A MOD TWO ANALOGUE OF A CONJECTURE OF COOKE

J. Aguadé, C. Broto and D. Notbohm

§0. Introduction.

The mod two cohomology of the three connective covering of S^{3} has the form

𝔽_{2}[x_{2n}] ⊗ E(Sq^{1}x_{2n})

where x_{2n} is in degree 2n and n = 2. If F denotes the fibre of the map S^{3} → B^{2}S^{1} of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod 2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove:

Theorem 1. There is a space X such that H^{∗}(X; 𝔽_{2}) ≅ 𝔽_{2}[x_{2n}] ⊗ E(Sq^{1}x_{2n}) with x_{2n} of degree 2n if and only if n = 1, 2, 4.

The “only if” part of theorem 1 can be considered, after [3] and [4], the mod two version of what we call there the Cooke conjecture. Its proof is similar to the proof for odd primes (which is contained in [3]), but one should be slightly more careful in small degrees. The most interesting goal of this paper is probably the construction of what we think to be a remarkable space X with H^{∗}(X; 𝔽_{2}) ≅ 𝔽_{2}[x_{8}] ⊗ E(Sq^{1}x_{8}) and degx_{8} = 8.

This paper represents a part of the p = 2 case of the general problem of the homotopy classification of spaces realizing cohomology algebras of the type 𝔽_{p}[x] ⊗ E(βx) where β denotes the Bockstein homomorphism. The case of p odd was completely elucidated in [3]. The homotopy classification in the case p = 2 should appear in [4]. In particular, as any reader familiar with [3] would expect, there are infinitely many different 2-complete spaces realizing each cohomology algebra in the theorem and the proof of the homotopy classification theorem will not be short. The present paper has been written in a self contained way, and can be read independently of [3] and [4], although it may be helpful to be familiar with the techniques discussed there.

Since the technical details of the construction of a space X with H^{∗}(X; 𝔽_{2}) ≅ 𝔽_{2}[x_{8}] ⊗ E(Sq^{1}x_{8}) may look involved, we provide now an outline of the way in which X can be reached. Let F be the fibre of the degree two map from S^{3} to B^{2}S^{1} (as noted above, F

Typeset by AMS-TEX

J. Aguadé and C. Broto are partially supported by DGICYT grant PB89-0321

turns out to provide a space as in the theorem for n = 1) and let W be the homotopy quotient of F by an involution coming from a degree −1 map of S^{3}. Then one has a map f: BD → W where D is the dihedral group of order 8. Using the techniques developed by Lannes ([8]) and other authors one proves that the function space map(BD, W)_{f} is mod 2 homotopy equivalent to Bℤ/2 × S^{2}. From this fact, one can construct a diagram which, up to homotopy, looks like

GL_{2}(𝔽_{2}) ⟳ BV × S^{2} ⟶ ⟶ ⟶ W

where V denotes the group of diagonal matrices in D. Let Y be the homotopy colimit of this diagram. Then the space X we are looking for can be taken as the 2-connective covering of Y.

The steps towards the construction of X appear in sections 1 to 5. Section 6 contains the proof of the “only if” part of t he theorem, i.e. the Cooke conjecture. A final section contains some inconclusive remarks related to the Arf invariant.

We work exclusively at the prime 2 and mod 2 coefficients are implicitelly assumed. When describing cohomology generators the subscripts usually denote the degrees of the generators.

The first and second author would like to thank the Sonderforschungsbereich 170 in Göttingen for the support during part of this joint work.

§1. The space W.

We start with the action of ℤ/2 on S^{1} given by the extension S^{1} → O(2) → ℤ/2. A functorial classifying space construction leads to a cellular action of ℤ/2 on BS^{1} and then on EBS^{1} and B^{2}S^{1}. Moreover the projection EBS^{1} → B^{2}S^{1} is equivariant and the action on BS^{1} is reproduced on the fibre over the base point of B^{2}S^{1}. The sphere S^{3} is the first cell of B^{2}S^{1} and inherits an action of ℤ/2 which turns out to be the two fold suspension of the original action on S^{1}. Hence it is easy to find a concrete degree two map S^{3} ⟶ B^{2}S^{1} which is equivariant. Now F is defined by the pull back diagram

(1)  \begin{matrix}BS^{1} & ⟶ & F & ⟶ & S^{3} \\ ∥ & & ↓ & & ↓^{2} \\ BS^{1} & ⟶ & EBS^{1} & ⟶ & B^{2}S^{1} \end{matrix}

so that it is given a ℤ/2 action that makes equivariant all maps in the diagram. Recall that H^{∗}(F) = 𝔽_{2}[x_{2}] ⊗ E(Sq^{1}x_{2}), where x_{2} is detected in BS^{1} and Sq^{1}x_{2} is the spherical class that comes from S^{3}.

Definition 2. Let W denote the homotopy quotient of F by the action of ℤ/2:

W = F_{hℤ/2} = Eℤ/2 ×_{ℤ/2} F .

The rest of the section is devoted to the computation of the cohomology of W. Observe that this space fits in the diagram

(2)  \begin{matrix}BS^{1} & ⟶ & F & ⟶ & S^{3} \\ ↓ & & ↓ & & ↓ \\ BO(2) & ⟶ & W & ⟶ & S_{hℤ/2}^{3} \\ ↓ & & ↓ & & ↓ \\ Bℤ/2 & & Bℤ/2 & & Bℤ/2 \end{matrix}

where the columns are fibrations and all of them have a section provided by the base point of BS^{1}, F or S^{3} which is fixed under the action of ℤ/2.

