A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN

RAN LEVI

Abstract. Let G be a finite p-superperfect group. A conjecture of F. Cohen suggests that ΩBG_{p}^{∧} is resolvable by finitely many fibrations over spheres and iterated loop spaces on spheres, where (−)_{p}^{∧} denotes the p-completion functor of Bousfield and Kan. We produce a counter-example to this conjecture and discuss some related aspects of the homotopy type of ΩBG_{p}^{∧}.

1. Introduction

Let p be a prime number. Recall that a group G is said to be p-perfect if H_{1}(BG; 𝔽_{p}) = 0 and p-superperfect if, in addition H_{2}(BG; 𝔽_{p}) = 0. The group G is said to be perfect (superperfect) if it is p-perfect (p-superperfect) with respect to any prime p. A conjecture of F. Cohen [2] suggests that if G is a finite superperfect group then ΩBG^{+} is spherically resolvable of finite weight, where (−)^{+} denotes the Quillen “plus” construction.

A simple observation due to Bousfield and Kan [1] shows that for any finite group G, the space BG^{+} is homotopy equivalent to the product (or wedge) of the p-completed classifying spaces BG_{p}^{∧} taken over all primes p dividing the order of G. Thus a more general version of Cohen’s conjecture appears in [5], in which the notion of “superperfect” is replaced by p-superperfect and the “plus” construction by the p-completion functor of Bousfield and Kan [1]. A considerable number of examples for the conjecture are given in [2, 5].

As we observe below, Cohen’s conjecture is related to questions of classical interest in homotopy theory. Unfortunately the conjecture turns out to be false. Indeed the purpose of this note is to prove the following

Theorem 1.1. Let r be a positive integer and let p ≥ 13 be a prime such that 4 divides p − 1. Then there exists a finite p-superperfect group D_{2}(p^{r}) such that ΩBD_{2}(p^{r})_{p}^{∧} is not spherically resolvable of finite weight.

The study of spaces of the form BG_{p}^{∧} for G finite and p- perfect appears to be related to various aspects of classical homotopy theory. As we shall observe, any example of a finite p-perfect group G, for which Cohen’s conjecture holds has the property that BG_{p}^{∧} has a global exponent in homotopy groups. This in turn produces a family of finite elliptic complexes, which satisfy the Moore finite exponents conjecture [4]. On the other hand an example of a finite p-perfect group G for which BG_{p}^{∧} admits no global homotopy exponent would produce a counterexample to Moore’s conjecture.

Our counter-example has a feature which appear mildly unusual. The groups G which are shown to fail Cohen’s conjecture turn out to have the property that the single loop space on a certain mod-p Moore space is a retract of ΩBG_{p}^{∧}. This observation is crucial in showing that those groups are indeed counter-examples for Cohen’s conjecture and in addition raises the question how much of the homotopy theory of Moore spaces can be retrieved by studying spaces of the form ΩBG_{p}^{∧} for finite p-perfect groups G. Notice that this stands in contrast to the fact that there are no essential maps from BG_{p}^{∧} to an iterated loop space on a finite complex by the Sullivan conjecture. We also remark that our example shows that in general spaces of the form ΩBG_{p}^{∧} do not have an H-space exponent as the same is true for the single loop space on a Moore space. Note however that in view of the remark above concerning the Moore conjecture, one might like to believe that the homotopy groups of BG_{p}^{∧} do have an exponent for every finite p-perfect group G. It might be the case that for every finite p-perfect group G, some iterated loop space Ω^{k}BG_{p}^{∧} has an H-space exponent.

Date: July 1 1994.

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

The author is supported by a DFG grant.

In previous study of spaces of the form ΩBG_{p}^{∧} [2, 5], several computational examples led to the question whether the mod-p loop space homology of BG_{p}^{∧} is a commutative algebra (or at least Lie nilpotent). Our example provides a negative answer to this question.

Throughout this article all space are assumed simply connected, p-complete and to have the homotopy type of a CW-complex. By H_{∗}(−) we shall always mean homology with coefficients in the prime field 𝔽_{p}.

