ON THE MOD p COHOMOLOGY OF BP U(p) ALE~S VAVPETI~C AND ANTONIO VIRUEL Abstract. We study the mod p cohomology of the classifying space of the p* *rojec- tive unitary group PU(p). We first proof that old conjectures due to J.F.* * Adams, and Kono and Yagita [16] about the structure of the mod p cohomology of c* *lassify- ing space of connected compact Lie groups held in the case of PU(p). Fina* *lly, we proof that the classifying space of the projective unitary group PU(p) is* * determined by its mod p cohomology as an unstable algebra over the Steenrod algebra * *for p > 3, completing previous works [10] and [6] for the cases p = 2, 3. 1.Introduction Compact Lie groups provide an example of one the classical mathematical maxim* *s: "the richer is the mathematical structure of an object, the more rigid the obje* *ct is". So for example all the rich mathematical structure associated to a connected co* *mpact Lie group is so intimately linked that it is completely recover (perhaps, up to* * local isomorphism) from some small data like the Dynkin diagram or the maximal torus normalizer [8]. In homotopy theory, this rigidity in the structure of a compact Lie group G is expected to be inherited by the classifying space BG and related structures. So* * for example, in the appropriate homotopical setting of p-compact groups [12], maxim* *al torus normalizers do characterize the isomorphic type of BG, at least at odd pr* *imes [3]. The aim of this work is to study the rigidity of the mod p cohomology of BG, namely H*(BG; Fp), proving several conjectures in the particular case of G bein* *g the projective unitary group P U(p), which is obtained as the quotient of the unita* *ry group of rank p, U(p), by the subgroup {Diag (ff, . .,.ff) | ff 2 S1} of diagonal mat* *rices. In [12, Theorem 1.1], it is shown that H*(BG; Fp) is a Noetherian algebra for* * any compact connected Lie group G, so by [29, Theorem 1.4] (or directly [28, Theorem 6.2]) we know that the kernel of the natural map (1) H*(BG; Fp) ____- -lim---H*(BE; Fp), Ap(G) where Ap(G) stands for the Quillen category of elementary abelian p-subgroups of G [28, 29, 17, 11], contains only nilpotent elements. For p > 2, a more strong* *er conjecture shows up ___________ Date: December 2, 2003. The first author is partially supported by the Ministry for Education, Scienc* *e and Sport of Republic of Slovenia Research Program no. 101-509. The second author is partial* *ly supported by the DGES-FEDER grant BFM2001-1825, and Junta de Andaluc'ia Grant FQM-0213. 1 2 ALE~S VAVPETI~C AND ANTONIO VIRUEL Conjecture 1.1 (J.F. Adams). Let G be a compact connected Lie group, and p be an odd prime. Then the mod p cohomology of BG is detected by elementary abelian p-subgroups [1, Definition 4.2], i.e. the natural map (1) is a monomorphism. Conjecture 1.1 trivially holds in the torsion free cases (see [3, Theorem 12.