Proposition 3. H^{∗}(S_{hℤ/2}^{3}) ≅ 𝔽_{2}[w_{1}] ⊗ E(y_{3}), with Sq^{1}(y_{3}) = w_{1}y_{3} and Sq^{i}(y_{3}) = 0 for i ≥ 2.

Proof. Our action of ℤ/2 on S^{1} is already the suspension of the action of ℤ/2 on itself. It follows that the fibration S^{3} → S_{hℤ/2}^{3} → Bℤ/2 is the sphere bundle of a four dimensional real bundle ξ over Bℤ/2, obtained as the Whitney sum of the universal line bundle

ℝ → Eℤ/2 ×_{ℤ/2} ℝ → Bℤ/2

and a three dimensional trivial bundle. Hence the Stiefel-Whitney classes of ξ are w_{1}(ξ) = w_{1} ∈ 𝔽_{2}[w_{1}] ≅ H^{∗}(Bℤ/2) and w_{i}(ξ) = 0 if i > 1.

Let T (ξ) ≃ Cofibre(S_{hℤ/2}^{3} → Bℤ/2) be the Thom space. Since the action of the Steenrod squares on the Thom class is given by multiplication by the Stiefel-Whitney classes, the complete structure of H^{∗}(S_{hℤ/2}^{3}) follows from the split exact sequence of H^{∗}(Bℤ/2)- U-modules

0 ⟶ H^{∗}(Bℤ/2) ⟶ H^{∗}(S_{hℤ/2}^{3}) ⟶^{Δ} H̃^{∗+1}(T (ξ)) ⟶ 0. □

Proposition 4. H^{∗}(W) ≅ 𝔽_{2}[w_{1}, w_{2}]⊗E(y_{3}) with Sq^{1}w_{2} = w_{1}w_{2}+y_{3}, Sq^{1}y_{3} = w_{1}y_{3} and Sq^{i}y_{3} = 0 if i ≥ 0. The maps induced in cohomology by the diagram (2) are as suggested by the names of the generators.

Proof. We compute H^{∗}(W) by means of the Serre spectral sequence for the fibration F → W → Bℤ/2. Since ℤ/2 can only act trivially on H^{∗}(F) we have

E_{2}^{∗,∗} ≅ H^{∗}(Bℤ/2) ⊗ H^{∗}(F) ≅ 𝔽_{2}[u] ⊗ 𝔽_{2}[x_{2}] ⊗ E(Sq^{1}x_{2}) ⇒ H^{∗}(W)

It is clear that x_{2} is transgressive and then so is Sq^{1}x_{2}. But the fibration has a section and therefore 𝔽_{2}[u] survives to E_{∞}^{∗,∗}. Hence the possible differentials on x_{2} and Sq^{1}x_{2} are zero and the spectral sequence collapses at the E_{2}-term. That is:

GrH^{∗}(W) ≅ E_{∞}^{∗,∗} ≅ E_{2}^{∗,∗} ≅ 𝔽_{2}[u, x_{2}] ⊗ E(Sq^{1}x_{2})

If we look at the Serre spectral sequences for the left and right columns fibrations of diagram (2) the above argument shows:

GrH^{∗}(BO(2)) ≅ 𝔽_{2}[u, x_{2}]
GrH^{∗}(S_{hℤ/2}^{3}) ≅ 𝔽_{2}[u] ⊗ E(Sq^{1}x_{2})

and the maps GrH^{∗}(BO(2)) ← GrH^{∗}(W) ← GrH^{∗}(S_{hℤ/2}^{3}) relate the generators with the same names. It is well known the structure of H^{∗}(BO(2)) and we know that of H^{∗}(S_{hℤ/2}^{3}) from Proposition 3. It follows the existence of polynomial generators w_{1} and w_{2} in H^{∗}(W) that map to the generators of H^{∗}(BO(2)) and an exterior generator y_{3} that comes from the one in H^{∗}(S_{hℤ/2}^{3}). Hence H^{∗}(W) = 𝔽_{2}[w_{1}, w_{2}] ⊗ E(y_{3}). Finally, the action of the Steenrod algebra on H^{∗}(W) is forced as well by the maps BO(2) → W → S_{hℤ/2}^{3} and F → W. □

§2. Preliminary results

Before starting the computations in which we are interested we recall in this section some results about general mapping spaces which will be needed in the next section.

2.1. Let P be a finite p group and G a compact Lie group. The classifying space construction induces a bijection ([7])

Rep(P, G) ≅ [BP, BG]

Furthermore, for a given representation ρ: P → G, if C_{ρ}(P) denotes the centralizer of the image of ρ, we have a homomorphism P × C_{ρ}(P) → G and BP × BC_{ρ}(P) → BG whose adjoint

BC_{ρ}(P) → map(BP, BG_{2})_{Bρ}

is a mod p equivalence.

