PAIRINGS OF p-COMPACT GROUPS AND H-STRUCTURES ON THE CLASSIFYING SPACES OF FINITE LOOP SPACES by Kenshi Ishiguro In [8], the author investigated certain pairing problems for classifying spaces of compact Lie groups. The main work in this paper can be regarded as a p-compact group version. Dwyer-Wilkerson [3] defined a p-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group at the prime p. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note of Moller [12] summarizes their work. Further development on the homotopy theory of p-compact groups can be seen, for example, in [4], [13] , [14] , [2] and [17]. We first recall some basic things about the p-compact groups and pairing problems, and then state our main results. A p-compact group, [3], is a loop space X such that X is Fp-finite and that its classifying space BX is Fp-complete. The p-completion of a compact Lie group G is a p-compact group if ss0(G) is a p-group. For an odd dimensional sphere S2n-1 , it is known that its p-completion has a loop structure if n divides p - 1. This is an example of p-compact groups other than compact Lie groups. More examples are known as Clark-Ewing p-compact groups, [12, x2]. For p-compact groups X and Y , a pointed map f : BX ____-BY is called a homomorphism. Let Y =X denote the homotopy fibre of f . The homomorphism f is called a monomorphism if Y =X is Fp-finite, and an epimorphism if the loop space (Y =X) is a p-compact group. The centralizer of f is the loop space of the component containing f of the mapping space of unpointed maps, denoted by map(BX; BY )f . A homo- morphism is called central if the evaluation map, ev : map(BX; BY )f ____- BY , is a homotopy equivalence. According to [4], any p-compact group X has a unique maximal central subgroups that is called the center of X and denoted by C(X). It is also shown in [4] that BC(X) ' map(BX; BX)id where id : BX ____-BX is the identity homomorphism. Typeset by AM S-TEX 1 2 Next we recall pairing problems for p-compact groups and compact Lie groups , [8] and [16]. Suppose that X, Y and Z are p-compact groups, and that ff : BX ____-BZ and f : BY ____-BZ are homomorphisms. The homotopy class of ff is said to be contained in the set of the homotopy classes of axes f ?(BX; BZ) if there is a map (called a pairing) : BX x BY ____- BZ with restrictions (axes) |BX ' ff and |BY ' f . In other words, if ff 2 f ?(BX; BZ), we have the following homotopy commutative diagram: BY | H H f |? HHj BX x BY _______-BZ 6| ff* | BX We note that f ?(BX; BZ) is a subset of the homotopy set [BX; BZ]. For a weak epimorphism f of the classifying spaces of connected compact Lie groups, the set of the homotopy classes of axes has been determined in [8]. In this paper we will obtain analogous results for p-compact groups. In [9], for connected compact Lie groups L and G, a map BL ____-BG or BL^p ____-BG^p is called a weak epimorphism, if there exists a fibration F ____-BL ____-BG or F ____-BL^p ____-BG^p such that H* (F ; Q) is a finite dimensional Q-module or that H* (F ; Z^p)Q is a finite dimensional Q^p-module, respectively. The second condition of the following theorem requires a similar assumption for a homomorphism of connected p-compact groups f : BY ____-BZ. By the way, the connectivity is not assumed in the first condition. Theorem 1. Suppose X is a p-compact group. If either (i)f : BY ____-BZ is an epimorphism of p-compact groups, or (ii)f : BY ____-BZ is a homomorphism of connected p-compact groups such that H* ((Z=Y ); Z^p) Q is a finite dimensional Q^p-vector space then the following hold: (1) If ff 2 f ?