2^{n}}, n ≥ 4, then H_{3}(X; ℤ) = ℤ/2^{n−1} and any map i : X → E_{0} which induces an isomorphism on H_{3} lifts to h : X → E_{3}. Furthermore, h induces an isomorphism on π_{i} for i ≤ 8. Thus there is an isomorphism of sets

h_{∗} : [SU(3), X] → [SU(3), E_{3}].

Since the Steenrod operations in this last theorem are useful, that information is listed below as in [2,8].

Lemma 1.6. If Γ is a 2-perfect group with a 2-Sylow subgroup given by SD_{2^{n}}, then H^{∗}(Γ; 𝔽_{2}) is isomorphic to 𝔽_{2}[u, v, z]/⟨u^{2}v = z^{2}⟩ where degree (u) = 3, β_{n−1}(u) = v and Sq^{2}(u) = z. Thus

(1) Sq^{1}(u) = 0, Sq^{2}(u) = z, and Sq^{2^{i}}(u) = 0, if i ≥ 2,

(2) Sq^{1}(v) = 0, Sq^{2}(v) = u^{2}, Sq^{4}(v) = v^{2}, and Sq^{2^{i}}(v) = 0 if i ≥ 3, and

(3) Sq^{1}(z) = u^{2}, Sq^{2}(z) = 0, Sq^{4}(z) = u^{3} + vSq^{2}u, and Sq^{2^{i}}(z) = 0 if i ≥ 3.

Recently, J. Martino and S. Priddy have proven that BM_{11}_{2}^{∧} and BSL(3, 𝔽_{3})_{2}^{∧} are homotopy equivalent; their result gives a stronger conclusion than Theorem 1.1 for these examples. In addition, A. Adem has shown the analogous result by constructing an analgamated free product ∏ together with homomorphisms ∏ → M_{11} and ∏ → SL(3, 𝔽_{3}) which induce mod-2 homology isomorphisms.

§2: Proof of Theorem 1.1 and Corollary 1.2.

Recall that the order of SL(3, 𝔽_{q}) is q^{3}(q^{3} − 1)(q^{2} − 1) and that the 2-Sylow subgroup of SL(3, 𝔽_{q}) is SD_{2^{n}} if q ≡ 3(mod 4) and n = ν_{2}((q^{3} − 1)(q^{2} − 1) the largest power of 2 which divides (q^{3} − 1)(q^{2} − 1). By [4], there is a 2-complete fibration

SU(3)_{2}^{∧} → BSL(3, 𝔽_{q})_{2}^{∧} → BSU(3)_{2}^{∧}.

6 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

Thus, ΩBSL(3, 𝔽_{q})_{2}^{∧} is the homotopy theoretic fibre of a map ψ : SU(3)_{2}^{∧} → SU(3)_{2}^{∧}.

The proof of Theorem 1.1 uses Theorem 1.5; the proof of this last theorem is held in abeyance until section 5. In any case, there are isomorphisms of sets

[SU(3), BSL(3, 𝔽_{q})_{2}^{∧}] → [SU(3), E_{3}] and
[SU(3), X] → [SU(3), E_{3}]

each of which is induced by maps

f : BSL(3, 𝔽_{q}) → E_{3} and
g : X → E_{3}.

Next consider the fibre sequence

ΩBSL(3, q)_{2}^{∧} → SU(3)_{2}^{∧} →^{ψ} SU(3)_{2}^{∧} →^{γ} BSL(3, 𝔽_{q})_{2}^{∧}

and the map α : SU(3) → X which corresponds to γ via Theorem 1.5. Thus α · ψ is null and there is a map of fibrations

ΩSU(3)_{2}^{∧} ⟶ ΩBSL(3, 𝔽_{q})_{2}^{∧} ⟶ SU(3)_{2}^{∧} ⟶^{ψ} SU(3)_{2}^{∧}
Ω(α)↓  ^{π}↓  ↓  ↓^{α}
ΩX ⟶^{1}  ΩX  ⟶ ∗ ⟶ X.

Notice that π may not necessarily be homotopy multiplicative, but π is a homotopy equivalence by the following remarks. Since α and γ induce epimorphisms on the level of third homotopy groups ℤ/2^{n−1}ℤ by construction, the fact that π is an equivalence is immediate from the next lemma.

Lemma 2.1. If X is a simply-connected space and H^{∗}(X; 𝔽_{2}) is isomorphic to H^{∗}(BSL(3, 𝔽_{q}); 𝔽_{2}), q ≡ 3(mod 4), as algebras over the Steenrod algebra together with the action of all Bocksteins, then any map

π : ΩBSL(3, 𝔽_{q})_{2}^{∧} → ΩX

which induces an isomorphism on the second homotopy group is a homotopy equivalence.

