LOOP SPACE HOMOLOGY ASSOCIATED TO THE MOD 2 DICKSON INVARIANTS

RAN LEVI AND NORA SEELIGER

Abstract. The spaces BG_{2} and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has the classifying spaces of the finite groups BG_{2}(q), for an odd prime power q, and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article we compute the loop space homology of BG_{2}(q)_{2}^{∧} and BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.

It is well known [5, 18, 11] that the mod 2 Dickson invariants of rank n,

H^{∗}(Bℤ/2)^{n}, 𝔽_{2})^{GL_{n}(𝔽_{2})} = P [d_{1}, d_{2} …, d_{n}], |d_{i}| = 2^{n} − 2^{n−i}, i = 1 … n,

where invariants are taken with respect to the natural action of the general linear group, are realisable as the mod 2 cohomology of a space if and only if n ≤ 4. For n = 2, 3 the corresponding spaces are the classifying spaces of the Lie groups SO(3) and G_{2} respectively. The rank 4 Dickson invariants are realisable by a space BDI(4) which was constructed by Dwyer and Wilkerson [5], and is exotic in the sense that it is not homotopy equivalent to the classifying space of any compact Lie group.

In 1994 Benson [2] introduced a family of spaces BSol(q), one for each odd prime power q, closely related to BDI(4), which he claimed realised the exotic fusion patterns studied by Solomon [19] 20 years earlier. He obtained this family of spaces by considering the pullback of the diagram

BDI(4) ^{ψ^{q}⊤1} → BDI(4) × BDI(4) ←^{Δ} BDI(4),

where ψ^{q} is the degree q unstable Adams operation constructed by Notbohm [15]. In [12] the first named author and Oliver showed that the patterns studied by Solomon form saturated fusion systems. They also showed that these fusion systems admit associated centric linking systems, and thus give rise to a family of 2-local finite groups (see [3]). The “classifying spaces” of these 2-local finite groups are also named BSol(q), and are shown to coincide with Benson’s family [12, Thm 4.5]. The family BSol(q) provides one of the most interesting collections of p-local finite groups, in that they are all exotic, and to date the only exotic systems known at the prime 2. The module structure of H^{∗}(BSol(q), 𝔽_{2}) was calculated by Benson in [2], and the algebra and A_{2}-module structure were determined by Grbic [7], who also computed the Bockstein spectral sequence for these spaces.

In this article we consider the spaces BG_{2}(q)_{2}^{∧} and BSol(q) for all odd prime powers q, and present a calculation of their mod 2 loop space homology. There are strong results known on the homotopy type of ΩBG_{p}^{∧} when G is a finite group [10], but not much is known on loop spaces of exotic classifying spaces. Furthermore, as we shall see these two families exhibit very systematic behaviour (see Remark 1 at the end of the introduction) which might be worth exploring further. This motivates our calculations.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

Key words and phrases. Classifying spaces, p local finite groups, Loop space Homology.

Throughout this paper H_{∗}(−) and H^{∗}(−) will mean mod 2 homology and cohomology respectively. Different coefficients will always be explicitly specified. Subscripts on homology or cohomology classes will always denote their degrees. The letters P, E, T and Γ will be used to denote polynomial, exterior, tensor and divided power algebras respectively. By convention we will always use the notation T [x] to denote the tensor algebra on a single odd-dimensional generator, although over 𝔽_{2} the tensor algebra on a single generator in any dimension is graded commutative, and thus isomorphic to the polynomial algebra on the same generator. The spectral sequences of Serre, Bockstein and Eilenberg-Moore will be used in our calculations and will be abbreviated as SSS, BSS and EMSS respectively.

For any integer n, let ν_{2}(n) denote the highest power of 2 dividing n. Our first result determines the mod 2 loop space homology of BG_{2}(q)_{2}^{∧}, for any odd prime power q.

Theorem A. Fix an odd prime power q. Then

H_{∗}(ΩBG_{2}(q)_{2}^{∧}) ≅ P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}] ⊗ E[x_{3}, x_{5}] ⊗ P [z_{6}]/(z_{6}^{2}),

as modules over H_{∗}(ΩG_{2}) ≅ P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}]. Furthermore:

• The relations which determine the algebra extension are given by x_{3}^{2} = x_{5}^{2} = z_{6}^{2} = 0, [a_{2}, z_{6}] = a_{4}^{2}, [a_{4}, z_{6}] = b_{10} + a_{2}a_{4}^{2}, and [b_{10}, z_{6}] = a_{4}^{4}. All other commutators of generators are trivial.

• The reduced coproduct is given by Δ(a_{4}) = a_{2} ⊗ a_{2}, Δ(z_{6}) = x_{3} ⊗ x_{3}, while all other generators are primitive.

• The action of the dual Steenrod algebra is determined by

a_{2} x_{3} a_{4} x_{5} z_{6} b_{10}
Sq_{∗}^{1} 0 0 0 0 x_{5} 0
Sq_{∗}^{2} 0 0 a_{2} x_{3} 0 a_{4}^{2}

and the Steenrod axioms.

• The homology Bockstein spectral sequences are determined by

\begin{matrix}q ≡ 1(4) & a_{2} & x_{3} & a_{4} & x_{5} & a_{2}x_{3} & z_{6} & b_{10} & x_{5}z_{6} \\ Sq_{∗}^{1} & 0 & 0 & 0 & 0 & 0 & x_{5} & 0 & 0 \\ β_{∗}^{r_{2}} & 0 & a_{2} & 0 & − & 0 & − & 0 & b_{10} \\ β_{∗}^{r_{2}+1} & − & − & 0 & − & a_{4} & − & − & − \end{matrix}

where r_{2} = ν_{2}(q^{2} − 1). The rubrics marked “−” mean that the corresponding element vanishes in the respective page of the BSS.

Next we have the analogous result for BSol(q).

Theorem B. Fix an odd prime power q. Then

H_{∗}(ΩBSol(q)) ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}] ⊗ E[y_{7}, y_{11}, y_{13}] ⊗ P [y_{14}]/(y_{14}^{2}),

as a module over H_{∗}(ΩDI(4)) ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}]. Furthermore:

• The relations which determine the algebra extension are given by y_{7}^{2} = y_{11}^{2} = y_{13}^{2} = y_{14}^{2} = 0, [a_{6}, y_{14}] = b_{10}^{2}, [b_{10}, y_{14}] = c_{12}^{2}, and [c_{12}, y_{14}] = e_{26} + a_{6}b_{10}^{2}, and [e_{26}, y_{14}] = b_{10}^{4}. All other commutators of generators are trivial.

• The reduced coproduct is given by Δ(c_{12}) = a_{6} ⊗ a_{6}, and Δ(y_{14}) = y_{7} ⊗ y_{7}. All other generators are primitive.

• The action of the dual Steenrod algebra is determined by

a_{6} y_{7} b_{10} y_{11} c_{12} y_{13} y_{14} e_{26}
Sq_{∗}^{1} 0 0 0 0 0 0 y_{13} 0
Sq_{∗}^{2} 0 0 0 0 b_{10} y_{11} 0 c_{12}^{2}
Sq_{∗}^{4} 0 0 a_{6} y_{7} 0 0 0 0

and the Steenrod axioms.

• The homology Bockstein spectral sequence is determined by the table

a_{6} y_{7} b_{10} y_{11} c_{12} y_{13} y_{14} a_{6}y_{7} y_{13}y_{14}
Sq_{∗}^{1} 0 0 0 0 0 0 y_{13} 0 0
β_{∗}^{r_{4}−1} 0 0 0 b_{10} 0 − − 0 e_{26}
β_{∗}^{r_{4}}  0 a_{6} − − 0 − − 0 −
β_{∗}^{r_{4}+1} − − − − 0 − − c_{12} −

where r_{4} = ν_{2}(q^{4} − 1). The rubrics marked “−” mean that the corresponding element vanishes in the respective page of the BSS.

