A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN RAN LEVI Abstract. Let G be a finite p-superperfect group. A conjecture of F. Co* *hen suggests that BG^pis resolvable by finitely many fibrations over sphere* *s and iterated loop spaces on spheres, where (-)^pdenotes the p-completion fu* *nctor of Bousfield and Kan. We produce a counter-example to this conjecture a* *nd discuss some related aspects of the homotopy type of BG^p. 1. Introduction Let p be a prime number. Recall that a group G is said to be p-perfect if H1(BG; Fp) = 0 and p-superperfect if, in addition H2(BG; Fp) = 0. The group G is said to be perfect (superperfect) if it is p-perfect (p-superperfect) with r* *espect to any prime p. A conjecture of F. Cohen [2] suggests that if G is a finite superp* *erfect group then BG+ is spherically resolvable of finite weight, where (-)+ denotes t* *he 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^ptaken over all primes p dividing the order of G. Thus a more general version of Cohen's conjecture appears in [5], in which t* *he notion of "superperfect" is replaced by p-superperfect and the "plus" construct* *ion 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 classic* *al 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 D2(pr) such th* *at BD2(pr)^pis not spherically resolvable of finite weight. The study of spaces of the form BG^pfor 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 fi* *nite exponents conjecture [4]. On the other hand an example of a finite p-perfect gr* *oup G for which BG^padmits no global homotopy exponent would produce a counter- example 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 ______________ Date: July 1 1994. 1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q* *52. The author is supported by a DFG grant. 1 2 RAN LEVI the single loop space on a certain mod-p Moore space is a retract of BG^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 BG^pfor finite p-perfect groups G. Notice that this stands in contrast to the fact that* * there are no essential maps from BG^pto 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 BG^pdo 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 homot* *opy groups of BG^pdo 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 kBG^p has an H-space exponent. In previous study of spaces of the form BG^p[2, 5], several computational examples led to the question whether the mod-p loop space homology of BG^pis 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 Fp. 2. homological rate of growth A space X is said to be spherically resolvable of weight r if there exist* *s a tower of principal fibrations: Xr -! Xr-1 -! . .-.! X2 -! X1 = X such that: 1. For each i 1 the fibration, Xi+1 - ! Xi, is induced from the path loop fibration over niSni+ki, ni 0, ki > 0 via a map aei : Xi -! niSni+ki. 2. Xr ' nrSnr+kr for some nr 0 and kr > 0. The main observation needed to produce a counter-example for Cohen's con- jecture is stated below and is proven in [6]. Theorem 2.1. Let X be a space which is finitely resolvable by fibrations ov* *er spheres and iterated loop spaces on spheres. Let (d) denote the coefficient of * *td in the poincare series for H*(X). Then for every real number > 1, dlim!1(d)_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 spectr* *al A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN 3 sequence to observe that exponential growth for the coefficients (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*(BG^p) grows exponentially in order to disprove Cohen's conjecture. For a finite group G, consider faithful representations of G in some unita* *ry group U(n). Any such representation ae gives rise to a fibration U(n)=ae -! BG -! BU(n) (1) with a simply-connected base space, where U(n)=ae denotes the orbit space of U(* *n) by the G-action via ae. Notice that p-completion respects