Proof. Since all spaces are simply-connected and of finite type over ℤ_{(2)}, it suffices to see that π induces an isomorphism on mod-2 homology.

Now observe that by 1.4 there are isomorphisms of Hopf algebras

H_{∗}(ΩX; 𝔽_{2}) ≅ H_{∗}(ΩBSL(3, 𝔽_{q})_{2}^{∧}; 𝔽_{2}) ≅ E[x_{2}, x_{3}] ⊗ 𝔽_{2}[x_{4}]

with Δx_{4} = x_{4} ⊗ 1 + x_{2} ⊗ x_{2} + 1 ⊗ x_{4}. Thus the module of primitives has basis {x_{2}, x_{3}, x_{4}^{2^{i}}|i ≥ 1}. By induction on i, the fact that π_{∗}(x_{2}) = x_{2}, and commutation with the coproduct Δ, one has π_{∗}(x_{2}x_{3}x_{4}^{2^{i}−1}) = x_{2}x_{3}x_{4}^{2^{i}−1} and thus π_{∗}(x_{4}^{2^{i}}) = x_{4}^{2^{i}} by commutation with the Bocksteins as β_{n}(x_{2}x_{3}x_{4}^{2^{i}−1}) = x_{4}^{2^{i}} where the 2-Sylow subgroup of SL(3, 𝔽_{q}) is SD_{2^{n}}.

Thus Theorem 1.1 follows. To end this section, observe that there are fibrations ΩX → SU(3) with fibre ΩSU(3). Thus there is a resolution

ΩX ⟵ Y_{1} ⟵ Y_{2} ⟵ ΩS^{3}
↓  ↓  ↓
S^{5}  S^{3}  ΩS^{5}

obtained from the map of fibrations (after completion at 2)

ΩSU(3)_{2}^{∧} ⟶  Y_{1}  ⟶ S^{3}
1↓  ↓  ↓
ΩSU(3)_{2}^{∧} ⟶ ΩBSL(3, 𝔽_{q})_{2}^{∧} ⟶ SU(3)_{2}^{∧}
↓  ↓  ↓
∗  ⟶  S^{5}  ⟶^{1} S^{5}.

Corollary 1.2 follows.

§3: The cohomology of ΩX, 1.4, and 1.6.

In this section it is shown that the cohomology of ΩX as Hopf algebra over the Steenrod algebra together with the action of the higher Bocksteins is completely determined by the cohomology of X provided X satisfies the hypotheses of Theorem 1.1. The action of the Steenrod operations on H_{∗}X is also recorded here. Proposition 1.4 was proven in [7] with the stated hypotheses that X = BΓ_{2}^{∧}. As the result and the same short proof holds with less restrictive hypotheses, Proposition 1.4 is also proven here.

8 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

Notice that if H^{∗}(X; 𝔽_{2}) = 𝔽_{2}[u, v, z]/⟨u^{2}v = z^{2}⟩ with degree(u) = 3, β_{n−1}(u) = v, and Sq^{2}(u) = z, there is an algebra extension

1 → 𝔽_{2}[u, v] → H^{∗}(X; 𝔽_{2}) → E[z] → 1.

By the change of rings spectral sequence [0, p. 276] there is a spectral sequence abutting to T or_{H^{∗}(X;𝔽_{2})}(𝔽_{2}, 𝔽_{2}) with E_{2} = T or_{E[z]}(𝔽, T or_{𝔽_{2}[u,v]}(𝔽_{2}, 𝔽_{2})) where T or_{𝔽_{2}[u,v]}(𝔽_{2}, 𝔽_{2}) is a trivial E[z]-module. The spectral sequence collapses. Dually, one has Ext^{H^{∗}(X;𝔽_{2})}(𝔽_{2}, 𝔽_{2}) = E[(−1, u)^{∗}, (−1, v)^{∗}] ⊗ 𝔽_{2}[(−1, z)^{∗}]. Thus in the Eilenberg-Moore spectral sequence abutting to H_{∗}(ΩX; 𝔽_{2}), E^{2} = E^{∞} and E^{0}H_{∗}(ΩX; 𝔽_{2}) is isomorphic to E[a, b] ⊗ 𝔽_{2}[c] with degree(a) = 2, degree(b) = 3, degree(c) = 4, β_{n−1}(b) = a, and Sq_{∗}^{2}(c) = a. This forces H_{∗}(ΩX; 𝔽_{2}) to be abelian with a and b primitive and Δ(c) = c ⊗ 1 + 1 ⊗ c + a ⊗ a.

