RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS JESPER MICHAEL MOLLER Abstract.A rational isomorphism is a p-compact group homomorphism in- ducing an isomorphism on rational cohomology. Finite covering homomor- phisms and nontrivial endomorphisms of simple p-compact groups are ratio* *nal isomorphisms. It is shown that rational isomorphisms of p-compact groups restrict to admissible rational isomorphisms of the maximal tori and the* * clas- sification of rational isomorphisms between connected p-compact groups is reduced to the simply connected case. The paper also contains a trivial* *ity criterion asserting that, on any connected p-compact group, only the tri* *vial homomorphism induces the trivial map in rational cohomology. 0.Introduction The notion of a p-compact group was introduced by Dwyer and Wilkerson [9] as a homotopy theoretic candidate for a replacement of compact Lie groups. Subsequent investigation in [18] and [8] strengthened the candidacy in finding that much of the internal structure of compact Lie groups does seem to be present also in p- compact groups. The process of gathering support for p-compact groups continues here where the outlining idea is to translate Baum's paper [3], describing local isomorphism systems of Lie groups, into the setting of p-compact groups. The rational isomorphisms of the title form the most prominent concept of this paper. However, for the sake of stressing the similarity with Lie groups, I sh* *all in this introduction restrict myself to the the particular rational isomorphisms called finite covering homomorphisms: A finite covering homomorphism between p-compact groups is an epimorphism whose kernel is a finite p-group [Definition* * 2]. It was shown in [18] that any connected p-compact group X admits a finite cover- ing homomorphism q :Y x S ! X where Y is a simply connected p-compact group and S is a p-compact torus; the homomorphism q can even be chosen to be what is here called a special finite covering homomorphism. Locally isomorphic p-compact groups [Definition 3] are characterized by having isomorphic finite covering gr* *oups of this kind [Proposition 1.5]. As an example of how these concepts behave as our experience with Lie groups tells us they should, the following theorem _ containing elements from Theorem * *3.3 and Corollary 3.5 _ is an almost mechanical translation of the basic lifting pr* *operty ____________ 1991 Mathematics Subject Classification. 55P35, 55S37. Key words and phrases. Rational isomorphism, covering homomorphism, p-compac* *t group, maximal torus, Weyl group, center, centralizer, local isomorphism system, simpl* *e p-compact group, trivial homomorphism. 1 2 JESPER MICHAEL MOLLER for covering homomorphisms exploited by Baum [3, Proposition 5] in computing local isomorphism systems. Theorem 0.1. Suppose that X1 and X2 are two locally isomorphic connected p- compact groups and q1: Y x S ! X1and q2: Y x S ! X2are special finite cover- ing homomorphisms. For any finite covering homomorphism f :X1 ! X2 there exist an automorphism g of Y and a finite self-covering homomorphism h of S such that the diagram of p-compact group homomorphisms Y x S_gxh_Y/x/S q1|| q2|| |fflffl |fflffl X1 ___f____X2// commutes up to conjugacy; g and h are uniquely determined up to conjugacy. Fur- thermore, f is an isomorphism if and only if both g and h are automorphisms. The formulation of Theorem 0.1 freely employs the basic definitions from the dictionary of [9] (and I shall continue this habit throughout this paper). The most notable tool used to derive Theorem 0.1 is an analog of the admissib* *le homomorphisms of Adams and Mahmud [1] or Adams and Wojtkowiak [2]. Theorem 0.2. Let X1 and X2 be two connected p-compact groups with maximal tori i1: T1 ! X1 and i2: T2 ! X2. For any homomorphism f :X1 ! X2 there exists a homomorphism ': T1 ! T2such that the diagram of p-compact group ho- momorphisms T1__'___T2// i1|| i2|| |fflffl |fflffl X1 __f__X2// commutes up to conjugacy; the conjugacy class of the homomorphism ' is unique up to the action of the Weyl group of X2. Assuming additionally that X1 and X2 are locally isomorphic, then, if f is a finite covering homomorphism, also ' is* * a finite covering homomorphism, and ' is an isomorphism if and only if f is an isomorphism. The above theorem is a reformulation of ingredients from Theorem 2.4 and The- orem 2.5. As the center [18, 8] of a connected p-compact group is a Weyl group invariant subgroup