fibrations with simpl* *y- connected base space. Now consider the case where G is p-superperfect and let a faithful represe* *nta- B"ae^p tion G -ae!U(n) be given. Then Bae^pcan be lifted to a map BG^p- ! BSU(n)^p, whose fibre which we denote (SU(n)=ae)^pis the 1-connected cover of (U(n)=ae)^p. Thus we get a sequence SU(n)^p- ! (SU(n)=ae)^p- ! BG^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 fa* *ithful representation ae : G -! U(n) 1. BG^pis spherically resolvable of finite weight if and only if (SU(n)=ae)^* *pis. 2. H*(BG^p) grows exponentially if and only if H*((SU(n)=ae)^p) does. 3. ss*BG^phas an exponent if and only if ss*(SU(n)=ae)^pdoes. Proof. The first statement follows at once from [5, II.3.0.2]; the second follo* *ws by inspection of the Serre spectral sequences of the two fibrations in 2 above; th* *e third statement follows by considering the long exact homotopy sequence for 2, taking into account the fact that SU(n)^pis spherically resolvable of finite weight an* *d_thus admits a homotopy exponent. |_* *_| Notice that (SU(n)=ae)^phas the homotopy type of a finite elliptic complex. This justifies the remark made in the introduction about the Moore conjecture. 3. the groups Dk(pr) and a related representation We assume all through that p is a prime such that 2k divides p-1. In that * *case the group Z=2k+1 Z operates on a free Z=prZ-module Tr of rank 2 as follows. Let* * i denote an element of multiplicative order 2k in the group of units (Z=prZ)*. Le* *t a and b denote a choice of generators for Tr. Let oe denote a generator for Z=2k+* *1 Z. Define an action of oe on Tr by oe(a) = b and oe(b) = ia. Define Dk(pr) to be t* *he semidirect product of Tr with Z=2k+1 Z with respect to the action given above. A presentation of Dk(pr) is given by r pr 2k+1 -1 -1 * * g Dk(pr) = ; 4 RAN LEVI where g is an integer such that g2k 1mod pr. Define a unitary representation aer : Dk(pr) - ! U(2k+1 ) as follows. Let denote a complex primitive root of 1 of order pr. Define aer(oe) to be the perm* *utation matrix whose i-th row is the standard unit vector ei+1 for 1 i 2k+1 - 1 and whose 2k+1 -st row is e1. Define 2k-1 aer(a) = diag(; 1; g; 1 . .;.g ; 1); 2 g2k-1 aer(b) = diag(1; g; 1; g ; . .;. ; 1; ): One easily verifies that the relations are satisfied and aer is evidently faith* *ful. Let T denote a 2k+1 -fold product of the 1-sphere S1 and let : T - ! U(2k+1 ) den* *ote the canonical inclusion. Fix the values of p, r and k and let G denote Dk(pr). Let OE : Tr -! G denote the inclusion. Then the restriction ae0rof aer to Tr fa* *ctors through T and we have ae0r= aerOE. Using this factorization one easily computes the Chern classes of aer. To fix our notation let H*(BT ) = P [u1; . .;.u2k+1]; |uj| = 2; H*(BTr) = P [v1; v2] E[x1; x2]; |vj| = 2; |xj| = 1: The following lemma is an easy exercise. Lemma 3.1. The total Chern class of aer restricts to k 2k 2k 1 - (v21 + v2 ) + (v1v2) in H*(BTr). Thus w2k(aer) = -(v2k1+ v2k2), w2k+1(aer) = (v1v2)2k and wj(aer) = 0 otherwise. Corollary 3.2. Let U(2k+1 )=aer denote the orbit space of U(2k+1 ) by Dk(pr) * *with respect to the representation aer. Then there is an isomorphism of algebras H*(U(2k+1 )=aer) ~=H*(BDk(pr))=(w2k; w2k+1) E; where E is an exterior algebra on 2k+1 - 2 generators, corresponding to the zero Chern classes of aer. Proof. This is an immediate consequence of the big collapse theorem of L. Smith* * __ [7]. * * |__| Next if the prime p in the definition of Dk(pr) is sufficiently large, the* *n there is an obvious map Y f : (U(2k+1 )=aer)^p- ! (S2i+1)^p; i6=2k-1-1;2k-1 realizing the factor E in H*((U(2k+1 )=aer)^p). Indeed one obtains this map by * *defin- ing it component by component, starting with a suitable skeleton and than exten* *d- ing, using the fact that possible obstructions vanish for large primes. Notice * *that the dimension of (U(2k+1 )=aer)^pis independent of p. Let Xk(pr) denote the homotopy fibre of f. The following lemma is obvious by inspection of the Eilenberg-Moore spectral sequence for the map f. A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN 5 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*(Xk(pr)) ~=H*(BDk(pr))=(w2k; w2k+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 n* *ext section we produce our counter-example based on the calculations carried out he* *re. We find it suitable to conjecture that results similar to those given below can* * be obtained for all Dk(pr). 