To compute the higher Bocksteins, one uses the algebra structure of H_{∗}(ΩX; 𝔽_{2}) given by E[a, b] ⊗ 𝔽_{2}[c] as shown in [7]. Thus β_{n−1}(ab) is defined and equals 0. But the classical Bockstein Lemma [11] applied to the cohomology Serre spectral sequence for the path-loop fibration over X implies that β_{n}(ab) is defined and equals c; the details of computation are omitted. Notice that the above gives that the Bockstein spectral sequence, E_{r}, for H_{∗}(ΩX; ℤ_{(2)}) is completely determined as follows:

(1) E_{1} ≅ H_{∗}(ΩX; 𝔽_{2}) = E[a, b] ⊗ 𝔽_{2}[c],

(2) E_{r} ≅ E_{n−1} if 1 ≤ r ≤ n − 1,

(3) E_{n} ≅ E[ab] ⊗ 𝔽_{2}[c] with β_{n}(ab) = c.

(4) E_{n+1} ≅ E_{∞} = 𝔽_{2}. Thus Proposition 1.4 follows from the known cohomology ring for X [8].

The next remark is an elementary exercise in undetermined coefficients and is useful later; a determination of the Steenrod operations in H^{∗}(X; 𝔽_{2}) is given:

1. β_{n−1}u = v and n ≥ 3, thus Sq^{1}(u) = 0.

2. Sq^{2}u = z by construction.
3. Sq^{2^{i}}u = 0 if i ≥ 2 by excess considerations.
4. Sq^{1}v = 0 because v = β_{n−1}(u).
5. Sq^{2}v = ε(u^{2}) for ε = 0, 1 Since u^{2}v = z^{2} and

6. Sq^{2}(u^{2}v) = u^{2}Sq^{2}v = ε(u^{4}) with Sq^{2}(z^{2}) = (Sq^{1}Sq^{2}u)^{2} = u^{4}. Thus ε = 1.

7. Sq^{4}v = v^{2} and Sq^{2^{i}}v = 0 if i ≥ 3 by excess considerations.
8. Sq^{1}z = Sq^{1,2}u = Sq^{3}u = u^{2}.

FINITE GROUPS OF 2-RANK TWO 9

9. Sq^{2}z = Sq^{2,2}z = Sq^{3,1}z = 0.

10. Sq^{4}z = ε(u^{3}) + δ(vz). That ε = δ = 1 follows next. Notice that Sq^{1}Sq^{4}z = Sq^{5}z = z^{2} = Sq^{1}{εu^{3}+δvz} = δvu^{2}. Thus δ = 1. Next observe that Sq^{2}Sq^{4}z = Sq^{2,4,2}u = (Sq^{6} + Sq^{5,1})(Sq^{2}u) = Sq^{6,2}u + Sq^{5}u^{2} = 0. But Sq^{2}Sq^{4}z = Sq^{2}(εu^{3} + vz) = εu^{2}z + u^{2}z and so ε = 1.

§4: Proof of Corollary 1.3.

The following information concerning π_{∗}SU(3) is well-known [10,12]; it can be deduced directly from [13] together with the fibration SU(3) → S^{5} with fibre S^{3} using compositions with η : S^{4} → S^{3}:

π_{i}SU(3) ≅ \{ \begin{matrix}0 if i = 1, 2, 4, 7, 9 \\ ℤ if i = 3, 5 \\ ℤ/2 if i = 6, 10 on 2 primary components \\ ℤ/4 if i = 8 on 2 primary components. \end{matrix}

To work out π_{∗}BSL(3, 𝔽_{q})_{2}^{∧}, consider the fibration, after completion at 2 with q ≡ 3(mod 4)

SU(3) →^{ψ} SU(3) → BSL(3, 𝔽_{q})_{2}^{∧}.

Thus

π_{i}BSL(3, 𝔽_{q})_{2}^{∧} ≅ \{ \begin{matrix}0 if i = 1, 2, 4 \\ ℤ/2^{n−1}ℤ if i = 3. \end{matrix}

Next observe that π_{5}SU(3) ≅ ℤ and that a choice of generator g has Hurewicz image equal to twice the fundamental class in H_{5}(S^{5}; ℤ) via the fibration SU(3) → S^{5}. As ψ is degree 2 on H_{5}( ; ℤ), it follows that ψ_{∗} : π_{5}SU(3) → π_{5}SU(3) is multiplication by 2 as π_{5}SU(3) injects in H_{5}(SU(3); ℤ). Thus

π_{5}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2ℤ for all q ≡ 3(mod 4).

