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. Moreove* *r, given that X is a loop space, one could ask whether the loop space structure is uniqu* *e. A celebrated example of this is given in the classical work of Rector [23] wher* *e it is shown that the 3-sphere S3 admits infinitely many non-equivalent loop struct* *ures (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 BX 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 t* *heory. Let Sm {d} denote the homotopy fibre of the degree d self map of Sm . Our aim i* *s to study the possible loop structures supported by Sm {d} for various values of m * *and d. We restrict attention to the case where, d = 2r. Our main theorem follows. Theorem 1.1. Let m > 1 be an integer. The space Sm {2r} is a loop space if a* *nd only if m = 3 and r > 2. Furthermore, for each value of r > 2, there is a uniqu* *e loop space structure on S3{2r}. The answer to the question, whether Sn{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 w* *here d is a power of 2 and will be handled below. As a consequence we obtain a compl* *ete answer to the problems considered here for this family of spaces. Corollary 1.2. Let m > 1 be an integer. Then the space Sm {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, Sm {d} admits at most one loop structure. ___________ 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q5* *2. Key words and phrases. Classifying spaces, loop spaces, cohomology, Steenrod * *squares, Bockstein spectral sequence. C. Broto is partially supported by DGICYT grant PB94-0725. 1 2 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 rela* *ted 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 -fu* *nctor technology, homology decomposition methods and various other homotopy theoretic tools. Many of the arguments are reminiscent of techniques, successfully used i* *n 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 BS3{2r} given by the 2-completed cla* *s- sifying space for the special linear group SL2(Fq), 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 BSL2(Fq)^2depends only on the mod-2 cohomology of BSL2(Fq) together with its Bockstein spectral sequence. Thus our problem amounts to show- ing that the specified cohomological information determines the homotopy type of BSL2(Fq)^2. Along the same lines we show that the homotopy type of BP SL2(Fq)^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 cert* *ain spaces with a given cohomological structure, as an algebra over the Steenrod al* *gebra and a specified Bockstein spectral sequence. From a purely algebraic point of v* *iew 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 spac* *e. 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 studi* *ed there is of the quaternion groups Q2n. It is shown that for n 5 the homotopy t* *ype of BQ2n 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 Q16on a classifying space for S3{2r} for an appropriate r. Thus one obtains a r* *elation 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 BQ2n is determined by its mod-2 cohomology for every n. An account of this, whe* *re 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 S1{d} is homotopy equivalent to the cyclic group on d elements. Furthermore,* * for m 2, it is easy to compute the integral homology of Sm {d} using the Wang long exact sequence or equivalently the Serre spectral sequence for the fibration de* *fining Sm {d}. Indeed the n-th homology of Sm {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 al* *l reduced homology groups of Sm {d} are torsion groups and thus by [3] there is a homotopy equivalence of Sm {d} to the product of all its p-primary localizations for pri* *mes p 3 dividing d. It is easy to see that for a prime p the localization (or completi* *on) of Sm {d} at p yields a space homotopy equivalent to Sm {pr}, where r is the highe* *st power of p dividing d. Hence there is a homotopy equivalence Yr ' m ji Sm {d} ____- S {pi }: i=1 Moreover, by naturality of the localization functors it is clear that Sm {d} is* * a loop space if and only if each factor on the right hand side of the above equation i* *s a loop space. Our problem is thus reduced to the study of possible loop structure* *s on Sm {pj}, 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 Sm {pr}* * 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 Sm {pr} the cohomology o* *f its classifying space is given as follows (1) H*(BSm {pr}; Fp) ~=P [um+1 ] E[xm ]; with an r-th order Bockstein connecting the generators. With this observation o* *ne 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 t* *he paper. The case p = 2 is as usual more complicated. However if Sm {2r} is a loop spa* *ce then the mod-2 cohomology of its classifying space is still given by Equation (* *1). A work of Aguade [1] puts strong restrictions on the possible values of m and r. * * It follows that if Sm {2r} is a loop space then m = 3 and r > 1. As mentioned abov* *e, for r 3 there is a space which gives a classifying space for S3{2r}, the 2-com* *pleted classifying space BSL2(Fq)^2. The 2-Sylow subgroup of BSL2(Fq) is a generalized quaternion group of order depending on q. Moreover for every r 3 there exists a q, such that Q2r is the 2-Sylow subgroup of SL2(Fq). The mod-2 cohomology of BSL2(Fq) is given by Equation (1) with m = 3 (cf. [15]) and an observation of F. Cohen [8] is that BSL2(Fq)^2is homotopy equivalent to S3{2r} for the appropriate r. The remaining problems are whether S3{4} is a loop space and moreover, whether the loop space structure on S3{2r} for r 3 and possibly on S3{4} is unique. Th* *is 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 4 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 respect* *ively. Let K denote the category of unstable algebras over the Steenrod algebra. By Kfiwe denote the category of unstable algebras together with a Bockstein spectr* *al sequence. A precise definition of this category appears in [4]. By H*fi(-) we* * mean mod-p cohomology as an object of Kfi. We shall abbreviate the phrases Serre spectral sequence, Eilenberg-Moore spec* *tral sequence and Bockstein spectral sequence by Sss, EMss and Bss respectively. We * *write P [xi1; xi2; . .].to denote a polynomial algebra on generators xijand E[xi1; xi* *2; . .].for an exterior algebra. The subscript on a generator will normally indicate its de* *gree, and may be omitted when no confusion can arise. 