4. the counter-example We specialize to the case k = 2 and calculate the cohomology algebra of X2(pr). Notice that X2(pr) can be constructed as above if p 13. Proposition 4.1. There is an isomorphism of algebras H*(BD2(pr))=(w4; w8) ~=P [a6; a06; b7; b07; d7; d07; t8; s15; s015; q16* *]=R; where R is the set of relations given by at = d0b0; ds = aq = s0b = d0b0t = a0t* *2; t3 = 0; all other possible products of generators except for those given above * *and in 6 RAN LEVI addition t2, b0t and d0t vanish. Thus H*(X2(pr)) is given in the cell diagram b* *elow. _____________________________ | ds = aq = s0b = d0b0t = a0t2| |____________________________| _____ ____|| |_q16|_ |_t2|_ @@I fir @@Ifir _____ @ _____0 ____ @ _____ |_s15_| ||s15_|| |_tb0|_ |_td0|_ @@Ifir ___________@ |_a0t_=_d0b0|_ ____ |_t8|_ fir ____ ____ ____ ____ | d | | b | | d0 | | b0 | |__7_| |__7_| |__7_| |__7_| fir fir ____ ____0 |_a6_| ||a6_|| ___ |_1_| Proof. Let Tr < D2(pr) denote the Sylow p-subgroup and write H*(BTr) ~=P [v1; v2] E[x1; x2]: Let i 2 Fp denote a primitive root of unity of order 4. Computing the algebra structure modulo the Chern classes w4 and w8, whose restrictions to H*(BTr) are given by -(v41+ v42) and v41v42respectively, one observes that the resulting al* *gebra is 22-dimensional. In fact obtaining an Fp vector space basis is easy by routi* *ne invariant calculation. Let res : H*(BD2(pr)) -! H*(BTr) 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 (w4; w8). 1. res(a6) = (v21- iv22)x1x2: 2. res(a06) = -v1v2x1x2: 3. res(b7) = (v31x2 + iv32x1) - (v21v2x1 + iv1v22x2): 4. res(b07) = v1v22x1 - v21v2x2: 5. res(d7) = (v31x1 + v32x2) + i(v21v2x2 - v1v22x1): A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN 7 6. res(d07) = v1v22x2 - iv21v2x1: 7. res(t8) = v1v32- iv31v2: 8. res(s15) = v31v42x1 + v41v32x2: 9. res(s015) = iv71x2 - v72x1: 10. res(q16) = iv71v2 - v1v72: * *|___| Proposition 4.2. There is a retract _ g f _ P 16(pr) S15 -! X2(pr) -! P 16(pr) S15; where P 16(pr) is the 16-dimensional mod-pr Moore space. Proof. Let X denote X2(pr) and recall our convention that all spaces are p-comp* *lete. First observe that there is a map on the 16- skeleton _ f0 : X(16)- ! P 16(pr) S15 given by pinching down the 8-skeleton together with the cells corresponding to * *the products on the right handWside of the diagram above. An obstructionWto extendi* *ng f0 to f : X -! P 16(pr) S15 might exist in ss21P 16(pr) S15, which vanishes* * for the primes under consideration. Next notice that _ _ _ X(8)' P 7(pr) P 7(pr) P 8(pr) S7: If the prime p is sufficiently large, computing ssiX(8) for i = 14 and 15 is st* *raight 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 cohomolo* *gy class. Since s; s0 and q are indecomposable, we conclude that the corresponding attaching maps are trivial and thus the desired map _ g : P 16(pr) S15 -! X is obtained. Finally notice that the composite f O g induces an isomorphism on mod-p __ cohomology and is thus a homotopy equivalence. |_* *_| Corollary 4.3. The homology algebra H*(X2(pr)) contains a tensor algebra on three generators. In particular H*(X2(pr)) grows exponentially and hence is not spherically resolvable of finite weight. We now consider BD2(pr)^p. Through the end of this paper let X denote as before X2(pr) and let G denote D2(pr). Then the Moore space P 16(pr) is a retra* *ct of X. Moreover, Let ss : X -! BG^pdenote the map obtained by the composite X -! (U(8)=ae2)^p- ! BG^p: One readily verifies that the homotopy fibre of ss is equivalent to (S7 x S15)^* *pand that the fibre inclusion map into X induces the zero map on mod-p cohomology. 