HOMOTOPY CLASSIFICATION OF SPACES

WITH INTERESTING COHOMOLOGY

AND A CONJECTURE OF COOKE,

PART I

J. Aguadé, C. Broto and D. Notbohm

1. Introduction. The title of this paper is reminiscent of the title of one of the last papers by George Cooke ([7]). In that paper, Cooke observes that if ΩX is a p-complete loop space then there is an action of [S^{1}_{p̂}, S^{1}_{p̂}] ≅ ℤ̂_{p} on ΩX. In particular, there is an action of the p − 1 roots of unity on ΩX and by taking the quotients of appropriate loop spaces by this action he obtains spaces with “interesting” cohomology, i.e. spaces whose cohomology algebras have quite few generators and relations and whose attaching maps represent interesting elements in the stable homotopy of spheres. By applying this technique to S^{3}⟨3⟩, the 3connective covering of S^{3}, and to the fibre of the map S^{3} → K(ℤ, 3) of degree p, Cooke constructs spaces realizing the cohomology algebras (subscripts denote degrees)

(1)  𝔽_{p}[x_{2n}] ⊗ E(βx_{2n}),

where n is any divisor of p(p − 1), β is the Bockstein homomorphism and E(y_{k}) denotes an exterior algebra on one generator y_{k} of degree k. This method was generalized in [5] to construct spaces whose mod p cohomology has the form P ⊗ E where P is a polynomial algebra and E is an exterior algebra on the Bocksteins of the generators of P. Cooke ends his paper by saying “I expect that the condition n|p(p − 1) is necessary as well” and this is the Cooke conjecture that we mention in the title of this paper. The conjecture was proved to hold true in some particular cases in [2] where cohomology algebras of the form 𝔽_{p}[x] ⊗ E(y) where studied by completely different methods to those used in the present paper.

Our purpose is to develop a study of spaces whose mod p cohomology has the form (1). In this part I of our work we consider the case of p odd and we prove the Cooke conjecture in full generality but we go further than that for we obtain a classification up to p-completion of all homotopy types with mod p cohomology of the form (1) above. When p = 2 both the results and the techniques involved in the proofs are significantly different and deserve a separate discusion which we plan to work out in part II of this work ([4]). In particular, the p = 2 version of the Cooke conjecture (which was certainly stated with only the case of p odd in mind) turns out to be wrong and additional fascinating families of spaces with “interesting cohomology” appear.

Typeset by AMS-TEX

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

We present in this introduction a rough overview of the main results of the paper. We start with a classification of unstable algebras over the Steenrod algebra of the form (1). We see that there are exactly two families of such algebras which we call B_{i,r} and A_{r} for i ≥ 0 and r dividing p − 1. As graded algebras, we have (subscripts denote degrees)

B_{i,r} ≅ 𝔽_{p}[x_{2p^{i}r}] ⊗ E(y_{2p^{i}r+1})

and the Steenrod algebra action is determined by β(x) = y and P^{p^{i}}(y) = (r−1)x^{s}y. A_{r} is isomorphic to B_{0,r} as graded algebras but the action of the Steenrod algebra is different in B_{0,r} and A_{r} for in A_{r} we have the relation P^{1}(y) = rx^{s}y. The algebras realized by Cooke in [7] are B_{0,r} and B_{1,r} while the algebras A_{r} seem to have remained unnoticed although their study will be fundamental in our classification of spaces realizing B_{i,r}. In this context, the conjecture of Cooke is stated as follows:

Theorem A. If H^{∗}(X; 𝔽_{p}) ≅ B_{i,r} as algebras over the Steenrod algebra, then i ≤ 1.

It is a natural question to ask about the realizability of the algebras A_{r}. It turns out that all algebras A_{r} are realizable as mod p cohomology of some appropriate spaces. More in general, for any k ≥ 0 we introduce the notation A_{r}^{(k)} to denote a cohomology algebra which looks like A_{r} except for the fact that the relation β(x) = y is replaced by the relation β_{(k+1)}(x) = y where β_{(k+1)} denotes the Bockstein homomorphism of order k + 1. In particular, A_{1}^{(0)} = A_{1}. Of course, β_{(k+1)} is not a Steenrod operation for k > 0 and so A_{r}^{(k)} is the same as A_{r}^{(k′)} as algebras over the Steenrod algebra for any k, k′ > 0. Nevertheless, it makes sense to say that the mod p cohomology of some space X is isomorphic to A_{r}^{(k)}.

Theorem B. For k ≥ 0 and r|(p − 1) there is a p-complete space X_{k}(r) such that H^{∗}(X_{k}(r); 𝔽_{p}) ≅ A_{r}^{(k)}.

The spaces in theorem B are constructed by first taking the quotient of (BS^{1})_{p} by some appropriate action of the p-adic integers and then killing the one dimensional skeleton.

Having established which algebras B_{i,r}, A_{r} are realizable, we consider the problem of classifying up to p-completion all homotopy types which realize these algebras. In the case of the algebras A_{r} we obtain that the spaces of part (1) of theorem B form a complete list of p-complete homotopy types realizing the algebras A_{r}^{(k)}.

Theorem C. Let H^{∗}(X; 𝔽_{p}) ≅ A_{r}^{(k)}. Then X̂_{p} ≃ X_{k}(r).

It is interesting to note that in proving this theorem we face the problem of computing the mod p cohomology of some component of map(Bℤ/p, X) in a case in which the appropriate T functor does not vanish in degree 1.

Finally, we consider the problem of classifying up to p-completion all homotopy types realizing B_{i,r}. Because of theorem A, we only need to deal with the cases of B_{0,r} and B_{1,r}. We obtain the amazing result that for each of these algebras there are infinitely many different p-complete spaces realizing it.

Theorem D. Let r|(p − 1) . There are spaces Y_{k,r} for 0 ≤ k ≤ ∞ and Z_{k,r} for 0 < k ≤ ∞ such that

H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ B_{1,r}
H^{∗}(Z_{k,r}; 𝔽_{p}) ≅ B_{0,r}

All these spaces are p-complete and have different homotopy type.

Here Y_{∞,r} and Z_{∞,r} are the p-completions of the spaces constructed by Cooke ([7]). In particular, Y_{∞,1} = S^{3}⟨3⟩_{p} and theorem D shows that there is an infinite family of “fake” S^{3}⟨3⟩, i.e. spaces with the same mod p cohomology as S^{3}⟨3⟩ but not homotopy equivalent to S^{3}⟨3⟩ even after p-completion. Among these spaces the true S^{3}⟨3⟩ is distinguished by being the only one which can carry an H-space structure.

Our next result shows that the spaces of theorem D form a complete list of p-complete homotopy types realizing the algebras B_{0,r} and B_{1,r}.

Theorem E. (1) If H^{∗}(X; 𝔽_{p}) ≅ B_{1,r} then there exists 0 ≤ k ≤ ∞ such that X̂_{p} ≃ Y_{k,r}.

(2) If H^{∗}(X; 𝔽_{p}) ≅ B_{0,r} then there exists 0 < k ≤ ∞ such that X̂_{p} ≃ Z_{k,r}.

Finally we study suspensions of all the spaces constructed. It turns out that even after an l-fold suspension all the “fake” spaces are not homotopy equivalent to the genuine ones; i.e. the spaces which were constructed out of S^{3}.

Theorem F. If k ≠ ∞, for all r|(p − 1) and for all 0 ≤ l < ∞, the l-fold suspensions Σ^{l}Y_{k,r} and Σ^{l}Y_{∞,r} are not homotopy equivalent.

The analogous statement is true for the spaces Z_{k,r}.

The method used to prove most of the theorems stated above is based on the study of the mapping spaces map(Bℤ/p, X), where X is a space whose cohomology is assumed to be of the form 𝔽_{p}[x]⊗E(y). Here the techniques developed by Lannes ([17]) play a fundamental role.

In order to show in a simplified way the main ideas in the proofs of theorems A to E above, we present now a rough description of the homotopy classification of spaces with the same cohomology as S^{3}⟨3⟩. This will also illustrate where the fake S^{3}⟨3⟩ come from. Imagine we have a p-complete space X with the same mod p cohomology as S^{3}⟨3⟩. Take Y = map(Bℤ/p, X)_{f} to be an appropriate component of the space of maps from Bℤ/p to X and compute, using the T functor, the mod p cohomology of Y. It turns out that Y is homotopy equivalent to X but the gain from X to Y is that Y exhibits a greater symmetry than X for Y belongs to a principal fibration

Bℤ/p → Y → Y (1).

The cohomology of Y (1) has the form 𝔽_{p}[x_{2}]⊗E(β(x_{2})) hence either H^{∗}(Y (1); 𝔽_{p}) ≅ B_{0,1} or H^{∗}(Y (1); 𝔽_{p}) ≅ A_{1} and it turns out that both cases are possible. Hence, we already have two possibilities for X: the true S^{3}⟨3⟩ obtained by taking Y (1) to be Cooke’s realization of B_{0,1} and a fake one obtained by taking Y (1) equal to the space X_{0}(1) of theorem B. If H^{∗}(Y (1); 𝔽_{p}) ≅ B_{0,1} then we can apply the same technique again and obtain a principal fibration

Bℤ/p → Y (1) → Y (2)

with again two possibilities for Y (2). At the end we obtain either an infinite sequence

X → Y → Y (1) → Y (2) → … → Y (j) → …

with all spaces having mod p cohomology of type B_{i,r}^{(k)} or a finite sequence

X → Y → Y (1) → Y (2) → … → Y (j),

where the last space has mod p cohomology of type A_{1}^{(k)}, stopping the inductive process because map(Bℤ/p, Y (j))_{f} will not be homotopy equivalent to Y (j). The first case forces X ≃ S^{3}⟨3⟩_{p} and in the second one we obtain an infinite family of fake S^{3}⟨3⟩. Moreover, the uniqueness of realizations of the algebras A_{r}^{(k)} yields the homotopy uniqueness of each of the fake spaces.

The paper is organized as follows. Section 2 deals with the algebraic problem of classifying unstable algebras over the Steenrod algebra of the form 𝔽_{p}[x] ⊗ E(y) with β(x) = y. There we introduce the algebras B_{i,r} and A_{r}. In section 3 we compute the T functor applied to these algebras, a computation that will be crucial for the rest of the paper. In section 4 we prove the Cooke conjecture, i.e. theorem A (cf. theorem 4.3). Section 5 is devoted to the construction of spaces whose cohomology is of the form A_{r}^{(k)}. Here we prove theorem B (cf. theorem 5.5). In section 6 we obtain the homotopy classification of the spaces of section 5, proving theorem C (cf. theorem 6.1). Section 7 deals with the construction of spaces realizing B_{i,r} and in particular we obtain the family of fake S^{3}⟨3⟩ and we prove theorem D (cf. propositions 7.1 and 7.7 and corollary 7.6). In section 8 we show that there are no more p-complete spaces realizing the algebras B_{i,r} beside those constructed in section 7, by proving theorem E (cf. theorem 8.2). In section 9 we study suspensions of all the constructed spaces and prove theorem F (c.f. corollary 9.13 and corollary 9.15) using the localization functor of [9]. A final section 10 contains some tables which may help the reader through the rather intricated notation we use to denote the spaces we are dealing with and their cohomology algebras.

The first and second author would like to thank the Sonderforschungsbereich 170 in Göttingen and specially L. Smith for the kind hospitality which made possible the joint work which has lead to the present paper. The third one would like to thank the Centre de Recerca Matemàtica in Barcelona for bringing together the authors again. All of us are grateful to Fred Cohen for many helpful discussions.

Warning. Throughout this paper p denotes an odd prime.

2. Some unstable algebras over the Steenrod algebra. Through this section we say that A is a P E-algebra if A is a commutative graded 𝔽_{p}-algebra which is the tensor product of a polynomial algebra on a generator of degree 2n and an exterior algebra on a generator of degree m. We say that A has type (2n, m). We will usually call x one polynomial generator and y one exterior generator.

Our first example of an unstable algebra over the Steenrod algebra is a P Ealgebra A of type (2, 3). We define an unstable action of the mod p Steenrod algebra over A by the Cartan formula and the identities

P^{1}x = x^{p} ; P^{i}x = 0, i > 1 ; βx = y,
P^{1}y = x^{p−1}y ; P^{i}y = 0, i > 1 ; βy = 0.

These formulas certainly define an unstable action of Ā over A, where Ā is the free associative algebra generated by P^{i}, i > 0, and β. In order to see that this action factors through the Steenrod algebra A we have to check that the Adem relations hold in A. This could be done directly using the techniques in [24] but it follows also from the following alternative description of A as a module over the Steenrod algebra.

Let H be the mod p cohomology of Bℤ/p and let P ⊂ H be the even dimensional subalgebra. P is a polynomial algebra on one generator v in degree 2. Let us denote by P^{+} the submodule of P formed by the elements of positive degree. Consider the diagram in U (the category of unstable modules over the Steenrod algebra):

ΣH ⟶^{π} \frac{ΣH}{ΣP^{+}} ⟵^{ϕ} P

where π is the natural projection and ϕ is the homomorphism given by

ϕ(v^{n}) = nσ(uv^{n−1})

where u is a one dimensional generator in H such that βu = v and σ denotes suspension. One can easily check that ϕ is an A-homomorphism. Actually, ϕ is the composition

P ⟶^{Δ} P ⊗ P ⟶^{k⊗1} Σ^{2}𝔽_{p} ⊗ P ≅ Σ^{2}P ⟶^{j} \frac{ΣH}{ΣP^{+}} ,

where Δ is the diagonal, k is the projection and j is an inclusion sending σ^{2}v^{n} to σ(uv^{n}). Then if Ā is the pull back of the above diagram, Ā is an unstable module over the Steenrod algebra and a straightforward computation shows that Ā ≅ A as Ā-modules. This shows that the Adem relations hold true in A since they hold true in Ā.

If λ is a unit in 𝔽_{p} then the map x ↦ λx induces an algebra automorphism of A which commutes with the Steenrod algebra action. Hence, for any r dividing p − 1 we have an action of the cyclic group of order r on A and the algebra of invariants of this action is also an unstable algebra over the Steenrod algebra. We call this algebra A_{r}. It is a P E-algebra of type (2r, 2r + 1) and the action of the Steenrod algebra is determined by:

P^{1}X = rX^{s+1},
βX = Y,
P^{1}Y = rX^{s}Y,

where s = (p − 1)/r.

Let i ≥ 0. Our second example of an unstable algebra over the Steenrod algebra is an algebra B which is a P E-algebra of type (2p^{i}, 2p^{i} + 1). We define an unstable action of the Steenrod algebra over B by the Cartan formula and the identities

P^{p^{i}}x = x^{p} , P^{k}x = 0, k ≠ 0, p^{i},
P^{k}y = 0 for any k > 0,
βx = y , βy = 0.

As before, a direct calculation as in [24] would check that the Adem relations hold, but we will instead use an alternative description of B as an unstable module over the Steenrod algebra. Let H be as before and let P (i) be the subalgebra of H generated by v^{p^{i}}. Let J(2) be the reduced mod p cohomology of S^{1} ∪_{p} e^{2}. Consider the diagram in U

Σ^{2p^{i}−1}J(2) ⊗ P (i) ⟶^{π} Σ^{2p^{i}}P (i) ⟵^{ϕ} P (i)

where the map π is the natural projection and ϕ is given by the composition

P (i) ⟶^{Δ} P (i) ⊗ P (i) ⟶ Σ^{2p^{i}}𝔽_{p} ⊗ P (i) ≅ Σ^{2p^{i}}P (i),

or, equivalently, by the formula

ϕ(v^{p^{i}}) = nσ^{2p^{i}}(v^{(n−1)p^{i}}).

Then if B̄ is the pull back of this diagram, B̄ is an unstable module over the Steenrod algebra such that B̄ ≅ B as Ā-modules. As before, this shows that B is an unstable algebra over the Steenrod algebra.

If λ is a unit in 𝔽_{p} then the map x ↦ λx induces an algebra automorphism of B which commutes with the Steenrod algebra action. This produces, in the same way as before, algebras B_{i,r} for any i ≥ 0 and any r dividing p − 1, with generators X and Y in degrees 2p^{i}r and 2p^{i}r + 1, respectively, such that

P^{p^{i}}X = rX^{s+1}, P^{p^{j}}X = 0, j ≠ i,
βX = Y,
P^{p^{i}}Y = (r − 1)X^{s}Y, P^{p^{j}}Y = 0, j ≠ i.

Notice that A_{r} and B_{0,r} are isomorphic as graded algebras, both being the tensor product of a polynomial algebra on one generator x in degree 2 and an exterior algebra on one generator y in degree 3. Moreover, the relation βx = y holds in both algebras. However, A_{r} and B_{0,r} are not isomorphic as algebras over the Steenrod algebra, as one can easily check.