of the maximal torus this theorem implies [Corollary 3.1] th* *at the center is a functor on the category of connected p-compact groups with fini* *te covering homomorphisms as morphisms. In particular X1 and X2 could be identical so that we are discussing endomor- phisms of some p-compact group X. By Theorem 0.2 any endomorphism f :X ! X restricts to an endomorphism ': T ! T of the maximal torus. If X is simple in t* *he sense [Definition 5] that ss2(BT ) Q is an irreducible representation of the W* *eyl group, then the induced homomorphism ss2(B') must be multiplication by some p-adic integer (and Bf = an Adams operation of exponent ); in particular, RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 3 f is a rational isomorphism if 6= 0. This observation leads to the following m* *ain result of Section 4. Theorem 0.3. Any nontrivial endomorphism f :X ! X of a connected simple p- compact group X is a rational isomorphism, and, if p divides the order of the W* *eyl group, even an automorphism. The proof of Theorem 0.3 is based on a triviality criterion valid for any p-c* *ompact group homomorphism f :X ! Y . By Theorem 5.1, f is trivial if f O x is trivial for any homomorphism x: Z=pn ! X of a cyclic p-group into X. In case X is_ connected it follows [Corollary 5.7] that f is trivial if the induced map H *Qp* *(Bf) in reduced rational cohomology is trivial. 1.Rational isomorphisms and finite coverings This section contains the basic definitions and a few auxiliary results to be* * used later. Let X, X1 and X2 be p-compact groups. A homomorphism f :X1 ! X2 of p-compact groups is [8, 3.1] a based map Bf :BX1 ! BX2 ; the homotopy fibre of Bf over the base point is denoted X2=fX1 or just X2=X1 when no confusing may arise [8, 3.2]. Let Rep(X1; X2) denote the set of conjugacy classes [9, 3.1* *] of p-compact group homomorphisms of X1 to X2, i.e. Rep(X1; X2) = [BX1; BX2] is the set of free homotopy classes of maps of BX1 to BX2. Denoting the identity component of Xj by X0j, j = 1; 2, X0j! Xj ! ss0(Xj) may serve as our first example of a short exact sequence [9, 3.2] of p-compact * *groups. Note that any homomorphism f :X1 ! X2 will respect this short exact sequence in the sense that f restricts to a homomorphism f0 :X01! X02between the identity components. The central concept of this paper is that of a rational isomorphism. Definition 1.The homomorphism f :X1 ! X2 is a rational isomorphism if the homomorphism H*Qp(Bf0): H*Qp(BX01) H*Qp(BX02), induced by the restric- tion f0 of f to the identity components, is an isomorphism. As in [9, 1.5], the cohomology theory H*Qp(-) is defined as H*(-; Z^p) Q . Let "Q(X1; X2) Rep(X1; X2) denote the set of conjugacy classes of rational isomorphisms from X1 to X2 and "Q(X) := "Q(X; X)the monoid of rational iso- morphisms from X to itself. Assume from now on that X, X1, and X2 are connected p-compact groups. Lemma 1.1. Let f :X1 ! X2 be a homomorphism between connected p-compact groups. Then the following are equivalent: (1) ss*(f) Q is an isomorphism. (2) H*Qp(f) is an isomorphism. (3) H*Qp(Bf) is an isomorphism. (4) ss*(Bf) Q is an isomorphism. 4 JESPER MICHAEL MOLLER Proof.By the Bott-Samelson theorem [4], the Hopf algebra HQp*(Xj), where HQp*(-) := H*(-; Z^p) Q , is naturally isomorphic to the universal enveloping algebra of the Lie algebra ss*(Xj) Q with the Samelson product determining the Lie bracket, j = 1; 2. Thus (1) ) (2). That (2) ) (3) follows from the (collaps* *ing) bar construction spectral sequence [13, Corollary 7.18] Qp Q TorH* (Xj)(Q p; Qp) ) H*p (BXj): If (3) holds, the Eilenberg-Moore spectral sequence [13, Theorem 7.1] TorH*Q (BX )(Q p; H*Q(BX1)) ) H*Q(X2=X1) p 2 p p shows that the fibre X2=X1 of Bf is a torsion space and (4) follows. Finally (4) , (1) is an immediate consequence of the homotopy exact sequence. __|_| A special class of rational isomorphisms are provided by the finite covering * *ho- momorphisms. Definition 2.A homomorphism f :X1 ! X2 between connected p-compact groups is a finite covering homomorphism if X2=X1 ' Bss for some finite p-group ss. Let Cov(X1; X2) "Q(X1; X2)denote the set of finite covering homomorphisms of X1 to X2 and Cov(X) := Cov(X; X) the monoid of finite covering homomor- phisms of X to itself. Obstruction theory and [17, Theorem 6.3] prove a lifting criterion similar to* * [19, Proposition 1.2]. Lemma 1.2. Let f :X1 ! X2 be a finite covering homomorphism and g :X ! X2 a homomorphism from the connected p-compact group