8 RAN LEVI This together with the assumption that the prime p is sufficiently large implie* *s that the composite S7 x S15 -! X -! P 16(pr) is null homotopic and thus yields a homotopy commutative diagram X _________BG^p- ______-S7 x S15 ________-X | | | | | | | | | | | | |? = |? |? |? P 16(pr) _____P-16(pr) _________*- _________-P 16(pr) Note that the map BG^p- ! P 16(pr) is not multiplicative in general, however the composite P 16(pr) -! X -! BG^p- ! P 16(pr) is evidently homotopic to the identity. Thus we have proven Proposition 4.4. The space P 16(pr) is a retract of BG^pfor p 13. Corollary 4.5. For p 13, the algebra H*(BG^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 BG^pis essential. Proof. By [4] any power map on a single loop space on a Moore space is essentia* *l.__ The result follows. * * |__| 5. speculations If G is a finite p-superperfect group than BG^p does not satisfy Cohen's conjecture unless possibly if the loop space homology H*(BG^p) does not grow exponentially. In view of the fact that the groups D2(pr) are by no means "path* *o- logical", it seems reasonable to wonder what group theoretic properties of G wo* *uld imply that the loop space homology of BG grows polynomialy or at least subexpo- nentially. We refer the reader to [2, 5] to inspect that, in fact, all the exam* *ples of groups G known to satisfy Cohen's conjecture have the property that H*(BG^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 gener* *al 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 BG^p might possibly shed new light on objects of classical interest in homotopy theo* *ry. One might wonder for example whether it is possible to obtain P n(pr) for any given values of n and p as a retract of BG^pfor some finite p-perfect group G. A COUNTER-EXAMPLE TO A CONJECTURE OF COHEN 9 Many interesting p-perfect groups G are not p-superperfect but in this case the 1-connected cover of BG^pis BG"^p, where "Gis the p-universal central exten* *sion of G and is finite and p-superperfect. It is easy to observe that if H*(BG^p) g* *row exponentially then so does H*(BG"^p). For instance BD1(3)^3' BSL2(F9)^3and is 3-perfect but not 3-superperfect. In this case the corresponding space X1(3)* * can be obtained at the prime 3 and is 10-dimensional. In addition there is a fibrat* *ion Y - ! BD1(3)^3- ! X1(3); where Y is spherically resolvable of weight 2. However the existence of a non-* *zero P 1 in H*(X1(3)) implies that a result corresponding to prop 4.2 for X1(3) fails to hold. It would be interesting to know whether or not the 1-connected cover of BD1(pr)^p is spherically resolvable of finite weight as this could in some sense provide a minimal counter-example (of order 4p2). Finally, the referee has suggested that the loop space on the space X1(3) might have the loop space of a 4-cell complex as a retract, where the 4-cell co* *mplex 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 p* *roducts in H*(X1(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 Localization* *s; LNM 304, (1972), Springer-Verlag. [2]F. R. Cohen; Remarks on the Homotopy Theory associated to Finite Perfect Gr* *oups ; 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 BG^p; to ap* *pear, [7]L. Smith; Homological Algebra and the Eilenberg- Moore Spectral Sequence; A* *MS Translatl. 129, (1967), 58-93. [8]G. Whitehead; Elements of Homotopy Theory; Springer-Verlag (1978). Mathematisches Institut, Universit"at Heidelberg, Im Neuenheimer Feld 288,* * 69120 Heidelberg, Germany E-mail address: rlvi@vogon.mathi.uni-heidelberg.de