THE A-MODULE STRUCTURE FOR THE COHOMOLOGY OF TWO STAGE SPACES.

LEIF KRISTENSEN AND ERIK KJÆR PEDERSEN

0. Introduction

The purpose of the present paper is to give a general method for determining the structure of the cohomology of stable two-stage space (for definitions see Section 1) as a module over the Steenrod algebra A. For convenience we only consider cohomology with coefficients in the integers in Z_{2}, the integers mod 2. The general part of our paper, however, works equally well with arbitrary coefficients. A two-stage space with stable k-invariant is an Hspace and the cohomology consequently a Hopf algebra. We also evaluate the diagonal in the cohomology.

The idea of the paper is to express the A-module structure of the cohomology of a twostage system in terms of Massey products in A. This is done in Section 3. A Massey product in A (definition in section 2)

< A, B, C >⊂ A,

is a coset defined for matrices A, B, and C with entries from A of type (1, s), (s, t) and (t, 1) such that AB = 0 and BC = 0. The indeterminacy is given by {AX + Y C} where X and Y run over all matrices of type (s, t) and (1, t). When dealing with Massey products in A one can usually avoid indeterminacy. This is also the case in the present paper. We shall always be dealing with a particular element from the coset.

The calculation of Massey products in general is not done in the present paper. It is, however, in principle possible to calculate any Massey product in A as follows. Kristensen and Madsen [6] worked out the reduced diagonal ψ : A → A ⊗ A of a Massey product. It contains Massey products of lower degrees together with some “primary” terms which were left undecided in Kristensen [4]. These terms, which arise from the primary terms in the Cartan formula for a secondary operation, were then computed in Kristensen [5].

Since ker ψ is spanned by Q_{i} = Sq^{(0,...,0,1)}, this together with evaluation in projective space enables us to determine < A, B, C >. This has been carried out in Kristensen [5].

As an example we give in section 5 a complete determination of the A-module structure of H^{∗}(E_{n}) (for definition of E_{n} see section 5) This has also been considered by J. Milgram. The theorem in Milgram [9] describing the A-module structure of H^{∗}(E_{n}), however, is incorrect. (He does not give enough relations.) In Milgram [8] the correct structure of H^{∗}(E_{2}) and H^{∗}(E_{3}) was found. The method used by Milgram is totally different from ours and does not work in the general case.

1. Two stage Postnikov systems

In the following sections we consider a fixed stable two stage Postnikov system

(1.1)  \xymatrix{ΩB \ar[d]^{h} \ar[r] & ΩB \ar[d] \\ E \ar[d]^{p} \ar[r] & P B \ar[d] \\ K(Z_{2}, n) \ar[r]^{ϕ} & B}

where

B = ∏ K(Z_{2}, n(j)), ϕ^{∗}(z_{n(j)}) = α_{j}(z_{n}),

z_{n(j)} the generator of H^{n(j)}K(Z_{2}, n(j)); Z_{2}), α_{j} ∈ A.

As an algebra H^{∗}(E) is isomorphic to

(1.2)  H^{∗}(K(Z_{2}, n))// im ϕ^{∗} ⊗ S

where S is a polynomial algebra, isomorphic under h^{∗} to the subalgebra of H^{∗}(ΩB) generated by σ(ker ϕ^{∗}), σ the suspension (see Kristensen [1],[2] and [3], Smith [10]).

2. Massey products in the Steenrod algebra

Let U and V be finite dimensional vector spaces over Z_{2}. Further, A(U, V) is the graded vector space of stable natural transformations H^{∗}(−, U) → H^{∗}(−, V) and O(U, V) is the graded vector space of cochain operations, i. e., natural transformations C^{♯}(−, U) → C^{♯}(−, V), C^{♯} the normalized chain functor.

In O(U, V) there is a differential

∇θ = δθ + θδ,

δ the coboundary in C^{♯}(−, U), C^{♯}(−, V), respectively. If we let ker ∇ = ZO(U, V), there is an exact sequence

(2.1)  O(U, V) → ZO(U, V) → A(U, V) → 0

(see Kristensen [2]), i. e.

