Bockstein Closed 2-Group Extensions and Cohomology of Quadratic Maps

Jonathan Pakianathan and Ergün Yalçın

April 27, 2012

Abstract

A central extension of the form E : 0 → V → G → W → 0, where V and W are elementary abelian 2-groups, is called Bockstein closed if the components q_{i} ∈ H^{∗}(W, 𝔽_{2}) of the extension class of E generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of G when E is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of G has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps Q : W → V associated to the extensions E of the above form.

2000 Mathematics Subject Classification. Primary: 20J06; Secondary: 17B56.

1 Introduction

Let G be a p-group which fits into a central extension of the form

E : 0 → V → G → W → 0

where V, W are 𝔽_{p}-vector spaces of dimensions n and m, respectively. E is called Bockstein closed if the components q_{i} ∈ H^{∗}(W, 𝔽_{p}) of the extension class of E generate an ideal which is closed under the Bockstein operator. We say E is p-power exact if the following three conditions are satisfied: (i) m = n, (ii) V is the Frattini subgroup of G, and (iii) the p-rank of G is equal to n. Associated to G there is a p-th power map ( )^{p} : W → V. When p is odd, the p-th power map is a homomorphism and if E is also p-power exact, then it is an isomorphism. Using this isomorphism, one can define a Bracket [ , ] : W × W → W on W which turns out to be a Lie Bracket if and only if the associated extension is Bockstein closed. This was studied by Browder-Pakianathan [3] who also used this fact to give a complete description of the Bockstein cohomology of G in terms of the Lie algebra cohomology of the associated Lie algebra. This theory was later used by Pakianathan [5] to give a counterexample to a conjecture of Adem [1] on exponents in the integral cohomology of p-groups for odd primes p.

In the case where p = 2, the 2-power map ( )^{2} : W → V is not a homomorphism, so the results of Browder-Pakianathan do not generalize to 2-groups in a natural way. In this case, the 2-power map is a quadratic map Q : W → V where the associated bilinear map B : W × W → V is induced by taking commutators in G. The 2-power exact condition is equivalent to the conditions: (i) m = n, (ii) the elements {Q(w) | w ∈ W } generate V, and (iii) if Q(w) = 0 for some w ∈ W, then w = 0. We studied the quadratic maps associated to Bockstein closed extensions in an earlier paper, and showed that an extension E is Bockstein closed if and only if there is a bilinear map P : V × W → V such that

P (Q(w), w′) = B(w, w′) + P (B(w, w′), w)  (1)

holds for all w, w′ ∈ W (see Theorem 1.1 in [7]). In some sense this is the Jacobi identity for the p = 2 case. If there is a quadratic map Q : W → W which satisfies this identity with P = B, then the vector space W becomes a 2-restricted Lie algebra with 2-power map defined by w^{[2]} = Q(w) + w for all w ∈ W. But in general there are no direct connections between Bockstein closed quadratic maps and mod-2 Lie algebras.

In this paper, we study the cohomology of Bockstein closed 2-power exact extensions. We calculate the mod-2 cohomology ring and give a description of the Bockstein spectral sequence. As in the case when p is odd, the Bockstein spectral sequence can be described in terms of a cohomology theory based on our algebraic data. In this case, the right cohomology theory is the cohomology H^{∗}(Q, U) of a Bockstein closed quadratic map Q : W → V. We define this cohomology using an explicit cochain complex associated to the quadratic map. The definition is given in such a way that the low dimensional cohomology has interpretation in terms of extensions of Bockstein closed quadratic maps. For example, H^{0}(Q, U) gives the Q-invariants of U and H^{1}(Q, U) ≅ Hom_{Quad}(Q, U) if U is a trivial Q-module. Also, H^{2}(Q, U) is isomorphic to the group of extensions of Q with abelian kernel U (see Proposition 4.4). The definition of H^{∗}(Q, U) is given in Section 4. To keep the theory more general, in the definition of H^{∗}(Q, U) we do not assume that the quadratic map Q is 2-power exact.

In Section 5, we calculate the mod-2 cohomology ring of a Bockstein closed 2-power exact group G using the Lyndon-Hochschild-Serre spectral sequence associated to the extension E. This calculation is relatively easy and it is probably known to experts in the field (see, for example, [4] or [9]). The mod-2 cohomology ring of G has a very nice expression given by

H^{∗}(G, 𝔽_{2}) ≅ A^{∗}(Q) ⊗ 𝔽_{2}[s_{1}, ..., s_{n}]

where s_{i}’s are some two dimensional generators and the algebra A^{∗}(Q) is given by

A^{∗}(Q) = 𝔽_{2}[x_{1}, …, x_{n}]/(q_{1}, …, q_{n})

where {x_{1}, …, x_{n}} forms a basis for H^{1}(W) and q_{i}’s are components of the extension class q ∈ H^{2}(W, V) with respect to a basis for V. The action of the Bockstein operator on this cohomology ring gives valuable information about the question of whether the extension can be uniformly lifted to other extensions. Also finding Bocksteins of generators of the mod-2 cohomology algebra is the starting point for calculating the integral cohomology of G. We prove the following:

Theorem 1.1. Let E : 0 → V → G → W → 0 be a Bockstein closed 2-power exact extension with extension class q and let β(q) = Lq. Then the mod-2 cohomology of G is in the above form and β(s) = Ls + η where s denotes a column matrix with entries in s_{i}’s and η is a column matrix with entries in H^{3}(W, 𝔽_{2}).

The proof of this theorem is given in Section 7 using the Eilenberg-Moore spectral sequence associated to the extension. The key property of the EM-spectral sequence is that it behaves well under the Steenrod operations. The Steenrod algebra structure of the EM-spectral sequence was studied by L. Smith [10], [11] and D. Rector [8] independently in a sequence of papers. Here we use only a special case of these results. More precisely, we use the fact that the first two vertical lines in the EM-spectral sequence are closed under the action of the Steenrod algebra. This is stated as Corollary 4.4 in [10].

The column matrix η of the formula β(s) = Ls + η defines a cohomology class [η] ∈ H^{3}(Q, L) where L is the Q-module associated to the matrix L. Recall that in the work of Browder-Pakianathan [3], there is a cohomology class lying in the Lie algebra cohomology H^{3}(L, ad) which is defined in a similar way and it is an obstruction class for lifting G uniformly twice. We obtain a similar theorem for uniform double lifting of 2-group extensions.

Theorem 1.2. Let E : 0 → V → G → W → 0 be a Bockstein closed 2-power exact extension with extension class q. Let Q be the associated quadratic map and L denote the Q-module defined by L in the equation β(q) = Lq. Then, the extension E has a uniform double lifting if and only if [η] = 0 in H^{3}(Q, L).

Another result we have is a description of the second page of the Bockstein spectral sequence in terms of the cohomology of Bockstein closed quadratic maps for the case where the extension has a uniform double lifting.

Theorem 1.3. Let E, G, Q, and L be as in Theorem 1.2. Assume that E has a uniform double lifting. Then, the second page of the Bockstein spectral sequence for G is given by

B_{2}^{∗}(G) = ⨁_{i=0}^{∞} H^{∗−2i}(Q, Sym^{i}(L))

where Sym^{i}(L) denotes the symmetric i-th power of L.

In the p odd case, the B-cohomology has been calculated in cases by comparing it to H^{∗}(𝔤, U(𝔤)^{∗}) where U(𝔤)^{∗} is the dual of the universal enveloping algebra of 𝔤 equipped with the dual adjoint action, where 𝔤 is an associated complex Lie algebra. This fundamental object has played a role in string topology (homology of free loop spaces) and is analogous to the (classical) ring of modular forms, identified by Eichler-Shimura as H^{∗}(SL_{2}(ℤ), Poly(V)) where V is the complex 2-dimensional canonical representation of SL_{2}(ℤ) (see [6] for more details). In string topology contexts, this is referred to as the Hodge decomposition and so the above can be thought of as a Hodge decomposition for the quadratic form Q. It describes the distribution of higher torsion in the integral cohomology of the associated group G.

As in the case of Lie algebras, it is possible to give a suitable definition of a universal enveloping algebra U(Q) for a quadratic map Q so that the representations of Q and representations of the universal algebra U(Q) can be identified in a natural way. However it is not clear to us how to find an isomorphism between the cohomology of the universal algebra U(Q) and the cohomology of the quadratic map Q. There is also the issue of finding analogies of the theorems on universal algebras of Lie algebras such as the Poincaré-Birkoff-Witt Theorem. We leave these as open problems.

The paper is organized as follows: In Section 2, we introduce the category of quadratic maps and show that it is naturally equivalent to the category of extensions of certain type. Then in Section 3, we give the definition of a Bockstein closed quadratic map. The definition of cohomology of Bockstein closed quadratic maps is given in Section 4. Sections 5, 6, and 7 are devoted to the mod-2 cohomology calculations using LHS- and EM-spectral sequences and the calculation of Bocksteins of the generators. In particular, Theorem 1.1 is proven in Section 7. In Section 8, we discuss the obstructions for uniform lifting and in Section 9, we explain the E_{2}-page of the Bockstein spectral sequence in terms of the cohomology of Bockstein closed quadratic maps.

2 Category of quadratic maps

Let E denote a central extension of the form

E : 0 → V → G → W → 0

where V and W are elementary abelian 2-groups. Associated to E, there is a cohomology class q ∈ H^{2}(W, V). Also associated to E there is a quadratic map Q : W → V defined by Q(w) = (ŵ)^{2}, where ŵ denotes an element in G that lifts w ∈ W. Similarly, the commutator induces a symmetric bilinear map B : W × W → V defined by B(x, y) = [x̂, ŷ] for x, y ∈ W where [g, h] = g^{−1}h^{−1}gh for g, h ∈ G. It is easy to see that B is the bilinear form associated to Q.

We have shown in [7] that the extension class q and the quadratic form Q are closely related to each other. In particular, we showed that we can take q = [f] where f is a bilinear factor set f : W × W → V satisfying the identity f(w, w) = Q(w) for all w ∈ W (see [7, Lemma 2.3]). We can write this correspondence more explicitly by choosing a basis {w_{1}, …, w_{m}} for W. Then,

f(w_{i}, w_{j}) = \begin{cases}B(w_{i}, w_{j}) & if i < j, \\ Q(w_{i}) & if i = j, \\ 0 & if i > j. \end{cases}

This gives a very specific expression for q. Let {v_{1}, …, v_{n}} be a basis for V, and let q_{k} be the k-th component of q with respect to this basis. Then,

q_{k} = ∑_{i} Q_{k}(w_{i})x_{i}^{2} + ∑_{i<j} B_{k}(w_{i}, w_{j})x_{i}x_{j}

where {x_{1}, …, x_{m}} is the dual basis of {w_{1}, …, w_{m}} and Q_{k} and B_{k} denote the k-th components of Q and B. This allows one to prove the following:

Proposition 2.1 (Corollary 2.4 in [7]). Given a quadratic map Q : W → V, there is a unique (up to equivalence) central extension

E(Q) : 0 → V → G(Q) → W → 0

with a bilinear factor set f : W × W → V satisfying f(w, w) = Q(w) for all w ∈ W.

This gives a bijective correspondence between quadratic maps Q : W → V and the central extensions of the form E : 0 → V → G → W → 0. We will now define the category of quadratic maps and the category of group extensions of the above type and then prove that the correspondence described above indeed gives a natural equivalence between these categories.