Next, observe that π_{6}SU(3) is ℤ/2ℤ with generator represented by S^{6} →^{ν′} S^{3} →^{i} SU(3). Since ψ_{∗} : π_{5}SU(3) → π_{5}SU(3) is a monomorphism,

π_{6}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2ℤ for all q ≡ 3(mod 4).

Since π_{7}SU(3) = 0, it follows that

π_{7}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2ℤ for all q ≡ 3(mod 4).

10 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

To determine the eighth homotopy group, recall that π_{8}SU(3) injects in π_{8}S^{5} and the image has index 2. Thus ψ_{∗} : π_{8}SU(3) → π_{8}SU(3) is multiplication by 2 and

π_{8}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2ℤ for all q ≡ 3(mod 4).

Furthermore, since π_{9}SU(3) = 0,

π_{9}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2ℤ for all q ≡ 3(mod 4).

Finally, since π_{9}S^{3} ≅ π_{10}S^{3} ≅ {0}, π_{10}SU(3) → π_{10}S^{5} ≅ ℤ/2ℤ is an isomorphism and

π_{10}BSL(3, 𝔽_{q})_{2}^{∧} ≅ ℤ/2 for all q ≡ 3(mod 4).

A similar analysis can be easily done for the periodic family of elements appearing in [10,12].

§5: The tower E_{3}.

Throughout this section consider Y = BΓ where Γ is a finite 2-perfect group with 2-Sylow subgroup SD_{2^{n}}, n ≥ 4. Thus by [2,8], the mod-2 cohomology algebra of Y is given by Lemma 1.6.

In this section, the tower E_{3} will be given. Unless otherwise stated all cohomology groups are taken with 𝔽_{2} coefficients.

Write K_{n} = K(ℤ/2, n) with fundamental class i_{n}. The space E_{3} fits in a tower E_{3}

↓
E_{2} ⟶^{e_{9}} K_{9}
↓
E_{1} ⟶^{ẽ_{7},e_{8}} K_{7} × K_{8}
P_{1}↓
E_{0} ⟶^{k_{0}} K_{6}

where E_{0} = K(ℤ/2^{n−1}, 3), n ≥ 4, with k_{0} = i_{3}^{2} + Sq^{2}ν and ν is given by the generator of H^{4}E_{0} ≈ 𝔽_{2}. There are choices of classes ẽ_{7} and e_{8} which require some discussion. Namely, ẽ_{7} will be determined uniquely by the cohomology ring of Y while e_{8} is taken to be an element in a 2-dimensional subspace of H^{8}E_{1}, which

FINITE GROUPS OF 2-RANK TWO 11

is determined by the cohomology ring of Y. The specific choice of e_{8} is of course determined by Y. After ẽ_{7} and e_{8} are discussed precisely, the class e_{9} and E_{3} will be given.

Proposition 5.1. There is a unique class e_{7} in H^{7}(E_{1}; 𝔽_{2}) such that (i) e_{7} restricts to Sq^{2}i_{5} in H^{7}(K_{5}; 𝔽_{2}) (ii) e_{7} is primitive in the coalgebra structure of H^{∗}(E_{1}; 𝔽_{2}) induced by loop

multiplication. Furthermore, there is a 2-dimensional subspace V ⊂ H^{8}(E_{1}; 𝔽_{2}) and a choice of element e_{8} in V such that

(iii) e_{8} restricts to Sq^{2,1}i_{5} in H^{8}(K_{5}; 𝔽_{2}),

(iv) the reduced coproduct of e_{8} is (p_{1}^{∗}ν) ⊗ (p_{1}^{∗}ν), and

(v) V is the linear span of e_{8} and e_{8} + p_{1}^{∗}(ν^{2}). The action of the Steenrod algebra satisfies

(vi) p_{1}^{∗}(Sq^{(4,2)}i_{3}) = Sq^{2}e_{7} + Sq^{1}e_{8}, and

(vii) p_{1}^{∗}(ν · Sq^{2}ν + Sq^{4,2}ν) = Sq^{2}(e_{8}).

Remark. In the above action of Sq^{1} and Sq^{2}, e_{8} can be replaced by e_{8} + p_{1}^{∗}(ν^{2}) without changing the formulae. In addition, p_{1}^{∗}Sq^{4,2}ν = p_{1}^{∗}Sq^{5,2}i_{3}.

Proof of 5.1. The proof is routine use of Massey-Peterson theory [5,6,9].

