FAKE THREE CONNECTED COVERINGS

OF LIE GROUPS

J. Aguadé, C. Broto and M. Santos

1. Introduction. In [1] it was proved that for each prime p there are infinitely many fake 3-connected coverings of S^{3}. By “fake” we mean spaces with the same mod p cohomology than S^{3}⟨3⟩ (as algebras over the Steenrod algebra) but different p-completed homotopy type. After that work was completed one could wonder if the existence of such fake spaces was a general phenomenon and, in particular, if one could use the same methods to produce fake three connected coverings of other Lie groups beside S^{3}. In this paper we prove that the results of [1] cannot be extrapolated since, indeed, there is homotopy uniqueness up to p-completion for 3-connected coverings of several compact connected Lie groups and p-compact groups.

If p is a regular prime for the compact connected Lie group G then S^{3} is a direct factor of G at the prime p and one can trivially construct infinitely many fake G⟨3⟩ out of the fake S^{3}⟨3⟩ constructed in [1]. If p is quasi-regular for G in the sense of [10] then G splits at p as a product of odd dimensional spheres and spaces B_{n}(p) that are sphere bundles over spheres. Hence, in this more general situation fake G⟨3⟩ would arise if there are fake B_{3}(p)⟨3⟩. The main result of this paper shows that there are no such fakes.

Theorem 1. Let B(p) denote the S^{3}-bundle over S^{2p+1} classified by a generator of the p-component of π_{2p}S^{3}. Let X be such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(B(p)⟨3⟩; 𝔽_{p}) as algebras over the Steenrod algebra. Then X̂_{p} ≃ B(p)⟨3⟩_{p}.

Corollary 2. Up to p-completion, there is only one space with the same mod p cohomology (as algebras over the Steenrod algebra) than G⟨3⟩ if G is i) SU(3) for p = 2; ii) Sp(2) for p = 3 or iii) G_{2} for p = 5.

The proof of the main result is obtained by an analysis similar to the one done in [1] for the case of S^{3}. Starting with a space X with the same mod p cohomology than G⟨3⟩ we construct an infinite tower

X → X_{1} → X_{2} → X_{3} → ⋯

such that X_{i+1} is obtained from X_{i} essentially by dividing by an action of the group Bℤ/p. At the far right end of the tower we get a space which up to p-completion coincides with B(p) and so we get a map X → B(p) which induces isomorphism in mod p cohomology between X and B(p)⟨3⟩. At each stage we need to compute the cohomology of X_{i+1}

Typeset by AMS-TEX

The authors are partially supported by DGICYT grant PB89-0321.

out from the cohomology of X_{i}. Like in the case of S^{3}⟨3⟩ discussed in [1] there is an indeterminacy in the computation of H^{∗}(X_{i+1}; 𝔽_{p}) but, in contrast to what happens in that case, only one of the two algebras which are algebraically compatible with the Serre spectral sequence can be the cohomology algebra of a space. Hence, the key point in the proof and also the point that explains the opposite behavior of S^{3} and B(p) is the proof of the non-realizability of a certain algebra over the Steenrod algebra as the cohomology of a space. This proof is obtained by showing that, for this particular algebra, the Lannes T functor ([9]) is not compatible with higher Bocksteins.

During the present paper, mod p coefficients are assumed unless otherwise stated. Most of the proofs are written for the case of p odd but they can be translated to the case of p = 2 by using the standard identifications Sq^{1} = β and Sq^{2i} = P^{i}. In a few cases in which there is enough difference between the odd prime case and the case of p = 2, we provide separate arguments. We denote by U and K the categories of unstable modules and unstable algebras over the Steenrod algebra, respectively. When describing graded algebras, we tend to use subscripts to denote the degree of an element. In some cases, however, in order to simplify the notation, we may omit these subscripts.

The second author would like to thank Fred Cohen for his hospitality and helpful comments during the preparation of this paper.

2. Uniqueness of B(p). The homotopy uniqueness of B(p) up to p-completion is a necessary step in the proof of theorem 1. We have the following result:

Proposition 3. If H^{∗}(X) ≅ H^{∗}(B(p)) as algebras over the Steenrod algebra then X_{p} ≃

B(p)_{p}.