2.1 Equivalence of categories

The category of quadratic maps Quad is defined as the category whose objects are quadratic maps Q : W → V where W and V are vector spaces over 𝔽_{2}. For quadratic maps Q_{1}, Q_{2}, a morphism f : Q_{1} → Q_{2} is defined as a pair of linear transformations f = (f_{W} , f_{V}) such that the following diagram commutes:

\begin{matrix}W_{1} & ⟶^{f_{W}} & W_{2} \\ ↓^{Q_{1}} & ↓^{Q_{2}} \\ V_{1} & ⟶^{f_{V}} & V_{2} . \end{matrix}  (2)

The composition of morphisms f = (f_{W} , f_{V}) and g = (g_{W} , g_{V}) is defined by coordinatewise compositions. The identity morphism is the pair (id_{W} , id_{V}). Two quadratic maps Q_{1} and Q_{2} are isomorphic if there are morphisms f : Q_{1} → Q_{2} and g : Q_{2} → Q_{1} such that f ∘ g = id_{Q_{2}} and g ∘ f = id_{Q_{1}}.

The category Ext is defined as the category whose objects are the equivalence classes of extensions of type

E : 0 → V → G → W → 0

where V and W are vector spaces over 𝔽_{2}, and the morphisms are given by a commuting diagram as follows:

\begin{matrix}E_{1} : & 0 & ⟶ & V_{1} & ⟶ & G_{1} & ⟶ & W_{1} & ⟶ & 0 \\ ↓^{f} & & & ↓^{f_{V}} & ↓^{f_{G}} & ↓^{f_{W}} \\ E_{2} : & 0 & ⟶ & V_{2} & ⟶ & G_{2} & ⟶ & W_{2} & ⟶ & 0 . \end{matrix}  (3)

Note that two extensions are considered equivalent if there is a diagram as above with f_{W} = id_{W} and f_{V} = id_{V}. All such morphisms are taken to be equal to identity morphism in our category. More generally, two morphisms f, g : E_{1} → E_{2} will be considered equal in Ext if f_{V} = g_{V} and f_{W} = g_{W}.

From the discussion at the beginning of this section, it is clear that the assignments Φ : Q → E(Q) and Ψ : E → Q_{E} give a bijective correspondence between the objects of Quad and Ext. We just need to extend this correspondence to a correspondence between morphisms. Given a morphism f : E_{1} → E_{2}, we take Ψ(f) to be the pair (f_{V} , f_{W}) : Q_{1} → Q_{2}. By commutativity of the diagram (3), it is easy to see that Q_{2}(f_{W} (w)) = f_{V} (Q_{1}(w)) holds for all w ∈ W_{1}. To define the image of a morphism f : Q_{1} → Q_{2} under Φ, we need to define a group homomorphism f_{G} : G(Q_{1}) → G(Q_{2}) which makes the diagram given in (3) commute. Note that once f_{G} is defined, we can define the morphism Φ(f) : E_{1} → E_{2} as a sequence of maps (f_{V} , f_{G}, f_{W}) as in diagram (3). It is clear that the composition Ψ ∘ Φ is equal to the identity transformation. The composition Φ ∘ Ψ is also equal to the identity in Ext although it may not be equal to identity on the middle map f_{G}. This follows from the fact that two morphisms f, g : E_{1} → E_{2} between two extensions are equal in Ext if f_{V} = g_{V} and f_{W} = g_{W}.

To define a group homomorphism f_{G} : G(Q_{1}) → G(Q_{2}) which makes the diagram given in (3) commute, first recall that for i = 1, 2, we can take G(Q_{i}) as the set V_{i} × W_{i} with multiplication given by (v, w)(v′, w′) = (v + v′ + f_{i}(w, w′), w + w′) where f_{i} : W_{i} × W_{i} → V_{i} is a bilinear factor set satisfying f_{i}(w, w) = Q_{i}(w) for every w ∈ W_{i} (see [7, Lem. 2.3]). Note that the choice of the factor set is not unique and if f_{i} and f′_{i} are two factor sets for E(Q_{i}) satisfying f_{i}(w, w) = f′_{i}(w, w) = Q_{i}(w) for all w ∈ W_{i}, then f_{i} + f′_{i} = δ(t) is a boundary in the bar resolution. When we apply this to the extension associated to the quadratic form Q_{2}f_{W} = f_{V} Q_{1} : W_{1} → V_{2}, we see that there is a function t : W_{1} → V_{2} such that

(δt)(w, w′) = t(w′) + t(w + w′) + t(w) = f_{2}(f_{W} (w), f_{W} (w′)) + f_{V} (f_{1}(w, w′)) (4)

for all w, w′ ∈ W_{1}. We define f_{G} : G(Q_{1}) → G(Q_{2}) by f_{G}(v, w) = (f_{V} (v) + t(w), f_{W} (w)) for all v ∈ V_{1} and w ∈ W_{1}. To check that f_{G} is a group homomorphism, we need to show that

f_{G}((v, w)(v′, w′))= f_{G}(v, w)f_{G}(v′, w′)

holds for all v, v′ ∈ V_{1} and w, w′ ∈ W_{1}. Writing this out in detail, one sees that this equation is equivalent to Equation 4, hence it holds. So, we obtain a group homomorphism f_{G} : G(Q_{1}) → G(Q_{2}) as desired. We conclude the following: Proposition 2.2. The categories Quad and Ext are equivalent.

An immediate consequence of this equivalence is the following: Corollary 2.3. Let Q_{1} and Q_{2} be quadratic maps with extension classes q_{1} ∈ H^{2}(W_{1}, V_{1}) and q_{2} ∈ H^{2}(W_{2}, V_{2}) respectively. If f : Q_{1} → Q_{2} is a morphism of quadratic maps, then

(f_{W})^{∗}(q_{2}) = (f_{V})_{∗}(q_{1})

in H^{2}(W_{1}, V_{2}). Proof. Let Q′ : W_{1} → V_{2} be the quadratic map defined by Q′ = f_{V} Q_{1} = Q_{2}f_{W}. Then, we have

W_{1} ⟶^{=} W_{1} ⟶^{f_{W}} W_{2}
↓^{Q_{1}}  ↓^{Q′}  ↓^{Q_{2}}
V_{1} ⟶^{f_{V}} V_{2} ⟶^{=} V_{2}

So, we find a factorization of f in Quad as Q_{1} ⟶^{f_{1}} Q′ ⟶^{f_{2}} Q_{2}. By Proposition 2.2, we obtain a factorization of the corresponding morphism in Ext, this gives the following commuting diagram:

E_{1} : 0 ⟶ V_{1} ⟶ G(Q_{1}) ⟶ W_{1} ⟶ 0
↓^{f_{V}}  ↓^{(f_{1})_{G}}  ∥
E′ : 0 ⟶ V_{2} ⟶ G̃ ⟶ W_{1} ⟶ 0
∥  ↓^{(f_{2})_{G}}  ↓^{f_{W}}
E_{2} : 0 ⟶ V_{2} ⟶ G(Q_{2}) ⟶ W_{2} ⟶ 0

Hence, we have (f_{W})^{∗}(q_{2}) = (f_{V})_{∗}(q_{1}) as desired.

2.2 Extensions and representations of quadratic maps

We now introduce certain categorical notions for maps between quadratic maps such as kernel and cokernel of a map and then give the definition of extensions of quadratic maps.

The kernel of a morphism f : Q_{1} → Q_{2} is defined as the quadratic map Q_{1}|_{ker f_{W}} : ker f_{W} → ker f_{V} . We denote this quadratic map as ker f. If ker f is the zero quadratic map, i.e., the quadratic map from a zero vector space to zero vector space, then we say f is injective. Similarly, we define the image of a quadratic map f : Q_{1} → Q_{2} as the quadratic map Q_{2}|_{Im f_{W}} : Im f_{W} → Im f_{V} . We denote this quadratic map by Im f and say f is surjective if Im f = Q_{2}. Given an injective map f : Q_{1} → Q_{2}, we say f is a normal embedding if

B_{2}(f_{W} (w_{1}), w_{2}) ∈ Im f_{V}

for all w_{1} ∈ W_{1} and w_{2} ∈ W_{2}. Given a normal embedding f : Q_{1} → Q_{2}, we can define the cokernel of f as the quadratic map coker f : coker f_{W} → coker f_{V} by the formula

(coker f)(w_{2} + Im f_{W}) = Q_{2}(w_{2}) + Im f_{V} .

We are now ready to define an extension of two quadratic maps.

Definition 2.4. We say a sequence of quadratic maps of the form

E : 0 → Q_{1} ⟶^{f} Q_{2} ⟶^{g} Q_{3} → 0  (5)

is an extension of quadratic maps if f is injective, g is surjective, and Im f = ker g.

Note that in an extension E as above, the first map f : Q_{1} → Q_{2} is a normal embedding because we have

B_{2}(Imf_{W} , W_{2}) = B_{2}(ker g_{W} , W_{2}) ⊆ ker g_{V} = Imf_{V} .

We say the extension E is a split extension if there is a morphism of quadratic maps s : Q_{3} → Q_{2} such that g ∘ s = id_{Q_{3}}. In this case we write Q_{3} ≅ Q_{1} ⋊ Q_{2}.

Later in this paper we consider the extensions where Q_{1} is just the identity map id_{U} : U → U of a vector space U. In this case, we denote the extension by

E : 0 → U ⟶^{i} Q̃ ⟶^{π} Q → 0,

and say E is an extension of Q with an abelian kernel U. In Section 4, we define obstructions for splitting such extensions and also give a classification theorem for such extensions in a subcategory of Quad where all the quadratic maps are assumed to be Bockstein closed.

Given an extension of quadratic map Q with an abelian kernel U, there is an action of Q on U induced from the bilinear form B̃ associated to Q̃. This action is defined as a homomorphism

ρ_{W} : W → Hom(U, U)

which satisfies i_{V} (ρ_{W} (w)(u)) = B̃(i_{W} (u), \overline{w}) where \overline{w} is vector in W̃ such that π_{W} (\overline{w}) = w.

In the definition of the representation of a quadratic map, we need the following family of quadratic maps: Let U be a vector space. We define

Q_{𝔤𝔩(U)} : End(U) → End(U)

to be the quadratic map such that Q_{𝔤𝔩(U)}(A) = A^{2}+A for all A ∈ End(U). (See Example 2.5 in [7].)

Definition 2.5. A representation of a quadratic form Q is defined as a morphism

ρ : Q → Q_{𝔤𝔩(U)}

in the category of quadratic maps. In other words, a representation is a pair of maps ρ = (ρ_{W} , ρ_{V}) such that the following diagram commutes

W ⟶^{ρ_{W}} End(U)
↓^{Q}  ↓^{Q_{𝔤𝔩(U)}}
V ⟶^{ρ_{V}} End(U).

If U is a k-dimensional vector space, then we say ρ is a k-dimensional representation of Q. Given a representation as above, we sometimes say U is a Q-module to express the fact that there is an action of Q on id_{U} : U → U via the representation ρ.

