LOOP STRUCTURES ON HOMOTOPY FIBRES OF SELF MAPS OF A SPHERE

CARLOS BROTO AND RAN LEVI

1. Introduction

The problem of deciding whether or not a given topological space X is homotopy equivalent to a loop space is of classical interest in homotopy theory. Moreover, given that X is a loop space, one could ask whether the loop space structure is unique. A celebrated example of this is given in the classical work of Rector [23] where it is shown that the 3-sphere S^{3} admits infinitely many non-equivalent loop structures (see also [21]) and of Dwyer, Miller and Wilkerson [11], where the authors show that all such loop structures collapse to a single one if the sphere is localized at a prime p.

The first problem is equivalent to the question whether given a space X, there exist a space BX and a homotopy equivalence of ΩBX to X. The second problem is equivalent to asking whether the homotopy type of the space BX is uniquely determined by the requirement that its loop space is homotopy equivalent to X.

In this paper we study a family of spaces of classical interest in homotopy theory. Let S^{m}{d} denote the homotopy fibre of the degree d self map of S^{m}. Our aim is to study the possible loop structures supported by S^{m}{d} for various values of m and d. We restrict attention to the case where, d = 2^{r}. Our main theorem follows.

Theorem 1.1. Let m > 1 be an integer. The space S^{m}{2^{r}} is a loop space if and only if m = 3 and r > 2. Furthermore, for each value of r > 2, there is a unique loop space structure on S^{3}{2^{r}}.

The answer to the question, whether S^{n}{d} is a loop space for d odd has been known for a while. The uniqueness question for d odd is simpler than the case where d is a power of 2 and will be handled below. As a consequence we obtain a complete answer to the problems considered here for this family of spaces.

Corollary 1.2. Let m > 1 be an integer. Then the space S^{m}{d} is a loop space if and only if

1. d is odd, m = 2n − 1 and n|p − 1 for every prime p dividing d or

2. 8|d and m = 3. Moreover, S^{m}{d} admits at most one loop structure.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q52.

Key words and phrases. Classifying spaces, loop spaces, cohomology, Steenrod squares, Bockstein spectral sequence.

C. Broto is partially supported by DGICYT grant PB94-0725.

The techniques by which Theorem 1.1 is proved and the relations with various other topics may be of independent interest. Our problem turns out to be closely related to the study of the homotopy type of p-completed classifying spaces of certain finite groups. The tools we use are chosen respectively. Indeed we employ Lannes T -functor technology, homology decomposition methods and various other homotopy theoretic tools. Many of the arguments are reminiscent of techniques, successfully used in the study of p-compact groups and p-completed classifying spaces of finite groups. The paper is thus a combination of various theoretical elements which produce what we regard as a torsion analogue of the results of [11].

For r ≥ 3 there is a well known model for BS^{3}{2^{r}} given by the 2-completed classifying space for the special linear group SL_{2}(𝔽_{q}), where q is chosen so that 2^{r} is the order of the 2-Sylow subgroup. Furthermore, it turns out that the homotopy type of the loop space ΩBSL_{2}(𝔽_{q})_{2}^{∧} depends only on the mod-2 cohomology of BSL_{2}(𝔽_{q}) together with its Bockstein spectral sequence. Thus our problem amounts to showing that the specified cohomological information determines the homotopy type of BSL_{2}(𝔽_{q})_{2}^{∧}. Along the same lines we show that the homotopy type of BP SL_{2}(𝔽_{q})_{2}^{∧}, where P SL means the projective special linear group, is determined by its mod-2 cohomology.

An interesting aspect of our study involves showing non-realizability of certain spaces with a given cohomological structure, as an algebra over the Steenrod algebra and a specified Bockstein spectral sequence. From a purely algebraic point of view the structures assumed on these spaces are by all means compatible with each other. Thus we are lead to employ homotopy theoretic constructions to show that the Steenrod algebra structure and Bockstein spectral sequence cannot exist in the same space.

In [4] the authors study the question to what extent does cohomology inform of the homotopy type of BG, where G is a finite p-group. One of the examples studied there is of the quaternion groups Q_{2^{n}}. It is shown that for n ≤ 5 the homotopy type of BQ_{2^{n}} is determined by its cohomology. The question of whether or not the same holds for higher values of n is reduced to asking if there is a free action of the group Q_{16} on a classifying space for S^{3}{2^{r}} for an appropriate r. Thus one obtains a relation between the current project and what may appear to be an unrelated problem. The authors actually expect that the results here will imply that the homotopy type of BQ_{2^{n}} is determined by its mod-2 cohomology for every n. An account of this, where the problem is discussed in a more general setting is to appear elsewhere.

1.1. Reduction of the problem and the known cases. Some observations are immediate. First notice that the problem we are looking at only makes sense for m ≥ 2 as S^{1}{d} is homotopy equivalent to the cyclic group on d elements. Furthermore, for m ≥ 2, it is easy to compute the integral homology of S^{m}{d} using the Wang long exact sequence or equivalently the Serre spectral sequence for the fibration defining S^{m}{d}. Indeed the n-th homology of S^{m}{d} is isomorphic to the cyclic group ℤ/d if n is non-zero and divisible by m − 1 and is trivial otherwise. In particular all reduced homology groups of S^{m}{d} are torsion groups and thus by [3] there is a homotopy equivalence of S^{m}{d} to the product of all its p-primary localizations for primes p dividing d. It is easy to see that for a prime p the localization (or completion) of S^{m}{d} at p yields a space homotopy equivalent to S^{m}{p^{r}}, where r is the highest power of p dividing d. Hence there is a homotopy equivalence

S^{m}{d} ^{≃}^{}∏_{i=1}^{r} S^{m}{p_{i}^{j_{i}}}.

Moreover, by naturality of the localization functors it is clear that S^{m}{d} is a loop space if and only if each factor on the right hand side of the above equation is a loop space. Our problem is thus reduced to the study of possible loop structures on S^{m}{p^{j}}, where p is a prime number.

The answer to our first question for odd primes is known. Also at p = 2 there exist partial answers in the literature. We summarize the state of knowledge in the subject.

The case where p is an odd prime was treated by Cejtin and Kleinerman [6] and a bit more generally by Kono and Oshita [19]. In that case they show that S^{m}{p^{r}} is a loop space if and only if m = 2n−1 and n divides p−1. It is not hard to see (Lemma 2.1) that independently of the loop space structure on S^{m}{p^{r}} the cohomology of its classifying space is given as follows

H^{∗}(BS^{m}{p^{r}}; 𝔽_{p}) ≅ P [u_{m+1}] ⊗ E[x_{m}], (1)

with an r-th order Bockstein connecting the generators. With this observation one is able to conclude that the homotopy type of a space with cohomology as above is unique up to p-completion. A proof of this is included as the last section of the paper.

The case p = 2 is as usual more complicated. However if S^{m}{2^{r}} is a loop space then the mod-2 cohomology of its classifying space is still given by Equation (1). A work of Aguadé [1] puts strong restrictions on the possible values of m and r. It follows that if S^{m}{2^{r}} is a loop space then m = 3 and r > 1. As mentioned above, for r ≥ 3 there is a space which gives a classifying space for S^{3}{2^{r}}, the 2-completed classifying space BSL_{2}(𝔽_{q})_{2}^{∧}. The 2-Sylow subgroup of BSL_{2}(𝔽_{q}) is a generalized quaternion group of order depending on q. Moreover for every r ≥ 3 there exists a q, such that Q_{2^{r}} is the 2-Sylow subgroup of SL_{2}(𝔽_{q}). The mod-2 cohomology of BSL_{2}(𝔽_{q}) is given by Equation (1) with m = 3 (cf. [15]) and an observation of F. Cohen [8] is that ΩBSL_{2}(𝔽_{q})_{2}^{∧} is homotopy equivalent to S^{3}{2^{r}} for the appropriate r.

The remaining problems are whether S^{3}{4} is a loop space and moreover, whether the loop space structure on S^{3}{2^{r}} for r ≥ 3 and possibly on S^{3}{4} is unique. This is answered in full by Theorem 1.1.

1.2. Notation and Terminology. By the word space we mean a simplicial set or a topological space of the homotopy type of a CW complex. By p-completion we mean the Bousfield-Kan p-completion. Throughout the paper we use the phrase Lannes theory to mean the theory of Lannes T functors and the homotopy theory of mapping spaces out of BV, where V is an elementary abelian p-group. We shall assume the reader is familiar with [20] and omit elementary calculation respectively.

Let K denote the category of unstable algebras over the Steenrod algebra. By K_{β} we denote the category of unstable algebras together with a Bockstein spectral sequence. A precise definition of this category appears in [4]. By H_{β}^{∗}(−) we mean mod-p cohomology as an object of K_{β}.

We shall abbreviate the phrases Serre spectral sequence, Eilenberg-Moore spectral sequence and Bockstein spectral sequence by Sss, EMss and Bss respectively. We write P [x_{i_{1}}, x_{i_{2}}, ⋯] to denote a polynomial algebra on generators x_{i_{j}} and E[x_{i_{1}}, x_{i_{2}}, ⋯] for an exterior algebra. The subscript on a generator will normally indicate its degree, and may be omitted when no confusion can arise.

1.3. Organization of the paper and Acknowledgements. The paper is organized as follows. In section 2 we give a proof of our main theorem based on sections to follow. Sections 3 and 4 are technical and include cohomological calculations and homology decompositions that will be required for proving homotopy uniqueness of the spaces under consideration. In Sections 5 and 6 we prove that the homotopy type of BP SL_{2}(𝔽_{q})_{2}^{∧}, q odd, is determined by its cohomology as an object of K_{β}. In Section 7 we study certain spherical fibrations with a special property. The results of section 7 allow us to prove in Section 8 the incompatibility of some algebras over the Steenrod algebra with the given Bss. This is used in the proof of Theorem 1.1 to conclude homotopy uniquenes of BS^{3}{2^{j}}, j ≥ 3. Section 9 deals with homotopy uniquenes of BS^{2n−1}{p^{r}}, when it exists for p odd. Some background material on homological algebra is written in an appendix.

We wish to thank Centre de Recerca Matemàtica and Northwestern University for providing several opportunities for the authors to meet.

2. Proof of the main theorem

We prove theorem 1.1. The essential tools we use in the course of doing so will be developed in the following sections.

Our first lemma is well known and has already been mentioned in the introduction. It determines the unique possible structure of the mod-p cohomology of BS^{m}{p^{r}} if it exists.

Lemma 2.1. Suppose that S^{m}{p^{r}}, m > 1, is a loop space. Then

H^{∗}(BS^{m}{p^{r}}; 𝔽_{p}) ≅ P [u_{m+1}] ⊗ E[x_{m}],

as an algebra with β_{r}(x) = u. Furthermore, if X is any p-complete space with this cohomological structure then ΩX is homotopy equivalent to S^{m}{p^{r}}.