Notice that there is a map f_{0} : Y → E_{0} which induces an isomorphism on H_{3}(; ℤ_{(2)}) by assumption. Next notice that there is a lift f_{1} : Y → E_{1} as u^{2} = Sq^{2}v in H^{∗}Y. Since f_{0}^{∗} is an epimorphism, so is f_{1}^{∗} where f_{1} is any lift of f_{0}. Since Sq^{2}z = 0 = Sq^{2,1}z in H^{∗}Y, all lifts f_{1} give the same values for f_{1}^{∗}(e_{7}) and f_{1}^{∗}(e_{8}) with e_{i} is given in Proposition 5.1.

To determine ẽ_{7} and e_{8} in a tower for Y, consider the diagram

K_{5} ⟶  E_{1}
↓
Y ⟶^{f_{0}} K(ℤ/2^{n−1}, 3) ⟶^{k_{0}} K_{6}

with f_{0}^{∗}(i_{3}) = u and f_{0}^{∗}(ν) = v.

12 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

Lemma 5.2.

(i) For any lift of f_{0} given by f_{1}, f_{1}^{∗}(e_{7}) ≠ 0 and f_{1}^{∗}(ẽ_{7}) = 0 where ẽ_{7} = e_{7} + p_{1}^{∗}(i_{3} · ν).

(ii) The kernel of f_{1}^{∗} in dimension 8 is one dimensional and lies in V the linear span of {e_{8}, e_{8} + p_{1}^{∗}(ν^{2})}.

Proof. In (i) assume that f_{1}^{∗}(e_{7}) = 0 and consider f_{1}^{∗}(e_{8}) which lies in the linear span of {ν^{2}, u · Sq^{2}u}. If f_{1}^{∗}(e_{8}) ≠ 0, then f_{1}^{∗}(e_{8}) = Aν^{2} + B(u · Sq^{2}u) and so f_{1}^{∗}(e′_{8}) = 0 where e′_{8} = e_{8} + A(p_{1}^{∗}ν)^{2} + Bp_{1}^{∗}(i_{3} · Sq^{2}i_{3}). In this case relation (vi) in Proposition 5.1 becomes

Sq^{2}e_{7} + Sq^{1}e′_{8} = p_{1}^{∗}(Sq^{4,2}i_{3} + Bi_{3}^{3}).

Since f_{1}^{∗}(e_{7}) = 0 = f_{1}^{∗}(e′_{8}), Sq^{4,2}u + Bu^{3} = 0 in H^{∗}Y. But Sq^{4,2}u = u^{3} + v · Sq^{2}u by Lemma 1.6 and thus Sq^{4,2}u and u^{3} are linearly independent in H^{∗}Y. This is a contradiction and so f_{1}^{∗}(e_{7}) ≠ 0. Furthermore, since H^{∗}Y is 1-dimensional with basis u · v, it follows that f_{1}^{∗}(e_{7}) = u · v and f_{1}^{∗}(ẽ_{7}) = 0 where ẽ_{7} = e_{7} + p_{1}^{∗}(i_{3} · ν).

Next notice that

Sq^{2}ẽ_{7} + Sq^{1}e_{8} = Sq^{2}e_{7} + p_{1}^{∗}Sq^{2}(i_{3} · ν) + Sq^{1}e_{8}
= p_{1}^{∗}(Sq^{4,2}i_{3} + i_{3}^{3} + ν · Sq^{2}i_{3})

by Proposition 5.1 (vi). If e_{8} is replaced by e_{8} + p_{1}^{∗}ν^{2}, the same relation holds.

To prove part (ii) consider Ker f_{1}^{∗} in dimension 8. By inspection H^{8}E_{1} is 3 dimensional with basis {e_{8}, p_{1}^{∗}(ν · Sq^{2}i_{3}), p_{1}^{∗}(ν^{2})}. Since f_{1}^{∗} is an epimorphism, Ker f_{1}^{∗} is one dimensional. If f_{1}^{∗}(e_{8}) = 0, then the lemma follows. Thus assume that f_{1}^{∗}(e_{8}) ≠ 0. Hence f_{1}^{∗}(e_{8}) = Av^{2} + B(u · Sq^{2}u). If B = 0, then either f_{1}^{∗}(e_{8}) or f_{1}^{∗}(e_{8} + p_{1}^{∗}ν^{2}) is zero and the lemma follows.

Thus assume that f_{1}^{∗}(e_{8}) = Av^{2} + u · Sq^{2}u. By varying e_{8} by p_{1}^{∗}v^{2}, assume that A = 0. Then one has f_{1}^{∗}(e_{8}) = u · Sq^{2}u. Let e”_{8} = e_{8} + p_{1}^{∗}(i_{3} · Sq^{2}i_{3}) to obtain the relation

Sq^{2}(ẽ_{7}) + Sq^{1}(e”_{8}) = Sq^{2}(ẽ_{7}) + Sq^{1}(e_{8}) + p_{1}^{∗}Sq^{1}(i_{3} · Sq^{2}i_{3})
= p_{1}^{∗}(Sq^{4,2}i_{3} + i_{3}^{3} + ν · Sq^{2}i_{3} + i_{3}^{3})
= p_{1}^{∗}(Sq^{4,2}i_{3} + νSq^{2}i_{3}).

