ON THE HOMOTOPY TYPE OF BG FOR CERTAIN FINITE 2-GROUPS G

CARLOS BROTO AND RAN LEVI

March 28 1995

1. Introduction

Let G and H be finite p-groups, which agree on some of their cohomological invariants. The question whether G and H are isomorphic groups has a negative answer in general. For instance mod-2 cohomology algebra cannot differ between dihedral 2-groups of various orders, even if the action of the Steenrod algebra is taken into account. A similar fact is known for various metacyclic 2-groups, among which one counts the quaternion and semidihedral groups [15]. In a recent paper, I. Leary exhibited examples of distinct 3-groups having isomorphic integral cohomology rings [10]. Further examples of distinct finite 2-groups, which mod-2 cohomology algebra cannot tell apart are given in [16]. However if one requires, in a sense that will be made precise later, that the given cohomological data on G includes its Bockstein spectral sequence then, although we are not able to prove the contrary, we are not aware of an example of two distinct p-groups which cohomology cannot tell apart.

The question becomes significantly more complicated when one considers a finite pgroup G and a p-complete space X such that BG and X have the same cohomological invariants and asks whether X and BG have the same homotopy type. This last question is reduced to the previous one if one can show for a particular example that X has the homotopy type of BH for “some” finite p-group H. One might thus wonder to what extent does cohomological data determine isomorphism type for finite p-groups G, or more generally, the homotopy type of BG.

In this note we shall mainly be interested in the second question. We restrict attention to the prime 2 and study some examples of finite 2-groups G, where the homotopy type of BG is determined by its cohomology. Further examples where the results are not as conclusive are also given.

Unless otherwise specified all spaces are assumed to have the homotopy type of a 2-complete CW complex. The cohomology of a space is considered as an object in the category K_{β}. It is, roughly speaking, the category of unstable algebras over the Steenrod algebra, in which the higher Bockstein operators are being taken into account. It is immediate, for instance, that every finite abelian 2-group G gives rise to a unique object of K_{β}, characterizing BG up to homotopy and hence the group G, up to isomorphism. We say that two spaces X and Y are comparable if X and Y give rise to isomorphic objects in K_{β}. We say that the homotopy type of a space X is determined by its cohomology if any space Y, comparable to X, is homotopy equivalent to it.

C. Broto is partially supported by DGICYT grant PB91-0467.

R. Levi is supported by a DFG grant

The first family of groups, we consider, is the dihedral 2-groups.

Theorem 1.1. Let D_{2^{n}} denote the dihedral group of order 2^{n}. Then the homotopy type of BD_{2^{n}} is determined by its cohomology.

A considerably larger family of 2-groups, containing the dihedral and quaternion groups of order 8, are the extra special 2-groups which we consider next.

Theorem 1.2. Let G denote an extra special 2-group. Then the homotopy type of BG is determined by its cohomology. In fact in almost all cases, the homotopy type of BG for an extra special 2-group G is determined by the cohomology algebra of BG.

The examples we present next appear less favorable. Indeed a basic question in the subject is whether in general cohomology forces at least the fundamental group of a space X, given that X is comparable to BG for some finite p-group G. In the following examples we were not able to determine a complete answer to this question. Thus one might suspect that an example of a space X comparable to BG yet not homotopy equivalent to it might arise from studying the groups G considered below.

Proposition 1.3. Let G = Q_{2^{n}} denote the generalized quaternion group of order 2^{n}. Then for n ≤ 5 the homotopy type of BG is determined by its cohomology. For n > 5 let X be a space comparable to BG. Then either π_{1}X ≅ Q_{2^{k}} for some 4 ≤ k ≤ n − 2 or X ≃ BG.

The question of whether or not the homotopy type of BQ_{2^{n}} is determined by its cohomology for n > 5 is reduced in section 5 below to understanding free actions of the group Q_{16} on certain simply-connected spaces Y such that ΩY is homotopy equivalent to S^{3}{2^{r}}, the homotopy fibre of a degree 2^{r} map on the sphere S^{3}. In particular showing that such actions do not exist would imply that the homotopy type of BQ_{2^{n}} is determined by its cohomology for all n.

Proposition 1.4. Let G = SD_{2^{n}} denote the semidihedral group of order 2^{n}. Let X be a nilpotent space, comparable to BG. Then π_{1}X ≅ SD_{2^{k}} for some k ≥ 4.

In section 2 below we give a precise definition of what it means for a space X to be determined by its cohomology. If one considers cohomology as a functor from the homotopy category to the category K_{β}, then the class of spaces which are determined by cohomology is understood to be the subclass of the homotopy category, on which the restriction of this functor is 1-1 when considered only as an operation on objects. Thus one might wonder whether this class is closed under simple operations such as products, wedges etc. It is hard to believe that one could produce successful answers to those questions in general. However in our more rigid context we are able to prove the following.

Proposition 1.5. Let G be a finite p-group such that the homotopy type of BG is determined by cohomology. Then for any elementary abelian p-group E, the same holds for B(G × E).

Next we restrict our attention to the isomorphism type of the groups under consideration.

Definition 1.6. Let D_{p} denote the class of finite p-groups which are determined among finite p-groups by their mod-p cohomology as objects of K_{β}. Namely if G ∈ D_{p} and π is another finite p-group, then comparability of BG and Bπ implies that G is isomorphic to π.

With this definition we are now able to state a corollary of our results.

Corollary 1.7. Finite abelian 2-groups, extra special 2-groups, dihedral quaternion and semidihedral groups belong to the class D_{2}. Furthermore, for any prime p, the class D_{p} is closed under direct products with finite elementary abelian p-groups.

It is not hard to observe that in fact, the isomorphism type of G, for G an extra special p-group, p odd, is determined by the cohomology algebra of BG. This result can be extended to any p-group of nilpotency class 2 under certain favorable hypotheses.

The main task of this paper is the observation that cohomology as an object of the category K_{β} can sometimes determine the homotopy type of BG for finite nonabelian 2-groups G and indeed, where considering it as an object in the categories K or U of unstable algebras or modules over the Steenrod algebra fails to do so. The reader is referred to section 8 for further discussion.

The authors are grateful to the Centre de Recerca Matemàtica, Barcelona, and the Mathematisches Institut der Universität Heidelberg for their kind hospitality and support while part of this research was done. We also thank Jaume Aguadé, Carles Casacuberta, Richard Kane, Ian Leary, Björn Schuster and Saïd Zarati for useful conversations.