* *1]). In the case of torsion, only a few examples have been worked out, all of them at p* * = 3: F4 [5, Teorema 5], E6 [23] and P U(3) [16, Theorem 3.3]. Our first result gener* *alizes the last reference proving, Theorem A. The group P U(p) verifies Conjecture 1.1 at the odd prime p, i.e. H*(BP U(p); Fp) is detected by elementary abelian p-subgroups, i.e. the natural* * map (1)is a monomorphism for the case G = P U(p). Proof.See Theorem 2.4 The knowledge of the structure of H*(BG; Fp) plays an important role when try* *ing to understand other generalized cohomologies of BG as it is shown in [16]. So * *for example, understanding Milnor primitive operations (see Section 3) is a crucial* * step in the use of the Atiyah-Hirzebruch spectral sequence [20, pag. 496]. A new conjec* *ture arises [16, Conjecture 5] Conjecture 1.2 (Kono-Yagita). If G is a connected compact Lie group, then for e* *ach odd dimensional element x 2 H*(BG; Fp), there is i such that such that Qm x 6= * *0 for all m i, where Qm are the Milnor primitive operators. Then our result generalizes the study of P U(3) carried out in [16] proving, Theorem B. The group P U(p) verifies Conjecture 1.2 for every odd prime p, i.e.* * for each odd dimensional element x 2 H*(BP U(p); Fp), there is i such that Qm x 6= * *0 for all m i, where Qm are the Milnor primitive operators. Proof.See Theorem 3.3. Remark 1.3. It is worth to remark that while the proof of Conjectures 1.1 and 1* *.2 in previous known cases is heavily based in a precise understanding of the cohomol* *ogy rings involved, i.e. generators and relations, the proofs of Theorems A and B i* *s done by geometrical methods and without using any information about the algebra structu* *re of H*BP U(p). So many restrictions on H*(BG; Fp) suggest that these algebras do not show up in nature very frequently. In other words, any space X whose mod p cohomology is isomorphic to that of BG, for a connected compact Lie group G, should be topolo* *gi- cally related with BG in some way. This idea is captured in the next conjecture* * [26, Conjecture 4.4] Conjecture 1.4. Let G be a compact connected Lie group, and let X be a p-comple* *te space such that H*(X; Fp) ~=H*(BG; Fp) as algebras over the mod p Steenrod alge* *bra Ap. Then X ' BG^p. The first result of this kind appeared in [9], where Dwyer, Miller and Wilker* *son proved Conjecture 1.4 for G = SU(2) = S3 at p = 2. In [10], the same authors considered the case when p does not divide the order of the Weyl group of G. No* *tbohm in [24] considered the case when p divides the order of the Weyl group of G, bu* *t BG ON THE MOD p COHOMOLOGY OF BPU(p) 3 has no torsion. For the case when torsion exists, there are only few known res* *ults [6, 30, 31, 32]. We prove, Theorem C. Let X be a p-complete space such that H*(X; Fp) ~=H*(BP U(p); Fp) as an unstable algebra over the Steenrod algebra Ap. Then X is homotopy equival* *ent to BP U(p)^p. Proof.If p = 2, then P U(2) = SO(3), and the theorem is known [9]. If p = 3 the theorem is proved in [6]. The case p 5 is consider in Section 4. Notation: Here all spaces are assumed to have the homotopy type of CW-complexes. Completion means Bousfield-Kan completion [4]. For a given space X, we write H*X for the mod p cohomology H*(X; Fp) and X^pfor Bousfield-Kan (Zp)1 -completion or p-completion of the space X. Given a group G and a ZG-module M, we write H*(G; M) for the cohomology of G with (twisted) coefficients in M. We assume th* *at the reader is familiar with the Lannes' theory [18]. 