2.2. For an elementary abelian p group V and a p-complete space of finite type, there is a bijection ([8])

[BV, X] ≅ Hom_{K}(H^{∗}(X), H^{∗}(BV))

where K denotes the category of unstable algebras over the Steenrod algebra. Furthermore, the functor T_{V}: K → K, left adjoint to tensoring with H^{∗}(BV) computes cohomology of mapping spaces. If R is an object of K we write T_{V} (R) ≅ ∐_{f∈Hom_{K}(H^{∗}(BV),R)} T_{V} (R, f) as a sum of connected components. Then for a given map f: BV → X, there is an isomorphism of algebras over the Steenrod algebra

T_{V} (H^{∗}(X), f^{∗}) ≅ H^{∗}(map(BV, X)_{f})

provided T_{V} (H^{∗}(X), f^{∗}) is of finite type and free in d egrees ≤ 2. Recall that for p = 2 Lannes called a n object R of K free in degrees ≤ 2 if the kernel of the multiplication R^{1} ⊗ R^{1} → R^{2} is generated by elements of the form x ⊗ y + y ⊗ x.

Moreover there is a coaugmentation ε → T_{V} that under the above isomorphism corresponds to the map in cohomology induced by the evaluation map(BV, X)_{f} → X.

Finally, we collect some concrete computations that we use later. The proof is along the lines of [4] section 3.

Lemma 5. (1) Let s^{∗}: H^{∗}(S_{hℤ/2}^{3}) ≅ 𝔽_{2}[w_{1}] ⊗ E(y_{3}) → H^{∗}(Bℤ/2) be the projection onto the polynomial part and g^{∗}: H^{∗}(S_{hℤ/2}^{3}) → H^{∗}(BV), V = ℤ/2 × ℤ/2, the composition of s^{∗} and the diagonal map, then

T_{ℤ/2}(H^{∗}(S_{hℤ/2}^{3}), s^{∗}) ≅ T_{V} (H^{∗}(S_{hℤ/2}^{3}), g^{∗}) ≅ 𝔽_{2}[w_{1}] ⊗ E(y_{2})

with Sq^{i}y_{2} = 0 for i ≥ 1. The coaugmentation is the inclusion given by w_{1} ↦ w_{1} and y_{3} ↦ w_{1}y_{2}

(2) Let f^{∗}: H^{∗}(W) → H^{∗}(BV), V = ℤ/2×ℤ/2, be the composition 𝔽_{2}[w_{1}, w_{2}]⊗E(y_{3}) → 𝔽_{2}[w_{1}, w_{2}] → 𝔽_{2}[t_{1}, t_{2}], then

T_{V} (H^{∗}(W), f^{∗}) ≅ 𝔽_{2}[t_{1}, t_{2}] ⊗ E(y_{2})

Moreover, the map T_{V} (H^{∗}(W), f^{∗}) → T_{V} (H^{∗}(BO(2)), Bi^{∗}) induced by BO(2) → W is the projection 𝔽_{2}[t_{1}, t_{2}] ⊗ E(y_{2}) → 𝔽_{2}[t_{1}, t_{2}]. □

§3. Some mapping spaces

From now on all spaces are assumed to be completed at the prime two.

Let D denote the dihedral group of order eight. We represent it as the subgroup i: D ↪ O(2) of O(2) generated by the matrices

( \begin{matrix}±1 & 0 \\ 0 & ±1 \end{matrix} )  and  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Let V denote the subgroup given by the diagonal matrices, and let Z := Z(O(2)) ≅ Z(D) ≅ ℤ/2 be the center of O(2) or of D.

We are interested in the mapping space map(BD, W)_{f} where f is the composition BD → BO(2) → W.

Interpreting the top row fibration in diagram (1), BS^{1} → F → S^{3}, as a pull back diagram and applying the Borel construction −×_{ℤ/2}Eℤ/2 yields another pull back diagram

(3)  \begin{matrix}BO(2) & ⟶ & Bℤ/2 \\ ↓ & & ↓^{s} \\ W & ⟶ & S_{hℤ/2}^{3} \end{matrix}

And then we apply the functor map(BD, −) to get the diagram

(4)

map(BD, BO(2))_{Bi} ⟶ map(BD, Bℤ/2)_{Bdet}
j↓  ↓^{k}
map(BD, W)_{f} ⟶ map(BD, S_{hℤ/2}^{3})_{g} .

To get a pull back diagram, one might have to take more components than just the one given by Bi in the upper left corner. Let Bj: BD ⟶ BO(2) be a lift of f (see 2.1). The restriction Bj|_{BV} is homotopic to Bi|_{BV}. This follows by 2.2 from looking at the induced maps in cohomology. There also exists only one extension of i : V ⟶ O(2) to D. Hence, Bj and Bi are homotopic and the above diagram is a pull back diagram.

The mapping spaces of the top row are calculated. In the composition

BZ → map(BD, BO(2))_{Bi} ⟶ map(BD, Bℤ/2)_{Bdet} ≃ Bℤ/2

the first arrow is a homotopy equivalence by 2.1. The second map is induced by the determinant and therefore trivial. The homotopy fibre is homotopy equivalent to ℤ/2 × Bℤ/2.