X into the base space X2. Then X1>> " " |f| " |fflffl X __g__X2// can be completed to a diagram commuting up to conjugacy if and only if g*ss1(X) f*ss1(X1). Moreover, if g factors through X1, then the space (X2=X1)h(gX)of all lifts of Bg is homotopy equivalent (by evaluation) to X2=X1; in particular, the factorization X ! X1 is unique up to conjugacy. Lemma 1.2 shows that if f O h is conjugate to g for some homomorphism h: X ! X1 , then there exists a fibration Bf_ X2=X1 -!map (BX; BX1)Bh --! map (BX; BX2)Bg of the corresponding mapping spaces. A finite covering homomorphism is an example of an epimorphism [9, 3.2] of p-compact groups. In fact, if cdFp(X1) = cdFp(X2) then f :X1 ! X2 is a finite covering homomorphism if and only if f is an epimorphism [9, Proposition 6.14]. Any finite covering homomorphism over a simply connected base space is an iso- morphism. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 5 The mod p cohomological dimension cdFp(X)occuring above is defined [9, Def- inition 6.13] to be the largest integer d such that Hd(X; Fp) 6= 0. Lemma 1.3. Let f :X1 ! X2 be a rational isomorphism. Then (1) f is an isomorphism if and only if f is a monomorphism. (2) f is a finite covering homomorphism if and only if f is an epimorphism. Proof.The first assertion is [18, Proposition 3.7]. If f is an epimorphic ratio* *nal iso- morphism, (X2=X1) is a p-compact group whose identity component has rational rank [9, 5.9] zero, hence [9, Lemma 5.10] a finite p-group. __|_| Thus a finite covering homomorphism is the same thing as an epimorphic ration* *al isomorphism. The endomorphism 3 of the 3-compact group SU(2)^3is an example of a rational isomorphism that is not an epimorphism, in particular not a finite cov* *ering homomorphism; see also Corollary 3.5 for more information about the relation between finite covering homomorphisms and rational isomorphisms. Recall [18, Lemma 3.3] that the universal cover X<1> of the connected p-compa* *ct group X is again a p-compact group. Proposition 1.4.The following conditions are equivalent: (1) X1 and X2 have identical mod p cohomological dimensions and isomorphic universal covering p-compact groups. (2) There exists a simply connected p-compact group Y , a p-compact torus S, and finite covering homomorphisms X1 Y x S ! X2. Proof.Assuming (1), both X1 and X2 are finitely covered [18, Theorem 5.4] by a p-compact group of the form Y x S where X1<1> ~= Y ~=X2<1> and S is p- compact torus of mod p cohomological dimension cdFp(Y )- cdFp(X1)= cdFp(Y )- cdFp(X2). __|_| Definition 3.The two connected p-compact groups X1 and X2 are locally isomor- phic if they satisfy either of the conditions in Proposition 1.4 We may elaborate a little on point (2) in Proposition 1.4. Let Y be a simply connected p-compact group with center [18, 8] Z(Y ) and let S be a p-compact torus. For any subgroup K < Z(Y ) and any homomorphism ' 2 Rep(K; S), define (inspired by and borrowing notation from [3]) the p-compact group Y x S=(K; ') by requiring (incl;') q (1) K _______Y/x/S _______Y/x/S=(K; ') to be an exact sequence of p-compact groups; see [9, Proposition 8.3] and [18, * *Propo- sition 4.6]. The finite covering homomorphism q of the short exact sequence (1) will be called a special finite covering homomorphism. Note that the finite cov* *ering homomorphism of [18, Theorem 5.4] occuring in point (2) of Proposition 1.4 is an example of such a special finite covering homomorphism. The following propositi* *on is therefore an immediate consequence of Proposition 1.4. 6 JESPER MICHAEL MOLLER Proposition 1.5.Let X be a connected p-compact group; let Y = X<1> denote the universal covering p-compact group of X and S = Z(X)0 the identity component of the center of X. Then any p-compact group locally isomorphic to X is isomorphic to ffi Y x S (K; ') for some subgroup K < Z(Y ) and some ' 2 Rep(K; S). Since Y is simply connected its center Z(Y ) is a finite abelian p-group [18, Theorem 5.3] and it follows in particular that the local isomorphism type of X contains only finitely many isomorphism classes of p-compact groups. Endomorphisms of p-compact tori have particularly nice properties. Let, for any endomorphism ': T ! T of a p-compact torus T , ':T ! T denote the discrete approximation [18, 2.7], [8, Proposition 3.2] to '. Lemma 1.6. Let T be a p-compact torus and ': T ! T an endomorphism of T . The following conditions are equivalent (1) ' is a rational isomorphism. (2) ss1(') is a monomorphism. (3) ' is an epimorphism. (4) ' is an epimorphism. If any of these conditions is satisfied, T='(T ) ' B(cokerss1(')) ' B(ker') and* * ' is a finite covering homomorphism. Proof.