2. The Adams' conjecture The aim of this section is to prove Adams' conjecture (Conjecture 1.1) for the group P U(p) at the prime p > 2. We start identifying some important subgroups * *of a compact connected Lie group G. Let T (G) G be a maximal torus and N(G) G its normalizer. Define Np(G) N(G), the p-normalizer of the maximal torus T (G* *), as the preimage of a p-Sylow subgroup in the Weyl group of G, WG = N(G)=T (G). Lemma 2.1. The groups Np(P U(p)) and Np(SU(p)) are isomorphic. Proof.Notice first that Np(P U(p)) = Np(SU(p))={Diag (ff, . .,.ff) | ff 2 S1}. * * Now, every element in Np(SU(p)) can be written in a unique way as Diag(z1, . .,.zp)P* * i, where P is the permutation matrix corresponding to the cyclic permutation (1, 2* *, . .,.p). Then ': Np(P U(p)) ____-Np(SU(p)) given by z1 zp-1 zp i '([Diag (z1, . .,.zp)P i]) = Diag(__, . .,.____, __)P z2 zp z1 provides the desired isomorphism. The isomorphism constructed above is very convenient as we can prove Conjectu* *re 1.1 for Np(SU(p)), Lemma 2.2. The cohomology H*BNp(SU(p)) is detected by elementary abelian sub- groups. Proof.Notice tha Np(U(p)) ~= S1 o Z=p, hence by [1, Theorem 4.3] we know that H*BNp(U(p)) is detected by elementary abelian subgroups. Moreover H*BNp(U(p)) is detected by just two subgroups, Vt = Z=p p T (U(p)) the maximal elementary abelian toral subgroup and Vn = Z=p Z=p = Z(U(p)) x Z=p \ SU(p) [1, Lemma 4.4], where Z(U(p)) is the center of the group U(p) Now the fibration S1 ____-BSU(p) ____-BU(p) 4 ALE~S VAVPETI~C AND ANTONIO VIRUEL gives rise to a fibration Bj S1 ____-BNp(SU(p)) ____-BNp(U(p)), whose the Gysin sequence is Bj* * d *-1 . . .___-H*BNp(U(p)) ____-H BNp(SU(p)) ____-H BNp(U(p)) ____- . . . Let x 2 H*BNp(SU(p)) and suppose d(x) 6= 0. Let V ____-BNp(U(p)) be an elemen- tary abelian group detecting d(x). Then V 0= \ BNp(SU(p)) is an el* *e- mentary abelian group, which appears in the fibration S1 ____-BV 0____-B, and detects the element x. If d(x) = 0, then x = Bj*(y) for some y 2 H*BNp(U(p)) and y is detected by Vt* *or Vn defined above, so the element x is detected by Vn or Vt\Np(SU(p)) ~=(Z=p)p-1. An easy consequence of the previous lemmas is Lemma 2.3. The mod p cohomology of BN(P U(p)) is detected by elementary abelian p-subgroups. Proof.Combining Lemmas 2.1 and 2.2 we obtain that H*BNp(P U(p)) is detected by elementary abelian p-subgroups. Then, because the index [N(P U(p)): Np(P U(p))]* * = (p - 1)! is nonzero in Fp, the transfer argument [33, Lemma 6.7.17] shows that H*BN(P U(p)) ____-H*BNp(P U(p)) is a monomorphism. Therefore H*BN(P U(p)) is also detected by elementary abelian p-subgroups. Finally, Theorem 2.4. The mod p cohomology of BP U(p) is detected by elementary abelian p-subgroups. Proof.According to [21, Theorem 1.2 & Lemma 3.1], H* P U(p)=N(P U(p)) is finite and its Euler characteristic, Ø P U(p)=N(P U(p)) , equals 1. Therefore, the tr* *ans- fer argument [12, Theorem 9.13] shows that H*BP U(p) _____-H*BN(P U(p)) is a monomorphism. As H*BN(P U(p)) is detected by elementary abelian subgroups by previous lemma, H*BP U(p) is so. 3. The Kono-Yagita conjecture In this very short section we provide a proof of Theorem B (see Theorem 3.3) * *by means of Theorem A. Recall that for an odd prime p, the Milnor primitive operat* *ors l pl j are inductively defined as Q0 = fi and Qn+1 = Pp Qn - QnP where fi and P are the Bockstein and the j-th Steenrod power respectively. As quoted above, we use Theorem A to prove Theorem B, hence we need some info* *r- mation about elementary abelian subgroups in P U(p). This information is collec* *ted in the following