2. Conventions and Terminology

All spaces considered are assumed to have the homotopy type of a CW complex and to be 2-complete in the sense of Bousfield and Kan [6]. We denote H^{∗}(−; 𝔽_{2}) by H^{∗}(−). Let A denote the mod-2 Steenrod algebra. Let K denote the category of unstable algebras over A. By “EMSS” we abbreviate the Eilenberg-Moore spectral sequence and by “SSS” the Serre spectral sequence, both for mod-2 cohomology, unless other coefficients are specified. The Bockstein spectral sequence for mod-2 cohomology is abbreviated by “BSS”.

Definition 2.1. Let K be an unstable algebra over A. A BSS for K is a spectral sequence {E_{i}(K), β_{i}}_{i=1}^{∞} of differential graded algebras where the differentials have degree one and such that (see [7, 12])

(1) E_{1}(K) = K and β_{1} = Sq^{1} is the primary Bockstein operator.

(2) If x ∈ K^{even} then β_{2}(x^{2}) = x Sq^{1}x + Sq^{|x|}Sq^{1}x, provided x^{2} represents a nonzero class in E_{2}(K).

(3) If x ∈ E_{i}(K)^{even}, i ≥ 2 then β_{i+1}(x^{2}) = x β_{i}(x), provided x^{2} represents a nonzero class in E_{i+1}(K).

Let K_{β} denote the category whose objects are pairs (K; {E_{i}(K), β_{i}}_{i=1}^{∞}), where K is an unstable A-algebra and {E_{i}(K), β_{i}}_{i=1}^{∞} is a BSS for K. We will frequently abbreviate an object (K; {E_{i}(K), β_{i}}_{i=1}^{∞}) of K_{β} by K_{β}. A morphism f : K_{β} ⟶ K′_{β} in K_{β} is a family of morphisms {f_{i}}_{i=1}^{∞}, where f_{1} : K ⟶ K′ is a morphism of Aalgebras and for each i ≥ 2, f_{i} : E_{i}(K) ⟶ E_{i}(K′) is a morphism of differential graded algebras, which as a morphism of graded algebras, is induced by f_{i−1}. The mod-2 cohomology of a space X together with its BSS is a typical object of the category K_{β} that will be denoted by H_{β}^{∗}(X). Thus mod-2 cohomology can be thought of as a functor from the category of spaces to the category K_{β}.

This category has been considered in [2] and [3]. Examples of spaces X with H^{∗}(X) ≅ P [x_{2}] ⊗ E[y_{3}], realizing every BSS, which could possibly be associated to this unstable algebra by our set of axioms are discussed in [2]. Thus we regard (1),

(2) and (3) above as a reasonable set of axioms for a BSS associated to an unstable algebra over the Steenrod algebra.

Definition 2.2. Two spaces X and Y are said to be comparable if there is an isomorphism

ϕ : (H^{∗}(X); {E_{i}(H^{∗}(X)), β_{i}}_{i=1}^{∞}) ⟶ (H^{∗}(Y); {E_{i}(H^{∗}(Y)), β_{i}}_{i=1}^{∞})

in K_{β}. We say that X is determined by cohomology if given a space Y comparable to X, there is a homotopy equivalence X ≃ Y.

Definition 2.3. Let K_{β} be an object in K_{β}. Let K be the underlying unstable algebra. We say that K_{β} is weakly generated by x_{1}, ⋯, x_{n} ∈ K if any endomorphism f of K_{β}, such that the restriction of f_{1} to the vector subspace of K spanned by x_{1}, ⋯, x_{n} is an isomorphism, is an isomorphism in K_{β}. In that case we shall say that x_{1}, ⋯, x_{n} is a set of generators for K_{β} in K_{β}.

Throughout the end of this note we shall be calculating the EMSS of certain fibrations such that the base space has a finite 2-group as a fundamental group. Thus we shall use the fact that the action of a finite 2-group on a finite dimensional 𝔽_{2}-vector space is always nilpotent and so the EMSS is granted to converge to the right target by [8].

Next we point out that in order to grant that a mod-2 cohomology isomorphism through dimension 2 induces an isomorphism on fundamental groups, one might be tempted to assume that all spaces considered are nilpotent. This assumption is not necessary in most cases. Indeed [6, I.6.2] gives a slightly more restricted result, still suitable for our purposes, without the nilpotency assumption. We will have to assume nilpotency however in section 5 below where this lemma of Bousfield and Kan cannot be applied. A final remark is that in our examples the 2-completeness assumption of the spaces X under consideration is not essential to the construction of maps from X to suitable classifying spaces BG. One can define the concept of comparability dropping the 2- completeness assumption and then, assuming that X is comparable to BG for some finite 2-group G, one might be able to construct a map from X to BG and then examine its cohomological properties. If the map happens to induce an isomorphism on cohomology then our assertions would hold for the 2-completion of X.

3. The Dihedral Groups

This section is devoted to the proof of Theorem 1.1. The dihedral group of order 8 is a particular case of an extra special 2-group. However, the calculation here applies to dihedral 2-groups of arbitrary order and is different in nature from the one performed in the proof of Theorem 1.2 below.

Let D_{2^{n}} denote a dihedral group of order 2^{n}.

D_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1; yxy = x^{−1}⟩.

Recall from [11] for n ≥ 3

H^{∗}(BD_{2^{n}}) ≅ P [x_{1}, y_{1}, w_{2}]/(x^{2} + xy),

where the degrees are given by the subscripts. The action of the Steenrod algebra is given by Sq^{1}(w) = wy and all the other operations are determined by the regular axioms. The Bockstein spectral sequence is determined by the requirement that

E_{2} ≅ E_{3} ≅ ⋯ ≅ E_{n−1} ≅ P [w^{2}] ⊗ E[xw]

and β_{n−1}(xw) = w^{2}.

Notice that the algebra generators for H^{∗}(BD_{2^{n}}) are the same as the generators in K_{β}. Thus for the proof of Theorem 1.1, it suffices to show that any space X comparable to BD_{2^{n}} admits a map X ⟶ BD_{2^{n}}, inducing an isomorphism on cohomology up to dimension 2.