A(U, V) = H(O(U, V), ∇).

Let O = O(Z_{2}, Z_{2}). For α ∈ ZO the exact sequence (2.1) gives the existence of a cochain operation

d(α; −, …, −) ∈ O(Z_{2}, ⊕ … ⊕ Z_{2}; Z_{2})

such that

∇d(α; x_{1}, …, x_{n}) = α(∑ x_{i}) + ∑ α(x_{i}).

We may assume d(α; −) normalized, i. e.

(2.2)  d(α; 0, …, 0, x, 0, …, 0) = 0.

In the sequel we need the following lemma about cochain operations. Consider a relation in A of the form

β Sq^{n+1+deg α} α + c,

where c is a sum of monomials in Sq^{i}, either admissible or of excess > n + 1. Let sq^{i}, \overline{β}, \overline{α} and \overline{c} be elements in ZO representing Sq^{i}, β, α and c respectively. (sq^{i} is further required to vanish on cocycles of dimension ≤ i − 1 and on cochains of dimension ≤ i − 2.) Then (of course)

\overline{β} sq^{n+1+deg α} \overline{α} + \overline{c}

in ZO is actually a boundary i. e. there exists θ ∈ O with

∇θ = \overline{β} sq^{n+1+deg α} \overline{α} + \overline{c}.

Lemma 2.1. θ may be chosen such that

θ(x) = 0 for dim x < n,
θ(x) = ∑ \overline{β}′\overline{α}(x)\overline{β}′′\overline{α}(x), dim x = n, δx = 0,

where

ψ(β) = ∑ β′ ⊗ β′′ + β̂ ⊗ β̂ + ∑ β′′ ⊗ β′,

ψ the diagonal in A

Proof. See Kristensen [2] □

This is called a \frac{1}{2} Cartan formula. There are pairings

A(V, W) ⊗ A(U, V) → A(U, W)
O(V, W) ⊗ O(U, V) → O(U, W)

given by the composition. The latter is left distributive, but because of the lack of additivity of cochain operations it is not right distributive.

Now we proceed to define Massey products. Assume given elements

A ∈ A(V; Z_{2}),  B ∈ A(U, V),  C ∈ A(Z_{2}, U)

such that AB = 0, BC = 0. Choose cochain operations \overline{A}, \overline{B}, \overline{C} to represent A, B, C. Then there exist ϑ ∈ O(U; Z_{2}), η ∈ O(Z_{2}, V) such that

∇ϑ = \overline{A}\overline{B}, ∇η = \overline{B}\overline{C}

and

M = ϑ\overline{C} + \overline{A}η + d(\overline{A}; ηδ, δη)

is (as one can easily check) in ZO, thus determines an element in A. (Note that by (2.2) d(\overline{A}; ηδ, δη) vanish on cocycles.) Varying ϑ and η we get a coset in A, the Massey product < A, B, C >. In what follows, however, ϑ and η will be chosen once and for all, so there will be no indeterminacy in the computations.

Since A(U, V) may be considered dim U × dim V matrices with entries from A, we may reformulate the concept of Massey product in terms of second order relations.

Let r ∈ A ⊗ A ⊗ A such that (μ ⊗ 1)(r) = (1 ⊗ μ)(r) = 0. Then r can be written

(2.3)  r = ∑_{i,j} α_{i} ⊗ β_{i,j} ⊗ γ_{j},

where ∑_{i} α_{i}β_{i,j}, ∑_{j} β_{i,j}γ_{j} are relations in A.

Choose cochain operations \overline{α}_{i}, \overline{β}_{i,j}, \overline{γ}_{j} to represent α_{i}, β_{i,j}, γ_{j} and also R_{i}, S_{j} in O such that

∇R_{j} = ∑_{i} \overline{α}_{i}\overline{β}_{i,j}, ∇S_{i} = ∑_{j} \overline{β}_{i,j}\overline{γ}_{j}.