By construction, we see that if t divides r then A_{r} is a subalgebra of A_{t} and B_{i,r} is a subalgebra of B_{i,t}. There are no further inclusions between these unstable algebras over the Steenrod algebra.

The next theorem proves that there are no more examples of P E-algebras which are unstable algebras over the Steenrod algebra and such that βx = y.

Theorem 2.1. Let A be a P E-algebra which is an unstable algebra over the Steenrod algebra and such that βx = y. Then A is isomorphic, as an algebra over the Steenrod algebra, to one of the algebras A_{r}, B_{i,r} constructed above.

Proof. Notice that the ideal generated by y in A is closed under the action of the Steenrod algebra. Hence A/⟨y⟩ ≅ 𝔽_{p}[x] should be an unstable algebra over the Steenrod algebra. It is well known (cf. [25]) that this implies that if A is of type (2n, 2n+1) then n = p^{i}r for some i ≥ 0 and some r dividing p−1. Put s = (p−1)/r. Then we can choose the generator x such that

P^{p^{i}}x = rx^{s+1}.

This well known fact admits a tedious elementary proof using the Adem relations and is also a trivial consequence of the Adams-Wilkerson embedding theorem ([1]). If i > 0 we can use the Adem relation

P^{1}βP^{p^{i}−1} = −βP^{p^{i}} + P^{p^{i}}β

to deduce

P^{p^{i}}y = (r − 1)x^{s}y.

By dimensional reasons and unstability, P^{p^{j}}x = 0 = P^{p^{j}}y for any j ≠ i. Hence A is isomorphic to B_{i,r}.

In the case i = 0 if we write P^{1}y = λx^{s}y then the Adem relation

2P^{1}βP^{1} = βP^{1}P^{1} + P^{1}P^{1}β

gives the following degree 2 equation for λ:

λ^{2} + (1 − 2r)λ + r(r − 1) = 0

whose roots are λ = r, r − 1. In the first case A is isomorphic to A_{r} and in the second one it is isomorphic to B_{i,r}. □

Some of these P E-algebras appear as the mod p cohomology of some spaces. Let S^{3}⟨3⟩ denote the 3-connective covering of S^{3}, i. e. the fibre of the degree one map S^{3} ⟶ K(ℤ, 3). Then one can easily deduce from the spectral sequence of the fibration

K(ℤ, 2) → S^{3}⟨3⟩ → S^{3}

and theorem 2.1 that

H^{∗}(S^{3}⟨3⟩; 𝔽_{p}) ≅ B_{1,1}

as algebras over the Steenrod algebra. Moreover, since S^{3}⟨3⟩ is a loop space, the p-completion of S^{3}⟨3⟩ carries an action of the cyclic group of order p − 1 and by taking the homotopy quotient of S^{3}⟨3⟩_{p} by the restriction of this action to the cyclic group of order r, for any r dividing p − 1, we obtain a space X_{r} such that H^{∗}(X_{r}; 𝔽_{p}) ≅ B_{1,r}. (See [7] for further details on this construction.)

If Y is the fibre of the map S^{3} → K(ℤ, 3) of degree p then

H^{∗}(Y; 𝔽_{p}) ≅ B_{0,1}

as algebras over the Steenrod algebra. Again, the p-completion of Y carries an action of the cyclic group of order p − 1 and this produces p-complete spaces Y_{r} for any r dividing p − 1 such that H^{∗}(Y_{r}; 𝔽_{p}) ≅ B_{0,r} as algebras over the Steenrod algebra. (See also [7] for details.)

Finally, in section 5 we will prove that all P E-algebras A_{r} are realizable.

Even if we are only interested in the P E-algebras with β(x) = y it will be necessary to consider P E-algebras where the Bockstein homomorphism acts trivially. They will indeed play an important role in the forthcoming sections. We denote by A′_{1} the P E-algebra of type (2, 3) with an unstable action of the Steenrod algebra given by β = 0 and P^{1}(y) = x^{p−1}y. Let A′_{r} for r dividing p − 1 be the algebra of invariants of A′_{1} by the action of the cyclic group of order r which sends x to λx and y to λy for λ an r-th root of unity. A′_{r} is a P E-algebra of type (2r, 2r +1) with an unstable action of the Steenrod algebra. Let B′_{i,1} be the same graded algebra B_{i,1} but with the Steenrod algebra action given by β = 0 and P^{t}(y) = 0 for t ≥ 0. Let B′_{i,r} for r dividing p − 1 be the algebra of invariants of B′_{i,1} under the action of the cyclic group of order r which sends x to λx and y to λy for λ an r-th root of unity.

We denote by C_{1} a P E-algebra of type (2, 1) with an unstable action of the Steenrod algebra given by β(x) = xz where x denotes a 2 dimensional generator and z denotes a one dimensional generator. Notice that A_{1} is isomorphic to a subalgebra of C_{1}. The cyclic group of order r for r dividing p − 1 acts on C_{1} leaving z and x^{r} invariants. We denote by C_{r} the algebra of invariants, which is a P E-algebra of type (2r, 1) with an unstable action of the Steenrod algebra. We also need a Bockstein-free version of these algebras which we denote by C′_{r}.

If any of the above algebras with trivial Bockstein appears as the mod p cohomology of some space X it makes sense to ask about the order of the higher Bockstein which connects the polynomial and the exterior part and we indicate this order as a superscript. In this way, the notation

H^{∗}(X; 𝔽_{p}) ≅ A_{r}^{(k)}

means that H^{∗}(X; 𝔽_{p}) ≅ A′_{r} as algebras over the Steenrod algebra and

β_{(i)}(x) = \{ \begin{matrix}0, & i ≤ k \\ y, & i = k + 1. \end{matrix}

where β_{(i)} denotes the i-th order Bockstein, i.e. the i-th differential in the mod p Bockstein spectral sequence of X. In the same way, we introduce the notations H^{∗}(X; 𝔽_{p}) = B_{i,r}^{(k)} and H^{∗}(X; 𝔽_{p}) = C_{r}^{(k)}.

For further reference, we summarize the algebras that we have considered so far in table 10.1.

3. Computing Lannes T functor. Let T denote the Lannes functor defined as left adjoint to H ⊗ − in the category U of unstable modules over the Steenrod algebra (see [17] for a full description of its properties.) Here H denotes the mod p cohomology of ℤ/p as in the previous section. 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 of R-U-modules and with values in the category of T_{f}(R)-U-modules (cf. [13].) 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.

The purpose of this section is to compute for various P E-algebras A constructed in the previous section, namely, A_{r} for r|(p − 1) and B_{i,r} for i ≥ 0 and r|(p − 1), the particular component of T (A) that corresponds to a map f: A → H that can be uniformly described as the composition

A ⟶^{h} 𝔽_{p}[x] ⟶^{k} H

where h: A → 𝔽_{p}[x] is the projection onto the polynomial part of A and k is the obvious inclusion of 𝔽_{p}[x] in the even part of H. Our result is as follows.

Theorem 3.1. (1) T_{f}(A_{1}) ≅ C_{1} and the natural map ε: A_{1} → T_{f}(A_{1}) is the inclusion of algebras given by ε(x) = x and ε(y) = xz.

(2) For all i ≥ 0 the natural map ε: B_{i,1} ⟶ T_{f}(B_{i,1}) is an isomorphism.

(3) For any r|(p − 1) and all i ≥ 0, the inclusions A_{r} → A_{1} and B_{i,r} → B_{i,1} induce isomorphisms T_{f}(A_{r}) ≅ T_{f}(A_{1}) ≅ C_{1} and T_{f}(B_{i,r}) ≅ T_{f}(B_{i,1}) ≅ B_{i,1}.

We will be using the following lemma that can be easily obtained:

Lemma 3.2. Let A and B be two unstable algebras over the Steenrod algebra and f: A → B a K-map that induces an isomorphism Hom_{K}(B, H) ≅ Hom_{K}(A, H). Then, for a K-map g: B → H and any B-U-module M

T_{g∘f}(M) ≅ T_{g}(M)

and the T_{g∘f}(A)-U-module structure of T_{g∘f}(M) is induced by T_{g∘f}(A) → T_{g}(B). Therefore, T_{g∘f}(M) is an A-U-module through A → T_{g∘f}(A) → T_{g}(B) or equivalently through A → B → T_{g}(B). □

Proof of 3.1.(1). Recall that A_{1} = 𝔽_{p}[x] ⊗ E[y], deg(x) = 2, β(x) = y, P^{1}(y) = yx^{p−1} and f has been defined as the composition k ∘ h where h is the projection A_{1} ⟶ 𝔽_{p}[x] and k identifies 𝔽_{p}[x] with the even part of H.

A_{1} sits in an exact sequence of A_{1}-U-modules

(1)  0 → y𝔽_{p}[x] → A_{1} → 𝔽_{p}[x] → 0

and T_{f} is exact so our first job will be the calculation of T_{f}(y𝔽_{p}[x]) and T_{f}(𝔽_{p}[x]). Both y𝔽_{p}[x] and 𝔽_{p}[x] can actually be considered as 𝔽_{p}[x]-U-modules, with the A_{1}- U-module structure induced by the projection h: A_{1} → 𝔽_{p}[x]. According to lemma 3.2 what we have to do is to compute T_{k}(y𝔽_{p}[x]) and T_{k}(𝔽_{p}[x]) as 𝔽_{p}[x]-U-modules. T (𝔽_{p}[x]) is well known (see [3]) and ε: 𝔽_{p}[x] → T_{k}(𝔽_{p}[x]) turns out to be an isomorphism. For y𝔽_{p}[x] we obtain:

Lemma 3.3. T_{k}(y𝔽_{p}[x]) ≅ z𝔽_{p}[x] with deg(z) = 1; that is, a 𝔽_{p}[x]-U-module on one generator of degree one on which the Steenrod operations act trivially. Moreover, ε: y𝔽_{p}[x] → T_{k}(𝔽_{p}[x]) is an 𝔽_{p}[x]-U-module map given by ε(y) = zx.

Proof. y𝔽_{p}[x] might be identified to Σx𝔽_{p}[x] as 𝔽_{p}[x]-U-module (Σ denotes the suspension). Since T_{k} commutes with suspensions we must calculate T_{k}(x𝔽_{p}[x]) and for this we use the following exact sequence of 𝔽_{p}[x]-U-modules:

0 → x𝔽_{p}[x] → 𝔽_{p}[x] → 𝔽_{p} → 0.

T_{k}(𝔽_{p}) is clearly trivial and we obtain T_{k}(x𝔽_{p}[x]) ≅ T_{k}(𝔽_{p}[x]) ≅ 𝔽_{p}[x]. It also follows that ε is the inclusion x𝔽_{p}[x] → 𝔽_{p}[x].

Finally we apply Σ and write Σ𝔽_{p}[x] as z𝔽_{p}[x] in order to get to the conclusion of the lemma. □

The above computation together with lemma 3.2 give us T_{f}(y𝔽_{p}[x]) and T_{f}(𝔽_{p}[x]) as T_{f}(A_{1})-U-modules and also as A_{1}-U-modules. Then, the exact sequence (1) induces a diagram of A_{1}-U-modules:

0 ⟶ y𝔽_{p}[x] ⟶ A_{1} ⟶ 𝔽_{p}[x] ⟶ 0
ε↓  ^{ε}↓  ^{ε}↓≅
0 ⟶ z𝔽_{p}[x] ⟶ T_{f}(A_{1}) ⟶ 𝔽_{p}[x] ⟶ 0

where the bottom row is an exact sequence of T_{f}(A_{1})-U-modules. This diagram implies that ε: A_{1} → T_{f}(A_{1}) is a K-monomorphism and T_{f}(A_{1}) ≅ 𝔽_{p}[x]⊗E[z] with ε determined by ε(x) = x and ε(y) = xz. □

Proof of 3.1.(2). Now we deal with the cases B_{i,1} ≅ 𝔽_{p}[x_{i}] ⊗ E[y_{i}], deg x_{i} = 2p^{i}, β(x_{i}) = y_{i} and P^{p^{i}}(y_{i}) = 0, so that B_{i,1} sits in an exact sequence of B_{i,1}-U-modules:

(2)  0 → y𝔽_{p}[x_{i}] ⟶ B_{i,1} ⟶ 𝔽_{p}[x_{i}] → 0

with y𝔽_{p}[x_{i}] isomorphic as 𝔽_{p}[x_{i}]-U-module to Σ^{2p^{i}+1}𝔽_{p}[x_{i}]. In these cases f = k ∘ h with h the projection B_{i,1} ⟶ 𝔽_{p}[x_{i}] and k: 𝔽_{p}[x_{i}] → H defined by k(x_{i}) = v^{p^{i}}, v a two dimensional generator of H.

Just as in the proof of the first part it is enough to compute T_{k}(y𝔽_{p}[x_{i}]) and T_{k}(𝔽_{p}[x_{i}]) and, in this case, both ε: y𝔽_{p}[x_{i}] ⟶ T_{k}(y𝔽_{p}[x_{i}]) and ε: 𝔽_{p}[x_{i}] ⟶ T_{k}(𝔽_{p}[x_{i}]) are isomorphisms, thus the sequence (2) gives rise to the diagram:

0 ⟶ y𝔽_{p}[x_{i}] ⟶ B_{i,1} ⟶ 𝔽_{p}[x_{i}] ⟶ 0
ε↓≅  ^{ε}↓  ^{ε}↓≅
0 ⟶ y𝔽_{p}[x_{i}] ⟶ T_{f}(B_{i,1}) ⟶ 𝔽_{p}[x_{i}] ⟶ 0

and this implies the desired result. □

Proof of 3.1.(3). We will work out only the case of A_{r} for the case of B_{i,r} is essentially the same. Recall that A_{r} is the subalgebra of invariants of A_{1} by the action of ℤ/r and T commutes with taking invariants. In fact, we obtain:

(1)  T_{f}(A_{r}) ≅ T_{f}(A_{1}^{ℤ/r}) ≅ (∏_{f_{λ}} T_{f_{λ}}A_{1})^{ℤ/r}

where f_{λ} runs through maps A_{1} → H that restrict as f to A_{r}; that is, f_{λ} is the composition A_{1} → 𝔽_{p}[x] ⟶^{k_{λ}} H, with k_{λ}(x) = λv, v a two dimensional generator of H and λ ∈ 𝔽_{p}, such that f_{λ}|A_{r} = f, and this equality holds if and only if λ^{r} = 1, i.e. λ ∈ ℤ/r ⊂ 𝔽_{p}^{∗}. Now it is clear that an element of ℤ/r induces a permutation of the factors in ∏_{λ∈ℤ/r} T_{f_{λ}}(A_{1}) and then

( ∏_{λ∈ℤ/r} T_{f_{λ}}(A_{1}))^{ℤ/r} ≅ T_{f}(A_{1}). □

In a forthcoming section we will need a few variants of theorem 3.1.

Proposition 3.4. Let c : B_{i,r} → H, c : A_{r} → H denote the homomorphisms which are zero in positive degrees. Then T_{c}(B_{i,r}) ≅ B_{i,r} and T_{c}(A_{r}) ≅ A_{r}.

Proof. The proof is completely analogous to the proof of 3.1 except for two differences: We have T_{c}(𝔽_{p}) = 𝔽_{p} and this implies T_{c}(A_{1}) = A_{1} and T_{c}(B_{i,1}) = B_{i,1} by the same argument as in 3.1(1) and (2). On the other hand, the product in (1) has only one factor in the case of the trivial homomorphism c. □

One can check that the proofs of 3.1 and 3.4 work also for the algebras A′_{r} and B′_{i,r} and we have:

Proposition 3.5. Let c and f be as in 3.1 and 3.4 respectively. Then T_{c}(B′_{i,r}) ≅ B′_{i,r}, T_{c}(A′_{r}) ≅ A′_{r} and T_{f}(A′_{r}) ≅ C′_{1}. □

4. Non-realizability of P E-algebras. In this section we prove the Cooke conjecture as stated in the introduction. The proof will follow from a study of the transgression in the Serre spectral sequence of some fibration. We start with a lemma describing an interesting relation in the action of the Steenrod algebra on the mod p cohomology of B^{2}ℤ/p. We use the notation

P^{Δ_{j}} = P^{p^{j}}P^{p^{j−1}}⋯ P^{1}.

Lemma 4.1. The following identities hold in H^{∗}(B^{2}ℤ/p; 𝔽_{p}):

(1) P^{t}P^{Δ_{r}}βι = 0 for 0 < t < p^{r+1}.

(2) P^{p^{j}}βP^{Δ_{j−1}}βι = βP^{Δ_{j}}βι ≠ 0 for j > 0.