Remark 1. For each n ≥ 3 consider the Hopf algebra Ω_{n} over 𝔽_{2}, given as follows:

Ω_{n} =^{def} P [a]/a^{2} ⊗ P [b_{2}, …, b_{n−1}, c] ⊗ E[x_{1}, … x_{n−1}] ⊗ P [y]/y^{2},

as a module over P [a]/a^{2} ⊗P [b_{2}, …, b_{n−1}, c], where |a| = 2^{n−1} −2, |b_{k}| = 2^{n} −2^{n−k} −2, for 2 ≤ k ≤ n − 1, |c| = 2^{n+1} − 6, |x_{j}| = 2^{n} − 2^{n−j} − 1, for 1 ≤ j ≤ n − 1, and |y| = 2^{n} − 2. The relations which determine the algebra structure are x_{j}^{2} = 0 for all j, y^{2} = 0, [a, y] = b_{2}^{2}, [b_{k}, y] = b_{k+1}^{2}, 2 ≤ k ≤ n − 2, [b_{n−1}, y] = c^{2} + ab_{2}^{2}, [c, y] = b_{2}^{4}. All other commutators of generators are trivial. The reduced coproduct is given by Δ(b_{2}) = a ⊗ a and Δ(y) = x_{1} ⊗ x_{1}, while all other generators are primitive.

Furnish Ω_{n} with a coaction of the dual Steenrod algebra as follows Sq_{∗}^{2^{n}−2}(b_{2}) = a, Sq_{∗}^{2^{n−k}}(b_{k}) = b_{k−1}, 3 ≤ k ≤ n − 1, Sq_{∗}^{2^{n−k}}(x_{j}) = x_{j−1}, 2 ≤ j ≤ n − 1, Sq_{∗}^{1}(y) = x_{n−1}, and Sq_{∗}^{2}(c) = b_{n−1}^{2}.

Then, with the exception of the BSS structure, which we do not attempt to write in this generality, Theorems A and B, are given as particular cases of Ω_{n} for n = 3 and 4 respectively.

The paper is organised as follows. In Section 1 we record some basic facts which form a basis for our calculation. The loop space homology of BG_{2}(q)_{2}^{∧} and BSol(q) are calculated in Sections 2 and 3 respectively.

Some of the calculations presented here may be possible to carry out more easily using the general methods developed recently by Daisuke Kishimoto and Akira Kono in [8]. In this paper the authors develop a clever method of calculating the cohomology of the twisted loop space of a space X with respect to a self map f. The pullback space in the Quillen-Friendlander fibre square (1) below is an example of such a twisted loop space. The authors are very grateful to Kono for the interest he showed in our results, and for pointing out an error in the calculation of the algebra structures in an earlier version of this paper.

1. Preliminaries

Recall the Quillen-Friedlander fibre square [6] for groups of Lie type. If G is a complex reductive Lie group, and G(q) is the corresponding algebraic group over the field of q elements, then after completion at a prime p not dividing q, there is a homotopy fibre square

(1)  \begin{matrix}BG(q)_{p}^{∧} & \overline{→} & BG_{p}^{∧} \\ BG_{p}^{∧} & ↓^{π_{q}} & \overline{^{1⊤ψ^{q}} →} & BG_{p}^{∧} × BG_{p}^{∧} & ↓^{Δ} \end{matrix}

where ψ^{q} is the q-th unstable Adams operation, and Δ is the diagonal map. In particular, since for any self map f: X → X, hofib(X ^{1⊤f}→ X × X) ≃ ΩX, one has a fibration sequence of loop spaces and loop maps:

(2)  ΩBG(q)_{p}^{∧} → G_{p}^{∧} ^{f^{q}}→ G_{p}^{∧},

obtained by backing up the fibration in the left column of the pull back diagram above twice. All p-compact groups, in particular DI(4), admit unstable Adams operations of degree q, where q is a p-adic unit. The pull back space arising from the construction of a fibre square analogous to Diagram (1) for the operation ψ^{q} on DI(4) is defined by Benson [2] to be BSol(q).

We next record three well known cohomology algebras, which will be used in our calculation. As a convention we will use Roman alphabet to denote classes in mod 2 homology and cohomology, and Greek letters to denote classes in integral homology and cohomology. A good reference for the cohomology of Lie groups and their classifying spaces is [14]. Throughout the paper we will assume the reader is familiar with the basics of Hopf algebras. An excellent reference is [13].

The Spaces BSU(3), SU(3) and ΩSU(3). Recall:

(3)  H^{∗}(BSU(3)) ≅ P [u_{4}, u_{6}], and H^{∗}(BSU(3), ℤ) ≅ P [γ_{4}, γ_{6}],

both as algebras, with Sq^{2}(u_{4}) = u_{6}. Recall also that

(4)  H_{∗}(SU(3), ℤ) ≅ E[χ_{3}, χ_{5}]

as a Hopf algebra. An elementary calculation, using the EMSS, yields

(5)  H_{∗}(ΩSU(3), ℤ) ≅ P [α_{2}, α_{4}]

as an algebra, with the Hopf algebra structure determined by Δ(α_{4}) = α_{2} ⊗ α_{2}, where Δ denotes the reduced diagonal. Since these algebras are torsion free,

H_{∗}(SU(3)) ≅ H_{∗}(SU(3), ℤ) ⊗ 𝔽_{2} ≅ E[x_{3}, x_{5}],

and

H_{∗}(ΩSU(3)) ≅ H_{∗}(ΩSU(3), ℤ) ⊗ 𝔽_{2} ≅ P [a_{2}, a_{4}]

as Hopf algebras, with Sq_{∗}^{2}(x_{5}) = x_{3} and Sq_{∗}^{2}(a_{4}) = a_{2}.

The Spaces BG_{2}, G_{2} and ΩG_{2} There is an algebra isomorphism

(6)  H^{∗}(BG_{2}) ≅ P [u_{4}, u_{6}, t_{7}],

with Sq^{2}(u_{4}) = u_{6} and Sq^{1}(u_{6}) = t_{7}. These are the rank 3 mod 2 Dickson invariants. The group SU(3) is a subgroup of G_{2} and the inclusion induces the obvious projection on mod 2 cohomology. Recall also that

(7)  H_{∗}(G_{2}) ≅ E[x_{3}, x_{5}, x_{6}]

with Sq_{∗}^{1}(x_{6}) = x_{5}, Sq_{∗}^{2}(x_{5}) = x_{3}. The Hopf algebra structure is determined by Δ(x_{6}) = x_{3} ⊗ x_{3}.

Using the BSS for H_{∗}(G_{2}) we see that

H_{i}(G_{2}, ℤ) ≅ \begin{cases}ℤ & i = 0, 3, 11, 14 \\ ℤ/2 & i = 5, 8 \end{cases} ,

while all other homology groups vanish.

The integral loop space homology of G_{2} as a Hopf algebra was computed by Bott [4]. Reduction mod 2 is immediate and gives

(8)  H_{∗}(ΩG_{2}) ≅ P [a_{2}, a_{4}]/(a_{2}^{2}) ⊗ P [b_{10}]

as an algebra. (Notice also that in integral homology the square of the 2-dimensional generator is twice the 4-dimensional generator.) The Hopf algebra structure of H_{∗}(ΩG_{2}) is determined by Δ̄(a_{4}) = a_{2} ⊗ a_{2}, while b_{10} can be chosen to be primitive (if some choice of b_{10} is not primitive, then b′_{10} = b_{10} + a_{4}^{2}a_{2} is). It follows that in cohomology the dual of a_{4} is the square of the dual of a_{2}, and so one has Sq_{∗}^{2}(a_{4}) = a_{2}.

Dually, one has

H^{∗}(ΩG_{2}) ≅ P [a_{2}]/(a_{2}^{4}) ⊗ Γ[a_{8}, b_{10}],

where a_{2} and b_{10} are the duals of a_{2} and b_{10} respectively, and a_{8} is dual to a_{4}^{2}.

Deciding the action of the homology Steenrod squares on b_{10} requires a calculation. The authors are grateful to Akira Kono for sketching for them the following argument, which follows the lines of [8]. Let G̃_{2} denote the 3-connected cover of G_{2}. Thus there is a principal fibration

K(ℤ, 2) → G̃_{2} → G_{2}.