(1) ) (2) is clear. If (2) holds, the homotopy exact sequence yields T='(T ) ' B(cokerss1(')) with cokerss1(') a finite p-group; i.e. ' is an epimor- phism (even a finite covering homomorphism). If (3) holds, (T='(T )) is a p- compact group and ss1(T='(T )) ~=cokerss1(') a finite p-group. Hence ss1(') Q is an automorphism and the induced endomorphism ' on ss1(T ) Q=ss1(T ) = T an epimorphism. If (4) holds, the Snake lemma tells that the Q p-vector space coker(ss1(')Q ) is a quotient of the countable abelian cokerss1('). Hence ss1('* *)Q must be an isomorphism which is the content of (1). __|_| In other words, Lemma 1.6 says that "Q(T )= Cov(T ). Finally, let G -j!U -q!V be a short exact sequence of p-compact groups. Let Y be any p-compact group and let ___ a Bq : map(BV; BY ) -! map (BU; BY )Bg g|G'0 denote precomposition with Bq; here, the disjoint union is indexed by the set of those g 2 Rep(U; Y ) for which Bg O Bj is nullhomotopic. In this situation we h* *ave a version of [19, Proposition 1.1]. ___ Lemma 1.7. The above map Bq is a homotopy equivalence. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 7 Proof.Inclusion of constant maps provide by the Sullivan conjecture for p-compa* *ct groups [8, Theorem 9.3] a homotopy equivalence BY ! map (BG; BY )0 inducing a homotopy equivalence BY hV-! map (BG; BY )hV0 which (as BY hV = map (BV; BY ) and map (BG; BY )hV0is a collection of compo- nents of map_(BG;_BY )hV = map (BU; BY ) [9, Lemma 10.5]) can be identified to the the map Bq. __|_| Corollary 1.8.Let g :U ! Y be a homomorphism from the total space U into some p-compact group Y . Then U __g__Y//>> " q|| " |fflffl" V can be completed to a diagram commuting up to conjugacy if and only if g|G is trivial. Moreover, if g factors through V , then the factorization V ! Y is uni* *que up to conjugacy. The factorization criterion, an analog of a very basic fact in (Lie) group_th* *eory,_ of Corollary 1.8 is obtained from Lemma 1.7 by applying the functor ss0 to Bq. 2. Centralizers of p-compact toral groups The key technical results of this paper, dealing with centralizers [9, 3.4] o* *f p- compact toral groups, are contained in this section. They lead to the consequen* *ce, analogous to the Lie group case [1, 2], that homomorphisms between p-compact groups lift to homomorphisms between the respective maximal tori and that these lifts are unique up to left action by the Weyl group of the target. Let G be a p-compact toral group, i.e. an extension [9, Definition 6.3] of a p-compact torus by a finite p-group, X a p-compact group with maximal torus i: T ! X, and CX (fG) the centralizer of some homomorphism f :G ! X ; the centralizer CX (fG) is [9, Proposition 5.1, Theorem 6.1] again a p-compact group and the natural homomorphism CX (fG) ! X is a monomorphism, in fact an example of a homomorphism of maximal rank [8, Definition 4.1, Proposition 4.3]. Assume that f lifts to a homomorphism ': G ! T into the maximal torus T ; i.e. that the space (X=T )hG of lifts of Bf is nonempty. This always is the case if * *[18, Lemma 3.13], [8, Proposition 2.14] the Weyl group order |WT(X) | is not divisi* *ble by p or if G is a p-compact torus. Mapping BG into Bi: BT ! BX , which we assume has been turned into a fibration, produces another fibration of mapping spaces (2) (X=T )hG -! map(BG; BT ) Bi_-!map(BG; BX) where the fibre over Bf is displayed. The Weyl space monoid [8, Definition 9.2]* * acts on this fibration with trivial action on the base space and the Weyl group WT(X) acts on the associated exact sequence ss0((X=T )hG) -!Rep (G; T ) i*-!Rep(G; X) 8 JESPER MICHAEL MOLLER of sets. Define WT(X) B' < WT(X) ' < WT(X) to be the isotropy subgroup at B' 2 ss0((X=T )hG), respectively ' 2 Rep (G; T ). Note that the set i-1*(f) Rep(G; T ) of conjugacy classes of lifts of f is a WT(X) -set. The essential idea of the proof of Theorem 2.1 is due to Bill Dwyer during a conversation at the Cech Centennial Homotopy Conference. Theorem 2.1. Let f :G ! X be a homomorphism from a p-compact toral group G into a p-compact group X with maximal torus i: T ! X. Let CX (fG)0 denote the identity component of the centralizer CX (fG) of f in X. Assume that (X=T )hG 6* *= ; and let ': G ! T be a lift of f. Then: (1) The homomorphism CT('G) ! CX (fG), induced by i, is a maximal