proposition ([7, Corollary 3.4] or [3, Theorem 9.1]) Proposition 3.1. The group P U(p) contains two conjugacy classes of maximal ele- mentary abelian subgroups corresponding to the conjugacy classes of the maximal* * toral elementary abelian and a rank two nontoral. ON THE MOD p COHOMOLOGY OF BPU(p) 5 In fact, those two subgroups of P U(p) already showed up along the proof of L* *emma 2.2 after the identification in Lemma 2.1. The first lemma in this section shows that Conjecture 1.2 holds for rank two * *ele- mentary abelian groups, Lemma 3.2. Let x be an odd dimensional element of H*B(Z=p)2 = E(x1, x2) Fp[y1, y2], then there exists an i such that Qm x is not trivial for all m > i. n Proof.First notice that Qnxi= ypi and Qnyi= 0. Now, if x is odd dimensional, th* *en x = x1f + x2g, where f, g 2 Fp[y1, y2]. If Qnx is nontrivial for all n, lemma h* *olds. So, r pr let r be an integer such that Qrx = 0. Then Qrxr= yp1f + y2rg = 0 and therefore there exists h 2 Fp[y1, y2] such that f = yp2h and g = -yp1h. For m > r we have that m pm pm pr pm pr pm-pr pm-pr pr pr Qm x = yp1f + y2 g = y1 y2 h - y2 y1 h = (y1 - y2 )y1 y2 h is nontrivial. Finally Theorem 3.3. For each odd dimensional element x 2 H*BP U(p), there is i such that such that Qm x 6= 0 for all m i. Proof.Let x be in H*BP U(p) an odd dimensional element. By Theorem A, Bj*(x) is nontrivial for some j :E ____-P U(p), where E is an elementary abelian p-gro* *up. If E is toral, then j factors trough maximal torus iT :T ____-P U(p). As H*BT is concentrated in even degrees, Bj* is trivial on elements of odd degree. Theref* *ore Bj*(x) is a non trivial odd dimensional element in H*BV for j :V ____-P U(p) the non toral elementary abelian subgroup which is of rank two by Proposition 3.1. * *By the previous lemma, there exists i such that for all m > i, Qm Bj*(x) = Bj*(Qm * *x) is nontrivial. Thus for all m > i, Qm x is nontrivial. 4. Cohomological uniqueness In this section we proceed to prove Theorem C in the case p > 3. So in what f* *ollows X is a p-complete space, such that there exists an isomorphism OE: H*BP U(p) ~=* *H*X as an unstable algebra over the Steenrod algebra Ap, for p > 3. The idea is to construct a homotopy equivalence BP U(p)^p_____-X by means of the cohomology decomposition of BP U(p) given by p-stubborn subgroups [14]. Recall that given a compact Lie group G, a subgroup P G is called p-stubborn [14, pag. 186] if the following conditions hold: - The connected component of P is a torus and ß0P is a p-group. - The quotient group NG (P )=P is finite and possesses no nontrivial norm* *al p-subgroups Then if Rp(G) denotes the full subcategory of the orbit category of G whose obj* *ects are the homogeneous spaces G=P where P G is p-stubborn, the natural map hocolim EG=P ____-BG G=P2Rp(G) 6 ALE~S VAVPETI~C AND ANTONIO VIRUEL induces an isomorphism of homology with Z(p)-coefficients [14, Theorem 4]. The p-stubborn subgroups of P U(p) are described in the next proposition. Proposition 4.1. The group P U(p) contains exactly three p-stubborn subgroups up to conjugation: (1) the maximal torus T , def (2) the p-normalizer Np:=Np(P U(p)) of the maximal torus, and (3) the group E2 = (Z=p)2 generated by the diagonal matrix Diag(1, i, . .,.i* *p-1), where i is a pthroot of the unit, and a permutation matrix which corresp* *onds to the cyclic permutation (1, 2, . .,.p). Proof.By [14, Proposition 1.6], P SU(p) is a p-stubborn subgroup if and only * *if P=(P \ Z) is a p-stubborn subgroup of P U(p), where Z ~=Z=p is the center of SU* *(p). Finally, [27, Theorems 6, 8 & 10] describe all the conjugacy classes of p-stubb* *orn groups in SU(p), what leads to the desired result. Let eRp(P U(p)) be the full subcategory of Rp(P U(p)) with only the three obj* *ects: P U(p)=T , P U(p)=Np, and P U(p)=E2. Then the strategy is to construct a homoto* *py commutative diagram (Lemma 4.3) fP {EG=P ' BP }PU(p)=P2Rep(PU(p))__-X such that it can be lifted to the topological category (after Proposition 4.5) * *so then we can recover BP U(p) (up to p-completion) as a hocolim. As every p-stubborn P P U(p) such that P U(p)=P 2 Rep(P U(p)) appears as a def subgroup of N :=N(P U(p)), we first construct a map BN ____-X. Theorem 4.2. There exists a map fN :BN ____-X such that the diagram H*BN * ` I@ * (2) BiN @ fN OE @ H*BP U(p) _________~-H*X = commutes. Proof.Let iE :E = (Z=p)p-1 ____-T ____-P U(p) be the maximal elementary abelian p-subgroup of P U(p). By Lannes' theory [18, Th'eor`eme 3.1.1.], there exists * *a map fE :BE _____-X such that f*E= Bi*EOE-1: H*X _____-H*BE. By [18, Proposition 3.4.6.], TBEi*EH*BP U(p)^p~=H* Map (BE, BP U(p)^p)BiE^p. Since Map (BE, BP U(p)^p)BiE^p' BCPU(p)(E)^p' BTp^, where CPU(p)(E) denotes the centralizer ([13],[25]), it follows that TfE*EH*X ~=TBEi*H*BP U(p) ~=H*BTp^. ON THE MOD p COHOMOLOGY OF BPU(p) 7 Because TfE*EH*X is zero in dimension 1, we can use [18, Th'eor`eme 3.2.1.] and* * obtain TfE*EH*X ~=H* Map (BE, X)fE. Therefore the mapping space Map (BE, X)fE has the same cohomology ring as BTp^. The mapping space Map (BE, X)fE is p-complete [18, Proposition 3.4.4], hence BT* *p^' Map (BE, X)fE. Now, the standard action of WPU(p)= p on T restricts to an action on E, which induces an action of p on Map (BE, X). If oe 2 p, then BiE ' BiE oe, and ther* *efore f*E= Bi*EOE-1 = oe*Bi*EOE-1 = oe*f*E, and by Lannes' theory [18, Th'eor`eme 3.1.1], fE ' fE oe. This means that p ac* *ts on Map (BE, X)fE. Consider now the space Y = Map (BE, X)fE x p E p which fits in the fibration Map (BE, X)fE ____-Y ____-B p. Fibrations with fiber Map (BE, X)fE and base B p are classified by Hn(B p; ßn(Map (BE, X)fE) = H2(B p; ß2(Map (BE, X)fE)). According to [2, Theorem 3.6], this group is trivial (recall that p 5) which * *shows that Y ' BNOp, the fiberwise p-completion of BN. Let fN : Map (BE, X)fE x p E p _____-X denote the evaluation map. We have to prove that the diagram (2) commutes, that is, that f*NOE = Bi*N. Let us def* *ine a = f*NOE. Since H*BN is detected by elementary abelian subgroups (Lemma 2.3), it is eno* *ugh to prove that Bj*Va = Bj*VBi*Nfor every elementary abelian p-group jV :V ____-N. In fact, the proof of Lemma 2.3 and Lemma 2.2 shows that it is enough to consid* *er E, the maximal toral elementary abelian subgroup, and Vn, the nontoral elementa* *ry abelian subgroup of rank two that coincides with E2 in Proposition 4.1. By construction of the map fN , the composition a * Bi* * H*BP U(p) ____-H BN ____-H BT is the same as Bi*T. Therefore Bj*Ea = Bj*EBi*Nfor V = E. Let now consider the case V = E2. As BjE factors through BT , BjE can detect only even dimensional elements in H*BN. Therefore, BjV detects HoddBN. In particular BjV