Using the mod-2 cohomology SSS for this fibration, an elementary computation shows that

H^{∗}(G̃_{2}) ≅ P [u_{8}] ⊗ E[y_{9}, z_{11}],

where u_{8} restricts to ι_{2}^{4} ∈ H^{8}(K(ℤ, 2)). The classes y_{9} and z_{11} correspond to the infinite cycles in the spectral sequence given by ι_{2}^{2}b_{5} and ι_{2}a_{3}^{3}, where a_{3} and b_{5} denote the generators of H^{∗}(G_{2}). By analysing the SSS for the fibration

G̃_{2} → G_{2} → K(ℤ, 3),

it is easy to see that Sq^{1}(u_{8}) = y_{9} and Sq^{2}(y_{9}) = z_{11}. Finally, using the spectral sequence for

ΩG_{2} → K(ℤ, 2) → G̃_{2}

one observes that ι_{2} restricts to a_{2}, and so ι_{2}^{4} restricts trivially, and is therefore the image of u_{8} under the inflation map. The rest of the spectral sequence is determined by letting y_{9} and z_{11} be the image of a_{8} and b_{10} under the transgression. In particular it follows that Sq^{2}(a_{8}) = b_{10}. Dually in homology we have Sq_{∗}^{2}(b_{10}) = a_{4}^{2}.

The Spaces BDI(4) and DI(4). Let BDI(4) denote the classifying space of the 2-compact group DI(4) [5]. Thus there is an algebra isomorphism

(9)  H^{∗}(BDI(4)) ≅ P [v_{8}, v_{12}, v_{14}, s_{15}],

with Sq^{4}(v_{8}) = v_{12}, Sq^{2}(v_{12}) = v_{14}, and Sq^{1}(v_{14}) = s_{15}. One also has

(10)  H_{∗}(DI(4)) ≅ E[y_{7}, y_{11}, y_{13}, y_{14}]

with Sq^{4}(y_{7}) = y_{11}, Sq^{2}(y_{11}) = y_{13}, Sq^{1}(y_{13}) = y_{14}. The Hopf algebra structure is determined by Δ(y_{14}) = y_{7} ⊗ y_{7}.

2. Loop space homology of BG_{2}(q)_{2}^{∧}.

Next, we calculate the mod 2 loop space homology of BG_{2}(q)_{2}^{∧}. To avoid an awkward notation, we use ˆ(−) to denote (−)_{2}^{∧} where it makes sense to do so.

To calculate the loop space homology of BG_{2}(q)_{2}^{∧}, consider first the Friedlander fibre square (1) for G_{2}. Taking iterated fibres on the left column of the square we get a sequence of fibrations

ΩĜ_{2} → ΩBG_{2}(q)_{2}^{∧} → Ĝ_{2} ^{f^{q}}→ Ĝ_{2} → BG_{2}(q)_{2}^{∧} → BĜ_{2},

where Ĝ_{2} denotes (G_{2})_{2}^{∧}. For the actual calculation, we use the SSS for the fibration

(11)  ΩĜ_{2} → ΩBG_{2}(q)_{2}^{∧} → Ĝ_{2}.

Thus, we start by calculating the map induced by f^{q} on homology.

Expanding the Friendlander fibre square, one sees that f^{q} is the composite

Ĝ_{2} ^{1⊤Ωψ^{q}} → Ĝ_{2} × Ĝ_{2} ^{−1⊤1} → Ĝ_{2} × Ĝ_{2} ^{μ}→ Ĝ_{2}.

There is an isomorphism of modules over P [u_{4}, u_{7}, u_{6}^{2}],

H^{∗}(BG_{2}) ≅ P [u_{4}, u_{7}, u_{6}^{2}] ⊗ E[u_{6}],

with Sq^{1}(u_{6}) = u_{7}, and Sq^{2}(u_{4}) = u_{6}. Considering this as a differential graded algebra with the differential given by Sq^{1}, and taking cohomology, we get the E_{2} term of the BSS for H^{∗}(BG_{2}),

E_{2} ≅ P [u_{4}, u_{6}^{2}],

which is concentrated in even degrees, and so E_{2} = E_{∞}. Hence the integral cohomology of BG_{2} is given by

H^{∗}(BG_{2}, ℤ) ≅ P [u_{4}, u_{7}, v_{12}]/(2u_{7}).

Notice that u_{4} and v_{12} are torsion free classes and (ψ^{q})^{∗}(u_{4}) = q^{2}u_{4}, while (ψ^{q})^{∗}(v_{12}) = q^{6}v_{12}. On the other hand, since the class u_{7} is of order 2, every element in the ideal it generates is of order 2, and since ψ^{q} is a mod-2 equivalence for q odd, the ideal generated by u_{7} is fixed under (ψ^{q})^{∗}.

Let χ_{i} denote a generator for H_{i}(G_{2}, ℤ) for those values of i, where the respective homology group is nontrivial. Using the known algebra structure of H_{∗}(G_{2}), it is easy to conclude that χ_{3}, χ_{5} and χ_{11} are indecomposable, and that χ_{3}χ_{5} = χ_{8} and χ_{3}χ_{11} = χ_{14}. Using the SSS for the path loop fibration over BG_{2} and naturality, we conclude that Ωψ_{∗}^{q}(χ_{3}) = q^{2}χ_{3} and Ωψ_{∗}^{q}(χ_{11}) = q^{6}χ_{11}, while Ωψ_{∗}^{q}(χ_{5}) = χ_{5} and Ωψ_{∗}^{q}(χ_{8}) = χ_{8}. In mod-2 homology both (ψ^{q})_{∗} and (Ωψ^{q})_{∗} behave like the identity.

Using the Künneth formula, we see that for n ≤ 11

H_{n}(G_{2} × G_{2}, ℤ) ≅ ⨁_{i+j=n} H_{i}(G_{2}, ℤ) ⊗ H_{j}(G_{2}, ℤ).

This, and the information about Ωψ_{∗}^{q} allows us to easily calculate f_{∗}^{q} on H_{∗}(G_{2}, ℤ). One has

f_{∗}^{q}(χ_{3}) = (q^{2} − 1)χ_{3}, f_{∗}^{q}(χ_{5}) = 0, and f_{∗}^{q}(χ_{11}) = (q^{6} − 1)χ_{11}.

Therefore on mod-2 homology f_{∗}^{q} is trivial.

Now consider fibration (11), which is induced from the path-loop fibration over G_{2} via the map f_{q}. Since f_{∗}^{q} is trivial on mod-2 homology, the SSS for (11) collapses at E^{2}, and it follows that for all odd q there is an isomorphism of modules over H_{∗}(ΩG_{2})

(12)  H_{∗}(ΩBG_{2}(q)_{2}^{∧}) ≅ {P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}]} ⊗ E[x_{3}, x_{5}] ⊗ P [z_{6}]/(z_{6}^{2}).

The structure of H_{∗}(ΩBG_{2}(q)_{2}^{∧}) as a module over the dual Steenrod algebra follows from the information we have about the two factors. Namely, Sq_{∗}^{2}(a_{4}) = a_{2}, Sq_{∗}^{2}(x_{5}) = x_{3}, Sq_{∗}^{2}(b_{10}) = a_{4}^{2}, and Sq_{∗}^{1}(z_{6}) = x_{5}.

The classes a_{2} and x_{3} are primitive with respect to the diagonal in H_{∗}(ΩBG_{2}(q)_{2}^{∧}) for dimensional reasons. The class x_{5} can be chosen to be primitive, since for any choice Δ(x_{5}) = A(a_{2} ⊗ x_{3} + x_{3} ⊗ a_{2}), for some A ∈ 𝔽_{2}, then x_{5} + Aa_{2}x_{3} is primitive, has the same action of Sq_{∗}^{2} as x_{5}, and represents the same class modulo H_{∗}(ΩG_{2}). One also has Δ(a_{4}) = a_{2} ⊗ a_{2}, since H_{∗}(ΩG_{2}) is a Hopf subalgebra. The class z_{6} can be chosen to have reduced diagonal x_{3} ⊗ x_{3}, since for any choice of representative, one has Δ(z_{6}) = x_{3}⊗x_{3}+B(a_{2}⊗a_{4}+a_{4}⊗a_{2}), and so z′_{6} = z_{6}+Ba_{2}a_{4} is congruent to z_{6} modulo H_{∗}(ΩG_{2}), has the same action of Sq_{∗}^{1} as z_{6}, and has the required diagonal. Finally the class b_{10} is primitive, since it is the image of the corresponding class in H_{∗}(ΩG_{2}).

Next, we compute the algebra extension. To do that, fix the representatives for x_{3}, x_{5} and z_{6} as above. Notice first that x_{3}^{2} = 0 and z_{6}^{2} = 0, since there are no primitives in the respective dimensions. Similarly, x_{5} is primitive, and so x_{5}^{2} = Ab_{10} for some A ∈ 𝔽_{2}. Applying Sq_{∗}^{2} to both sides we see that A = 0, so x_{5}^{2} = 0.