Next, we calculate map(BD, S_{hℤ/2}^{3})_{g}. S^{2} is the fix point set of S^{3} under the action of ℤ/2, hence the section s extends to an inclusion l: Bℤ/2 × S^{2} → S_{hℤ/2}^{3}. Now we can use

the factorization of g: BD ⟶ S_{hℤ/2}^{3} as BD ⟶^{B det} Bℤ/2 ⟶^{s} S_{hℤ/2}^{3} in order to define a map

BD × Bℤ/2 × S^{2} ⟶^{B det} Bℤ/2 × Bℤ/2 × S^{2} ⟶^{m} Bℤ/2 × S^{2} ⟶^{l} S_{hℤ/2}^{3}

which coincides with g when restricted to BD and with l when restricted to Bℤ/2 × S^{2}. Let

h: Bℤ/2 × S^{2} → map(BD, S_{hℤ/2}^{3})_{g}

be the adjoint of that composition. Then we have:

Proposition 6. (1) h: Bℤ/2 × S^{2} → map(BD, S_{hℤ/2}^{3})_{g} is a homotopy equivalence.

(2) The evaluation map(BD, S_{hℤ/2}^{3})_{g} ⟶ S_{hℤ/2}^{3} induces in cohomology the map given by ω_{1} ↦ ω_{1} and y_{3} ↦ ω_{1}y_{2}. Here, we write H^{∗}(Bℤ/2) ≅ 𝔽_{2}[w_{1}] and H^{∗}(S^{2}) ≅ E(y_{2}).

Proof. Since the dihedral group sits in a split extension

V → D → Σ_{2}

the computation of map(BD, S_{hℤ/2}^{3}) reduces to the computation of the homotopy fixed

point set by the action of Σ_{2} of map(B̃V, S_{hℤ/2}^{3}), where B̃V = ED/V (see [10], remark 3.12):

map(BD, S_{hℤ/2}^{3}) ≃ map(B̃V, S_{hℤ/2}^{3})^{hΣ_{2}}

Recall that for a G-space X the homotopy fixed point set X^{hG} is defined as the space of equivariant maps map_{G}(EG, X) and coincides with the space of lifts of the fibration X ×_{G} EG → BG.

So map(BD, S_{hℤ/2}^{3})_{f} is homotopic to a component of map(B̃V, S_{hℤ/2}^{3})^{hΣ_{2}}. Observe that

g: B̃V ⟶ S_{hℤ/2}^{3}, defined as the composition B̃V → BD ⟶^{g} S_{hℤ/2}^{3}, also factors as

B̃V ⟶^{B det} Bℤ/2 ⟶^{s} S_{hℤ/2}^{3}

and B det is Σ_{2}-equivariant with trivial action of Σ_{2} on Bℤ/2. We have therefore a Σ_{2}- equivariant map

map(Bℤ/2, S_{hℤ/2}^{3})_{s} ⟶ map(B̃V, S_{hℤ/2}^{3})_{g}

with trivial action of Σ_{2} on the left hand side. Furthermore, by 2.2 this map is a homotopy equivalence and so it induces a homotopy equivalence

map(Bℤ/2, S_{hℤ/2}^{3})_{s} ⟶ map(B̃V, S_{hℤ/2}^{3})_{g}^{hΣ_{2}}.

On the other hand, l: Bℤ/2 × S^{2} → S_{hℤ/2}^{3} induces

Bℤ/2 × S^{2} ⟶ map(Bℤ/2, Bℤ/2 × S^{2})_{i_{1}} ⟶ map(Bℤ/2, S_{hℤ/2}^{3})_{s}

where i_{1}: Bℤ/2 → Bℤ/2 × S^{2} is the inclusion of the first factor so that s = k ∘ i_{1}. Observe that the first arrow is a homotopy equivalence and also the second one by 2.2. Finally, one checks that the composition

Bℤ/2 → map(Bℤ/2, S_{hℤ/2}^{3})_{s} → map(B̃V, S_{hℤ/2}^{3})_{g}^{Σ_{2}} ≃ map(BD, S_{hℤ/2}^{3})_{g}

is the map h. This proves (1). Part (2) of the proposition follows by 2.2. □

This statement calculates the mapping space in the lower right corner of diagram (4). Collecting all the facts we get a mod–2 pull back diagram

(5)  \begin{matrix}Bℤ/2 × ℤ/2 & ⟶ & Bℤ/2 & ⟶^{const} & Bℤ/2 \\ ∥ & & ^{j}↓ & & ↓^{k} \\ Bℤ/2 × ℤ/2 & ⟶ & map(BD, W)_{f} & ⟶ & Bℤ/2 × S^{2} & . \end{matrix}

from which we can calculate map(BD, W)_{f}. Observe that k: Bℤ/2 → Bℤ/2 × S^{2} is the inclusion of the first factor. This follows by construction of the equivalences Bℤ/2 ⟶^{≃} map(BD, Bℤ/2)_{B det} (cf. 2.1) and h: Bℤ/2 × S^{2} ⟶^{≃} map(BD, S_{hℤ/2}^{3})_{g}.

Proposition 7. The mapping space map(BD, W)_{f} is homotopy equivalent to Bℤ/2×S^{2}.

Proof. First we claim that j in diagram (5) is not nullhomotopic. If we restrict the maps from BD to BV we obtain a diagram

(6)

Bℤ/2 ≃ map(BD, BO(2))_{Bi} ⟶^{Δ} map(BV, BO(2))_{Bi} ≃ Bℤ/2 × Bℤ/2
j↓  ↓j̃
map(BD, W)_{f} ⟶ map(BV, W)_{f}

where we still denote by Bi and f the restrictions of Bi: BD → BO(2) and f: BD → W to BV and Δ: Bℤ/2 → Bℤ/2 × Bℤ/2 is the diagonal map.