FINITE GROUPS OF 2-RANK TWO 13

However, f_{1}^{∗}p_{1}^{∗}(Sq^{4,2}i_{3} + νSq^{2}i_{3}) = (u^{3} + vSq^{2}u) + vSq^{2}u = u^{3} and f_{1}^{∗}Sq^{2}(ẽ_{7}) = 0 = Sq^{1}(e”_{8}). This is a contradiction and so f_{1}^{∗}(e_{8}) = Av^{2} and the lemma follows.

Thus there are choices of k-invariants ẽ_{7} and e_{8} such that f_{1}^{∗}(ẽ_{7}) = 0 = f_{1}^{∗}(e_{8}). Hence there is a lift as follows:

Y ⟶^{f_{2}} E_{2} ⟶^{e_{9}} K_{9}
1↓  ^{p_{2}}↓
Y ⟶^{f_{1}} E_{1} ⟶^{(ẽ_{7},e_{8})} K_{7} × K_{8}
1↓  ^{p_{1}}↓
Y ⟶^{f_{0}} E_{0} ⟶^{K_{0}} K_{6}

Lemma 5.3. The image of p_{2}^{∗} in H^{∗}E_{2} is isomorphic to H^{∗}Y in dimensions ≤ 13.

Proof. A direct check gives that the algebra

H^{∗}K(ℤ/2^{n−1}, 3)/I

where I is the ideal generated by (i) i_{3}^{2} + Sq^{2}ν, (ii) Sq^{4,2}i_{3} + i_{3}^{3} + ν · Sq^{2}i_{3}, and (iii) Sq^{4,2}ν + νSq^{2}ν is isomorphic to 𝔽_{2}[u, v, z]/⟨z^{2} = u^{2}v⟩ through dimension 13.

Lemma 5.4. There exist classes e_{9} in H^{9}E_{2} and e_{10} in H^{10}E_{2} such that (i) e_{9} restricts to Sq^{3}i_{6} + Sq^{2}i_{7} in H^{9}(K_{6} × K_{7}) and Sq^{1}e_{9} restricts to Sq^{3}i_{7}, (ii) e_{10} restricts to Sq^{3,1}i_{6} + Sq^{2,1}i_{7} in H^{10}(K_{6} × K_{7}) (iii) f_{2}^{∗}e_{9} = 0 = f_{2}^{∗}e_{10}, and (iv) e_{9} and e_{10} are primitive in the principal action K_{6} × K_{7} × E_{2} → E_{2}.

Proof. Consider 1. Sq^{2}ẽ_{7} + Sq^{1}e_{8} = p_{1}^{∗}(Sq^{4,2}i_{3} + i_{3}^{3} + ν · Sq^{2}i_{3}) to get

2. Sq^{3}ẽ_{7} = p_{1}^{∗}(Sq^{5,2}i_{3} + ν · i_{3}^{2}).

In addition, by 5.1 (vii)

3. Sq^{2}e_{8} = p_{1}^{∗}(ν · Sq^{2}ν + Sq^{4,2}ν). Since p_{1}^{∗}(i_{3}^{2} + Sq^{2}ν) = 0 in H^{∗}E_{1}, it follows

that p_{1}^{∗}Sq^{5,2}i_{3} = Sq^{4,2}v and p_{1}^{∗}(ν · i_{3}^{2}) = p_{1}^{∗}(ν · Sq^{2}ν). Thus one has

4. Sq^{3}ẽ_{7} + Sq^{2}e_{8} = 0 in H^{∗}E_{1}. Thus there exists a non-zero class e_{9} in H^{∗}E_{2} which restricts to Sq^{3}i_{6} + Sq^{2}i_{7} in H^{9}(K_{6} × K_{7}) and, of course, Sq^{1}e_{9}

restricts to Sq^{3}i_{7}.

14 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

5. Next, notice that

Sq^{3,1}ẽ_{7} + Sq^{2,1}e_{8} = Sq^{2}P_{1}^{∗}(Sq^{2}ẽ_{7} + Sq^{1}e_{8})
= p_{1}^{∗}Sq^{2}(Sq^{4,2}i_{3} + i_{3}^{3} + ν · Sq^{2}i_{3})
= p_{1}^{∗}(i_{3}^{2} · Sq^{2}i_{3} + Sq^{2}ν · Sq^{2}i_{3}) = 0

Thus there exists a class e_{10} in H^{10}E_{2} which restricts to Sq^{3,1}i_{6} + Sq^{2,1}i_{7}.