Next we systematically examine all the commutators involving x_{3}, x_{5} and z_{6}, as listed in the following table.

5  7  8  9  10  11  13  15 16
[a_{2}, x_{3}] [a_{2}, x_{5}] [a_{2}, z_{6}] [a_{4}, x_{5}] [a_{4}, z_{6}] [x_{5}, z_{6}] [x_{3}, b_{10}] [x_{5}, b_{10}] [z_{6}, b_{10}]
[x_{3}, a_{4}] [x_{3}, x_{5}] [x_{3}, z_{6}]

We will show that

[a_{2}, z_{6}] = a_{4}^{2}, [a_{4}, z_{6}] = b_{10} + a_{4}^{2}a_{2}, and [b_{10}, z_{6}] = a_{4}^{4},

while all the other commutators in the table vanish.

Observe first that every non-primitive class among the generators of H_{∗}(ΩBG_{2}(q)_{2}^{∧}) has a reduced diagonal consisting of a single element. The commutator of two primitives is always primitive, and if a is a primitive and Δ(b) = c ⊗ c, then

(13)  Δ([a, b]) = c ⊗ [a, c] + [a, c] ⊗ c.

Thus the commutators [a_{2}, x_{3}], [a_{2}, x_{5}], [x_{3}, x_{5}], [x_{3}, b_{10}] and [x_{5}, b_{10}] are automatically primitive.

Since [a_{2}, x_{3}] = Ax_{5}. Applying Sq_{∗}^{2} to both sides, it follows that A = 0, so a_{2} and x_{3} commute. Next, one has Δ(a_{4}) = a_{2} ⊗ a_{2}, Δ(z_{6}) = x_{3} ⊗ x_{3}, and [a_{2}, x_{3}] = 0. Applying (13) to [x_{3}, a_{4}] and [a_{2}, z_{6}], one sees that these commutators are also primitive. The only other primitive in dimension 7 is [a_{2}, x_{5}], and so [x_{3}, a_{4}] = A[a_{2}, x_{5}], for some A ∈ 𝔽_{2}. Similarly

[a_{2}, z_{6}] = B[x_{3}, x_{5}] + Ca_{4}^{2},

as [x_{3}, x_{5}] and a_{4}^{2} are the only other primitives in dimension 8. But Sq_{∗}^{1} applied to both sides yields 0 on the right hand side, and [a_{2}, x_{5}] on the left hand side. Hence a_{2} commutes with x_{5}, and consequently a_{4} commutes with x_{3}.

By a similar method we analyse [a_{4}, x_{5}], [x_{3}, z_{6}], [x_{3}, x_{5}] and [x_{5}, z_{6}]. First, by direct calculation, and the results already listed above,

Δ([a_{4}, z_{6}]) = a_{2} ⊗ [a_{2}, z_{6}] + [a_{2}, z_{6}] ⊗ a_{2}.

Since we do not yet know whether a_{2} commutes with z_{6}, we write the commutator [a_{4}, z_{6}] in general form as [a_{4}, z_{6}] = Ab_{10} + Ba_{4}^{2}a_{2}, for some A, B ∈ 𝔽_{2}. Applying Sq_{∗}^{1} to both sides we have [a_{4}, x_{5}] = 0. Now, since x_{3} and x_{5} commute, [x_{5}, z_{6}] is primitive. Since x_{5} commutes with both a_{2} and a_{4}, there are no other nonzero primitives in dimension 11, and so [x_{5}, z_{6}] = 0. Applying Sq_{∗}^{2} and then Sq_{∗}^{1}, we conclude that [x_{3}, z_{6}] and [x_{3}, x_{5}] both vanish.

Next, notice that [z_{6}, b_{10}] is a primitive class. Hence [z_{6}, b_{10}] = Aa_{4}^{4} for some A ∈ 𝔽_{2}. Applying Sq_{∗}^{1} and then Sq_{∗}^{2} to both sides, and using the fact that x_{5} and a_{4} commute, we conclude that b_{10} commutes with x_{5} and x_{3}.

It remains to analyse the commutators [a_{2}, z_{6}], [a_{4}, z_{6}] and [b_{10}, z_{6}]. To do that, recall from [7],

H^{∗}(BG_{2}(q)) = P [u_{4}, u_{6}, t_{7}, y_{3}, y_{5}]/(y_{5}^{2} + y_{3}t_{7} + y_{3}^{2}u_{4}, y_{3}^{4} + y_{5}t_{7} + y_{3}^{2}u_{6}).

Denote classes in H_{∗}(BG_{2}(q)) by adding a bar to the corresponding cohomology class, and consider the cobar spectral sequence for H_{∗}(ΩBG_{2}(q)_{2}^{∧}). Thus

E^{2} ≅ Cotor^{H_{∗}(BG_{2}(q)_{2}^{∧})}(𝔽_{2}, 𝔽_{2}) ≅ H_{∗}(T (Σ^{−1}(H_{∗}(BG_{2}(q)_{2}^{∧})), d_{E})),

where d_{E} is the differential on T (Σ^{−1}(H_{∗}(BG_{2}(q)_{2}^{∧}))) induced by the reduced diagonal [1]. If x, y ∈ H_{∗}(BG_{2}(q)_{2}^{∧}) are any classes, we denote the corresponding elements of the tensor algebra by [x] [y] etc., and their product in the tensor algebra structure by standard bar notation [x|y].

Consider the homology classes y_{5}^{2}, y_{3}t_{7} and y_{3}^{2}u_{4}. The corresponding reduced diagonals are y_{5} ⊗y_{5}, y_{3} ⊗t_{7} +t_{7} ⊗y_{3} and y_{3} ⊗y_{3}u_{4} +y_{3}u_{4} ⊗y_{3} +y_{3}^{2} ⊗u_{4} +u_{4} ⊗y_{3}^{2} respectively. Hence

d_{E}([y_{5}^{2}]) = [y_{5}]^{2}, d_{E}([y_{3}t_{7}]) = [[y_{3}], [t_{7}]], and d_{E}([y_{3}^{2}u_{4}]) = [[y_{3}], [y_{3}u_{4}]] + [[y_{3}^{2}], [u_{4}]].

On the other hand, since y_{5}^{2} + y_{3}t_{7} + y_{3}^{2}u_{4} = 0 in H^{∗}(BG_{2}(q)), we have

[y_{5}]^{2} + [[y_{3}], [t_{7}]] = [[y_{3}], [y_{3}u_{4}]] + [[y_{3}^{2}], [u_{4}]].

Furthermore, [y_{5}] and [y_{3}] and [t_{7}] are all cycles, which are permanent for dimensional reasons and hence represent a_{4}, a_{2} and z_{6} respectively in loop space homology. The equations above show that the expression [y_{5}]^{2} + [[y_{3}], [t_{7}]] is a boundary, and so we obtain the relation [a_{2}, z_{6}] = a_{4}^{2}. Next, notice that Sq_{∗}^{2}([a_{4}, z_{6}]) = [a_{2}, z_{6}] = a_{4}^{2}, while Δ([a_{4}, z_{6}]) = a_{2} ⊗ a_{4}^{2} + a_{4}^{2} ⊗ a_{2}. Hence we conclude that

[a_{4}, z_{6}] = b_{10} + a_{4}^{2}a_{2} = b_{10} + [a_{2}, z_{6}]a_{2}.

Finally, since b_{10} = [a_{4}, z_{6}] + a_{4}^{2}a_{2}, we calculate directly,

[b_{10}, z_{6}] = [[a_{4}, z_{6}] + a_{4}^{2}a_{2}, z_{6}] = [[a_{4}, z_{6}], z_{6}] + [a_{4}^{2}a_{2}, z_{6}] =

[a_{4}, z_{6}^{2}] + [[a_{2}, z_{6}]a_{2}, z_{6}] = 0 + [a_{2}, z_{6}]^{2} = a_{4}^{4}.