Proof. Recall that H^{∗}(B^{2}ℤ/p; 𝔽_{p}) is a free graded–commutative algebra on free generators P^{I}ι where I is an admissible sequence of excess ≤ 2:

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

Hence βP^{Δ_{j}}βι is an indecomposable in H^{∗}(B^{2}ℤ/p; 𝔽_{p}). We prove first (1). If r = 0 we have P^{t}P^{1}βι = λP^{t+1}βι = 0 by unstability. We proceed then by induction:

P^{t}P^{Δ_{r}}βι = P^{t}P^{p^{r}}P^{Δ_{r−1}}βι = ∑_{s=0}^{[t/p]} λ_{s}P^{p^{r}+t−s}P^{s}P^{Δ_{r−1}}βι.

In the right hand expression the term for s = 0 vanishes by unstability and all other terms are zero by the induction hypothesis. The lemma follows now from the Adem relation for P^{p^{j}}βP^{p^{j−1}} in the following way

P^{p^{j}}βP^{p^{j−1}}P^{Δ_{j−2}}βι = ∑_{t=0}^{p^{j−1}} λ_{t}βP^{p^{j}+p^{j−1}−t}P^{t}P^{Δ_{j−2}}βι

+ ∑_{t=0}^{p^{j−1}−1} μ_{t}P^{p^{j}+p^{j−1}−t}βP^{t}P^{Δ_{j−2}}βι.

(If j = 1, delete P^{Δ_{j−2}} in this formula.) By (1) and the unstability condition, the right hand term reduces to λ_{p^{j−1}}βP^{p^{j}}P^{Δ_{j−1}}βι and the proof ends by checking λ_{p^{j−1}} = 1. □

Proposition 4.2. Assume H^{∗}(X; 𝔽_{p}) ≅ B_{i+1,1}, i ≥ 0, as algebras over the Steenrod algebra. Assume also that there is a fibration

X ⟶ E ⟶^{g} B^{2}ℤ/p

such that x transgresses to P^{Δ_{i}}βι plus decomposables, where ι ∈ H^{2}(B^{2}ℤ/p; 𝔽_{p}) is the fundamental class. Then i = 0.

Proof. Consider the spectral sequence of the fibration X → E → B^{2}ℤ/p. Since τ(x) = P^{Δ_{i}}βι + d this element has to be killed by g^{∗}. Hence g^{∗}(βP^{Δ_{i}}βι + βd) = 0. If we assume i > 0 we can apply the lemma and obtain

0 = g^{∗}(P^{1}P^{p^{i}}βP^{Δ_{i−1}}βι+P^{1}βd) = g^{∗}(P^{p^{i}+1}βP^{Δ_{i−1}}βι+P^{1}βd) = g^{∗}([βP^{Δ_{i−1}}βι]^{p}+P^{1}βd).

Notice that βP^{Δ_{i−1}}βι is an even dimensional indecomposable in H^{∗}(B^{2}ℤ/p; 𝔽_{p}). Using 4.1 we have the equalities

P^{1}P^{Δ_{j}}βι = 0, j ≥ 0,
P^{1}βP^{Δ_{j}}βι = \{ \begin{matrix}[βP^{Δ_{j−1}}βι]^{p}, j > 0 \\ 0, j = 0 \end{matrix}

and an elementary argument shows that P^{1}βd cannot contain the term [βP^{Δ_{i−1}}βι]^{p}. Hence, there is some differential in the spectral sequence, coming from an element in total degree 2p^{i+1} + 2p − 1 which kills [βP^{Δ_{i−1}}βι]^{p} + other terms. By inspecting the E_{2}-term of the spectral sequence in total degree 2p^{i+1} + 2p − 1 we see that only x ⊗ ι^{p−2}βι and y ⊗ ι^{p−1} may eventually kill this element.

We know by hypothesis that the first non vanishing differential maps x to P^{Δ_{i}}βι + d. Hence it does not map x ⊗ ι^{p−2}βι to [βP^{Δ_{i−1}}βι]^{p} + other terms. Since βx = y, the next differential maps y to βP^{Δ_{i}}βι + β(d) and so it cannot kill [βP^{Δ_{i−1}}βι]^{p} + other terms. In any case, [βP^{Δ_{i−1}}βι]^{p} + other terms survives, a contradiction that can only be avoided if i = 0. □

Notice that the “elementary argument” mentioned in the above proof fails if p = 2. This fact gives rise to a manifold of fascinating phenomena which will be studied in [4].

Theorem 4.3. If H^{∗}(X; 𝔽_{p}) ≅ B_{i,r} as algebras over the Steenrod algebra, then i ≤ 1.

Proof. Since H^{1}(X; 𝔽_{p}) = 0 we can assume, without loss of generality, that X is p-complete. Let f : B_{i,r} → H^{∗}(Bℤ/p; 𝔽_{p}) be the non trivial homomorphism considered in the last section. Then by [17; 3.1.1] there is a map ϕ : Bℤ/p → X inducing f in mod p cohomology. By theorem 3.1 we have

T_{f}B_{i,r} ≅ B_{i,1},

where T denotes the T functor with respect to V = ℤ/p. Then, [17; 3.2.1] shows that

H^{∗}(map(Bℤ/p, X)_{ϕ}; 𝔽_{p}) ≅ B_{i,1}

where map(Bℤ/p, X)_{ϕ} is the space of all maps Bℤ/p → X homotopic to ϕ. 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 Y is the homotopy quotient of map(Bℤ/p, X)_{ϕ} by this action, we have a fibration

(1)  map(Bℤ/p, X)_{ϕ} → Y → B^{2}ℤ/p.

If we denote by i the induced map Bℤ/p → map(Bℤ/p, X)_{ϕ} and by e : map(Bℤ/p, X)_{ϕ} → X the evaluation map at the base point of Bℤ/p (which is the unit of Bℤ/p as a simplicial group) then one sees easily that e ∘ i = ϕ. In particular, i^{∗}(x) = v^{p^{i}} = P^{Δ_{i−1}}βu where x ∈ B_{i,1} is the class in degree 2p^{i}. Hence, the class x in the mod p cohomology of the fibre of (1) transgresses to P^{Δ_{i−1}}βι plus decomposables and proposition 4.2 shows that i ≤ 1. □

Remark 4.4. This method can be applied to give a new short proof of the well known fact that if H^{∗}(X; 𝔽_{p}) ≅ 𝔽_{p}[x_{2p^{i}}] then i = 0 for p odd and i = 0, 1 for p = 2. In this case there exists also a fibration sequence

Bℤ/p ⟶ X ⟶ \overline{X} ⟶^{g} B^{2}ℤ/p.

Then τ(x) = P^{Δ_{i}}βι_{2} + d where d is a decomposable and the result follows from 0 = g^{∗}(βP^{Δ_{i}}βι_{2} + βd).

5. Spaces realizing A_{r}. Let us denote by π the additive group of the p-adic integers. Let G be the automorphism group of π. G is isomorphic to the multiplicative group of the invertible elements in the ring structure of π and this is the direct product of the cyclic group of (p − 1)th-roots of unity by the group U_{1} = 1 + pπ. The group U_{1} is torsion free and there is a monomorphism

ϕ : π → G

given by ϕ(x)(y) = exp(px)y where the product is taken in the ring structure of the p-adic integers. Moreover, after identifying G with the invertibles of π, ϕ maps onto U_{1}, the logarithm providing an inverse.

We obtain therefore a precise description of all possible actions of π on π, namely, all these actions are obtained by composing ϕ with multiplication by a p-adic number α. We will denote by ϕα the one defined by α and by π^{ϕα} the additive group of p-adic integers endowed with the action defined by ϕα. Among them the ones of most interest for us correspond to α = p^{k} for k ≥ 0 and we will abbreviate ϕp^{k} as ϕ_{k}.

Realizing A_{r}. We will construct a space realizing A_{r} as well as other related spaces. We suggest to consider tables 10.2 and 10.3 in the appendix as a quick reference guide to the spaces introduced in this section. For this aim we consider π^{ϕ_{k}}, the p-adics endowed with the action defined by ϕ_{k} for k ≥ 0. B^{2}π^{ϕ_{k}} inherits the action and we define spaces

E_{k} = E_{ϕ_{k}} = B^{2}π^{ϕ_{k}} ×_{π} EG

for all k ≥ 0. (π acts on G through ϕ.) Let us compute the mod p cohomology of E_{k}. From the obvious fibration

B^{2}π^{ϕ_{k}} → E_{k} → EG/π ≃ Bπ

we get a spectral sequence

H^{∗}(π; H^{∗}(B^{2}π^{ϕ_{k}}; 𝔽_{p})) ⇒ H^{∗}(E_{k}; 𝔽_{p}).

Notice that B^{2}π ≃ BS^{1}_{p} and Bπ ≃ S^{1}_{p}. Since π is q-divisible for any q ≠ p, π can only act trivially on H^{∗}(B^{2}π^{ϕ_{k}}; 𝔽_{p}) which is either trivial or one-dimensional in each degree. Hence the spectral sequence yields immediately that for any k ≥ 0

H^{∗}(E_{k}; 𝔽_{p}) ≅ E(z) ⊗ 𝔽_{p}[x]

where z and x are classes in degrees 1 and 2, respectively. The Steenrod algebra should act trivially on z and the Steenrod powers act on 𝔽_{p}[x] as they do in H^{∗}(BS^{1}; 𝔽_{p}). It only remains to determine the action of the Bockstein homomorphism on the class x. This will distinguish E_{0} from E_{k} for k ≥ 1. More in general, we will show that the action of the higher Bocksteins on x implies that all these spaces are different.

Proposition 5.1. H^{∗}(E_{k}; 𝔽_{p}) ≅ C_{1}^{(k)} for k ≥ 0.

Proof. To prove this, we will compute the cohomology of E_{k} with p-adic coefficients in low dimensions by means of the Serre spectral sequence. We need the following results on the homology of the p-adic integers.

Lemma 5.2. (1) H_{1}(π; ℤ) = π and for j ≥ 2, H_{j}(π; ℤ) is a ℚ-vector space. (2) H^{1}(π; π) ≅ π and for j ≥ 2, H^{j}(π; π) = 0 (trivial coefficients).

(3) The cohomology of π with twisted p-adics coefficients is

H^{0}(π; π^{ϕα}) = 0,
H^{1}(π; π^{ϕα}) ≅ ℤ/p^{ν(α)+1},
H^{j}(π; π^{ϕα}) = 0, j ≥ 2

where ν(α) denotes the biggest power of p dividing α.

Proof. We obtain H_{1}(π; ℤ) ≅ π by the Hurewicz theorem. Since H^{j}(π; ℤ/q) = 0 for all j ≥ 2 and all primes q the universal coefficient formula implies that Hom(H_{j}(π; ℤ), ℤ/q) ≅ Ext(H_{j}(π; ℤ), ℤ/q) ≅ 0 for all j ≥ 2 and all primes q, hence statement (1) follows.

The statement (2) follows by the universal coefficient formula because Hom(π, π) ≅ π, Hom(A, π) = 0 if A is p-divisible and Ext(A, π) = 0 if A is torsion free.

To prove (3) note first that zero is the only invariant element of π^{ϕα} under the action of π so H^{0}(π; π^{ϕα}) = 0. Next, we consider the well known description of the first cohomology group through derivations:

H^{1}(π; π^{ϕα}) ≅ Der(π, π^{ϕα})/Ider(π, π^{ϕα}).

A derivation π → π^{ϕα} is determined by the image of 1 ∈ π. In fact, for a given derivation d: π → π^{ϕα}, if x ∈ π then d(x) + exp(pαx)d(1) = d(x + 1) = d(1 + x) = d(1) + exp(pα)d(x) and this equation has a unique solution for d(x) once d(1) is fixed.

Moreover, the formula

d_{a}(x) = a \frac{exp(pαx) − 1}{p^{ν(α)+1}}

defines a derivation π → π^{ϕα} for any a ∈ π. This derivation is inner precisely when a ≡ 0 (p^{ν(α)+1}) and therefore H^{1}(π; π^{ϕα}) ≅ ℤ/p^{ν(α)+1}.

It remains to compute H^{j}(π, π^{ϕα}) for j ≥ 2. We will see that H^{∗}(π; π^{ϕα}) is isomorphic to H^{∗}(ℤ; π^{ϕα}) with the action induced by restriction and then the result will follow because ℤ is free.

The isomorphism that we claim is induced by the inclusion ℤ → π and it is proved in degrees 0 and 1 by direct computation. It would be also clear if the coefficients were ℤ/p^{r} for any r > 1. Then the Lyndon-Hochschild-Serre spectral sequence for ℤ → π → π/ℤ shows first that H̃^{∗}(π/ℤ; ℤ/p^{r}) = 0 and since π/ℤ can only act trivially on ℤ/p^{r}, also that H^{∗}(π; π^{ϕα}) ≅ H^{∗}(ℤ; π^{ϕα}). □

As a consequence of this lemma, in the spectral sequence of the fibration B^{2}π^{ϕ_{k}} → E_{k} → Bπ with coefficients in ℤ̂_{p} the only term that can contribute to H^{3}(E_{k}; ℤ̂_{p}) is H^{1}(π; H^{2}(B^{2}π; ℤ̂_{p})) ≅ ℤ/p^{k+1}. This finishes the proof of proposition 5.1. □

Notice now that for any r dividing p−1 there is an embedding of ℤ/r in G which gives an action of ℤ/r on B^{2}π and EG. Since G is isomorphic to ℤ/p − 1 × π we obtain an induced free action of ℤ/r on E_{k}. Notice that since β_{(k+1)}(x) = xz the action has to be trivial on z ∈ H^{1}(E_{k}; 𝔽_{p}). Let E_{k}(r) be the quotient of E_{k} by this action. Since r is prime to p, it is clear that

H^{∗}(E_{k}(r); 𝔽_{p}) = H^{∗}(E_{k}; 𝔽_{p})^{ℤ/r} = E(z) ⊗ 𝔽_{p}[u]

where u corresponds to x^{r} in H^{∗}(E_{k}; 𝔽_{p}) and β_{(k+1)}(u) = ruz. Hence,

Proposition 5.3. H^{∗}(E_{k}(r); 𝔽_{p}) ≅ C_{r}^{(k)} for k ≥ 0, r|(p − 1). □

Finally, let us consider the composition

Bπ ⟶^{f} E_{k} ⟶ E_{k}(r)

where f is a section of the fibration B^{2}π → E_{k} → Bπ. If E′_{k}(r) denotes the cofibre of this composition then we have H^{∗}(E′_{k}(r); 𝔽_{p}) ≅ 𝔽_{p}[u] ⊗ E(w) with deg u = 2r, deg w = 2r + 1, β_{(k+1)}(u) = w and P^{1}(w) = ru^{s}w, where s = (p − 1)/r as is usual in this paper. In particular

H^{∗}(E′_{k}(r); 𝔽_{p}) ≅ A_{r}^{(k)}.

Definition 5.4. For k ≥ 0, r|p − 1, we define X_{k}(r) as the p-completion of E′_{k}(r).

The next theorem establishes some properties of these spaces.

Theorem 5.5. For any r dividing p − 1 and k ≥ 0,

(1) X_{k}(r) is a simply connected p-complete space whose homotopy groups are finite p-groups.

(2) H^{∗}(X_{k}(r); 𝔽_{p}) ≅ A_{r}^{(k)}.

Proof. Will be based in the following two propositions.

Proposition 5.6. Let R = ℤ̂_{p} or ℤ and X a space with cohomology of finite type over R. If H^{∗}(X; 𝔽_{p}) = A_{r}^{(k)} then in the R-cohomology Bockstein spectral sequence {B_{l}, d_{l}} for X

(1) the first non-trivial differential is d_{k+1} and d_{k+1}(u) = w,

(2) u^{p^{n}} survives to B_{n+k+1} and d_{n+k+1}([u^{p^{n}}]) = [u^{p^{n}−1}w] and

(3) B_{∞} = 0.

Proof. This is a direct consequence of known results about the differential in the Bockstein spectral sequence (cf. [16; pag. 102]). □

The next proposition might be of independent interest and we establish it for any prime number, either two or odd.

Proposition 5.7. Let p be any prime and X a 1-connected, p-complete space, then:

(1) The following conditions are equivalent:

(i) H^{j}(X; 𝔽_{p}) is finite for all j.

(ii) π_{j}(X) is a finitely generated ℤ̂_{p}-module for all j. (iii) H^{j}(X; ℤ̂_{p}) is a finitely generated ℤ̂_{p}-module for all j.

(2) The following conditions are equivalent:

(i) H̃^{j}(X; ℤ̂_{p}) is a finite p-group for all j. (ii) π_{j}(X) is a finite p-group for all j.