By comparison with H^{∗}Y, there are choices of e_{9} and e_{10} such that f_{2}^{∗}e_{9} = 0 = f_{2}^{∗}e_{10}. The proof of statement (iv) is again routine use of Massey-Peterson theory [5,6,9].

Thus by Lemma 5.4, there is a lift f_{3} : Y → E_{3} giving a homotopy commutative diagram

Y ⟶^{f_{3}} E_{3}
1↓  ↓
Y ⟶^{f_{2}} E_{2}

where f_{3}^{∗} gives an isomorphism in dimensions ≤ 9. This concludes the construction of E_{3}.

§6: Theorem 1.5.

In this section assume that X is a jezzailchi for Y = BΓ_{2}^{∧} given in Section 5. The purpose of this section is to prove the next theorem of which Theorem 1.5 is an immediate consequence.

Theorem 6.1. There exists a map g : X → E_{3} which induces a mod-2 cohomology isomorphism in dimension ≤ 9.

The point of this section is that the jezzailchi condition allows a comparison of X and Y one stage further in the tower than one would ordinarily expect for purely cohomological reasons. This information is enough to prove 6.1. The first observation is as follows.

Lemma 6.2. (a) If X is a jezzelchai for Y, then there is an equivalence of infinite

FINITE GROUPS OF 2-RANK TWO 15

loop spaces Ω^{∞}(θ) : QX → QY together with a homotopy commutative diagram

A ⟶^{k} X ⟶^{E} QX
j↓ ↓^{Ω^{∞}(θ)}
Y ⟶_{E} QY ⟶^{1} QY.

(b) If α : QY → B is any map of infinite loop spaces, then there is a map α′ : QX → B of infinite loop spaces together with a homotopy commutative diagram

A ⟶^{k} X ⟶^{E} QX
j↓ ↓_{α′}
Y ⟶_{E} QY ⟶_{α} B

Proof. Let θ = (Σ^{∞}j) ∘ σ where σ is given in definition 1.0. Then Σ^{∞}j = (Σ^{∞}j) ∘ σ ∘ (Σ^{∞}k) because X is a jezzailchi for Y and part (b) follows by naturality and the definition α′ = α ∘ Ω^{∞}(θ).

Notice that E_{1} is an infinite loop space and p_{1} : E_{1} → E_{0} is an infinite loop map. Thus, by 6.2, there is a homotopy commutative diagram

A ⟶^{k} X ⟶^{E} QX
j↓  ↓_{α′}
Y ⟶^{E} QY ⟶^{F_{1}} E_{1}

where F_{1} is the infinite loop extension of f_{1} : Y → E_{1}. Furthermore f_{1}^{∗}(ẽ_{7}) = 0 = f_{1}^{∗}(e_{8}) and k^{∗} is a monomorphism. Thus let g_{1} = α′ ∘ E and notice that g_{1}^{∗}(ẽ_{7}) = 0 = g_{1}^{∗}(e_{8}) and there is a lift

g_{2} : X → E_{2}

which fits in a homotopy commutative diagram

X ⟶^{g_{2}} E_{2}
1↓ ↓
X ⟶^{g_{1}} E_{1}
1↓ ↓
X ⟶^{g_{0}} E_{0}

with g_{0∗} an isomorphism on H_{3}(; ℤ_{(2)}). Furthermore, the composites A →^{j_{Y}} Y →^{f_{2}} E_{2} and A →^{j_{X}} X →^{g_{2}} E_{2} are homotopic in E_{1}. The next lemma is a restatement of this fact.

16 F.R. COHEN^{∗}, J.R. HARPER, AND R. LEVI

Lemma 6.3. Let A = BSD_{2^{n}}, n ≥ 3. There is a map d : A → K_{6} × K_{7} such that f_{2} ∘ j_{Y} is homotopic to d ⊕ (g_{2} ∘ j_{X}) where “⊕” denotes the principal action.

Write d componentwise as (d_{6}, d_{7}). Since f_{2}^{∗}(e_{9}) = 0, the next equation follows at once from 6.3:

0 = Sq^{3}d_{6} + Sq^{2}d_{7} + j_{X}^{∗}g_{2}^{∗}(e_{9}).

Lemma 6.4. If there is a lift g_{2} of g_{1} such that g_{2}^{∗}(e_{9}) = 0, then there is a lift g_{3} : X → E_{3} which induces a mod-2 cohomology isomorphism in dimension ≤ 9.