(BX; BZ), then the map ff factors through the classifying space of the center of Z, denoted by C(Z), up to homotopy. (2) Moreover, we have f ?(BX; BZ) = [BX; BC(Z)] . Here we make a remark analogous to the one in [8]. Taking Y = Z and f = id , our problem asks possible BX-actions on BY . A consequence 3 of Theorem 1 shows that such an action under ff exists if and only if the orbit map ff : BX ____-BY is central. We see, for instance, that there are no nontrivial BX-actions on B(S2n-1 )^p for n 3, since the center C((S2n-1 )^p) is contractible. A connected p-compact group Y is called semi-simple if ss1(Y ) is finite, [13]. In this case, the center C(Y ) is a finite abelian p-group, [14]. If X is connected and Y is semi-simple, the homotopy set [BX; BC(Y )] is trivial. Consequently, there are likewise no nontrivial BX-actions on BY . Furthermore, if we take X = Y = Z and f = ff = id, the problem now asks whether BX is an H-space. A pairing : BX x BX ____-BX could be the H-multiplication. Before stating our result, recall that a p-compact group X is called abelian if ev : map(BX; BX)id ____-BX is a equivalence. Any abelian p-compact group is equivalent to the product of a p-compact torus and a finite abelian p-group, [4] and [14]. Corollary 2 stated in x2 implies that BX is an H-space if and only if X is abelian. This result holds when a p-compact group X is replaced by a finite loop space. Theorem 2. Suppose X is a finite loop space. If its classifying space BX is an H-space, then X is equivalent to the product of a torus and a finite abelian group. The above result is a generalization of Corollary 2.4 in [8]: If G is a compact Lie group and BG is an H-space, then G is an abelian group. Theorem 3 in x2 will give the p-completed version of this result. Namely, if (BG)^p is an H-space, then G is p-nilpotent in the sense of [6]. The group G need not be abelian. We can find, however, an abelian compact Lie group A such that (BG)^p' BA. 1. Mapping spaces and Proof of Theorem 1. We will prove Theorem 1 in this section. To do so, we need a few basic results about p-compact groups. The following lemma translates a setting of groups to a homotopy setting of p-compact groups. Lemma 1. Suppose j : BX ____-BY and q : BY ____-BZ are homo- morphisms of p-compact groups. If the composite map q . j is a homotopy equivalence (isomorphism), then j is a monomorphism and q is an epimor- phism. 4 Sketch of Proof. We sketch the proof. From our assumption, one can show that Y ' (Z=Y ) x Z and (Z=Y ) ' Y =X. Thus Y =X is Fp-finite, and (Z=Y ) is a p-compact group. Therefore j is a monomorphism and q is an epimorphism. We recall [3, Theorem 9.7] that if a p-compact group X is connected, the cohomology algebra H* (BX; Z^p) Q is a polynomial ring over Q^pcon- centrated in even degree. The number of the generators of the polynomial algebra is called rank of X and denoted by rank(X). If n = rank(X), it is known that the maximal torus of X is equivalent to (BT n)^p. * * It is also known that H* (BX; Z^p) Q is isomorphic to the invariant ring (H* (BT n; Z^p) Q)W (X) , where W (X) is the Weyl group of X. Proposition 1. Suppose either (i)X, Y and Z are p-compact groups, i : BX ____-BZ is a monomor- phism and f : BY ____-BZ is an epimorphism, or (ii)X, Y and Z are connected p-compact groups, i : BX ____-BZ is a monomorphism and f : BY ____-BZ is a homomorphism such