Proof. Since H^{1}(X; 𝔽_{p}) = 0 the fundamental group of X is p-perfect and so ([4], VII.3) X is p-good, X̂_{p} is simply connected and H^{∗}(X̂_{p}) ≅ H^{∗}(B(p)). Hence, we can assume, without loss of generality, that X is p-complete and simply connected. By proposition 5.7 in [1] we know that the homotopy groups of X are finitely generated over ℤ̂_{p} and also that H^{∗}(X; ℤ̂_{p}) is of finite type over ℤ̂_{p}. Now we can apply theorem 3.1 in [3] to conclude that X ≃ W_{p}where W is a CW-complex of finite type. Clearly, W can be chosen to be a three-cell complex

W = S^{3} ∪_{ϕ_{1}} e^{2p+1} ∪_{ϕ_{2}} e^{2p+4}.

Since the 3-dimensional class is linked to the (2p + 1)-dimensional class through a P^{1}, the map ϕ_{1} is congruent modulo p to the generator of π_{2p}(S^{3})_{(p)} and the same is true in B(p). Hence, one has the diagram:

S^{2p+3}
↓
S^{3} ∪_{ϕ_{1}} e^{2p+1} ⟶^{i} B(p)_{(p)}
j↓
W

According to [12], the homotopy group π_{2p+3}B(p)_{(p)} vanishes. Hence, we have a map W → B(p)_{(p)} inducing isomorphism in homology with coefficients in ℤ̂_{p}. This implies

X ≃ B(p)_{p}. □

Remark. The above result will be used in the proof of theorem 1 and, actually, proposition 3 is a consequence of theorem 1. To see this, let L denote the localization functor with respect to the map Bℤ/p → ∗ as defined in [5]. Then a result of Neisendorfer ([11], also cf. lemma 9.9 in [1]) implies that L(B(p)⟨3⟩_{p}) and B(p) are homotopy equivalent after p-completion. Hence, if X is a space with the same mod p cohomology than B(p) then, by theorem 1 we have B(p)⟨3⟩_{p} ≃ X⟨3⟩_{p}. Applying now the functor L we get X_{p}≃ B(p)_{p}.

3. The algebras K and L and the T functor. Let us denote by L the unstable algebra over the Steenrod algebra L = H^{∗}(B(p)⟨3⟩). The structure of L is well known ([12]):

L ≅ 𝔽_{p}[x_{2p^{2}}] ⊗ E(y_{2p+1}, z_{2p^{2}+1})

(subscripts denote degrees) and the Steenrod algebra action is given by the formulas β(x) = P^{p}(y) = z. The action of the Steenrod algebra on L can be easily deduced from these formulas and the Adem relation. One obtains

P^{1}(x) = P^{p}(x) = β(y) = P^{1}(y) = β(z) = P^{1}(z) = P^{p}(z) = P^{p^{2}}(z) = 0.

The vanishing of all these Steenrod operations follows from dimensional reasons, except for the last two equalities which can be proved using the Adem relations

Sq^{4}Sq^{4} = Sq^{7}Sq^{1} + Sq^{6}Sq^{2}
Sq^{8}Sq^{1} = Sq^{4}Sq^{5} + Sq^{9} + Sq^{7}Sq^{2}
P^{p^{i}}β = P^{1}βP^{p^{i}−1} + βP^{p^{i}}, i > 0.

Let us define now K to be the graded algebra

K = 𝔽_{p}[x_{2}] ⊗ E(y_{3}, z_{2p+1}).

We want to study actions of the Steenrod algebra on K which turn K into an unstable algebra over the Steenrod algebra. In particular, we are interested in actions of the Steenrod algebra on K subject to the conditions β(y) = 0 and P^{1}(y) = z. The following proposition gives a complete classification of such actions.

Proposition 4. Assume K is an unstable algebra over the Steenrod algebra with β(y) = 0 and P^{1}(y) = z. Then the action of the Steenrod algebra on K is determined by β(x) and P^{p}(z). Moreover, P^{p}(z) is either zero or x^{p(p−1)}z.

Proof. It is obvious that the action of the Steenrod algebra on K is determined by unstability and the values of

β(x), β(y), β(z), P^{1}(y), P^{1}(z), P^{p}(z).

Since 2P^{2} = P^{1}P^{1} one obtains P^{1}(z) = 0 (use Sq^{2}Sq^{2} = Sq^{3}Sq^{1} if p = 2). The Adem relation 2P^{1}βP^{1} = βP^{1}P^{1} + P^{1}P^{1}β gives β(z) = 0 (use Sq^{1}Sq^{2} = Sq^{3} if p = 2).