1.3. Organization of the paper and Acknowledgements. The paper is orga- nized as follows. In section 2 we give a proof of our main theorem based on sec* *tions to follow. Sections 3 and 4 are technical and include cohomological calculation* *s 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 SL2(Fq)^2, q odd, is determined by its cohomology as an object of Kf* *i. In Section 7 we study certain spherical fibrations with a special property. The re* *sults of section 7 allow us to prove in Section 8 the incompatibility of some algebra* *s over the Steenrod algebra with the given Bss. This is used in the proof of Theorem * *1.1 to conclude homotopy uniquenes of BS3{2j}, j 3. Section 9 deals with homotopy uniquenes of BS2n-1{pr}, when it exists for p odd. Some background material on homological algebra is written in an appendix. We wish to thank Centre de Recerca Matematica 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 wi* *ll be developed in the following sections. Our first lemma is well known and has already been mentioned in the introduct* *ion. It determines the unique possible structure of the mod-p cohomology of BSm {pr}* * if it exists. Lemma 2.1. Suppose that Sm {pr}, m > 1, is a loop space. Then H*(BSm {pr}; Fp) ~=P [um+1 ] E[xm ]; as an algebra with fir(x) = u. Furthermore, if X is any p-complete space with * *this cohomological structure then X is homotopy equivalent to Sm {pr}. Proof.The first statement follows at once by inspection of the Serre spectral s* *equence for the path-loop fibration over BSm {pr}. For the second, notice that assumin* *g p- completeness, X is simply connected. Thus there is a map fl : (Sm )^p____-X, wh* *ich induces an isomorphism in mod-p homology up to dimension m. The homotopy fibre of fl is easily seen to have the mod-p cohomology of an m-sphere and since it i* *s p- complete, it is equivalent to the p-completion of an m-sphere. Thus X is equiva* *lent 5 to the fibre of a self map of (Sm )^p. One easily computes the degree of this s* *elf map_ to be precisely pr, 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 BS3{2r} exists then (2) H*(BS3{2r}) ~=P [u4] E[x3]; with fir(x) = u. Let X be a 2-complete space, whose cohomology agrees with the above. Then the* *re is a morphism in K ff : H*(X) ____-H*(BZ=2) with ff(x) = 0 and ff(u) = z4, where z 2 H*(BZ=2) is the generator. By Lannes theory there exists a map g : BZ=2 ____-X, inducing ff on cohomology. The objects H*(X) and H*(BSL2(Fq)), for some q, are isomorphic in K. Moreover, ff coincides with the the map induced on cohomology by the inclusion i of Z=2 a* *s the center of SL2(Fq). Thus TZ=2(H*(X); g*) ~=TZ=2(H*(BSL2(Fq)); Bi*) ~=H*(BSL2(Fq)): Since the right hand side vanishes in dimension 1, it follows that the T funct* *or 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 g ss (3) BZ=2 ____-Map(BZ=2; X)g x EBZ=2 ____-Map(BZ=2; X)g xBZ=2EBZ=2: Proposition 2.2. Let X be a 2-complete space such that H*(X) ~= P [u4] E[x3] with fir(x) = u. Let Er be a space such that there is a principal fibration g ss BZ=2 ____-X ____-Er and the map g is essential. Then r 3 and the cohomology of E3 is given as an object of Kfiby structure 1 below. Furthermore, for r 4 the cohomology of Er * *is isomorphic in Kfito one of the objects given in 2 and 3. 1. P [w2; w3; x3]=(x23+ w23+ w3x3 + w32), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = w22and Sq2x3 = w2w3 + w2x3. 2. P [w2; w3; x3]=(x23+ w3x3), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = 0, Sq2x3 = w2x3 and fir-2x3 = w22. 3. P [w2; w3; x3]=(x23), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = Sq2x3 =* * 0 and fir-2x3 = w22. Moreover, let Y be any simply connected space with either one of the cohomolog* *ical structures given above and let Y <2> denote its 2-connected cover. Then Y <2> ' S3{2r}. 6 Structure 1 in the proposition is isomorphic to H*(BP SL2(Fq)), q 3 (mod 8), or equivalently to H*(BA4), where A4 is the alternating group on four letters. * *Struc- ture 2 is isomorphic to H*(BP SL2(Fq)), q 1 (mod 8), r > 3 [22]. Structure 3 * *is isomorphic as an object of K to H*(BSO(3) x S3). The proof of the proposition w* *ill be carried out in a separate section. Corollary 2.3. The space S3{4} is not a loop space. Proof.If S3{4} is a loop space, its classifying space X = BS3{4} fits in the pr* *incipal fibration (3), with base space E2 = Map(BZ=2; X)g xBZ=2 EBZ=2, but then, the * * __ cohomology of X described in (2) contradicts Proposition 2.2. * * |__| Proof of Theorem 1.1.Assume that BSm {2r} is a classifying space for Sm {2r}, m* * > 1. According to Lemma 2.1, H*(BSm {pr}; Fp) ~= P [um+1 ] E[xm ], as an algebra with fir(x) = u. In [1] it is proved that a necessary condition for this algebr* *a 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 BSL2(Fq)^2, for var* *ious odd prime powers q, provide enough models for classifying spaces of Sm {2r}. T* *his 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 BS3{2r} is determined by its cohomology as an object of Kf* *i. In the sequel we will show that if a 2-complete space Y has the cohomology of BP SL2(Fq) as objects in the category Kfithen Y ' BP SL2(Fq)^2(see Theorem 6.2). Now, suppose that X is a space such that H*fi(X) ~=H*fi(BS3{2r}). Construct t* *he space Er with respect to X as described above. Then H*fi(Er) is isomorphic to o* *ne of the three possible structures specified in Proposition 2.2. By the remarks foll* *owing the proposition either 1. H*fi(Er) ~=H*fi(BP SL(Fq)) for an appropriate odd prime power q or 2. H*fi(Er) coincides with structure 3 of Proposition 2.2. Assume the first case. Then Er ' BP SL2(Fq)^2for a suitable value of q and X is homotopy equivalent to its 2-connected cover, given by BSL2(Fq)^2. Homotopy uniqueness of BSm {2r} 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 S3{2r}. Assume * *we are given a principal fibration which up to homotopy has the form g ss (4) BZ=2 ____-X ____-Er with non trivial g. Our aim is computing the