This completes the computation of the Hopf algebra structure. To summarise, we have shown that

H_{∗}(ΩBG_{2}(q)_{2}^{∧}) ≅ P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}] ⊗ E[x_{3}, x_{5}] ⊗ P [z_{6}]/(z_{6}^{2}),

as modules over H_{∗}(ΩBG_{2}). The relations which determine the algebra extension are given by x_{3}^{2} = x_{5}^{2} = z_{6}^{2} = 0, [a_{2}, z_{6}] = a_{4}^{2}, [a_{4}, z_{6}] = b_{10} + a_{2}a_{4}^{2}, and [b_{10}, z_{6}] = a_{4}^{4}. All other commutators of generators are trivial. The coproduct is given by Δ(a_{4}) = a_{2}⊗a_{2}, Δ(z_{6}) = x_{3} ⊗ x_{3}, and all other generators are primitive.

It remains to compute the Bockstein spectral sequence for H_{∗}(ΩBG_{2}(q)_{2}^{∧}). This is done by calculating the integral SSS for the fibration in the top row of the diagarm

\begin{aligned}ΩĜ_{2} & → & ΩBG_{2}(q)_{2}^{∧} & \overline{→} & Ĝ_{2} \\ ΩĜ_{2} & ‖ & \overline{→} & ∗ & ↓ & \overline{→} & Ĝ_{2} & ↓^{f^{q}} \end{aligned}.

We use naturality and the action of f_{∗}^{q} computed above. First, analyse the SSS for the bottom row, using the same notation we have been using before. One has d_{3}(χ_{3}) = a_{2}, and since in integral homology a_{2}^{2} = 2a_{4}, d_{3}(a_{2}χ_{3}) = 2a_{4}. Hence d_{3}(a_{4}^{k}χ_{3}) = a_{4}^{k}a_{2}, and d_{3}(a_{4}^{k}a_{2}χ_{3}) = 2a_{4}^{k+1}. Since χ_{8} = χ_{3}χ_{5}, it follows that d_{3}(χ_{8}) = a_{2}χ_{5}, but d_{3}(a_{2}χ_{8}) = 0. Similarly, d_{3}(χ_{14}) = a_{2}χ_{11}. In addition one must have d_{3}(χ_{11}) = a_{2}χ_{8}, since otherwise a_{2}χ_{8} will be an infinite cycle. This determines d_{3}. The next nontrivial differential is d_{5}, which takes χ_{5} isomorphically to a_{4}, which in E^{5} is a class of order 2. Finally d_{11} takes χ_{11} to b_{10}, and E^{12} = E^{∞}. Now, using naturality of the spectral sequence and our knowledge of f_{∗}^{q}, it follows that in the SSS for the top row in the diagram, d_{3}(χ_{3}) = (q^{2} − 1)a_{2}, d_{5}(χ_{5}) = 0, and d_{11}(χ_{11}) = (q^{6} − 1)b_{10}. This information suffices for the computation of the BSS. The integral calculation yields in particular the observation that a_{2} is a class of order (q^{2} − 1), while b_{10} has order (q^{6} − 1).

Now, consider H_{∗}(ΩBG_{2}(q)_{2}^{∧}) as a module over H_{∗}(ΩG_{2}) as in (12):

H_{∗}(ΩBG_{2}(q)_{2}^{∧}) ≅ {P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}]} ⊗ E[x_{3}, x_{5}] ⊗ P [z_{6}]/(z_{6}^{2}).

Taking the Sq_{∗}^{1} homology, one has

E^{2} ≅ P [a_{2}]/(a_{2}^{2}) ⊗ P [a_{4}, b_{10}] ⊗ E[x_{3}, h_{11}].

where the class h_{11} is represented by the Sq_{∗}^{1} cycle x_{5}z_{6}. Notice that for any odd q, r_{6} = ν_{2}(q^{6} − 1) = ν_{2}(q^{2} − 1) = r_{2}. Hence the next nontrivial Bockstein operator is β_{∗}^{r_{2}}(x_{3}) = a_{2}, and β_{∗}^{r_{2}}(h_{11}) = b_{10}. It now follows that β_{∗}^{r_{2}+1}(a_{2}x_{3}) = a_{4}, and E^{r_{2}+2} = E^{∞}.

The results are summarised in the following table.

\begin{matrix}q ≡ 1(4) & a_{2} & x_{3} & a_{4} & x_{5} & a_{2}x_{3} & z_{6} & b_{10} & x_{5}z_{6} \\ Sq_{∗}^{1} & 0 & 0 & 0 & 0 & 0 & x_{5} & 0 & 0 \\ β_{∗}^{r_{2}} & 0 & a_{2} & 0 & − & 0 & − & 0 & b_{10} \\ β_{∗}^{r_{2}+1} & − & − & 0 & − & a_{4} & − & − & − \end{matrix}

This completes the proof of Theorem A.

3. Loop space homology of BSol(q)

For any odd prime power q, the 2-local finite group Sol(q) is defined in [12]. The starting point is a family of saturated fusion systems F_{Sol(q)} over the Sylow 2-subgroup of Spin_{7}(q). These fusion systems were originally defined by Ron Solomon [19] as a part of his contribution to the classification of finite simple groups. He did not use the language of fusion systems, but essentially presented the entire family and studied its general behaviour. In [2], Benson gave a construction of a family of spaces BSol(q), which he claimed realise the fusion patterns defined by Solomon. These spaces are given as the pullback spaces in the diagram

(14)  \begin{matrix}BSol(q) & \overline{→} & BDI(4) \\ BDI(4) & ↓ & \overline{^{ψ^{q}⊤1}→} & BDI(4)^{×2} & ↓^{Δ} \end{matrix}

where Δ is the diagonal and ψ^{q} is the degree q unstable Adams operation on BDI(4) constructed by Notbohm [15]. The paper [12] unifies the two constructions. On one hand it is shown that the fusion patterns defined by Solomon are indeed saturated fusion systems, each of which admits an associated centric linking system L_{Sol(q)}, and on the other hand that the classifying spaces of the corresponding 2-local finite groups BSol(q) =^{def} |L_{Sol(q)}|_{2}^{∧} coincide with the spaces constructed by Benson, whose approach allows a calculation of the mod 2 cohomology of BSol(q), as demonstrated in [7]. We shall also utilize Benson’s pullback diagram in the current work.

In what follows we will denote ΩBDI(4) by DI(4) (not to be confused with the notation DI(n) which is sometimes used to denote the rank n algebra of Dickson invariants).

As before, one has a fibration of loop spaces and loop maps

ΩDI(4) → ΩBSol(q) → DI(4),

resulting from looping the left hand side column in Benson’s pull back diagram (14). Thus our first task is to compute the loop space homology of DI(4).

Proposition 3.1. There is an isomorphism of Hopf algebras

H_{∗}(ΩDI(4)) ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}],

where a_{6}, b_{10} and e_{26} are primitive, and Δ(c_{12}) = a_{6} ⊗ a_{6}. The action of the dual Steenrod algebra is determined by Sq_{∗}^{4}(b_{10}) = a_{6}, Sq_{∗}^{2}(c_{12}) = b_{10}, and Sq_{∗}^{2}(e_{26}) = c_{12}^{2}.

Proof. Consider the homology EMSS for the path-loop fibration over DI(4). The E^{2} term is given by

Cotor^{H_{∗}(DI(4))}(𝔽_{2}, 𝔽_{2}) ≅ Ext_{H^{∗}(DI(4))}(𝔽_{2}, 𝔽_{2}).