By dimensional reasons we have P^{p}(z) = λx^{p(p−1)}z + μx^{p^{2}−1}y. The relation P^{1}P^{p} = P^{p+1} shows that μ = 0. If p = 2 there is nothing more to prove. If p is odd and P^{p}(z) = λx^{p(p−1)}z then the Adem relation

P^{p}P^{p} = 2P^{2p} + P^{2p−1}P^{1}

shows that λ = 0, 1. □

We introduce now the following family of unstable algebras over the Steenrod algebra. We define K_{λ}^{(r)}, r > 0, λ = 0, 1 as

K_{λ}^{(r)} = 𝔽_{p}[x_{2}] ⊗ E(y_{3}, z_{2p+1}),

with the action of the Steenrod algebra given by P^{1}(y) = z, β(y) = β(z) = P^{1}(z) = 0 and P^{p}(z) = λx^{p(p−1)}z. We also assume that β_{(r)}(x) = y where β_{(r)} denotes the r-th order higher Bockstein. Since β_{(r)} for r > 1 is not in the Steenrod algebra, it is clear that K_{λ}^{(r)} and K_{λ}^{(s)} for r, s > 1 are the same object of K. However, it makes sense to ask if K_{λ}^{(r)} is the cohomology of some space and the notation K_{λ}^{(r)} will be a convenient one. We will also use the notation K′_{λ} to indicate the generic unstable algebra over the Steenrod algebra represented by any K_{λ}^{(r)} with r > 1.

Let us denote by B(p)⟨3; p^{r}⟩ the homotopy fibre of the map B(p) → K(ℤ, 3) of degree p^{r}. There is a fibration

B(p)⟨3⟩ ⟶ B(p)⟨3; p^{r}⟩ ⟶ B^{2}ℤ/p^{r}.

It is an easy exercise to show that H^{∗}(B(p)⟨3; p^{r}⟩) ≅ K_{0}^{(r)}. This shows that K_{0}^{(r)} is a genuine algebra over the Steenrod algebra. It would not be difficult to show that K_{1}^{(r)}is also an unstable algebra over the Steenrod algebra. However, a key point in the proof of theorem 1 is to show that K_{1}^{(r)} is not the cohomology of any space (proposition 8).

There are homomorphisms of unstable algebras over the Steenrod algebra

f : L → H^{∗}(Bℤ/p)
f : K_{λ}^{(r)} → H^{∗}(Bℤ/p)

which vanish on y and z and such that f(x) is non trivial.

Let T denote the Lannes functor defined as left adjoint to H^{∗}(Bℤ/p) ⊗ − in the category U of unstable modules over the Steenrod algebra (see [9] for a full description of its properties.) When R is an unstable algebra over the Steenrod algebra then so is T (R) and T becomes a functor in the category K of unstable algebras over the Steenrod algebra.

Given a K-map f : R → H, its adjoint restricts to a K-map T^{0}(R) → 𝔽_{p}, where T^{0}(R) is the subalgebra of T (R) of all elements of degree zero. We define the connected component of T (R) corresponding to f as:

T_{f}(R) = T (R) ⊗_{T^{0}(R)} 𝔽_{p}.

Furthermore, T_{f} may be thought as a functor defined on the category U(R) of R-U-modules and with values in the category of T_{f}(R)-U-modules (cf. [8].) We can also consider T_{f}(M) as an R-U-module induced by the natural K-map ϵ : R → T_{f}(R) and then ϵ : M → T_{f}(M) becomes a natural transformation of R-U-modules. We recall from [8] that T_{f} is exact and commutes with suspensions, tensor products and direct limits.

We need to compute the value of T_{f} on the algebras L and K_{λ}^{(r)}, where f denotes the homomorphisms that we have just described. The results are summarized in the following proposition:

Proposition 5. (1) ϵ : K → T_{f}(K) is an isomorphism for K = K_{0}^{(1)}, K′_{0}, L.

(2) T_{f}(K_{1}^{(1)}) ≅ 𝔽_{p}[x_{2}] ⊗ E(y_{3}, w_{1}) with β(x_{2}) = y_{3} and P^{1}(y_{3}) = x_{2}^{p}w_{1}. x_{2} ∈ K_{1}^{(1)} is sent to x_{2} ∈ T_{f}(K_{1}^{(1)}) by ϵ. A similar result (with β(x_{2}) = 0) holds for K′_{1}.