Proof. The first statement follows at once by inspection of the Serre spectral sequence for the path-loop fibration over BS^{m}{p^{r}}. For the second, notice that assuming pcompleteness, X is simply connected. Thus there is a map γ : (S^{m})_{p}^{∧} ^{}X, which induces an isomorphism in mod-p homology up to dimension m. The homotopy fibre of γ is easily seen to have the mod-p cohomology of an m-sphere and since it is pcomplete, it is equivalent to the p-completion of an m-sphere. Thus ΩX is equivalent to the fibre of a self map of (S^{m})_{p}^{∧}. One easily computes the degree of this self map to be precisely p^{r}, up to a p-adic unit, and the proof is complete.

We now restrict attention to the case where m = 3 and p = 2. From this point and on H^{∗}(−) will denote mod-2 cohomology. Thus if BS^{3}{2^{r}} exists then

H^{∗}(BS^{3}{2^{r}}) ≅ P [u_{4}] ⊗ E[x_{3}], with β_{r}(x) = u. (2)

Let X be a 2-complete space, whose cohomology agrees with the above. Then there is a morphism in K

α : H^{∗}(X) ^{}H^{∗}(Bℤ/2)

with α(x) = 0 and α(u) = z^{4}, where z ∈ H^{∗}(Bℤ/2) is the generator. By Lannes theory there exists a map g : Bℤ/2 ^{}X, inducing α on cohomology.

The objects H^{∗}(X) and H^{∗}(BSL_{2}(𝔽_{q})), for some q, are isomorphic in K. Moreover, α coincides with the the map induced on cohomology by the inclusion i of ℤ/2 as the center of SL_{2}(𝔽_{q}). Thus

T_{ℤ/2}(H^{∗}(X), g^{∗}) ≅ T_{ℤ/2}(H^{∗}(BSL_{2}(𝔽_{q})), Bi^{∗}) ≅ H^{∗}(BSL_{2}(𝔽_{q})).

Since the right hand side vanishes in dimension 1, it follows that the T functor computes the cohomology of the mapping space and so the evaluation map

Map(Bℤ/2, X)_{g} ^{}X

is a homotopy equivalence. Thus we have obtained a model of X with an action of Bℤ/2. Consider the associated principal fibration

Bℤ/2 ^{g}^{}Map(Bℤ/2, X)_{g} × EBℤ/2 ^{π}^{}Map(Bℤ/2, X)_{g} ×_{Bℤ/2} EBℤ/2. (3)

Proposition 2.2. Let X be a 2-complete space such that H^{∗}(X) ≅ P [u_{4}] ⊗ E[x_{3}] with β_{r}(x) = u. Let E_{r} be a space such that there is a principal fibration

Bℤ/2 ^{g}^{}X ^{π}^{}E_{r}

and the map g is essential. Then r ≥ 3 and the cohomology of E_{3} is given as an object of K_{β} by structure 1 below. Furthermore, for r ≥ 4 the cohomology of E_{r} is isomorphic in K_{β} to one of the objects given in 2 and 3.

1. P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2} + w_{3}^{2} + w_{3}x_{3} + w_{2}^{3}), Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = 0, Sq^{2}w_{3} = w_{2}w_{3}, Sq^{1}x_{3} = w_{2}^{2} and Sq^{2}x_{3} = w_{2}w_{3} + w_{2}x_{3}.

2. P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2} + w_{3}x_{3}), Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = 0, Sq^{2}w_{3} = w_{2}w_{3}, Sq^{1}x_{3} = 0, Sq^{2}x_{3} = w_{2}x_{3} and β_{r−2}x_{3} = w_{2}^{2}.

3. P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2}), Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = 0, Sq^{2}w_{3} = w_{2}w_{3}, Sq^{1}x_{3} = Sq^{2}x_{3} = 0 and β_{r−2}x_{3} = w_{2}^{2}.

Moreover, let Y be any simply connected space with either one of the cohomological structures given above and let Y ⟨2⟩ denote its 2-connected cover. Then ΩY ⟨2⟩ ≃ S^{3}{2^{r}}.

Structure 1 in the proposition is isomorphic to H^{∗}(BP SL_{2}(𝔽_{q})), q ≡ ±3 (mod 8), or equivalently to H^{∗}(BA_{4}), where A_{4} is the alternating group on four letters. Structure 2 is isomorphic to H^{∗}(BP SL_{2}(𝔽_{q})), q ≡ ±1 (mod 8), r > 3 [22]. Structure 3 is isomorphic as an object of K to H^{∗}(BSO(3) × S^{3}). The proof of the proposition will be carried out in a separate section.

Corollary 2.3. The space S^{3}{4} is not a loop space.

Proof. If S^{3}{4} is a loop space, its classifying space X = BS^{3}{4} fits in the principal fibration (3), with base space E_{2} = Map(Bℤ/2, X)_{g} ×_{Bℤ/2} EBℤ/2, but then, the cohomology of X described in (2) contradicts Proposition 2.2.

Proof of Theorem 1.1. Assume that BS^{m}{2^{r}} is a classifying space for S^{m}{2^{r}}, m > 1. According to Lemma 2.1, H^{∗}(BS^{m}{p^{r}}; 𝔽_{p}) ≅ P [u_{m+1}] ⊗ E[x_{m}], as an algebra with β_{r}(x) = u. In [1] it is proved that a necessary condition for this algebra to be the cohomology of a space is that m = 3 and r > 1. By Corollary 2.3 we rule out the case r = 2 as well. For the remaining cases, the spaces BSL_{2}(𝔽_{q})_{2}^{∧}, for various odd prime powers q, provide enough models for classifying spaces of S^{m}{2^{r}}. This completes the proof of the first statement of the theorem.

By Lemma 2.1, the second part of the theorem amounts to proving that for r ≥ 3 the homotopy type of BS^{3}{2^{r}} is determined by its cohomology as an object of K_{β}.

In the sequel we will show that if a 2-complete space Y has the cohomology of BP SL_{2}(𝔽_{q}) as objects in the category K_{β} then Y ≃ BP SL_{2}(𝔽_{q})_{2}^{∧} (see Theorem 6.2).

Now, suppose that X is a space such that H_{β}^{∗}(X) ≅ H_{β}^{∗}(BS^{3}{2^{r}}). Construct the space E_{r} with respect to X as described above. Then H_{β}^{∗}(E_{r}) is isomorphic to one of the three possible structures specified in Proposition 2.2. By the remarks following the proposition either

1. H_{β}^{∗}(E_{r}) ≅ H_{β}^{∗}(BP SL(𝔽_{q})) for an appropriate odd prime power q or

2. H_{β}^{∗}(E_{r}) coincides with structure 3 of Proposition 2.2.

Assume the first case. Then E_{r} ≃ BP SL_{2}(𝔽_{q})_{2}^{∧} for a suitable value of q and X is homotopy equivalent to its 2-connected cover, given by BSL_{2}(𝔽_{q})_{2}^{∧}. Homotopy uniqueness of BS^{m}{2^{r}} follows by showing the non-realizability of structure 3 in Proposition 2.2. This is done in section 8, Proposition 8.1, where it is shown that the proposed Steenrod algebra action is incompatible with the specified higher order Bockstein operations.

3. Proof of Proposition 2.2

Fix a positive integer r and let X be a classifying space for S^{3}{2^{r}}. Assume we are given a principal fibration which up to homotopy has the form

Bℤ/2 ^{g}^{}X ^{π}^{}E_{r} (4)

with non trivial g. Our aim is computing the cohomology of E_{r} as an object of K_{β}. Consider the Sss for the delooping of (4)

X ^{π}^{}E_{r} ^{w_{2}}^{}K(ℤ/2, 2). (5)

The E_{2} page takes the form

E_{2}^{p,q} ≅ H^{p}(K(ℤ/2, 2)) ⊗ H^{q}(X) ≅ (P [ι_{2}, Sq^{1}ι_{2}, Sq^{2,1}ι_{2}, ⋯])^{p} ⊗ (P [u_{4}] ⊗ E[x_{3}])^{q}.

We claim that the spectral sequence is determined by the requirement that x_{3} is a permanent cycle and d_{5}(u_{4}) = Sq^{2,1}ι_{2} + ι_{2}Sq^{1}ι_{2}.

The first differential, d_{2}, is determined by its value on the class u_{4}. Assume it is non trivial; that is, d_{2}(u_{4}) = ι_{2}x_{3}. It would follow that x_{3} is a permanent cycle, for d_{3}(x_{3}) = ι_{2}^{2} would imply d_{3}(ι_{2}x_{3}) = ι_{2}^{3} ≠ 0 in the E_{3}-term of the spectral sequence. But ι_{2}x_{3} = 0 in E_{3}, which gives a contradiction. This calculation suffices to conclude the structure of H^{∗}(E_{r}) in low dimensions if this were the case. Namely, we would have the classes represented by ι_{2} in degree 2, Sq^{1}ι_{2} and x_{3} in degree 3, ι^{2} in degree 4 and ι_{2}Sq^{1}ι_{2} and Sq^{2,1}ι_{2} in degree 5. Using naturality of the Bss, one finds that x_{3} would in that case be a permanent cycle in the Bss for E_{r}, thus a non torsion class in the 2-adic cohomology of E_{r}. This in turn is impossible, since E_{r} is the total space in a fibration with base and fibre, which are torsion spaces. Hence d_{2} is trivial.

d_{3} is zero by degree reasons. The next possibly non-trivial differential is d_{4}, which may be given by d_{4}(x_{3}) = ι_{2}^{2}. But if x_{3} transgresses to ι_{2}^{2} then Sq^{2}ι_{2}^{2} = (Sq^{1}ι_{2})^{2} should also be in the image of the transgression, which is of course impossible. Hence d_{4} is also trivial.

Next we compute d_{5}. Notice that u_{4} is now transgressive. Since it restricts nontrivially to the cohomology of Bℤ/2 by hypothesis, it follows by naturality that d_{5}(u_{4}) = Sq^{2,1}ι_{2} + Aι_{2}Sq^{1}ι_{2}, for some A ∈ 𝔽_{2}. But Sq^{1,2,1}ι_{2} = (Sq^{1}ι_{2})^{2}, which would imply a contradiction if A = 0. Hence A = 1 and d_{5}(u_{4}) = Sq^{2,1}ι_{2} + ι_{2}Sq^{1}ι_{2}. It now follows at once that each element of the form u_{4}^{2^{k}} is transgressive to a non-zero class and so the E_{∞} term of the spectral sequence is isomorphic to P [ι_{2}, Sq^{1}ι_{2}] ⊗ E[x_{3}] as an algebra. We obtain an isomorphism of P [w_{2}, w_{3}]-modules

H^{∗}(E_{r}) ≅ P [w_{2}, w_{3}] ⊗ E[x_{3}],

with Sq^{1}w_{2} = w_{3}.