Proof. Let F be the fibre of the rationalization X → X_{0}. We can first obtain some general facts about F. H^{∗}(F; 𝔽_{p}) ≅ H^{∗}(X; 𝔽_{p}) and F → X is the p-completion of F. Also, H^{∗}(F; ℤ̂_{p}) ≅ H^{∗}(X; ℤ̂_{p}) and π_{j}(F) is a p-group for all j. Moreover, since F is the fibre of a map between 1-connected spaces, the fundamental group of F is abelian and acts trivially on the homology and homotopy of the universal cover of F (cf. [15]). Hence the mod C Hurewicz theorem can be applied to F and we obtain that H̃_{j}(F; ℤ) is a p-group (i.e. a group all of whose elements are p-power torsion) for any j.

Now, the proof of part (1) of the proposition will consist in the following sequence of statements. Claim 5.7.1: H^{j}(X; 𝔽_{p}) is finite for all j if and only if H̃_{j}(F; ℤ) is a finitely cogenerated p-group for all j.

Let us write H̃_{j}(F; ℤ) as an extension of a divisible p-group D_{j} by a pure subgroup P_{j} which is a direct sum of cyclic p-groups. Then one easily deduces that the mod p homology of F is of finite type over 𝔽_{p} if and only if both P_{j} and D_{j} contain finitely many summands for all j. Since a bounded pure subgroup is a direct summand this means that the (reduced) integral homology groups of F are a direct sum of finitely many cyclic p-groups and finitely many groups ℤ/p^{∞}; that is, they are finitely cogenerated p-groups. Claim 5.7.2: H̃_{j}(F; ℤ) is a finitely cogenerated p-group for all j if and only if π_{j}(F) is so.

Since π_{1}(F) is abelian π_{1}(F) ≅ H_{1}(F; ℤ) and then since the class of finitely cogenerated abelian p-groups is an acyclic ring of abelian groups this claim follows by the mod C Hurewicz theorem. Claim 5.7.3: π_{j}(F) is a finitely cogenerated p-group for all j if and only if π_{j}(X) is a finitely generated ℤ̂_{p}-module for all j.

¿From the homotopy exact sequence for the fibration F → X → X_{0} we obtain short exact sequences

0 → π_{j+1}(X) ⊗ ℚ/ℤ → π_{j}(F) → Tor(π_{j}(X), ℚ/ℤ) → 0

and then one of the implications. On the other hand, since X is the p-completion of the nilpotent space F we also have short split exact sequences ([6; VI.5.1])

0 → Ext(ℤ/p^{∞}, π_{j}(F)) → π_{j}(X) → Hom(ℤ/p^{∞}, π_{j−1}(F)) → 0,

hence the implication in the other direction is also true. Claim 5.7.4: H^{j}(X; 𝔽_{p}) is finite for all j if and only if H^{j}(X; ℤ̂_{p}) is a finitely generated ℤ̂_{p}-module for all j.

It suffices to show this equivalence for F, and this follows easily by the universal coefficients formula using claim 5.7.1.

This finishes the proof of part (1). Let us turn to the proof of part (2). With the same argument as in Claim 5.7.4 we obtain that H̃^{j}(X; ℤ̂_{p}) is a finite p-group for all j if and only if H̃_{j}(F; ℤ) is a finite p-group for all j. Again by the mod C Hurewicz theorem this is equivalent to π_{j}(F) to be a finite p-group for all j and finally the same argument of Claim 5.7.3 shows that if the homotopy groups of either X or F are finite p-groups, then F → X is actually a homotopy equivalence. □

We can now finish the proof of Theorem 5.5. The space X_{k}(r) was defined as the p-completion of E′_{k}(r). By construction we have π_{1}(E_{k}(r)) ≅ ℤ/r × π and then by the van Kampen theorem π_{1}(E′_{k}(r)) ≅ ℤ/r. Hence ([6; p. 206]) E′_{k}(r) is ℤ/p-good and X_{k}(r) is p-complete, simply connected and has the same mod p cohomology as E′_{k}(r). So we have proved part (2) of the theorem.

By Proposition 5.7(1) X_{k}(r) is of finite type over ℤ̂_{p} hence the Bockstein spectral sequence applies and by Proposition 5.6 the cohomology groups H̃^{j}(X_{k}(r); ℤ̂_{p}) are actually finite p-groups, hence by proposition 5.7(2) we obtain part (1) of the theorem. □

Remark 5.8. From Proposition 5.6 we can derive the integral cohomology of the spaces X_{k}(r):

H̃^{i}(X_{k}(r); ℤ̂_{p}) ≅ H̃^{i}(X_{k}(r); ℤ) ≅ \{ \begin{matrix}ℤ/p^{k+1+ν(j)} & i = 2rj + 1, j ≥ 1 \\ 0 & otherwise. \end{matrix}

Final remarks. Here is the reason for which we have been dealing with a certain collection among all possible actions of π on π.

Proposition 5.9. Let π^{ξ} be the additive group of the p-adic integers together with a π action defined by a non trivial homomorphism ξ: π → Aut(π) = G and define

E_{ξ} = B^{2}π^{ξ} ×_{π} EG.

Then E_{ξ} is homotopy equivalent to a space E_{k} = E_{ϕ_{k}}

Proof. ¿From our discussion of the possible actions of π on π at the beginning of this section, ξ is of the form ϕα for a p-adic integer α. That is:

ξ(x)(y) = e^{pαx}y.

Now, α might be written as α = p^{ν(α)}w where w ∈ 1 + pπ. Since w is invertible it determines an automorphism

w: π → π.

Now, the identity B^{2}π → B^{2}π is w-equivariant if we consider the action given by ξ on the source an by ϕ_{ν(α)} on the target:

ξ(x)(y) = e^{pαx}y = e^{pwp^{ν(α)}x}y = ϕ_{k}(wx)(y).

In this way we get a map E_{ξ} → E_{ν(α)} which is in fact a homotopy equivalence because w is invertible. □

Remark 5.10. Observe that until now all our constructions could be performed using Bℤ instead of Bπ as base space of our fibrations, with actions of ℤ on π induced by restriction from the actions of π on π that we used. Also, we could use Bℤ/p^{∞} instead of B^{2}π. However the above proposition would not be true in that case. We would need to complete our spaces before proving such a result.

6. Uniqueness of spaces realizing A_{r}. In section 5 we have constructed the spaces X_{k}(r) whose mod p cohomologies realize A_{r}^{(k)} for r dividing p − 1 and k ≥ 0 (theorem 5.5). In this section we show that up to p-adic completion these spaces are the only ones which realize the algebras with higher Bocksteins A_{r}^{(k)}.

Theorem 6.1. Let X be such that H^{∗}(X; 𝔽_{p}) ≅ A_{r}^{(k)} for some r|(p−1) and k ≥ 0. Then X̂_{p} ≃ X_{k}(r).

Proof. Let X be a space satisfying the hypothesis of the theorem. Since H_{1}(X; 𝔽_{p}) = 0 we have that π_{1}(X) is ℤ/p−perfect. Hence ([6; p. 206]) X̂_{p} is a simply connected p-complete space with the same mod p cohomology as X itself. By 5.6 and 5.7 the homotopy groups of X̂_{p} are finite p−groups.

Let f : Bℤ/p → X̂_{p} be a map such that f^{∗} is non-trivial in degree 2r and trivial in degree 2r + 1 and let Y denote the component of the mapping space map(Bℤ/p, X̂_{p}) containing the map f. There is an evaluation map e : Y → X̂_{p}. The next step in the proof of 6.1 will be to show that Y is homotopy equivalent to the space E_{k} of section 5. According to the computation of the T functor in section 3, T_{f}(H^{∗}(X; 𝔽_{p})) ≅ E(z) ⊗ 𝔽_{p}[w] with deg(z) = 1, deg(w) = 2 and

β(w) = \{ \begin{matrix}zw, & k = 0 \\ 0, & k > 0. \end{matrix}

Notice that T_{f}(H^{∗}(X; 𝔽_{p})) is only an algebra over the Steenrod algebra and so higher Bocksteins do not make any sense in T_{f}(H^{∗}(X; 𝔽_{p})) unless we show that it is the cohomology of some space.

The computed value of the functor T_{f} is interpreted by [10] as follows. Let P_{n}X̂_{p} denote the n-th stage of the Postnikov decomposition of X̂_{p}. Then { map(Bℤ/p, P_{n}X̂_{p})_{f_{n}}} is a tower with

Y ≃ \underleftarrow{lim}_{n} map(Bℤ/p, P_{n}X̂_{p})_{f_{n}}

and

\underrightarrow{lim}_{n} H^{∗}(map(Bℤ/p, P_{n}X̂_{p})_{f_{n}}; 𝔽_{p}) ≅ \{ \begin{matrix}C′_{1}, & k > 0, \\ C_{1}, & k = 0. \end{matrix}

The natural homomorphism H^{∗}(X; 𝔽_{p}) → \underrightarrow{lim}_{n} H^{∗}(map(Bℤ/p, P_{n}X̂_{p})_{f_{n}}; 𝔽_{p}) sends x to w^{r} and y to w^{r}z.

Some information about the homotopy of the spaces map(Bℤ/p, P_{n}X̂_{p})_{f_{n}} is provided by results of Thom ([26], revisited in [21]). The principal fibration P_{n}X̂_{p} → P_{n−1}X̂_{p} gives rise to a principal fibration

map(Bℤ/p, P_{n}X̂_{p})_{f_{n}} → map(Bℤ/p, P_{n−1}X̂_{p})_{f_{n−1}}

with fibre a union of components of map(Bℤ/p, K(π_{n}X̂_{p}, n)). But each of these components has the homotopy type of a product of Eilenberg-MacLane spaces

K(H^{n−j}(Bℤ/p, π_{n}X̂_{p}), j), 1 ≤ j ≤ n.

Since the homotopy groups of X̂_{p} are finite p-groups and so are the homotopy groups of map(Bℤ/p, P_{n−1}X̂_{p})_{f_{n−1}} by induction, those of map(Bℤ/p, P_{n}X̂_{p})_{f_{n}} should also be finite p-groups.

Let us write Y_{n} = map(Bℤ/p, P_{n}X̂_{p})_{f_{n}}.

The class z in degree one in \underrightarrow{lim}_{n} H^{∗}(Y_{n}; 𝔽_{p}) is represented by a class z_{n} ∈ H^{∗}(Y_{n}; 𝔽_{p}) for some n and also by its images in H^{∗}(Y_{n+i}; 𝔽_{p}). We fix such a sequence {z_{n}}. This sequence provides a map of towers:

{Y_{n}} ⟶^{{z̄_{n}}} {K(ℤ/p^{α(n)}, 1)}.

We will prove that we can choose the sequence α(n) to be unbounded. In fact, any of these maps is a lifting of the classifying map Y_{n} ⟶ K(ℤ/p, 1) of the class z_{n}. Suppose by induction that z_{n} is classified by z̄_{n}: Y_{n} ⟶ K(ℤ/p^{α(n)}, 1), that is z̄_{n}^{∗}(ι) = z_{n} if ι is the fundamental class of H^{∗}(K(ℤ/p^{α(n)}, 1); 𝔽_{p}). Observe that we can as well assume that α(n) is the maximum possible such that this lifting exists. This is because all of the homotopy groups of Y_{n} are finite and then z_{n} should be dual to a torsion homology class. Now we look at the class z_{n+i}, i ≥ 1. This is classified by Y_{n+i} ⟶ Y_{n} ⟶ K(ℤ/p^{α(n)}, 1) and the obstructions for the existence of a lifting

Y_{n+i} ⟶ K(ℤ/p^{α(n+i)}, 1)

with α(n + i) > α(n) are some higher Bocksteins. But no higher Bockstein can be non trivial on z_{n} for all big enough n because if this happens then w^{r} is in the image of some higher Bockstein, contradicting the fact that x ∈ H^{∗}(X; 𝔽_{p}) has a non trivial Bockstein of order k + 1. Hence, lim_{n→∞} α(n) = ∞.

Consider now the inverse system of fibrations

… ⟶ F_{n}  ⟶ F_{n−1}  ⟶ …
↓  ↓
… ⟶  Y_{n}  ⟶  Y_{n−1}  ⟶ …
↓  ↓
… ⟶ Bℤ/p^{α(n)} ⟶ Bℤ/p^{α(n−1)} ⟶ …

Notice that since the maps Y_{n} ⟶ Bℤ/p^{α(n)} are liftings of non-trivial maps Y_{n} ⟶ Bℤ/p, they induce epimorphisms between fundamental groups and so the spaces F_{n} are connected.

All homotopy groups involved in the above inverse system of fibrations are finite p-groups, hence, in the limit, we get a fibration:

F ⟶ Y ⟶ Bπ

where F = \underleftarrow{lim}_{n} F_{n} and π denotes as usual the additive group of the p-adic integers. Note that in all these fibrations the base space is not simply connected. Nevertheless, at any stage the Eilenberg-Moore spectral sequence of [12] starts with

E_{2}^{∗,∗} ≅ Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p})

and converges strongly to the mod p cohomology of the fibre F_{n} because ([12]) the fundamental group of the base is a p-group and thus it acts nilpotently on the mod p cohomology of the fibre. In the limit we have a spectral sequence

E_{2}^{∗,∗} ≅ \underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p})

converging to \underrightarrow{lim}_{n} H^{∗}(F_{n}; 𝔽_{p}).

Lemma 6.3. \underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p}) ≅ 𝔽_{p}[w], deg(w) = 2 and so therefore \underrightarrow{lim}_{n} H^{∗}(F_{n}; 𝔽_{p}) ≅ 𝔽_{p}[w].

Proof. Tor is covariant with respect to any of its three variables and \underrightarrow{lim} is an exact functor. Hence one can easily derive a commutation formula for Tor and \underrightarrow{lim} which shows that

\underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p}) ≅ Tor_{E(z)}^{∗,∗} (E(z) ⊗ 𝔽_{p}[w], 𝔽_{p}) ≅ 𝔽_{p}[w].

Alternatively, one can directly compute the \underrightarrow{lim} as follows.

Let us denote

K_{n} = ker \{H^{∗}(Y_{n}; 𝔽_{p}) → \underrightarrow{lim}_{n} H^{∗}(Y_{n}; 𝔽_{p}) ≅ E(z) ⊗ 𝔽_{p}[w]\}.

K_{n} is an ideal of H^{∗}(Y_{n}; 𝔽_{p}) and therefore a sub-H^{∗}(Bℤ/p^{α(n)}; 𝔽_{p})-module. The induced H^{∗}(Bℤ/p^{α(n)}; 𝔽_{p})-module structure of E(z) ⊗ 𝔽_{p}[w] factors:

H^{∗}(Bℤ/p^{α(n)}; 𝔽_{p}) ⟶ E(z_{n})
↓  ↓
K_{n} ⟶  H^{∗}(Y_{n}; 𝔽_{p})  ⟶ E(z) ⊗ 𝔽_{p}[w]

Observe that H^{∗}(Bℤ/p^{α(n)}; 𝔽_{p}) ≅ E(z_{n})⊗𝔽_{p}[a_{n}], deg(a_{n}) = 2. As a consequence we have:

(1) Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (E(z) ⊗ 𝔽_{p}[w], 𝔽_{p})

≅ Tor_{E(z_{n})}^{∗,∗} (E(z_{n}) ⊗ 𝔽_{p}[w], 𝔽_{p}) ⊗ Tor_{𝔽_{p}[a_{n}]}^{∗,∗}(𝔽_{p}, 𝔽_{p})
≅ 𝔽_{p}[w] ⊗ Tor_{𝔽_{p}[a_{n}]}(𝔽_{p}, 𝔽_{p})

There is an exact sequence:

(2) … → Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{r,s} (K_{n}, 𝔽_{p}) → Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{r,s} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p}) →

→ Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{r,s} (E(z) ⊗ 𝔽_{p}[w], 𝔽_{p}) → Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{r−1,s} (K_{n}, 𝔽_{p}) → …

which is natural with respect to maps Y_{n+i} → Y_{n}. Observe that no element of K_{n} survives to the limit \underrightarrow{lim} H^{∗}(Y_{n}; 𝔽_{p}). Since K_{n} is finite dimensional there is a large enough i such that K_{n} → K_{n+i} is zero, hence so is

Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (K_{n}, 𝔽_{p}) → Tor_{H^{∗}(ℤ/p^{α(n+i)};𝔽_{p})}^{∗,∗} (K_{n+i}, 𝔽_{p}).

Then (2) implies

\underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{r,s} (H^{∗}(Y_{n}; 𝔽_{p}), 𝔽_{p}) ≅ \underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (E(z) ⊗ 𝔽_{p}[w], 𝔽_{p}).

For similar reasons (1) implies:

\underrightarrow{lim}_{n} Tor_{H^{∗}(Bℤ/p^{α(n)};𝔽_{p})}^{∗,∗} (E(z) ⊗ 𝔽_{p}[w], 𝔽_{p}) ≅ 𝔽_{p}[w]. □

Lemma 6.4. Let {Z_{n}} be a tower of fibrations of pointed connected p-complete spaces with mod p cohomology of finite type. If \underrightarrow{lim}_{n} H^{∗}(Z_{n}; 𝔽_{p}) is a polynomial algebra on one generator w in degree 2 then \underleftarrow{lim}_{n} Z_{n} ≃ B^{2}π.