Using Lannes T functor we obtain

H^{∗}(map(BV, W)_{f}) ≅ 𝔽_{2}[t_{1}, t_{2}] ⊗ E(y_{2})

and j̃ induces in mod 2 cohomology the projection of 𝔽_{2}[t_{1}, t_{2}] ⊗ E(y_{2}) onto 𝔽_{2}[t_{1}, t_{2}]. Hence j̃ is non trivial at the level of the fundamental group. And then j is non trivial either.

Standard arguments about fibrations show that

map(Bℤ/2, map(BD, W)_{f})_{j} ≃ map(BD, W)_{f}.

A quick look at the exact sequences of homotopy groups of the fibrations in diagram (5) shows the existence of a section of the map map(BD, W)_{f} → S^{2} that we can think of as being a map S^{2} ⟶ map(Bℤ/2, map(BD, W)_{f})_{j}. Taking the adjoint produces a diagram of fibrations

(7)  \begin{matrix}Bℤ/2 × ℤ/2 & ⟶ & Bℤ/2 × S^{2} & ⟶ & Bℤ/2 × S^{2} \\ ∥ & & ↓ & & ∥ \\ Bℤ/2 × ℤ/2 & ⟶ & map(BD, W)_{f} & ⟶ & Bℤ/2 × S^{2} \end{matrix}

where the middle vertical arrow is a mod–2 equivalence. This proves the proposition. □

Corollary 8. Let BD × S^{2} ⟶ W be the adjoint of S^{2} ⟶ map(BD, W)_{f}. Then the restriction BV × S^{2} ⟶ W is a map that in cohomology is

H^{∗}(W) ⟶ H^{∗}(BV) ⊗ H^{∗}(S^{2})
w_{1} ⟼ σ_{1}
w_{2} ⟼ σ_{2} + y_{2}
y_{3} ⟼ σ_{1}y_{2}

Proof. The composition BV ⟶ BV × S^{2} ⟶ W is the obvious map and factors over BO(2) ⟶ W Hence, w_{1} ↦ σ_{1} and w_{2} ↦ σ_{2} + λy_{2}. The commutative diagram

(8)  \begin{matrix}BV × S^{2} & ⟶ & W \\ ^{B det ×id}↓ & & ↓ \\ Bℤ/2 × S^{2} & ⟶ & S_{hℤ/2}^{3} \end{matrix}

shows that y_{3} ↦ σ_{1}y_{2}. The relation Sq^{1}(w_{2}) = w_{1}w_{2} + y_{3} finally shows that w_{2} ↦ σ_{2} + y_{2}. □

§4. The space Y.

Let I be the category associated to the symmetric group Σ_{3} and the subgroup Σ_{2} in the following way. I has two objects 0 and 1 and the morphisms sets are

Hom(0, 0) = Σ_{3}
Hom(0, 1) = ∅
Hom(1, 1) = {1}
Hom(1, 0) = Σ_{3}/Σ_{2}

with the obvious compositions given by the product in Σ_{3} and the action of Σ_{3} on Σ_{3}/Σ_{2}. This category was analized in the general case of a group G and a subgroup H in [2]. Let us consider the contravariant functor M: I → U given by

M_{0} = H^{∗}(BV × S^{2}) ≅ 𝔽_{2}[t_{1}, t_{2}] ⊗ E(y_{2})
M_{1} = H^{∗}(W) ≅ 𝔽_{2}[w_{1}, w_{2}] ⊗ E(y_{3})

with the natural action of Σ_{3} = GL_{2}𝔽_{2} on M_{0} and the homomorphism φ: M_{1} → M_{0} induced by the inclusion of V as diagonal matrices in D. From the previous section we know that φ is given by

φ(w_{1}) = σ_{1}
φ(w_{2}) = σ_{2} + y_{2}
φ(y_{3}) = σ_{1}y_{2}

Proposition 9.

\underleftarrow{lim}_{I}^{i}M = \{ \begin{matrix}0, i > 0 \\ 𝔽_{2}[σ_{1}^{2} + σ_{2} + y_{2}, σ_{1}σ_{2}] ⊗ E(σ_{1}σ_{2}y_{2}), i = 0 \end{matrix}

Proof. The computation of the derived functors \underleftarrow{lim}_{I}^{i} is done in [2]. For i ≥ 1 there is a long exact sequence

⋯ → H^{i}(Σ_{3}; M_{0}) → H^{i}(Σ_{2}; M_{0}) → \underleftarrow{lim}_{I}^{i+1}M → H^{i+1}(Σ_{3}; M_{0}) → ⋯

Since the restriction H^{i}(Σ_{3}; A) → H^{i}(Σ_{2}; A) is an isomorphism for any i ≥ 1 and any 𝔽_{2}[Σ_{3}]-module A we obtain the vanishing of \underleftarrow{lim}_{I}^{i+1} M for any i ≥ 1. To compute \underleftarrow{lim}_{I}^{0} M =

{x ∈ M_{1} | φ(x) ∈ M_{0}^{Σ_{3}}} notice that

𝔽_{2}[σ_{1}^{2} + σ_{2} + y_{2}, σ_{1}σ_{2}] ⊗ E(σ_{1}σ_{2}y_{2}) ⊂ \underleftarrow{lim}_{I}^{0}M