Since E_{r} is a torsion space, there must exist some positive integer j such that β_{j}(x_{3}) = w_{2}^{2}. This enables an easy calculation of the Bss for E_{r} in low dimensions. In particular it follows that H^{4}(E_{r}; ℤ_{2}^{∧}) is cyclic and that H^{5}(E_{r}; ℤ_{2}^{∧}) vanishes. In turn, this implies that in the 2-adic Sss for (5), the transgression

d_{5} : E_{5}^{0,4} ≅ H^{4}(X; ℤ_{2}^{∧}) ≅ ℤ/2^{r} ^{}E_{5}^{5,0} ≅ H^{5}(K(Z/2, 2); ℤ_{2}^{∧}) ≅ ℤ/4

is an epimorphism with a non-trivial kernel. It then follows that r ≥ 3 and

H^{4}(E_{r}; ℤ_{2}^{∧}) ≅ ℤ/2^{r−2}.

Assume r = 3 so that Sq^{1}(x_{3}) = w_{2}^{2}. Write Sq^{2}(w_{3}) = Aw_{2}w_{3} + Bw_{2}x_{3}, A, B ∈ 𝔽_{2}. Then

w_{3}^{2} = Sq^{1,2}(w_{3}) = Aw_{3}^{2} + B(w_{3}x_{3} + w_{2}^{3}).

Hence A = 1, B = 0 and Sq^{2}(w_{3}) = w_{2}w_{3}. Similarly write Sq^{2}(x_{3}) = Aw_{2}w_{3}+Bw_{2}x_{3}. Then as before

x_{3}^{2} = Aw_{3}^{2} + B(w_{3}x_{3} + w_{2}^{3}).

One has Sq^{2}(x_{3}^{2}) = w_{2}^{4} on one hand and on the other hand

Sq^{2}(x_{3}^{2}) = Sq^{2}(Aw_{3}^{2} + B(w_{3}x_{3} + w_{2}^{3})) = B(w_{2}w_{3}x_{3} + w_{3}Sq^{2}(x_{3}) + w_{3}^{2}w_{2} + w_{2}^{4}).

Manipulating with this equation one obtains A = B = 1 so Sq^{2}(x_{3}) = w_{2}x_{3} + w_{2}w_{3} and x_{3}^{2} = w_{3}^{2} +w_{3}x_{3} +w_{2}^{3}. This completes the description of H^{∗}(E_{r}) in the case where r = 3, namely case 1 of Proposition 2.2.

Assume next r > 3. Then Sq^{1}(x_{3}) = 0 and β_{r−2}(x_{3}) = w_{2}^{2}. One checks as before that Sq^{2}w_{3} = w_{2}w_{3} and writes Sq^{2}(x_{3}) = Aw_{2}w_{3} + Bw_{2}x_{3}. Then x_{3}^{2} = Aw_{3}^{2} + Bw_{3}x_{3}. Applying Sq^{2} one obtains that either A or B must be 0. If A = 1 then change the basis and take x̄_{3} = x_{3} + w_{3}. Then Sq^{2}x̄_{3} = 0 and we get the structure of case 3. Hence we may assume without loss of generality that to begin with either A = 0, B = 1 or A = B = 0. These two cases correspond to cases 2 and 3 of Proposition 2.2 respectively.

To complete the proof we need to calculate the cohomology of the 2-connected covers for spaces with the cohomological structures given in the proposition. Consider the principal fibration

Bℤ/2 ^{}Y ⟨2⟩ ^{}Y, (6)

where Y such that H^{∗}(Y) ≅ P [w_{2}, w_{3}]⊗E[x_{3}] as a P [w_{2}, w_{3}]-module with Sq^{1}w_{2} = w_{3}, Sq^{2}w_{3} = w_{2}w_{3} and β_{r−2}(x_{3}) = w_{2}^{2}. Then in the Sss for (6) one has d_{2}(z) = w_{2}, d_{3}(z^{2}) = w_{3} and d_{5}(z^{4}) = 0. The spectral sequence thus collapses at the E_{6} page without any extension problems and so H^{∗}(Y ⟨2⟩) ≅ P [u_{4}] ⊗ E[x_{3}]. Since all spaces involved are torsion there must exist a positive integer t, such that β_{t}(x_{3}) = u_{4}. The integral Sss now gives at once that t = r and so by Lemma 2.1, Y ⟨2⟩ is a classifying space for S^{3}{2^{r}}.

4. Homology decomposition of BP SL_{2}(𝔽_{q})

Let P SL_{2}(𝔽_{q}) denote the projective special linear group of rank 2 over the field of q elements 𝔽_{q}. Preparing to prove homotopy uniqueness of BP SL_{2}(𝔽_{q})_{2}^{∧}, we study homology decompositions of this space as in [17, 18, 10]. For the terminology we use, the reader is referred to [10].

Let q be an odd prime power. Then the order of P SL_{2}(𝔽_{q}) is \frac{q(q^{2}−1)}{2}. Notice that the highest power of 2 dividing \frac{q(q^{2}−1)}{2} is precisely 4 if q ≡ ±3 (mod 8) and that if q ≡ ±1 (mod 8) then \frac{q(q^{2}−1)}{2} is divisible by 2^{s} for s ≥ 3 and, depending on q, every such s may appear. We will write 2^{s}∥n if 2^{s} is the highest power of 2 dividing n. The 2-Sylow subgroup of P SL_{2}(𝔽_{q}) is known to be the dihedral group D_{2^{s}} if 2^{s}∥\frac{q(q^{2}−1)}{2}. The respective cohomology algebras are also known [22] and are described below.

All groups P SL_{2}(𝔽_{q}) have the same mod-2 cohomology algebra and only elementary torsion in their integral cohomology if q ≡ ±3 (mod 8):

H^{∗}(BP SL_{2}(𝔽_{q})) ≅ P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2} + w_{3}^{2} + w_{3}x_{3} + w_{2}^{3}),

with Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = 0, Sq^{2}w_{3} = w_{2}w_{3}, Sq^{1}x_{3} = w_{2}^{2} and Sq^{2}x_{3} = w_{2}w_{3} + w_{2}x_{3}. As we observe later every 2-complete space with this cohomology is homotopy equivalent to (BA_{4})_{2}^{∧}, where A_{4} is the alternating group on 4 letters.

The case q ≡ ±1 (mod 8) is a bit more complicated. Here too, the mod-2 cohomology as an algebra over the Steenrod algebra is the same in all case. The Bss though varies according to the highest power of 2 dividing the order of P SL_{2}(𝔽_{q}). Indeed if q ≡ ±1 (mod 8) and 2^{s}∥\frac{q(q^{2}−1)}{2} then

H^{∗}(P SL_{2}(𝔽_{q})) ≅ P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2} + w_{3}x_{3}),

with Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = 0, Sq^{2}w_{3} = w_{2}w_{3}, Sq^{1}x_{3} = 0, Sq^{2}x_{3} = w_{2}x_{3} and β_{s−1}x_{3} = w_{2}^{2}.

Below we prove that among 2-complete spaces there is a unique homotopy type corresponding to the above cohomology structure for every s ≥ 3. Hence, for our purposes, we will only need to consider prime powers q with q ≡ +1 (mod 8), for, as a consequence of a theorem of Dirichlet’s, these are enough to cover all cases.

Lemma 4.1. For every s ≥ 3 there is an odd prime q with q ≡ +1 (mod 8) such that 2^{s}∥(q − 1) and hence 2^{s}∥\frac{q(q^{2}−1)}{2}.

Proof. For s ≥ 3 consider the arithmetic progression n_{s,j} = 2^{s−2}j − 2^{s−3}. Let

q_{s,j} = 8n_{s,j} + 1 = 2^{s+1}j − (2^{s} − 1).

Since the greatest common divisor of 2^{s+1} and 2^{s} − 1 is 1, Dirichlet’s theorem [16] implies the existence of infinitely many primes of the form q_{s,j}. Notice that each such prime is congruent to 1 modulo 8 and that

q_{s,j} − 1 = 8n_{s,j} = 2^{s}(2j − 1).

Thus for integers j such that q_{s,j} is a prime, the highest power of 2 dividing the order of P SL_{2}(𝔽_{q_{s,j}}) is 2^{s}.

Throughout this section we restrict attention to the case of odd prime powers q, where q ≡ 1 (mod 8) and consider certain collection of subgroups of P SL_{2}(𝔽_{q}). Let s be such that 2^{s}∥q − 1 and let ε be a primitive root of unity in 𝔽_{q} of order 2^{s}. Then the matrices X = ( \begin{matrix}ε & 0 \\ 0 & ε^{−1} \end{matrix} ) and Y = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) satisfy the relation

X^{2^{s−1}} = Y^{2} = (XY)^{2} = 1,

modulo the center of SL_{2}(𝔽_{q}), generated by the matrix −I. Hence the classes [X] and [Y] of these matrices in P SL_{2}(𝔽_{q}) generate a dihedral group of order 2^{s}, which gives the 2-Sylow subgroup.

Notice that if q ≡ ±1 (mod 8) the 2-Sylow subgroup is of order at least 8. Consider the following elementary abelian 2-subgroups of P SL_{2}(𝔽_{q}).

Z = ⟨[X]^{2^{s−2}}⟩ ≅ ℤ/2
V = ⟨[X]^{2^{s−2}}, [Y]⟩ ≅ ℤ/2 × ℤ/2
W = ⟨[X]^{2^{s−2}}, [X][Y]⟩ ≅ ℤ/2 × ℤ/2

According to [17, Thm. 7.7], the set E_{2} = {Z, V, W } is an ample collection of elementary abelian 2-subgroups of P SL_{2}(𝔽_{q}) (which, roughly speaking means that it can be used to obtain a homology decomposition). The associated conjugacy category A_{E_{2}} has E_{2} as objects, with morphisms among them induced by inclusions and conjugations in P SL_{2}(𝔽_{q}).

Our next statement describes the category A_{E_{2}}. First, let G_{1} and G_{2} be finite groups with a specific common subgroup H. Consider the category

A(G_{1}, G_{2}, H) = G_{2} ⟳ 2 [figure: G_{2}/H] 0 [figure: G_{1}/H] 1 ⟳G_{1} .

It has three objects 0 , 1 and 2 . The automorphism group of i is given by G_{i} for i = 1, 2 whereas the automorphism group of 0 is trivial. Each morphism set from 0 to i , i = 1, 2, admits a natural G_{i} action and is isomorphic as a G_{i}-set to G_{i}/H.