Proof. An argument similar to one used above shows that there is a map of towers:

(1)  {Z_{n}} ⟶^{{g_{n}}} {K(ℤ/p^{γ(n)}, 2)}

with γ(n) an unbounded sequence. Here, each g_{n} detects a class in degree two of H^{∗}(Z_{n}; 𝔽_{p}) that represents w in the limit, hence {g_{n}} induces an isomorphism

\underrightarrow{lim}_{n} H^{∗}(Z_{n}; 𝔽_{p}) ≅ \underrightarrow{lim}_{n} H^{∗}(K(ℤ/p^{γ(n)}, 2); 𝔽_{p}) ≅ 𝔽_{p}[w]

and dually

\underleftarrow{lim}_{n} H_{∗}(Z_{n}; 𝔽_{p}) ≅ \underleftarrow{lim}_{n} H_{∗}(K(ℤ/p^{γ(n)}, 2); 𝔽_{p})

because all relevant (co)homology groups are finite. By the same reason, this implies that the induced map of towers:

{H_{∗}(Z_{n}; 𝔽_{p})} → {H_{∗}(K(ℤ/p^{γ(n)}, 2); 𝔽_{p})}

is a pro-isomorphism.

Now, according to [6; III.6.6, pg. 88] the map of towers {R_{n}Z_{n}} → {R_{n}K(ℤ/p^{γ(n)}, 2)} is a weak pro-homotopy equivalence, where {R_{n}X} is the tower which defines Bousfield-Kan p-completion. Hence,

\underleftarrow{lim}_{n} Z_{n} = \underleftarrow{lim}_{n} R_{∞}Z_{n} = \underleftarrow{lim}_{n} R_{n}Z_{n} ≃ \underleftarrow{lim}_{n} R_{n}K(ℤ/p^{γ(n)}, 2) = B^{2}π. □

This lemma applies immediately to the tower {F_{n}} because the homotopy groups of each F_{n} are finite p-groups and a space whose homotopy groups are finite p-groups is necessarily p-complete, nilpotent and of finite mod p type. We have, therefore, obtained a fibration

(2)  B^{2}π → Y → Bπ.

Since π can only act trivially on H_{∗}(B^{2}π; 𝔽_{p}), Y is p-complete (cf. [6; mod-ℤ/p fibre lemma]). The Serre spectral sequence and the injectivity of H^{∗}(X; 𝔽_{p}) → \underrightarrow{lim} H^{∗}(Y_{n}; 𝔽_{p}) show that the natural map

H^{∗}(Y; 𝔽_{p}) → \underrightarrow{lim}_{n} H^{∗}(Y_{n}; 𝔽_{p}) ≅ T_{f}(H^{∗}(X; 𝔽_{p}))

is an isomorphism. By naturality of the Bockstein homomorphisms, we deduce that H^{∗}(Y; 𝔽_{p}) = C_{1}^{(k)}. We want to deduce from here that the fibration (2) is fibre homotopy equivalent to the fibration

B^{2}π → E_{k} → Bπ

of section 5. Fibrations with base space Bπ and fibre B^{2}π are classified by the homotopy set:

[Bπ, B Aut(B^{2}π)]

where Aut(B^{2}π) is the topological monoid of the self homotopy equivalences of B^{2}π. According to [22] there is a fibration

B^{2}π → B Aut(B^{2}π) → B Aut(π)

having a section B Aut(π) → B Aut(B^{2}π). Then

[Bπ, B Aut(B^{2}π)] ≅ [Bπ, B Aut(π)] ≅ Hom(π, Aut(π)).

Therefore, any fibration B^{2}π → Z → Bπ is determined by an action of π on π. All these actions were considered in section 5. From such classification we obtain an equivalence of fibrations

B^{2}π ⟶ Y ⟶ Bπ
∥  ^{≃}↑^{g}  ^{≃}↑^{w}
B^{2}π ⟶ E_{k} ⟶ Bπ

There is an action of ℤ/r on E_{k} considered in section 5 and also an action of ℤ/r on Y defined in the following way. We have Y = map(Bℤ/p, X̂_{p})_{f} and ℤ/r acts on Bℤ/p. Since f^{∗} commutes with this action, we get an action on Y such that the evaluation map e : Y → X̂_{p} is equivariant. Naturality of T shows that on mod p cohomology this action leaves z fixed and sends w to λw where λ is an r-th root of unity. In this form we obtain a map l : E_{k} → X̂_{p}

Lemma 6.5. l is homotopic to an equivariant map.

Proof. First of all we notice that it is enough to prove that l is equivariant up to homotopy for Wojtkowiak proved in [27] that when a finite group of order prime to p acts freely on a space and the target space is p-complete, nilpotent and of finite type over ℤ̂_{p} then a map equivariant up to homotopy is homotopic to an equivariant map.

E_{k} is a two stage Postnikov system and there is an exact sequence of Didierjan ([8]) for the group of homotopy classes of self homotopy equivalences of E_{k}:

1 → H^{2}(π; π^{φ_{k}}) → E(E_{k}) → Aut(π) ⊕ Aut(π).

Since H^{2}(π; π^{φ_{k}}) = 0, this shows that a homotopy self equivalence of E_{k} is determined up to homotopy by its action on π_{1}(E_{k}) and π_{2}(E_{k}). The lemma is proved if we show that ℤ/r acts on π_{1}(Y) and π_{2}(Y) as it does on π_{1}(E_{k}) and π_{2}(E_{k}).

The action of ℤ/r on π_{1}(Y) = π is determined by the action on H^{1}(Y; 𝔽_{p}) which can only be trivial. Similarly, the action of ℤ/r on π_{2}(Y) = π is determined by the action on H^{2}(Y; ℤ̂_{p}) ≅ ℤ̂_{p} and this action is determined by the action on H^{2}(Y; 𝔽_{p}). □

Hence we obtain a map

h : E_{k}(r) = E_{k} ×_{ℤ/r} Eℤ/r → X̂_{p}.

Let now k : Bπ → E_{k}(r) be the map considered in section 5. If kh is trivial we obtain a map E′_{k}(r) → X̂_{p} which induces isomorphism in mod p cohomology and the theorem is proved. But X̂_{p} is simply connected, its homotopy groups are finite p-groups and H^{i}(π; P) = 0 for i > 1 and any finitely generated ℤ̂_{p}-module P with trivial action. Hence, by obstruction theory, any map Bπ → X̂_{p} is trivial. This ends the proof of theorem 6.1. □

7. Spaces realizing B_{i,r}. In section 2 we constructed for each algebra B_{i,r} a topological realization. It turns out that these are not the only ones. In this section we will construct several families of spaces, some of which will have cohomology isomorphic to B_{i,r}. We suggest using tables 10.2 and 10.3 in the appendix as a quick reference guide to all these spaces.

Let X_{k} = X_{k}(1) be the spaces introduced in section 5. Theorem 5.5 proves that

H^{∗}(X_{k}; 𝔽_{p}) ≅ A_{1}^{(k)}.

Hence, the two dimensional class x in H^{∗}(X_{k}; 𝔽_{p}) can be represented by a map

(1)  X_{k} → B^{2}ℤ/p^{k+1}.

Let Y_{k} be the fibre of this map. Y_{k} is a p-complete space with finite homotopy groups. From the construction of X_{k} we see that there is an action of the cyclic group of order r on X_{k}, for any r|p − 1. By [27] we can assume that the map (1) is equivariant with respect to this action and the natural action on B^{2}ℤ/p^{k+1}. This yields an action of the cyclic group of order r on Y_{k} and we define Y_{k,r} as the p-completion of the homotopy quotient of Y_{k} by this action:

Y_{k,r} = (Eℤ/r ×_{ℤ/r} Y_{k})_{p}.

Proposition 7.1. H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ B_{1,r} as algebras over the Steenrod algebra.

Proof. We consider the sequence of fibrations

Bℤ/p^{k+1} → Y_{k,1} → X_{k} → B^{2}ℤ/p^{k+1}.

In the spectral sequence of the first three terms, the classes u and v in H^{∗}(Bℤ/p^{k+1}; 𝔽_{p}) are transgressive and are mapped onto the classes x and y of H^{∗}(X_{k}; 𝔽_{p}). By degree reasons it follows that these are the only non vanishing differentials. Therefore, H^{∗}(Y_{k,1}; 𝔽_{p}) is a P E-algebra of type (2p, 2p + 1).

We prove now H^{2p+1}(Y_{k,1}; ℤ̂_{p}) ≅ ℤ/p.

We have seen that Y_{k,1}, being a p-complete space, is (2p − 1)-connected. All homology groups of Y_{k,1} are torsion groups, and therefore, H^{2p}(Y_{k,1}; ℤ̂_{p}) = 0. The long exact sequence of cohomology groups associated to the fibration Y_{k,1} → X_{k} → B^{2}ℤ/p^{k+1} contains

0 ⟶ H^{2p+1}(B^{2}ℤ/p^{k+1}; ℤ̂_{p}) ⟶ H^{2p+1}(X_{k}; ℤ̂_{p})

⟶ H^{2p+1}(Y_{k,1}; ℤ̂_{p}) ⟶ H^{2p+2}(B^{2}ℤ/p^{k+1}; ℤ̂_{p}) ⟶ H^{2p+2}(X_{k}; ℤ̂_{p}) = 0.

The last group vanishes because of remark 5.8. The first two groups are isomorphic, because both measure which high order Bockstein acts nontrivially on ι_{2}, the generator of H^{∗}(B^{2}ℤ/p^{k+1}; ℤ/p), or on x^{p}. In both cases this is β_{(k+2)}. Thus, we have to calculate H^{2p+2}(B^{2}ℤ/p^{k+1}; ℤ̂_{p}). In dimension 2p + 1, the mod-p cohomology of B^{2}ℤ/p^{k+1} is generated by ι_{2}^{p−1}β_{(k+1)}(ι_{2}) and P^{1}β_{(k+1)}(ι_{2}). All higher order Bocksteins vanish on ι_{2}^{p−1}β_{(k+1)}(ι_{2}), which therefore comes from an integral class, and βP^{1}β_{(k+1)}(ι_{2}) ≠ 0. Thus H^{2p+2}(B^{2}ℤ/p^{k+1}; ℤ̂_{p}) ≅ ℤ/p and H^{2p+1}(Y_{k,1}; ℤ̂_{p}) ≅ ℤ/p as claimed.

Hence, the two generators of H^{∗}(Y_{k,1}; 𝔽_{p}) are connected via the Bockstein. The only algebra over the Steenrod algebra of this type is B_{1,1} (theorem 2.1).

For r|p−1, the space Y_{k,r} fits into the fibration Y_{k,1} → Y_{k,r} → Bℤ/r. A spectral sequence argument establishes the isomorphisms

H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ H^{∗}(Y_{k,1}; 𝔽_{p})^{ℤ/r} ≅ B_{1,1}^{ℤ/r} = B_{1,r}. □

For any map f : BA → Y, A an abelian group, the connected group BA acts on the mapping space map(BA, Y)_{f}. The Borel construction

Bor(Y, f) := EBA ×_{BA} map(BA, Y)_{f}

sits in a sequence of fibrations

BA → map(BA, Y)_{f} → Bor(Y, f) → B^{2}A.

Lemma 7.2. Let f : BA → Y be a map, A a compact abelian group. Then, e_{Y} : map(BA, Y)_{f} ≃ Y if and only if there exists a principal fibration BA ⟶^{f} Y ⟶ \overline{Y} and e\overline{Y} : map(BA, \overline{Y})_{c} ≃ \overline{Y} where c denotes the constant map. Moreover, \overline{Y} ≃ Bor(Y, f).

Proof. First let us assume that there exists a principal fibration BA ⟶^{f} Y ⟶ \overline{Y}. By ‘Thom–theory’ ([26], revisited in [21]) this principal fibration establishes a diagram of principal fibrations

(*)

map(BA, BA)_{c} ⟶ map(BA, Y)_{f} ⟶ map(BA, \overline{Y})_{c}
e_{BA}↓  ^{e_{Y}} ↓  ^{e\overline{Y}} ↓
BA  ⟶  Y  ⟶  \overline{Y}

The product h·g of two maps h : BA ⟶ BA and g : BA ⟶ Y is given by the action of BA on Y. In general the fiber in the top row consists of all maps h : BA ⟶ BA such that h · f ≃ f, in particular it contains the component of the constant map. The fundamental group π_{1}(\overline{Y}) acts on BA via maps homotopic to the identity. The fundamental group π_{1}(map(BA, \overline{Y})_{c}) acts on the fiber via this action, which therefore also acts via maps homotopic to the identity. Because the total space of the fibration is connected, this action also must permute the components of the fiber, which is therefore connected and consists only of the component of the constant map. Moreover, because A is a compact abelian group the map e_{BA} is an equivalence.

If e\overline{Y} also is an equivalence, then e_{Y} is also an equivalence, which proves one half of the statement.

Now we assume that map(BA, Y)_{f} ≃ Y. The space BA acts on map(BA, Y)_{f} with homotopy orbit \overline{Y} := Bor(Y, f). This establishes the desired principal fibration

BA ⟶ map(BA, Y)_{f} ≃ Y ⟶ \overline{Y} .

Applying ‘Thom-theory’ again, yields the diagram (∗) of principal fibrations. This time the first two vertical maps are equivalences and so is the third one. Moreover, the equivalence of both rows in (∗) proves that \overline{Y} ≃ Bor(Y, f). □

The following lemma may also be found in [20].

Lemma 7.3. Let K → G → H be an exact sequence of topological groups. If the evaluation map map(BK, Y)_{c} → Y is an equivalence, then

map(BH, Y) → ∐_{g|_{BK}≃c} map(BG, Y)_{g}

is an equivalence, where c indicates a constant map.

Proof. H acts on B̃K := EG/K ≃ BK freely, and on Y trivially. The canonical map Y → map(BK, Y)_{c} is equivariant and an equivalence. Therefore

map(BH, Y) ≃ Y^{hH} ≃ (map(B̃K, Y)_{c})^{hH} ≃ ∐_{g|_{BK}≃const} map(BG, Y)_{g}.

Here Y^{hH} denotes the homotopy fixed point set. The last equivalence follows from [14]. □

Now we can state and prove the main result of this section. For j ≤ k + 1, let Y_{k}(j) denote the homotopy fibre of the composition

X_{k} ⟶ B^{2}ℤ/p^{k+1} ⟶^{B^{2}q} B^{2}ℤ/p^{k+1−j},

where q : ℤ/p^{k+1} → ℤ/p^{k+1−j} is the projection. These spaces fit into a sequence

Y_{k} := Y_{k}(0) → Y_{k}(1) → ⋯ → Y_{k}(k + 1) = X_{k}.

The realization Bℤ/p → X_{k} of the composition H^{∗}(X_{k}; 𝔽_{p}) → 𝔽_{p}[x] → H^{∗}(Bℤ/p; 𝔽_{p}) can be lifted to a map Bℤ/p^{k+2} → Y_{k}.

Table 10.4 in the appendix displays the above sequences. In table 10.5 one can read the cohomology algebras of the spaces involved in table 10.4.

Proposition 7.4. (1)

H^{∗}(Y_{k}(j); 𝔽_{p}) ≅ \{ \begin{matrix}B_{1,1}, & j = 0 \\ B_{0,1}, & j = 1 \\ B′_{0,1}, & 2 ≤ j ≤ k \\ A′_{1}, & j = k + 1. \end{matrix}

(2) The spaces fit into fibrations

Bℤ/p^{l} ⟶^{g_{j,l}} Y_{k}(j) ⟶^{f_{j,j+l}} Y_{k}(j + l) ⟶^{a_{2}} B^{2}ℤ/p^{l},

where l ≤ k − j + 1. The last map classifies the two dimensional class x and is an H^{2}( ; 𝔽_{p})-isomorphism. The first map is a realization of the composition

H^{∗}(Y_{k}(j); 𝔽_{p}) → 𝔽_{p}[x] → H^{∗}(Bℤ/p^{l}; 𝔽_{p}).

(3) For l ≤ k − j + 1, the evaluation e : map(Bℤ/p^{l}, Y_{k}(j))_{g_{j,l}} → Y_{k}(j) is a homotopy equivalence. Moreover, Y_{k}(j + l) ≃ Bor(Y_{k}(j), g_{j,l}).

(4) For l = k − j + 2, there is a fibration

Bℤ/p^{l−1} → map(Bℤ/p^{l}, Y_{k}(j))_{g_{j,l}} → E_{k} → B^{2}ℤ/p^{l−1}.