Let X be a space comparable to BD_{2^{n}}. Let D_{4} denote an elementary abelian 2group of rank 2. There is a map ϕ_{2} : X ⟶ BD_{4}, classifying the classes x, y ∈ H^{1}(X). Consider the tower of principal fibrations

⋯ ⟶ BD_{2^{n}} ⟶^{π_{n−1}} BD_{2^{n−1}} ⟶^{π_{n−2}} ⋯ ⟶^{π_{3}} BD_{8} ⟶^{π_{2}} BD_{4}. (3.1)

Each projection π_{i} corresponds to the central extension

0 ⟶ ℤ/2 ⟶ D_{2^{i+1}} ⟶ D_{2^{i}} ⟶ 1

classified by the w ∈ H^{2}(BD_{2^{i}}) if i ≥ 3 and by x^{2} + xy ∈ H^{2}(BD_{4}) if i = 2.

Lemma 3.1. The map ϕ_{2} : X ⟶ BD_{4} factors through BD_{2^{n}}.

Proof. Clearly the composite

X ⟶^{ϕ_{2}} BV ⟶^{x^{2}+xy} K(ℤ/2, 2)

is null-homotopic. Hence ϕ_{2} factors through BD_{8}. Thus we must show that the composite

X ⟶^{ϕ_{n−k}} BD_{2^{n−k}} ⟶^{w} K(ℤ/2, 2)

is null-homotopic for k ≥ 1.

Indeed it is easy to see by applying naturality of Sq^{1} that ϕ_{n−k}^{∗}(w) = aw for some a ∈ 𝔽_{2}. If a = 1 then ϕ_{n−k}^{∗} is an isomorphism, contradicting the existence of different Bockstein operators in H^{∗}(X) and H^{∗}(BD_{2^{n−k}}). Thus ϕ_{n−k}^{∗}(w) = 0 as required.

To complete the proof of Theorem 1.1 we must show that ϕ_{n}^{∗}w = w. Suppose ϕ_{n}^{∗}w = 0. An easy EMSS calculation for the fibration

F ⟶^{j} X ⟶^{ϕ_{n}} BD_{2^{n}}

shows that

H^{∗}(F) ≅ P [v_{2}] ⊗ E[α_{1}],

as a P [v]-module, where v = j^{∗}(w) ∈ H^{2}(X).

Since β_{n−1}(xw) = w^{2} in H_{β}^{∗}(X) and j^{∗}(x) = 0, it follows by naturality of the BSS that β_{r}(αv) = v^{2} in H_{β}^{∗}(F) for some r < n − 1, which in turn easily implies that β_{r}(α) = v.

This determines the homotopy type of F to be that of Bℤ/2^{r}. Hence X ≃ BH, where H fits in a group extension of the form

0 ⟶ ℤ/2^{r} ⟶ H ⟶ BD_{2^{n}} ⟶ 1.

We argue that H ≅ D_{2^{n+r}} and having done so, we obtain a contradiction to our hypothesis, for if X ≅ BD_{2^{n+r}}, r ≥ 1 then H^{∗}(X) gives rise to a different object of K_{β} than the one assumed.

Indeed, consider the composite

Bℤ/2^{r} ⟶ X ⟶^{ϕ_{n}} BD_{2^{n}}.

Since ϕ_{n}^{∗}w = 0, it follows that ϕ_{n} lifts to a map ϕ_{n+1} : X ⟶ BD_{2^{n+1}}. One observes immediately that the homotopy fibre of ϕ_{n+1} is Bℤ/2^{r−1}. The same argument as above implies that ϕ_{n+1}^{∗}w = aw for some a ∈ 𝔽_{2}. But if a = 1 then ϕ_{n+1} is an equivalence which essentially means r = 1 and we get the desired contradiction. Thus if r > 1 then a = 0 and one proceeds inductively. Notice that the process must stop for ϕ_{n+r} : X ⟶ BD_{2^{n+r}} has to be an equivalence. This completes the proof of Theorem 1.1.

4. The extra special 2-Groups

An extra special 2-group is a central extension of an elementary abelian 2-group by a cyclic group of order 2. The cohomology of extra special 2-groups was computed by Quillen [14]. We start by recalling Quillen’s result. A good general reference for the subject is [4].

Let G be an extra special 2-group. Thus G fits in a central extension

0 ⟶ ℤ/2 ⟶ G ⟶ E ⟶ 0. (4.1)

where E is an elementary abelian 2-group of rank n. Write

H^{∗}(BE) ≅ P [x_{1}, x_{2}, ⋯, x_{n}].

Then the extension 4.1 is classified by some quadratic form q ∈ H^{2}(BE).

Theorem 4.1 (Quillen). Let G be as above. Then

H^{∗}(BG) ≅ P [x_{1}, x_{2}, ⋯, x_{n}]/(q_{0}, q_{1}, ⋯, q_{h}) ⊗ P [ζ]

with q_{0} = q, q_{1} = Sq^{1}q_{0} and q_{i} = Sq^{2^{i−1}}q_{i−1}. The elements x_{1}, x_{2}, ⋯, x_{n} are degree 1 elements inflated from H^{1}(BE). The number 2^{h} can be interpreted as the index of a maximal elementary abelian subgroup of G. The element ζ has degree 2^{h} and can be chosen to be any element of this degree, restricting non-trivially to the kernel in the extension 4.1 above.

The following theorem deals with the Bockstein spectral sequence for H^{∗}(BG) and is proven in [4].

Theorem 4.2. Let G be an extra special 2-group with h ≥ 2. Then the E_{2} page of the Bockstein spectral sequence for H^{∗}(BG) is given by

E_{2}^{∗} = E[γ_{1}, γ_{3}, γ_{4} ⋯, γ_{h}] ⊗ P [ζ]

with |γ_{1}| = 3, |γ_{j}| = 2^{j−1} for j ≥ 3 and |ζ| = 2^{h}.

For 3 ≤ r ≤ h we have,

E_{r}^{∗} = E[γ_{1}γ_{3} ⋯ γ_{r}, γ_{r+1}, ⋯, γ_{h}] ⊗ P [ζ].

Finally, E_{h+1}^{∗} = E_{h}^{∗} and E_{r}^{∗} = 𝔽_{2} for r ≥ h + 2.

The differentials are given by β_{2}(γ_{1}) = γ_{3}, β_{r}(γ_{1}γ_{3} ⋯ γ_{r}) = γ_{r+1}, β_{r}(γ_{i}) = 0 for r < i, β_{r}(ζ) = 0 for all r and β_{h+1}(γ_{1}γ_{3} ⋯ γ_{h}) = ζ.

Notice that the case h = 0 corresponds to elementary abelian 2-groups and h = 1 to ℤ/4 × E or D_{8} × E where E is elementary abelian. The first case is abelian and thus obvious. We will remark about D_{8} × E later.