Proposition 4.2. For q ≡ 1 (mod 8), the set E_{2} = {Z, V, W } is an ample collection of elementary abelian 2-subgroups of P SL_{2}(𝔽_{q}) and the E_{2}-conjugacy category A_{E_{2}} is the image of the functor from the category A(Σ_{3}, Σ_{3}, ℤ/2) to the category of elementary abelian 2-groups taking 0 to Z, 1 to W and 2 to V.

Proof. Let G denote BP SL_{2}(𝔽_{q}) with q ≡ 1 (mod 8). Notice that the automorphisms of a subgroup U in the conjugacy category are given by the quotient N_{G}(U)/C_{G}(U) of the normalizer by the centralizer of U in G. One easily verifies the following.

1. C_{G}(Z) = N_{G}(Z) = ⟨( \begin{matrix}ζ & 0 \\ 0 & ζ^{−1} \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )⟩ ≅ D_{2q−2}, where ζ is generator of 𝔽_{q}^{∗} and

D_{2q−2} is the dihedral group of order 2q − 2. Notice that D_{2^{s}} < D_{2q−2} and the mod-2 restriction map is an isomorphism.

2. C_{G}(V) = V and N_{G}(V)/V ≅ Σ_{3} is generated by the classes [A] and [B] in G, of A = \frac{1}{i−1}( \begin{matrix}1 & 1 \\ i & −i \end{matrix} ) and B = ( \begin{matrix}0 & \sqrt{i} \\ i\sqrt{i} & 0 \end{matrix} ) respectively. Here i = ε^{2^{s−2}}, so by \sqrt{i} we

mean ε^{2^{s−3}} . Conjugation by A sends [X]^{2^{s−2}} to [Y ] and [Y ] to [X]^{2^{s−2}}[Y ],

whereas conjugation by B flips [Y ] and [X]^{2^{s−2}}[Y ] and fixes [X]^{2^{s−2}}.

3. C_{G}(W) = W and N_{G}(W)/W ≅ Σ_{3} generated by the classes [C] and [D] in G of C = \frac{1}{i−1}( \begin{matrix}1 & ε \\ iε^{−1} & −i \end{matrix} ) and D = ( \begin{matrix}0 & −\sqrt{i}ε \\ −i\sqrt{i}ε^{−1} & 0 \end{matrix} ) respectively. Conjugation by C

sends [X]^{2^{s−2}} to [X][Y ] and [X][Y ] to [X]^{2^{s−2}+1}[Y ], whereas conjugation by

D flips [X][Y ] and [X]^{2^{s−2}+1}[Y ] and fixes [X]^{2^{s−2}}. Thus, the automorphisms in A_{E_{2}} are as described, and the other morphisms follow

immediately. This completes the proof of the proposition.

We can draw the E_{2}-conjugacy category as

A_{E_{2}} : Σ_{3} ⟳ W [figure: Σ_{3}/Σ_{2}] Z [figure: Σ_{3}/Σ_{2}] V ⟳ Σ_{3} .

The centralizer diagram

α_{E_{2}} : A_{E_{2}}^{op} ^{}Spaces

is the functor defined in [17], which associates with each object U a model for the classifying space of its centralizer in G. Namely, to an object U it associates the orbit space EG ×_{G} (G/C_{G}(V)). Up to homotopy and 2-completion the diagram in our case is given by

Σ_{3} ⟳ BW [figure] BD_{2^{s}} [figure] BV ⟳ Σ_{3}.

Notice that C_{P SL_{2}(𝔽_{q})}(Z) = D_{(q−1)}, but (BD_{(q−1)})_{2}^{∧} = BD_{2^{s}}.

One checks immediately that this diagram satisfies the conditions of [17, Thm 7.7] and so, using the terminology of [9], it is a sharp diagram. Specifically, this means that the natural map

a_{E_{2}} : hocolim α_{E_{2}} ^{}BP SL_{2}(𝔽_{q})

induces a mod-2 cohomology isomorphism. Moreover,

\begin{matrix}lim \\ ←------- \end{matrix}^{0}_{A_{E_{2}}} H^{∗} ∘ α_{E_{2}} ≅ H^{∗}(BP SL_{2}(𝔽_{q})),

whereas all the higher limits vanish. We summarize this in the following

Proposition 4.3. For q ≡ 1 (mod 8) there is a homotopy equivalence

(hocolim_{A_{E_{2}}} α_{E_{2}})_{2}^{∧} ≃ BP SL_{2}(𝔽_{q})_{2}^{∧}

and an isomorphism

\begin{matrix}lim \\ ←------- \end{matrix}^{0}_{A_{E_{2}}} H^{∗} ∘ α_{E_{2}} ≅ H^{∗}(BP SL_{2}(𝔽_{q}))

while \begin{matrix}lim \\ ←------- \end{matrix}^{i}_{A_{E_{2}}} H^{∗} ∘ α_{E_{2}} = 0 for all i ≥ 1.

5. p-Sylow subgroups

A question that could be considered of general interest is the following. Let X be a p-complete space. Is there an appropriate concept of a p-Sylow subgroup for X? For general spaces X, it is not even clear what these objects ought to satisfy or whether they could exist. However, if X has the mod-p cohomology of a finite group G, then a sensible definition for a p-Sylow subgroup for X appears to be a map f : Bπ ^{}X, where π is the p-Sylow subgroup of G, which realizes the restriction map in p-local cohomology.

One way to attack this problem starts by finding an elementary abelian p-subgroup V contained in π, the p-Sylow of G, such that C_{G}(V) = π. Of course, this subgroup V does not always exist, which is the main reason why, in our specific case, we work first with P SL_{2}(𝔽_{q}) rather than directly with SL_{2}(𝔽_{q}). In those favorable cases where one can find a subgroup as above, one may approach further technical difficulties. The main complication is in the use of Lannes’ T functor, which is not guaranteed to do its work. This makes it essential to use an interpretation of T due to E. Farjoun and J. Smith [14]. We define a class of groups which behaves nicely with respect to this interpretation of the T functor and call those groups T_{p}-groups. We show that cyclic p-groups and dihedral 2-groups are T_{p}-groups, a fact which is then used to obtain a 2-Sylow subgroup for a space X with the same cohomology as P SL_{2}(𝔽_{q}).

Definition 5.1. Let π be a finite p-group. We say that π is a T_{p}-group if for any tower of fibrations {F_{s}} such that

1. π_{i}(F_{s}) is a finite p-group for every i and s and

2. \underrightarrow{lim} H_{β}^{∗}(F_{s}; 𝔽_{p}) ≅ H_{β}^{∗}(π; 𝔽_{p})

there is a homotopy equivalence \underleftarrow{lim} F_{s} ≃ Bπ.

We now give some elementary examples of groups which are T_{p}. We know of further examples, which are irrelevant to this paper. One might like to ask whether every finite p-group is T_{p} as a positive result might have interesting implications.

Proposition 5.2. Every cyclic p-group is a T_{p}-group. Also, every dihedral 2-group is a T_{2}-group.

Proof. The proof for the case of a cyclic p-group is identical to that of Lemma 6.4 in [2] and we omit it. For dihedral 2-groups, our statement is a stronger version of [4, Thm. 1.1]. The proof combines methods from [4, §3] and [2, §6].

Recall that H_{β}^{∗}(BD_{2^{n}}) ≅ P [x_{1}, y_{1}, w_{2}]/(x^{2} + xy), where degrees are given by subscripts, Sq^{1}(w) = wy and β_{n−1}(xw) = w^{2}.

Let {Y_{s}} be a tower of fibrations satisfying 1 and 2 in Definition 5.1 with respect to D_{2^{n}}. Let π_{s} : Y_{s} ^{}Y_{s−1} denote the projection. For a sufficiently large s there are elements y_{1,s} ∈ H^{1}(Y_{s}) such that

1. π_{s+1}^{∗}(y_{1,s}) = y_{1,s+1} and

2. the sequence \{y_{1,s}\} represents the element y_{1} ∈ \underrightarrow{lim}_{s}H^{1}(Y_{s}) ≅ H^{∗}(BD_{2^{n}}).

Let φ_{s} : Y_{s} ^{}Bℤ/2 be a map classifying y_{1,s}. Then there is the following homotopy commutative diagram of fibrations.

[figure: …; F_{s+2}; F_{s+1}; F_{s}]

[figure: …; Y_{s+2}; j_{s+2}; π_{s+2}; Y_{s+1}; j_{s+1}; π_{s+1}; Y_{s}; j_{s}]

[figure: …; = = =; Bℤ/2; φ_{s+2}; = = = =; Bℤ/2; φ_{s+1}; = = = =; Bℤ/2; φ_{s}]

Since the homotopy groups of all spaces involved are finite, it follows that the sequence

\underleftarrow{lim} F_{s}  ^{}\underleftarrow{lim} Y_{s}  ^{}BD_{2^{n}}

is a quasi fibration, namely it gives rise to a long exact sequence in homotopy.

The Eilenberg-Moore spectral sequence for each fibration in the diagram collapses at its E_{2} page, being a fibration over Bℤ/2. Thus

H^{∗}(F_{s}) ≅ T or_{H^{∗}(Bℤ/2)}(𝔽_{2}, H^{∗}(Y_{s})).

Since T or commutes with direct limits, we have that

(7) P [v_{2}] ⊗ E[α_{1}] ≅ T or_{H^{∗}(Bℤ/2)}(𝔽_{2}, H^{∗}(D_{2^{n}})) ≅ ≅ T or_{H^{∗}(Bℤ/2)}(𝔽_{2}, \underrightarrow{lim}_{s}H^{∗}(Y_{s})) ≅ \underrightarrow{lim}_{s}H^{∗}(F_{s})

as P [v_{2}]-modules. Notice that the edge homomorphism

\underrightarrow{lim} H_{β}^{∗}Y_{s} → \underrightarrow{lim} H_{β}^{∗}F_{s}

is a morphism in K_{β} which at the level of the first page of the Bss is an epimorphism

P [x_{1}, y_{1}, w_{2}]/(x^{2} + xy) [figure] P [v_{2}] ⊗ E[α_{1}]

sending x_{1} to α_{1}, y_{1} to zero and w_{2} to v_{2}.

Next, we compute \underrightarrow{lim} H_{β}^{∗}F_{s} as an object of K_{β}. From the description of the edge homomorphism above it follows that Sq^{1} acts trivially on P [v_{2}] ⊗ E[α_{1}] and the morphism induced on the second page of the Bss

P [w_{2}^{2}] ⊗ x_{1}w_{2}P [w_{2}^{2}]^{}^{}P [v_{2}] ⊗ E[α_{1}]

is monic.