The Massey product corresponding to (2.3) will then be represented by

∑_{j} R_{j}\overline{γ}_{j} + ∑_{i} \overline{α}_{i}S_{i} + ∑_{i} d(\overline{α}_{i}; δS_{i}, S_{i}δ) + ∑_{i} d(\overline{α}_{i}; \overline{β}_{i,1}\overline{γ}_{1}, \overline{β}_{i,2} …).

Putting r_{j} = ∑ α_{i}β_{i,j}, s_{i} = ∑_{j} β_{i,j}γ_{j} we normally will write the second order relation as

∑ r_{j}γ_{j} + ∑ α_{i}s_{i}.

In the definition of the Massey product there is indeterminacy arising from the choice of R_{j}, S_{i}. However, in this paper the Massey product is evaluated on a class z annihilated by γ_{j}. Hence the indeterminacy arising from the choice of R_{j} disappears.

We shall also consider secondary cohomology operations

Qu(s_{i}) : H^{∗} → H^{∗+deg(s_{i})−1}

associated with the relations s_{i}. Let \overline{z} be a cocycle representing z in H^{∗}. Then Qu(s_{i}) is defined on z if γ_{j}(z) = 0. For a given choice of S_{i}, Qu(s_{i})(z) is the class (actually coset) represented by

S_{i}(\overline{z}) + ∑ \overline{β}_{i,j}(W_{j}),

where δw_{j} = \overline{γ}_{j}(\overline{z}). If we alter S_{i} by adding an element \overline{ϵ} ∈ ZO, S′_{i} = S_{i} + \overline{ϵ} then we get a new secondary operation Qu(s_{i})′, and

Qu(s_{i}) − Qu(s_{i})′ = ϵ, ϵ = cls \overline{ϵ} ∈ A.

From this it follows that once we decide a choice of secondary cohomology operation Qu(s_{i}) we have a fixed S_{i} up to a boundary (ϵ = 0 iff \overline{ϵ} = ∇\overline{ϵ}′). Hence there is no indeterminacy on our Massey products at all when we decide a choice of Qu(s_{i}).

3. H^{∗}(E) as a module over A

Let S′ = h^{∗}(S) = h^{∗}H^{∗}(E) (see (1.1). Then S′ is a left A-module, and we have the following

Theorem 3.1. Let S′ be generated as left A-module by

{∑_{i} β_{i}^{j}(z_{n(i)−1})}_{j}

and let a generating set of relations be given by

{∑_{j} λ_{j}^{s}(∑_{i} β_{i}^{j}(z_{n(i)−1}))}_{s}.

Then as a module over the Steenrod algebra H^{∗}(E) is generated by z, B_{j}(z), where z = p^{∗}z_{n}, and B_{j} is a secondary operation associated with the relation (3.1) below. A generating set of relations is given by

∑_{j} λ_{j}^{s}B_{j}(z) = M^{s}(z) + cls(Q(z)),

where M^{s} is the Massey product associated with the second order relation (3.4) below. The element cls(Q(z)) is of the form ∑ α(z)β(z), α, β ∈ A. It is associated with the relation q in (3.5) below and is specifically computable using Lemma 2.1.

Proof. The proof will consist of a sequence of lemmas. Let

∑ β_{i}z_{n(i)−1} ∈ S′.

Then since S′ is generated on σ ker ϕ^{∗}), we have

ϕ^{∗}(∑_{i} β_{i}z_{n(i)}) = ∑ β_{i}α_{i}z_{n} = 0.

Hence ∑ β_{i}α_{i} is an unstable relation in A, that is, there exists a unique sum c of admissible monomials in A of excess larger than n such that

(3.1)  ∑ β_{i}α_{i} + c

is a relation. Let B be the corresponding unstable secondary operation. Since α_{j}(z) = 0, the operation B is defined on z ∈ H^{∗}(E), and we have

Lemma 3.2. H^{∗}(B(z)) = ∑_{i} β_{i}(z_{n(i)−1}).