Lemma 4.3. Let G be an extra special 2-group with h ≥ 2. Then H_{β}^{∗}(BG) is weakly generated by H^{1}(BG).

Proof. Let K_{β} denote H_{β}^{∗}(BG). Let f be an endomorphism of K_{β}, such that the corresponding endomorphism f_{1} of K, restricted to dimension 1, is an isomorphism. We must show that f is an isomorphism.

Since f_{1}x_{i} = x_{i} for all i, it follows that f_{r} maps the exterior generators in E_{r}^{∗}(K) to themselves for all r. Thus we get at the h + 1 page of the spectral sequence

f_{h+1}(ζ) = f_{h+1}β_{h+1}(γ_{1}γ_{3} ⋯ γ_{h}) = β_{h+1}f_{h+1}(γ_{1}γ_{3} ⋯ γ_{h}) = ζ.

On the other hand f_{j} is induced by f_{j−1}. Thus we must have f_{1}(ζ) = ζ + p, for some polynomial p = p(x_{1}, ⋯, x_{n}). By Theorem 4.1 the self map of K which takes x_{i} to itself for every i and ζ to ζ + p is an isomorphism, hence the proof is complete.

We are now ready to prove Theorem 1.2.

Assume that X is a space with

H^{∗}(X) ≅ H^{∗}(BG)

as algebras, where G is an extra special 2-group of order 2^{n+1} classified by the quadratic form q and with maximal elementary abelian 2-subgroup of index 2^{h}. We will show:

(1) If h ≥ 3, X ≃ BG.

(2) if h ≥ 1 and H_{β}^{∗}(X) ≅ H_{β}^{∗}(BG), then X ≃ BG.

We start by proving (1). Suppose h ≥ 3 and let X be a space such that H^{∗}(X) ≅ H^{∗}(BG) as algebras. Let G be given by the central extension 4.1. The argument above yields that any map X ⟶^{p} BE inducing an isomorphism on H^{1}(−), lifts to a map X ⟶^{ψ} BG such that ψ^{∗} is an isomorphism up to dimension 2^{h}−1. Furthermore, it is possible to choose ψ such that under a suitable change of basis, ψ^{∗}(ζ) = ϵζ for some ϵ ∈ 𝔽_{2}. If ϵ = 1 then ψ^{∗} is an isomorphism. Thus assume that ϵ = 0.

The EMSS for the fibration

F ⟶ X ⟶^{ψ} BG

is easily shown to collapse at E_{2} and

E_{2} ≅ E_{∞} ≅ P [v] ⊗ E[α]

with |v| = 2^{h} and |α| = 2^{h} − 1, as a P [v]-module. By [1] it follows that either |v| = 2 or |v| = 4. Hence we are in the case h ≤ 2, contradicting our hypothesis.

Next we turn to the proof of (2). Consider the case h ≥ 2. Indeed, there is a map X ⟶^{p} BE corresponding to the projection BG ⟶^{π} BE. Since the composite

X ⟶^{p} BE ⟶^{q} K(ℤ/2, 2)

is null-homotopic, there is a map X ⟶^{ψ} BG, lifting p. Moreover, since p induces an isomorphism in 1-dimensional cohomology and h ≥ 2, it follows that any such lift induces an isomorphism in cohomology up to dimension 2^{h} − 1 ≥ 3. The result follows by Lemma 4.3.

For h = 1 it remains to consider the case, where G ≅ D_{8} × E with E elementary abelian. This case is a consequence of Theorem 1.1 proved in the previous section and Proposition 1.5 which is proved in section 7.

This completes the proof of Theorem 1.2.

Remark 4.4. Notice that the conclusion X ≃ BG is not true for h = 1 unless H_{β}^{∗}(X) ≅ H_{β}^{∗}(BG), for dihedral groups of different orders produce counter examples. We are not aware of a suitable example for h = 2.

5. The Quaternion Groups Q_{2^{n}}

The quaternion groups are given as follows

Q_{2^{n}} = ⟨x, y|x^{2^{n−2}} = y^{2}; x^{2^{n−1}} = 1; yxy^{−1} = x^{−1}⟩.

Recall the cohomological structure of BQ_{2^{n}}.

H^{∗}(BQ_{2^{n}}) ≅ P [x_{1}, y_{1}, v_{4}]/(α, β)

where α = x^{2} + xy + y^{2}, β = x^{2}y + y^{2}x if n = 3 and α = x^{2} + xy, β = y^{3} if n ≥ 4. The Bockstein spectral sequence is determined by the regular axioms for Sq^{1} and the requirement that β_{n}(c) = v, where c ∈ H^{3}(BQ_{2^{n}}) is the unique generator. Notice that the n − th Bockstein is defined here with no indeterminacy and that the class c is a product of 1-dimensional generators. Thus H^{∗}(BQ_{2^{n}}) is weakly generated by x and y as an object of K_{β}.

The central extension

0 ⟶ ℤ/2 ⟶ Q_{2^{n}} ⟶ D_{2^{n−1}} ⟶ 0

is characterized by the class q_{2} ∈ H^{∗}(BD_{2^{n−1}}), where q_{2} = x^{2} + xy + y^{2} if n = 3 and q_{2} = w + y^{2} if n ≥ 4.

The group Q_{8} is an extra special 2-group. Thus we will assume n ≥ 4.

Proposition 5.1. Suppose X is comparable to BQ_{2^{n}} for some n ≥ 4. Then π_{1}X is isomorphic to Q_{2^{k}} for some 4 ≤ k ≤ n. In particular if n = k then X ≃ BQ_{2^{n}}.

Proof. There is a map ϕ_{2} : X ⟶ BD_{4}, where D_{4} = ℤ/2 ⊕ ℤ/2, classifying x, y ∈ H^{∗}(X). Consider the tower of principal fibrations

⋯ ⟶ BD_{2^{n}} ⟶^{π_{n−1}} BD_{2^{n−1}} ⟶^{π_{n−2}} ⋯ ⟶^{π_{3}} BD_{8} ⟶^{π_{2}} BD_{4}. (5.1)

Recall that the map π_{2} is classified by the class x^{2} + xy ∈ H^{2}(BD_{4}) and that each map π_{i} for i ≥ 3 is classified by w ∈ H^{2}(BD_{2^{i}}).