Proof. Consider first the case of K_{0}^{(1)}. We work in the category U(K_{0}^{(1)}). There is an exact sequence in this category

0 → zK_{0}^{(1)} → K_{0}^{(1)} → B → 0

where B ≅ 𝔽_{p}[x_{2}] ⊗ E(y_{3}) is the algebra denoted by B_{0,1} in [1]. The exactness of T_{f} yields a commutative diagram of K_{0}^{(1)}-U-modules with exact rows:

0 ⟶ zK_{0}^{(1)} ⟶ K_{0}^{(1)} ⟶ B ⟶ 0
ϵ_{1}↓  ^{ϵ_{2}}↓  ^{ϵ_{3}}↓
0 ⟶ T_{f}(zK_{0}^{(1)}) ⟶ T_{f}(K_{0}^{(1)}) ⟶ T_{f}(B) ⟶ 0

Theorem 3.1 in [1] gives B ≅ T_{h}(B) where h : B → H is the obvious map. By lemma 3.2 in [1] we also have B ≅ T_{f}(B) because f = h ∘ g with g : K_{0}^{(1)} → B and the hypothesis of lemma 3.2 in [1] is trivially satisfied. Hence, ϵ_{3} is an isomorphism.

On the other side, since the Steenrod algebra acts trivially on z ∈ K_{0}^{(1)} there is an isomorphism zK_{0}^{(1)} ≅ Σ^{2p+1}B. Since T_{f} commutes with suspensions, ϵ_{1} is an isomorphism and so is ϵ_{2}. The case of K′_{0} is analogous.

Let now M denote the algebra M = 𝔽_{p}[x_{2p^{2}}] ⊗ E(y_{2p+1}) with the Steenrod algebra acting trivially on y_{2p+1}. There is an exact sequence in U(L):

0 ⟶ zL ⟶ L ⟶ M ⟶ 0.

Notice that zL ≅ Σ^{2p^{2}+1}M. Since T_{f} commutes with suspensions, to show that ϵ : L → T_{f}L is an isomorphism it suffices to prove that ϵ : M → T_{f}M is an isomorphism. We can work now in the category U(𝔽_{p}[x_{2p^{2}}]) and observe that T_{f}M ≅ T_{g}M where g is the non trivial map g : 𝔽_{p}[x_{2p^{2}}] → H^{∗}(Bℤ/p) ([1], 3.2). We consider the exact sequence

0 ⟶ yM ⟶ M ⟶ 𝔽_{p}[x_{2^{2}}] ⟶ 0

and notice that yM ≅ Σ^{2p+1}𝔽_{p}[x_{2p^{2}}] and T_{g}𝔽_{p}[x_{2p^{2}}] ≅ 𝔽_{p}[x_{2p^{2}}]. This concludes the analysis for the algebra L.

For K_{1}^{(1)} we start in the same way as with K_{0}^{(1)} by considering the diagram of K_{1}^{(1)}-U-modules

(1)  \begin{aligned}0 & ⟶ & zK_{1}^{(1)} & ⟶ & K_{1}^{(1)} & ⟶ & B & ⟶ & 0 \\ & & ^{ϵ_{1}}↓ & & ^{ϵ_{2}}↓ & & ^{ϵ_{3}}↓ \\ 0 & ⟶ & T_{f}(zK_{1}^{(1)}) & ⟶ & T_{f}(K_{1}^{(1)}) & ⟶ & T_{f}(B) & ⟶ & 0 \end{aligned}

ϵ_{3} is an isomorphism and we can observe that zK_{1}^{(1)} ≅ Σx^{p}B. To compute T_{f}(x^{p}B) we consider the exact sequence

0 ⟶ x^{p}B ⟶ B ⟶ B/x^{p}B ⟶ 0

Multiplication by x^{p} annihilates the right hand module. Hence, by proposition 2.3 in [8], T_{f}(B/x^{p}B) = 0 and so T_{f}(x^{p}B) ≅ B and the map x^{p}B → T_{f}(x^{p}B) ≅ B is the natural inclusion. If we input now this fact into diagram (1) we easily get the structure of T_{f}K_{1}^{(1)}. The case of K′_{1} is analogous. □

4. Spectral sequence computations. In this section we compute the mod p cohomology of a space which is obtained from a space X by dividing by an action of the abelian group Bℤ/p, in the cases in which the cohomology of X is of the form either L or K_{λ}^{(r)}. We use the notation

P^{Δ_{j}} = P^{p^{j}} ⋯ P^{p}P^{1}, j ≥ 0,
Sq^{Δ_{j}} = Sq^{2^{j}} ⋯ Sq^{2}Sq^{1}, j ≥ 0.