The isomorphism holds since H_{∗}(DI(4)) is of finite type. To calculate the right hand side, consider the differential graded Hopf algebra

P_{∗} =^{def} (P [x_{7}]/(x_{7}^{4}) ⊗ E[y_{11}, z_{13}]) ⊗ (P [â_{6}]/(â^{4}) ⊗ Γ[b̂_{10}, t̂_{24}, ê_{26}]),

where the left factor (which is isomorphic to H^{∗}(DI(4))), is primitively generated, and the right factor is the dual of the Hopf algebra P [a_{6}]/(a^{2}) ⊗ P [b_{10}, c_{12}, e_{26}], where all generators but c_{12} are primitive, and Δ(c_{12}) = a_{6}⊗a_{6}. We denote by γ_{k}(b̂) the generator of Γ[b̂_{10}] in dimension 10k, where γ_{1}(b̂) = b_{10}, and γ_{0}(b̂) = 1. We use similar notation for the generators corresponding to ĉ_{12} and ê_{26}. Thus as an 𝔽_{2}-algebra Γ[b̂_{10}, t̂_{24}, ê_{26}] can be written as

⨂_{n≥0} E[γ_{2^{n}}(b̂), γ_{2^{n}}(t̂), γ_{2^{n}}(ê)],

where we omit subscripts for short. The differential on P_{∗} is given on generators by

• d(x) = d(y) = d(z) = 0,
• d(â) = x,
• d(γ_{2^{n}}(b̂)) = yγ_{2^{n}−1}(b̂) ,

• d(â^{2}) = z,

• d(γ_{2^{n}}(t̂)) = zâ^{2}γ_{2^{n}−1}(ĉ) , and

• d(γ_{2^{n}}(ê)) = x^{3}âγ_{2^{n}−1}(ê),

and is required to satisfy the Leibniz rule on products. Notice that the differential is, in particular, a map of graded algebras over H^{∗}(DI(4)) = P [x_{7}]/(x_{7}^{4}) ⊗ E[y_{11}, z_{13}]. In particular P_{∗} is a free differential graded H^{∗}(DI(4))-module. Furthermore, as a chain complex it is split as the tensor product of the following acyclic subcomplexes

{P [x_{7}]/(x_{7}^{4}) ⊗ P [â_{6}]/(â_{6}^{4}) ⊗ E[z_{13}] ⊗ Γ[t̂_{24}, ê_{26}]} ⊗ {E[y_{11}] ⊗ Γ[b̂_{10}]}.

Hence P_{∗} is a free H^{∗}(DI(4))-resolution of 𝔽_{2}.

Since P_{∗} is a free differential graded H^{∗}(DI(4))-module, it is immediate that

(15) E^{2} = Ext_{H^{∗}(DI(4))}(𝔽_{2}, 𝔽_{2}) = H^{∗}(Hom_{H^{∗}(DI(4))}(P_{∗}, 𝔽_{2})) ≅

Hom_{𝔽_{2}}(P [â_{6}]/(â_{6}^{4}) ⊗ Γ[b̂_{10}, t̂_{24}, ê_{26}], 𝔽_{2}) ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}].

Since this module is concentrated in even degrees, there are no possible nontrivial differentials, so E^{2} = E^{∞}. By inspection of the SSS for the path-loop fibration over DI(4), one easily obtains Sq_{∗}^{4}(b_{10}) = a_{6} and Sq_{∗}^{2}(c_{12}) = b_{10}.

To calculate further Steenrod operations, we use a similar trick to the one used to ΩG_{2}. Let X denote the 7-connected cover of DI(4). Thus there is a fibration

X → DI(4) ^{x_{7}}→ K(ℤ, 7).

To calculate H^{∗}(X) we use Smith’s Big Collapse Theorem [17]. Notice that x_{7}^{∗} is onto, and its kernel consists of the ideal generated by all the polynomial generators of H^{∗}(K(ℤ, 7)), different from ι_{7}, Sq^{4}ι_{7}, and Sq^{6}ι_{7}, along with ι_{7}^{4}, (Sq^{6}ι_{7})^{2}, and (Sq^{4}ι_{7})^{2}. This collection of generators forms a regular sequence in H^{∗}(K(ℤ, 7)), and so the conditions of Smith’s theorem are satisfied, and H^{∗}(X) can be written additively as the exterior algebra on infinitely many generators, corresponding in a 1-1 fashion to generators listed above, but with a dimension shift one down. Let ϵ_{27}, τ_{25}, and σ_{21} be the elements in H^{∗}(X) corresponding to ι_{7}^{4}, (Sq^{6}ι_{7})^{2}, and (Sq^{4}ι_{7})^{2} respectively. Notice that Sq^{2}(τ_{25}) = ϵ_{27} and Sq^{4}(σ_{21}) = τ_{25}. Write

(16)  H^{∗}(X) ≅ E[σ_{21}, τ_{25}, ϵ_{27}] ⊗ E[k^{2}, k^{3}, …, k^{I}, …],

where k^{I} in dimension |Sq^{I}ι_{7}| − 1 stands for the exterior generator corresponding to Sq^{I}ι_{7}, for each I such that Sq^{I}ι_{7} ∈ Ker(x_{7}^{∗}).

Now, consider the cohomology SSS for the principal fibration

ΩDI(4) ^{j}→ K(ℤ, 6) ^{π}→ X.

Notice first that by naturality of the spectral sequence, the second factor in (16) injects into H^{∗}(K(ℤ, 6)) via π^{∗}, while the classes σ_{21}, τ_{25} and ϵ_{27} are all in Ker(π^{∗}). Furthermore, one has j^{∗}(ι_{6}) = a_{6}, and so j^{∗}(Sq^{4}ι_{6}) = b̂_{10}. The bottom dimensional class in H^{∗}(ΩDI(4)) which is not hit by j^{∗}, is γ_{2}(b̂_{10}), which is therefore transgressive. Hence d(γ_{2}(b̂_{10})) = σ_{21}, and it follows that d(Sq^{4}γ_{2}(b̂_{10})) = Sq^{4}(σ_{21}) = τ_{25}, while d(Sq^{6}γ_{2}(b̂_{10})) = d(Sq^{2,4}γ_{2}(b̂_{10})) = Sq^{2}τ_{25} = ϵ_{27}. Hence Sq^{4}γ_{2}(b̂_{10}) = t̂_{24}, and Sq^{2}t̂_{24} = ê_{26}. Dually, in homology, Sq_{∗}^{2}(e_{26}) = c_{12}^{2}, and Sq_{∗}^{4}(c_{12}^{2}) = b_{10}^{2}.

Next, we work out the Pontryagin algebra structure. Since b_{10}, c_{12} and e_{26} are elements of infinite height in E^{∞}, they represent elements of infinite height in homology. Hence, it remains only to check whether a_{6}^{2} = c_{12}. But in cohomology one has ĉ_{12} = Sq^{2}(b̂_{10}) = Sq^{2}Sq^{4}(â_{6}) = â_{6}^{2}. Hence in homology Δ(c_{12}) = a_{6} ⊗ a_{6}. But since a_{6} is primitive, so is a_{6}^{2}, and since H_{12}(ΩDI(4)) is 1-dimensional, it follows that a_{6}^{2} = 0. This completes the calculation of the algebra structure.

The classes a_{6} and b_{10} are primitive for dimension reasons, and we have already computed the reduced diagonal of c_{12}. Thus it remains to compute the reduced diagonal of e_{26}. Notice that H_{26}(ΩDI(4)) is 2-dimensional, generated additively by e_{26} and a_{6}b_{10}^{2}, and that e_{26} can be modified by an additive summand of a_{6}b_{10}^{2} without changing the algebra structure. For any choice of e_{26} one has

Δ(e_{26}) = A(a_{6}b_{10} ⊗ b_{10} + b_{10} ⊗ a_{6}b_{10}) + B(a_{6} ⊗ b_{10}^{2} + b_{10}^{2} ⊗ a_{6}),

for some A, B ∈ 𝔽_{2}. But B is the coefficient of Δ(a_{6}b_{10}^{2}), and so by modifying the choice of e_{26} if necessary, we may assume that B = 0. Furthermore, if A ≠ 0, then H_{26}(ΩDI(4)) contains no primitive class, and so dually every class in H^{26}(ΩDI(4)) is decomposable, which is clearly impossible. Hence A = 0, and there is a choice for the class e_{26} which is primitive.

This completes the calculation of H_{∗}(ΩDI(4)) as a Hopf algebra and hence the proof of the proposition. □

Dually, the cohomology Hopf algebra is given by

H^{∗}(ΩDI(4)) ≅ P [â_{6}]/(â_{6}^{4}) ⊗ Γ[b̂_{10}, t̂_{24}, ê_{26}].

We are now ready to start the calculation of H_{∗}(ΩBSol(q)). Consider the fibration

ΩDI(4) → ΩBSol(q) → DI(4).

The homology SSS associated to this fibration is a spectral sequence of Hopf algebras over H_{∗}(ΩDI(4)), whose E^{2}-page has the form

E_{∗,∗}^{2} = H_{∗}(ΩDI(4)) ⊗ H_{∗}(DI(4)) ≅

{P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}]} ⊗ {E[y_{7}, y_{11}, y_{13}] ⊗ P [y_{14}]/(y_{14}^{2})} .