detects H3BN H3BP U(p) fiH2BP U(p) which is nontrivial (notice that H2(BP U(p), Z) ~=ß1P U(p) = Z=p so the Universal Coefficient Theor* *em for cohomology [19, Theorem 4.3 in pag. 163] and the description of the Bockste* *in morphism [20, pag. 455] imply the statement). As f*TH2X 6= 0 by construction, t* *hen f*NH3X fiH2X 6= 0 and fN BjV detects H3X as well. Finally, by Lemma 3.1 the group P U(p) has only one nontoral elementary p-subgroup, hence by Lannes' theo* *ry _ * there exists just one morphism of unstable algebras H*BP U(p) _____-H BV^ such that ^Vis a nontrivial elementary abelian, H*BV^ is a finite module over H*BP U* *(p) (via _) and _HoddBP U(p) 6= 0, thus ^V = V = E2 and _ = Bj*VBi*N, as well as _ = Bj*VOEf*N= Bj*Va, thus Bj*VBi*N= j0V*Bi*N= Bj*Va also in this case. 8 ALE~S VAVPETI~C AND ANTONIO VIRUEL fN Define maps fP :EP U(p)=P ' BP ____-BN ____-X for P = T , Np, and E2. This gives rise to a diagram fP (3) {EG=P ' BP }PU(p)=P2Rep(PU(p))__-X Next lemma shows diagram (3)commutes up to homotopy. Lemma 4.3. For every two objects P and Q in Rep(P U(p)) and morphism cg 2 Mor (P, Q) the diagram fP BP ____-Xw | ww Bcg || ww ?| fQ BQ ____-X commutes. Proof.Because every morphism in eRp(P U(p)) is a composition of an automorphism and an inclusion, it is enough to prove that the diagram fP BP ____-Xw | ww Bcg || ww ?| fP BP ____-X commutes for every object P U(p)=P in eRp(P U(p)). If P = T , then the element * *g is in the normalizer N, hence the diagram fN BP -___-BNw ____-Xw | ww ww Bcg || ww ww ?| fN BP -___-BN ____-X commutes. Let P = Np. Since cg(Np) = Np, and T is the connected component of Np, also cg(T ) = T , hence g 2 N. Again we get a commutative diagram as in the previous case. Let P = E2. Then Bi*E2= Bc*gBi*E2, since BiE2 ' BiE2Bcg, and therefore f*E2= Bi*E2OE-1 = Bc*gBi*E2OE-1 = Bc*gf*E2. By Lannes' theory [18, Th'eor`eme 3.1.1.], fE2 ' fE2Bcg, which finishes the pro* *of. The diagram (3) commutes only up to homotopy, hence we do not know if the collection of maps {fP}PU(p)=P2Rep(PU(p))induces a map hocolim EP U(p)=P ____-X. PU(p)=P2Rep(PU(p)) ON THE MOD p COHOMOLOGY OF BPU(p) 9 The obstructions lie in the groups ---limi-----ßj(Map (BP, X)fP), Rep(PU(p)) where limiis the i-th derived functor of the inverse limit functor ([4] and [34* *]). Now we will prove that all obstruction groups are trivial. Let Xj, PU(p)j:eRp(P U(p)) ____-Ab be functors defined by Xj(P U(p)=P ) = ßj(Map (BP, X)fP), PU(p)j(P U(p)=P ) = ßj(Map (BP, BP U(p)^p)(BiP)^p), where Ab is the category of abelian groups. Note that Map (BP, BP U(p)^p)(BiP)^* *p~= BZ(P )^2[14, Theorem 3.2] therefore PU(p)1(P U(p)=P ) is well defined and, by * *the next lemma, also X1(P U(p)=P ) is well defined. Lemma 4.4. There exists a natural transformation T : PU(p)j___- Xjwhich is an equivalence. Proof.Let P be the maximal torus T of the p-normalizer Np, and let E ~=(Z=p)p-1 be the maximal toral elementary abelian subgroup in N. We apply Lannes' T funct* *or to the diagram H*BN * ` I@ * (4) BiN @ fN @ H*BP U(p) H*X and get TBEi*EH*BN ` I@ @@ TBEi*EH*BP U(p) TfE*EH*X By [18, Th'eor`eme 3.4.5], it follows that TBEi*EH*BN ~=H*BCT(E) = H*BT, TBEi*EH*BP U(p) ~=H*BCPU(p)(E) = H*BT, and the left map in the above diagram is an isomorphism. Because TfE*EH*X ~= TBEi*EH*BP U(p) ~=H*BT , it is zero in the degree 1, hence by [18, Th'eor`eme 