Proof. Let \overline{z}_{n} be the basic cocycle of K(Z_{2}, n), \overline{z} = p^{∗}(\overline{z}_{n}), and let \overline{β}_{i}, \overline{α}_{i}, \overline{c} be cochain operations representing β_{i}, α_{i}, c. Choose cochains u_{i} such that δu_{i} = \overline{α}_{i}(\overline{z}) and a cochain operation R such that ∇R = ∑_{i} \overline{β}_{i}\overline{α}_{i} + \overline{c}. Then

R(\overline{z}) + ∑_{i} \overline{β}_{i}(u_{i})

is a cocycle representing B(z). We get

h^{♯}(R(\overline{z}) + ∑ \overline{β}_{i}(u_{i})) = R((ph)^{♯}(\overline{z}_{n})) + ∑_{i} \overline{β}_{i}(h^{♯}u_{i}).

In the Serre spectral sequence of the fibration

ΩB → E → K(Z_{2}, n)

we get from diagram (1.1) by naturality the differentials

d_{n(j)}(z_{n(j)−1}) = α_{j}(z_{n}).

Since δu_{i} = \overline{α}_{i}(\overline{z}) and d_{n(j)} is induced by the coboundary in C^{♯}(E), we get

cls(h^{♯}u_{i}) = z_{n(i)−1}.

With notation as in Theorem 3.1, Lemma 3.2 gives

h^{∗}(∑_{j} λ_{j}B_{j}(z)) = ∑_{j} λ_{j}(∑_{i} β_{i}^{j}(z_{n(i)−1})) = 0,

and therefore

∑_{j} λ_{j}β_{i}^{j}(z_{n(i)−1}) = 0

for all i. Hence e_{i} ∈ A such that e_{i} is a sum of admissible monomials, exc e_{i} > n(i) − 1 and

(3.2)  ∑_{j} λ_{j}β_{i}^{j} + e_{i}

is a relation.

We write e_{i} as a sum of two terms, Sq^{p} γ_{i} ( for dimensional reasons p does not depend on i) of excess n(i) and f_{i} of excess > n(i):

(3.3)  e_{i} = Sq^{p} γ_{i} + f_{i},  p − deg(γ_{i}) = n(i).

From (3.2) and (3.3) we get

(∑ γ_{i}α_{i}z_{n})^{2} = (∑ γ_{i}ϕ^{∗}z_{n(i)})^{2} = ϕ^{∗}(∑ Sq^{p} γ_{i}z_{n(i)}) = ϕ^{∗}(∑_{i} e_{i}z_{n(i)})

= ϕ^{∗}(∑_{j} λ_{j}(∑_{i} β_{i}^{j}z_{n(i)}) = ∑_{l} λ_{j}ϕ^{∗}(∑_{i} β_{i}^{j}z_{n(i)}) = 0.

THE A-MODULE STRUCTURE FOR THE COHOMOLOGY OF TWO STAGE SPACES. 7

Consequently

∑ γ_{i}α_{i}(z_{n}) = 0

and there is a unique sum g of admissible monomials of excess larger than n such that

∑ γ_{i}α_{i} + g = 0

Consider the second order relation

(3.4)  ∑_{j} λ_{j}r_{j} + ∑_{i} s_{i}α_{i} + Sq^{p} t + q,

\begin{aligned}r_{j} & = ∑_{i} β_{i}^{j}α_{i} + c_{j}, \\ s_{i} & = ∑_{j} λ_{j}β_{i}^{j} + Sq^{p} γ_{i} + f_{i}, \\ t & = ∑_{i} γ_{i}α_{i} + g, \\ q & = Sq^{p} g + ∑_{j} λ_{j}c_{j} + ∑_{i} f_{i}α_{i}, \end{aligned} ; (3.5)

and choose cochain operations R_{i}, S_{i}, T, Q such that

∇R_{i} = r_{i}, ∇S_{i} = s_{i}, ∇T = t, ∇Q = q.

Lemma 3.3. Let notation be as above and let M denote the Massey product corresponding to (3.4). Then

∑ λ_{j}B_{j}(z) = M(z) + cls(Q(\overline{z})).