Since ϕ_{2}^{∗}(x^{2} + xy) = 0, ϕ_{2} lifts to ϕ_{3} : X ⟶ BD_{8} inducing isomorphism in 1dimensional cohomology. Suppose that for some i ≥ 3 there is a map ϕ_{i} : X ⟶ BD_{2^{i}} factoring ϕ_{2}. Then ϕ_{i}^{∗} is an isomorphism in dimension 1 and by applying naturality of Sq^{1} one easily observes that ϕ_{i}^{∗}(w) = by^{2} for some b ∈ 𝔽_{2}. If b = 0 then ϕ_{i} lifts further to a map ϕ_{i+1} : X ⟶ BD_{2^{i+1}} and the argument may be iterated. On the other hand if for some k ≥ 4, ϕ_{k−1}^{∗}(w) = y^{2} then ϕ_{k−1}^{∗}(w + y^{2}) = 0 and the composite

X ⟶^{ϕ_{k−1}} BD_{2^{k−1}} ⟶^{w+y^{2}} K(ℤ/2, 2)

is null-homotopic. Thus ϕ_{k−1} lifts to a map λ_{k}: X → BQ_{2^{k}}, inducing an isomorphism on 1,2 and 3-dimensional cohomology. Hence in this case π_{1}(X) ≅ Q_{2^{k}} for some k ≥ 4 ([6, I,6.2].) It remains to show that the lifting process actually finishes at some k ≤ n. Thus suppose that it doesn’t. Then ϕ_{i} is defined for every i ≥ 2 and ϕ_{i}^{∗}(w) = 0. Define F_{i} to be the homotopy fibre of ϕ_{i}, thus obtaining a sequence of fibrations

[figure: ⋯; F_{k−1}; ⋯; F_{i}; j; ⋯; F_{3}; F_{2}; ⋯; X; ϕ_{k−1}; ⋯; X; ϕ_{i}; ⋯; X; ϕ_{3}; X; ϕ_{2}; =; =; =; =; =; =; ⋯; BD_{2^{k−1}}; ⋯; BD_{2^{i}}; ⋯; BD_{8}; BD_{4}]

An EMSS calculation yields

H^{∗}F_{i} ≅ E[a_{1}, a_{2}] ⊗ P [v_{4}]

as P [v_{4}]-modules, where v_{4} is the restriction of v_{4} ∈ H^{4}(X). Moreover by naturality of the EMSS, the map F_{i+1} ⟶ F_{i} induces an isomorphism in even dimensional cohomology and it is trivial in odd dimensions.

Now consider each fibration separately. Since β_{n}(x^{3}) = v in H_{β}^{∗}(X), we obtain that β_{r_{i}}(a_{1}a_{2}) = v in H^{∗}(F_{i}) for some r_{i} < n and therefore β_{s_{i}}(a_{1}) = a_{2} for some s_{i} ≤ r_{i}. If s_{i} = r_{i} for some i then F_{i} is comparable to Bℤ/2^{r_{i}} and thus F ≃ Bℤ/2^{r_{i}}. In that case a similar argument to the one presented in the proof of Theorem 1.1 shows that X ≃ BQ_{2^{i+r_{i}}}, which in turn implies that i + r_{i} = n by considering the action of the higher Bockstein operators, distinguishing between classifying spaces of different quaternion groups. Assume now s_{i} < r_{i} for all i. Comparing cohomology of F_{i+1} and F_{i}, we observe that r_{i+1} < r_{i}. Hence, at the (k − 1)-st step we have a sequence of inequalities:

1 ≤ s_{k−1} < r_{k−1} < ⋯ < r_{3} < r_{2} < n

from which it follows that the lifting process has to stop at some k, 4 ≤ k ≤ n.

Corollary 5.2. The homotopy type of BQ_{16} is determined by cohomology.

Proposition 5.3. Let X be a space comparable to BQ_{2^{n}} with π_{1}X ≅ Q_{2^{k}} for some 4 < k < n, and let X̃ be its universal cover, then

(a) H_{β}^{∗}(X̃) = E[z_{3}] ⊗ P [v_{4}] with β_{n−k}(z_{3}) = v_{4}

(b) The action of Q_{2^{k}} on H^{∗}(X̃, ℤ) is trivial.

(c) The quotient space X̃/Q_{2^{k−1}} is comparable to BQ_{2^{n−1}}.

Proof. The EMSS for X̃ ⟶ X ⟶ BQ_{2^{k}} shows that

H^{∗}X̃ = E[z_{3}] ⊗ P [v_{4}]

where v_{4} is the restriction of v_{4} ∈ H^{4}(X). Furthermore, by naturality of the BSS, β_{t}(z_{3}) = v_{4} for some t < n . It follows that X̃ is of finite type [2, prop 5.7] and that the rings H^{∗}(X̃; ℤ) and ℤ/2^{t}[v̄_{4}] have isomorphic augmentation ideals. The generator v̄_{4} reduces to v_{4} mod 2. We proceed in proving (c). The proofs of (a) and (b) will follow.

There is a homomorphism Q_{2^{n}} ⟶ ℤ/2 given by sending x to the generator of ℤ/2 and y to 1. The kernel of this homomorphism is a quaternion subgroup of Q_{2^{n}} of order 2^{n−1}. Let X ⟶^{α} BQ_{2^{k}} denote the map classifying the universal cover for X. Then there is a commutative diagram

[figure: X̃; X̃; =; (A); X_{1}; α_{1}; X; α; Bℤ/2; =; (B); BQ_{2^{k−1}}; BQ_{2^{k}}; Bℤ/2; x]

Calculating the EMSS for (A) and (B) and using naturality, it follows that H^{∗}(X_{1}) ≅ H^{∗}(BQ_{2^{n−1}}) as an algebra. Obviously π_{1}X_{1} ≅ Q_{2^{k−1}}. Moreover by naturality of the BSS, β_{r}(x^{3}) = v_{4} in H^{∗}(X_{1}), for some r ≤ n − 1. Thus X_{1} is comparable to BQ_{2^{r}}.

Next consider the integral SSS for (A).

E_{2}^{p,q} ≅ H^{p}(ℤ/2; H^{q}(X_{1}; ℤ)).

In particular E_{2}^{p,2} ≅ H^{p}(ℤ/2; ℤ/2⊕ℤ/2) and by inspection of the spectral sequence it follows that the action of ℤ/2 is the twisting action. Thus E_{2}^{0,2} ≅ ℤ/2 and E_{2}^{p,2} = 0 if p ≥ 1. Consequently E_{2}^{0,4} ≅ (H^{4}(X_{1}; ℤ))^{ℤ/2} ≅ ℤ/2^{n−1} and since H^{4}(X_{1}, ℤ) ≅ ℤ/2^{r}, r ≤ n − 1, we get both that r = n − 1 and that the action is trivial. This completes the proof of part (c).