3* *.2.1.], 10 ALE~S VAVPETI~C AND ANTONIO VIRUEL TfE*EH*X ~=H* Map (BE, X)fE and the right map in the diagram is an isomorphism. We conclude that in the diagram Map (BE, BNOp)(BiE)^p @ @@R Map (BE, BP U(p)^p)(BiE)^p Map (BE, X)fE both maps are equivariant mod p equivalences. Taking homotopy fixed points we obtain the following diagram Map (BE, BNOp)h(P=E)(BiE)^p @ @@R Map (BE, BP U(p)^p)h(P=E)(BiE)^p Map (BE, X)h(P=E)fE. where both maps are mod p equivalences, since an equivariant mod p equivalence between 1-connected spaces induces a mod p equivalence between the homotopy fix* *ed- point sets. Using Map (BP, .) ' Map (BE, .)h(P=E), we obtain mod p equivalences Map (BP, BNOp)(BiP)^p @ @ @R Map (BP, BP U(p)^p)(BiP)^p Map (BP, X)fP. Let us consider the remaining case P = E2 ~=(Z=2)2. Applying Lannes' functor * *to diagram (4) gives TBPi*PH*BN ` I@ @@ TBPi*PH*BP U(p) TfP*PH*X By [18, Th'eor`eme 3.4.5], we get TBPi*PH*BN ~=H*BCN (P ) = H*BP, TBPi*PH*BP U(p) ~=H*BCPU(p)(P ) = H*BP, and the left map is an isomorphism. Since TfP*PH*X is free in dimension 2, it* * follows by [18, Th'eor`eme 3.2.4] that TfP*PH*X ~=H* Map (BP, X)fP and also the right m* *ap is an isomorphism. So, in the diagram Map (BP, BNOp)BjP @ @@R Map (BP, BP U(p)^p)BiP Map (BP, X)fP both maps are mod p equivalences. ON THE MOD p COHOMOLOGY OF BPU(p) 11 We have shown that in all cases (P = Np, T , or E2) the map Map (BP, BP U(p)^p)(BiP)^p___- Map (BP, X)fP is a mod p equivalence. Because Map (BP, BP U(p)^p)(BiP)^pand Map (BP, X)fP are p-complete spaces [18, Proposition 3.4.4.], above map is a homotopy equivalence* *. To see that this homotopy equivalence is natural, we have to prove that it commute* *s with maps induced by conjugation, which means that we have to show that the diagram Map (BP, BNOp)(BiP)^pBcg @ @@R Map (BP, BP U(p)^p)(BiP)^poe_Map (BP, BNOp)(BiP)^p__-Map (BP, X)fP commutes. This follows from two commutative diagrams, BCPU(p)(P|)^p=======================================BCN (P )^p || @@' '| || @R ?|| | Map (BP, BP U(p)^p)(BiP)^poe____Map (BP, BNOp)(BiP)^p || ww Bcg | ww | | || Map (BP, BP U(p)^p)(BiP)^pBcgoe_Map (BP, BNOp)(BiP)^pBcg | | ' ` '6| ?|| || BCPU(p)(cg(P ))^p===================================BCN (cg(P ))^p and BCN (P )^p=============================BCPU(p)(P|)^p |' ' || ?|| || Map (BP, BNOp)(BiP)^p_____- Map (BP,wX)fP || ww | ww Bcg || | Map (BP, BNOp)(BiP)^pBcg____-Map(BP, X)fPBcg || | 6'| I@@' | || @ ?|| BCN (cg(P ))^p=========================BCPU(p)(cg(P ))^p which can be glued together. Proposition 4.5. For all i, j 1, ---limi-----ßj(Map (BP, X)fP) = 0. Rep(PU(p)) 12 ALE~S VAVPETI~C AND ANTONIO VIRUEL Proof.By the previous lemma, --limi------ßj(Map (BP, X)fP) = --limi------ßj(Map (BP, BP U(p)^p)(BiP)^* *p) eRp(PU(p)) eRp(PU(p)) and the right side is 0 [14, Theorem 4.8]. Because all obstructions vanish, there exists a map f :BP U(p)^p_____-X. 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Wojtkowiak, On maps from holim F to Z, in Algebraic Topology, Barcelona * *1986, SLNM 1298, 227-236. 14 ALE~S VAVPETI~C AND ANTONIO VIRUEL Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska 19, SI-1111 Ljubljana, Slovenia E-mail address: ales.vavpetic@FMF.Uni-Lj.Si Dpto de 'Algebra, Geometr'ia y Topolog'ia, Universidad de M'alaga, Apdo correos 59, E29080 M'alaga, Spain E-mail address: viruel@agt.cie.uma.es