cohomology of Er as an object of K* *fi. Consider the Sss for the delooping of (4) ss w2 (5) X ____-Er ____-K(Z=2; 2): 7 The E2 page takes the form Ep;q2~=Hp(K(Z=2; 2)) Hq(X) ~=(P [2; Sq12; Sq2;12; . .].)p (P [u4] E[x3])q: We claim that the spectral sequence is determined by the requirement that x3 is* * a permanent cycle and d5(u4) = Sq2;12 + 2Sq12. The first differential, d2, is determined by its value on the class u4. Assum* *e it is non trivial; that is, d2(u4) = 2x3. It would follow that x3 is a permanent cycl* *e, for d3(x3) = 22would imply d3(2x3) = 326= 0 in the E3-term of the spectral sequence. But 2x3 = 0 in E3, which gives a contradiction. This calculation suffices to co* *nclude the structure of H*(Er) in low dimensions if this were the case. Namely, we wou* *ld have the classes represented by 2 in degree 2, Sq12 and x3 in degree 3, 2 in de* *gree 4 and 2Sq12 and Sq2;12 in degree 5. Using naturality of the Bss, one finds that* * x3 would in that case be a permanent cycle in the Bss for Er, thus a non torsion c* *lass in the 2-adic cohomology of Er. This in turn is impossible, since Er is the total * *space in a fibration with base and fibre, which are torsion spaces. Hence d2 is trivi* *al. d3 is zero by degree reasons. The next possibly non-trivial differential is d* *4, which may be given by d4(x3) = 22. But if x3 transgresses to 22then Sq222= (Sq12)2 sh* *ould also be in the image of the transgression, which is of course impossible. Hence* * d4 is also trivial. Next we compute d5. Notice that u4 is now transgressive. Since it restricts n* *on- trivially to the cohomology of BZ=2 by hypothesis, it follows by naturality that d5(u4) = Sq2;12 + A2Sq12, for some A 2 F2. But Sq1;2;12 = (Sq12)2, which would imply a contradiction if A = 0. Hence A = 1 and d5(u4) = Sq2;12 + 2Sq12. It now k follows at once that each element of the form u24 is transgressive to a non-zer* *o class and so the E1 term of the spectral sequence is isomorphic to P [2; Sq12] E[x3* *] as an algebra. We obtain an isomorphism of P [w2; w3]-modules H*(Er) ~=P [w2; w3] E[x3]; with Sq1w2 = w3. Since Er is a torsion space, there must exist some positive integer j such th* *at fij(x3) = w22. This enables an easy calculation of the Bss for Er in low dimens* *ions. In particular it follows that H4(Er; Z^2) is cyclic and that H5(Er; Z^2) vanishes.* * In turn, this implies that in the 2-adic Sss for (5), the transgression d5: E0;45~=H4(X; Z^2) ~=Z=2r ____-E5;05~=H5(K(Z=2; 2); Z^2) ~=Z=4 is an epimorphism with a non-trivial kernel. It then follows that r 3 and H4(Er; Z^2) ~=Z=2r-2: Assume r = 3 so that Sq1(x3) = w22. Write Sq2(w3) = Aw2w3+ Bw2x3, A; B 2 F2. Then w23= Sq1;2(w3) = Aw23+ B(w3x3 + w32): Hence A = 1; B = 0 and Sq2(w3) = w2w3. Similarly write Sq2(x3) = Aw2w3+Bw2x3. Then as before x23= Aw23+ B(w3x3 + w32): 8 One has Sq2(x23) = w42on one hand and on the other hand Sq2(x23) = Sq2(Aw23+ B(w3x3 + w32)) = B(w2w3x3 + w3Sq2(x3) + w23w2 + w42): Manipulating with this equation one obtains A = B = 1 so Sq2(x3) = w2x3 + w2w3 and x23= w23+ w3x3+ w32. This completes the description of H*(Er) in the case w* *here r = 3, namely case 1 of Proposition 2.2. Assume next r > 3. Then Sq1(x3) = 0 and fir-2(x3) = w22. One checks as before that Sq2w3 = w2w3 and writes Sq2(x3) = Aw2w3+ Bw2x3. Then x23= Aw23+ Bw3x3. Applying Sq2 one obtains that either A or B must be 0. If A = 1 then change the basis and take x3 = x3 + w3. Then Sq2x3 = 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. C* *onsider the principal fibration (6) BZ=2 ____-Y <2> ____-Y; where Y such that H*(Y ) ~=P [w2; w3]E[x3] as a P [w2; w3]-module with Sq1w2 = * *w3, Sq2w3 = w2w3 and fir-2(x3) = w22. Then in the Sss for (6) one has d2(z) = w2, d3(z2) = w3 and d5(z4) = 0. The spectral sequence thus collapses at the E6 page without any extension problems and so H*(Y <2>) ~=P [u4] E[x3]. Since all spac* *es involved are torsion there must exist a positive integer t, such that fit(x3) =* * u4. The integral Sss now gives at once that t = r and so by Lemma 2.1, Y <2> is a class* *ifying space for S3{2r}. 4. Homology decomposition of BP SL2(Fq) Let P SL2(Fq) denote the projective special linear group of rank 2 over the f* *ield of q elements Fq. Preparing to prove homotopy uniqueness of BP SL2(Fq)^2, we study homology decompositions of this space as in [17, 18, 10]. For the terminology w* *e use, the reader is referred to [10]. 2-1) Let q be an odd prime power. Then the order of P SL2(Fq) is q(q___2. Notice t* *hat 2-1) the highest power of 2 dividing q(q___2is precisely 4 if q 3 (mod 8) and that* * if 2-1) s q 1 (mod 8) then q(q___2is divisible by 2 for s 3 and, depending on q, every such s may appear. We will write 2skn if 2s is the highest power of 2 dividing * *n. The 2-* *1) 2-Sylow subgroup of P SL2(Fq) is known to be the dihedral group D2s if 2skq(q__* *_2. The respective cohomology algebras are also known [22] and are described below. All groups P SL2(Fq) have the same mod-2 cohomology algebra and only elementa* *ry torsion in their integral cohomology if q 3 (mod 8): H*(BP SL2(Fq)) ~=P [w2; w3; x3]=(x23+ w23+ w3x3 + w32); 9 with Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = w22and Sq2x3 = w2w3 + w2x3. As we observe later every 2-complete space with this cohomology is homoto* *py equivalent to (BA4)^2, where A4 is the alternating group on 4 letters. The case q 1 (mod 8) is a bit more complicated. Here too, the mod-2 cohomol- ogy as an algebra over the Steenrod algebra is the same in all case. The Bss th* *ough varies according to the highest power of 2 dividing the order of P SL2(Fq). Ind* *eed if 2-1) q 1 (mod 8) and 2skq(q___2then H*(P SL2(Fq)) ~=P [w2; w3; x3]=(x23+ w3x3); with Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = 0, Sq2x3 = w2x3 and fis-1x3 = w22. 