Proof. Clearly g_{3}^{∗} is an epimorphism. Furthermore H^{i}X and H^{i}E_{3} are vector spaces of the same (finite) dimension if i ≤ 9. The lemma follows.

The next lemma gives that such a lift exists and thus Theorem 6.1 follows. It is in the proof of this lemma where specific knowledge of the k-invariant e_{9} is used.

Lemma 6.5. If g_{2}^{∗}(e_{9}) ≠ 0, then g_{2}^{∗}(e_{9}) = u^{3} + vz.

Furthermore there is a map d′ : X → K_{6} × K_{7} with (d′)^{∗}(i_{6}) = 0, (d′)^{∗}(i_{7}) = uv and g′_{2} = d′ ⊕ g_{2} satisfies (g′_{2})^{∗}(e_{9}) = 0.

Proof. If g_{2}^{∗}(e_{9}) ≠ 0, consider the equation

(i)  0 = Sq^{3}d_{6} + Sq^{2}d_{7} + j_{X}^{∗}g_{2}^{∗}(e_{9}).

Since j_{x}^{∗} is a monomorphism, it will suffice to see how this last equation can be satisfied in H^{∗}A = H^{∗}BSD_{2^{n}}; the mod-2 cohomology ring is given in [2,8] as follows.

H^{∗}A = 𝔽_{2}[x, y, ū, p]/J

where degree (x) = 1 = degree(y), degree(ū) = 3, degree(p) = 4 and J is the ideal generated by

x^{2} + xy, x^{3}, xū, and ū^{2} + p(x^{2} + y^{2}).

The Steenrod operations are specified by

Sq^{1}ū = 0, Sq^{2}u = y^{2}ū + px + py, Sq^{3}ū = ū^{2} = Sq^{2}p,
Sq^{1}p = 0, and Sq^{4}p = p^{2}.

FINITE GROUPS OF 2-RANK TWO 17

By inspecting H^{6}A one observes that it is four dimensional with basis given by {x^{2}p, y^{2}p, ūy^{3}, y^{6}} and Sq^{3}H^{6}A = 0. Furthermore H^{7}A is four dimensional with basis {y^{3}p, ūy^{4}, y^{7}, ūp} and Sq^{2}y^{3}p. Finally H^{9}A is six dimensional with basis {xp^{2}, y^{5}p, y^{6}ū, y^{9}, y^{2}ūp, yp^{2}}. Thus equation (i) becomes

(ii)  0 = Sq^{2}d_{7} + j_{X}^{∗}g_{2}^{∗}(e_{9}).

By the above remarks g_{2}^{∗}e_{9} = b_{1}u^{3} + b_{2}vz for some b_{i} ≠ 0 and d_{7} = a_{1}, ūy^{4} + a_{2}y^{7} + a_{3}up for some a_{i} ≠ 0. Thus the next equation follows from (ii).

(iii) 0 = a_{1}(y^{5}p + y^{6}ū) + a_{2}(y^{9}) + a_{3}(yp^{2}) + b_{1}(ū^{3}) + b_{2}(p + y^{4})(y^{2}ū + px + py).

But ū^{3} = ūpx^{2} + ūpy^{2} = ūpy^{2} and (p + y^{4})(y^{2}ū + px + py) = py^{2}ū + xp^{2} + yp^{2} + y^{6}ū + y^{5}p as y^{4}px = y^{2}px^{3} = 0. Thus

(iv) 0 = a_{1}(y^{5}p+y^{6}ū)+a_{2}(y^{9})+a_{3}(yp^{2})+b_{1}(ūpy^{2})+b_{2}(py^{2}ū+xp+yp^{2}+y^{6}ū+y^{5}p)

and b_{1} = b_{2}.

Since g_{2}^{∗}(e_{9}) is assumed to be non-zero, it is thus equal to u^{3} + vz = Sq^{2}(uv). Define d′ : X → K_{6} × K_{7} by (d′)^{∗}(i_{6}) = 0 and (d′)^{∗}(i_{7}) = uv. Recall that e_{9}^{∗}(i_{9}) restricts to Sq^{3}(i_{6}) + Sq^{2}(i_{7}) in H^{∗}(K_{6} × K_{7}). Thus

(d′ ⊕ g_{2})^{∗}(e_{9}) = (d′)^{∗}(Sq^{3}i_{6} + Sq^{2}i_{7}) + g_{2}^{∗}(e_{9})

as e_{9} restricts to Sq^{3}i_{6} + Sq^{2}i_{7} by Lemma 5.4(i). But (d′)^{∗}(Sq^{3}i_{6} + Sq^{2}i_{7}) = Sq^{2}(uv) = u^{3} + vz and thus the lemma follows.

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