3 Bockstein closed quadratic maps

Let E(Q) be a central extension of the form 0 → V → G(Q) → W → 0 associated to a quadratic map Q : W → V, where V and W are 𝔽_{2}-vector spaces. Let q ∈ H^{2}(W, V) denote the extension class of E. Choosing a basis {v_{1}, …, v_{n}} for V, we can write q as a tuple

q = (q_{1}, …, q_{n})

where q_{i} ∈ H^{2}(W, 𝔽_{2}) for all i. The elements {q_{i}} generate an ideal I(Q) in the cohomology algebra H^{∗}(W, 𝔽_{2}). It is easy to see that the ideal I(Q) is independent of the basis chosen for V, and hence is completely determined by Q.

Definition 3.1. We say Q : W → V is Bockstein closed if I(Q) is invariant under the Bockstein operator on H^{∗}(W; 𝔽_{2}). A central extension E(Q) : 0 → V → G(Q) → W → 0 is called Bockstein closed if the associated quadratic map Q is Bockstein closed.

The following was proven in [7] as Proposition 3.3.

Proposition 3.2. Let Q : W → V be a quadratic map, and let q ∈ H^{2}(W, V) be the corresponding extension class. Then, Q is Bockstein closed if and only if there is a cohomology class L ∈ H^{1}(W, End(V)) such that β(q) = Lq.

We often choose a basis for W and V (dim W = m and dim V = n) and express the formula β(q) = Lq as a matrix equation. From now on, let us assume W and V have some fixed basis and let {x_{1}, …, x_{m}} be the dual basis for W. Then, each component q_{k} is a quadratic polynomial in variables x_{i} and L is an n × n matrix with entries given by linear polynomials in x_{i}’s. If we express q as a column matrix whose i-th entry is q_{i}, then β(q) = Lq makes sense as a matrix formula where Lq denotes the matrix multiplication. In general, we can have different matrices, say L_{1} and L_{2}, such that β(q) = L_{1}q = L_{2}q. It is known that L is unique when E is 2-power exact. (See Proposition 8.1 in [7].)

Example 3.3. Let G be the kernel of the mod 2 reduction map GL_{n}(ℤ/8) → GL_{n}(ℤ/2). It is easy to see that G fits into a central short exact sequence

0 → 𝔤𝔩_{n}(𝔽_{2}) → G → 𝔤𝔩_{n}(𝔽_{2}) → 0

with associated quadratic map Q_{𝔤𝔩_{n}} where 𝔤𝔩_{n}(𝔽_{2}) is the vector space of n × n matrices with entries in 𝔽_{2} and the quadratic map Q_{𝔤𝔩_{n}} : 𝔤𝔩_{n}(𝔽_{2}) → 𝔤𝔩_{n}(𝔽_{2}) is defined by Q_{𝔤𝔩_{n}}(𝔸) = 𝔸^{2} + 𝔸 (see [7, Ex. 2.5 and 2.6] for more details). We showed in [7, Cor. 3.9] that this extension and its restrictions to suitable subspaces such as 𝔰𝔩_{n}(𝔽_{2}) or 𝔲_{n}(𝔽_{2}) are Bockstein closed. Here 𝔰𝔩_{n}(𝔽_{2}) denotes the subspace of 𝔤𝔩_{n}(𝔽_{2}) formed by matrices of trace zero and 𝔲_{n}(𝔽_{2}) denotes the subspace of strictly upper triangular matrices. Note that the extension for 𝔲_{n}(𝔽_{2}) is also a 2-power exact extension (see [7, Ex. 9.6]).

We now consider the question of when an extension of two Bockstein closed quadratic maps is also Bockstein closed. The equations that we find in the process of answering this question will give us the motivation for the definition of the cohomology of Bockstein closed quadratic maps.

Let

E : 0 → U ⟶^{i} Q̃ ⟶^{π} Q → 0

be an extension of the quadratic map Q with abelian kernel U. We can express this as a diagram of quadratic maps as follows:

U ⟶^{i_{W}} U ⊕ W ⟶^{π_{W}} W
↓^{id}_{U}  ↓Q̃  ↓^{Q}
U ⟶^{i_{V}} U ⊕ V ⟶^{π_{V}} V.

Note that we have Q̃(u, 0) = (u, 0) and π_{V} Q̃(0, w) = Q(w) for all u ∈ U and w ∈ W. Also, there is an action of Q on U given by the linear map ρ_{W} : W → End(U) defined by the equation

i_{V} (ρ_{W} (w)u) = B̃((u, 0), (0, w)).

Hence, we can write

Q̃(u, w) = (u + ρ_{W} (w)u + f(w), Q(w))

where f : W → U is a quadratic map called factor set. We will study the conditions on f and ρ_{W} which make Q̃ a Bockstein closed quadratic map.

Let k = dim U. Choose a basis for U, and let {z_{1}, …, z_{k}} be the associated dual basis for U^{∗}. Then, we can express the extension class q̃ of Q̃ as a column matrix

q̃ = [\begin{matrix}q \\ β(z) + Rz + f \end{matrix}]

where f and q denote the column matrices for the quadratic maps f : W → U and Q : W → V respectively. Here z is the column matrix with i-th entry equal to z_{i} and R is a k ×k matrix with entries in x_{i}’s which is associated to ρ_{W} : W → End(U). Applying the Bockstein operator, we get

β(q̃) = [\begin{matrix}β(q) \\ β(R)z + Rβ(z) + β(f) \end{matrix}] .

Note that Q̃ is Bockstein closed if we can find L̃ such that β(q̃) = L̃q̃. Since β(q) = Lq for some L, we can take L̃ as

L̃ = [\begin{matrix}L & 0 \\ L_{2,1} & L_{2,2} \end{matrix}] .

Note that assuming the top part of the matrix L̃ is in a special form does not affect the generality of the lower part. So, under the assumption that Q is Bockstein closed, the quadratic map Q̃ is Bockstein closed if only if there exist L_{2,1} and L_{2,2} satisfying

β(R)z + Rβ(z) + β(f) = L_{2,1}q + L_{2,2}(β(z) + Rz + f).  (6)

We have

L_{2,2} = ∑_{i=1}^{k} L_{2,2}^{(i)}z_{i} + ∑_{j=1}^{m} L_{2,2}^{(j)}x_{j}

where L_{2,2}^{(i)} and L_{2,2}^{(j)} are scalar matrices, so we can write L_{2,2} = L_{2,2}^{z} + L_{2,2}^{x} where L_{2,2}^{z} is the first sum and L_{2,2}^{x} is the second sum in the above formula.

Equation (6) gives L_{2,2}^{z}β(z) = 0 which implies L_{2,2}^{z} = 0. Writing L_{2,2} = L_{2,2}^{x} in (6), we easily see that we must have L_{2,2} = R. Putting this into (6), we get

[β(R) + R^{2}]z + [β(f) + Rf] = L_{2,1}q.  (7)

As we did for L_{2,2}, we can write L_{2,1} also as a sum L_{2,1} = L_{2,1}^{z} +L_{2,1}^{x} where the entries of L_{2,1}^{z} are linear polynomials in z_{i}’s and the entries of L_{2,1}^{x} are linear polynomials in x_{i}’s. So, Equation 7 gives two equations:

\begin{aligned}[β(R) + R^{2}]z & = L_{2,1}^{z}q \\ β(f) + Rf & = L_{2,1}^{x}q. \end{aligned}  (8)

From now on, let us write Z = L_{2,1}^{z}. Note that Z is a k × n matrix (k = dim U and n = dim V) with entries in the dual space U^{∗}, so it can be thought of as a linear operator Z : U → Hom(V, U). Viewing this as a bilinear map U × V → U, and then using an adjoint trick, we obtain a linear map ρ_{V} : V → Hom(U, U) = End(U). As a matrix, let us denote ρ_{V} by T. The relation between Z and T can be explained as follows: If {u_{1}, …, u_{k}} are basis elements for U dual to the basis elements {z_{1}, …, z_{k}} of U^{∗} and if {v_{1}, …, v_{k}} is the basis for V dual to the basis elements {t_{1}, …, t_{k}} of V^{∗}, then Z(u_{i})(v_{j}) = T (v_{j})(u_{i}) for all i, j. So, if Z = ∑_{i=1}^{k} Z(i)z_{i} and T = ∑_{j=1}^{n} T (j)t_{j}, then we have Z(i)e_{j} = T (j)e_{i} where e_{i} and e_{j} are i-th and j-th unit column matrices. This implies, in particular, that

Zq = T (q)z

where T (q) is the matrix obtained from T by replacing t_{i}’s with q_{i}’s. So, the first equation in (8) can be interpreted as follows:

Lemma 3.4. Let ρ_{W} : W → End(U) and ρ_{V} : V → End(U) be two linear maps with corresponding matrices R and T. Let Z denote the matrix for the adjoint of ρ_{V} in Hom(U, Hom(V, U)). Then, the equation

[β(R) + R^{2}]z = Zq

holds if and only if ρ = (ρ_{W} , ρ_{V}) : Q → Q_{𝔤𝔩(U)} is a representation.

Proof. Note that the diagram

W ⟶^{ρ_{W}} End(U)
↓^{Q}  ↓^{Q_{𝔤𝔩(U)}}
V ⟶^{ρ_{V}} End(U)

commutes if and only if

β(R) + R^{2} = T (q)

where T (q) is the k × k matrix obtained from T by replacing t_{i}’s with q_{i}’s. We showed above that Zq = T (q)z, so β(R) + R^{2} = T (q) holds if and only if

[β(R) + R^{2}]z = T (q)z = Zq.

This completes the proof.

As a consequence of the above lemma, we can conclude that if the action of W on End(U) comes from a representation ρ : Q → Q_{𝔤𝔩(U)}, then the only obstruction for a quadratic map Q̃ to be Bockstein closed is the second equation

β(f) + Rf = L_{2,1}^{x}q

given in (8). Note that both R and L_{2,1}^{x} are matrices with entries in x_{i}’s, so this equation can be interpreted as saying that β(f) + Rf = 0 in A^{∗}(Q) = 𝔽_{2}[x_{1}, …, x_{m}]/(q_{1}, …, q_{n}). In the next section, we define the cohomology of a Bockstein closed quadratic map using this interpretation.

4 Cohomology of Bockstein closed quadratic maps

Let Q : W → V be a Bockstein closed quadratic map where W and V are 𝔽_{2}-vector spaces of dimensions m and n, respectively. Let U be a k-dimensional Q-module with associated representation ρ : Q → Q_{𝔤𝔩(U)}. We will define the cohomology of Q with coefficients in U as the cohomology of a cochain complex C^{∗}(Q, U). We now describe this cochain complex.

Let A(Q)^{∗} denote the 𝔽_{2}-algebra

A^{∗}(Q) = 𝔽_{2}[x_{1}, …, x_{m}]/(q_{1}, …, q_{n})

as before. The algebra A^{∗}(Q) is a graded algebra where the grading comes from the usual grading of the polynomial algebra. We define p-cochains of Q with coefficients in U as

C^{p}(Q, U) = A(Q)^{p} ⊗ U.

We describe the differentials using a matrix formula. Choosing a basis for U, we can express a p-cochain f as a k × 1 column matrix with entries f_{i} ∈ A^{p}(Q). Let R ∈ H^{1}(W, End(U)) be the cohomology class associated to ρ_{W}. We can express R as a k × k matrix with entries in x_{i}’s. We define the boundary maps

δ : C^{p}(Q, U) → C^{p+1}(Q, U)

by

δ(f) = β(f) + Rf.