(5) There exists a map B^{2}π ≃ BS^{1}_{p} → Y_{k}, which is a realization of the com-

position H^{∗}(Y_{k}; 𝔽_{p}) → 𝔽_{p}[x] → H^{∗}(BS^{1}_{p}; 𝔽_{p})

Proof. For j ≥ 1, a Serre spectral sequence argument for the fibrations

Bℤ/p^{k+1−j} → Y_{k}(j) → X_{k} → B^{2}ℤ/p^{k+1−j}

shows that H^{∗}(Y_{k}(j); 𝔽_{p}) is a P E-algebra of type (2,3). The action of the Steenrod algebra will be calculated later.

(2) follows from the commutativity of the diagram

Bℤ/p^{l} ⟶ Bℤ/p^{k+1−j} ⟶ Bℤ/p^{k+1−j−l} ⟶ B^{2}ℤ/p^{l}
∥  ↓  ↓  ∥
Bℤ/p^{l} ⟶  Y_{k}(j)  ⟶ Y_{k}(j + l) ⟶ B^{2}ℤ/p^{l}
↓  ↓  ↓  ↓
∗ ⟶  X_{k}  X_{k}  ⟶ ∗
↓  ↓  ↓  ↓
B^{2}ℤ/p^{l} ⟶ B^{2}ℤ/p^{k+1−j} ⟶ B^{2}ℤ/p^{k+1−j−l} ⟶ B^{3}ℤ/p^{l}

The conditions on the maps can be easily obtained by looking at the differentials of the associated Serre spectral sequences.

The classifying map Y_{k}(j) → B^{2}ℤ/p^{j} of the fibration Y_{k}(0) → Y_{k}(j) is an H^{∗}( ; 𝔽_{p})-isomorphism in low dimensions. This proves that β(x) = β(y) = 0 for j ≥ 2 and that β(x) = y for j = 1, which determines one part of the Steenrod algebra action.

(3) follows from (2) and lemma 7.2. We only have to show that map(Bℤ/p^{l}, Y_{k}(j))_{c} ≃ Y_{k}(j) for all l and j. For l = 1 this is a consequence of theorem 3.4 and [17]. Now, lemma 7.3 and an induction over l proves the statement.

To prove (4), we use again lemma 7.3. In this case, it is l = k − j+2 and i = k − j+1,

map(Bℤ/p^{k−j+2}, Y_{k}(k + 1))_{f_{j,k+1}g_{j,k−j+2}} ≃ map(Bℤ/p, Y_{k}(k + 1))_{g_{k+1,1}},

because f_{j,k+1}g_{j,k+2}|_{Bℤ/p^{k−j+1}} ≃ c, Y_{k}(k +1) ≃ X_{k} and map(Bℤ/p, X_{k})_{g_{k+1,1}} ≃ E_{k} (see the proof of 6.1). We can apply the results of section 6 and get a principal fibration

map(Bℤ/p^{k−j+2}, Bℤ/p^{k−j+1})_{q} → map(Bℤ/p^{k−j+2}, Y_{k}(j))_{g_{k−j+2,j}}
→ map(Bℤ/p, Y_{k}(k + 1))_{g_{k+1,1}} ≃ E_{k}.

The first mapping space is equivalent to Bℤ/p^{k−j+1}. This establishes the fibration of (4).

The composition

B^{2}π ≃ BS^{1}_{p} ⟶^{p^{k+1}} BS^{1}_{p} ⟶ E_{k} ≃ map(Bℤ/p^{,}X_{k})_{g_{k+1,1}} ⟶ X_{k}

can be lifted to BS^{1}_{p} → Y_{k}. Obviously, this map induces the desired map in mod p cohomology of (5).

To complete the proof of (1), we finally have to calculate P^{1}(y). For j = k + 1, there is nothing to show. P^{1}(y) ≠ 0, for j ≤ k, contradicts the fact that map(Bℤ/p, Y_{k}(j))_{g_{j,1}} ≃ Y_{k}(j), as theorem 3.1 shows. □

To get a complete picture, we define X_{∞} := S^{3}_{p} and Y_{∞} := S^{3}⟨3⟩_{p}. Then,

BS^{1}_{p} → Y_{∞} → X_{∞} → B^{2}S^{1}_{p} ≃ (B^{2}ℤ/p^{∞})_{p}

are fibrations. There exists a long sequence of maps

Y_{∞} → Y_{∞}(1) → Y_{∞}(2) → ⋯,

where Y_{∞}(j) is the homotopy fibre of the map S^{3} ⟶^{p^{j}} B^{2}S^{1}_{p} of degree p^{j}. Moreover, proposition 7.4 holds for Y_{∞}. The proof is analogous.

Corollary 7.5. For every k ≤ ∞ and every 0 ≤ j ≤ k, the homotopy type of Y_{k}(j) determines the homotopy type of every space in the sequence associated to Y_{k}.

Proof. Proposition 7.4 (2) and (3). □

Corollary 7.6. The spaces Y_{k,r} are of different homotopy type.

Proof. For r = 1, this follows from proposition 7.4 (3) and (4). For r > 1, the map Y_{k,1} = Y_{k} → Y_{k,r} induces an equivalence Y_{k} ≃ map(Bℤ/p, Y_{k})_{g_{0,1}} ≃ map(Bℤ/p, Y_{k,r})_{g}, where g : Bℤ/p ⟶^{g_{0,1}} Y_{k} ⟶ Y_{k,r}. This follows from theorem 3.1 and [17]. □

Next we construct a list of spaces realizing the algebras B_{0,r}, for r|p − 1. Let s ∈ ℤ/r ⊂ (ℤ/p)^{∗} ≅ Aut(ℤ/p) be a generator. The diagram

ℤ/p × ℤ/p ⟶ ℤ/p
s×s↓  ^{s}↓
ℤ/p × ℤ/p ⟶ ℤ/p

commutes, because s is given by a multiplication. The horizontal arrows are given by addition. Passing to classifying spaces and mapping spaces and taking adjoints yields a commutative diagram

Bℤ/p × map(Bℤ/p, Y_{k,r})_{g} ⟶ map(Bℤ/p, Y_{k,r})_{g}
s×map(s,id)↓  ^{map(s,id)}↓
Bℤ/p × map(Bℤ/p, Y_{k,r})_{g} ⟶ map(Bℤ/p, Y_{k,r})_{g}.

Here, g denotes the composition Bℤ/p ⟶^{g_{0,1}} Y_{k} ⟶ Y_{k,r}. Because H^{∗}(Y_{k,r}; 𝔽_{p}) is a P E-algebra of type (2pr, 2pr + 1), the component of g is fixed by the ℤ/r-action. Thus, we get a ℤ/r-action on the quotient Bor(Y_{k,r}, g) ≃ Bor(Y_{k}, g_{0,1}) ≃ Y_{k}(1), k ≥ 0. The first equivalence follows from theorem 3.1 and [17] and the second equivalence is from lemma 7.2.

Now, we define Z′_{k,r} := Eℤ/r ×_{ℤ/r} Y_{k}(1) and Z_{k,r} := (Z′_{k,r})_{p}. We also put Z_{k} = Z_{k,1}.

Proposition 7.7. For 0 < k ≤ ∞ and r|p − 1, all the spaces Z_{k,r} are pairwise not homotopy equivalent, and H^{∗}(Z_{k,r}; 𝔽_{p}) ≅ B_{0,r} as algebras over the Steenrod algebra.

Proof. For r = 1, Z_{k,r} ≃ Y_{k}(1), and H^{∗}(Y_{k}(1); 𝔽_{p}) ≅ B_{0,1} by proposition 7.4 (1). For r > 1, the calculation of H^{∗}(Z_{k,r}; 𝔽_{p}), is analogous to the calculation of H^{∗}(Y_{k,r}; 𝔽_{p}) in the proof of proposition 7.1. Let f : Bℤ/p → Z_{k,1} → Z_{k,r} be the obvious composition. By theorem 3.1 and [17],

map(Bℤ/p, Z_{k,r})_{f} ≃ Z_{k,1} ≃ Y_{k}(1).

Now, the statement follows from corollary 7.5. □

8. Classification of spaces realizing B_{i,r}. In this section we classify up to pcompletion the possible homotopy types of spaces realizing B_{i,r}. By theorem 4.3 we only have to consider the cases i = 0, 1. Let Y_{k,r} and Z_{k,r} be the spaces constructed in section 7 with

H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ B_{1,r}
H^{∗}(Z_{k,r}; 𝔽_{p}) ≅ B_{0,r}

We will show that these spaces form a complete list of p-complete homotopy types realizing B_{1,r} and B_{0,r}, respectively.

The next proposition is an immediate consequence of well know properties of the Bockstein spectral sequence (see section 5).

Proposition 8.1. Let X be a space with p-adic cohomology of finite type over ℤ̂_{p}. If H^{∗}(X; 𝔽_{p}) = B_{i,r} then in the ℤ̂_{p}-cohomology Bockstein spectral sequence {B_{l}, d_{l}} for X we have B_{∞} = 0. □

Theorem 8.2. (1) If H^{∗}(X; 𝔽_{p}) ≅ B_{1,r} then there exists 0 ≤ k ≤ ∞ such that X̂_{p} ≃ Y_{k,r}.

(2) If H^{∗}(X; 𝔽_{p}) ≅ B_{0,r} then there exists 0 < k ≤ ∞ such that X̂_{p} ≃ Z_{k,r}.

Proof. (cf. tables 10.4 and 10.5.) Let Y be the p-completion of a space X realizing B_{1,r}. Then Y is 1-connected, p-complete and realizes also B_{1,r}. By 5.7 the p-adic cohomology of Y is of finite type over ℤ̂_{p} and by 8.1 and 5.7 all homotopy groups of Y are finite p-groups.

Consider first the case of B_{1,1}. We will construct a sequence of maps

Y := Y (0) → Y (1) → Y (2) → ⋯,

such that H^{∗}(Y (j); 𝔽_{p}) ≅ 𝔽_{p}[a_{j}]⊗E(b_{j}) isomorphic to either A_{1}^{(j−1)} or B_{0,1}^{(j−1)}, j ≥ 1,

and such that there exists a fibration sequence Bℤ/p^{j} ⟶ Y (0) ⟶ Y (j) ⟶^{a_{j}} B^{2}ℤ/p^{j}, where the last map is algebraically given as in Proposition 7.4 (2).

Let us assume that we already constructed Y (j), j ≥ 1, with H^{∗}(Y (j); 𝔽_{p}) ≅ 𝔽_{p}[a_{j}] ⊗ E(b_{j}) isomorphic to either A_{1}^{(j−1)} or B_{0,1}^{(j−1)}. Let g_{j,1} : Bℤ/p → Y (j) be the realization of the composition H^{∗}(Y (j); 𝔽_{p}) → 𝔽_{p}[a_{j}] → H^{∗}(Bℤ/p; 𝔽_{p}). Then the computation of the T functor on the algebras A and B in 3.1 and 3.5 and the results of [17] imply that a necessary and sufficient condition for

map(Bℤ/p, Y (j))_{f_{j}} ≃ Y (j)

is P^{1}(b_{j}) = 0, i.e. H^{∗}(Y (j); 𝔽_{p}) = B_{0,1}^{(j−1)}. Hence, for P^{1}(b_{j}) = 0, there exists a principal fibration (lemma 7.2)

Bℤ/p → Y (j) → Bor(Y (j), f_{j}) =: Y (j + 1) → B^{2}ℤ/p,

which fits into the commutative diagram

∗ ⟶ Y (0)  Y (0) ⟶ ∗
↓  ↓  ↓  ↓
Bℤ/p ⟶^{f_{j}} Y (j) ⟶ Y (j + 1) ⟶ B^{2}ℤ/p
∥  ^{a_{j}} ↓  ^{a_{j+1}}↓  ∥
Bℤ/p ⟶^{a_{j}f_{j}} B^{2}ℤ/p^{j} ⟶ B^{2}ℤ/p^{j+1} ⟶ B^{2}ℤ/p .

A short calculation shows that Bor(B^{2}ℤ/p^{j}, a_{j}f_{j}) ≃ B^{2}ℤ/p^{j+1}. This establishes the map a_{j+1}. The differentials in the Serre spectral sequence of the fibration given by the right three terms in the middle row are given by the equations d^{2}(a_{j}) = β(ι) and d^{2}(b_{j}) = 0. The equations follow from a comparison with the spectral sequence of the fibration in the bottom row. Now a straightforward calculation shows that H^{∗}(Y (j+1); 𝔽_{p}) ≅ 𝔽_{p}[a_{j+1}]⊗E(b_{j+1}) is a P E–algebra of type (2, 3) with the relation β_{j+1}(a_{j+1}) = b_{j+1}; i.e. H^{∗}(Y (j + 1); 𝔽_{p}) is isomorphic to either A_{1}^{(j)} or B_{0,1}^{(j)}. The relation on the Bockstein follows from the fact H_{2}(Y (j + 1); ℤ) ≅ π_{2}(Y (j + 1)) ≅ π_{2}(B^{2}ℤ/p^{j+1}) ≅ ℤ/p^{j+1}. This finishes the induction step.

The construction of Y (1) does not fit into this picture, but it is done in the obvious way by starting with a map Bℤ/p → Y (0) which also is algebraically given as in proposition 7.4.

We can continue as long as P^{1}(b_{j}) = 0. Let us first assume that we can construct only a finite sequence of spaces and let Y (k + 1) denote the last space. Then H^{∗}(Y (k+1); 𝔽_{p}) ≅ 𝔽_{p}[x]⊗E(y) is a P E-algebra of type (2, 3) such that β_{(k+1)}(x) = y and P^{1}(y) ≠ 0. Since P^{2}(y) = 0 by unstability, the only possibility for P^{1}(y) is P^{1}(y) = x^{p−1}y and so H^{∗}(Y (k + 1); 𝔽_{p}) ≅ A_{1}^{(k)}. This implies that Y (k + 1) ≃ X_{k} (theorem 6.1) and Y = Y (0) ≃ Y_{k} = Y_{k,1} (corollary 7.5).

If the sequence is infinite, we define Y (∞) := hocolim Y (j). In the Milnor sequence

1 ⟶ \underleftarrow{lim}^{1}H^{∗+1}(Y (j); 𝔽_{p}) ⟶ H^{∗}(Y (∞); 𝔽_{p}) ⟶ \underleftarrow{lim} H^{∗}(Y (j); 𝔽_{p}) ⟶ 1 ,

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

\underleftarrow{lim} H^{∗}(Y (j); 𝔽_{p}) ≅ H^{∗}(S^{3}; 𝔽_{p}).

Hence Y (∞)_{p} ≃ S^{3}_{p}. Let F be the homotopy fibre of Y (0) → Y (∞). Since the direct limit of a directed system of fibrations is again a fibration, the fibration F → Y (0) → Y (∞) is the direct limit of the fibrations

Bℤ/p^{j} → Y (0) → Y (j).

Hence, F ≃ \underrightarrow{lim}_{j} Bℤ/p^{j} = Bℤ_{p^{∞}} and by taking the p-completion we obtain a fibration

BS^{1}_{p} ⟶ Y (0) ⟶ S^{3}_{p}

classified by a map S^{3}_{p} ⟶ B Aut(BS^{1}_{p}) into the classifying space of the monoid of self homotopy equivalences of BS^{1}_{p} ([23]), which lifts to a map

S^{3}_{p} ⟶ BS Aut(BS^{1}_{p}) ≃ B^{2}S^{1}_{p} ≃ K(π, 3) .

where S Aut denotes the self homotopy equivalences which are homotopic to the identity. This map is classified by degree and the p adic units are the only possible ones that produce the right cohomology of Y (0) and therefore Y = Y (0) ≃ Y_{∞}.

Now let Y′ be the p-completion of a space realizing B_{1,r}. Let f : Bℤ/p ⟶ Y′ be a realization of the composition B_{1,r} ⟶ B_{1,1} ⟶ H^{∗}(Bℤ/p; 𝔽_{p}). By the computation of the T functor in theorem 3.1 and the results of [17], map(Bℤ/p, Y′)_{f} is a realization of B_{1,1}. Thus, there exists an equivalence h : map(Bℤ/p, Y′)_{f} ≃ Y_{k} for some 0 ≤ k ≤ ∞. The space map(Bℤ/p, Y′)_{f} inherits a ℤ/r-action from the ℤ/r-action on Bℤ/p. The component of f is fixed under this action because H^{∗}(Y′; 𝔽_{p}) ≅ B_{1,r}. h induces an equivariant map in H^{∗}( ; 𝔽_{p}) because the canonical map Bℤ/p → map(Bℤ/p, Y′)_{f} is equivariant. By lemma 8.3 below, we can replace h by an equivariant equivalence. Taking homotopy orbits gives equivalences

Y′ ≃ (Eℤ/r ×_{ℤ/r} map(Bℤ/p, Y′)_{f})_{p} ≃ (Eℤ/r ×_{ℤ/r} Y_{k})_{p} ≃ Y_{k,r}.