that H* ((Z=Y ); Z^p) Q is a finite dimensional Q^p-vector space. If there is a map (extension) fe: BY ____-BX with f ' i . ef, BX efj3 |i j j |? BY _______-fBZ then BX is equivalent to BZ under the map i. Proof. First assume the condition (i). It suffices to show that i : BX ____- BZ is an epimorphism. Recall that f : BY ____-BZ lifts to fe if and only if the homotopy fixed point (Z=X)hY is nonempty, [3, x 3.3]. Since f : BY ____-BZ is an epimorphism, by definition, the loop space (Z=Y ) is a p-compact group. Let U = (Z=Y ) so that BU ____-BY ____-BZ is a fibration of p-compact groups. Then (Z=X)hY is homotopy equivalent to ((Z=X)hU )hZ . Notice here that the action of U on Z=X is trivial. Since the Sullivan conjecture for p-compact groups holds, [4, Theorem 9.3], we see (Z=X)hU ' Z=X. Consequently (Z=X)hY ' (Z=X)hZ . This means that (Z=X)hZ is nonempty, and therefore the identity map 1BZ : BZ ____-BZ lifts to a map r : BZ ____-BX so that i . r ' 1BZ . 5 BX r j3 |i j j |? BZ _______-1BZBZ From Lemma 1 the monomorphism i is also an epimorphism. Hence i is an isomorphism. Next assume the condition (ii). Since H* ((Z=Y ); Z^p) Q is finite dimensional, we see that H* (Z=Y ; Z^p)Q is a finitely generated polynomial algebra, and hence we have H* (BY ; Z^p) Q ~= (H* (Z=Y ; Z^p) Q) (H* (BZ; Z^p) Q) Thus we can find a homomorphism (left inverse) of polynomial algebras r : H* (BY ; Z^p) Q ____-H* (BZ; Z^p) Q with r . f *= id. Consequently r . ef *. i* = id, since f ' i . ef. Hence i* is injective. We claim that i* is surjective and hence this homomorphism is bijective. It's enough to show that the composition ' = i* . r . ef *is bijective. H* (BX; Z^p) Q _______-'H* (BX; Z^p) Q fe*| 6|i* |? | H* (BY ; Z^p) Q _______-rH* (BZ; Z^p) Q Since i : BX ____-BZ is a monomorphism and i* is injective, we see rank(X) = rank(Z). Hence the Krull dimension of the image of ' is equal to rank(X). Thus, at each degree, ' is an injective linear self-map of a finite dimensional Q^p-vector space, and therefore this linear map is bijective. Consequently the monomorphism i is a rational isomorphism. According to [13, Lemma 2.5(1)], we see that BX is equivalent to BZ under the map i. Proof of Theorem 1. (1) : We will show that if ff 2 f ?(BX; BZ), the composite map BX _____-ffBZ ____-B(Z=C(Z)) , say qff, is null homotopic. BC(Z) | |? BX _________-ffBZ H H qffHHj ||? B(Z=C(Z)) 6 Using a result of Moller [13, Theorem 6.1], it's enough to prove that qff. ' 0 for any homomorphism : BZ=pn ____-BX and any n 1. Since ff 2 f ?(BX; BZ), according to [8, Proposition 1.1], we see ff . is contained in f ?(BZ=pn ; BZ). So f factors through map(BZ=pn ; BZ)ff., which is the classifying space of the centralizer of ff . . A result of Dwyer-Wilkerson [3], [12, Theorem 5.1] shows that map(BZ=pn ; BZ)ff. is a p-compact group and ev : map(BZ=pn ; BZ)ff. ____-BZ is a monomorphsim, since Z=pn is a p-compact toral group. If : BXxBY ____-BZ is a pairing with restrictions (axes) |BX ' ff and |BY ' f , then the map f : BY ____-BZ is expressed as the following composition: __ BY _______-map(BZ=pn ; BZ)ff. P P P f P PPq ||?ev BZ where __ is induced by the adjoint map. In fact, for any y 2 BY , we see ev O __(y) = __(y)(*) = ((*); y) ' f (y). Since ev is a monomorphsim, by the assumption of f , Proposition 1 implies: map(BZ=pn ; BZ)ff. ' BZ Thus ff . is central. Hence the map qff : BX ____-B(Z=C(Z)) is null homotopic. Consequently, the map ff : BX ____-BZ factors through BC(Z). (2) : Using [4, Theorem 9.3], one can show that the map of homotopy sets [BX; BC(Z)] ____-[BX; BZ] is injective, since its kernel [BX; Z=C(Z)] is trivial. The image of the map is included in f ?