Note that

δ^{2}(f) = β(R)f + Rβ(f) + Rβ(f) + R^{2}f = [β(R) + R^{2}]f = 0

in A^{∗}(Q) because β(R) + R^{2} = T (q) by the argument given in the proof of Lemma 3.4. So, C^{∗}(Q, U) with the above boundary maps is a cochain complex.

Note that although the definition of δ only uses R, i.e., ρ_{W}, the existence of ρ_{V} is needed to ensure that δ^{2} = 0. Thus both maps in the structure of U as a Q-module play a role in establishing δ as a differential. Also note that we need the quadratic map Q to be Bockstein closed for the well-definedness of the differential δ.

Definition 4.1. The cohomology of a Bockstein closed quadratic form Q with coefficients in a Q-module U is defined as

H^{∗}(Q, U) := H^{∗}(C^{∗}(Q, U), δ)

where C^{∗}(Q, U) = A^{∗}(Q) ⊗ U and the boundary maps δ are given by δ(f) = β(f) + Rf.

Let U be the one dimensional trivial Q-module, i.e., ρ = (ρ_{W} , ρ_{V}) = (0, 0). In this case we write U = 𝔽_{2}. Then, H^{∗}(Q, 𝔽_{2}) is just the cohomology of the complex A(Q)^{∗} and the boundary map δ is equal to the Bockstein operator. In this case, the cohomology group H^{∗}(Q, 𝔽_{2}) also has a ring structure coming from the usual multiplication of polynomials. Note that given two cocycles f, g ∈ A^{∗}(Q), we have β(fg) = β(f)g +fβ(g) = 0 modulo I(Q). So, we define the product of two cohomology classes [f], [g] ∈ H^{∗}(Q, 𝔽_{2}) by

[f][g] = [fg]

where fg denotes the usual multiplication of polynomials.

Given a morphism φ : Q_{1} → Q_{2}, we have (φ_{W})^{∗}(q_{2}) = (φ_{V})_{∗}(q_{1}) by Corollary 2.3. This shows that (φ_{W})^{∗} : H^{∗}(W_{2}, 𝔽_{2}) → H^{∗}(W_{1}, 𝔽_{2}) takes the entries of q_{2} into the ideal I(Q_{1}). Thus, φ_{W} induces an algebra map φ^{∗} : A^{∗}(Q_{2}) → A^{∗}(Q_{1}) which gives a chain map C^{∗}(Q_{2}, U_{2}) → C^{∗}(Q_{1}, U_{1}) where U_{2} is representation of Q_{2} and U_{1} is a representation of Q_{1} induced by φ. So, φ induces a homomorphism

φ^{∗} : H^{∗}(Q_{2}, U_{2}) → H^{∗}(Q_{1}, U_{1}).

If U_{1} = U_{2} = 𝔽_{2}, the induced map is also an algebra map.

In the rest of the section, we discuss the interpretations of low dimensional cohomology, H^{i}(Q, U) for i = 0, 1, 2, in terms of extension theory. First we calculate H^{0}(Q, U). Note that C^{0}(Q, U) = U and given u ∈ C^{0}(Q, U), we have δ(u)(w) = ρ_{W} (w)u. So, H^{0}(Q, U) = U^{Q} where

U^{Q} = {u | ρ_{W} (w)(u) = 0 for all w ∈ W }.

Note that this is analogous to Lie algebra invariants

U^{𝔤} = {u | x · u = 0 for all x ∈ 𝔤}.

We refer to the elements of U^{Q} as Q-invariants of U.

Now, we consider H^{1}(Q, U). Note that C^{1}(Q, U) ≅ Hom(W, U). Let d_{W} : W → U be a 1-cochain. By our definition of differentials, d_{W} is a derivation if and only if δ(d_{W}) = β(d_{W}) + Rd_{W} = 0 in A(Q)^{2} ⊗ U. The last equation can be interpreted as follows: There is a linear map d_{V} : V → U such that

(1 + ρ_{W} (w))d_{W} (w) + d_{V} (Q(w)) = 0.

A trivial derivation will be a derivation d_{W} : W → U of the form d_{W} (w) = ρ_{W} (w)u for some u ∈ U. Note that when U is a trivial module, d_{W} : W → U is a derivation if and only if there is a linear map d_{V} : V → U such that the following diagram commutes

W ⟶^{d_{W}} U
↓^{Q}  ↓^{id}
V ⟶^{d_{V}} U .

So, when U is a trivial Q-module, we have

H^{1}(Q, U) ≅ Hom_{Quad}(Q, U).

If U = 𝔽_{2}, then

H^{1}(Q, 𝔽_{2}) = ker{β : H^{1}(W, 𝔽_{2}) → A^{2}(Q)}.

So, we have the following:

Proposition 4.2. Let Z(Q) be the vector space generated by k-invariants q_{1}, …, q_{n} and let Z(Q)^{β} = {z ∈ Z(Q) | β(z) = 0}. Then, H^{1}(Q, 𝔽_{2}) ≅ Z(Q)^{β}.

We refer to the elements of Z(Q)^{β} as the Bockstein invariants of Q. For an arbitrary Q module U, we have the following:

Proposition 4.3. There is a one-to-one correspondence between H^{1}(Q, U) and the splittings of the split extension 0 → U → U ⋊ Q → Q → 0.

Proof. Observe that s : Q → U ⋊ Q is a morphism in Quad if and only if Q̃(s_{W} (w)) = s_{V} (Q(w)). Since πs = id, we can write s_{W} (w) = (d_{W} (w), w) and s_{V} (v) = (d_{V} (v), v). So, s is a morphism in Quad if and only if d_{W} (w) + ρ(w)d_{W} (w) = d_{V} (Q(w)), i.e., d_{W} is a derivation. It is easy to see that trivial derivation corresponds to a splitting which is trivial up to an automorphism of U ⋊ Q.

We now consider extensions of a Bockstein closed quadratic map Q with an abelian kernel U, and show that H^{2}(Q, U) classifies such extensions up to an equivalence. If there is a diagram of quadratic maps of the following form

\begin{aligned}E_{1} : & 0 & ⟶ & U & ⟶ & Q_{1} & ⟶ & Q & ⟶ & 0 \\ & & & ∥ & & ↓^{φ} & & ∥ \\ E_{2} : & 0 & ⟶ & U & ⟶ & Q_{2} & ⟶ & Q & ⟶ & 0 , \end{aligned}  (9)

then we say E_{1} is equivalent to E_{2}. Let Ext(Q, U) denote the set of equivalence classes of extensions of the form 0 → U → Q̃ → Q → 0 with abelian kernel U where Q̃ and Q are Bockstein closed. We can define the summation of two extensions as it is done in group extension theory. So, Ext(Q, U) is an abelian group. We prove the following:

Proposition 4.4. H^{2}(Q, U) ≅ Ext(Q, U).

Proof. Note that we already have a fixed decomposition for the domain and the range of Q̃, so we will write our proof using these fixed decompositions. We skip some of the details which are done exactly as in the case of group extensions.

First we show there is a 1-1 correspondence between 2-cocycles and Bockstein closed extensions. Recall that a 2-cocycle is a quadratic map f : W → U such that β(f)+Rf = 0 in A(Q)^{∗}. Consider the extension E : 0 → U → Q̃ → Q → 0 where

Q̃(u, w) = (u + ρ_{W} (w)u + f(w), Q(w)).

We have seen earlier that Q̃ is Bockstein closed if and only if β(f) + Rf = 0 in A(Q)^{∗}. So, E is an extension of Bockstein closed quadratic maps if and only if f is a cocycle.

Now, assume that E_{1} and E_{2} are two equivalent extensions. Let φ : Q_{1} → Q_{2} be a morphism which makes the diagram (9) commute. Then, we can write φ_{W} (u, w) = (u + a(w), w) and φ_{V} (u, v) = (u + b(v), v). Let f_{1} and f_{2} be the cocycles corresponding to extensions E_{1} and E_{2} respectively. Then, the identity Q_{2}(φ_{W} (u, w)) = φ_{V} (Q_{1}(u, w)) gives

(u + a(w) + ρ(w)(u + a(w)) + f_{2}(w), Q(w)) = (u + ρ(w)u + f_{1}(w) + b(Q(w)), Q(w)).

So, we have

f_{2}(w) + f_{1}(w) = (1 + ρ(w))a(w) + b(Q(w)).  (10)

Thus f_{1} + f_{2} = δ(a) in A(Q)^{∗}. Conversely, if f_{1} + f_{2} = δ(a) in A(Q)^{∗}, then there is a b : V → U such that equation (10) holds, so we can define the morphism φ : Q_{1} → Q_{2} as above so that the diagram (9) commutes.

5 LHS-spectral sequence for 2-power exact extensions

In this section we study the Lyndon-Hochschild-Serre (LHS) spectral sequence associated to a Bockstein closed 2-power exact extension. We first recall the definition of a 2-power exact extension.

Definition 5.1. A central extension of the form

E(Q) : 0 → V → G(Q) → W → 0

with corresponding quadratic map Q : W → V is called 2-power exact if the following conditions hold:

(i) dim(V) = dim(W),

(ii) the extension is a Frattini extension, i.e., image of Q generates V, and

(iii) the extension is effective, i.e., Q(w) = 0 if and only if w = 0.

In this section, we calculate the mod-2 cohomology of G(Q) using a LHS-spectral sequence when E(Q) is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring structure of 2-power exact groups has a simple form and it is not very difficult to obtain once certain algebraic lemmas are established. Similar calculations were given by Rusin [9, Lemma 8] and Minh-Symonds [4].

We first prove an important structure theorem concerning the k-invariants of Bockstein closed 2-power exact extensions.

Proposition 5.2. Let E(Q) : 0 → V → G(Q) → W → 0 be a Bockstein closed, 2-power exact extension with dim(W) = n. Then, the k-invariants q_{1}, …, q_{n}, with respect to some basis of V, form a regular sequence in H^{∗}(W, 𝔽_{2}) = 𝔽_{2}[x_{1}, …, x_{n}] and A^{∗}(Q) = 𝔽_{2}[x_{1}, …, x_{n}]/(q_{1}, …, q_{n}) is a finite dimensional 𝔽_{2}-vector space.

Proof. We have shown in [7, Proposition 7.8] that the k-invariants q_{1}, …, q_{m} form a regular sequence in H^{∗}(W; 𝔽_{2}). This sequence is regular in any order. To show the second statement, let K denote the algebraic closure of 𝔽_{2}. Since the dimension of the variety associated to I(Q) = (q_{1}, …, q_{m}) is zero, the (projective) Nullstellensatz shows that A^{∗}(Q) is a nilpotent algebra (elements u in positive degree have u^{k} = 0 for some k, depending on u). However since A^{∗}(Q) is a finitely generated and commutative algebra, this shows that A^{∗}(Q) is finite dimensional as a vector space over K.