torus for CX (fG). (2) The Weyl group of CX (fG)0 is isomorphic to WT(X) B' . (3) WT(X) acts transitively on the set ss0((X=T )hG). Proof.Put C := CX (fG) and C0 = CX (fG)0. The first assertion is contained in the proof of [8, Proposition 4.3]. Note t* *hat the factorization BT ' BCT('G) ! BC ! BX of Bi: BT ! BX and the fact [9, Theorem 9.7] that Weyl groups are faithfully represented in maximal tori, allow us to consider WT(C) as a subgroup of WT(X) . Computing the cardinality of the set of components of the space (X=T )hG of l* *ifts of Bf is the first step. This space of lifts occurs as the fibre of the fibrati* *on a (3) (X=T )hG ! BCT( G) ! BC 2i-1*(f) obtained by restricting fibration (2) to the connected component BC of the base space map (BG; BX). Thus we may describe the fibre a (4) (X=T )hG ' C=CT( G) 2i-1*(f) as a disjoint union of homogeneous spaces. Replacing G by G= ker' if necessary, we may assume (as CX (G= ker') ' CX (G) by [9, Lemma 7.5] and (X=T )hG ' (X=T )h ker'hG= ker'' (X=T )hG= ker'by [9, Lemma 10.5] and the Sullivan conjecture [14]) that ' is a monomorphism [9, x7] * *and hence [18, Proposition 3.4] that G has a discrete approximation which is a subg* *roup of (Z=p1 )r. Using [9, Theorem 4.7, Proposition 5.7, Theorem 6.1, Proposition 6* *.7] we compute the Euler characteristic O((X=T )hG) = O(X=T ) = |WT(X) | where the last equality is [9, Proposition 8.10, Proposition 9.5]. As each disjoint summ* *and C=CT( G) of the right hand side of (4) has Euler characteristic |WT(C)|, the or* *der of the index set is |i-1*(f)| = |WT(X) :WT(C)|. Hence |ss0((X=T )hG)| = |WT(X) :WT(C)| . |ss0(C)| = |WT(X) :WT(C0)| because the index of the Weyl group of C0 in the Weyl group of C equals the number of components |ss0(C)| of C by [18, Proposition 3.8] or [8, Remark 2.11]. Observe that WT(X) ' < WT(C) and that WT(X) B' < WT(C0): If w 2 WT(X) ', the two maps wOB'; B': BG ! BT are homotopic so composition with RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 9 w is a self-map w_of BCT('G) = map(BG; BT )B' over BC. Thus w represents an element of the Weyl group of C. If w 2 WT(X) B', the fibre map w_preserves the component containing ' of the fibre C=CT('G) (X=T )hG; i.e. is homotopic over BC to a based map, meaning [18, Proposition 3.8] that w 2 WT(C0) < WT(C). The final step is a counting argument. Consider the orbit WT(X) . B' ss0((X=T )hG). As |ss0((X=T )hG)| |WT(X) . B'| = |WT(X) :WT(X) B' | |WT(X) :WT(C0)| = |ss0((X=T )hG)| we have WT(X) . B' = ss0((X=T )hG) and WT(X) B' = WT(C0). __|_| Corollary 2.2.Let f :G ! X and ': G ! T be as in Theorem 2.1. Then (1) The Weyl group of CX (fG) is isomorphic to the isotropy subgro* *up WT(X) '. (2) WT(X) acts transitively on the set i-1*(f). (3) (X=T )hG is homotopy equivalent to a disjoint union of |WT(X) :WT(X) '| copies of the homogeneous space CX (fG)=CT('G). Proof.According to the proof of Theorem 2.1, |i-1*(f)| = |WT(X) :WT(C)| and WT(X) ' < WT(C). Consider the orbit WT(X) . ' i-1*(f) Rep (G; T ). A counting argument similar to the one in the proof of Theorem 2.1 shows that WT(X) . ' = i-1*(f) * *and WT(X) ' = WT(C). The final assertion is the homotopy equivalence (4) from the proof of Theorem 2.1. __|_| The p-compact toral group G could in particular be a p-compact torus. Corollary 2.3.Let f :S ! X be a homomorphism from a p-compact torus S into a p-compact group X with maximal torus i: T ! X. Then (1) There exists a homomorphism ': S ! T such that Bi O B' = Bf. (2) WT(X) acts transitively on the set ss0((X=T )hS) of vertical homotopy classes of lifts of Bf. (3) If X is connected, ss0((X=T )hS) = i-1*(f), i.e. two lifts B'1; B'2: BS ! BT of Bf are homotopic over Bf if (and only if) they are homotopic. Proof.The existence of ' is [9, Proposition 8.11]. If X is connected, the centr* *alizer CX (fS) of S in X is also connected [18, Proposition 3.11], so the homotopy exa* *ct sequence for the fibration (3) shows that ss0((X=T )hS) = i-1*(f) Rep(S; T ). * * __|_| Thus the maximal torus homomorphism i: T ! X induces a bijection of sets ~= WT(X)\ Rep(S; T ) -! Rep(S; X) for any p-compact torus S and any connected p-compact group X. The material of Theorem 2.1 and Corollary 2.3 establishes the analogs to the main results of the papers [1, 2] by Adams and Mahmud, respectively Adams and Wojtkowiak. 