Consider the commutative diagram

[figure: X̃; X̃; =; (C); X_{2}; α_{2}; X_{1}; α_{1}; Bℤ/2; =; (D); Bℤ/2^{k−2}; BQ_{2^{k−1}}; Bℤ/2; y]

Again the EMSS is employed to compute H^{∗}(X_{2}). By naturality we get

H^{∗}(X_{2}) ≅ P [v_{4}] ⊗ E[x_{1}] ⊗ P [u_{2}]/(u^{2})

where v_{4} is restriction of the corresponding class in H^{∗}(X_{1}) and x_{1} and u_{2} are inflated from H^{∗}(Bℤ/2^{k−1}) via α_{2}^{∗}. The structure given above is easily seen to be both the P [v_{4}]-module structure and the H^{∗}(Bℤ/2^{k−1}) ≅ E[x_{1}] ⊗ P [u_{2}]-module structure. Using naturality of the BSS, it follows that β_{k−1}(x) = u and β_{s}(xu) = v, for some s < n − 1. We proceed in computing the Bockstein spectral sequence for H^{∗}(X_{2}). Consider SSS with integral coefficients for the fibration (C) above.

E_{2}^{p,q} ≅ H^{p}(Bℤ/2; H^{q}(X_{2}; ℤ) ⇒ H^{p+q}(X_{1}; ℤ)

In total degree two we have

E_{2}^{2,0} = H^{2}(Bℤ/2; H^{0}(X_{2}; ℤ)) ≅ ℤ/2
E_{2}^{0,2} = H^{0}(Bℤ/2; H^{2}(X_{2}; ℤ)) ≅ H^{0}(Bℤ/2; ℤ/2^{k−1}) ≅ (ℤ/2^{k−1})^{ℤ/2}.

Since H^{2}(X_{1}; ℤ) ≅ ℤ/2 ⊕ ℤ/2, it follows that ℤ/2 acts non trivially on ℤ/2^{k−1} and E_{2}^{0,2} ≅ ℤ/2.

In total degrees three and four we have:

E_{2}^{1,2} = H^{1}(Bℤ/2; H^{2}(X_{2}; ℤ)) = H^{1}(Bℤ/2; ℤ/2^{k−1})
E_{2}^{4,0} = H^{4}(Bℤ/2; H^{0}(X_{2}; ℤ)) = H^{4}(Bℤ/2; ℤ)
E_{2}^{2,2} = H^{2}(Bℤ/2; H^{2}(X_{2}; ℤ)) = H^{2}(Bℤ/2; ℤ/2^{k−1})
E_{2}^{0,4} = H^{0}(Bℤ/2; H^{4}(X_{2}; ℤ)) = H^{0}(ℤ/2; ℤ/2^{r}) = (ℤ/2^{s})^{ℤ/2}

Since H^{3}(X_{1}; ℤ) = 0 it follows that E_{3}^{1,2} ≅ ℤ/2 and d_{3} : E_{3}^{1,2} ⟶ E_{3}^{4,0} is an isomorphism.

Next notice that E_{3}^{0,4} = (ℤ/2^{s})^{ℤ/2}, where s < n − 1 and that 2 annihilates the cohomology of Bℤ/2 with any coefficients. Since H^{4}(X_{1}; ℤ) ≅ ℤ/2^{n−1}, it follows that E_{3}^{0,4} = ℤ/2^{n−2} and E_{3}^{2,2} = ℤ/2. Thus s = n − 2 and the action of ℤ/2 on H^{4}(X_{2}; ℤ) is trivial. This completes the calculation of the BSS for H^{∗}(X_{2}).

Now, we are ready to compute the BSS for H^{∗}(X̃). Again we use the SSS with integral coefficients for the maps α, α_{1} and α_{2} in the diagrams above. In total degree four, these spectral sequences give the following extensions:

α: ℤ/2^{k} → ℤ/2^{n} → (ℤ/2^{t})^{Q_{2^{k}}}.

α_{1}: ℤ/2^{k−1} → ℤ/2^{n−1} → (ℤ/2^{t})^{Q_{2^{k−1}}}.

α_{2}: ℤ/2^{k−2} → ℤ/2^{n−2} → (ℤ/2^{t})^{ℤ/2^{k−2}}. It follows that

(ℤ/2^{t})^{Q_{2^{k}}} ≅ (ℤ/2^{t})^{Q_{2^{k−1}}} ≅ (ℤ/2^{t})^{ℤ/2^{k−2}} ≅ ℤ/2^{n−k},

where the groups ℤ/2^{k−2} ⊂ Q_{2^{k−1}} operate on ℤ/2^{t} as subgroups of Q_{2^{k}}. But Aut(ℤ/2^{t}) is abelian and then the action of Q_{2^{k}} factors through its abelianization: (Q_{2^{k}})_{ab} ≅ ℤ/2⊕ℤ/2. Since ℤ/2^{k−2}, generated by x^{2} ∈ Q_{2^{k}}, is contained in the kernel of the abelianization, its action is trivial. But the action of Q_{2^{k}} on ℤ/2^{t} has the same invariants as the action of ℤ/2^{k−2}. Hence t = n − k and Q_{2^{k}} operates trivially. This completes the proof of (a) and (b)

Corollary 5.4. Let X be comparable to BQ_{2^{n}} with π_{1}X ≅ Q_{2^{k}} for some 4 < k < n. Then for every 1 ≤ j ≤ k − 4, there exists a space X_{j} comparable to BQ_{2^{n−j}} with π_{1}X_{j} ≅ Q_{2^{k−j}}.

Corollary 5.5. Let X be comparable to BQ_{2^{n}} for some n ≥ 5. Then π_{1}X ≠ Q_{2^{n−1}}. Consequently the homotopy type of BQ_{32} is determined by its cohomology.

Proof. By Proposition 5.3 assuming the contrary implies that the universal cover X̃ of X satisfies

H^{∗}(X̃) ≅ P [t_{4}] ⊗ E[x_{3}]

with Sq^{1}x = t. By an argument due to Aguadé [1], a space with such cohomology does not exist.