Recall that a regular sequence in a polynomial algebra is always algebraically independent (see [12, Prop. 6.2.1]). So, if E(Q) is a Bockstein closed 2-power extension, then the subalgebra generated by the k-invariants q_{1}, …, q_{n} is a polynomial algebra. We denote this subalgebra by 𝔽_{2}[q_{1}, …, q_{n}]. We have the following:

Proposition 5.3. Let E(Q) : 0 → V → G(Q) → W → 0 be a Bockstein closed, 2-power exact extension with dim(W) = n. Then, H^{∗}(W; 𝔽_{2}) is free as a 𝔽_{2}[q_{1}, …, q_{n}]-module. As 𝔽_{2}[q_{1}, …, q_{n}]-modules H^{∗}(W; 𝔽_{2}) ≅ 𝔽_{2}[q_{1}, …, q_{n}] ⊗ A^{∗}(Q) where A^{∗}(Q) is given the trivial module structure.

Proof. Let P = 𝔽_{2}[q_{1}, …, q_{n}] be the subalgebra generated by q_{1}, …, q_{n} in H^{∗}(W; 𝔽_{2}). Since A^{∗}(Q) is finite dimensional, H^{∗}(W; 𝔽_{2}) is finitely generated over P. For example, if we take Â^{∗}(Q) a 𝔽_{2}-vector subspace of H^{∗}(W; 𝔽_{2}) mapping 𝔽_{2}-isomorphically to A^{∗}(Q) under the projection H^{∗}(W; 𝔽_{2}) → A^{∗}(Q), then a homogeneous 𝔽_{2}-basis of Â^{∗}(Q) gives a set of generators of H^{∗}(W; 𝔽_{2}) as a P -module.

To prove this, one inducts on the degree of α ∈ H^{∗}(W; 𝔽_{2}). Let {a_{i} | i ∈ I} be a 𝔽_{2}-basis of Â^{∗}(Q). We want to show α is in the P -span of the a_{i}. For degree of α equal to one or two, this is immediate since the degree of q_{i} is 2 for all i. In general, by subtracting a suitable linear combination of a_{i}’s from α, we get an element β in the ideal (q_{1}, …, q_{n}). To show α is in the P -span of the a_{i}, it is enough to show β is. However β = ∑_{i=1}^{n} β_{i}q_{i} where the degree of the β_{i} is 2 less than the degree of β. By induction, each β_{i} and hence β is in the P -span of the a_{i} and so we are done.

Since H^{∗}(W; 𝔽_{2}) is a polynomial algebra, it is trivially Cohen-Macaulay. Since H^{∗}(W; 𝔽_{2}) is a finitely generated P -module, the fact that H^{∗}(W; 𝔽_{2}) is a free P -module follows from the fact that P is a polynomial algebra (see Theorem 5.4.10 of [2] for example). Thus H^{∗}(W; 𝔽_{2}) is a free P -module.

Finally if {b_{j} | j ∈ J} is a basis for the free P -module H^{∗}(W; 𝔽_{2}), then every element α ∈ H^{∗}(W; 𝔽_{2}) can be written uniquely in the form ∑_{j∈J} α_{j}b_{j} where α_{j} ∈ 𝔽_{2}[q_{1}, …, q_{n}]. Projecting to A^{∗}(Q), this shows every element of A^{∗}(Q) can be written uniquely as a span of the corresponding images of the b_{j}. In other words, the {b_{j} | j ∈ J} projects to a basis of A^{∗}(Q). Thus one can define a map of 𝔽_{2}[q_{1}, …, q_{n}]-modules

𝔽_{2}[q_{1}, …, q_{n}] ⊗ A^{∗}(Q) → H^{∗}(W; 𝔽_{2})

which is an isomorphism.

Now, consider the LHS-spectral sequence in mod-2 coefficients

E_{2}^{p,q} = H^{p}(W, H^{q}(V, 𝔽_{2})) ⇒ H^{p+q}(G(Q), 𝔽_{2})

associated to the extension E(Q). Let {x_{1}, …, x_{n}} denote a basis for the dual of W and let {t_{1}, …, t_{n}} be a basis for the dual of V.

Lemma 5.4. For each 1 ≤ k ≤ m, we have d_{2}(t_{k}) = q_{k}.

Proof. From standard theory, the central extension E(Q) corresponds to a principal BV -bundle BV → BG(Q) → BW with classifying map f : BW → BBV where BBV = K(V, 2). Also H^{2}(BBV, V) ≅ Hom(V, V) and q = f^{∗}(id) where id ∈ Hom(V, V) is the identity map.

In the LHS-spectral sequence for the fibration BV → EBV → K(V, 2), we have d_{2} : H^{1}(BV, V) → H^{2}(BBV, V) is an isomorphism (since EBV is contractible). In fact, it identifies H^{1}(BV, V) = H^{2}(BBV, V) = Hom(V, V), so d_{2}(id) = id. This identity pulls back to our fibration BV → BG(Q) → BW as d_{2}(id) = f^{∗}(id) = q. From this the lemma easily follows after taking a basis for V.

Now we are ready for some computations.

Theorem 5.5. Let E(Q) : 0 → V → G(Q) → W → 0 be a Bockstein closed, 2-power exact extension with dim(W) = n. Then

H^{∗}(G(Q); 𝔽_{2}) ≅ 𝔽_{2}[s_{1}, …, s_{n}] ⊗ A^{∗}(Q)

as graded algebras, where deg(s_{i}) = 2 for i = 1, …, n.

Proof. Consider the LHS-spectral sequence for the (central) extension E(Q). The E_{2}- page has the form

E_{2}^{∗,∗} = H^{∗}(W, 𝔽_{2}) ⊗ H^{∗}(V, 𝔽_{2}) = 𝔽_{2}[x_{1}, …, x_{n}] ⊗ 𝔽_{2}[t_{1}, …, t_{n}].

We have previously seen that d_{2}(t_{k}) = q_{k}. Here we have chosen basis for W,V and their duals as done previously. Note that d_{2}(t_{k}^{2}) = 0 and d_{3}(t_{k}^{2}) = β(q_{k}) by a standard theorem of Serre. Let

∧^{∗}(t_{1}, …, t_{n})

be the 𝔽_{2}-subspace of H^{∗}(V, 𝔽_{2}) generated by the monomials t_{1}^{ϵ_{1}} … t_{n}^{ϵ_{n}} where ϵ_{i} = 0, 1 for each i = 1, …, n. Then,

H^{∗}(V, 𝔽_{2}) ≅ 𝔽_{2}[t_{1}^{2}, …, t_{k}^{2}] ⊗ ∧^{∗}(t_{1}, …, t_{n})

as 𝔽_{2}-vector spaces. (Not as algebras!) Using Proposition 5.3, we also write

H^{∗}(W, 𝔽_{2}) ≅ 𝔽_{2}[q_{1}, …, q_{n}] ⊗ A^{∗}(Q)

as vector spaces. Thus as a differential graded complex, (E_{2}^{∗,∗}, d_{2}) splits as a tensor product

E_{2}^{∗,∗} ≅ 𝔽_{2}[t_{1}^{2}, …, t_{n}^{2}] ⊗ A^{∗}(Q) ⊗ (∧^{∗}(t_{1}, …, t_{n}) ⊗ 𝔽_{2}[q_{1}, …, q_{n}], d_{2})

where the differential on the first two tensor summands is trivial. By Künneth’s theorem,

E_{3}^{∗,∗} ≅ 𝔽_{2}[t_{1}^{2}, …, t_{n}^{2}] ⊗ A^{∗}(Q) ⊗ H^{∗}(∧^{∗}(t_{1}, …, t_{n}) ⊗ 𝔽_{2}[q_{1}, …, q_{n}], d_{2}).

Again by Künneth’s theorem, the final term is just H^{∗}(pt, 𝔽_{2}) since it breaks up as the tensor of

H^{∗}(∧^{∗}(t_{j}) ⊗ 𝔽_{2}[q_{j}], d_{2}(t_{j}) = q_{j})

which is the cohomology of a point. Thus

E_{3}^{∗,∗} = 𝔽_{2}[t_{1}^{2}, …, t_{n}^{2}] ⊗ A^{∗}(Q)

with d_{3}(t_{j}^{2}) = β(q_{j}). Since Q is Bockstein closed, β(q_{j}) = 0 in A^{∗}(Q) and so d_{3}(t_{j}^{2}) = 0. Thus E_{3}^{∗,∗} = E_{4}^{∗,∗}. By dimensional considerations, there can be no further differentials in the spectral sequence and so we see E_{∞}^{∗,∗} = E_{3}^{∗,∗}.

This says, in particular, that there are elements s_{i} ∈ H^{∗}(G, 𝔽_{2}) such that

res_{V}^{G}(s_{i}) = t_{i}^{2}.

Since the t_{i}’s are algebraically independent in H^{∗}(V, 𝔽_{2}), we conclude that the s_{i}’s are algebraically independent in H^{∗}(G, 𝔽_{2}). Thus when we define an 𝔽_{2}-vector space homomorphism

𝔽_{2}[s_{1}, …, s_{n}] ⊗ A^{∗}(Q) → H^{∗}(G, 𝔽_{2})

using the inclusion map on the first factor and the inflation map on the second, we will have a well-defined map of algebras. (Note that the inflation map is always an algebra map.) Finally by the structure of E_{∞}^{∗,∗}, it is clear that this map is onto, and since our algebras have finite type, this means that it is an isomorphism of algebras as desired.

Remark 5.6. Note that although we assumed that the extension E(Q) is 2-power exact in the above calculation, we only use the fact that the k-invariants q_{1} …, q_{n} form a regular sequence. If E(Q) : 0 → V → G(Q) → W → 0 is a Bockstein closed extension with dim V = n, dim W = m such that the k-invariants q_{1} …, q_{n} form a regular sequence, then the k-invariants will be algebraically independent in 𝔽_{2}[x_{1}, …, x_{m}], in particular, they will be linearly independent. This shows that in this case, E(Q) is a Frattini extension, i.e., {Q(w) | w ∈ W } generates V. Also, we must have n ≤ m for dimension reasons. The assumption that Q is Bockstein closed will imply that the variety of I(Q) is 𝔽_{2}-rational, so W must have a m − n dimensional subspace W′ such that Q restricted to W′ is zero. But it may happen that this W′ has nontrivial commutators with the rest of the elements in W. So, we can not conclude that G(Q) splits as G(Q) ≅ G′ × ℤ/2. Hence, these groups are still interesting and the calculation above shows that the mod-2 cohomology of these groups is also in the form H^{∗}(G, 𝔽_{2}) ≅ A^{∗}(Q) ⊗ 𝔽_{2}[s_{1}, ..., s_{n}].

The group extensions associated to the strictly upper triangular matrices 𝔲_{n}(𝔽_{2}) are Bockstein closed 2-power exact extensions (see Example 3.3). So, we have a complete calculation for the mod-2 cohomology of these groups. To calculate the Bockstein’s of the generators of mod-2 cohomology of a 2-power exact extension, we need to consider also the Eilenberg-Moore spectral sequence associated to the extension.

6 Eilenberg-Moore Spectral Sequence

In this section we study the Eilenberg-Moore Spectral Sequence of a Bockstein closed 2-power exact extension

E(Q) : 0 → V → G(Q) → W → 0

associated to a quadratic map Q : W → V. Although we have already found the algebra structure of the mod-2 cohomology of G(Q) as

H^{∗}(G(Q); 𝔽_{2}) ≅ 𝔽_{2}[s_{1}, …, s_{n}] ⊗ A^{∗}(Q)

using the LHS-spectral sequence, we find the EM-spectral sequence more useful in studying the Steenrod algebra structure of H^{∗}(G(Q); 𝔽_{2}). In fact, the theorems laid out in Larry Smith’s paper [10] directly compute the behavior of the EM-spectral sequence in our case and give us the Steenrod structure we desire. In order to give a fuller picture of the underlying topology, we summarize some essentials of the EM-spectral sequence here.