10 JESPER MICHAEL MOLLER Theorem 2.4. Let X1 and X2 be p-compact groups with maximal tori i1: T1 ! X1, i2: T2 ! X2, and Weyl groups W1, W2. Let f :X1 ! X2 be a homo- morphism of X1 into X2. Then (1) There exists a homomorphism ': T1 ! T2such that the diagram BT1 __B'__BT2// Bi1|| Bi2|| |fflffl |fflffl BX1 _Bf__BX2// commutes. (We say that B' covers Bf.) (2) The Weyl group W2 acts transitively on the set ss0((X2=T2)hT1) of vertic* *al homotopy classes of maps covering Bf. (3) If X2 is connected, two maps B'1; B'2: BT1 ! BT2 that cover Bf are vertical homotopic if they are homotopic. Proof.Apply Corollary 2.3 to the homomorphism f O i1: T1 ! X1. __|_| In case f is a rational isomorphism a stronger version is possible, cfr. [11,* * Propo- sition 1.2]. If there exists a rational isomorphism between the p-compact group* *s X1 and X2 then they have the same rational rank [8, 5.9] so [9, Theorem 9.7] there* * ex- ists a p-compact torus T with homomorphisms i1: T ! X1and i2: T ! X2making T into a maximal torus for both p-compact groups. Theorem 2.5. Let X1 and X2 be p-compact groups with maximal tori i1: T ! X1, i2: T ! X2and Weyl groups W1; W2. Let f :X1 ! X2 be a rational isomorphism of X1 into X2. Then (1) There exists a finite covering homomorphism ': T ! T such that B' cov- ers Bf, i.e. such that the diagram BT __B'__BT// Bi1|| Bi2|| |fflffl |fflffl BX1 _Bf__BX2// commutes. (2) The space (X2=T )hT of all maps covering Bf is homotopically discrete [9, Remark 6.15] and W2 acts simply and transitively on its set of components ss0((X2=T )hT). (3) There exists a homomorphism WT('): W1 ! W2 , such that B' O w = WT(')(w) O B' holds in ss0((X2=T )hT) for all w 2 W1; WT(') takes the Weyl group W10< W1 of the identity component of X1 isomorphically onto the Weyl group W20< W2 of the identity component of X2. (4) If X2 is connected, two maps B'1; B'2: BT ! BT that cover Bf are vertically homotopic if they are homotopic. Under the additional assumption that ss0(f): ss0(X1) ! ss0(X2) is an isomor- phism, the homomorphism WT('): W1 ! W2 in point (3) is an isomorphism and the homomorphism ' in point (1) is an isomorphism if and only if f is an isomor- phism. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 11 Proof.(1) We only need to show that ' is a rational isomorphism for then ' is actually a finite covering homomorphism by Lemma 1.6. Note that by replacing X1 and X2 by their respective identity components it represents no loss of gene* *rality to assume that X1 and X2 are connected p-compact groups. Let w1 2 W1 be represented by a map w1: BT ! BT over BX. Since B' O w1 covers Bf, B' O w1 is [Theorem 2.4.(2)] (vertically) homotopic to w2O B' for so* *me w2 2 W2. Hence the image '*(V ) V of V := H2Qp(BT ) under '* := H2Qp(B') is W1-invariant. Moreover, as H*Qp(Bf) is an isomorphism, the endomorphism S('*) of the symmetric algebra S(V ) = H*Qp(BT ) restricts to an isomorphism S(V )W1 S(V )W2 of invariant algebras [8, Theorem 9.7]; thus S(V )W1 = S('*)(S(V )W2 ) = S('*(V ))W1 which, as any nontrivial W1-invariant complement to '*(V ) would contribute ef- fectively to S(V )W1 , implies that '*(V ) = V , i.e. that '* is an isomorphism. (2) Since ' is an epimorphism, C := CX2(fi1T ) = CX2(i2'T ) = CX2(i2T ) by [8, Lemma 7.5]. Thus C is a p-compact toral group with C0 = T as its identity component [8, x8]. By Corollary 2.2, a (X2=T )hT ' C=T is homotopically discrete and the W2-action on the set of components is transit* *ive and simple by Theorem 2.1. (3) The order of Wj0equals the rank of H*Qp(BT ) as an H*Qp(BXj)-module [8, Theorem 9.7], [5, Ch. 5, x5, no 2] so since H*Qp(B') and H*Qp(Bf0) are isomorph* *isms [Definition 1], |W10| = |W20|. Since W2 acts simply and transitively on ss0((X2=T )hT), the equation B' O w = WT(')(w) O B' does define a homomorphism between the Weyl groups. Naturality of this construction implies that that WT(') is in fact a homomorphism 1_____W10//____W1//____ss0(X1)//__1// ~=|| WT(')|| ss0(f)|| |fflffl |fflffl |fflffl 1_____W20//____W2//____ss0(X2)//__1// of the short exact sequences relating the full Weyl groups to those of the iden* *tity components [18, Proposition 3.8]. Here, the restriction WT(')|W10:W10! W20 is a monomorphism because ss2(B') Q is an isomorphism and the subgroups Wj0 are faithfully represented [9, Theorem 9.7] in ss2(BT ) Q . (Alternatively one* * may view the Weyl groups W10and W20as Galois groups of field extensions connected by isomorphisms induced by ' and the restriction f0 of f to the identity component* *s.) (4) This is point (3) of Corollary 2.3. Now assume additionally that ss0(f): ss0(X1) ! ss0(X2)is an isomorphism. By the