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 2-1) that 2sk(q - 1) and hence 2skq(q___2. Proof.For s 3 consider the arithmetic progression ns;j= 2s-2j - 2s-3: Let qs;j= 8ns;j+ 1 = 2s+1j - (2s- 1): Since the greatest common divisor of 2s+1 and 2s - 1 is 1, Dirichlet's theorem * *[16] implies the existence of infinitely many primes of the form qs;j. Notice that e* *ach such prime is congruent to 1 modulo 8 and that qs;j- 1 = 8ns;j= 2s(2j - 1): Thus for integers j such that qs;jis a prime, the highest power of 2 dividing t* *he_order of P SL2(Fqs;j) is 2s. * *|__| 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 SL2(Fq). * *Let s be such that 2skq - 1 and let " be a primitive root of unity in Fq of order 2* *s. Then the matrices X = "00"-1and Y = -0110satisfy the relation s-1 2 2 X2 = Y = (XY ) = 1; modulo the center of SL2(Fq), generated by the matrix -I. Hence the classes [X] and [Y ] of these matrices in P SL2(Fq) generate a dihedral group of order 2s, * *which gives the 2-Sylow subgroup. Notice that if q 1 (mod 8) the 2-Sylow subgroup is of order at least 8. Con* *sider the following elementary abelian 2-subgroups of P SL2(Fq). s-2 Z = <[X]2 > ~=Z=2 s-2 V = <[X]2 ; [Y ]> ~=Z=2 x Z=2 s-2 W = <[X]2 ; [X][Y ]> ~=Z=2 x Z=2 10 According to [17, Thm. 7.7], the set E2 = {Z; V; W } is an ample collection o* *f el- ementary abelian 2-subgroups of P SL2(Fq) (which, roughly speaking means that it can be used to obtain a homology decomposition). The associated conjugacy cate- gory AE2 has E2 as objects, with morphisms among them induced by inclusions and conjugations in P SL2(Fq). Our next statement describes the category AE2. First, let G1 and G2 be finite groups with a specific common subgroup H. Consider the category __oe_______ ______-_ A(G1; G2; H) = G2 |_2|oe___oe_|_0|__-_-|_1|G1: G2=H G1=H __ __ __ __ It has three objects |_0|, |_1|and |_2|. The_automorphism group of |_i|is given* * by_Gi_for i =_1;_2 whereas the automorphism group of |_0|is trivial. Each morphism set fr* *om |_0| to |_i|, i = 1; 2, admits a natural Gi action and is isomorphic as a Gi-set to * *Gi=H. Proposition 4.2. For q 1 (mod 8), the set E2 = {Z; V; W } is an ample collec- tion of elementary abelian 2-subgroups of P SL2(Fq) and the E2-conjugacy catego* *ry AE2 is the image of the functor from_the category_A(3; 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 SL2(Fq) with q 1 (mod 8). Notice that the automorphisms of a subgroup U in the conjugacy category are given by the quotient NG (U)=CG (* *U) of the normalizer by the centralizer of U in G. One easily verifies the followi* *ng. i 0 * 1. CG (Z) = NG (Z) = < 0i-1 ; -0110> ~=D2q-2, where i is generator of Fq a* *nd D2q-2is the dihedral group of order 2q - 2. Notice that D2s < D2q-2and the mod-2 restriction map isfanfisomorphism.i 2. CG (V ) = V and NG (V ) V ~= 3 is generated by the classes A and B in * *G, p_ s-2 p _ of A = _1_i-111i-iand B = i0p_ii0respectively. Here i = "2 , so by iwe s-3 2s-2 2s-2 mean "2 . Conjugation by A sends X to Y and Y to X Y , 2s-2 2s-2 whereas conjugation by BfflipsfiY and X Y and fixes X . 3. CG (W ) = W and NG (W ) W ~= 3 generated by the classes C and D in 1 " p_ G of C = _1_i-1i"-1-iand D = -ip0_i"--i"10respectively. Conjugation by C 2s-2 2s-2+1 sends X to X Y and X Y to X Y , whereas conjugation by 2s-2+1 2s-2 D flips X Y and X Y and fixes X . Thus, the automorphisms in AE2are as described, and the other morphisms follo* *w __ immediately. This completes the proof of the proposition. * * |__| We can draw the E2-conjugacy category as oe____ _____- AE2: 3 W oe____oe_Z _____-_-V 3 : 3=2 3=2 The centralizer diagram ffE2: AopE2___-Spaces 11 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 xG (G=CG (V )). Up to homotopy and 2-completion the diagram in our case is given by ____- oe___ 3 BW ____-_-BD2s oe___oe_BV 3: Notice that CPSL2(Fq)(Z) = D(q-1), but (BD(q-1))^2= BD2s. 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 aE2: hocolimffE2 ____-BP SL2(Fq) induces a mod-2 cohomology isomorphism. Moreover, lim-------0H* O ffE2~= H*(BP SL2(Fq)); AE2 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 ffE2 ' BP SL2(Fq)2 AE2 2 and an isomorphism lim-------0H* O ffE2~= H*(BP SL2(Fq)) AE2 * * __ while lim-------iAEH* O ffE2= 0 for all i 1. * * |__| 2 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 w* *hether they could exist. However, if X has the mod-p cohomology of a finite group G, t* *hen a sensible definition for a p-Sylow subgroup for X appears to be a map f : Bss __* *__-X, where ss 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-subg* *roup V contained in ss, the p-Sylow of G, such that CG (V ) = ss. Of course, this su* *bgroup V does not always exist, which is the main reason why, in our specific case, we w* *ork first with P SL2(Fq) rather than directly with SL2(Fq). In those favorable cases wher* *e 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 t* *o do its work. This makes it essential to use an interpretation of T due to E. Farjo* *un 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 Tp-groups. We show that c* *yclic p-groups and dihedral 2-groups are Tp-groups, a fact which is then used to obta* *in a 2-Sylow subgroup for a space X with the same cohomology as P SL2(Fq). 12 Definition 5.1. Let ss be a finite p-group. We say that ss is a Tp-group if fo* *r any tower of fibrations {Fs} such that 1. ssi(Fs) is a finite p-group for every i and s and 2. lim-!H*fi(Fs; Fp) ~=H*fi(ss; Fp) there is a homotopy equivalence lim-Fs ' Bss. We now give some elementary examples of groups which are Tp. We know of furth* *er examples, which are irrelevant to this paper. One might like to ask whether ev* *ery finite p-group is Tp as a positive result might have interesting implications. Proposition 5.2. Every cyclic p-group is a Tp-group. Also, every dihedral 2-gr* *oup is a T2-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 fromf[4,fx3]iand [2, x6]. Recall that H*fi(BD2n) ~=P [x1; y1; w2] (x2 + xy), where degrees are given by* * sub- scripts, Sq1(w) = wy and fin-1(xw) = w2. Let {Ys} be a tower of fibrations satisfying 1 and 2 in Definition 5.1 with r* *espect to D2n. Let sss : Ys ____-Ys-1 denote the projection. For a sufficiently large * *s there are elements y1;s2 H1(Ys) such that 1. ss*s+1(y1;s) = y1;s+1and 2. the sequence {y1;s} represents the element y1 2 lim-!sH1(Ys) ~=H*(BD2n). Let 's : Ys ____-BZ=2 be a map classifying y1;s. Then there is the following ho* *motopy commutative diagram of fibrations. : :_:__-Fs+2_____- Fs+1______-Fs | | | js+2|| js+1|| js|| ?| sss+2 |? sss+1 ?