Consider a pullback square of spaces

X ×_{B} Y ⟶ Y
↓  ↓^{p}
X ⟶^{f} B

where p : Y → B is a fibration and B is simply-connected. The space X ×_{B} Y is given as

X ×_{B} Y = {(x, y) ∈ X × Y | f(x) = p(y)}

and can be viewed as an amalgamation of X and Y over B. Fix k a field, and let C^{∗}(X) denote the dga (differential graded algebra) given by the cochain complex of X with coefficients in k. One can show that the natural map α : C^{∗}(X) ⊗_{C^{∗}(B)} C^{∗}(Y) → C^{∗}(X ×_{B} Y) yields an isomorphism

Tor_{C^{∗}(B)}(C^{∗}(X), C^{∗}(Y)) ≅ H^{∗}(X ×_{B} Y)

of algebras. (Though one should be careful in interpreting the algebra structure of the Tor term.) For more details on this isomorphism see [10, Thm 3.2].

One then can show algebraically, through various filtrations of resolutions computing the Tor term above, that there is a spectral sequence starting at

E_{2}^{∗,∗} = Tor_{H^{∗}(B)}(H^{∗}(X), H^{∗}(Y))

converging to Tor_{C^{∗}(B)}(C^{∗}(X), C^{∗}(Y)) ≅ H^{∗}(X ×_{B} Y).

Explicitly, this can be constructed as a second quadrant spectral sequence in the (p, q)-plane using the bar resolution. The picture of the E_{1}-page using the bar resolution is as follows: on the p = −n line, one has the algebra

H^{∗}(X) ⊗ H̄^{∗}(B) ⊗ ⋯ ⊗ H̄^{∗}(B) ⊗ H^{∗}(Y)

where all tensor products are over k, H̄^{∗} denotes the positive degree elements of H^{∗}, and n factors of H̄^{∗}(B) are used in the above tensor product. We place the graded complex above on the p = −n line in such a way that the elements of total degree q are placed at the (−n, q) lattice point in the (p, q)-plane. Let [a|b_{1}| … |b_{n}|c] be short hand for a ⊗ b_{1} ⊗ ⋯ ⊗ b_{n} ⊗ c. The differential d_{1} is horizontal moving one step to the right and is given explicitly in characteristic 2 (we will only use this case and we do this also to avoid stating the signs!) by:

d_{1}([a|b_{1}| … |b_{n}|c]) = [af^{∗}(b_{1})|b_{2}| … |b_{n}|c]
+Σ_{i=1}^{n−1}[a|b_{1}| … |b_{i}b_{i+1}| … |b_{n}|c]
+[a|b_{1}| … |b_{n−1}|p^{∗}(b_{n})c]

for all a ∈ H^{∗}(X), b_{i} ∈ H̄^{∗}(B), c ∈ H^{∗}(Y).

The power of the EM-spectral sequence is the availability of this geometric resolution to represent its E_{1}-term. For example the p = −1 line can be interpreted as a portion of H^{∗}(X × B × Y). If A is the kernel of d_{1} on this line, then A is the set of elements of the form [a|b|c] satisfying

[af^{∗}(b)|c] = [a|p^{∗}(b)c],

and the elements of A are permanent cycles in the EM-spectral sequence. As shown in [10], there is a Steenrod module structure on the p = −1 and p = 0 lines via the natural identification of them inside H^{∗}(X × B × Y) and H^{∗}(X × Y) respectively. Furthermore, this Steenrod module structure persists through all pages of the spectral sequence. Finally the associated filtration on H^{∗}(X ×_{B} Y) is a filtration of Steenrod modules and components of the associated graded module agree with the Steenrod module structure on the E_{∞}-page on the p = 0 and p = −1 lines of the EM-spectral sequence with the bar resolution. In fact various authors have shown that the whole EM-spectral sequence has a natural structure of a module over the Steenrod algebra (using the bar resolution model) with “vertical” and “diagonal” Steenrod operations (corresponding to the fact that the EM-spectral sequence is not an unstable module over A_{2}). We won’t need these here - we will only need the structure on the p = 0 and p = −1 lines which was already laid out in [10] very naturally.

The bar resolution mentioned above is a nice natural resolution that can be used to compute E_{2}^{∗,∗} = Tor_{H^{∗}(B)}(H^{∗}(X), H^{∗}(Y)) in the EM-Spectral sequence and has the benefits of being “natural” and “geometric” and hence can be used to study natural operations on the spectral sequence. However typically this resolution is too big to carry out computations directly. In the case where H^{∗}(B) is a polynomial algebra, a smaller Koszul resolution is available to compute this tor-term and the E_{2}-page is more tractable. Finally if the ideal generated by f^{∗}(H^{∗}(B)) in H^{∗}(X) is nice enough, then a change of rings isomorphism can also be used to simplify the calculations immensely. For details see [10]. We will summarize the main result below after a necessary definition.

Definition 6.1. Let k be a field and Λ be a graded commutative algebra over k. An ideal I in Λ is called a Borel ideal if there is a regular sequence {x_{1}, x_{2}, …, x_{n}, … } (either finite or infinite) that generates the ideal I. Recall that a regular sequence {x_{1}, x_{2}, …, x_{n} … } is one such that x_{1} is not a zero divisor in Λ and x_{i+1} is not a zero divisor of Λ/(x_{1}, …, x_{i}) for all i ≥ 1.

Theorem 6.2 (Collapse Theorem). Let

X ×_{B} Y ⟶ Y
↓  ↓^{p}
X ⟶^{f} B

be a pullback square of spaces, with B simply connected, p a Serre fibration. Furthermore, assume that

(i) p^{∗} : H^{∗}(B) → H^{∗}(Y) is onto, (ii) ker(p^{∗}) is a Borel ideal of H^{∗}(B), and (iii) Im(f^{∗}) generates a Borel ideal J of H^{∗}(X). Then, the associated Eilenberg-Moore spectral sequence collapses at the E_{2}-page and we have

E_{2}^{∗,∗} = E_{∞}^{∗,∗} ≅ (H^{∗}(X)/J) ⊗ Λ^{∗}(u_{1}, …) ≅ H^{∗}(X ×_{B} Y)

where Λ^{∗}(u_{1}, …) is an exterior algebra on generators all of which lie on the p = −1 line of the spectral sequence.

Proof. See Theorem 3.1 in [10].

Remark 6.3. Note that in the above theorem, the final isomorphism is an isomorphism of k-vector spaces in general due to lifting issues when lifting the algebra structure over the associated filtrations. We will come back to this in our specific example later.

In our case we are interested in studying the cohomology of groups G(Q) given by central extensions

E(Q) : 0 → V → G(Q) → W → 0

where V and W are elementary abelian. We have seen that the category of such extensions is naturally equivalent to the category of quadratic maps from W to V. However it is basic in group cohomology that the equivalence classes of extensions of this form are uniquely specified by a cohomology class in H^{2}(W, V). By the representability of cohomology, H^{2}(W, V) is itself isomorphic to [BW, BBV] where [−, −] denotes homotopy classes of maps, BG denotes the classifying space of a monoid G, and BBV = K(V, 2).

Under this correspondence we will denote the homotopy class representing a specific quadratic form Q : W → V by BQ : BW → BBV. Thus we have a pullback square

BG(Q) ⟶ EBV
↓  ↓^{p}
BW ⟶^{BQ} BBV

for which we have an EM-spectral sequence

E_{2}^{∗,∗} = Tor_{H^{∗}(BBV)}(H^{∗}(BW), 𝔽_{2}) ⇒ H^{∗}(G(Q), 𝔽_{2}).

In fact the same pullback square above exhibits BG(Q) as the homotopy fiber of the map BQ. We record these observations:

Proposition 6.4. Let V, W be elementary abelian 2-groups. Then there is a bijective correspondence between equivalence classes of quadratic forms Q : W → V and [BW, BBV]. If BQ : BW → BBV is the homotopy class of maps associated to a quadratic form Q : W → V then BG(Q) = HF (BQ) where HF stands for homotopy fiber.

Similarly, there is a natural equivalence between n-ary forms W → V and homotopy classes of maps [BW, B^{n}V] where B^{n}V = K(V, n). However in this case the homotopy fiber is not an Eilenberg-MacLane space but has a two stage Postnikov tower.

It is well-known that if dim(V) = 1 then the characteristic element κ ∈ H^{n}(K(V, n)) is a generator of mod-2 cohomology H^{∗}(K(V, n)) as a free module over the Steenrod algebra A_{2}. As an algebra H^{∗}(K(V, n)) is a polynomial algebra on all “permissable Steenrod operation sequences” on κ of excess less than n. The dim(V) > 1 case follows similarly from Künneth’s theorem. Using these facts, one sees that conditions (i) and (ii) in Theorem 6.2 hold trivially in this case. Thus it remains to consider condition (iii).

When Q is Bockstein closed, it is clear that the ideal J generated by Im(BQ)^{∗} is exactly the ideal I(Q) generated by the components of Q. This is because the Bockstein is Sq^{1} and all other Steenrod squares on these components are determined as they have degree 2 and automatically lie in I(Q). Thus in the case of a Bockstein closed quadratic form, J is a Borel ideal if and only if I(Q) is. This yields the following theorem immediately.

Theorem 6.5 (EM-SS collapse for G(Q)). Let E(Q) : 0 → V → G(Q) → W → 0 be a Bockstein closed extension associated to the quadratic map Q : W → V. Assume that I(Q) is a Borel ideal, i.e., {q_{1}, …, q_{n}} is a regular sequence in H^{∗}(BW). This holds for example when Q is Bockstein closed and 2-power exact. Then, the EM-spectral sequence collapses at the E_{2}-page and

E_{2}^{∗,∗} = E_{∞}^{∗,∗} = A^{∗}(Q) ⊗ Λ^{∗}(u_{ij}).

This lifts to give H^{∗}(G(Q), 𝔽_{2}) ≅ A^{∗}(Q) ⊗ Λ^{∗}(û_{ij}) as 𝔽_{2}-vector spaces (not as algebras). Here A^{∗}(Q) = H^{∗}(W)/I(Q) and the u_{ij} are exterior Koszul generators corresponding to the polynomial subalgebra 𝔽_{2}[κ_{ij} | i = 1, …, n, j ≥ 0] of H^{∗}(K(V, 2)). Specifically, κ_{ij} = Sq^{2^{j}} ⋯ Sq^{2}Sq^{1}κ_{i} were κ_{i} is the i-th characteristic element in H^{2}(K(V, 2)). Thus u_{ij} lies in bidegree (−1, 2^{j+1} + 1) in the EM-spectral sequence.

Proof. See the discussion above. The only thing that was not explained was the selection of the exterior elements. This comes from the fact that (BQ)^{∗}(κ_{i}) = q_{i} and a change of rings isomorphism. For details see [10].