above diagram, WT('): W1 ! W2 is then an isomorphism. Suppose ': T ! T is an isomorphism and that B' covers Bf as in point (1). Re- placing i2 by i2O ' we may even assume that ' is the identity, i.e. that i2: T * *! X2 admits the factorization i2 = f O i1. Since also W1 = W2, this implies that the 12 JESPER MICHAEL MOLLER p-normalizer [9, Definition 9.8] of T in X1, Np, equals the p-normalizer of T in X2 and that the natural monomorphism Np ! X1 followed by f is the natural monomorphism Np ! X2. Thus f|Np is a monomorphism and so is f by [18, The- orem 2.17]. Consequently, f is a monomorphic rational isomorphism inducing an isomorphism on the groups of components, i.e. an isomorphism by Lemma 1.3. __|* *_| In the language of [21, Definition 1.1], the homomorphism ' from point (1) of Theorem 2.5 induces a WT(')-admissible monomorphism ss1('): ss1(T ) ! ss1(T ) (provided ss0(f) is an isomorphism). 3.Classification of rational isomorphisms The aim of this section is to relate the set of rational isomorphisms between two locally isomorphic p-compact groups to the monoid of rational automorphisms of their universal covering p-compact group. We begin by showing that the cen- ter construction is a functor on the category of p-compact groups with rational isomorphisms. Let X1 and X2 be two connected p-compact groups with maximal tori i1: T ! X1 and i2: T ! X2 of the same rank; let W1 and W2 denote the associated Weyl groups. Rational isomorphisms between p-compact groups behave like epimorphisms be- tween Lie groups in that they preserve centrality. Lemma 3.1. Let f :X1 ! X2 be a rational isomorphism and z :Z ! X1 a central [9, 3.4] homomorphism from some p-compact toral group Z into X1. Then the composition f O z :Z ! X2 is central. Proof.Choose [18, Lemma 4.1] a homomorphism y :Z ! T such that z = i1Oy and [Theorem 2.5] a finite covering homomorphism T (f): T ! Tsuch that Bi2OT (f) = Bf O Bi1. The commutative diagram T(f) Z AAy__T_//____T// AA | | z AAi1||fflffli2||fflffl X1 _f___X2// induces another commutative diagram CT(yZ) _______C1//_'__X1// | | | | | |f |fflffl |fflffl|fflffl CT(T (f)yZ)_____C2//____X2// where C1 := CX1(zZ) and C2 := CX2(fzZ) have CT(yZ) ~=T and CT(T (f)yZ) ~= T as maximal tori [Theorem 2.1]. Note that C2 here can be replaced by its ident* *ity component C02leading to a factorization of i2 which allows us to view the Weyl group of C02as a subgroup of W2; in the language of [8, x4] C02! X2 is a subgro* *up RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 13 of maximal rank. Applying the functor H*Qp(-) leads to a third commutative diagram H*Qp(BT )oo___H*Qp(BC1)oo~=_H*Qp(BX1) OO OO OO H*Qp(B')~=|| || ~=H*Qp(Bf)|| | | | H*Qp(BT )oo___H*Qp(BC02)oo__H*Qp(BX2) in cohomology; here C2 has been replaced by C02. This diagram shows that the ra* *nk, |WT(C02)|, of H*Qp(BT ) as an H*Qp(BC02)-module equals the rank, |W1|, of H*Qp(* *BT ) as an H*Qp(BX1)-module. Since |W1| = |W2| [Theorem 2.5], we have WT(C02) = W2 so C02! X2 is both a monomorphism and a rational isomorphism [8, Theorem 9.7], hence [Lemma 1.3] an isomorphism. Now C2 ! X is both an epimorphism and a monomorphism, i.e. [9, 3.2] an isomorphism, meaning that f O z :Z ! X2 is central. __|_| Functoriality of Z is now an easy consequence. Corollary 3.2.For any rational isomorphism f 2 "Q(X1; X2)there exists a uniquely determined rational isomorphism Z(f) 2 "Q(Z(X1); Z(X2))such that Z(X1) _Z(f)Z(X2)_// | | | | |fflfflT(f)|fflffl T _________T// i1|| i2|| |fflffl |fflffl X1 ___f____X2// commutes up to conjugacy for any finite covering homomorphism T (f) that covers f. Proof.In order to prove existence, let T(f): T ! T be the discrete approximation to any finite covering homomorphism T (f): T ! T that covers f. By Lemma 3.1, T(f) restricts to a homomorphism Z(f): Z(X1) ! Z(X2) between the p-discrete centers [18, Definition 4.3] of X1 and X2. Define Z(f): Z(X1) ! Z(X2) to be the closure [9, Proposition 6.9] of Z(f). Then the above diagram commutes and Z(f) is a rational isomorphism because [18, Corollary 5.2] Z(Xj) ! Xj induces an isomorphism on ss1(-) Q. There is [18, Lemma 4.8] a homotopy equivalence (X2=Z(X2))hZ(X1)' X2=Z(X2) between X2=Z(X2) and the space of all lifts to BZ(X2) of the central [Lemma 3.1] homomorphism Z(X1) ! X1 ! X2. In particular, the space of lifts is connected meaning that all lifts are vertically homotopic. This proves uniqueness of Z(f)* *. __|_| The uniqueness clause ensures that Z(f2Of1) = Z(f2)OZ(f1) whenever the ratio- nal isomorphisms f1 and f2 are composable; in particular, Z(f) is an isomorphism for any isomorphism f. This center functor plays a central role in the computat* *ion of the set of rational isomorphisms between locally isomorphic p-compact groups. 