Furthermore since exc q > n, Q(\overline{z}) is given by a \frac{1}{2} Cartan formula. Proof. ∑_{j} λ_{j}B_{j}(z) is represented by

∑_{j}(\overline{λ}_{j}(R_{j}(\overline{z}) + ∑_{i} \overline{β}_{i}^{j}u_{i}))

= ∑_{j} \overline{λ}_{j}R_{j}(\overline{z}) + ∑_{i} ∑_{j} \overline{λ}_{j}\overline{β}_{i}^{j}u_{i} + ∑_{j} ∇d(\overline{λ}_{j}, R_{j}(\overline{z}) + ∑_{i} \overline{β}_{i}^{j}u_{i}, \overline{β}_{1}^{j}, …).

This expression is related to the Massey product. From (3.5) we have

∑_{i} ∑_{j} \overline{λ}_{j}\overline{β}_{i}^{j}u_{i} = ∑_{i} ∇S_{i}u_{i} + ∑_{i} sq^{p} \overline{γ}_{i}u_{i} + ∑_{i} \overline{f}_{i}u_{i}

and

\overline{f}u_{i} = 0 since dim u_{i} < exc f_{i} − 1.

∑_{i} sq^{p} \overline{γ}_{i}u_{i}

= sq^{p}(T (\overline{z}) + ∑_{i} \overline{γ}_{i}u_{i}) + sq^{p}(T \overline{z}) + ∇d(sq^{p}, T (\overline{z}) + ∑_{i} \overline{γ}_{i}u_{i}, \overline{γ}_{1}u_{1}, …)

and T (\overline{z}) + ∑ \overline{γ}_{i}u_{i} is a cocycle of dimension p − 1 such that

∑_{i} sq^{p} \overline{γ}_{i}u_{i} = sq^{p}(T \overline{z}) + ∇d(sq^{p}; T (\overline{z}) + ∑_{i} \overline{γ}_{i}u_{i}, \overline{γ}_{1}u_{1}, …).

Furthermore

\begin{aligned}∇d(\overline{λ}_{j}; R_{j}(\overline{z}) + ∑_{i} \overline{β}_{i}^{j}u_{i}, β_{1}^{j}u_{1}, …) & ∼ d(\overline{λ}_{j}; \overline{c}_{j}(\overline{z}), \overline{β}_{1}^{j}\overline{α}_{1}\overline{z}, …), \\ ∇d(sq^{p}; T (\overline{z}) + ∑ \overline{γ}_{i}u_{i}, \overline{γ}_{1}u_{1}, …) & ∼ d(sq^{p}; \overline{g}(\overline{z}), \overline{γ}_{1}\overline{α}_{1}\overline{z}_{1}, …), \\ ∑_{i} ∇S_{i}u_{i} & ∼ ∑ S_{i}\overline{α}_{i}(\overline{z}). \end{aligned}

Hence

∑_{j} \overline{λ}_{j}(R_{j}(\overline{z}) + ∑_{i} \overline{β}_{i}^{j}u_{i}) ∼ \overline{M}(\overline{z}) + Q(\overline{z}).

□

This ends the proof of Theorem 3.1.

4. Diagonal in H^{∗}(E)

Since E is an H-space, there is a Hopf-algebra structure on H^{∗}(E) with diagonal ψ^{∗}, where ψ is the multiplication. By Theorem 1 a generator of H^{∗}(E) is of the form B(z), where B is a secondary operation corresponding to a relation of the form

∑_{i} β_{i}α_{i} + c

where exc c > n. To compute ψ^{∗}(B(z)) write c = Sq^{j_{i}} Sq^{J_{i}} +d, where j_{i} − deg J_{i} = n + 1, exc d > n + 1. We have

ψ^{∗}(z) = z × 1 + 1 × z,

and by naturality

ψ^{∗}(B(z)) = B(z × 1 + 1 × z)

= B(z × 1) + B(1 × z) + d(c; z × 1, 1 × z)

= 1 × B(z) + B(z) × 1 + ∑_{i} Sq^{J_{i}}(z × 1) Sq^{J_{i}}(1 × z)

= 1 × B(z) + B(z) × 1 + ∑_{i} Sq^{J_{i}}(z) × Sq^{J_{i}}(z).

In this computation we have used Kristensen [2].