The reader might have noticed that when we compare the result of the EM-spectral sequence calculation with the result previously obtained with the LHS-spectral sequence, they look different. The LHS-spectral sequence gave under the same conditions that

H^{∗}(G(Q), 𝔽_{2}) ≅ A^{∗}(Q) ⊗ 𝔽_{2}[s_{1}, …, s_{n}]

as algebras. However the EM-spectral sequence above shows that

H^{∗}(G(Q), 𝔽_{2}) ≅ A^{∗}(Q) ⊗ Λ^{∗}(û_{ij})

as vector spaces. The point is this last isomorphism is not of algebras and the mentioned exterior algebra is isomorphic to the polynomial algebra 𝔽_{2}[s_{1}, …, s_{n}] as 𝔽_{2}-vector spaces. Indeed note that û_{ij} lives in degree (2^{j+1} + 1) − 1 = 2^{j+1}. The vector space isomorphism comes from mapping û_{ij} to s_{i}^{2^{j}} and extending to square free powers of the û_{ij} as if making an algebra map. To see this is indeed an isomorphism of graded vector spaces just note that a power of s_{i} say s_{i}^{n} can be expressed uniquely as a (square free) product of 2-power powers of s_{i} using the 2-adic expansion of n. For example s_{3}^{5} = s_{3}^{4}s_{3}^{1} and hence would be associated to û_{32}û_{30} since 2^{2} = 4 and 2^{0} = 1. We record this vector space isomorphism for reference.

Proposition 6.6 (Exterior-Polynomial vector space isomorphism). If

Λ^{∗} = Λ^{∗}(û_{ij} | i = 1, …, n, j ≥ 0)

and P^{∗} = 𝔽_{2}[s_{1}, …, s_{n}] with |s_{i}| = 2, |û_{ij}| = 2^{j+1} then Λ^{∗} ≅ P^{∗} as graded vector spaces via the isomorphism that takes û_{ij} to s_{i}^{2^{j}} extended to the canonical vector space basis of Λ^{∗} in the obvious way.

7 Proof of Theorem 1.1

In this section we prove Theorem 1.1 using the Eilenberg-Moore spectral sequence calculations given in the previous section.

Before the proof, we first discuss the Steenrod algebra structure of the EM-spectral sequence. For this purpose we will want to use the bar resolution and not the smaller Koszul resolutions used previously as it is more “geometric”. This bar resolution has an induced Steenrod module structure on the p = −1 and p = 0 lines that induces the Steenrod module structure on H^{∗}(G(Q)). This is proven as Corollary 4.4 in [10]. In fact, we will work out representatives of the u_{i0} in the bar resolution and work out a formula for Sq^{1} on them. Just for dimensional reasons, if we throw the u_{i0} into a vector u_{0}, one has Sq^{1}(u_{0}) = Ru_{0}+ terms in lower filtration where R ∈ Hom(W, End(V)), or intuitively, is a matrix with entries in A^{1}(Q). By results in [10], this lifts to the identity Sq^{1}(s) = Rs+term in A^{3}(Q). Thus, if the calculation for Sq^{1}(u_{0}) gives R = L, then we will have the proof of Theorem 1.1.

Let us first set out finding explicit representatives of the u_{i0} in the E_{1} term of the bar resolution. Since we know they survive to E_{∞}^{−1,3} we know they will have to be permanent cycles.

In our case, the p = −1 line of the EM-spectral sequence (in the bar resolution) is H^{∗}(BW) ⊗ H̄^{∗}(BBV) ⊗ k. Let us write H^{∗}(BW) = 𝔽_{2}[x_{1}, …, x_{n}]. Then Sq^{1}(q_{i}) = Σ_{j}L_{ij}q_{j} where L_{ij} ∈ H^{1}(W) since our quadratic form is Bockstein closed. Using the usual bar notation, let us define elements v_{i} = [1|κ_{i0}|1] + Σ_{j}[L_{ij}|κ_{j}|1] ∈ E_{1}^{−1,3} where κ_{i0} = Sq^{1}(κ_{i}) and κ_{i} is the i-th characteristic element of H^{∗}(BBV). Let us compute d_{1}(v_{i}). Let p : EBV → BBV be the path loop fibration then

d_{1}(v_{i}) = ([1(BQ)^{∗}(κ_{i0})|1] + [1|p^{∗}(κ_{i0})1]) + Σ_{j}([L_{ij}(BQ)^{∗}(κ_{j})|1] + [L_{ij}|p^{∗}(κ_{j})1])
= [Sq^{1}(q_{i})|1] + Σ_{j}[L_{ij}q_{j}|1]
= [Sq^{1}(q_{i}) + Σ_{j}L_{ij}q_{j}|1]
= 0.

Since v_{i} lies on the p = −1 line, all further differentials on v_{i} must vanish for dimensional reasons and so indeed the v_{i} are permanent cycles. Also note that since E_{1}^{−2,3} = 0 for dimensional reasons, the vector space spanned by the v_{i} is not the image of d_{1}. Similarly for dimensional reasons it cannot be the image of any d_{r}. (This is because one has to use s tensors of H̄^{∗}(BBV) in the line p = −s and hence the line vanishes below q = 2s.) Thus the space spanned by the v_{i} embeds into E_{∞}^{−1,3}. However comparing with the Koszul resolutions E_{∞}^{−1,3} term we see then that the span of the v_{i} must be the same as the span of the u_{i0} in E_{∞}. Thus, without loss of generality we may assume (by changing our basis for the span of the s_{i}) that v_{i} represents u_{i0}. Thus it remains to compute Sq^{1}(v_{i}) to figure out Sq^{1}(s_{i}) up to terms of lower filtration.

We now carry out the computation of Sq^{1}(v_{i}) using Larry Smith’s work showing that Sq^{1} on H^{∗}(BW) ⊗ H̄^{∗}(BBV) ⊗ 𝔽_{2} naturally induced by viewing that p = −1 line as H^{∗}(BW × BBV × pt) is compatible with the Sq^{1} action on the cohomology of the total space H^{∗}(BG(Q), 𝔽_{2}). Note v_{i} = [1|κ_{i0}|1] + Σ_{j}[L_{ij}|κ_{j}|1]. Here recall that [a|b|c] can be identified with the cross product a × b × c in H^{∗}(BW × BBV × pt). Using the Cartan formula for Steenrod squares on cross products, we see that Sq^{1} will operate like a derivation and so we get

\begin{aligned}Sq^{1}(v_{i}) & = [1|Sq^{1}(κ_{i0})|1] + Σ_{j}([Sq^{1}(L_{ij})|κ_{j}|1] + [L_{ij}|Sq^{1}(κ_{j})|1]) \\ & = Σ_{j}([L_{ij}^{2}|κ_{j}|1] + [L_{ij}|κ_{j0}|1]) \end{aligned}

since Sq^{1}(κ_{i0}) = Sq^{1}(Sq^{1}(κ_{i})) = 0 and Sq^{1}(L_{ij}) = L_{ij}^{2} since the degree of L_{ij} is one. Since the p = −1 line is a module over the p = 0 line, we can write

[L_{ij}|κ_{j0}|1] = L_{ij}[1|κ_{j0}|1] = L_{ij}(v_{j} + ∑_{k}[L_{jk}|κ_{k}|1]).

Plugging this into the above equations and simplifying, one gets

Sq^{1}(v_{i}) = ∑_{j} L_{ij}v_{j} + (∑_{j} L_{ij}^{2}[1|κ_{j}|1] + ∑_{j} ∑_{k} L_{ij}L_{jk}[1|κ_{k}|1]).

Let v be the column vector with i-th coordinate v_{i}, and let L be the matrix with (i, j)- entry L_{ij} and c be the column vector with i-th coordinate [1|κ_{i}|1]. Then the above equation becomes

Sq^{1}(v) = Lv + (Sq^{1}(L) + L^{2})c

where Sq^{1} on a matrix just means apply it on each entry. In the next theorem, we will show that the second term (Sq^{1}(L) + L^{2})c in the above equation is zero in E_{2}^{∗,∗} = E_{∞}^{∗,∗} by showing that it is a boundary under d_{1} and this then shows that Sq^{1}(v) = Lv in E_{∞}^{∗,∗} which lifts to show Sq^{1}(s) = Ls + η with η a column matrix with entries in A^{3}(Q).

Theorem 7.1 (Steenrod Structure). Let Q : W → V be a Bockstein closed quadratic map whose components q_{i} form a regular sequence in H^{∗}(W) (for example if it is 2-power exact). Let dim(V) = n, then H^{∗}(G(Q)) ≅ k[s_{1}, …, s_{n}]⊗A^{∗}(Q) as algebras. If we write β(q) = Sq^{1}(q) = Lq where q is the column vector with entries the components of Q (we can write this as Q is Bockstein closed) then we have:

β(s) = Ls + η

where η is a column vector with entries in A^{3}(Q) and s is the column vector with entries the s_{i}. Since A^{∗}(Q) is the image of H^{∗}(BW) → H^{∗}(BG(Q)), this determines the structure of H^{∗}(G(Q)) over the Steenrod algebra completely as all Steenrod operations can be determined from this information and the axioms of the Steenrod algebra.

Proof. In the paragraph before the statement of this theorem, it was shown in the bar resolution that

Sq^{1}(v) = Lv + (Sq^{1}(L) + L^{2})c

in E_{1}^{−1,4} of the bar resolution model of the Eilenberg-Moore spectral sequence. Furthermore we showed that v consists of permanent cycles that survive and represent s. Thus to get the main formula of the theorem it is sufficient to show (Sq^{1}(L) + L^{2})c is zero in E_{2}^{−1,4} = E_{∞}^{−1,4} as this will give

Sq^{1}(v) = Lv

which will lift to

Sq^{1}(s) = Ls + η

where η’s components have to live in lower filtration, i.e., on E_{∞}^{0,3} = A^{3}(Q). This will prove the theorem as the other statements have been proven in previous sections. Thus it remains to show that (Sq^{1}(L) + L^{2})c is a boundary under d_{1}.

Note that by the regularity of the sequence q_{1}, …, q_{n}, the equality [β(L) + L^{2}]q = 0 gives that

β(L) + L^{2} = ∑_{i} T_{i}q_{i}

for some scalar matrices T_{1}, .., T_{n} (see Proposition 8.2 in [7]). Note that [B(L)+L^{2}]q = 0 gives that for all k,

∑_{i} ∑_{j} T_{i}(k, j)q_{i}q_{j} = 0.

Hence, T_{i}(k, i) = 0 for every k and i, and T_{i}(k, j) = T_{j}(k, i) for every 1 ≤ i, j ≤ n. Here we are using the fact that the set

{q_{i}q_{j} | 1 ≤ i < j ≤ n}

is linearly independent in H^{∗}(W) as the q_{i} are algebraically independent since they form a regular sequence. Let us call this common entry

a_{k}(i, j) = T_{i}(k, j) = T_{j}(k, i).

The a_{k} are then (skew) symmetric scalar matrices with zeros down the diagonal as a_{k}(i, i) = T_{i}(k, i) = 0. Since we are in characteristic 2, we can view these as skew symmetric matrices which will be useful in what follows.