The next and last non-trivial differential in the Bss of the source is determined by β_{n−1}(x_{1}w_{2}) = w_{2}^{2}. Thus in the Bss of the target we must have β_{n−1}(α_{1}v_{2}) = v_{2}^{2}. By the P [v_{2}]-module structure of \underrightarrow{lim}_{s}H^{∗}(F_{s}), it follows that β_{n−1}(α_{1}) = v_{2} and so β_{n−1}(α_{1}v_{2}^{k}) = v_{2}^{k+1}. Moreover, since homotopy groups of Y_{s} are finite for every s, it follows that the homotopy groups of each F_{s} are finite and since cyclic 2-groups are T_{2} by the first part of the proposition, we have holim F_{s} ≃ Bℤ/2^{n−1}.

We have thus obtained a quasi fibration

Bℤ/2^{r}  ^{}\underleftarrow{lim} Y_{s}  ^{}BD_{2^{n}}. (8)

In particular Y = \underleftarrow{lim} Y_{s} is the classifying space of a finite 2-group and so (8) is in fact a fibration. It follows that the map of towers

{Y } ^{}{Y_{s}},

where {Y } is the constant tower, is a weak pro-homotopy equivalence. Consequently the map of towers {H_{n}(Y)} ^{}{H_{n}Y_{s}} is a pro-isomorphism by [3, III, 3.4] and so

H_{β}^{∗}Y ≅ \underrightarrow{lim} H_{β}^{∗}Y_{s} ≅ H_{β}^{∗}BD_{2^{n}}.

Now [4, Thm. 1.1] applies,

Y ≃ BD_{2^{n}}

and the proof is complete.

Lemma 5.3. Let X be a p-complete space with H_{β}^{∗}(X) ≅ H_{β}^{∗}(BG) where G is a finite p-perfect group. Then:

i) π_{i}(X) is a finite p-group for all i > 0.

ii) For any elementary abelian p-group V, and an n-th Postnikov section P_{n}X of X, the homotopy groups of any component of Map(BV, P_{n}X) are finite p-groups.

Proof. By Hypothesis H^{1}(X) = 0 and X is p-complete. Hence X is 1-connected and [2, 5.7 (1)] applies so that H^{j}(X; ℤ̂_{p}) is a finitely generated ℤ̂_{p}-module for all j. Furthermore, by our assumption on the Bss all p-adic cohomology groups are finite p-groups, hence the first statement follows by [2, 5.7 (2)].

The second statement follows as well by an argument of [2, §6]. For an Eilenberg-MacLane space we have

Map (BV, K(π, n)) ≃ ∏_{1≤j≤n} K (H^{n−j}(BV, π), j) .

so if π is a finite p-group, then the homotopy groups of Map(BV, K(π, n)) have the same property. Proceed by induction.

We are now ready to produce a 2-Sylow subgroup for a space X with the same cohomology as P SL_{2}(𝔽_{q}). More precisely, we have the following

Proposition 5.4. Let X be a 2-complete space with

H_{β}^{∗}(X) ≅ H_{β}^{∗}(BP SL_{2}(𝔽_{q})).

Let D_{2^{s}} ^{i}^{}P SL_{2}(𝔽_{q}) be the inclusion of a 2-Sylow subgroup. Then there is a map

f: BD_{2^{s}} ^{}X

such that f^{∗}, coincides with the composition

H_{β}^{∗}(X; 𝔽_{2}) ≅ H_{β}^{∗}(BP SL_{2}(𝔽_{q})) ^{(Bi)^{∗}}^{}H_{β}^{∗}(BD_{2^{s}}).

Proof. If q ≡ ±3 (mod 8) then the 2-Sylow subgroup of P SL_{2}(𝔽_{q}) is D_{4}, an elementary abelian 2-group of rank 2. Then our claim follows directly from Lannes theory [20]. Namely, in this case one has

[BD_{4}, X] ≅ [BD_{4}, BP SL_{2}(𝔽_{q})_{2}^{∧}] ≅ Hom_{K}(H^{∗}(BP SL_{2}(𝔽_{q})), H^{∗}(BD_{2^{s}})).

Suppose q ≡ ±1 (mod 8). Because of Lemma 4.1 it is enough to consider the case q ≡ 1 (mod 8). Let s satisfy 2^{s}∥(q−1) so in particular s > 2. Fix a 2-Sylow subgroup π ≅ D_{2^{s}} and let Z ≅ ℤ/2 be its center. Let j denote the inclusion of Z in P SL_{2}(𝔽_{q}).

By Lannes theory there is a map f_{Z}: BZ ^{}X, inducing the composition

H^{∗}(X) ≅ H^{∗}(BP SL_{2}(𝔽_{q})) ^{(Bj)^{∗}}^{}H^{∗}(BZ)

on cohomology.

Consider the evaluation map Map(BZ, X)_{f_{Z}} ^{ev}^{}X. We proceed by showing that Map(BZ, X)_{f_{Z}} ≃ BD_{2^{s}} and that the map thus obtained has the correct cohomological effect.

We start by computing the Lannes T functor. By hypothesis on the cohomology of X and the calculations of section 4,

(9) T (H^{∗}(X); f_{Z}^{∗}) ≅ T (H^{∗}(BP SL_{2}(𝔽_{q}); (Bj)^{∗}) ≅

≅ H^{∗} (BC_{P SL_{2}(𝔽_{q})}(Z)) ≅ H^{∗}(BD_{(q−1)}) ≅ H^{∗}(BD_{2^{s}})

Furthermore, one obtains as well that the augmentation

ε: H^{∗}(X) ^{}T (H^{∗}(X); f_{Z}^{∗}) (10)

coincides with (Bi)^{∗}: H^{∗}(BP SL_{2}(𝔽_{q})) ^{}H^{∗}(D_{2^{s}}) under the identification of the sources given in terms of the assumed isomorphism. It is then, explicitly described as

[figure: ε; :; P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2} + w_{3}x_{3}); P [x_{1}, y_{1}, w_{2}]/(x_{1}^{2} + x_{1}y_{1}); w_{2}; w_{2} + y_{1}^{2}; w_{3}; w_{2}y_{1}; x_{3}; w_{2}x_{1}] (11)

Let {P_{n}X}_{n} be the Postnikov Tower for X and f_{Z}(n): BZ ^{}P_{n}X be the factorization of f_{Z} through the tower. Applying the functor Map(BV, −), we get a tower, {Map(BV, P_{n}X)_{f_{Z}(n)}}_{n} and a homotopy equivalence

(12) holim Map(BZ, P_{n}X)_{f_{Z}(n)} ≃ Map(BZ, X)_{f_{Z}}.

Since the space X is 2-complete nilpotent and of finite type, the Farjoun-Smith theorem states

T (H^{∗}(X); f_{z}^{∗}) ≅ \begin{matrix}lim \\ -------→ \end{matrix}_{n} H^{∗} (Map(BZ, P_{n}X)_{f_{Z}(n)}) . (13)

The map in the theorem is given as the obvious map

(14)  \begin{matrix}lim \\ -------→ \end{matrix}_{n} T (H^{∗}(P_{n}X), f_{Z}^{∗}(n)) ^{}\begin{matrix}lim \\ -------→ \end{matrix}_{n} H^{∗} (Map(BZ, P_{n}X)_{f_{Z}(n)}) ,

using the fact that for a space X as above the map \begin{matrix}lim \\ -------→ \end{matrix}_{n} H^{∗}(P_{n}(X)) ^{}H^{∗}(X) is an isomorphism and T commutes with direct limits.

Notice that in our particular case the left hand side in (13) is isomorphic to H^{∗}(BD_{2^{s}}) as an A_{2}^{∗} algebra.

There are homomorphisms H^{∗}(P_{n}X) ^{}H^{∗}(Map(BZ, P_{n}X)_{f_{Z}(n)}) induced by evaluation maps. Taking the limit on both sides and using the isomorphism (14), the resulting map

H^{∗}(X) ^{ev^{∗}}^{}\begin{matrix}lim \\ -------→ \end{matrix}_{n} H^{∗}(Map(BZ, P_{n}X)_{f_{Z}(n)}) (15)

coincides with the augmentation map given in (11). But being induced by maps of spaces, this monomorphism is compatible with higher Bockstein operations. One observes at once that it should induce an isomorphism in the second page of the Bss and therefore there is only one structure of H^{∗}(BD_{2^{s}}) as an object of K_{β} that is compatible with such a monomorphism, namely

\begin{matrix}lim \\ -------→ \end{matrix}_{n} H_{β}^{∗} (Map(BZ, P_{n}X)_{f_{Z}(n)}) ≅ H_{β}^{∗}(BD_{2^{s}}) .

Now, since X is 2-complete and H_{β}^{∗}(X) ≅ H_{β}^{∗}(P SL_{2}(𝔽_{q})), Lemma 5.3 imply that π_{j}(X) is a finite 2-group for all j and also that the homotopy groups of the mapping spaces π_{j}(Map(BZ, P_{n}X)_{f_{Z}(n)}) are finite 2-groups for all j and n.

By Proposition 5.2 the dihedral group D_{2^{s}} is a T_{2}-group and so we conclude that holim_{n} Map(BZ, P_{n}X)_{f_{Z}(n)} ≃ BD_{2^{s}}. This combined with (12) gives

Map(BZ, X)_{f_{Z}} ≃ BD_{2^{s}}.

Taking the evaluation, we obtain a map

f: BD_{2^{s}} ^{}X,

such that the induced homomorphism on cohomology coincides with the composition

H_{β}^{∗}(X) ≅ H_{β}^{∗}(BP SL_{2}(𝔽_{q})) ^{(Bi)^{∗}}^{}H_{β}^{∗}(BD_{2^{s}}).

6. Homotopy type of the two completion of BP SL_{2}(𝔽_{q})

In this section we prove that if X is a 2-complete space with the mod-2 cohomology of BP SL_{2}(𝔽_{q}) as an object of K_{β} then X ≃ BP SL_{2}(𝔽_{q})_{2}^{∧}.

Proposition 6.1. Let X be a 2-complete space such that

H_{β}^{∗}(X) ≅ H_{β}^{∗}(BP SL_{2}(𝔽_{q}))

Let s be the highest power of 2 dividing the order of P SL_{2}(𝔽_{q}). Assume s > 2 and let

f: BD_{2^{s}} ^{}X

be the map constructed in Proposition 5.4. Then, the induced map

Bℤ/2 ≃ Map(BD_{2^{s}}, BD_{2^{s}})_{id} ^{f_{♯}}^{}Map(BD_{2^{s}}, X)_{f}

is a homotopy equivalence.

Proof. Without loss of generality, we can assume that q ≡ 1 (mod 8). Consider the sequence of maps

BD_{4} ⊂ BD_{8} ⊂ … ⊂ BD_{2^{r}} ⊂ … ⊂ BD_{2^{s}},

induced by the natural inclusions. Denote f_{r} = f ∘ (Bi_{r}): BD_{2^{r}} → BD_{2^{s}} → X. We prove by induction that f induces homotopy equivalences

f_{♯}: Map(BD_{2^{r}}, BD_{2^{s}})_{Bi_{r}} ^{}Map(BD_{2^{r}}, X)_{f_{r}} .