\underleftarrow{lim}_{I}^{0}M = [𝔽_{2}[σ_{1}, σ_{2} + y_{2}] ⊗ E(σ_{1}y_{2})] ∩ [𝔽_{2}[σ_{1}^{2} + σ_{2}, σ_{1}σ_{2}] ⊗ E(y_{2})]

𝔽_{2}[σ_{1}, σ_{2}] ⊗ E(y_{2}) = [𝔽_{2}[σ_{1}, σ_{2} + y_{2}] ⊗ E(σ_{1}y_{2})] + [𝔽_{2}[σ_{1}^{2} + σ_{2}, σ_{1}σ_{2}] ⊗ E(y_{2})]

Hence, the computation of \underleftarrow{lim}_{I}^{0} M can be reduced to an easy Poincaré series checking. To

obtain \underleftarrow{lim}_{I}^{1} M we use the exact sequence ([2])

0 → \underleftarrow{lim}_{I}^{0}M → M_{1} → Z → \underleftarrow{lim}_{I}^{1}M → 0

where Z is isomorphic to the quotient M_{0}^{Σ_{2}}/M_{0}^{Σ_{3}}. Hence, the vanishing of \underleftarrow{lim}_{I}^{1} M can also be easily obtained by checking the agreement of some Poincaré series. □

Let Y: I → S_{∗} be the covariant functor to the category of pointed spaces given by

Y_{0} = [ED ×_{D} D/V] × S^{2} ≃ BV × S^{2}
Y_{1} = W,

with the maps considered in the section before. Y is, up to homotopy, the diagram mentioned in the introduction:

GL_{2}(𝔽_{2}) ⟳ BV × S^{2} ⟶ ⟶ ⟶ W.

Let Y be the homotopy colimit of the functor Y. Then the spectral sequence of Bousfield-Kan ([5]) and the vanishing of the higher derived functors of lim allow the computation of the cohomology of Y.

Corollary 10. H^{∗}(Y) ≅ 𝔽_{2}[z_{2}, z_{3}] ⊗ E(y_{5}) and the map ϵ: W → Y is such that ϵ^{∗}(z_{2}) = w_{1}^{2} + w_{2}, ϵ^{∗}(z_{3}) = w_{1}w_{2} + y_{3}, ϵ^{∗}(y_{5}) = w_{2}y_{3}. □

In particular, the action of the Steenrod algebra on H^{∗}(Y) is given by

Sq^{1}z_{2} = z_{3}
Sq^{2}z_{3} = z_{2}z_{3} + y_{5}
Sq^{2}y_{5} = z_{2}y_{5}
Sq^{1}y_{5} = Sq^{4}y_{5} = 0.

§5. The space X.

Let now X be the homotopy fibre of an essential map Y → B^{2}ℤ/2.

Theorem 11. H^{∗}(X) ≅ 𝔽_{2}[x_{8}] ⊗ E(Sq^{1}x_{8}).

Proof. Notice that the map k: X → Y is trivial in cohomology. From this it is easy to determine all the differentials in the Serre spectral sequence of the fibration

Bℤ/2 → X → Y

and one sees ea that the spectral sequence collapses at E_{5} and

E_{∞} = E_{5} ≅ 𝔽_{2}[x_{8}] ⊗ E(y_{9}).

It only remains to prove that Sq^{1}x_{8} ≠ 0. Consider the Serre spectral sequence of the fibration X → Y → B^{2}ℤ/2 and notice that

α = Sq^{4}Sq^{2}Sq^{1}ι + ι^{2}Sq^{2}Sq^{1}ι + (Sq^{1}ι)^{3} ∈ H^{∗}(B^{2}ℤ/2)

maps to zero in H^{∗}(Y). Hence, x_{8} ∈ H^{∗}(X) transgresses to α. Then Sq^{1}x_{8} transgresses to Sq^{1}α which is non-trivial in H^{∗}(B^{2}ℤ/2). □

§6. Non-realizability of P E-algebras

In this section we prove the main theorem stated in the introduction.

Let us assume 𝔽_{2}[x_{2n}] ⊗ E(Sq^{1}x) is an unstable algebra over the Steenrod algebra. The classification of these algebras is done exactly as in the odd prime case ([3]). First one obtains the existence of the algebras

A ≅ 𝔽_{2}[x_{2}] ⊗ E(y_{3}), Sq^{1}(x) = y, Sq^{2}(y) = xy

and

B_{i} ≅ 𝔽_{2}[x_{2^{i+1}}] ⊗ E(y_{2^{i+1}+1}), Sq^{1}(x) = y, Sq^{2^{i+1}}(y) = 0

as algebras over the Steenrod algebra and then one proves that there are no more.

Proposition 12. Let A be a P E-algebra which is an unstable algebra over the Steenrod algebra and such that Sq^{1}(x) = y. Then A is isomorphic, as an algebra over the Steenrod algebra, to one of the algebras A, B_{i} as defined above.

Proof. Since the ideal generated by y is closed under the Steenrod algebra action the degree of x should be a power of two.