Proposition 6. Let Bℤ/p ⟶^{i} X ⟶^{k} Y be a principal fibration with H^{∗}(X) ≅ K_{0}^{(r)} and i^{∗}(x_{2}) ≠ 0. Then H^{∗}(Y) ≅ K_{λ}^{(r+1)} for some λ ∈ {0, 1} and k^{∗}(x_{2}) = 0, k^{∗}(y_{3}) = y_{3} and k^{∗}(z_{2p+1}) = z_{2p+1}.

Proof. Let us consider the Serre spectral sequence of the fibration X → Y → B^{2}ℤ/p. Recall that

H^{∗}(B^{2}ℤ/p) = \{ \begin{matrix}𝔽_{p}[ι, βP^{1}βι, …, βP^{Δ_{j}}βι, …] ⊗ E(βι, P^{1}βι, …, P^{Δ_{j}}βι, …), p > 2 \\ 𝔽_{2}[ι, Sq^{1}ι, …, Sq^{Δ_{j}}ι, …], p = 2. \end{matrix}

Observe that x_{2} ∈ H^{∗}(X) has to be transgressive to βι modulo units. Then, since the transgression commutes with the Steenrod operations, there are differentials

d(x^{p^{j}}) = \{ \begin{matrix}P^{Δ_{j−1}}βι, j ≥ 1, p > 2 \\ Sq^{Δ_{j−1}}ι, j ≥ 1, p = 2 \end{matrix}

In the case of p odd, Kudo’s transgression theorem implies that there are differentials

d(x^{p^{j}(p−1)}P^{Δ_{j−1}}βι) = βP^{Δ_{j}}βι.

The spectral sequence with coefficients in ℤ_{(p)} shows that y_{3} has to survive. Using this, one can compute all differentials in the spectral sequence and conclude that

E_{∞} = 𝔽_{p}[ι_{2}] ⊗ E(y_{3}, z_{2p+1}).

The spectral sequence with coefficients in ℤ_{(p)} yields also that β_{(r+1)} has to be non trivial on the two dimensional class of Y. From here one concludes

H^{∗}(Y) ≅ 𝔽_{p}[x_{2}] ⊗ E(y_{3}, z_{2p+1})

and the class z_{2p+1} can be chosen such that P^{1}(y_{3}) = z_{2p+1}. In the case of p = 2 one needs to prove that y_{3} and z_{2p+1} are indeed exterior generators. This follows easily by the Bockstein spectral sequence. The conclusion follows from proposition 4. □

The next spectral sequence that we have to study needs a more involved analysis.

Proposition 7. Let Bℤ/p ⟶^{i} X ⟶^{k} Y be a principal fibration with H^{∗}(X) ≅ L and i^{∗}(x_{2p^{2}}) ≠ 0. Then H^{∗}(Y) ≅ K_{λ}^{(1)} for some λ ∈ {0, 1} and k^{∗} is trivial.

Proof. Let us consider the Serre spectral sequence of X ⟶ Y ⟶^{l} B^{2}ℤ/p. Obviously, y_{2p+1} ∈ H^{∗}(Y) has to be transgressive and our first goal is to compute the transgression of y_{2p+1}. By dimensional reasons,

τ(y_{2p+1}) = \{ \begin{matrix}δβP^{1}βι + μι^{p+1}, p > 2 \\ δ(Sq^{1}ι)^{2} + μι^{3}, p = 2 \end{matrix}

If we notice now that β(y_{2p+1}) = 0 and that the Steenrod operations commute with the transgression, we easily get μ = 0. We want to prove that δ ≠ 0. Assume that δ = 0 and so y_{2p+1} survives to E_{∞}. In this case, the first differential which may be non trivial appears in total degree 2p^{2}. Hence, we know H^{∗}(Y) up to dimension 2p^{2} − 1 and it turns out to be generated by

x_{2} = l^{∗}(ι), βx_{2}, P^{1}βx_{2}, βP^{1}βx_{2}, ȳ.