This gives a complete description of the diagonal.

5. examples

Let us consider the two-stage Postnikov system

\xymatrix{KZ_{2}, 2n − 1) \ar[d] \ar[r] & K(Z_{2}, 2n − 1) \ar[d] \\ E_{n} \ar[d] \ar[r] & P (Z_{2}, n − 1) \ar[d] \\ K(Z_{2}, n) \ar[r]^{ϕ} & K(Z_{2}, 2n)}

where ϕ^{∗}(z_{2n}) = z_{n}^{2}.

Let ζ be the dual of the squaring homomorphism A^{∗} → A^{∗}. Then ζ(Sq^{2i}) = Sq^{i}, ζ(Sq^{2i+1}) = 0, and we have for α ∈ A,

ϕ^{∗}(αz_{2n}) = αϕ^{∗}z_{2n} = αz_{n}^{2} = ((ζα)(z_{n}))^{2}.

So wee see that

H^{∗}(E_{n}) = ⋀[Sq^{I}(z), Iadm, exc I < n] ⊗ S,

where z = p^{∗}(z_{n}) and S is isomorphic to the subalgebra of H^{∗}(K(Z_{2}, 2n − 1)) generated by α(z_{2n−1}), α ∈ ker ζ.

By Theorem 3.1 we need to find generators and relations for S′h^{∗}(S) as a left A-module.

Lemma 5.1. As a left A-module S′ is generated by t_{1}, …, t_{2i+1}, …, t_{2n−1}, where t_{2i+1} = Sq^{2i+1}(z_{2n−1}). A generating set of relations is

r^{p,q} = Sq^{2p+1} t_{2q+1} + ∑_{j=0}^{[p− \frac{1}{2} ]} \binom{q − 1 − j}{p − 1 − 2j} Sq^{2p+2q+1−2j} t_{2j+1}, p ≤ 2q < n

and

\begin{aligned}s^{p,q} = Sq^{4p+2} t_{2q+1} & + Sq^{2(p+q+1} t_{2j+1}, \\ & + ∑_{j+q+1}^{2p+q+1} \binom{p − 1 − j}{−q + j} Sq^{2(2p+q+1−j)} t_{2j+1}, p < q < n. \end{aligned}

Remark. The binomial coefficient \binom{n}{i} is the coefficient to t^{i} in the power series

(1 + t)^{n} ∈ Z_{2}[[t]], n, i ∈ Z

Proof. In Milgram [8] it is proved that Sq^{t} z_{2n−1}, t = 2^{i} − 1, generate S′ as a left A-module. For convenience we take more generators, so the first assertion is obvious. To prove we have enough relations, remember that S′ is a polynomial algebra, so we will prove that Sq^{i} t_{2j+1} can be written as a sum of polynomial generators by means of the relations r^{p,q}, s^{p,q}. Using Adem relations it is easy to show that we can take

{Sq^{i} t_{2q+1}|i ≥ 4q + 2 or i ≡ 0 mod 4}

as some of the polynomial generators, so assume i < 4q + 2 and i = 2p + 1 or i = 4p + 2. If i = 2p + 1, then r^{p,q} immediately expresses Sq^{i} t_{2q+1} as a sum of polynomial generators, and if i = 4p + 2, then repeated application of s^{p,q} will express Sq^{i} t_{2q+1} as a sum of polynomial generators. □

Remark. Note that the relations r^{p,q} and s^{p,q} are derived from the Adem relation

Sq^{2p+1} Sq^{2q+1} + ∑ \binom{2q − j}{2p + 1 − 2j} Sq^{2p+2q+2−j} Sq^{j}

and from the relation r(2(p + q + 1), 2p + 1; 4p + 2q + 3) in Kristensen [4]

By Theorem 3.1 we have that H^{∗}(E_{n}) is generated by

z, B_{1}(z), …, B_{2i+1}(z), …, B_{2n−1}(z),

where B_{2i+1} is a secondary operation associated with the relation

Sq^{2i+1} Sq^{n} +(∑ \binom{n − 1 − j}{2i + 1 − 2j} Sq^{n+2i+1−j} Sq^{j}).