2. homological rate of growth

A space X is said to be spherically resolvable of weight ≤ r if there exists a tower of principal fibrations:

X_{r} ⟶ X_{r−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X

such that:

1. For each i ≥ 1 the fibration, X_{i+1} ⟶ X_{i}, is induced from the path loop fibration over Ω^{n_{i}}S^{n_{i}+k_{i}}, n_{i} ≥ 0, k_{i} > 0 via a map ρ_{i} : X_{i} ⟶ Ω^{n_{i}}S^{n_{i}+k_{i}}.

2. X_{r} ≃ Ω^{n_{r}}S^{n_{r}+k_{r}} for some n_{r} ≥ 0 and k_{r} > 0.

The main observation needed to produce a counter-example for Cohen’s conjecture is stated below and is proven in [6].

Theorem 2.1. Let X be a space which is finitely resolvable by fibrations over spheres and iterated loop spaces on spheres. Let θ(d) denote the coefficient of t^{d} in the poincaré series for H_{∗}(X). Then for every real number λ > 1,

lim_{d→∞} \frac{θ(d)}{λ^{d}} = 0.

The theorem is proven by first considering iterated loop spaces on sphere, for which the homological structure is well known [3], and showing that the theorem holds for those spaces. The general statement follows by using the Serre spectral sequence to observe that exponential growth for the coefficients θ(d) cannot be obtained for a space which is finitely resolvable.

As a consequence of theorem 2.1, it suffices to find an example of a finite p-perfect group G such that H_{∗}(ΩBG_{p}^{∧}) grows exponentially in order to disprove Cohen’s conjecture.

For a finite group G, consider faithful representations of G in some unitary group U(n). Any such representation ρ gives rise to a fibration

U(n)/ρ ⟶ BG ⟶ BU(n)  (1)

with a simply-connected base space, where U(n)/ρ denotes the orbit space of U(n) by the G-action via ρ. Notice that p-completion respects fibrations with simplyconnected base space.

Now consider the case where G is p-superperfect and let a faithful representa-

tion G ⟶^{ρ} U(n) be given. Then Bρ_{p}^{∧} can be lifted to a map BG_{p}^{∧} ⟶^{Bρ̃_{p}^{∧}} BSU(n)_{p}^{∧}, whose fibre which we denote (SU(n)/ρ)_{p}^{∧} is the 1-connected cover of (U(n)/ρ)_{p}^{∧}. Thus we get a sequence

ΩSU(n)_{p}^{∧} ⟶ Ω(SU(n)/ρ)_{p}^{∧} ⟶ ΩBG_{p}^{∧} ⟶ SU(n)_{p}^{∧},  (2)

in which every consecutive pair of maps is a fibration sequence up to homotopy.

Lemma 2.2. Let G be a finite p-superperfect group. Then for every unitary faithful representation ρ : G ⟶ U(n)

1. ΩBG_{p}^{∧} is spherically resolvable of finite weight if and only if Ω(SU(n)/ρ)_{p}^{∧} is.

2. H_{∗}(ΩBG_{p}^{∧}) grows exponentially if and only if H_{∗}(Ω(SU(n)/ρ)_{p}^{∧}) does.

3. π_{∗}BG_{p}^{∧} has an exponent if and only if π_{∗}(SU(n)/ρ)_{p}^{∧} does.

Proof. The first statement follows at once from [5, II.3.0.2]; the second follows by inspection of the Serre spectral sequences of the two fibrations in 2 above; the third statement follows by considering the long exact homotopy sequence for 2, taking into account the fact that SU(n)_{p}^{∧} is spherically resolvable of finite weight and thus admits a homotopy exponent.

Notice that (SU(n)/ρ)_{p}^{∧} has the homotopy type of a finite elliptic complex. This justifies the remark made in the introduction about the Moore conjecture.