| : :_:__-Ys+2______-Ys+1______-Ys | 's+2 || 's+1|| 's|| ?| |? ?| : :=:==BZ=2 ==== BZ=2 ==== BZ=2 Since the homotopy groups of all spaces involved are finite, it follows that * *the sequence lim-Fs ____- lim-Ys ____-BD2n 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 colla* *pses at its E2 page, being a fibration over BZ=2. Thus H*(Fs) ~=T orH*(BZ=2)(F2; H*(Ys)): 13 Since T or commutes with direct limits, we have that (7) P [v2] E[ff1] ~=T orH*(BZ=2)(F2; H*(D2n)) ~= ~=T orH*(BZ=2)(F2; lim H*(Y )) ~=lim H*(F ) -! s s -! s s as P [v2]-modules. Notice that the edge homomorphism lim-!H*fiYs ! lim-!H*fiFs is a morphism in Kfiwhich at the level of the first page of the Bss is an epimo* *rphism ffi2 P [x1; y1; w2] (x + xy) ___--P [v2] E[ff1] sending x1 to ff1, y1 to zero and w2 to v2. Next, we compute lim-!H*fiFs as an object of Kfi. From the description of the* * edge homomorphism above it follows that Sq1 acts trivially on P [v2] E[ff1] and the morphism induced on the second page of the Bss P [w22] x1w2P [w22]-____-P [v2] E[ff1] is monic. The next and last non-trivial differential in the Bss of the source is determ* *ined by fin-1(x1w2) = w22. Thus in the Bss of the target we must have fin-1(ff1v2) =* * v22. By the P [v2]-module structure of lim-!sH*(Fs), it follows that fin-1(ff1) = v2* * and so fin-1(ff1vk2) = vk+12. Moreover, since homotopy groups of Ys are finite for eve* *ry s, it follows that the homotopy groups of each Fs are finite and since cyclic 2-group* *s are T2 by the first part of the proposition, we have holimFs ' BZ=2n-1. We have thus obtained a quasi fibration (8) BZ=2r ____- lim-Ys ____-BD2n: In particular Y = lim-Ys 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 } ____-{Ys}; where {Y } is the constant tower, is a weak pro-homotopy equivalence. Consequen* *tly the map of towers {Hn(Y )} ____-{HnYs} is a pro-isomorphism by [3, III, 3.4] an* *d so H*fiY ~= lim-!H*fiYs ~=H*fiBD2n: Now [4, Thm. 1.1] applies, Y ' BD2n __ and the proof is complete. |_* *_| Lemma 5.3. Let X be a p-complete space with H*fi(X) ~=H*fi(BG) where G is a f* *inite p-perfect group. Then: i) ssi(X) is a finite p-group for all i > 0. ii)For any elementary abelian p-group V , and an n-th Postnikov section PnX o* *f X, the homotopy groups of any component of Map (BV; PnX) are finite p-groups. 14 Proof.By Hypothesis H1(X) = 0 and X is p-complete. Hence X is 1-connected and [2, 5.7 (1)] applies so that Hj(X; ^Zp) is a finitely generated ^Zp-module * *for all j. Furthermore, by our assumption on the Bss all p-adic cohomology groups are fini* *te p-groups, hence the first statement follows by [2, 5.7 (2)]. The second statement follows as well by an argument of [2, x6]. For an Eilenb* *erg- MacLane space we have Y Map (BV; K(ss; n))' K Hn-j(BV; ss); j: 1jn so if ss is a finite p-group, then the homotopy groups of Map (BV; K(ss; n)) ha* *ve_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 SL2(Fq). More precisely, we have the following Proposition 5.4. Let X be a 2-complete space with H*fi(X) ~=H*fi(BP SL2(Fq)): i Let D2s____-P SL2(Fq) be the inclusion of a 2-Sylow subgroup. Then there is a m* *ap f :BD2s____-X such that f*, coincides with the composition (Bi)* * H*fi(X; F2) ~=H*fi(BP SL2(Fq)) ____-Hfi(BD2s): Proof.If q 3 (mod 8) then the 2-Sylow subgroup of P SL2(Fq) is D4, an elemen- tary abelian 2-group of rank 2. Then our claim follows directly from Lannes the* *ory [20]. Namely, in this case one has [BD4; X] ~=[BD4; BP SL2(Fq)^2] ~=Hom K (H*(BP SL2(Fq)); H*(BD2s)): Suppose q 1 (mod 8). Because of Lemma 4.1 it is enough to consider the case q 1 (mod 8). Let s satisfy 2sk(q -1) so in particular s > 2. Fix a 2-Sylow su* *bgroup ss ~=D2sand let Z ~=Z=2 be its center. Let j denote the inclusion of Z in P SL2* *(Fq). By Lannes theory there is a map fZ :BZ ____-X, inducing the composition (Bj)* * H*(X) ~=H*(BP SL2(Fq)) ____-H (BZ) on cohomology. ev Consider the evaluation map Map (BZ; X)fZ ____-X. We proceed by showing that Map (BZ; X)fZ ' BD2sand 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 SL2(Fq); (Bj)*)~= ~=H* BCPSL2(Fq)(Z) ~=H*(BD(q-1)) ~=H*(BD2s) 15 Furthermore, one obtains as well that the augmentation (10) ": H*(X) ____-T (H*(X); f*Z) coincides with (Bi)*: H*(BP SL2(Fq)) ____-H*(D2s) under the identification of t* *he sources given in terms of the assumed isomorphism. It is then, explicitly descr* *ibed as ffi ffi " : P [w2; w3; x3] (x23+ w3x3)___- P [x1; y1; w2] (x21+ x1y1) w2 |___- w2 + y21 (11) w3 |___- w2y1 x3 |___- w2x1 Let {PnX}n be the Postnikov Tower for X and fZ(n): BZ _____-PnX be the factorization of fZ through the tower. Applying the functor Map (BV; -), we get* * a tower, {Map (BV; PnX)fZ(n)}n and a homotopy equivalence (12) holimMap (BZ; PnX)fZ(n)' Map (BZ; X)fZ: Since the space X is 2-complete nilpotent and of finite type, the Farjoun-Smi* *th theorem states (13) T (H*(X); f*z) ~=lim-------!H* Map (BZ; PnX)fZ(n) : n The map in the theorem is given as the obvious map (14) !lim-------T (H*(PnX); f*Z(n)) ____-!lim-------H* Map (BZ; PnX)fZ(n)* * ; n n using the fact that for a space X as above the map!lim-------nH*(Pn(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*(BD2s) as an A*2algebra. There are homomorphisms H*(PnX) _____-H*(Map (BZ; PnX)fZ(n)) induced by evaluation maps. Taking the limit on both sides and using the isomorphism (14),* * the resulting map ev* * (15) H*(X) ____-!lim-------H (Map (BZ; PnX)fZ(n)) n 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*(BD2s) as an object of Kfithat is compatible with such a monomorphism, namely * !lim-------H*fiMap (BZ; PnX)fZ(n) ~=Hfi(BD2s) : n Now, since X is 2-complete and H*fi(X) ~=H*fi(P SL2(Fq)), Lemma 5.3 imply that ssj(X) is a finite 2-group for all j and also that the homotopy groups of the m* *apping spaces ssj(Map (BZ; PnX)fZ(n)) are finite 2-groups for all j and n. 16 By Proposition 5.2 the dihedral group D2s is a T2-group and so we conclude th* *at holimnMap (BZ; PnX)fZ(n)' BD2s. This combined with (12) gives Map (BZ; X)fZ ' BD2s: Taking the evaluation, we obtain a map f :BD2s____-X; such that the induced homomorphism on cohomology coincides with the composition (Bi)* * __ H*fi(X) ~=H*fi(BP SL2(Fq)) _____-Hfi(BD2s): |__| 6. Homotopy type