The case r = 2, follows by Lannes theory [20]. Since D_{4} ≅ (ℤ/2)^{2} and is selfcentralizing in both D_{2^{s}} and P SL_{2}(𝔽_{q}), a T functor computation shows that BD_{4} ≃ Map(BD_{4}, BD_{2^{s}})_{Bi_{2}} ≃ Map(BD_{4}, X)_{f_{2}}.

By induction assume the conclusion is true for r − 1. The space Map(BD_{2^{r}}, X) may be written as the homotopy fix point space Map(ED_{2^{r}}/D_{2^{r−1}}, X)^{hℤ/2}. The corresponding construction holds for BD_{2^{s}} replacing X. Now, the map

f_{♯} : Map(ED_{2^{r}}/D_{2^{r−1}}, BD_{2^{s}})_{Bi_{r−1}} ^{}Map(ED_{2^{r}}/D_{2^{r−1}}, X)_{f_{r−1}}

is a ℤ/2-equivariant homotopy equivalence, so we obtain,

(16) Map(BD_{2^{r}}, BD_{2^{s}})ḡ ≃ Map(ED_{2^{r}}/D_{2^{r−1}}, BD_{2^{s}})_{Bi_{r−1}}^{hℤ/2} ≃

≃ Map(ED_{2^{r}}/D_{2^{r−1}}, X)_{f_{r−1}}^{hℤ/2} ≃ Map(BD_{2^{r}}, X)f̄ where ḡ = {g | g_{|BD_{2^{r−1}}} ≃ Bi_{r−1}} and f̄ = {f | f_{|BD_{2^{r−1}}} ≃ f_{r−1}}. This completes the induction step.

Finally, notice that for r = s, the left hand side of the equation above is connected. Hence the same hold for the right end side and the component of the mapping space there is essentially the one of f. This completes the proof.

Theorem 6.2. Let X be a 2-complete space with

H_{β}^{∗}(X) ≅ H_{β}^{∗}(BP SL_{2}(𝔽_{q})).

Then there is a homotopy equivalence

X ≃ BP SL_{2}(𝔽_{q})_{2}^{∧}

Proof. If q ≡ ±3 (mod 8) then the 2-Sylow subgroup of P SL_{2}(𝔽_{q}) is D_{4}, i.e. elementary abelian of rank 2. In this case one has by Lannes theory a map

f : BD_{4} ^{}X

inducing the restriction map under the given identification of the algebras H^{∗}(X) and H^{∗}(BP SL_{2}(𝔽_{q})). The T functor is now employed to compute the cohomology of Map(BD_{4}, X)_{f}. Indeed, since D_{4} is the 2-Sylow subgroup, it is immediate that H^{∗}(Map(BD_{4}, X)_{f}) ≅ H^{∗}(BD_{4}) as A_{2}^{∗} algebras and so Map(BD_{4}, X)_{f} is homotopy equivalent to BD_{4}. The evaluation map thus gives a map BD_{4} ^{}X, which is equivariant with respect to any automorphism of D_{4}, in particular with respect to the natural ℤ/3 action. This gives a map

f̃: BD_{4} ×_{ℤ/3} Eℤ/3 ^{}X,

which one easily checks to induce an isomorphism on mod-2 cohomology and so a homotopy equivalence after 2-completion,

X ≃ (BD_{4} ×_{ℤ/3} Eℤ/3)_{2}^{∧} ≃ (BA_{4})_{2}^{∧}.

Next restrict attention to the case where the 2-Sylow subgroup is non-abelian. In particular q ≡ ±1 (mod 8). And we can restrict our attention to the case q ≡ 1 (mod 8) without loss of generality. The discussion above gives a homotopy commutative diagram

[figure: Σ_{3} ⟳; BW; Σ_{3}/Σ_{2}; BD_{2^{s}}; \underline{Σ_{3}/Σ_{2}}; BV; ⟳ Σ_{3}; f_{2}; f_{1}; X; f_{0}] (17)

One can regard this diagram as a natural transformation, defined only up to homotopy, from the diagram α_{E_{2}} for BP SL_{2}(𝔽_{q}) (see section 4) to the constant diagram on our space X

f: α_{E_{2}} ^{}X.

The map f_{0} : BD_{2^{s}} → X is given by Proposition 5.4. The maps f_{1} : BV → X, and f_{2} : BW → X, as well as homotopy commutativity in the diagram are provided by Lannes theory [20]. Indeed one only needs to check that these maps can be defined on the level of cohomology (which of course they can since they exist for X = BP SL_{2}(𝔽_{q})).

By Proposition 4.2 we can identify A_{E_{2}} with A(Σ_{3}, Σ_{3}, ℤ/2). To simplify the notation let A denote A(Σ_{3}, Σ_{3}, ℤ/2) and let α denote the diagram α_{E_{2}}.

Diagram (17) defines a map from the 0-skeleton of hocolim_{A^{op}} α to X. The obstructions to extending this map to the actual homotopy colimit lie in the groups

lim_{A}^{j+1}π_{j} (Map(α, X)_{f})

for j ≥ 1 [26]. By Lemmas 6.3 and 6.4 below these obstruction groups vanish. Thus we obtain a map

(BP SL_{2}(𝔽_{q}))_{2}^{∧} ≃ hocolim_{A^{op}} α ^{f}^{}X

where the homotopy equivalence on the left was stated in Proposition 4.3. Moreover, the diagram

BD_{2^{s}}
[figure: f_{0}]
[figure: (BP SL_{2}(𝔽_{q}))_{2}^{∧}; Bi; f; X]

commutes up to homotopy. It follows at once that f induces an isomorphism in mod-2 cohomology and is therefore a homotopy equivalence.

To complete the proof of the theorem we need to prove vanishing of the obstruction groups. For each j ≥ 1 define a functor Π_{j}: A ^{}Ab by

Π_{j}( i ) = π_{j} Map (α( i ), X)_{f_{i}} .

Lemma 6.3. The values of the functors defined above are given as follows. 1. Π_{j}( 0 ) = π_{j}(Bℤ/2)

2. Π_{j}( 1 ) = π_{j}(BV)

3. Π_{j}( 2 ) = π_{j}(BW)

Proof. First notice that

Π_{j}( 0 ) = π_{j} Map (α( 0 ), X)_{f_{0}} = π_{j} (Map(BD_{2^{s}}, X)_{f_{0}}) = π_{j}(Bℤ/2)

by Proposition 6.1.

Next since V is self centralizing in P SL_{2}(𝔽_{q}) it follows from Lannes theory that

Π_{j}( 1 ) = π_{j} (Map(α( 1 ), X)_{f_{1}}) = π_{j} (Map(BV, X)_{f_{1}}) = π_{j}(BV).

The third case follows by the same argument.

Lemma 6.4. lim_{A}^{i} Π_{j} = 0 for all i ≥ 0, j ≥ 1 except for i = j = 1 in which case lim_{A}^{1} Π_{1} = ℤ/2.

Proof. We only need to take care of the case j = 1. The functor Π_{1} is

Σ_{3} ⟳ V [figure: Σ_{3}/Σ_{2}] ℤ/2 [figure: Σ_{3}/Σ_{2}] W ⟳ Σ_{3}

and according to Proposition 10.3 we have an exact sequence

0 \overline{}lim_{A}^{0}Π_{1}  ^{}ℤ/2 ^{}V^{Σ_{2}} ⊕ W^{Σ_{2}}  \overline{}lim_{A}^{1}Π_{1}  ^{}0

and lim_{A}^{i} Π_{1} = 0 if i > 1. The result is forced by the fact that V^{Σ_{2}} ⊕ W^{Σ_{2}} ≅ ℤ/2 ⊕ ℤ/2.

7. Spherical fibrations

Our argument in the proof of non-existence of a space with the cohomological structure, given in 3 of Proposition 2.2, involves investigations of certain spherical fibrations. We say that the fibration

F ^{}E ^{}B

is cohomologically trivial if the associated mod-2 Sss collapses at its E_{2} page and

H^{∗}(E) ≅ H^{∗}(B) ⊗ H^{∗}(F)

as algebras over the Steenrod algebra. By a spherical fibration we mean a Serre fibration, where the fibre is a sphere. We shall assume as always that all spaces are 2-complete.

The case of interest to us is when the base space of the given fibration is BO(2) and the fibre is a 3-sphere. this section is devoted to the proof of the following statement.

Theorem 7.1. Any 2-complete, cohomologically trivial spherical fibration

ξ = S^{3} ^{f}^{}E(ξ) ^{g}^{}BO(2)

is trivial.

Proof. Let i: ℤ/2^{}^{}O(2) be the inclusion of the central element of order two in O(2). The quotient is homeomorphic to O(2) and we have an induced principal fibration

Bℤ/2 ^{Bi}^{}BO(2) ^{Bπ}^{}BO(2) (18)

which is classified by the second Stiefel-Whitney class BO(2) ^{w_{2}}^{}B^{2}ℤ/2. Thus

Bπ^{∗}(w_{1}) = w_{1} and Bπ^{∗}(w_{2}) = 0 (19)

in mod-2 cohomology, while Bi^{∗}(w_{1}) = 0 and Bi^{∗}(w_{2}) = z^{2}, the class z being the 1-dimensional generator of H^{∗}(Bℤ/2).

Furthermore, since the fibration ξ is cohomologically trivial, one has by Lannes theory [20] that there is a lifting of Bi to a map

h: Bℤ/2 ^{}E(ξ),

which is unique up to homotopy. Thus the map

g_{∗} : Map(Bℤ/2, E(ξ))_{h} ^{}Map(Bℤ/2, BO(2))_{Bi}

is equivariant with respect to the action of the topological group Bℤ/2, acting in both mapping spaces by translation in the source.

Since ℤ/2 ^{i}^{}O(2) is central, the evaluation Map(Bℤ/2, BO(2))_{Bi} ^{ev}^{}BO(2) is a mod-2 equivalence. Also, an easy computation of the respective Lannes T functor shows that Map(Bℤ/2, E(ξ))_{Bi} ^{ev}^{}E(ξ) is a mod-2 equivalence. Denote

E(ξ_{1}) = Map(Bℤ/2, E(ξ))_{h} ×_{Bℤ/2} EBℤ/2.

Taking the respective Borel constructions one gets a diagram of principal fibrations

[figure: Bℤ/2 = = = Bℤ/2; S^{3}; E(ξ); h; BO(2); Bi; S^{3}; ‖; E(ξ_{1}); BO(2); Bπ; B^{2}ℤ/2; {= \atop =}; B^{2}ℤ/2; w_{2}] (20)

where all spaces are assumed 2-complete. Let ξ_{1} denote the quotient spherical fibration S^{3} ^{}E(ξ_{1}) ^{}BO(2).