We input now this information on H^{∗}(Y) into the Serre spectral sequence of the fibration Bℤ/p → X → Y. Obviously, u ∈ H^{1}(Bℤ/p) has to be transgressive to x_{2} ∈ H^{2}(Y) (up to units). Then, v ∈ H^{2}(Bℤ/p) transgresses to βx_{2}, v^{p} transgresses to P^{1}βx_{2}, v^{p^{2}} transgresses to P^{p}P^{1}βx_{2} and, in the case of p odd, Kudo’s formula determines differentials on v^{p−1}βx_{2} and v^{p(p−1)}P^{1}βx_{2}. In this form, one is able to fully determine the differentials in this spectral sequence up to total degree 2p^{2} − 1. Notice that v^{p^{2}} ∈ H^{∗}(Bℤ/p) is in the image of H^{∗}(X), hence it has to survive and so we get that P^{p}P^{1}βx_{2} ∈ H^{∗}(Y) has to be decomposable in terms of x_{2}, βx_{2}, P^{1}βx_{2}, βP^{1}βx_{2} and ȳ. Hence, the same is true for βP^{p}P^{1}βx_{2} ∈ H^{∗}(Y).

Let us return now to the Serre spectral sequence of X → Y → B^{2}ℤ/p. Since, as we have just proved, βP^{p}P^{1}βx_{2} has to be decomposable in H^{∗}(Y), there should be a differential in this spectral sequence starting at total degree 2p^{2} + 1. hence, z_{2p^{2}+1} has to support a non-trivial differential. This is a contradiction since z_{2p+1} = P^{p}y_{2p+1} and y_{2p+1} is asumed to be transgressive to zero. This contradiction proves that y_{2p+1} ∈ H^{∗}(X) cannot survive and, indeed, up to units we have

τ(y_{2p+1}) = \{ \begin{matrix}βP^{1}βι, p > 2 \\ (Sq^{1}ι)^{2}, p = 2 \end{matrix}

with zero indeterminacy. From here we get that z_{2p^{2}+1} ∈ H^{∗}(X) is also transgressive and

τ(z_{2p^{2}+1}) = \{ \begin{matrix}βP^{p}P^{1}βι, p > 2 \\ (Sq^{2}Sq^{1}ι)^{2}, p = 2 \end{matrix}

modulo the indeterminacy. Let us study now the differentials of the spectral sequence of X → Y → B^{2}ℤ/p when applied to x_{2p^{2}} ∈ H^{∗}(X). Notice that this element cannot survive because z does not. If p ≠ 2, the y_{2p+1}-row has disappeared when the first non trivial differential on x_{2p^{2}} appears because τ(y_{2p+1}) is not a zero divisor in H^{∗}(B^{2}ℤ/p) and so x_{2p^{2}} is transgressive. By naturality and up to units we have

(1)  τ(x_{2p^{2}}) = P^{p}P^{1}βι + d

If p = 2 the analysis needs to be slightly different but one also concludes that x_{2p^{2}} is transgressive and formula (1) holds.

Now, the knowledge about the transgressions of x_{2p^{2}}, y_{2p+1} and z_{2p^{2}+1} as well as the commutativity of the transgression with the Steenrod operations and Kudo’s transgression formula, allow us to fully determine the differentials in the spectral sequence of the fibration X → Y → B^{2}ℤ/p. The conclusion is that

H^{∗}(B^{2}ℤ/p) → H^{∗}(Y)

is an epimorphism and H^{∗}(Y) is generated by the images of ι, βι and P^{1}βι. In the case of p = 2, (Sq^{1}ι)^{2} and (Sq^{2}Sq^{1}ι)^{2} are the transgressions of y_{5} and z_{9}, respectively. Hence, we have, in any case

H^{∗}(Y) ≅ 𝔽_{p}[x_{2}] ⊗ E(β(x_{2}), P^{1}β(x_{2}))

and so, by proposition 4, H^{∗}(Y) ≅ K_{λ}^{(1)} for some λ ∈ {0, 1}. □

5. Non-realizability of K_{1}^{(r)}. In this section we will prove that K_{1}^{(r)} cannot be the mod p cohomology of any space. Roughly speaking, this non-realizability result will follow from the fact that the homomorphism K_{1}^{(r)} → T_{f}K_{1}^{(r)} is not compatible with the Bockstein spectral sequence.

Proposition 8. There is no space X such that H^{∗}(X) ≅ K_{1}^{(r)} for any r > 0.

Proof. Let X be a space such that H^{∗}(X) ≅ K_{1}^{(r)}. By the same argument as in proposition 3 we can assume, without loss of generality, that X is p-complete and simply connected. Let f : K_{1}^{(r)} → H^{∗}(Bℤ/p) be the non trivial homomorphism considered before. Then by [9; 3.1.1] there is a map ϕ : Bℤ/p → X inducing f in mod p cohomology. By proposition 5 we have T_{f}K_{1}^{(r)} ≅ 𝔽_{p}[x_{2}] ⊗ E(y_{3}, w_{1}).