of the two completion of BP SL2(Fq) In this section we prove that if X is a 2-complete space with the mod-2 cohom* *ology of BP SL2(Fq) as an object of Kfithen X ' BP SL2(Fq)^2. Proposition 6.1. Let X be a 2-complete space such that H*fi(X) ~=H*fi(BP SL2(Fq)) Let s be the highest power of 2 dividing the order of P SL2(Fq). Assume s > 2 a* *nd let f :BD2s____-X be the map constructed in Proposition 5.4. Then, the induced map f] BZ=2 ' Map (BD2s; BD2s)id____- Map (BD2s; X)f is a homotopy equivalence. Proof.Without loss of generality, we can assume that q 1 (mod 8). Consider the sequence of maps BD4 BD8 : : :BD2r : : :BD2s; induced by the natural inclusions. Denote fr = f O (Bir): BD2r! BD2s! X. We prove by induction that f induces homotopy equivalences f]: Map (BD2r; BD2s)Bir____- Map (BD2r; X)fr: The case r = 2, follows by Lannes theory [20]. Since D4 ~= (Z=2)2 and is sel* *f- centralizing in both D2sand P SL2(Fq), a T functor computation shows that BD4 ' Map (BD4; BD2s)Bi2' Map (BD4; X)f2. By induction assume the conclusion is true for r - 1. The space Map (BD2r; X) may be written as the homotopy fix point space Map (ED2r=D2r-1; X)hZ=2. The cor- responding construction holds for BD2sreplacing X. Now, the map f] : Map (ED2r=D2r-1; BD2s)Bir-1____- Map (ED2r=D2r-1; X)fr-1 is a Z=2-equivariant homotopy equivalence, so we obtain, (16) Map (BD2r; BD2s)g ' Map (ED2r=D2r-1; BD2s)hZ=2Bir-1' ' Map (ED2r=D2r-1; X)hZ=2fr-1' Map (BD2r; X)f 17 where g = {g | g|BD2r-1' Bir-1} and f = {f | f|BD2r-1' fr-1}. This completes the induction step. Finally, notice that for r = s, the left hand side of the equation above is c* *onnected. Hence the same hold for the right end side and the component of the mapping spa* *ce __ there is essentially the one of f. This completes the proof. * * |__| Theorem 6.2. Let X be a 2-complete space with H*fi(X) ~=H*fi(BP SL2(Fq)): Then there is a homotopy equivalence X ' BP SL2(Fq)^2 Proof.If q 3 (mod 8) then the 2-Sylow subgroup of P SL2(Fq) is D4, i.e. eleme* *n- tary abelian of rank 2. In this case one has by Lannes theory a map f : BD4 ____-X inducing the restriction map under the given identification of the algebras H*(* *X) and H*(BP SL2(Fq)). The T functor is now employed to compute the cohomology of Map (BD4; X)f. Indeed, since D4 is the 2-Sylow subgroup, it is immediate th* *at H*(Map (BD4; X)f) ~=H*(BD4) as A*2algebras and so Map (BD4; X)f is homotopy equivalent to BD4. The evaluation map thus gives a map BD4 _____-X, which is equivariant with respect to any automorphism of D4, in particular with respect * *to the natural Z=3 action. This gives a map f": BD4 xZ=3EZ=3 ____-X; which one easily checks to induce an isomorphism on mod-2 cohomology and so a homotopy equivalence after 2-completion, X ' (BD4 xZ=3EZ=3)^2' (BA4)^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 comm* *u- tative diagram ______3=2______- oe________3=2___ 3 BW _______________-__________-BD2soe_______oe_______BV 3 HH | HH | (17) HH |f0 f2 HHj ?| ss f1 X One can regard this diagram as a natural transformation, defined only up to hom* *o- topy, from the diagram ffE2 for BP SL2(Fq) (see section 4) to the constant diag* *ram on our space X f :ffE2 ____-X: The map f0: BD2s ! X is given by Proposition 5.4. The maps f1: BV ! X, and f2: 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 18 defined on the level of cohomology (which of course they can since they exist f* *or X = BP SL2(Fq)). By Proposition 4.2 we can identify AE2 with A(3; 3; Z=2). To simplify the no- tation let A denote A(3; 3; Z=2) and let ff denote the diagram ffE2. Diagram (17) defines a map from the 0-skeleton of hocolimAopff to X. The ob- structions to extending this map to the actual homotopy colimit lie in the grou* *ps limj+1ssj(Map (ff; X)f) A for j 1 [26]. By Lemmas 6.3 and 6.4 below these obstruction groups vanish. Thus we obtain a map f (BP SL2(Fq))^2' hocolimff ____-X Aop where the homotopy equivalence on the left was stated in Proposition 4.3. Moreo* *ver, the diagram BD2s P | PP P | PP Pf0 Bi | P PP ?| P Pq (BP SL2(Fq))^2__________________-X f commutes up to homotopy. It follows at once that f induces an isomorphism in mo* *d-2_ cohomology and is therefore a homotopy equivalence. |* *__| To complete the proof of the theorem we need to prove vanishing of the obstru* *ction groups. For each j 1 define a functor j: A ____-Ab by __ __ j(|_i|) = ssjMap (ff(|_i|);fX)i: Lemma 6.3. The values of the functors defined above are given as follows. __ 1. j(|_0|)_= ssj(BZ=2) 2. j(|_1|)_= ssj(BV ) 3. j(|_2|) = ssj(BW ) Proof.First notice that __ __ j(|_0|) = ssjMap (ff(|_0|);fX)0= ssj(Map (BD2s; X)f0) = ssj(BZ=2) by Proposition 6.1. Next since V is self centralizing in P SL2(Fq) it follows from Lannes theory * *that __ __ j(|_1|) = ssj(Map (ff(|_1|); X)f1)= ssj(Map (BV; X)f1) = ssj(BV ): * *__ The third case follows by the same argument. |* *__| Lemma 6.4. limiAj = 0 for all i 0; j 1 except for i = j = 1 in which case lim1A1 = Z=2. Proof.We only need to take care of the case j = 1. The functor 1 is oe____ _____- 3 V oe____oe_Z=2 _____-_-W 3 3=2 3=2 19 and according to Proposition 10.3 we have an exact sequence 0 ____-lim01 ____-Z=2 ____-V 2 W 2 ____-lim11 ____-0 A A and limiA1 = 0 if i > 1. The result is forced by the fact that V 2 W 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 sp* *herical fibrations. We say that the fibration F ____-E ____-B is cohomologically trivial if the associated mod-2 Sss collapses at its E2 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 spac* *es are 2-complete. The case of interest to us is when the base space of the given fibration is B* *O(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 f g = S3 ____-E() ____-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 Bi Bss (18) BZ=2 ____-BO(2) ____-BO(2) w2 2 which is classified by the second Stiefel-Whitney class BO(2) ____-B Z=2. Thus (19) Bss*(w1) = w1 and Bss*(w2) = 0 in mod-2 cohomology, while Bi*(w1) = 0 and Bi*(w2) = z2, the class z being the 1-dimensional generator of H*(BZ=2). Furthermore, since the fibration is cohomologically trivial, one has by Lann* *es theory [20] that there is a lifting of Bi to a map h: BZ=2 ____-E(); which is unique up to homotopy. Thus the map g* : Map (BZ=2; E())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. 