In the case when i > 0 the Adem relation

Sq^{2^{i}}Sq^{2^{i}+1} = ∑_{j=0}^{2^{i−1}} μ_{j} Sq^{2^{i+1}+1−j}Sq^{j}

gives Sq^{2^{i+1}}y = 0 when applied to x. Hence A should be isomorphic to B_{i}. If i = 0 then either Sq^{2}y = 0 or Sq^{2}y = xy and A is isomorphic to B_{0} and A, respectively. □

Let X be the space constructed in the previous sections. We have H^{∗}(X) ≅ B_{2}. Let F be the space considered in section 1. Then

H^{∗}(F) ≅ 𝔽_{2}[x_{2}] ⊗ E(Sq^{1}x_{2}) = B_{0}
H^{∗}(S^{3}⟨3⟩) ≅ 𝔽_{2}[x_{4}] ⊗ E(Sq^{1}x_{4}) = B_{1}

Hence the “if” part of the main theorem is proved.

Remark. The spaces F, S^{3}⟨3⟩ and X are not the unique ones realizing B_{0}, B_{1} and B_{2} respectively, not even up to 2-completion. In [4] we construct infinite families of spaces which realize these cohomology algebras. Actually, the goal of [4] is a complete classification of all 2-complete homotopy types realizing B_{0}, B_{1} and B_{2}.

It only remains to prove

Theorem 13. (Cooke conjecture for p = 2 [6], [3].) Assume that there is a space X with H^{∗}(X) ≅ B_{i}. Then i = 0, 1 or 2.

Proof. The proof is similar to the proof of the Cooke conjecture in [3] except for some details in small dimensions. We can as well assume that X is 2-complete because H^{1}(X) = 0. From [8] we know that there is a map

ϕ: Bℤ/2 → X

inducing the non trivial f_{0}: B_{i} → H^{∗}(Bℤ/2) and such that, according to the computation of the value of the T functor on the algebra B_{i} done in [3], the evaluation

map(Bℤ/2, X)_{ϕ} → X

is a homotopy equivalence. Now Bℤ/2 acts on this mapping space and if Y denotes the homotopy quotient, we obtain a sequence of fibrations:

Bℤ/2 ⟶^{j} map(Bℤ/2, X)_{ϕ} → Y → B^{2}ℤ/2

where j: Bℤ/2 → map(Bℤ/2, X)_{ϕ} composed with the evaluation becomes ϕ.

Our results arise from the analysis of the Serre spectral sequence for computing the cohomology of Y. This gives us the necessary restrictions on the dimensions of the generators of the cohomology of X.

Recall that H^{∗}(B^{2}ℤ/2) is a polynomial algebra on the generators

ι, Sq^{1}ι, Sq^{2}Sq^{1}ι, …, Sq^{Δ_{j}}ι, …

where we use the notation Sq^{Δ_{j}} = Sq^{2^{j}}Sq^{2^{j−1}} … Sq^{1}. Observe that the Adem relations and instability imply that the following relations hold in H^{∗}(B^{2}ℤ/2):

Sq^{1}Sq^{Δ_{j}}ι = (Sq^{Δ_{j−1}}ι)^{2}  j > 0
Sq^{2}Sq^{Δ_{j}}ι = 0  j > 1

The first observation is that since the evaluation is an equivalence j looks like ϕ in cohomology: j^{∗}(x) = u^{2^{i+1}} and therefore in the Serre spectral sequence for map(Bℤ/2, X)_{ϕ} → Y → B^{2}ℤ/2 the class x transgresses to Sq^{Δ_{i}} plus decomposables:

τ(x) = Sq^{Δ_{i}}ι + d.

Assume now i > 2. In the Serre spectral sequence we have the transgressions:

τ(x) = Sq^{Δ_{i}}ι + d
τ(y) = Sq^{1}(Sq^{Δ_{i}}ι) + Sq^{1}d = (Sq^{Δ_{i−1}}ι)^{2} + Sq^{1}d

Now, if we apply Steenrod operations to either τ(x) or τ(y) we obtain new elements in the bottom line of the spectral sequence that have to be killed by a differential. Let us try with Sq^{2} applied to τ(y). Since Sq^{2}(Sq^{Δ_{i−1}}ι)^{2} = (Sq^{Δ_{i−2}}ι)^{4} we obtain that Sq^{2}τ(y) = (Sq^{Δ_{i−2}}ι)^{4}+Sq^{2}Sq^{1}d has to be zero modulo elements killed by a differential. Let us denote α: = τ(x) and β: = τ(y). Then our equation is

(9)  0 = (Sq^{Δ_{i−2}}ι)^{4} + Sq^{2}Sq^{1}d  mod (α, β)

Notice that d can be written as

d = aSq^{Δ_{i−1}}ι + bSq^{Δ_{i−2}}ι + c

where a, b and c are polynomials in ι, Sq^{1}ι, …, Sq^{Δ_{i−3}}ι. Hence, a straightforward argument shows that equation (1) cannot be solved if i > 3. If i = 3 then the equation (1) admits several solutions because in that case Sq^{2}Sq^{1} applied to either Sq^{Δ_{0}}ι·Sq^{Δ_{1}}ι, Sq^{Δ_{2}}ι or ι(Sq^{Δ_{1}}ι)^{3} gives a term (Sq^{Δ_{1}}ι)^{4}. Nevertheless, d should also satisfy other equations obtained by applying Sq^{2^{k}} to α and β. In particular we obtain:

Sq^{2}(d) = Sq^{4}(d) = 0 mod (α, β)

and a long and boring but straightforward computation shows that no one of the solutions of (1) can satisfy these new equations. □

Remark. In [4] we prove that A is also realizable as the mod two cohomology of some space.