Lemma 7.2. ξ_{1} is cohomologically trivial.

Proof. The Thom space for ξ_{1}, T (ξ_{1}), is defined as the homotopy cofibre of the projection E(ξ_{1}) ^{}BO(2). Its mod-2 cohomology is determined by the Thom isomorphism theorem

H̃^{∗}(T (ξ_{1})) ≅ U_{ξ_{1}} · H^{∗}(BO(2))

as H^{∗}(BO(2)) modules, where U_{ξ_{1}} ∈ H̃^{∗}(T (ξ_{1})) is the Thom class. The action of the Steenrod squares on the Thom class defines the Stiefel-Whitney classes of ξ_{1} by the equations

Sq^{i}(U_{ξ_{1}}) = U_{ξ_{1}} · w_{i}(ξ_{1}).

The action of the Steenrod algebra on the cohomology of T (ξ_{1}) is thus determined by its action on the Thom class, given above, the action on H^{∗}(BO(2)) and the Cartan formula.

We have then a long exact sequence of modules over both H^{∗}(BO(2)) and the Steenrod algebra

… ^{}H̃^{q}(T (ξ_{1})) ^{j^{∗}}^{}H^{q}(BO(2)) ^{p^{∗}}^{}H^{q}(E(ξ_{1})) ^{δ}^{}H̃^{q+1}(T (ξ_{1})) ^{}… (21)

where j^{∗} is determined by its value on U_{ξ_{1}}, j^{∗}(U_{ξ_{1}}) = e_{ξ_{1}}, the Euler class, which coincides with the top Stiefel-Whitney class.

We proceed by computing the Stiefel-Whitney classes for ξ_{1}.

Claim 7.3. The Stiefel-Whitney classes of ξ_{1} are trivial.

Proof. Write w(ξ_{1}) = 1 + w_{1}(ξ_{1}) + w_{2}(ξ_{1}) + w_{3}(ξ_{1}) + w_{4}(ξ_{1}) with w_{i}(ξ_{1}) ∈ H^{i}(BO(2)), for the total Stiefel-Whitney class of ξ_{1}. Since Bπ^{∗}(ξ_{1}) = ξ and ξ is cohomologically trivial Bπ^{∗}(w(ξ_{1})) = 1.

The action of the Steenrod squares on the Stiefel-Whitney classes of any spherical fibration is given by the Wu formula. This fact is exploited in [5], where possible total Stiefel-Whitney classes in certain invariant algebras are classified. In the particular case, where the base of a spherical fibration is BO(2), only 1, 1 + w_{1}, 1 + w_{1} + w_{2} and their products can form total Stiefel-Whitney classes (see [5, section 2]). So, w(ξ_{1}) = (1 + w_{1})^{r}(1 + w_{1} + w_{2})^{s} with r, s ≥ 0. One then checks that the only possibility that satisfies Bπ^{∗}(w(ξ_{1})) = 1 is r = s = 0 (see (19)). Hence the total Stiefel-Whitney class for ξ_{1} is trivial.

It follows that the long exact sequence (21) collapses to a short exact sequence

0 ^{}H^{∗}(BO(2)) ^{p^{∗}}^{}H^{∗}(E(ξ_{1})) ^{δ}^{}U_{ξ_{1}} · H^{∗+1}(BO(2)) ^{}0 .

There is a 3-dimensional class x_{3} in H^{∗}(E(ξ_{1})) with δ(x_{3}) = U_{ξ_{1}}. We now compute the Steenrod algebra action on x_{3}.

Since Sq^{i}(U_{ξ_{1}}) = U_{ξ_{1}} · w_{i}(ξ_{1}) = 0, we have Sq^{i}(x_{3}) ∈ p^{∗}(H^{∗}(BO(2))). Identifying H^{∗}(BO(2)) with its image under p^{∗}, write Sq^{2}(x_{3}) = λ_{1}w_{1}^{5} + λ_{2}w_{1}^{3}w_{2} + λ_{3}w_{1}w_{2}^{2}, λ_{i} ∈ ℤ/2.

Without loss of generality λ_{1} = λ_{3} = 0 or otherwise define x̄_{3} = x_{3}+λ_{1}w_{1}^{3}+λ_{3}w_{1}w_{2}. Then Sq^{2}(x̄_{3}) = λw_{1}^{3}w_{2} with λ = λ_{2} + λ_{3} and δ(x̄_{3}) = U_{ξ_{1}}.

It follows that (x_{3})^{2} = Sq^{3}(x_{3}) = Sq^{1}Sq^{2}x_{3} = 0 and so (Sq^{1}x_{3})^{2} = Sq^{2}(x_{3}^{2}) = 0. Hence Sq^{1}x_{3}, being considered as an element of H^{∗}(BO(2)), must be zero. Finally, Sq^{2}Sq^{2}x_{3} = Sq^{1}Sq^{2}Sq^{1}x_{3} = 0, and since w_{1}^{3}w_{2}^{2} ≠ 0, it follows that λ = 0.

We have thus obtained a choice of x_{3} ∈ H^{3}(E(ξ_{1})) with δ(x_{3}) = U_{ξ_{1}} and Sq^{i}(x_{3}) = 0 for all i ≥ 1 and therefore H^{∗}(E(ξ_{1})) ≅ H^{∗}(BO(2)) ⊗ H^{∗}(S^{3}) as algebras over the Steenrod algebra and the proof of the lemma is complete.

Let j_{n}: D_{2^{n}} ^{}O(2) denote the canonical inclusion of the dihedral group of order 2^{n} in O(2). Notice that j_{n} factors through j_{n+1} and the direct limit with respect to the j_{i}’s gives a discrete approximation for BO(2) at the prime 2 (in the sense of [12]).

Lemma 7.4. For every n ≥ 2, the pull-back fibration (Bj_{n})^{∗}(ξ) is trivial.

Proof. Recall that the class j_{n}^{∗}(w_{2}) = w ∈ H^{∗}(BD_{2^{n}}) classifies the central extension ℤ/2 ^{}D_{2^{n}} ^{}D_{2^{n−1}} and so the square

[figure: BD_{2^{n}}; Bj_{n}; BO(2); BD_{2^{n−1}}; Bj_{n−1}; BO(2); Bπ]

is a pull-back diagram. Notice that this means that the orbit spaces O(2)/D_{2^{n}} are the same for all n. Indeed, an easy diagram chase argument show that these orbit spaces are, one and all, homotopy equivalent to S^{1}.

Observe that the fibration ξ_{1} over BO(2) satisfies the same condition as ξ, namely it is cohomologically trivial. Hence the construction can be repeated inductively to obtain a sequence of spherical fibrations

[figure: ξ_{i}; S^{3}; E(ξ_{i}); BO(2); ξ_{i+1}; S^{3}; ‖; E(ξ_{i+1}); BO(2); Bπ] (22)

where each ξ_{i} is a pull-back of ξ_{i+1}.

Pulling back Diagram (22) along the successive maps Bj_{n}, we get the following diagram, where every square on the right is a pull-back square.

[figure: S^{3}; E((Bj_{n})^{∗}(ξ)); BD_{2^{n}}]

[figure: S^{3}; ‖; E((Bj_{n−1})^{∗}(ξ_{1})); BD_{2^{n−1}}]
⋮  ⋮  ⋮
[figure: S^{3}; E((Bj_{1})^{∗}(ξ_{n−1})); Bℤ/2]

[figure: S^{3}; ‖; E((Bj_{0})^{∗}(ξ_{n})); ∗]

Obviously, (Bj_{0})^{∗}(ξ_{n}) is a trivial fibration. Since (Bj_{n})^{∗}(ξ) is a successive pull-back of (Bj_{0})^{∗}(ξ_{n}) it is trivial as well.

To end the proof of the theorem notice that BO(2) is, up to completion, the homotopy colimit of the sequence … ^{}BD_{2^{m}} ^{}BD_{2^{m}+1} ^{}. . .. Thus in the category of fibrations and commutative diagrams between them, the fibration ξ is the homotopy colimit of … ^{}(Bj_{m})^{∗}(ξ) ^{}(Bj_{m+1})^{∗}(ξ) ^{}. . .. But we have seen that all these fibrations are trivial and furthermore, it is easy to observe that the product decomposition for the respective total spaces commute with the inclusions. Thus ξ is a trivial fibration and the proof is complete.

8. homotopy uniqueness of BS^{3}{2^{r}}

We are now ready to prove that any 2-complete space with the mod-2 cohomology of BS^{3}{2^{r}} coincides with it up to homotopy, whence the second part of Theorem 1.1.

The proof boils down to showing that structure 3 of Proposition 2.2 is not realizable as the cohomology of a space.

Proposition 8.1. There does not exist a space W with mod-2 cohomology

H^{∗}(W) ≅ P [w_{2}, w_{3}, x_{3}]/(x_{3}^{2}),

with Sq^{1}w_{2} = w_{3}, Sq^{2}w_{3} = w_{2}w_{3} and with β_{r−2}x_{3} = w_{2}^{2}.

Proof. Let W be a space with this cohomological structure. Thus H^{∗}(W) is isomorphic to H^{∗}(BSO(3)) ⊗ H^{∗}(S^{3}) as an algebra over the Steenrod algebra, with the additional higher Bockstein. Let i : ℤ/2 ^{}SO(3) be the map sending the generator to the diagonal matrix diag(−1, −1, 1). Then the centralizer of the image of i in SO(3) is O(2). By [20]

T_{ℤ/2}(H^{∗}(BSO(3)), Bi^{∗}) ≅ H^{∗}(BO(2))

and furthermore, by [13],

Map(Bℤ/2, BSO(3)) ≃ BO(2).

By Lannes theory there is a map f : Bℤ/2 ^{}W inducing the composite

H^{∗}(W) ^{proj}^{}H^{∗}(BSO(3)) ^{Bi^{∗}}^{}H^{∗}(Bℤ/2).

Thus we have

(23) T_{ℤ/2}(H^{∗}(W), f^{∗}) ≅

T_{ℤ/2}(H^{∗}(BSO(3)), Bi^{∗}) ⊗ T_{ℤ/2}(H^{∗}(S^{3}), ∗) ≅ H^{∗}(BO(2)) ⊗ H^{∗}(S^{3})

as algebras over the Steenrod algebra. Furthermore, since there are no relations among 1-dimensional elements in the object computed above, it follows that the calculation gives the cohomology of the mapping space

X = Map(Bℤ/2, W)_{f}.