Since BSol(q) is 6-connected, π_{6}(ΩBSol(q)) ≅ H_{7}(BSol(q), ℤ). Thus H_{7}(BSol(q), ℤ) is a finite 2-group, and so d_{7}(y_{7}) = 0. Thus d_{7} vanishes on all the y_{i} (y_{14} by considering the dual cohomology spectral sequence, and the other generators by dimension reasons). The next possible nonvanishing differential is d_{11}. Considering the dual cohomology SSS, d_{11}(b̂_{10}) = d_{11}(Sq^{4}â_{6}) = Sq^{4}d_{7}(â_{6}) = 0. Hence in homology d_{11}(y_{11}) = 0. Similarly d_{13}(y_{13}) = 0. Hence the spectral sequence collapses at E^{2}, and

(17) H_{∗}(ΩBSol(q)) ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}] ⊗ E[y_{7}, y_{11}, y_{13}] ⊗ P [y_{14}]/(y_{14}^{2})

as a module over H_{∗}(ΩDI(4)).

The loop space homology H_{∗}(ΩDI(4)) is contained in H_{∗}(ΩBSol(q)) as a Hopf subalgebra. The classes y_{7} and y_{11} are primitive for dimension reason. For y_{13} one has Δ(y_{13}) = A(a_{6} ⊗y_{7} +y_{7} ⊗a_{6}) = AΔ(a_{6}y_{7}) for some A ∈ 𝔽_{2}. Hence y_{13} can be chosen to be primitive. Finally Δ(y_{14}) = y_{7} ⊗ y_{7}. This completes the description of the coalgebra structure on H_{∗}(ΩBSol(q)).

Since y_{7} and y_{11} and y_{13} are primitive, so are their squares. Since there are no nontrivial primitives in the respective dimensions, except for e_{26}, we conclude at once that y_{7}^{2} = y_{11}^{2} = 0, while y_{13}^{2} = Ae_{26} for some A ∈ 𝔽_{2}. But Sq_{∗}^{2}(e_{26}) = c_{12}^{2}, while Sq_{∗}^{2}(y_{13}^{2}) = 0, so A = 0, and therefore y_{13}^{2} = 0. Finally, y_{14}^{2} is primitive, and since there are no nontrivial primitives in dimension 28, it follows that y_{14}^{2} = 0.

Next, we calculate all the commutators involving the classes y_{i}. The results are summarised in the following table, while the calculations are below. Each entry in the table stands for the commutator [Column, Row].

a_{6} y_{7} b_{10} y_{11}  c_{12}  y_{13} y_{14} e_{26}
y_{7} 0 0 0 0  0  0 0 0
y_{11} 0 0 0 0  0  0 0 0
y_{13} 0 0 0 0  0  0 0 0
y_{14} b_{10}^{2} 0 c_{12}^{2} 0 e_{26} + a_{6}b_{10}^{2} 0 0 b_{10}^{4}

Since a_{6} and y_{7} are both primitive, so is their commutator, and so [a_{6}, y_{7}] = Ay_{13}, for some A ∈ 𝔽_{2}. Applying Sq^{2} to both side, we see that A = 0, and so a_{6} and y_{7} commute. Thus it follows that [a_{6}, y_{14}] is primitive, and so must be a multiple of b_{10}^{2}. Applying successive dual Steenrod squares

[a_{6}, y_{14}] ^{Sq_{∗}^{1}}→ [a_{6}, y_{13}] ^{Sq_{∗}^{2}}→ [a_{6}, y_{11}] ^{Sq_{∗}^{4}}→ [a_{6}, y_{7}] ,

we conclude that all these commutators, with the possible exception of [a_{6}, y_{14}] itself, vanish.

The class y_{7} clearly commutes with itself, and its commutators with all other y_{i} are primitive. This implies at once that [y_{7}, y_{11}] and [y_{7}, y_{14}] vanish, and [y_{7}, y_{13}] = Sq_{∗}^{1}[y_{7}, y_{14}] = 0 as well.

The class b_{10} commutes with y_{7} for dimension reasons, and so [b_{10}, y_{14}] is primitive, and is therefore a multiple of c_{12}^{2}. Applying dual Steenrod squares we have

[b_{10}, y_{14}] ^{Sq_{∗}^{1}}→ [b_{10}, y_{13}] ^{Sq_{∗}^{2}}→ [b_{10}, y_{11}] ,

which show that [b_{10}, y_{13}] and [b_{10}, y_{11}] vanish.

Since y_{11} commutes with y_{7}, the commutator [y_{11}, y_{14}] is primitive and Sq_{∗}^{1}[y_{11}, y_{14}] = [y_{11}, y_{13}]. But there are no nontrivial primitives in dimension 25, and so both commutators vanish.

The classes c_{12} and y_{7} commute, since there are no 19 dimensional nonzero primitives, and so [y_{14}, c_{12}] is primitive. Thus [y_{14}, c_{12}] is a multiple of e_{26}. As before, we have

[c_{12}, y_{14}] ^{Sq_{∗}^{1}}→ [c_{12}, y_{13}] ^{Sq_{∗}^{2}}→ [c_{12}, y_{11}] ,

which show that all commutators involving c_{12}, except possibly [c_{12}, y_{14}] vanish.

We already established that y_{13} commutes with y_{7} and y_{11}, and it commutes with y_{14} as well since the commutator [y_{13}, y_{14}] is primitive. This also shows that all commutators with y_{14} with other y_{i} vanish.

Finally, [e_{26}, y_{7}] vanishes for lack of primitives in dimension 33. Thus [e_{26}, y_{14}] is primitive and one has a chain of operations

[e_{26}, y_{14}] ^{Sq_{∗}^{1}}→ [e_{26}, y_{13}] ^{Sq_{∗}^{2}}→ [e_{26}, y_{11}] .

The only nonzero primitive in dimension 40 is b_{10}^{4}, and so [e_{26}, y_{14}] = Ab_{10}^{4} for some A ∈ 𝔽_{2}. Applying Sq_{∗}^{1} and Sq_{∗}^{2,1} to b_{10}^{4}, we conclude that [e_{26}, y_{13}] and [e_{26}, y_{11}] vanish.

It remains to evaluate the commutators of y_{14} with the generators of H_{∗}(ΩDI(4)). To do that, we consider the cobar spectral sequence for H_{∗}(ΩBSol(q)), with

E^{2} = Cotor^{H_{∗}(BSol(q))}(𝔽_{2}, 𝔽_{2}) ≅ H_{∗}(T (Σ^{−1}H_{∗}(BSol(q))), d_{E}),

where d_{E} is the external differential on the cobar construction, induced by the reduced diagonal in H_{∗}(BSol(q)).

Recall from [7]

H^{∗}(BSol(q)) ≅ P [u_{8}, u_{12}, u_{14}, u_{15}, t_{7}, t_{11}, t_{13}]/I,

where I is the ideal generated by the polynomials r_{1} = t_{11}^{2} + u_{8}t_{7}^{2} + u_{15}t_{7}, r_{2} = t_{13}^{2} + u_{12}t_{7}^{2} + u_{15}t_{11} and r_{3} = t_{7}^{4} + u_{14}t_{7}^{2} + u_{15}t_{13}.

As for BG_{2}(q), we denote classes in H_{∗}(BSol(q)) by its dual cohomology class decorated by a bar. If a ∈ H^{∗}(BSol(q)) is any class, then the corresponding tensor algebra generator will be denoted by [a], while products of these generators will be written using the usual bar notation [a_{1}|a_{2}| ⋯ |a_{n}]. Thus the relation r_{1} translate to the following equation in the E^{1} page of the cobar spectral sequence.

0 = d_{E}([t_{11}^{2}] + [u_{8}t_{7}^{2}] + [u_{15}t_{7}]) = [t_{11}]^{2} + d_{E}([u_{8}t_{7}^{2}]) + [[u_{15}], [t_{7}]].

The classes [t_{11}], [u_{15}] and [t_{7}] are easily seen to be the permanent cycles in the spectral sequence corresponding to b_{10}, y_{14} and a_{6} respectively. Hence we obtain the relation

[a_{6}, y_{14}] = b_{10}^{2}.

Next, notice that one has

[c_{12}, y_{14}] ^{Sq_{∗}^{2}}→ [b_{10}, y_{14}] ^{Sq_{∗}^{4}}→ [a_{6}, y_{14}] .

The commutator [b_{10}, y_{14}] is primitive, while

Δ([c_{12}, y_{14}]) = a_{6} ⊗ [a_{6}, y_{14}] + [a_{6}, y_{14}] ⊗ a_{6} = a_{6} ⊗ b_{10}^{2} + b_{10}^{2} ⊗ a_{6}.