A generating set of relations is given by Massey products of certain second-order relations. These Massey products will be computed now.

First consider the second order relation

sq^{2p+1} τ_{2q+1} + ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} sq^{2p+2q+1−2j} τ_{2j+1} + r^{p,q} sq^{n} +q′

where

τ_{2j+1} = sq^{2j+1} sq^{n} + ∑_{m} \binom{n − 1 − m}{2j + 1 − 2m} sq^{2j+1+n−m} sq^{m},

r^{p,q} = sq^{2p+1} sq^{2q+1} + ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} sq^{2p+2q+1−2j} sq^{2j+1}

and

q = sq^{2p+1}(∑_{m} \binom{n − 1 − m}{2q + 1 − 2m} sq^{2q+1+n−m} sq^{m})+

+ ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} sq^{2p+2q−2j+1}(∑_{m} \binom{n − 1 − m}{2j + 1 − 2m} sq^{m}),

and denote the corresponding Massey product by M^{p,q}.

Write M^{p,q} = a + b, where a is a sum of admissible monomials each of excess less than or equal to n − 1 and exc b ≥ n, then since z^{2} = 0 in H^{∗}(E_{n}),

M^{p,q}(z) = a(z)

We can compute a by evaluating M^{p,q} in H^{∗}(K(Z_{2}, n−1)). Let T_{2j+1}, R^{p,q} and Q be cochain operations such that ∇T_{2j+1} = τ_{2j+1}, ∇R^{p,q} = r^{p,q} and ∇Q = q, chosen as in Lemma 2.1. Let \overline{z}_{n−1} ∈ C^{n−1}(K(Z_{2}, n − 1)) be the basic cocycle. Then

\overline{M}^{p,q}(\overline{z}_{n−1}) = sq^{2p+1} T_{2q+1}(\overline{z}_{n−1}) + ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} sq^{2p+2q+1−2j} T_{2j+1}(\overline{z}_{n−1})+

+ R^{p,q} sq^{n}(\overline{z}_{n−1}) + Q(\overline{z}_{n−1})

= sq^{2p+1}(∑_{i≤q} sq^{2j+1−i} \overline{z}_{n−1} sq^{i} \overline{z}_{n−1})+

+ ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} sq^{2p+2q+1−2j}(∑_{i≤j} sq^{2j+1−i} \overline{z}_{n−1} sq^{i} \overline{z}_{n−1}).

(5.1)

Since M^{p,q} is stable, this must be equal to

(α(\overline{z}_{n−1}))^{2} = sq^{p+q+n} α(\overline{z}_{n−1}),

for some α in A. (An alternate argument for this is that suspension and reduced diagonal both vanish).

Now we consider the second order relation

sq^{4p+2} τ_{2q+1} + sq^{2(p+q+1)} τ_{2p+1} + ∑_{j>q} \binom{p − 1 − j}{−q + j} sq^{2(2p+q+1−j)} τ_{2j+1} + s^{p,q} sq^{n} +q,

where

τ_{2j+1} = sq^{2j+1} sq^{n} + ∑_{k} \binom{n − 1 − k}{2j + 1 − 2k} sq^{n+2j+1−k} sq^{k},

s^{p,q} = sq^{4p+2} sq^{2p+1} + sq^{2(p+q+1)} sq^{2p+1} + ∑_{j>q} \binom{p − 1 − j}{−q + j} sq^{2(2p+q+1−j)} sq^{2j+1},

and

q = sq^{4p+2}(∑_{k} \binom{n − 1 − k}{2p + 1 − 2k} sq^{n+2q+1−k} sq^{k})

+ sq^{2(p+q+1)}(∑_{k} \binom{n − 1 − k}{2p + 1 − 2k} sq^{n+2p+1−k} sq^{k})

+ ∑_{j>q} \binom{p − 1 − j}{−q + j} sq^{2(2p+q+1−j)}(∑_{k} \binom{n − 1 − k}{2j + 1 − 2k} sq^{n+2j+1−k} sq^{k}),

and denote the corresponding Massey product by N^{p,q}. Again we get N^{p,q}(z_{n−1}) written as a sum of products of primary operations on z_{n−1}, but now dim N^{p,q}(z_{n−1} is odd so N^{p,q}(z_{n−1}) = 0 Hence N^{p,q}(z) = 0. The only thing missing is a determination of cls Q(z). This, however, is given by a \frac{1}{2} Cartan formula (Lemma 2.1).