20 i ev Since Z=2 ____-O(2) is central, the evaluation Map (BZ=2; BO(2))Bi ____-BO(2) is a mod-2 equivalence. Also, an easy computation of the respective Lannes T fu* *nctor ev shows that Map (BZ=2; E())Bi ____-E() is a mod-2 equivalence. Denote E(1) = Map (BZ=2; E())h xBZ=2EBZ=2: Taking the respective Borel constructions one gets a diagram of principal fib* *rations BZ=2 === BZ=2 h|| ||Bi ?| ?| S3w____-E() ____-BO(2) ww (20) ww || ||Bss ?| ?| S3 ____-E(1) ___-BO(2) | | | |w2 ?| ?| B2Z=2 == B2Z=2 where all spaces are assumed 2-complete. Let 1 denote the quotient spherical fi* *bra- tion S3 ____-E(1) ____-BO(2). Lemma 7.2. 1 is cohomologically trivial. Proof.The Thom space for 1, T (1), is defined as the homotopy cofibre of the pr* *o- jection E(1) _____-BO(2). Its mod-2 cohomology is determined by the Thom iso- morphism theorem "H*(T (1)) ~=U1 . H*(BO(2)) as H*(BO(2)) modules, where U1 2 "H*(T (1)) is the Thom class. The action of the Steenrod squares on the Thom class defines the Stiefel-Whitney classes of 1 by * *the equations Sqi(U1 ) = U1 . wi(1): The action of the Steenrod algebra on the cohomology of T (1) is thus determine* *d by its action on the Thom class, given above, the action on H*(BO(2)) and the Cart* *an formula. We have then a long exact sequence of modules over both H*(BO(2)) and the Steenrod algebra (21) j* q p* q ffi q+1 : : :___-H"q(T (1)) ____-H (BO(2)) ____-H (E(1)) ____-H" (T (1)) ____- : : : where j* is determined by its value on U1 , j*(U1 ) = e1 , the Euler class, whi* *ch coincides with the top Stiefel-Whitney class. We proceed by computing the Stiefel-Whitney classes for 1. Claim 7.3. The Stiefel-Whitney classes of 1 are trivial. 21 Proof.Write w(1) = 1 + w1(1) + w2(1) + w3(1) + w4(1) with wi(1) 2 Hi(BO(2)), for the total Stiefel-Whitney class of 1. Since Bss*(1) = and is cohomologica* *lly trivial Bss*(w(1)) = 1. The action of the Steenrod squares on the Stiefel-Whitney classes of any sphe* *rical fibration is given by the Wu formula. This fact is exploited in [5], where poss* *ible total Stiefel-Whitney classes in certain invariant algebras are classified. In the pa* *rticular case, where the base of a spherical fibration is BO(2), only 1, 1 + w1, 1 + w1 * *+ w2 and their products can form total Stiefel-Whitney classes (see [5, section 2]).* * So, w(1) = (1 + w1)r(1 + w1 + w2)s with r; s 0. One then checks that the only possibility that satisfies Bss*(w(1)) = 1 is r = s = 0 (see (19)). Hence the t* *otal_ Stiefel-Whitney class for 1 is trivial. * * |__| It follows that the long exact sequence (21) collapses to a short exact seque* *nce p* * ffi *+1 0 ____-H*(BO(2)) ____-H (E(1)) ____-U1 . H (BO(2)) ____-0 : There is a 3-dimensional class x3 in H*(E(1)) with ffi(x3) = U1 . We now compute the Steenrod algebra action on x3. Since Sqi(U1 ) = U1 . wi(1) = 0, we have Sqi(x3) 2 p*(H*(BO(2))). Identifying H*(BO(2)) with its image under p*, write Sq2(x3) = 1w51+ 2w31w2 + 3w1w22, i2 Z=2. Without loss of generality 1 = 3 = 0 or otherwise define x3= x3+1w31+3w1w2. Then Sq2(x3) = w31w2 with = 2 + 3 and ffi(x3) = U1 . It follows that (x3)2 = Sq3(x3) = Sq1Sq2x3 = 0 and so (Sq1x3)2 = Sq2(x23) = 0. Hence Sq1x3, being considered as an element of H*(BO(2)), must be zero. Finally, Sq2Sq2x3 = Sq1Sq2Sq1x3 = 0, and since w31w226= 0, it follows that = 0. We have thus obtained a choice of x3 2 H3(E(1)) with ffi(x3) = U1 and Sqi(x3)* * = 0 for all i 1 and therefore H*(E(1)) ~= H*(BO(2)) H*(S3) as algebras over the * * __ Steenrod algebra and the proof of the lemma is complete. * *|__| Let jn: D2n ____-O(2) denote the canonical inclusion of the dihedral group of* * order 2n in O(2). Notice that jn factors through jn+1 and the direct limit with respe* *ct to the ji'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 (Bjn)*() is trivial. Proof.Recall that the class j*n(w2) = w 2 H*(BD2n) classifies the central exten* *sion Z=2 ____-D2n ____-D2n-1and so the square Bjn BD2n ______-BO(2) | | | | |Bss ?| Bjn-1 ?| BD2n-1 _____-BO(2) is a pull-back diagram. Notice that this means that the orbit spaces O(2)=D2n * *are the same for all n. Indeed, an easy diagram chase argument show that these orb* *it spaces are, one and all, homotopy equivalent to S1. 22 Observe that the fibration 1 over BO(2) satisfies the same condition as , nam* *ely it is cohomologically trivial. Hence the construction can be repeated inductive* *ly to obtain a sequence of spherical fibrations i S3w_____-E(i) ____-BO(2) ww (22) ww || ||Bss ?| ?| i+1 S3 ____-E(i+1) ___-BO(2) where each i is a pull-back of i+1. Pulling back Diagram (22) along the successive maps Bjn, we get the following diagram, where every square on the right is a pull-back square. S3w_____-E((Bjn)*()) _____-BD2n ww | ww || || ?| ?| S3 ____-E((Bjn-1)*(1)) ___-BD2n-1 .. . . . .. .. S3w____-E((Bj1)*(n-1)) ___-BZ=2 ww | ww || | ?| ?|| S3 _____-E((Bj0)*(n)) ______-* Obviously, (Bj0)*(n) is a trivial fibration. Since (Bjn)*() is a successive p* *ull-back_ of (Bj0)*(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 : : :___-BD2m ____-BD2m+1 ____- : :.:Thus in the category of fibrations and commutative diagrams between them, the fibration* * is the homotopy colimit of : :_:__-(Bjm )*() ____-(Bjm+1 )*() ____- : :.:But we ha* *ve seen that all these fibrations are trivial and furthermore, it is easy to obser* *ve that the product decomposition for the respective total spaces commute with the inclusio* *ns.__ Thus is a trivial fibration and the proof is complete. * * |__| 8. homotopy uniqueness of BS3{2r} We are now ready to prove that any 2-complete space with the mod-2 cohomology of BS3{2r} 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 re* *alizable as the cohomology of a space. Proposition 8.1. There does not exist a space W with mod-2 cohomology H*(W ) ~=P [w2; w3; x3]=(x23); with Sq1w2 = w3, Sq2w3 = w2w3 and with fir-2x3 = w22. 