Let {P_{s}X}_{s>0} denote the Postnikov tower of X. Then, the main theorem in [6] implies that

T_{f}K_{1}^{(r)} ≅ \underrightarrow{lim} H^{∗}(map(Bℤ/p, P_{s}X)_{ϕ_{s}}).

For a given s, consider the map induced in cohomology by the evaluation map

e^{∗} : H^{∗}(P_{s}X) → H^{∗}(map(Bℤ/p, P_{s}X)_{ϕ_{s}}).

In the limit when s → ∞ this is the homomorphism K_{1}^{(r)} → 𝔽_{p}[x_{2}] ⊗ E(y_{3}, w_{1}) given by x ↦ x, y ↦ y, z ↦ x^{p}w. It is clear that in any argument which involves only finitely many cohomology classes and finitely many primary and secondary operations, by taking s large enough we can substitute H^{∗}(P_{s}X) by K_{1}^{(r)} and H^{∗}(map(Bℤ/p, P_{s}X)_{ϕ}) by 𝔽_{p}[x_{2}] ⊗ E(y_{3}, w_{1}).

Let us analyze now the Bockstein spectral sequence of H^{∗}(X) ≅ K_{1}^{(r)} up to degree 2p + 3. We have β_{(r)}(x) = y and so E_{r+1} only contains x^{p}, x^{p−1}y and z up to that degree. If p is odd or p = 2 and r > 1 it is well known that β_{(r+1)}(x^{p}) = x^{p−1}y. If p = 2 and r = 1 then β_{(2)}(x^{2}) = xy + z. In any case, z is a permanent cycle.

In the Bockstein spectral sequence of map(Bℤ/p, P_{s}X)_{ϕ} for s big enough we have by naturality β_{i}(w) = 0 for i < N and N arbitrarily large and also β_{(r)}(x) = y. Hence,

β_{(r+1)}(x^{p}w) = x^{p−1}yw ≠ 0

and this is a contradiction since z maps to x^{p}w by the evaluation map. □

Remark. One could define a suitable category of unstable algebras over the Steenrod algebra “with higher Bocksteins”. An object in this category B would be a spectral sequence (E_{r}, d_{r}) of graded algebras over 𝔽_{p} such that E_{1} ∈ K, d_{1} = β, d_{i} are derivations and such that the usual formula for the higher Bockstein of x^{p} holds. In particular, the cohomology algebra of a space is an object in this category. There is a forgetful functor from B to K. Then the above proof shows that the T functor cannot be lifted to B.

Remark. The above proof is a further example of the usefulness of the T functor to deal with the classical problem of realizability of unstable algebras over the Steenrod algebra as cohomology rings. Other examples that we would like to mention here are the celebrated theorem by Dwyer-Miller-Wilkerson on the non-realizability of inseparable subalgebras of H^{∗}(BT^{n}) for p odd ([7]) and also the proof of the Cooke conjecture ([1], [2]) on the non-realizability of algebras of the form 𝔽_{p}[x] ⊗ E(β(x)).

6. Proof of theorem 1. Like in proposition 3, we can assume, without loss of generality, that X is p-complete and simply connected. Let f : L → H^{∗}(Bℤ/p) be the non trivial homomorphism considered before. Then by [9; 3.1.1] there is a map ϕ : Bℤ/p → X inducing f in mod p cohomology. By proposition 5 we have T_{f}L ≅ L. Then, [9; 3.2.1] shows that

H^{∗}(map(Bℤ/p, X)_{ϕ}) ≅ L

where map(Bℤ/p, X)_{ϕ} is the space of all maps Bℤ/p → X homotopic to ϕ. Hence, up to p-completion, map(Bℤ/p, X)_{ϕ} is homotopy equivalent to X. Observe now that Bℤ/p is a connected abelian simplicial group and the action of Bℤ/p on itself by right translations induces an action of Bℤ/p on the space map(Bℤ/p, X)_{ϕ}. If X_{1} is the homotopy quotient of map(Bℤ/p, X)_{ϕ} by this action, we have a fibration

Bℤ/p ⟶^{ϕ} X ⟶ X_{1}.