Finally we remark that proving that the homotopy type of BQ_{2^{n}} is determined by cohomology for all n amounts to showing the non-existence of a faithful action of Q_{16} on spaces of the form Y_{r}, r ≥ 2, where Y_{r} is a simply-connected 2-local space such that

H^{∗}(Y_{r}) ≅ P [t_{4}] ⊗ E[x_{3}]

with β_{r}x = t. For r = 2 it is not clear whether or not a space such as Y_{2} exists. Of course non-existence of Y_{2} would imply that if X is comparable to BQ_{2^{n}} then π_{1}X ≠ Q_{2^{n−2}} and hence the homotopy type of BQ_{64} could be determined by its cohomology. For r ≥ 3 spaces of the form Y_{r} arise from the 2-completion of BSL_{2}(𝔽_{q}) for q odd. However there is no suitable free action of Q_{16} on BSL_{2}(𝔽_{q}) because in this case the orbit space would be the classifying space of a finite group and comparable to BQ_{2^{n}} for some n. This implies a contradiction for groups with periodic mod-2 cohomology are completely classified. It might still be the case that the 2-completion BSL_{2}(𝔽_{q})_{2}^{∧} admits a free Q_{16}-action. Nonetheless we find it reasonable to conjecture that the homotopy type of BQ_{2^{n}} is determined by cohomology for all n.

6. The Semidihedral Groups

The semidihedral group SD_{2^{n}} is given as follows

SD_{2^{n}} = ⟨x, y|x^{2^{n−1}} = y^{2} = 1, yxy = x^{2^{n−2}−1}⟩.

Recall from [9]

H^{∗}(BSD_{2^{n}}) ≅ P [x, y, u, t]/(x^{2} + xy, xu, x^{3}, u^{2} + (x^{2} + y^{2})t)

where |x| = |y| = 1, |u| = 3 and |t| = 4. The action of the Steenrod algebra is given by Sq^{1}u = Sq^{1}t = 0, Sq^{2}u = (x + y)(t + yu), Sq^{2}t = u^{2} and Sq^{2^{i}}u = Sq^{2^{i}}t = 0 for i ≥ 2. The Bockstein spectral sequence is determined by the regular axioms for Sq^{1} together with the requirement that β_{n−1}u = t in E_{n−1}. Thus H^{∗}(BSD_{2^{n}}) is weakly generated by x, y and u.

Consider the central extension

0 ⟶ ℤ/2 ⟶ SD_{2^{n}} ⟶ D_{2^{n−1}} ⟶ 1 (6.1)

characterized by the class w + x^{2} ∈ H^{2}(BD_{2^{n−1}}). Consider the tower of principal fibrations

⋯ ⟶ BD_{2^{n}} ⟶^{π_{n−1}} BD_{2^{n−1}} ⟶^{π_{n−2}} ⋯ ⟶^{π_{3}} BD_{8} ⟶^{π_{2}} BD_{4} (6.2)

Define BD_{∞} as the inverse limit of the tower 6.2. It turns out to be an extension

Bℤ̂_{2} → BD_{∞} → BD_{4} (6.3)

and by comparison with any other Bℤ/2^{n−2} → BD_{2^{n}} → BD_{4} it follows that the only possible differential in the SSS for 6.3 is d_{2}(u) = x^{2} + xy, where u is the generator of the mod 2 cohomology of Bℤ̂_{2}, and therefore

H^{∗}(BD_{∞}) ≅ P [x, y]/(x^{2} + xy)

Let X be a space comparable to BSD_{2^{n}}. There is an obvious map ϕ_{2} : X → BD_{4} classifying the classes x, y ∈ H^{1}(X). Lifting ϕ_{2} through the tower 6.2 and 6.1 we will obtain a map f: X → BG, where G is either semidihedral or D_{∞}, such that the induced map in mod 2 cohomology is an isomorphism in dimensions one and two.

Recall from [5] that a map f is called an Hℤ/2-equivalence whenever the induced map on mod 2 cohomology is an isomorphism in dimension 1 and a monomorphism in dimension 2 . There exists a localization theory E^{ℤ/2} on the category of groups that invert those maps. Finite 2-groups and inverse limits of finite 2-groups are examples of E^{ℤ/2}-local groups.

Proposition 6.1. Let X be a space comparable to BSD_{2^{n}}. Then there is an Hℤ/2equivalence from X to either BSD_{2^{k}} for some k or BD_{∞}.

Proof. The map π_{2} is classified by x^{2} + xy ∈ H^{2}(BD_{4}). But x^{2} + xy = 0 in H^{2}(X). Hence there is a map ϕ_{3} : X ⟶ BD_{8} with π_{2}ϕ_{3} = ϕ_{2}. Assume that there is a map ϕ_{k} : X ⟶ BD_{2^{k}} lifting ϕ_{3}. Clearly ϕ_{k}^{∗}x = x, ϕ_{k}^{∗}y = y. Using naturality of Sq^{1} we find that ϕ_{k}^{∗}w = ax^{2} for some a ∈ 𝔽_{2}.

If a = 1 then the composite

X ⟶^{ϕ_{k}} BD_{k} ⟶^{w+x^{2}} K(ℤ/2, 2)

is null homotopic. Thus ϕ_{k} lifts to a map

ψ_{k+1} : X ⟶ BSD_{2^{k+1}},

which induces an isomorphism on cohomology up to dimension 2. Thus it is an Hℤ/2equivalence. On the other hand if a = 0 then ϕ_{k} can be lifted further. If ϕ_{k}^{∗}w = 0 for all k we get a map X ⟶ BD_{∞} inducing as well isomorphism on cohomology up to dimension 2, which is therefore an Hℤ/2-equivalence.

Lemma 6.2. Let f: X → BG be an Hℤ/2-equivalence where G and its subgroups are Hℤ/2-local. Then the induced map f_{#} : π_{1}X → G is an epimorphism. Moreover, if G is a finite 2-group then π_{1}X ≅ G.

Proof. The induced map f_{#} : π_{1}X → G is an Hℤ/2-equivalence as well, and therefore E^{ℤ/2}(π_{1}X) ≅ G. Let H denote the image of f_{#}. Since H is E^{ℤ/2}-local by hypothesis, the above isomorphism factors through H and therefore H = G; that is, f_{#} is an epimorphism.

Assume now, that G is a finite 2-group. In particular the previous hypotheses are satisfied. Let F be the fibre of f and

E_{2}^{∗,∗} ≅ H^{∗}(BG; H^{∗}(F; 𝔽_{2})) ⟹ H^{∗}(X; 𝔽_{2})

the SSS for the fibration F → X → BG. Since f induces isomorphism on mod 2 cohomology in dimension one and a monomorphism in dimension two, every element of E_{2}^{1,0} and E_{2}^{2,0} must be a permanent cycle and E_{2}^{0,1} ≅ H^{1}(F; 𝔽_{2})^{G} must be trivial. Being this last the group of invariants of an 𝔽_{2} vector space by the action of a finite 2-group, it is trivial if and only if H^{1}(F; 𝔽_{2}) is itself trivial. But this means that π_{1}F = 0 because F is 2-complete. It follows that π_{1}X ≅ G.