(BX; BZ). We have just seen in part (1) that [BX; BC(Z)] maps onto f ?(BX; BZ). Consequently, f ?(BX; BZ) = [BX; BC(Z)]. As seen in [8, Proposition 1.1], there is a strong relationship between pairing problems and mapping spaces. The following result shows that, for the homomorphism f : BY ____-BZ in Theorem 1, no p-compact groups find a difference between BC(Z) and map(BY; BZ)f . The proof uses the uniqueness of the pairing in our case. Corollary 1. Let f : BY ____-BZ be as in Theorem 1. For any p-compact group X, the map of homotopy sets [BX; BC(Z)] ____-[BX; map(BY; BZ)f ] 7 is bijective, where the above map is induced by the canonical map BC(Z) = map(BZ; BZ)id ____-map(BY; BZ)f : Proof. First notice that there is a map j : [BX; map(BY; BZ)f ] ____-f ?(BX; BZ) induced by adjoints. In fact, a map BX ____-map(BY; BZ)f induces a pairing BX x BY ____-BZ, and one of its axes is contained in f ?(BX; BZ). Thus we get the following commutaive diagram: [BX; BC(Z)] _______-[BX; map(BY; BZ)f ] P P P P PPq ||?j f ?(BX; BZ) By [4, Lemma 5.3], for ff 2 f ?(BX; BZ), there is a unique pairing : BX x BY ____-BZ with |BX ' ff and |BY ' f . Hence j is bijective. Theorem 1 shows [BX; BC(Z)] ____-f ?(BX; BZ) is bijective. Therefore the desired result holds. Remark. This result seems to indicate that map(BY; BZ)f can be homo- topy equivalent to BC(Z) for such an f . For instance, if map(BY; BZ)f were shown to be a p-compact group, the statement would be true. When f : BY ____-BZ is an epimorphism, a result of Dwyer-Wilkerson [4, Lemma 10.3] implies map(BY; BZ)f ' BC(Z). 2. H-structures on the classifying spaces. In this section we will prove Theorem 2 using the following result, which is an easy consequence of Theorem 1. Corollary 2. Suppose X is a p-compact group. If BX is an H-space, then X is abelian. Proof. Since BX is an H-space, we see (1BX )? (BX; BX) = [BX; BX]. Because, if m : BX x BX ____-BX is the H-multiplication, for any ff 2 [BX; BX], a pairing is given by the composite map m O (ff x 1BX ). Taking ff = 1BX in Theorem 1, we see that the identity map of BX factors through BC(X). Proposition 1 implies BX ' BC(X), and therefore X is abelian. 8 Remark. A double loop space is homotopy commutative, and McGibbon [11] shows that G^pis homotopy commutative if p > 2nr where G is a simply- connected compact Lie group and G '0 S2n1-1 x . .x.S2nr-1 with n1 . . . nr. The twice deloopability or the existence of an H-structure on the classifying space is, however, far different from the homotopy commutativity. Corollary 2 implies BG^pis an H-space if and only if G is a torus. We note here a theorem of Hubbuck [7]; Namely T n is the only nontrivial finite connected homotopy commutative H-space. Proof of Theorem 2. First consider a connected finite loop space X. At any prime p, the p-completion X^p is a p-compact group, and BX^p is an H- space. Corollary 2 says that there is a torus T n such that BX^p ' (BT n)^p, where n = rank(X). Hence BX ' BT n. Next consider the general case so that we begin with the fibration X0 ____- X ____-ss0X where X0 denotes the identity component of X. Since BX is an H-space, then ss0X = ss1BX is abelian. Consequently, we have a fibration BT n ____-BX ____-Bss0X. Notice [1] that this fibration is principal so that it is preserved by the p-completion. So the