The results concerning the cohomology of E_{n} are summarized in

Theorem 5.2. As a module over the Steenrod algebra, H^{∗}(E_{n}) is generated by

z, B_{1}(z), …, B_{2j+1}(z), …, B_{2n+1}(z),

where B_{2j+1} is a secondary operation associated with the relation

Sq^{2j+1} Sq^{n} + ∑_{k} \binom{n − 1 − k}{2j + 1 − 2k} Sq^{n+2j+1−k} Sq^{k} .

The diagonal is given by

ψ^{∗}(B_{2j+1}(z)) = B_{2j+1}(z) × 1 + (n − 1 − j) Sq^{j}(z) × Sq^{j}(z) + 1 × B_{2j+1}(z).

A generating set of relations is given by

Sq^{2p+1} B_{2q+1}(z) + ∑_{j} \binom{q − 1 − j}{p − 1 − 2j} Sq^{2p+2q+1−2j} B_{2j+1}(z)

= Sq^{p+q+n} α(z) + (n − 1 − q)(∑_{k≤p} Sq^{2p+1−k} Sq^{q}(z) Sq^{k} Sq^{q}(z))+

+ ∑_{j} \binom{q − 1 − j}{p − 1 − 2j}(n − 1 − j)( ∑_{k≤p+q−j} Sq^{2p+2q+1−2j−k} Sq^{j}(z) Sq^{k} Sq^{j}(z)),

p ≤ 2q ≤ n; α as in (5.1), and

Sq^{4p+2} B_{2q+1}(z) + Sq^{2(p+q+1)} B_{2p+1}(z) + ∑_{j>q} \binom{p − 1 − j}{−q + j} Sq^{2(2p+q+1−j)} B_{2j+1}(z)

= (n − 1 − q)(∑_{k≤2p} Sq^{4p+2−k} Sq^{q}(z) Sq^{k} Sq^{q}(z))+

+ (n + 1 − p)( ∑_{k≤p+q} Sq^{2(p+q+1)−k} Sq^{p}(z) Sq^{k} Sq^{p}(z))+

+ ∑_{j>q} \binom{p − 1 − j}{−q + j}(n − 1 − j)( ∑_{k<2p+q−j} Sq^{2(2p+q+1−j)−k} Sq^{j}(z) Sq^{k} Sq^{j}(z)),

p < q < n.

References

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[2] , On secondary operations, Math. Scand. 12 (1963), 57–82.

[3] , On the cohomology of spaces with two non-vanishing homotopy groups, Math. Scand. 12 (1963), 83–105.

[4] , On secondary cohomology operations, II, Algebraic Topology, (Univ. of Illinois at Chicago Circle), 1968, pp. 117–133.

[5] , Massey products in Steenrods algebra, Proc. Adv. Study Inst., (Aarhus), 1970, pp. 240–255.

[6] L. Kristensen and I. Madsen, On the structure of the operation algebra for certain cohomology theories, Algebraic Topology, (Univ. of Illinois at Chicago Circle), 1968, pp. 134–160.

[7] W. Massey and F. Peterson, Cohomology of certain fibre spaces I, Topology 4 (1965), 47–65.

[8] J. Milgram, Steenrod squares and higher Massey Products, Bol. Soc. Mat. Mexicana (2) 13 (1968), 32–57.

[9] , The structure over the Steenrod algebra of some twostage Postnikov systems, Quart. J. Math. Oxford Ser. (2) 20 (1969), 161–169.

[10] L. Smith, Cohomology of stable two stage Postnikov systems, Illinois J. Math. 11 (1967), 310–329.

Aarhus Universitet, Aarhus Denmark

University of Chicago, U.S.A.