3. the groups D_{k}(p^{r}) and a related representation

We assume all through that p is a prime such that 2^{k} divides p−1. In that case the group ℤ/2^{k+1}ℤ operates on a free ℤ/p^{r}ℤ-module T_{r} of rank 2 as follows. Let ζ denote an element of multiplicative order 2^{k} in the group of units (ℤ/p^{r}ℤ)^{∗}. Let a and b denote a choice of generators for T_{r}. Let σ denote a generator for ℤ/2^{k+1}ℤ. Define an action of σ on T_{r} by σ(a) = b and σ(b) = ζa. Define D_{k}(p^{r}) to be the semidirect product of T_{r} with ℤ/2^{k+1}ℤ with respect to the action given above. A presentation of D_{k}(p^{r}) is given by

D_{k}(p^{r}) = ⟨a, b, σ|a^{p^{r}} = b^{p^{r}} = σ^{2^{k+1}} = [a, b] = 1; σaσ^{−1} = b; σbσ^{−1} = a^{g}⟩,

where g is an integer such that g^{2^{k}} ≡ 1modp^{r}.

Define a unitary representation ρ_{r} : D_{k}(p^{r}) ⟶ U(2^{k+1}) as follows. Let θ denote a complex primitive root of 1 of order p^{r}. Define ρ_{r}(σ) to be the permutation matrix whose i-th row is the standard unit vector e_{i+1} for 1 ≤ i ≤ 2^{k+1} − 1 and whose 2^{k+1}-st row is e_{1}. Define

ρ_{r}(a) = diag(θ, 1, θ^{g}, 1 ⋯, θ^{g^{2^{k}} −1}, 1),
ρ_{r}(b) = diag(1, θ^{g}, 1, θ^{g^{2}}, ⋯, θ^{g^{2^{k}} −1}, 1, θ).

One easily verifies that the relations are satisfied and ρ_{r} is evidently faithful. Let T denote a 2^{k+1}-fold product of the 1-sphere S^{1} and let ψ : T ⟶ U(2^{k+1}) denote the canonical inclusion. Fix the values of p, r and k and let G denote D_{k}(p^{r}). Let ϕ : T_{r} ⟶ G denote the inclusion. Then the restriction ρ′_{r} of ρ_{r} to T_{r} factors through T and we have ψρ′_{r} = ρ_{r}ϕ. Using this factorization one easily computes the Chern classes of ρ_{r}. To fix our notation let

H^{∗}(BT) = P [u_{1}, ⋯, u_{2^{k+1}}], |u_{j}| = 2,
H^{∗}(BT_{r}) = P [v_{1}, v_{2}] ⊗ E[x_{1}, x_{2}], |v_{j}| = 2, |x_{j}| = 1.

The following lemma is an easy exercise.

Lemma 3.1. The total Chern class of ρ_{r} restricts to

1 − (v_{1}^{2^{k}} + v_{2}^{2^{k}}) + (v_{1}v_{2})^{2^{k}}

in H^{∗}(BT_{r}). Thus w_{2^{k}}(ρ_{r}) = −(v_{1}^{2^{k}} + v_{2}^{2^{k}}), w_{2^{k+1}}(ρ_{r}) = (v_{1}v_{2})^{2^{k}} and w_{j}(ρ_{r}) = 0 otherwise.

Corollary 3.2. Let U(2^{k+1})/ρ_{r} denote the orbit space of U(2^{k+1}) by D_{k}(p^{r}) with respect to the representation ρ_{r}. Then there is an isomorphism of algebras

H^{∗}(U(2^{k+1})/ρ_{r}) ≅ H^{∗}(BD_{k}(p^{r}))/(w_{2^{k}}, w_{2^{k+1}}) ⊗ E,

where E is an exterior algebra on 2^{k+1} − 2 generators, corresponding to the zero Chern classes of ρ_{r}.

Proof. This is an immediate consequence of the big collapse theorem of L. Smith [7].

Next if the prime p in the definition of D_{k}(p^{r}) is sufficiently large, then there is an obvious map

f : (U(2^{k+1})/ρ_{r})_{p}^{∧} ⟶ ∏_{i≠2^{k−1}−1,2^{k}−1}(S^{2i+1})_{p}^{∧},

realizing the factor E in H^{∗}((U(2^{k+1})/ρ_{r})_{p}^{∧}). Indeed one obtains this map by defining it component by component, starting with a suitable skeleton and than extending, using the fact that possible obstructions vanish for large primes. Notice that the dimension of (U(2^{k+1})/ρ_{r})_{p}^{∧} is independent of p.