By proposition 7 we have H^{∗}(X_{1}) ≅ K_{λ}^{(1)} for some λ ∈ {0, 1}. But proposition 8 shows that K_{1}^{(1)} cannot be the cohomology of any space. Hence, H^{∗}(X_{1}) ≅ K_{0}^{(1)}. Consider now the homomorphism f : K_{0}^{(1)} → H^{∗}(Bℤ/p). If we substitute X_{1} by its p-completion we can find a map ϕ : Bℤ/p → X_{1} inducing f in cohomology. Let us consider the mapping space map(Bℤ/p, X_{1})_{ϕ}. According to [9; 3.2.1] the cohomology of this mapping space is isomorphic to T_{f}K_{0}^{(1)} ≅ K_{0}^{(1)} (proposition 5). Hence, the evaluation map gives a homotopy equivalence between the p-completion of this mapping space and X_{1}. In the same way as before, this allows us to construct a fibration

Bℤ/p ⟶^{ϕ} X_{1} ⟶ X_{2}

and proposition 6 plus the non realizability of K_{1}^{(r)} (proposition 8) gives H^{∗}(X_{2}) ≅ K_{0}^{(2)}. It is clear that this process can be repeated again and again producing a sequence of spaces and maps

X → X_{1} → X_{2} → X_{3} → ⋯

with H^{∗}(X_{r}) ≅ K_{0}^{(r)}, r > 0. Let X_{∞} be the homotopy colimit of this sequence of spaces. The cohomology of X_{∞} can be calculated by the classical short exact sequence

1 ⟶ \underleftarrow{lim}^{1}H^{∗+1}(X_{j}) ⟶ H^{∗}(X_{∞}) ⟶ \underleftarrow{lim} H^{∗}(X_{j}) ⟶ 1 .

The first term vanishes because all the groups are finite, and

\underleftarrow{lim} H^{∗}(X_{j}) ≅ H^{∗}(B(p)).

Hence, by proposition 3 (X_{∞})_{p} ≃ B(p)_{p}. Let F be the homotopy fibre of the map X_{1} → X_{∞}. An elementary spectral sequence argument shows that H^{∗}(F) ≅ H^{∗}(BS^{1}). Hence, F ≃ (BS^{1})_{p} = K(ℤ̂_{p}, 2) and the fibration is principal because B(p)_{p} is simply connected. Then X_{1} is the homotopy fibre of a map B(p)_{p} → K(ℤ̂_{p}, 3). The cohomology of X_{1} forces this map to be of degree pα where α is a unit in ℤ̂_{p}. Hence, X_{1} ≃ B(p)⟨3; p⟩_{p}. Now X is the fibre of the map B(p)⟨3; p⟩_{p} → B^{2}ℤ/p classified by the two dimensional class x_{2}. Hence, X ≃ B(p)⟨3⟩_{p} and the proof is complete. □

It is obvious from the above proof that the same arguments show the homotopy uniqueness of the spaces B(p)⟨3; p^{r}⟩:

Theorem 9. Let X be such that H^{∗}(X) ≅ K_{0}^{(r)}. Then X̂_{p} ≃ B(p)⟨3; p^{r}⟩_{p}. □

References

3. [1] J. Aguadé, C. Broto and D. Notbohm, Homotopy Classification of Spaces With Interesting Cohomology and a Conjecture of Cooke, Part I, Topology 33 (1994), 455–492.

[2] J. Aguadé, C. Broto and D. Notbohm, A mod 2 analogue of a conjecture of Cooke, (submitted).

[3] V. Belfi and C. Wilkerson, Some Examples in the Theory of P -completions, Indiana U. Math. J. 25 (1976), 565–576.

[4] A. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer Verlag, 1972.

[5] E. Dror Farjoun, Homotopy localization and v_{1}-periodic spaces, Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Lecture Notes in Math. 1509, Springer Verlag, 1992.

[6] E. Dror Farjoun and J. Smith, A geometric interpretation of Lannes’ functor T, Astérisque 191 (1990), 87–95.

[7] W.G.Dwyer, H. Miller and C.W. Wilkerson, Homotopical Uniqueness of classifying spaces, Topology 31 (1992), 29–45.

[8] W.G. Dwyer and C.W. Wilkerson, Smith theory and the functor T, Comm. Math. Helv. 661 (1991), 1–17.

[9] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Publ. Math. IHES 75 (1992), 135–244.

[10] M. Mimura and H. Toda, Cohomology operations and the homotopy of compact Lie groups-I, Topology 9 (1970), 317–336.

[11] J.A. Neisendorfer, Localization and Connected Covers of Finite Complexes, to appear in Contemp. Math.

[12] H. Toda, On homotopy groups of S^{3}-bundles over spheres, J. Math. Kioto Univ. 2-2 (1963), 193–207.

file: SU3.TEX date: 8.9.1994

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