loop space BX^p is a p-compact group. Corollary 2 says that there is a finite abelian p-group flp suchQthat BX^p ' (BT n)^px Bflp. We notice Bflp = (Bss0X)^p so that ss0X = p flp, since ss0X is a finitely generated abelian group. Considering the fiber square, Q BX __________- p(BX)^p | | |? Q |? (BX)0 _______-( p(BX)^p)0 we see that the splitting of each BX^p induces a section for the fibration BT n ____-BX ____-Bss0X. Since this fibration is principal, the classifying space BX also splits. Consequently BX ' BT nx Bss0X. If a compact Lie group G is connected and the p-completion of the clas- sifying space (BG)^pis an H-space, then G must be abelian. When G is not connected, however, the analogous result does not hold. A counter-example is given by a p-nilpotent group. A finite group ss is called p-nilpotent, if the subgroup of ss generated by all elements of order prime to p does not contain any p-torsion element. It is known that ss is the semidirect product o ssp where ssp is the p- Sylow subgroup. Consequently, if ssp is abelian, the p-completed space 9 (Bss)^p ' Bssp is an H-space (actually, an infinite loop space). Henn [6] provides a generalized definiton of the p-nilpotence for compact Lie groups. Theorem 3. Suppose G is a compact Lie group and the p-completion of the classifying space (BG)^pis an H-space. Then G is the product of a torus T and a finite p-nilpotent group oe whose p-Sylow subgroup oep is abelian, and hence (BG)^p' (BT )^px Boep. Proof. Suppose P is a maximal p-toral subgroup of G, [10]. The H- structure on (BG)^p induces a group homomorphism P x P ____-P which makes BP an H-space, [5] and [15]. (BG)^px (BG)^p _______-(BG)^p 6| 6| | | BP x BP _ _________-BP According to [8, Corollary 2.4], we see that P is an abelian group. Let N P be the normalizer of P in G and let W = N P=P . Since the maximal p-toral subgroup P is abelian, the mod p cohomology H* ((BG)^p; Fp) is isomorphic to the ring of invariants H* (BP ; Fp)W = H* (BN P ; Fp) and therefore (BG)^p' (BN P )^p. Consequently (BN P )^p has an H-structure: : (BN P )^px (BN P )^p____-(BN P )^p and we obtain the following diagram __ (BN P )^p _______-map(BP; (BN P )^p)Bi P P P id P PPq ||?ev (BN P )^p Notice [5] and [15] that map(BP; (BN P )^p)Bi ' BP , since the classifying space of the centralizer of P in N P = P o W is p-equivalent to BP . Consequently (BN P )^p ' BP and hence (BG)^p ' BP . This implies that the compact Lie group G is p-nilpotent in the sense of [6]. By [6, Proposition 1.3 and Theorem 2.5], we can show the desired result. 10 References 1. A. BOUSFIELD and D. KAN, Homotopy limits, completions and localisations, LNM 304 (1972). 2. C. BROTO and A. VIRUEL, Homotopy Uniqueness of BP U(3), Preprint. 3. W.G. DWYER and C.W. 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McGIBBON, Homotopy commutativity in localized group, Amer. J. Math. 106 (3) (1984), 665-687. 12. J. MOLLER, Homotopy Lie groups, Bull. of AMS 32 (4) (1995), 413-428. 13. J. MOLLER, Rational isomorphisms of p-compact groups, Topology 35 (1) (1996* *), 201-225. 14. J. MOLLER and D. NOTBOHM, Centers and finite coverings of finite loop space* *s, J. reine angew. Math. 456 (1994), 99-133. 15. D. NOTBOHM, Maps between classifying spaces, Math. Z. 207 (1991), 153-168. 16. N. ODA, The homotopy set of the axes of pairings, Canad. J. Math. 42 (1990), 856-868. 17. A. VIRUEL, Homotopy Uniqueness of BG2, Preprint. Fukuoka University, Fukuoka 814-80, Japan e-mail: kenshi@ ssat.fukuoka-u.ac.jp