Let X_{k}(p^{r}) denote the homotopy fibre of f. The following lemma is obvious by inspection of the Eilenberg-Moore spectral sequence for the map f.

Lemma 3.3. Let p and k be chosen so that the map f defined above exists. Then there is an isomorphism of algebras

H^{∗}(X_{k}(p^{r})) ≅ H^{∗}(BD_{k}(p^{r}))/(w_{2^{k}}, w_{2^{k+1}}).

We remark that the assumption that the prime p is sufficiently large makes calculations easier but seems not to play an important role otherwise. In the next section we produce our counter-example based on the calculations carried out here. We find it suitable to conjecture that results similar to those given below can be obtained for all D_{k}(p^{r}).

4. the counter-example

We specialize to the case k = 2 and calculate the cohomology algebra of X_{2}(p^{r}). Notice that X_{2}(p^{r}) can be constructed as above if p ≥ 13.

Proposition 4.1. There is an isomorphism of algebras

H^{∗}(BD_{2}(p^{r}))/(w_{4}, w_{8}) ≅ P [a_{6}, a′_{6}, b_{7}, b′_{7}, d_{7}, d′_{7}, t_{8}, s_{15}, s′_{15}, q_{16}]/R,

where R is the set of relations given by at = d′b′; ds = aq = s′b = d′b′t = a′t^{2}; t^{3} = 0; all other possible products of generators except for those given above and in addition t^{2}, b′t and d′t vanish. Thus H^{∗}(X_{2}(p^{r})) is given in the cell diagram below.

[figure: ds = aq = s′b = d′b′t = a′t^{2}; q_{16}; t^{2}; s_{15}; s′_{15}; β_{r}; tb′; td′; β_{r}; a′t = d′b′; β_{r}; t_{8}; d_{7}; b_{7}; d′_{7}; β_{r}; b′_{7}; a_{6}; β_{r}; a′_{6}; β_{r}; 1]

Proof. Let T_{r} < D_{2}(p^{r}) denote the Sylow p-subgroup and write

H^{∗}(BT_{r}) ≅ P [v_{1}, v_{2}] ⊗ E[x_{1}, x_{2}].

Let ζ ∈ 𝔽_{p} denote a primitive root of unity of order 4. Computing the algebra structure modulo the Chern classes w_{4} and w_{8}, whose restrictions to H^{∗}(BT_{r}) are given by −(v_{1}^{4} + v_{2}^{4}) and v_{1}^{4}v_{2}^{4} respectively, one observes that the resulting algebra is 22-dimensional. In fact obtaining an 𝔽_{p} vector space basis is easy by routine invariant calculation.

Let res : H^{∗}(BD_{2}(p^{r})) ⟶ H^{∗}(BT_{r}) denote the restriction. We conclude the proof by spelling out the restrictions of the specified generators and leave it for the reader to verify that all the promised relations hold modulo the ideal (w_{4}, w_{8}).

1. res(a_{6}) = (v_{1}^{2} − ζv_{2}^{2})x_{1}x_{2}.

2. res(a′_{6}) = −v_{1}v_{2}x_{1}x_{2}.

3. res(b_{7}) = (v_{1}^{3}x_{2} + ζv_{2}^{3}x_{1}) − (v_{1}^{2}v_{2}x_{1} + ζv_{1}v_{2}^{2}x_{2}).

4. res(b′_{7}) = v_{1}v_{2}^{2}x_{1} − v_{1}^{2}v_{2}x_{2}.

5. res(d_{7}) = (v_{1}^{3}x_{1} + v_{2}^{3}x_{2}) + ζ(v_{1}^{2}v_{2}x_{2} − v_{1}v_{2}^{2}x_{1}).

6. res(d′_{7}) = v_{1}v_{2}^{2}x_{2} − ζv_{1}^{2}v_{2}x_{1}.

7. res(t_{8}) = v_{1}v_{2}^{3} − ζv_{1}^{3}v_{2}.