Let

α_{k} = ∑_{i,j} a_{k}(i, j)[1|κ_{i}|κ_{j}|1]

in E_{1}^{−2,4}. Then,

d_{1}(α_{k}) = ∑_{i,j} a_{k}(i, j)([q_{i}|κ_{j}|1] + [1|κ_{i}κ_{j}|1]) = ∑_{i,j} a_{k}(i, j)[q_{i}|κ_{j}|1]

as the second term in the first sum is zero due to the skew-symmetry of the matrices a_{k}. Thus using that the p = −1 line is a module over the p = 0 line with an action that is purely in the left slot we get

d_{1}(α_{k}) = ∑_{ij} a_{k}(i, j)q_{i}[1|κ_{j}|1].

Using that ∑_{i} a_{k}(i, j)q_{i} = ∑_{i} T_{i}(k, j)q_{i} is the (k, j)-entry of B(L) + L^{2} we get:

d_{1}(α_{k}) = ∑_{j}(B(L) + L^{2})_{k,j}[1|κ_{j}|1].

Writing c to be the column vector with components [1|κ_{j}|1] as before and α to be the column vector with entries the α_{k}, then this reads as

d_{1}(α) = (B(L) + L^{2})c

which shows that indeed the desired term is a d_{1} boundary and hence completes the proof of the theorem.

8 An obstruction class for uniform double lifting

Let E(Q) : 0 → V → G(Q) → W → 0 be a Bockstein closed 2-power exact extension with associated quadratic map Q : W → V. Recall that this means dim W = dim V = n, and the quadratic map Q is Frattini and effective. Note that in this case k-invariants q_{1}, …, q_{n} form a regular sequence, and as a consequence there is a unique L ∈ Hom(W, End(V)) such that β(q) = Lq (see Proposition 8.1 of [7]).

First we make a simple observation about L. Note that applying the Bockstein operator to the equation β(q) = Lq gives

0 = β(L)q + Lβ(q) = [β(L) + L^{2}]q.

Since q_{1}, …, q_{n} forms a regular sequence, the entries of β(L) + L^{2} must be linear combinations of components of q (see Proposition 8.2 in [7]). This means that linear map L : W → End(V) extends to a morphism ρ_{L} : Q → Q_{𝔤𝔩(V)} in Quad. So, L defines a representation of Q. Let us denote this representation also with L.

Now, we consider the cohomology ring of the group G(Q). By Theorem 5.5, we have

H^{∗}(G(Q); 𝔽_{2}) ≅ 𝔽_{2}[s_{1}, …, s_{n}] ⊗ A^{∗}(Q)

as graded algebras, where deg(s_{i}) = 2 for i = 1, …, n. In the previous section, we calculated the images of Bockstein operator on s_{i}’s. We obtained that

β(s) = Ls + η

where η is a column vector whose entries are in H^{3}(W, 𝔽_{2}). Applying the Bockstein operator again, we get

0 = β(L)s + Lβ(s) + β(η) = [β(L) + L^{2}]s + [β(η) + Lη].

Since, we already know that β(L) + L^{2} = 0 in A^{∗}(Q), we obtain that

β(η) + Lη = 0

in A(Q)^{∗}. This shows that shows that η is a 3-dimensional cocycle in the chain complex C^{∗}(Q, L). So, it defines a three dimensional cohomology class [η] ∈ H^{3}(Q, L). We will now show that this class is an obstruction class for uniformly lifting the extension E(Q) twice.

Let E′ : 0 → V → Γ → G → 0 denote the extension with extension class s ∈ H^{2}(G, V) where G = G(Q). Since res_{V}^{G} s_{i} = t_{i}^{2} for all i, the restriction of this extension to V ⊆ G gives the extension 0 → V → M → V → 0 where M ≅ (ℤ/4)^{n} as an abelian group. Let 0 → V → M → V → 0 be the ℤ/4W -lattice whose logarithm is L : W → End(V). By this we mean that the representation for M is of the form w → I + 2L(w) mod 4 (see Lemma 4.3 [7] for details). By Theorem 4.6 in [7], E′ lifts to an extension

E′′ : 0 → M → Γ̃ → G → 0

if and only if β(s) = Ls. So, η = β(s) + Ls is the obstruction for such a lifting. But, the extension E′ is determined by our choices of generators s_{i}. We can replace each s_{i} with

s′_{i} = s_{i} + ξ_{i}

for some ξ_{i} ∈ H^{1}(W, 𝔽_{2}) and the resulting extension E′ will still satisfy the above condition. It may happen that for this new extension E′, the obstruction for lifting it to E′′ is zero. Let us calculate the new obstruction

η′ = β(s′) + Ls′ = β(s + ξ) + L(s + ξ) = η + (β(ξ) + Lξ).

Note that the class β(ξ) + Lξ is equal to δ(ξ) where δ is the boundary map of the chain complex C^{∗}(Q, L). So, we proved that in the set of all possible extensions E′ : 0 → V → Γ → G → 0 whose restriction to V ⊆ G gives the extension 0 → V → M → V → 0, there is at least one extension E′ that lifts to E′′ of the form

E′′ : 0 → M → Γ̃ → G → 0

if and only if η is a coboundary, i.e., the cohomology class [η] = 0 in H^{3}(Q, L). So, the proof of Theorem 1.2 is complete.

9 The Bockstein spectral sequence

Fix a prime p and let 𝔽_{p} be the field with p elements. The Bockstein spectral sequence is a well-known technique of relating the mod-p cohomology of a space with its integral cohomology. It arises from the Massey exact triple

\xymatrix{H^{∗}(X; ℤ) \ar[rr]^{·p} &  & H^{∗}(X; ℤ) \ar[dl]^{φ} \\  & B_{1}^{∗} = H(X, 𝔽_{p}) \ar@{-->}[ul]^{β̂}}

associated to the sequence of coefficients 0 → ℤ → ℤ → 𝔽_{p} → 0. Here β̂ is the integral Bockstein operator and φ is the map induced by mod p reduction. The composition φ∘β̂ yields the differential given by the standard Bockstein β on H^{∗}(X; 𝔽_{p}). If we denote the cohomology of this differential by B_{2}^{∗} then the Massey exact triple above yields an exact triple

\xymatrix{pH^{∗}(X; ℤ) \ar[rr]^{·p} &  & pH^{∗}(X; ℤ) \ar[dl]^{φ_{2}} \\  & B_{2}^{∗} \ar[ul]^{β̂_{2}}}

and by iteration we get Massey exact triples of the form

\xymatrix{p^{r}H^{∗}(X; ℤ) \ar[rr]^{·p} &  & p^{r}H^{∗}(X; ℤ) \ar[dl]^{φ_{r+1}} \\  & B_{r+1}^{∗} \ar[ul]^{β̂_{r+1}}}

where φ_{r+1}∘β̂_{r+1} = β_{r+1} gives a differential on B_{r+1}^{∗} called the (r+1)-st higher Bockstein. The collection of the Massey exact triangles above is referred to as the Bockstein spectral sequence for X at the prime p.

Furthermore when X is the classifying space of a finite p-group, it is known that B_{∞}^{∗} = H^{∗}(pt, 𝔽_{p}) is the limit of the B_{r}^{∗}. Note if s > 0 and exp(H^{s}(X; ℤ)) = p^{r+1}, then the multiplication by p in this last diagram is zero. Hence φ_{r+1} gives an injection 0 ≠ p^{r}H^{s}(X; ℤ) → B_{r+1}^{s} and hence B_{r+1}^{s} ≠ 0. On the other hand it is clear that B_{r+2}^{s} = 0 from the next Massey triangle. Thus it is not hard to see that exp(H^{s}(X; ℤ)) = p^{r+1} if and only if B_{r+1}^{s} ≠ 0 but B_{r+2}^{s} = 0. So, by seeing how quickly any level of the Bockstein spectral sequence converges to zero, we can determine the exponent of the integral cohomology of the p-group at that level.

Now we have seen that if Q : W → V is a quadratic map associated to a Bockstein closed 2-power exact sequence E(Q) : 0 → V → G(Q) → W → 0 which lifts uniformly twice then

H^{∗}(G(Q); 𝔽) ≅ 𝔽[s_{1}, …, s_{n}] ⊗ A^{∗}(Q)

where n = dim(W). As Q is Bockstein closed, we may write βq = Lq for L ∈ Hom(W, End(V)) as usual and we have seen in this case we can choose s so that βs = Ls also. Thus (B_{1}^{∗}(G(Q)), β_{1}) is completely known from the matrix L given from the Bockstein closed quadratic form Q.

To compute B_{2}^{∗}, we decompose the polynomial algebra 𝔽[s_{1}, …, s_{n}] = ⊕_{i=0}^{∞}S^{i} into its homogeneous components S^{i}. We then get a decomposition of differential algebras:

(B_{1}^{∗}(G(Q)), β_{1}) = ⊕_{i=0}^{∞}(S^{i} ⊗ A^{∗}(Q), β)

and thus

B_{2}^{∗}(G(Q)) = ⊕_{i=0}^{∞}H^{∗}(S^{i} ⊗ A^{∗}(Q), β).

Note that B_{2}^{∗}(G(Q)) contains the information for the distribution of all the higher torsion (of exponent greater than p) for H^{∗}(G(Q); ℤ). Note that S^{0} = 𝔽_{2}, so the term corresponding to i = 0 is just the cohomology of A^{∗}(Q) under β which is H^{∗}(Q; 𝔽_{2}).

Note that for the summand coming from S^{1}, we have a differential given by δ(f) = βf +L(f) and so the cohomology of that term corresponds to H^{∗}(Q; L) where L denotes the Q-module associated to the matrix L. However since the elements of S^{1} are degree 2 elements, the contribution of H^{∗}(Q; L) will be in degree ∗ + 2. Since S^{i} is just the i-fold symmetric product of S^{1}, the complexes for these terms are derived from those for S^{1}, and we will denote the corresponding matrix and Q-module by Sym^{i}(L). Note also that since the elements of S^{i} have degree 2i, their contribution will be shifted in degree by 2i. By these observations we can conclude the following:

Theorem 9.1. Let G(Q) be a group associated to a Bockstein closed, 2-power exact quadratic form with β(q) = Lq. Furthermore suppose Q : W → V can be lifted uniformly twice. Then

H^{∗}(G(Q); 𝔽_{2}) ≅ 𝔽_{2}[s_{1}, …, s_{n}] ⊗ A^{∗}(Q)

where n = dim(W) and the higher torsion is given by

B_{2}^{∗} ≅ ⊕_{i=0}^{∞}H^{∗−2i}(Q, Sym^{i}(L)).

Thus, the higher torsion in G(Q) is computable from the cohomology of the quadratic form Q in coefficients given by various symmetric powers of the defining module L.

Note that for each j ≥ 0, the j-th term B_{2}^{j} in the above formula is given by a finite direct sum since there are only a finite number of nonzero coefficient modules Sym^{i}(L) in the sum. Furthermore, note that the cochain complex used to compute H^{∗}(Q, Sym^{i}(L)) has length always given by the length of the cochain complex A^{∗}(Q) which is finite. Thus the equation above does yield an algorithm to compute B_{2}^{j} for every j ≥ 0.

Acknowledgement. We thank the referee for careful reading of the paper and for many helpful comments.

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Dept. of Mathematics University of Rochester, Rochester, NY 14627 U.S.A. E-mail address: jonpak@math.rochester.edu

Dept. of Mathematics Bilkent University Ankara, 06800, Turkey. E-mail address: yalcine@fen.bilkent.edu.tr