Corollary 6.3. Let X be a nilpotent space comparable to BSD_{2^{n}}. Then π_{1}X is isomorphic to SD_{2^{k}} for some k. Moreover, the natural map ψ_{k} : X ⟶ BSD_{2^{k}} classifying the universal cover for X is either a homotopy equivalence or satisfies ψ_{k}^{∗}(u) = ψ_{k}^{∗}(t) = 0.

Proof. In Proposition 6.1 we constructed an Hℤ/2-equivalence f: X → BG, where G is either semidihedral of D_{∞}. It follows from lemma 6.2 that the induced map π_{1}X → G is an epimorphism. Then, G cannot be D_{∞} for in that case X would not be nilpotent. Hence G is semidihedral, which is a finite 2-group, so again by lemma 6.2 f induces π_{1}X ≅ G.

Finally one uses the action of the higher Bockstein operators as well as the action of the Steenrod algebra in order to show that ψ_{k} could only have the cohomological effect described in the proposition, in case it fails to be a homotopy equivalence.

7. products with elementary abelian groups

This section is motivated by the will to understand our subject matter from a different point of view. Namely, we have exhibited examples of finite non-abelian 2-groups G such that the homotopy type of BG is determined by its cohomology. A natural question at that point is whether one can construct new examples of groups G with this property out of old ones. More precisely, given two finite 2-groups (or more generally p-groups) G and H such that the homotopy type of BG and BH is determined by cohomology, can one perform certain group theoretic constructions on G and H, thus obtaining a new group K, possibly infinite, whose classifying space BK has the same property. One might have in mind constructions such as Cartesian products, amalgamated products, fibre products etc. The simplest possible construction is of course the Cartesian product. Thus one might wonder whether the homotopy type of B(H × G) is determined by its cohomology if the same holds for each one of the factors. Theorem 1.5 handles the simplest possible version of this question. We now turn into its proof. Assume that E is an elementary abelian p-group and suppose that X is a space comparable to B(G × E), that is

H_{β}^{∗}(X) ≅ H_{β}^{∗}(B(G × E)) ≅ H_{β}^{∗}(BG) ⊗ H_{β}^{∗}(BE).

We can obtain a classifying map for one dimensional classes

ϕ: X → BE

inducing the obvious inclusion H^{∗}(BE) ↪ H^{∗}(BG) ⊗ H^{∗}(BE). Let F be the fibre of ϕ. Since H^{∗}(X) is free as a module over H^{∗}(BE), the EMSS for this fibration is concentrated in the zero vertical line and hence collapses with no extension problems. Thus H^{∗}(F) ≅ H^{∗}(BG). By naturality of the BSS and the structure of H^{∗}(X) it follows at once that F is comparable to BG. By hypothesis, G is determined by cohomology, hence F ≃ BG and therefore X is the classifying space of a 2-group K, given as an extension

1 → G → K → E → 1.

Central elements of order 2 in K are cohomologically detectable by means of the Lannes’ T functor (see [13]). Thus we obtain a section s: E → K of the map ϕ, such that s(E) is central in K. Hence K ≅ G × E and X ≃ BK ≃ B(G × E).

8. Discussion

It remains unclear whether either the isomorphism type of finite p-groups or the homotopy type of their classifying spaces is determined by cohomology in the sense discussed above. We end this paper by making some further remarks.

Let C denote one of the following categories. The category Alg of 𝔽_{p}-algebras, the category U of unstable A-modules, the category K of unstable A-algebras or the category K_{β} described above. The cohomology functor can be viewed as a functor from the homotopy category of spaces to C for either one of those categories. The homotopy type of a space X is determined by cohomology as an object of C if and only if the cohomology of X can be pulled back to a unique object in the homotopy category, namely the object represented by X itself. Thus the question which are the spaces which are determined by their cohomology as objects of C is equivalent to the question what is the sub class of objects in the homotopy category on which the cohomology functor is 1-1. Our aim in this note was to exhibit examples of spaces of the form BG, where G is a finite 2-group, which are determined by cohomology as objects of K_{β}. We also observed that most of the extra special 2-groups are determined by cohomology as objects in Alg.

One might thus like to restrict attention to one of the algebraic categories above and to the homotopy category of classifying spaces BG for finite p-groups. Since a finite group uniquely characterizes the homotopy type of its classifying space it make sense to ask what is the family of finite p-groups G for which the homotopy type of BG is characterized by cohomology as an object of C. This question might be mildly too general to admit a successful answer. Thus as pointed out above, one might wonder whether certain constructions on groups G with the property that BG is determined by cohomology yield new objects with the same property. An example of a partial answer to this question is given in Theorem 1.5. However the theorem in fact demonstrates the difficulty in providing a more general answer. It is natural for instance to replace E by an arbitrary finite abelian p-group and ask whether the result of Theorem 1.5 remains valid.

Those questions motivate the following definition. We restrict attention to one of the categories C above and to p-complete spaces of finite cohomological type for a fixed prime p.

Definition 8.1. The homotopy type of a space X is said to be strongly determined by cohomology if for a for any space Y the natural map of sets

[Y; X] ⟶ Hom_{C}(H^{∗}(X); H^{∗}(Y))

is an epimorphism.

In the categories C = K or K_{β} the tensor product is a categorical sum and the functor Hom_{C}(−; K) from C to Sets turns sums into products. Thus it is clear that the category of spaces which are strongly determined by cohomology with C as above, admits finite products. It is also clear that spaces of the form BE (for C = K and E elementary abelian) and BA (for C = K_{β} and A abelian), are strongly determined by cohomology. Thus it makes sense to ask what other group theoretic constructions are possible inside those categories.

On the other extreme, the authors find it reasonable to conjecture that the category K_{β} does not characterize the homotopy type of every classifying space BG, where G is a finite p-group. Thus one might like to find an example of an exotic BG, namely a space X comparable to BG yet of a different homotopy type.

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Dept. de Matemàtiques, Univeritat Autònoma de Barcelona, E-08193 Bellaterra, Spain

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

Mathematisches Institut der Universität Heidelberg, INF288 Heidelberg 69120, Germany

E-mail address: rlvi@vogon.mathi.uni-heidelberg.de