§7. The Arf invariant

In [1], [3], [4] and the present paper we have considered spaces X whose mod p cohomology has the form 𝔽_{p}[x_{2n}] ⊗ E(βx) where β denotes the Bockstein homomorphism. One sees immediately a relation between these spaces and the Arf invariant elements. For instance, (see also [1]) assume that X is 2-complete and

H^{∗}(X) ≅ B_{i−1} = 𝔽_{2}[x_{2^{i}}] ⊗ E(Sq^{1}x_{2^{i}}).

Then the cell structure of X looks like

X = S^{2^{i}} ∪_{2} e^{2^{i}+1} ∪_{θ} e^{2^{i+1}} ∪ ⋯

Here we use the standard notation in which θ ∈ π_{2^{i+1}−1}(S^{2^{i}+1}) is the map obtained from the attaching map of e^{2^{i+1}} by collapsing the 2^{i}-skeleton. Let X′ be the Spanier-Whitehead dual of the 2^{i+1}-skeleton of X. The cohomology of X′ has generators a, b and c linked by Steenrod operations in the way Sq^{2^{i}}a = Sq^{1}b = c. Then, for i > 3 the Adams decomposition of Sq^{2^{i}} through secondary operations implies that a and b have to be linked by the secondary operation corresponding to the Arf invariant elements in the stable stem. Hence, a realization of B_{i}, i > 2, would immediately provide an Arf invariant element. But theorem 13 in the present paper shows that B_{i}, i > 2 is not realizable as the cohomology of any space. Therefore, although Arf invariant elements have been constructed up to the 62-stem, only η^{2} and ν^{2} could be used to build a space with cohomology B_{i}. The siapplying is similar to what happens with the Hopf invariant one elements. Any space whose cohomology is a polynomial algebra on one generator should have the second cell attached to the first one through a map of Hopf invariant one but, although η, ν and σ have Hopf invariant one, only η and ν can be the starting block of such a space.

If X realizes B_{1} or B_{2} then the above argument cannot be used to show that the third cell is attached by the Arf invariant element, because Sq^{4} and Sq^{8} are not decomposable through secondary operations. Actually, we do not know what is the attaching map of the third cell of the space X constructed in section 5. In the case of S^{3}⟨3⟩ the attaching map for the third cell is indeed η^{2}.

Proposition 14. S^{3}⟨3⟩_{2} = S^{4} ∪_{2} e^{5} ∪_{η^{2}} e^{8} · · ·.

Proof. It suffices to prove that the 8-cell is attached non-trivially to the 5-cell. Let us first recall the few first stable homotopy groups of BS^{1} at the prime two (cf. [9]):

i =  1 2 3 4  5  6  7
π_{∗}^{S} = 0 ℤ 0 ℤ ℤ/2 ℤ ℤ/2
generator  β_{1}  2β_{2} β_{1}ν 2β_{3} β_{2}ν

We then use the homology Serre spectral sequence of the fibration BS^{1} → S^{3}⟨3⟩ → S^{3} to compute the stable homotopy of S^{3}⟨3⟩:

E_{p,q}^{2} = H_{p}(S^{3}; π_{q}^{S}(BS^{1})) ⇒ π_{p+q}^{S}(S^{3}⟨3⟩).

From this one deduces that

π_{7}^{S}(BS^{1}) → π_{7}^{S}(S^{3}⟨3⟩)

is an isomorphism. Since the generator of π_{7}^{S}(BS^{1}/S^{2}) comes from the fourth-dimensional cell, we obtain that π_{7}^{S}S^{4} = ℤ/8ν projects to π_{7}^{S}(S^{3}⟨3⟩). Consider the diagram

S^{4}
↓
S^{7} ⟶^{f} S^{4} ∪_{2} e^{5} ⟶^{h} S^{3}⟨3⟩
g↓
S^{5}

We want to prove that gf is essential. If not, then f lifts stably to S^{4} but, since hf is trivial, f is divisible by two in π_{7}^{S}S^{4} and so it is stably trivial in π_{7}^{S}(S^{4} ∪_{2} e^{5}). Since f is the attaching map of the seven-dimensional cell of S^{3}⟨3⟩, this contradicts the existence of a non-trivial Sq^{4} in S^{3}⟨3⟩. □

We finish with some open questions which deserve further study.

The above proof relies on the map S^{3}⟨3⟩ → S^{3} and so it does not work if S^{3}⟨3⟩ is replaced by a space with the same cohomology but a different mod 2 homotopy type. All such spaces are classified in [4] and it would be interesting to know if the attaching map of the third cell of such a space should always be η^{2}.

In the present paper we have constructed a space

X = S^{8} ∪_{2} e^{9} ∪_{θ} e^{16} ∪ ⋯

whose cohomology is B_{2}. Is θ = ν^{2}? In a similar way as in the case of S^{3}⟨3⟩, the mod 2 homotopy types of all spaces with the same cohomology than X are classified in [4]. If X′ is now one of these spaces, is also the third cell of X′ attached through the Arf invariant map?

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain.

Mathematisches Institut der Georg August Universität Göttingen, Bunsenstrasse 3, 37073 Göttingen, Germany.