8. res(s_{15}) = v_{1}^{3}v_{2}^{4}x_{1} + v_{1}^{4}v_{2}^{3}x_{2}.

9. res(s′_{15}) = ζv_{1}^{7}x_{2} − v_{2}^{7}x_{1}.

10. res(q_{16}) = ζv_{1}^{7}v_{2} − v_{1}v_{2}^{7}.

Proposition 4.2. There is a retract

P^{16}(p^{r}) ⋁ S^{15} ⟶^{g} X_{2}(p^{r}) ⟶^{f} P^{16}(p^{r}) ⋁ S^{15},

where P^{16}(p^{r}) is the 16-dimensional mod-p^{r} Moore space.

Proof. Let X denote X_{2}(p^{r}) and recall our convention that all spaces are p-complete. First observe that there is a map on the 16- skeleton

f′ : X^{(16)} ⟶ P^{16}(p^{r}) ⋁ S^{15}

given by pinching down the 8-skeleton together with the cells corresponding to the products on the right hand side of the diagram above. An obstruction to extending f′ to f : X ⟶ P^{16}(p^{r}) ⋁ S^{15} might exist in π_{21}P^{16}(p^{r}) ⋁ S^{15}, which vanishes for the primes under consideration.

Next notice that

X^{(8)} ≃ P^{7}(p^{r}) ⋁ P^{7}(p^{r}) ⋁ P^{8}(p^{r}) ⋁ S^{7}.

If the prime p is sufficiently large, computing π_{i}X^{(8)} for i = 14 and 15 is straight forward by using the Hilton-Milnor theorem [8]. In particular one observes that the homotopy in those dimensions is generated by primary Whitehead products and thus a non-trivial attaching map results in “attaching” a decomposable cohomology class. Since s, s′ and q are indecomposable, we conclude that the corresponding attaching maps are trivial and thus the desired map

g : P^{16}(p^{r}) ⋁ S^{15} ⟶ X

is obtained.

Finally notice that the composite f ∘ g induces an isomorphism on mod-p cohomology and is thus a homotopy equivalence.

Corollary 4.3. The homology algebra H_{∗}(ΩX_{2}(p^{r})) contains a tensor algebra on three generators. In particular H_{∗}(ΩX_{2}(p^{r})) grows exponentially and hence is not spherically resolvable of finite weight.

We now consider ΩBD_{2}(p^{r})_{p}^{∧}. Through the end of this paper let X denote as before X_{2}(p^{r}) and let G denote D_{2}(p^{r}). Then the Moore space P^{16}(p^{r}) is a retract of X. Moreover, Let π : X ⟶ BG_{p}^{∧} denote the map obtained by the composite

X ⟶ (U(8)/ρ_{2})_{p}^{∧} ⟶ BG_{p}^{∧}.

One readily verifies that the homotopy fibre of π is equivalent to (S^{7} × S^{15})_{p}^{∧} and that the fibre inclusion map into X induces the zero map on mod-p cohomology.

This together with the assumption that the prime p is sufficiently large implies that the composite

S^{7} × S^{15} ⟶ X ⟶ P^{16}(p^{r})

is null homotopic and thus yields a homotopy commutative diagram

[figure: ΩX; ΩBG_{p}^{∧}; S^{7} × S^{15}; X; ΩP^{16}(p^{r}); ΩP^{16}(p^{r}); ∗; P^{16}(p^{r}); =]

Note that the map ΩBG_{p}^{∧} ⟶ ΩP^{16}(p^{r}) is not multiplicative in general, however the composite

ΩP^{16}(p^{r}) ⟶ ΩX ⟶ ΩBG_{p}^{∧} ⟶ ΩP^{16}(p^{r})

is evidently homotopic to the identity. Thus we have proven

Proposition 4.4. The space ΩP^{16}(p^{r}) is a retract of ΩBG_{p}^{∧} for p ≥ 13.

Corollary 4.5. For p ≥ 13, the algebra H_{∗}(ΩBG_{p}^{∧}) contains a tensor algebra on two generators and thus grows exponentially

Corollary 4.5 combined with theorem 2.1 completes the proof of theorem 1.1. In addition we have

Corollary 4.6. For p ≥ 13, any power map on ΩBG_{p}^{∧} is essential.