23 Proof.Let W be a space with this cohomological structure. Thus H*(W ) is isomor- phic to H*(BSO(3)) H*(S3) as an algebra over the Steenrod algebra, with the additional higher Bockstein. Let i : Z=2 ____-SO(3) be the map sending the gene* *ra- tor to the diagonal matrix diag(-1; -1; 1). Then the centralizer of the image o* *f i in SO(3) is O(2). By [20] TZ=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 proj * Bi* * H*(W ) ____-H (BSO(3)) ____-H (BZ=2): Thus we have (23) TZ=2(H*(W ); f*) ~= TZ=2(H*(BSO(3)); Bi*) TZ=2(H*(S3); *) ~=H*(BO(2)) H*(S3) as algebras over the Steenrod algebra. Furthermore, since there are no relatio* *ns 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) x S3 ____-BSO(3) x S3 and it follows that fir-2(x3) = w22in H*(X) up to the usual indeterminacy. Lemma 8.2. The space X defined above fits in a fibration S3 ____-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 struct* *ure implies that ss1(X) ~= Z=2. It is easy to calculate the cohomology of the univ* *ersal cover, Y of X from the fibration Y ____-X ____-BZ=2: Thus H*(Y ) ~= P [w2] E[x3], with fir(x3) = w22. In particular the second int* *egral cohomology of Y is torsion free. Since X is a torsion space, inspection of th* *e Sss for the fibration above gives that the Z=2 action on H2(Y ; Z) is given by invo* *lution. Thus Z=2 acts on ss2(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 res* *ult_ follows. |__| 24 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*(S3) as algebras over the Steenrod algebra. By Theorem 7.1 X must be homotopy equivalent to a product S3 x BO(2), up to completion. This is of course impossible since b* *oth S3 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 S2n-1{pr}. Thus n|p-1 and H*(X; Fp) is given by Equation (1). In particular this cohomology alg* *ebra appears as the cohomology of a finite group. Indeed let G denote the semi dire* *ct product of Z=pr by Z=n, where the last acts on Z=pr as a subgroup of its automo* *rphism group. Then H*fi(X; Fp) ~=H*fi(BG; Fp). Conversely, Proposition 9.1. Let X be a space with H*fi(X; Fp) ~= H*fi(BG; Fp), where G ~= Z=pr o 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, s* *o we indicate the proof without going into great detail. Lannes theory gives that there is a map f : BZ=p ____-X; * *OE such that f* coincides with the restriction map induced by the inclusion Z=p __* *__-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=pr. This * *is done along the lines of the proof of Proposition 5.4 using the fact that Z=pr i* *s a Tp-group (Proposition 5.2). It follows that Map(BZ=p; X) ' BZ=pr: Moreover the evaluation map ev* : H*(X) _____-H*(Z=pr) 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 componen* *t, since it does so if X is replaced by BG. Furthermore, one has that the evaluati* *on map is equivariant with respect to the trivial action of Z=n on X. Thus the evalua* *tion map extends to a map on the Borel construction f : Map (BZ=p; X)f xZ=nEZ=n ____-X; and it is a triviality that this map induced a mod-p cohomology isomorphism. No* *tice_ 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 compu* *te higher derived functors of the inverse limit functor. These techniques are esse* *ntially due to Aguade [1] and are adapted here to serve our purposes. 25 Lemma 10.1 (Aguade). 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 sh* *ort exact sequence K ____-R[G=H] ____-R : Then for any R[G]-module M there is a long exact sequence (24) 0 ! MG ! MH ! Hom G(K; M) ! H1(G; M) ! H1(H; M) ! : : : : :!:Hi(H; M) ! ExtiG(K; M) ! Hi+1(G; M) ! Hi+1(H; M) ! : : : For the proof the reader is referred to [1]. Consider the following categories: __ oe______ ______-__ A = G2 |_2|oe_____oe|__|_|__||_||_||0|_____-____-|_1|G1 G2=H G1=H and __ ______-__ B = |_0|_____-____-|_1|G1 G1=H Here G1 and_G2_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 Gi_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 Gi action and are isomorphic_as Gi-sets_to Gi=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 CR-mod denote the functor_category_from C to R-mod . Define the functor L 2 AR-mod which takes |_0|to R and |_i|to R[Gi=H]. The action of L on morphisms is taken * *to be the obvious one. __ For any other M 2 AR-mod , let Mi denote M(|_i|), i = 0; 1; 2. Then Hom AR-mod(L; M) ~=Hom R (R; M0) ~=M0: hence L is projective in AR-mod . Let R 2 AR-mod denote the constant functor w* *ith value R. Then there is an exact sequence in AR-mod : 0 ____-K ____-L ____-R ____-0 where K(0) = 0 and K(i) = Ki = ker(R[Gi=H] ____-R) for i = 1; 2. This gives an exact sequence: (25) 0 ! Hom AR-mod(R; M) ! Hom AR-mod(L; M) ! Hom AR-mod(K; M) ! ! Ext1AR-mod(R; M) ! 0 and isomorphisms: (26) ExtiAR-mod(K; M) ~=Exti+1AR-mod(R; M) ; i 1 Lemma 10.2. M ExtnAR-mod(K; M) ~= ExtnR[Gi](Ki; Mi) i=1;2 26 Proof.Let #i : AR-mod ! R[Gi]-mod , i = 1; 2, be the forgetful functors defined by #i(M) = Mi. There are functors going_the other direction si : R[Gi]-mod ! AR-mod , i = 1; 2, defined by si(N)(|_j|) = N if i = j and 0 if i 6= j. The fun* *ctors si are clearly exact and left adjoint to the forgetful functors. Thus Hom AR-mod(siN; M) ~=Hom R[Gi](N; Mi): It follows that the functors #i preserve injectives; that is, if I 2 AR-mod is* * injective then Ii is an injective R[Gi]-module for i = 1; 2. So, given_an object of AR-mod , M and an injective resolution M ! I* we have that I(|_i|)* is an injective resolution for Mi and this together with the defi* *nition of the functor K, given above, implies: (27) Ext*AR-mod(K; M) := H (Hom AR-mod(K; I*))~= i M j M ~=H Hom R[Gi]; (Ki; I(__|_i|)*) ~= Ext *R[G(K ; M )* * : |___| i]i i i=1;2 i=1;2 For a small category C and a functor F 2 CR-mod the n-th cohomology group of* * C with coefficients in F is defined to be the group ExtnCR-mod(R; F ), where R 2 * *CR-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: M (28) 0 ! H0(A; M) ! M0 ! MHi=MGii! H1(A; M) ! i=1;2 M M ! H1(Gi; Mi) ! H1(H; Mi) ! H2(A; M) ! i=1;2 i=1;2 M M ! H2(Gi; Mi) ! H2(H; Mi) ! : : : i=1;2 i=1;2 and for the cohomology of B: (29) 0 ! H0(B; M) ! M0 ____-MH1=MG11! H1(B; M) ! ! H1(G1; M1) ! H1(H; M1) ! H2(B; M) ! 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Kane, eds.), Lecture Notes in Math., vol. 1298, Springer-Verlag, 1987* *, 227-236 Departament de Matematiques, Univeritat Autonoma de Barcelona, E-08193 Bel- laterra, 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