Hence we conclude that

[b_{10}, y_{14}] = c_{12}^{2} and [c_{12}, y_{14}] = e_{26} + a_{6}b_{10}^{2} = e_{26} + a_{6}[a_{6}, y_{14}].

Finally, by the previous calculations,

[e_{26}, y_{14}] = [[c_{12}, y_{14}], y_{14}] + [a_{6}[a_{6}, y_{14}], y_{14}] = [a_{6}, y_{14}]^{2} = b_{10}^{4}.

This completes the calculation of H_{∗}(ΩBSol(q)) as a Hopf algebra over the dual Steenrod algebra.

Our final task is the calculation of the BSS for ΩBSol(q). Using the known structure of H^{∗}(BDI(4)) and H^{∗}(DI(4)) and the corresponding BSS, we conclude that

H^{∗}(BDI(4), ℤ) ≅ P [υ_{8}, υ_{12}, υ_{15}, υ_{28}]/(2υ_{15}),

which allows us to conclude that (ψ^{q})^{∗}(υ_{2i}) = q^{i}υ_{2i}. Also, similarly to the corresponding computation for G_{2},

H_{i}(DI(4), ℤ) ≅ \begin{cases}ℤ & i = 0, 7, 11, 18, 27, 34, 38, 45 \\ ℤ/2 & i = 13, 20, 24, 31 \end{cases} ,

while all other homology groups vanish. Denote homology classes by χ_{i}, where i corresponds to the dimension. The by inspection of the mod 2 homology structure, it is easy to conclude that χ_{7}, χ_{11}, and χ_{27} are the indecomposable among the torsion free classes, while the only torsion indecomposable class is χ_{13}.

Consider the fibration

ΩDI(4) → ΩBSol(q) → DI(4).

The integral homology SSS calculation of this fibration is similar to the one done for G_{2}. Since H_{∗}(ΩDI(4), ℤ) is torsion free, the E^{2} page of the spectral sequence is the tensor product of the homologies of base and fibre. One has a commutative diagram of fibrations

\begin{aligned}ΩDI(4) & → & ΩBSol(q) & → & DI(4) \\ ΩDI(4) & ‖ & \overline{→} & ∗ & ↓ & \overline{→} & DI(4) & ↓^{f^{q}} \end{aligned}

where f^{q} is any map in the homotopy class of the composite

DI(4) ^{Δ}→ DI(4) × DI(4) ^{Ωψ^{q}×(−1)} → DI(4) × DI(4) ^{μ}→ DI(4).

It is easy to verify that

H_{n}(DI(4) × DI(4), ℤ) ≅ ⨁_{i+j=n} H_{i}(DI(4), ℤ) ⊗ H_{j}(DI(4), ℤ)

for n ≤ 32. Hence it follows that for i = 4, 6, 14, one has (f^{q})_{∗}(χ_{2i−1}) = (q^{i}−1)χ_{2i−1}. In the SSS for the path-loop fibration over DI(4), one has d_{7}(χ_{7}) = a_{6}, d_{7}(a_{6}χ_{7}) = 2c_{12}, d_{11}(χ_{11}) = b_{10}, d_{13}(χ_{13}) = c̄_{12} (the class of c_{12} modulo Im(d_{7})), and d_{27}(χ_{27}) = e_{26}. Thus by commutativity of the diagram above, and naturality of the SSS one has in the integral homology SSS for the top row, d_{7}(χ_{7}) = (q^{4} − 1), d_{11}(χ_{11}) = (q^{6} − 1)b_{10}, and d_{27}(χ_{27}) = (q^{14} − 1)e_{26}. Setting q = 4k ± 1 and r_{i} = ν_{2}(q^{i} − 1), and performing the necessary arithmetics, we see that r_{6} = r_{14} = ν_{2}(k) + 3 = r_{4} − 1.

We are now ready to compute the Bockstein spectral sequence for H_{∗}(ΩBSol(q)), which is a spectral sequence of modules over H_{∗}(ΩDI(4)), and so we use the module structure given by (17) in the calculation. The first page of the spectral sequence is determined by Sq_{∗}^{1}(y_{14}) = y_{13}. Thus

E^{2} ≅ P [a_{6}]/(a_{6}^{2}) ⊗ P [b_{10}, c_{12}, e_{26}] ⊗ E[y_{7}, y_{11}, h_{27}],

where h_{27} corresponds to the infinite cycle in E^{1} given by y_{13}y_{14}. By the integral homology calculation above, the next nontrivial differential is β_{∗}^{r_{4}−1}, and one has β^{r_{4}−1}(y_{11}) = c_{10}, while β^{r_{4}−1}(h_{27}) = e_{26}. Next, we have β_{∗}^{r_{4}}(y_{7}) = a_{6}, and since Δ(c_{12}) = a_{6} ⊗ a_{6}, it follows that β_{∗}^{r_{4}+1}(a_{6}y_{7}) is defined and is equal to c_{12}, provided that a_{6}y_{7} ≠ 0 in E^{r_{4}+1}, which is obvious since already E^{2} does not contain a nonzero class in dimension 14. This completes the calculation of the BSS for ΩBSol(q), which takes the form

\begin{aligned}& a_{6} & y_{7} & b_{10} & y_{11} & c_{12} & y_{13} & y_{14} & a_{6}y_{7} & y_{13}y_{14} \\ Sq_{∗}^{1} & 0 & 0 & 0 & 0 & 0 & 0 & y_{13} & 0 & 0 \\ β_{∗}^{r_{4}−1} & 0 & 0 & 0 & b_{10} & 0 & − & − & 0 & e_{26} \\ β_{∗}^{r_{4}} & 0 & a_{6} & − & − & 0 & − & − & 0 & − \\ β_{∗}^{r_{4}+1} & − & − & − & − & 0 & − & − & c_{12} & − \end{aligned} .

References

[1] J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci., USA, 42 (1956) 409-412.

[2] D. J. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometry and cohomology in group theory (Durham 1994), LMS Lect. Notes Series 252, (1998) 10–23.

[3] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, Jour. Amer. Math. Soc. 16 (2003), 779–856.

[4] R. Bott, The space of loops on a Lie group, Michigan Math. J., 5 (1958), 35-61.

[5] W. Dwyer, C. Wilkerson, A new finite loop space at the prime two, Jour. Amer. Math. Soc. 6 (1993), no. 1, 37–64.

[6] E. Friedlander, Etale homotopy theory of simplicial schemes, Ann. Math. Stud. 104, Princeton University Press, (1982).

[7] J. Grbic, The cohomology of certain 2-local finite groups, Manuscripta Math. 120 (2006) 307–318.

[8] D. Kishimoto, A. Kono, On the cohomology of free and twisted loop spaces, Jour. Pure and App. Alg. (2009) doi: 10.1016/j.jpaa.2009.07.006

[9] A. Kono, K, Kozima, The mod 2 homology of the space of loops on the exceptional Lie group, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 187–202.

[10] R. Levi, On finite groups and homotopy theory, Mem. Amer. Math. Soc., Vol 118, no. 567 (1995).

[11] Lin J. P., Williams, F., On 14-connected finite H-spaces, Israel J. Math. 66 (1989), no. 1-3, 274288.

[12] R. Levi, B. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. and Top. 6 (2002) 917–990.

[13] J. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 1965 211–264.

[14] M. Mimura, H. Toda, Topology of Lie groups I, II, Translations of Math. Mono. 91, Amer. Math. Soc., (1991).

[15] D. Notbohm, On the 2-compact group DI(4), J. Reine Angew. Math 555, (2003) 163–165.

[16] D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. Math. 2nd Ser., Vol. 96, no. 3 (1972) 552–586.

[17] L. Smith, Homological Algebra and the Eilenberg Moore Spectral Sequence, Amer. Math. Soc. Transl., No. 129, (1967), pp. 58-93.

[18] L. Smith, R. M. Switzer Realizability and nonrealizability of Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), no. 2, 303313.

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

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, U.K.

  E-mail address: r.levi@abdn.ac.uk

L.A.G.A. – Laboratoire de Mathématiques, Institut Galilée, 99 Avenue. J.B. Clément, 93430 Villetaneuse, France

  E-mail address: seeliger@math.univ-paris13.fr