Proof. By [4] any power map on a single loop space on a Moore space is essential. The result follows.

5. speculations

If G is a finite p-superperfect group than ΩBG_{p}^{∧} does not satisfy Cohen’s conjecture unless possibly if the loop space homology H_{∗}(ΩBG_{p}^{∧}) does not grow exponentially. In view of the fact that the groups D_{2}(p^{r}) are by no means “pathological”, it seems reasonable to wonder what group theoretic properties of G would imply that the loop space homology of BG grows polynomialy or at least subexponentially. We refer the reader to [2, 5] to inspect that, in fact, all the examples of groups G known to satisfy Cohen’s conjecture have the property that H_{∗}(ΩBG_{p}^{∧}) grows polynomialy.

Although theorem 1.1 shows that Cohen’s conjecture is false as stated in [5], it is still conceivable by results in [2, 4] that the conjecture holds in general if one drops the finiteness requirement on the length of a resolution.

Another point to be emphasized is the significance of proposition 4.4. This result suggests that understanding the homotopy type of spaces of the from ΩBG_{p}^{∧} might possibly shed new light on objects of classical interest in homotopy theory. One might wonder for example whether it is possible to obtain ΩP^{n}(p^{r}) for any given values of n and p as a retract of ΩBG_{p}^{∧} for some finite p-perfect group G.

Many interesting p-perfect groups G are not p-superperfect but in this case the 1-connected cover of BG_{p}^{∧} is BG̃_{p}^{∧}, where G̃ is the p-universal central extension of G and is finite and p-superperfect. It is easy to observe that if H_{∗}(ΩBG_{p}^{∧}) grow exponentially then so does H_{∗}(ΩBG̃_{p}^{∧}). For instance BD_{1}(3)_{3}^{∧} ≃ BSL_{2}(𝔽_{9})_{3}^{∧} and is 3-perfect but not 3-superperfect. In this case the corresponding space X_{1}(3) can be obtained at the prime 3 and is 10-dimensional. In addition there is a fibration

Y ⟶ ΩBD_{1}(3)_{3}^{∧} ⟶ X_{1}(3),

where Y is spherically resolvable of weight 2. However the existence of a non-zero P^{1} in H^{∗}(X_{1}(3)) implies that a result corresponding to prop 4.2 for X_{1}(3) fails to hold. It would be interesting to know whether or not the 1-connected cover of ΩBD_{1}(p^{r})_{p}^{∧} is spherically resolvable of finite weight as this could in some sense provide a minimal counter-example (of order 4p^{2}).

Finally, the referee has suggested that the loop space on the space X_{1}(3) might have the loop space of a 4-cell complex as a retract, where the 4-cell complex supports a non-trivial P^{1}. We remark that such a retract, if it exists cannot be multiplicative, i.e. it does not exist before looping due to the existence of products in H^{∗}(X_{1}(3)). The question whether or not it exists after looping once remains unsolved.

References

[1] A. K. Bousfield and D. M. Kan; Homotopy Limits Completions and Localizations; LNM 304, (1972), Springer-Verlag.

[2] F. R. Cohen; Remarks on the Homotopy Theory associated to Finite Perfect Groups ; LNM 1509 (1992) Springer- Verlag.

[3] F. R. Cohen, T. J. Lada and J. P. May; The Homology of Iterated Loop Spaces ; LNM 533, (1976) Springer-Verlag.

[4] F. R. Cohen, J. C. Moore and J. Neisendorfer; Exponents in Homotopy Theory; Ann. of Math. Studies, 133 (1987), 3-34.

[5] R. Levi; On Finite Groups and Homotopy Theory; to appear in the Memoirs of the A.M.S.

[6] R. Levi; On Homological Rate of Growth and the Homotopy Type of ΩBG_{p}^{∧}; to appear,

[7] L. Smith; Homological Algebra and the Eilenberg- Moore Spectral Sequence; AMS Translatl. 129, (1967), 58-93.

[8] G. Whitehead; Elements of Homotopy Theory; Springer-Verlag (1978).

Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

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