This finishes the proof of part (1).

To prove (2), let Z be the p-completion of a space X realizing the algebra B_{0,1}. The homotopy fibre of the classifying map x : Z ⟶ B^{2}ℤ/p of the 2-dimensional class x is a realization of B_{1,1} and hence, equivalent to some Y_{k}, k ≥ 1. By corollary 7.5 it follows that Z ≃ Z_{k}.

If the p-complete space Z′ realizes the algebra B_{0,r}, we can proceed as in the case of B_{1,r}. We have an equivalence map(Bℤ/p, Z′)_{f} ≃ Z_{k} for some k and for a suitable map f : Bℤ/p ⟶ Z′. Now all the above arguments go through with minor changes. This shows that Z′ ≃ Z_{k,r} and finishes the proof. □

Lemma 8.3. (1) Let Y be a space equipped with a ℤ/r action, and let h : Y → Y_{k} be an equivalence, such that H^{∗}(h; 𝔽_{p}) is equivariant. Then, there exists an equivalence h′ : Y → Y_{k}, which is equivariant.

(2) Let Z be a space equipped with a ℤ/r action, and let h : Z → Z_{k} be an equivalence, such that H^{∗}(h; 𝔽_{p}) is equivariant. Then, there exists an equivalence h′ : Z → Z_{k}, which is equivariant.

Proof. We only prove (1), the proof of (2) is analogous. There exists a map g : Bℤ/p^{k} ⟶ Y and an equivalence f : Y ≃ hofib(a : Bor(Y, g) → B^{2}ℤ/p^{k}), where hofib denotes the homotopy fibre. Analogously to the construction of the spaces Y_{k,r}, the ℤ/r action on Y passes to Bor(Y, f) and the map a is equivariant up to homotopy. ℤ/r acts on ℤ/p^{k} via the inclusion ℤ/r ⊂ ℤ/p−1 ⊂ ℤ̂_{p}^{∗}. By [27] (see lemma 6.5), we can replace a by an equivariant map. This induces a ℤ/r-action on the homotopy fibre. The homotopy equivalence f is equivariant up to homotopy. This construction can be applied to Y_{k} as well.

The Borel construction yields an equivalence Bor(Y, g) ≃ Bor(Y_{k}, g_{0,k}) which is also equivariant in mod-p cohomology. Both spaces are equivalent to X_{k}. As in lemma 6.5 we can replace this equivalence by an equivariant equivalence. Taking homotopy fibers produces an equivalence Y → Y_{k}, which is equivariant up to homotopy. Again, the Wojtkowiak argument establishes an equivalence Y → Y_{k} which is equivariant. □

9. Homotopy properties of the constructed spaces.

In [9], for any map f : A → B between spaces, Dror Farjoun constructed a localisation functor

L_{f} : Spaces → Spaces .

Here, Spaces means the category of topological spaces, of CW –complexes, or the simplicial category. In this section we will, among other things, compute the value of this functor when applied to some of the spaces constructed in the previous sections, in the particular case in which f is the map Bℤ/p → ∗. The functor L_{f} is coaugmented, homotopically idempotent, and takes values among the f–local spaces. A space Y is called f–local, if the map

f^{∗}: map(B, Y) → map(A, Y)

is a homotopy equivalence. Moreover, the coaugmentation l : X → L_{f}X into the localisation L_{f}X is homotopically universal, i.e. for any map X → Z into a f–local space Z, there exists a map L_{f}X → Z, unique up to homotopy, such that

\begin{matrix}X & & X \\ ^{l}↓ & & ↓ \\ L_{f}X & ⟶ & Z \end{matrix}

commutes up to homotopy. Actually, l induces a homotopy equivalence

l^{∗}: map(L_{f}X, Z) ⟶^{≃} map(X, Z)

for any f-local space Z. Such functors satisfy several properties by general nonsense arguments; e.g we have Lemma 9.1. Let g : X → Y be a map between spaces, then the following statements are equivalent:

(1) g induces a homotopy equivalence L_{f}X ≃ L_{f}Y.

(2) For any f-local space Z, g^{∗}: [Y, Z] ⟶ [X, Z] is a bijection.

(3) For any f-local space Z, g^{∗}: map(Y, Z) ⟶ map(X, Z) is a homotopy equivalence. □

Lemma 9.2. (1) For any small category C and for any functor F : C → Spaces,

L_{f}(hocolim_{C} F) ≃ L_{f}(hocolim_{C}(L_{f} ∘ F)) .

(2) The homotopy inverse limit over any small category of f-local spaces is f-local.

Proof. For an f-local space Z, the map map(hocolim_{C} L_{f}∘F, Z) ⟶ map(hocolim_{C} F, Z) can be factored as

map(hocolim_{C} L_{f} ∘ F, Z) ≃ holim_{C} map(L_{f} ∘ F, Z) ≃ ≃ holim_{C} map(F, Z) ≃ map(hocolim_{C} F, Z).

Then, (1) is a consequence of 9.1. The proof of statement (2) is similar. □

We will also use some facts about L_{f} that we quote from [11]:

Lemma 9.3. Assume that F ⟶ E ⟶ B is a fibration.

(1) If L_{f}F ≃ ∗ then L_{f}E ≃ L_{f}B.

(2) For f: W → ∗, if B is f-local, then L_{f} preserves the fibration. □

For a space W, we denote the localisation with respect to the map W → ∗ by L_{W}. Then, a space X is W -local if and only if map(W, X) ≃ X or equivalently, for X connected, if and only if map_{∗}(W, X) ≃ ∗. In this section we are interested in the localisation with respect to Bℤ/p.

Some elementary calculations of L_{Bℤ/p} are provided by the next two results.

Lemma 9.4. Let π denote a discrete group, (1) K(π, 1) is Bℤ/p-local if and only if π is p-torsion free.

(2) K(π, n) is Bℤ/p-local for all n ≥ 1 if and only if π is a uniquely p-divisible abelian.

(3) If π is a p-group then L_{Bℤ/p}(K(π, n)) ≃ ∗ if n > 1 or π is finite.

Proof. In general a direct computation of homotopy groups shows that the connected component containing the constant map of map(Bℤ/p, K(π, 1)) is homotopy equivalent to K(π, 1). Now the set of components of map(Bℤ/p, K(π, 1)) is Rep(ℤ/p, π), hence there is a unique component if and only if π is p-torsion free. This proves (1).

¿From a computation of homotopy groups it follows that K(π, n), n ≥ 2 is Bℤ/plocal if and only if H^{r}(ℤ/p; π) = 0 for 1 ≤ r ≤ n, that is, if and only if π is uniquely p divisible. This is (2).

Finally we prove (3). Clearly L_{Bℤ/p}(Bℤ/p) ≃ ∗, then we use induction on the order of π and n in order to get the result for any finite p-group. A general p-group is direct limit of its finite subgroups, hence the result follows by 9.2(1). □

Remark 9.5. The following explicit calculations will be useful later.

(1) For any n ≥ 2, L_{Bℤ/p}K(ℤ, n) ≃ K(ℤ[\frac{1}{p}], n), and

(2) For any n ≥ 2, L_{Bℤ/p}K(ℤ̂_{p}, n) ≃ K(ℚ̂_{p}, n).

This is computed using the exact sequences 0 → ℤ → ℤ[\frac{1}{p}] → ℤ/p^{∞} → 0 and 0 → ℤ̂_{p} → ℚ̂_{p} → ℤ/p^{∞} → 0 and then applying 9.3 and 9.4.

Examples of Bℤ/p-local spaces are provided by the Sullivan conjecture:

Lemma 9.6.

(1) Any finite CW-complex is Bℤ/p-local. (This is the Sullivan conjecture: [19].)

(2) Let X be a connected nilpotent space with H^{∗}(X; 𝔽_{p}) of finite type, then X is Bℤ/p-local if and only if H^{∗}(X; 𝔽_{p}) is locally finite as module over the Steenrod algebra. ([18]) □

The next lemma lists several properties of Bℤ/p-local spaces.

Lemma 9.7.

(1) If Z is Bℤ/p-local, then Z is Bℤ/p^{k}-local for every 1 ≤ k ≤ ∞. If, in addition, Z is p-complete, then Z is also BS^{1}-local.

(2) Let Z be a connected space and Z̃ → Z the universal covering. If Z is Bℤ/p–local, then Z̃ is also Bℤ/p–local. Reciprocally, if π_{1}(Z) is p-torsion free and Z̃ is Bℤ/p-local, then Z is also Bℤ/p-local.

The proof is based on the following lemma of Zabrodsky [28] (see also [19]).

Lemma 9.8. Let G be a toplogical group and G → E → B a principal fibration. If, for a space X, map(G, X)_{const} ≃ X, then

map(B, X) ≃ map(E, X)_{f|_{G}≃const} .

The mapping space map(E, X)_{f|_{G}≃const} consists of the components of all maps f : E → X, whose restriction f|_{G} is homotopic to the constant map.

Proof of lemma 9.7. (1). The principal fibration Bℤ/p → Bℤ/p^{k+1} → Bℤ/p^{k}, the Zabrodsky lemma and an induction prove (1) for k < ∞. Moreover, for a Bℤ/p–local space Z, the canonical map map(Bℤ/p^{k+1}, Z) → map(Bℤ/p^{k}, Z) is a homotopy equivalence. Therefore,

map(Bℤ/p^{∞}, Z) ≃ \underleftarrow{lim}_{k} map(Bℤ/p^{k}, Z) ≃ \underleftarrow{lim}_{k} Z ≃ Z ,

and Z is Bℤ/p^{∞}–local. If Z is also p–complete map(BS^{1}, Z) ≃ map(Bℤ/p^{∞}, Z) ≃ Z, which shows that Z is BS^{1}–local and finishes the proof of (1).

(2). Assume that Z is connected and Bℤ/p-local. Let Z → K := K(π_{1}(Z), 1) be the classifying map of the universal covering Z̃ → Z. Applying the functor map(Bℤ/p, ) establishes a commutative diagram of fibrations

map(Bℤ/p, Z̃) ⟶ map(Bℤ/p, Z) ⟶ map(Bℤ/p, K)_{const}
e_{Z}↓  ^{e_{Z}} ↓  ^{e_{K}} ↓
Z̃  ⟶  Z  ⟶  K .

The vertical arrows are given by the evaluation. In the upper middle term we do not have to consider particular components, because Z is Bℤ/p local; i.e. there is only the component of the constant map and e_{Z} is a homotopy equivalence. Since e_{K} is also a homotopy equivalence, this is also true for eZ̃. That is to say that Z̃ is Bℤ/p-local.

Finally, if π_{1}(Z) is p-torsion free, Bπ_{1}(Z) is Bℤ/p-local by 9.4(1) and then, according to 9.3(2) Z is Bℤ/p-local if and only if Z̃ is Bℤ/p-local. This finishes the proof of the second statement. □

Now we are prepared to start with the calculation of the localisations of the spaces we constructed in the previous sections.

The next result is actually a particular case of a more general result of Neisendorfer.

Lemma 9.9. S^{3}, L_{Bℤ/p}S^{3}⟨3⟩ and L_{Bℤ/p}(S^{3}⟨3⟩_{p}) are homotopy equivalent after completion.

Proof. By lemma 9.6, S^{3} and S^{3}_{p} are Bℤ/p-local. So, by lemma 9.3 L_{Bℤ/p} preserves the fibration BS^{1} → S^{3}⟨3⟩ → S^{3} as well as its p-completion, hence, by lemma 9.5 we obtain fibrations:

K(ℤ[\frac{1}{p}], 2) → L_{Bℤ/p}S^{3}⟨3⟩ → S^{3}

and

K(ℚ̂_{p}, 2) → L_{Bℤ/p}(S^{3}⟨3⟩_{p}) → S^{3}_{p}

and the p-completion of those gives the result. □

Recall that in section 5 the space E_{k} was defined as the total space of certain fibration

BS^{1}_{p} → E_{k} → S^{1}_{p}

that has a section s: S^{1}_{p} → E_{k}. Then E′_{k} is the homotopy cofibre of this section. Finally, X_{k} was defined as the p completion of E′_{k}. For the localization of that spaces we obtain:

Lemma 9.10. (1) (L_{Bℤ/p}E_{k})_{p} ≃ S^{1}_{p}.

(2) (L_{Bℤ/p}E′_{k})_{p} ≃ ∗.

(3) L_{Bℤ/p}X_{k} ≃ ∗.

Proof. Since S^{1}_{p} is Bℤ/p-local, L_{Bℤ/p} preserves the above fibration and we obtain a fibration

K(ℚ̂_{p}, 2) → L_{Bℤ/p}E_{k} → S^{1}_{p} .

The p-completion of this fibration proves (1).

This fibration has also a section s: S^{1}_{p} → L_{Bℤ/p}E_{k}. Let C be the homotopy cofibre of this section. Then C is simply connected and mod p acyclic, hence Bℤ/plocal by lemma 9.6. Now, since a homotopy cofibre is a special sort of homotopy colimit, by 9.2 we obtain

L_{Bℤ/p}E′_{k} ≃ L_{Bℤ/p}C ≃ C

and therefore (L_{Bℤ/p}E′_{k})_{p} ≃ ∗. This is statement (2).

According to the next lemma, (3) follows from (2) because X_{k} is 1-connected and H^{i}(X_{k}; ℤ̂_{p}) is finite for all i > 0 (see 5.5 and 5.8.) □

Lemma 9.11. Let X be a space for which X̂_{p} is 1-connected. If H^{i}(X̂_{p}; ℤ̂_{p}) is finite for all i > 0 and (L_{Bℤ/p}X)_{p} ≃ ∗, then L_{Bℤ/p}(X̂_{p}) ≃ ∗.

Proof. We want to show that for any Bℤ/p-local space Z, any map X̂_{p} → Z factors through a point. that is [X̂_{p}, Z] = ∗ for any connected Bℤ/p-local space Z.

If Z is Bℤ/p-local, the universal covering Z̃, is also Bℤ/p-local and since X̂_{p} is 1-connected

[X̂_{p}, Z] ≅ [X̂_{p}, Z̃]/π_{1}(Z)

hence it is enough to show that [X̂_{p}, Z] = ∗ for all Z which is 1-connected and Bℤ/p-local.

If Z is 1-connected and Bℤ/p-local, so is Ẑ_{p}. Since H^{i}(X̂_{p}; ℤ̂_{p}) if finite for all i > 0 and X̂_{p} is 1-connected, 5.7 implies that the homotopy groups of X̂_{p} are finite p-groups. Then, the arithmetic fracture lemma shows that [X̂_{p}, Ẑ_{p}] ≅ [X̂_{p}, Z] and

[X̂_{p}, Z] ≅ [X̂_{p}, Ẑ_{p}] ≅ [L_{Bℤ/p}X, Ẑ_{p}] ≅ [(L_{Bℤ/p}X)_{p}, Ẑ_{p}] ≅ [∗, Ẑ_{p}] = ∗ □

The spaces Y_{k}(j) for k ≥ 1 and 0 ≤ j ≤ k + 1, were constructed out of X_{k} in section 7 in such a way that they fit in sequences of fibrations with fibre Bℤ/p:

Y_{k}(0) → Y_{k}(1) → … → Y_{k}(k + 1) = X_{k}.

Here Y_{k} = Y_{k}(0) is the kth fake S^{3}⟨3⟩ and Y_{∞} = S^{3}⟨3⟩_{p} also fits in one such sequence Y_{∞} → Y_{∞}(1) → Y_{∞}(2) → … with Y_{∞}(∞) = hocolim_{j} Y_{∞}(j) and Y_{∞}(∞)_{p} ≃ S^{3}_{p}.

Theorem 9.12. For all 0 ≤ j ≤ k + 1 ≤ ∞,

L_{Bℤ/p}Y_{k}(j) ≃ L_{Bℤ/p}Y_{k}(0) = L_{Bℤ/p}Y_{k} ≃ \{ \begin{matrix}Y_{∞}(∞) & for k = ∞ \\ ∗ & for k < ∞ \end{matrix}

Remark. Compare with 9.9 for k = ∞.

Proof. The principal fibrations Bℤ/p → Y_{k}(j) → Y_{k}(j +1) and lemma 9.2 establish equivalences L_{Bℤ/p}Y_{k}(j) ≃ L_{Bℤ/p}Y_{k}(j + 1).

Therefore, if k < ∞ we have L_{Bℤ/p}Y_{k}(j) ≃ L_{Bℤ/p}X_{k} ≃ ∗ by 9.10.