Also, the evaluation map e : X ^{}W induces the same map on cohomology as the obvious inclusion

BO(2) × S^{3} ^{}BSO(3) × S^{3}

and it follows that β_{r−2}(x_{3}) = w_{2}^{2} in H^{∗}(X) up to the usual indeterminacy.

Lemma 8.2. The space X defined above fits in a fibration

S^{3} ^{}X ^{}BO(2).

Proof. We will show that the second Moore-Postnikov section of X is equivalent to BO(2). Indeed, since X is a 2-complete torsion space, its cohomological structure implies that π_{1}(X) ≅ ℤ/2. It is easy to calculate the cohomology of the universal cover, Y of X from the fibration

Y ^{}X ^{}Bℤ/2.

Thus H^{∗}(Y) ≅ P [w_{2}] ⊗ E[x_{3}], with β_{r}(x_{3}) = w_{2}^{2}. In particular the second integral cohomology of Y is torsion free. Since X is a torsion space, inspection of the Sss for the fibration above gives that the ℤ/2 action on H^{2}(Y; ℤ) is given by involution. Thus ℤ/2 acts on π_{2}(X) by involution. This determines the homotopy type of the second Moore-Postnikov section of X to be the 2-completion of BO(2) and the result follows.

Lemma 8.2 gives an immediate contradiction. On one hand X is a torsion space, but on the other hand we have established above that H^{∗}(X) ≅ H^{∗}(BO(2)) ⊗ H^{∗}(S^{3}) as algebras over the Steenrod algebra. By Theorem 7.1 X must be homotopy equivalent to a product S^{3} × BO(2), up to completion. This is of course impossible since both S^{3} and BO(2) have non-trivial rational cohomology. This completes the proof of the proposition and with that the proof of Theorem 1.1.

9. Homotopy Uniqueness for the odd prime case

Let p be an odd prime and let X be a classifying space for S^{2n−1}{p^{r}}. Thus n|p−1 and H^{∗}(X; 𝔽_{p}) is given by Equation (1). In particular this cohomology algebra appears as the cohomology of a finite group. Indeed let G denote the semi direct product of ℤ/p^{r} by ℤ/n, where the last acts on ℤ/p^{r} as a subgroup of its automorphism group. Then H_{β}^{∗}(X; 𝔽_{p}) ≅ H_{β}^{∗}(BG; 𝔽_{p}). Conversely,

Proposition 9.1. Let X be a space with H_{β}^{∗}(X; 𝔽_{p}) ≅ H_{β}^{∗}(BG; 𝔽_{p}), where G ≅ ℤ/p^{r} ⋊ ℤ/n. Then X ≃ BG up to p-completion.

Sketch of proof. The argument in this proof is similar to that of Theorem 6.2, so we indicate the proof without going into great detail.

Lannes theory gives that there is a map

f : Bℤ/p ^{}X,

such that f^{∗} coincides with the restriction map induced by the inclusion ℤ/p ^{ϕ}^{}G on mod-p cohomology. Consider the mapping space Map(Bℤ/p, X)_{f}.

We have to show how to recognize this as a classifying space for ℤ/p^{r}. This is done along the lines of the proof of Proposition 5.4 using the fact that ℤ/p^{r} is a T_{p}-group (Proposition 5.2). It follows that Map(Bℤ/p, X) ≃ Bℤ/p^{r}. Moreover the evaluation map ev^{∗} : H^{∗}(X) ^{}H^{∗}(ℤ/p^{r}) coincides with the restriction to the p-Sylow subgroup.

Now, one has an action of ℤ/n on Map(Bℤ/p, X)_{f}, which preserves the component, since it does so if X is replaced by BG. Furthermore, one has that the evaluation map is equivariant with respect to the trivial action of ℤ/n on X. Thus the evaluation map extends to a map on the Borel construction

f̄: Map(Bℤ/p, X)_{f} ×_{ℤ/n} Eℤ/n ^{}X,

and it is a triviality that this map induced a mod-p cohomology isomorphism. Notice that the source is homotopy equivalent to BG. This completes the proof.

10. Appendix: Some homological algebra

In this appendix we review the techniques used in section 6 in order to compute higher derived functors of the inverse limit functor. These techniques are essentially due to Aguadé [1] and are adapted here to serve our purposes.

Lemma 10.1 (Aguadé). Let G be a finite groups and let H < G be a subgroup. Let R be a commutative ring with a unit. Let K be the R[G] module defined by the short exact sequence

K ^{}R[G/H] ^{}R .

Then for any R[G]-module M there is a long exact sequence

(24) 0 → M^{G} → M^{H} → Hom_{G}(K, M) → H^{1}(G; M) → H^{1}(H; M) → …

… → H^{i}(H; M) → Ext_{G}^{i}(K, M) → H^{i+1}(G; M) → H^{i+1}(H; M) → …

For the proof the reader is referred to [1].

Consider the following categories:

A = [figure: G_{2} ⟳ 2; \underline{G_{2}/H}; 0; G_{1}/H; 1 ⟳G_{1}]

and

B =  0 [figure: G_{1}/H] 1 ⟳G_{1}

Here G_{1} and G_{2} are groups and H a common subgroup. The category A has three objects 0 , 1 and 2 . The automorphism groups of i is given by G_{i} for i = 1, 2 whereas the automorphism group of 0 is trivial. The morphism set from 0 to i , i = 1, 2, admit a natural G_{i} action and are isomorphic as G_{i}-sets to G_{i}/H. The category B is a full sub category of A with objects 0 and 1 .

Let R-mod denote the category of R-modules. For a small category C let C_{R}-mod denote the functor category from C to R-mod. Define the functor L ∈ A_{R}-mod which takes 0 to R and i to R[G_{i}/H]. The action of L on morphisms is taken to be the obvious one.

For any other M ∈ A_{R}-mod, let M_{i} denote M( i ), i = 0, 1, 2. Then

Hom_{A_{R}-mod}(L, M) ≅ Hom_{R}(R, M_{0}) ≅ M_{0}.

hence L is projective in A_{R}-mod. Let R ∈ A_{R}-mod denote the constant functor with value R. Then there is an exact sequence in A_{R}-mod:

0 ^{}K ^{}L ^{}R ^{}0

where K(0) = 0 and K(i) = K_{i} = ker(R[G_{i}/H] ^{}R) for i = 1, 2. This gives an exact sequence:

(25) 0 → Hom_{A_{R}-mod}(R; M) → Hom_{A_{R}-mod}(L; M) → Hom_{A_{R}-mod}(K; M) →

→ Ext_{A_{R}-mod}^{1}(R; M) → 0

and isomorphisms:

(26) Ext_{A_{R}-mod}^{i}(K; M) ≅ Ext_{A_{R}-mod}^{i+1}(R; M) ; i ≥ 1

Lemma 10.2.

Ext_{A_{R}-mod}^{n}(K, M) ≅ ⨁_{i=1,2} Ext_{R[G_{i}]}^{n}(K_{i}, M_{i})

Proof. Let #_{i} : A_{R}-mod → R[G_{i}]-mod, i = 1, 2, be the forgetful functors defined by #_{i}(M) = M_{i}. There are functors going the other direction s_{i} : R[G_{i}]-mod → A_{R}-mod, i = 1, 2, defined by s_{i}(N)( j ) = N if i = j and 0 if i ≠ j. The functors s_{i} are clearly exact and left adjoint to the forgetful functors. Thus

Hom_{A_{R}-mod}(s_{i}N, M) ≅ Hom_{R[G_{i}]}(N, M_{i}).

It follows that the functors #_{i} preserve injectives; that is, if I ∈ A_{R}-mod is injective then I_{i} is an injective R[G_{i}]-module for i = 1, 2.

So, given an object of A_{R}-mod, M and an injective resolution M → I_{∗} we have that I( i )_{∗} is an injective resolution for M_{i} and this together with the definition of the functor K, given above, implies:

(27) Ext_{A_{R}-mod}^{∗}(K, M) := H (Hom_{A_{R}-mod}(K, I_{∗})) ≅

≅ H(⨁_{i=1,2} Hom_{R[G_{i}]}, (K_{i}, I( i )_{∗})) ≅ ⨁_{i=1,2} Ext_{R[G_{i}]}^{∗}(K_{i}, M_{i}) .

For a small category C and a functor F ∈ C_{R}-mod the n-th cohomology group of C with coefficients in F is defined to be the group Ext_{C_{R}-mod}^{n}(R, F), where R ∈ C_{R}-mod is the constant functor. These cohomology groups are usually interpreted as the higher derived functors of the inverse limit of the functor F over the category C. We are now ready to prove the following

Proposition 10.3. There is a long exact sequence:

(28) 0 → H^{0}(A; M) → M_{0} → ⨁_{i=1,2} M_{i}^{H}/M_{i}^{G_{i}} → H^{1}(A; M) →

→ ⨁_{i=1,2} H^{1}(G_{i}; M_{i}) → ⨁_{i=1,2} H^{1}(H; M_{i}) → H^{2}(A; M) →

→ ⨁_{i=1,2} H^{2}(G_{i}; M_{i}) → ⨁_{i=1,2} H^{2}(H; M_{i}) → …

and for the cohomology of B:

(29) 0 → H^{0}(B; M) → M_{0} ^{}M_{1}^{H}/M_{1}^{G_{1}} → H^{1}(B; M) →

→ H^{1}(G_{1}; M_{1}) → H^{1}(H; M_{1}) → H^{2}(B; M) → H^{2}(G_{1}; M_{1}) → …

Proof. Start by adding the long exact sequences (24) for the pairs of groups (G_{1}, H) and (G_{2}, H) and use the isomorphisms (26) and Lemma 10.2 to obtain the long exact sequence of the bottom row in the diagram:

[figure: 0; H^{0}(A; M); M_{0}; Hom_{A_{R}-mod}(K, M)]

[figure: 0; ⨁_{i=1,2} M_{i}^{H}/M_{i}^{G_{i}}; Hom_{A_{R}-mod}(K, M); ‖]

[figure: H^{1}(A, M); 0; ⨁_{i=1,2} H^{1}(G_{i}, M_{i}); ⨁_{i=1,2} H^{1}(H, M_{i}); H^{2}(A; M); …]

Observe that the top row is a rewriting of 25 and the vertical arrows follows because M_{0} → Hom_{A_{R}-mod}(K, M) ≅ ⊕_{i=1,2} Hom_{G_{i}}(K_{i}, M_{i}) clearly factors through ⊕_{i=1,2}M_{i}^{H}/M_{i}^{G_{i}}. The sequence (28) follows by diagram chasing. The proof for (29) is similar.

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Departament de Matemàtiques, Univeritat Autònoma de Barcelona, E-08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Dept. of Mathematics, Northwestern University, 2033 Sheridan Rd. Evanston, IL 60208

  E-mail address: ran@math.nwu.edu