For k = ∞ we have first that Y_{∞}(∞) is 1-connected and its mod p cohomology is finite hence it is Bℤ/p-local by 9.6(2). Then the result follows from 9.1 because for any Bℤ/p-local space Z the map Y_{∞}(j) → hocolim_{j} Y_{∞}(j) = Y_{∞}(∞) induces

map(Y_{∞}(∞), Z) ⟶^{≃} holim_{j} map(Y_{∞}(j), Z) ⟶^{≃}

⟶^{≃} holim_{j} map(L_{Bℤ/p}Y_{∞}(j), Z) ⟶^{≃} map(L_{Bℤ/p}Y_{∞}(j), Z) . □

Corollary 9.13. If k ≠ ∞, the l-fold suspensions of Y_{∞}(j) and of Y_{k}(j) are not homotopy equivalent for all l.

Proof. By theorem 9.12, Σ^{l}L_{Bℤ/p}Y_{∞}(j) ≃ Σ^{l}Y_{∞}(∞) and this is 1-connected with finite mod p cohomology hence Bℤ/p-local. The suspensions are homotopy colimits. By lemma 9.2 we have

L_{Bℤ/p}Σ^{l}Y_{k}(j) ≃ L_{Bℤ/p}Σ^{l}L_{Bℤ/p}Y_{k}(j) ≃ L_{Bℤ/p}Σ^{l}Y_{∞}(∞) ≃ Σ^{l}Y_{∞}(∞) .

But for k < ∞, the same argument proves (L_{Bℤ/p}Σ^{l}Y_{k}(j)) ≃ ∗. □

Similiar arguments as in the proof of theorem 9.5 show that L_{Bℤ/p}(Y_{∞}(j) ×_{ℤ/r} Eℤ/r) ≃ Y_{∞}(∞) ×_{ℤ/r} Eℤ/r, where ℤ/r acts canonically on Y_{∞}(∞). This space is mod–p acyclic and using lemma 9.11 we deduce that L_{Bℤ/p}Y_{∞,r} ≃ ∗. For k < ∞, one also can prove that L_{Bℤ/p}Y_{k,r} ≃ ∗. Thus, the above application of the localisation functor does not see any difference between the suspensions of the different realisations of B_{i,r}. But there is a way to distinguish between these spaces.

Let g : Bℤ/p → Y_{k,r} be the map of section 7, and let h : Y_{k} × Bℤ/p → Y_{k,r} be the adjoint of the equivalence Y_{k} ≃ map(Bℤ/p, Y_{k,r})_{g}. ℤ/(p − 1) acts on Y_{k}. For every a ∈ ℤ/p^{∗} we have a map

Σ^{l}Y_{k} × Bℤ/p ⟶^{a×id} Σ^{l}Y_{k} × Bℤ/p ⟶ Σ^{l}(Y_{k} × Bℤ/p) ⟶^{Σ^{l}h} Σ^{l}Y_{k,r} .

which has as adjoint a map

f_{a} : Σ^{l}Y_{k} ⟶ map(Bℤ/p, Σ^{l}Y_{k,r}) .

If a and b differ by a r-th power, the two associated maps f_{a} and f_{b} are homotopic, because Y_{k,r} is the homotopy orbit of the ℤ/r–action on Y_{k}. There is also an obvious map Σ^{l}Y_{k,r} → map(Bℤ/p, Σ^{l}Y_{k,r}), which is the standard section of the evaluation.

Let s = (p − 1)/r. Then, ℤ/s ⊂ ℤ/p^{∗} consists of the congruence classes modulo r-th powers. All these maps together fit into a map

f : Σ^{l}Y_{k,r} ∨ ⋁_{a∈ℤ/s} Σ^{l}Y_{k} → map(Bℤ/p, Σ^{l}Y_{k,r}) .

Theorem 9.14. The map

f : Σ^{l}Y_{k,r} ∨ ⋁_{a∈ℤ/s} Σ^{l}Y_{k} → map(Bℤ/p, Σ^{l}Y_{k,r})

is a homotopy equivalence.

Proof. We have to calculate the mapping space using the T –functor The T –functor is exact and commutes with suspensions. Hence,

T (H^{∗}(Σ^{l}Y_{k,r}; 𝔽_{p})) ≅ T (H̃^{∗}(Σ^{l}Y_{k,r}; 𝔽_{p}) ⊕ T (ℤ/p)

≅ Σ^{l}T (H^{∗}(Σ^{l}Y_{k,r}; 𝔽_{p})) ⊕ ℤ/p

≅ ( ⨁_{a∈ℤ/s} Σ^{l}H^{∗}(Y_{k}; 𝔽_{p})) ⊕ Σ^{l}H^{∗}(Y_{k,r}; 𝔽_{p}) ⊕ ℤ/p

≅ H^{∗}(( ⋁_{a∈ℤ/s} Σ^{l}Y_{k}) ∨ Σ^{l}Y_{k,r}; 𝔽_{p}) .

All the isomorphism are obvious, but the second last. This one follows from theorem 3.1, the identity T_{const}H^{∗}(Y_{k,r}; ℤ/p) ≅ H^{∗}(Y_{k,r}; ℤ/p), the fact that every map H^{∗}(Y_{k,r}; 𝔽_{p}) → H^{∗}(Bℤ/p; 𝔽_{p}) factors over H^{∗}(Y_{k,}; ℤ/p) and that every two factorizations differ by an r-th power. By construction, this series of isomorphisms is just the map induced by f, which shows that f is a mod–p equivalence [17]. The integral homology of (⋁_{a∈ℤ/s} Σ^{l}Y_{k}) ∨ Σ^{l}Y_{k,r} consists of finite p-torsion in each dimension. Moreover, the space is 3-conected. The mod C Hurewicz theorem for the class of finite p-groups shows that π_{∗}((⋁_{a∈ℤ/s} Σ^{l}Y_{k}) ∨ Σ^{l}Y_{k,r}) consists of finite p-torsion in each dimension. Hence, by [6] (⋁_{a∈ℤ/s} Σ^{l}Y_{k}) ∨ Σ^{l}Y_{k,r} is p-complete. Because Y_{k,r} is p–complete, f is an equivalence ([17]). □

Corollary 9.15. If k ≠ ∞, the l–fold suspensions Σ^{l}Y_{∞,r} and Σ^{l}Y_{k,r} are not homotopy equivalent for all l.

Proof. Applying localisation to the mapping space gives

L_{Bℤ/p}(map(Bℤ/p, Σ^{l}Y_{k,r})) ≃ L_{Bℤ/p}(Σ^{l}Y_{k,r} ∨ ⋁_{a∈ℤ/s} Σ^{l}Y_{k})

≃ L_{Bℤ/p}(L_{Bℤ/p}(Σ^{l}Y_{k,r}) ∨ ⋁_{a∈ℤ/s} L_{Bℤ/p}(Σ^{l}Y_{k}))

≃ L_{Bℤ/p}(Σ^{l}L_{Bℤ/p}(Y_{k,r}) ∨ ⋁_{a∈ℤ/s} Σ^{l}L_{Bℤ/p}(Y_{k})) .

The last two equivalences follow from lemma 9.1 and lemma 9.2. By proposition 9.13 and the following remarks, for k = ∞ and k < ∞, these spaces cannot be homotopy equivalent. □

Remark 9.16. Using the same methods and ideas, one can also distinguish between the l–fold suspensions of Z_{k,r}, k < ∞, and Z_{∞,r}.

Finally, we discuss the question of which of these spaces are H–spaces. The spaces Y_{∞} = S^{3}⟨3⟩_{p} and Z_{∞} = Y_{∞}(1) are loop spaces, in particuliar H-spaces. This follows easily from the construction. But these are the only ones among the spaces Y_{k,r} and Z_{k,r} which are H-spaces, as the following proposition shows.

Proposition 9.17. For k < ∞ or r > 1, the spaces Y_{k,r} and Z_{k,r} cannot carry an H-space structure.

Proof. If the algebras B_{0,r} and B_{1,r} have the structure of a Hopf algebra, they are primitively generated. For r > 1, the Steenrod power P^{p^{i}}, i = 0, 1, maps the primitive 2p^{i}r-dimensional class on a nonprimitive class. Hence, B_{0,r} and B_{1,r} are Hopf algebras only for r = 1.

Now, we assume that Y_{k}(j), 0 ≤ j ≤ k, is an H-space. We consider the diagram

(3)  \begin{matrix}Bℤ/p × Bℤ/p & ⟶^{μ} & Bℤ/p \\ ^{g_{j,1}×g_{j,1}}↓ & & ^{g_{j,1}}↓ \\ Y_{k}(j) × Y_{k}(j) & ⟶^{μ} & Y_{k}(j) \\ ↓ & & ^{f_{j,j+1}}↓ \\ Y_{k}(j + 1) × Y_{k}(j + 1) & & Y_{k}(j + 1) \end{matrix}

where μ denotes the multiplication. The upper square commutes in mod-p cohomology, because in H^{∗}(Y_{k}(j); 𝔽_{p}) the 2-dimensional class for j ≥ 1 and the 2pdimensional class for j = 0 are primitive. Thus, the upper square commutes up to homotopy. The obvious composition f_{j,j+1}μ(g_{j,1} ×g_{j,1}) : Bℤ/p×Bℤ/p → Y_{k}(j +1) is homotpic to the constant map and, by theorem 3.1 and taking the adjoint,

map(Bℤ/p × Bℤ/p, Y_{k}(j + 1))_{c} ≃ map(Bℤ/p, map(Bℤ/p, Y_{k}(j + 1))_{c})_{c}
≃ map(Bℤ/p, Y_{k}(j + 1))_{c}
≃ Y_{k}(j + 1).

Both vertical columns in (3) are principal fibrations. We can apply lemma 9.3, which establishes a map

μ : Y_{k}(j + 1) × Y_{k}(j + 1) → Y_{k}(j + 1)

making the lower square commutative up to homotopy. As easily shown, μ is an H-space structure on Y_{k}(j + 1).

If Y_{k} = Y_{k}(0) is an H-space, the above induction procedure shows that Y_{k}(k + 1) = X_{k} carries also an H-space structure. But this is a contradiction, because the Steenrod power P^{1} maps the ‘primitive’ 3-dimensional class of H^{∗}(X_{k}; 𝔽_{p}) onto a nonprimitive class. □

10. Appendix. Through this paper we have introduced several families of algebras over the Steenrod algebra A_{r}, B_{i,r}, C_{r}, etc. as well as several families of spaces E_{k}(r), X_{k}(r), Y_{k,r}, etc. We think that the reader would find helpful to have the definitions of these algebras and spaces and the relationships between them displayed in a set of tables. In this appendix we include the following tables: Table 10.1 contains the definitions of the algebras over the Steenrod algebra introduced in section 2. Table 10.2 contains the definitions and the mod p cohomology of the spaces introduced in sections 5 and 7. In table 10.3 we list some fibrations between these spaces. Tables 10.4 and 10.5 display the spaces used in the proof of theorem 8.2 and their cohomology algebras.

A_{r} = A_{r}^{(0)} = 𝔽_{p}[x] ⊗ E(y)  deg(x) = 2r  β(x) = y
deg(y) = 2r + 1  P^{1}(y) = rx^{s}y
B_{i,r} = B_{i,r}^{(0)} = 𝔽_{p}[x] ⊗ E(y)  deg(x) = 2p^{i}r  β(x) = y
deg(y) = 2p^{i}r + 1  P^{p^{i}}(y) = (r − 1)x^{s}y
C_{r} = C_{r}^{(0)} = 𝔽_{p}[x] ⊗ E(z) deg(x) = 2r β(x) = xz

deg(z) = 1

A′_{r}  same as A_{r} but with β(x) = β(y) = 0
B′_{i,r} same as B_{i,r} but with β(x) = β(y) = 0
C′_{r}  same as C_{r} but with β(x) = β(z) = 0
A_{r}^{(k)} equal to A′_{r} as algebras over the Steenrod algebra but with β_{(j)}(x) = 0
for j ≤ k and β_{(k+1)}(x) = y
B_{i,r}^{(k)} equal to B′_{i,r} as algebras over the Steenrod algebra but with β_{(j)}(x) = 0
for j ≤ k and β_{(k+1)}(x) = y
C_{r}^{(k)} equal to C′_{r} as algebras over the Steenrod algebra but with β_{(j)}(x) = 0
for j ≤ k and β_{(k+1)}(x) = xz

Table 10.1

HOMOTOPY CLASSIFICATION 39

E_{k} = EG ×_{π} B^{2}π^{ϕ_{k}}, k ≥ 0  H^{∗}(E_{k}; 𝔽_{p}) ≅ C_{1}^{(k)}, (5.1)
E E_{k}(r) = E_{k}/ℤ/r, k ≥ 0, r|p − 1 H^{∗}(E_{k}(r); 𝔽_{p}) ≅ C_{r}^{(k)}, (5.3)
E′_{k}(r) = Cofibre (Bπ → E_{k}(r)), k ≥ 0, r|p − 1  H^{∗}(E′_{k}(r); 𝔽_{p}) ≅ A_{r}^{(k)}

X_{k}(r) = (E′_{k}(r))_{p}, k ≥ 0, r|p − 1

X X_{k} = X_{k}(1), k ≥ 0 H^{∗}(X_{k}(r); 𝔽_{p}) ≅ A_{r}^{(k)}, (5.5)

X_{∞} ≅ (S^{3})_{p}

Y_{k} = Fibre (X_{k} → B^{2}ℤ/p^{k+1}), k ≥ 0

Y_{∞} = (S^{3}⟨3⟩)_{p}  H^{∗}(Y_{k,r}; 𝔽_{p}) ≅ B_{1,r}, (7.1)

Y_{k,r} = (Eℤ/r ×_{ℤ/r} Y_{k}))_{p}, 0 ≤ k ≤ ∞, r|p − 1

Y  \begin{matrix}Y_{k}(j) = Fibre (X_{k} → B^{2}ℤ/p^{k+1−j}) \\ k ≥ 0, 0 ≤ j ≤ k + 1 \end{matrix}  H^{∗}(Y_{k}(j); 𝔽_{p}) ≅ \{ \begin{matrix}B_{1,1}, & j = 0 \\ B_{0,1}, & j = 1 \\ B′_{0,1}, & 2 ≤ j ≤ k \\ A′_{1}, & j = k + 1 \end{matrix}
Y_{∞}(j) = Fibre (S^{3} ⟶^{p^{j}} (B^{2}S^{1})_{p}), j ≥ 0  (7.4)

Y_{k}(0) = Y_{k,1} = Y_{k} Y_{k}(k + 1) = X_{k}

Z Z_{k,r} = (Eℤ/r ×_{ℤ/r} Y_{k}(1))_{p}, 0 < k ≤ ∞, r|p − 1 H^{∗}(Z_{k,r}; 𝔽_{p}) ≅ B_{0,r}, (7.7)

Z_{k} = Z_{k,1} = Y_{k}(1), 0 < k ≤ ∞

Table 10.2 B^{2}π → E_{k} → Bπ E_{k} → E_{k}(r) → Bℤ/r Y_{k} → X_{k} → B^{2}ℤ/p^{k+1} Y_{k} → Z_{k} → B^{2}ℤ/p Z_{k} → X_{k} → B^{2}ℤ/p^{k} Y_{k}(j) → Y_{k}(j + l) → B^{2}ℤ/p^{l}, l ≤ k − j + 1 Y_{k}(j) → X_{k} → B^{2}ℤ/p^{k+1−j}, j ≥ 1

Y_{∞} → X_{∞} → (B^{2}S^{1})_{p}

Table 10.3

S^{3}⟨3⟩_{p} = Y_{∞}→ Y_{∞}(1) = Z_{∞}→ Y_{∞}(2)  → Y_{∞}(3)  → ⋯  ⋯ → S^{3}_{p}
·
·
·
·
Y_{k} → Y_{k}(1) = Z_{k} → Y_{k}(2) → Y_{k}(3) → ⋯ → Y_{k}(k + 1) = X_{k}
·
·
·
Y_{3} → Y_{3}(1) = Z_{3} → Y_{3}(2)  → Y_{3}(3)  → Y_{3}(4) = X_{3}
Y_{2} → Y_{2}(1) = Z_{2} → Y_{2}(2)  → Y_{2}(3) = X_{2}
Y_{1} → Y_{1}(1) = Z_{1} → Y_{1}(2) = X_{1}
Y_{0} → Y_{0}(1) = X_{0}

Table 10.4

B_{1,1} B_{0,1} B_{0,1}^{(1)} B_{0,1}^{(2)} · · · · · · · · · · · ·
B_{1,1} B_{0,1} B_{0,1}^{(1)} B_{0,1}^{(2)} ⋯ A_{1}^{(k)} · · ·
B_{1,1} B_{0,1} B_{0,1}^{(1)} B_{0,1}^{(2)} A_{1}^{(3)}
B_{1,1} B_{0,1} B_{0,1}^{(1)} A_{1}^{(2)}
B_{1,1} B_{0,1} A_{1}^{(1)}
B_{1,1} A_{1}

Table 10.5

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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.