14 JESPER MICHAEL MOLLER Assume from now on that X1 and X2 are locally isomorphic, i.e. [Proposition 1* *.5] that there exist special finite covering homomorphisms (as in the short exact s* *e- quence (1) of Section 1) Kj (incl;'Yjx)S_//_qj__Xj/=/Y x S=(Kj; 'j) for some simply connected p-compact group Y , some p-compact torus S, some fini* *te subgroups Kj < Z(Y ), and some homomorphisms 'j:Kj ! S , j = 1; 2. The aim is to relate "Q(X1; X2)to "Q(Y )and "Q(S). Let f :X1 ! X2 be any rational isomorphism. The composite maps B(incl) Bqj BY _______/BY/x BS = B(Y x S) _______/BXj;/ j = 1; 2; induce homotopy equivalences BY ! BXj<2> of 2-connected covers. Let fY be the rational isomorphism corresponding to Bf<2>. Then Y __fY__Y// (5) q1|Y|| q2|Y|| |fflffl |fflffl X1 __f__X2// commutes up to conjugacy. The composite homomorphisms Z(qj) S _inclZ(Y_)/x/S = Z(Y x S)_____Z(Xj);// j = 1; 2; take S isomorphically to the identity component Z(Xj)0 of the center [18, Corol- lary 5.5]. Let fS :S ! S be the homomorphism corresponding to the finite cover- ing [Lemma 1.6] homomorphism Z(f)0: Z(X1)0 ! Z(X2)0of identity components. Then S __fS__S// (6) q1|S|| q2|S|| |fflffl |fflffl X1 __f__X2// commutes up to conjugacy [Corollary 3.2]. Now put (f) := (fY ; fS). The following main result is the analog of [3, Coro- rollary 6] for Lie groups and [16, Theorem 0.3] for classifying spaces of Lie g* *roups. Theorem 3.3. Let X1 = Y x S=(K1; '1) and X2 = Y x S=(K2; '2) be two locally isomorphic connected p-compact groups. (1) The map : "Q(X1; X2)! "Q(Y )x "Q(S) is injective. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 15 (2) For f 2 "Q(X1; X2), g 2 "Q(Y ), and h 2 "Q(S), (f) = (g; h) if and only if the diagram Y x S _gxh_Y/x/S (7) q1|| |q2| |fflffl |fflffl X1 ____f___X2// commutes up to conjugacy. (3) The image of consists of all pairs (g; h) 2 "Q(Y )x "Q(S) for which Z(g)(K1) < K2 and the diagram K1 _'1__S// (8) Z(g)|K1|| h|| |fflffl|fflffl K2 _'2__S// commutes up to conjugacy. Proof.The first step is to see that diagram (7) with g = fY and h = fS commutes up to conjugacy: Diagram (6) shows that both Bq2 O (BfY x BfS) and Bf O Bq1 have adjoints belonging to the mapping space map BY; map(BS; BX2)BfOB(q1|S) ' map(BY; BX2) where the homotopy equivalence is by centrality [Lemma 3.1] of f O (q1|S); furt* *her- more diagram (5) shows that both adjoints land in the component of Bf O B(q1|Y ) of map (BY; BX2) under this homotopy equivalence. Thus the adjoints, and with them also the original maps, are homotopic. In other words, fY x fS is a lift of f O q1 or f is a factorization of q2O (f* *Y x fS); as lifts are unique by Lemma 1.2, point (2) follows, and as factorizations are * *unique by Corollary 1.8, point (1) follows. Suppose (g; h) = (f) for some rational isomorphism f. The functor Z applied to the commutative diagram (7) determines a homomorphism K1 _(incl;'Z(Y1))x_S//Z(q1)Z(X1)_// | | | | Z(g)xh| |Z(f) |fflffl |fflffl |fflffl K2 _(incl;'Z(Y2))x_S//Z(q2)Z(X2)_// between short exact sequences of p-compact group homomorphisms; the horizontal short exact sequences come from [18, Corollary 5.5]. Commutativity of the left square in this diagram means that the pair (g; h) satifies the conditions of po* *int (3). Suppose, conversely, that (g; h) 2 "Q(Y )x "Q(S)satisfies the conditions in p* *oint (3) meaning that the left square in the above diagram exists or, equivalently, * *that 16 JESPER MICHAEL MOLLER the left square in the commutative diagram BK1 (B(incl);B(YBx'S)1)_//Bq1___BX1//O | | O | B(gxh)| O |fflffl |fflffl fflffl BK2 (B(incl);B(YBx'S)2)_//Bq2___BX2// exists. According to Corollary 1.8, this commutative diagram can be completed by some homomorphism f :X1 ! X2. Obviously, f is a rational isomorphism and (f) = (g; h) since, by construction, diagram (7) commutes. __|_| In the situation of Theorem 3.3 there is a simple relation between the mapping space components containing f, g, and h. Proposition 3.4.Suppose that the homomorphism f :X1 ! X2 is covered by a product homomorphism g x h: Y x S ! Y x S. Then there exists a fibration of the form BK2 ! map(BY; BY )Bg x map(BS; BS)Bh ! map(BX1; BX2)Bf relating the centralizers of f, g, and h. Proof.Recall that for any two p-compact groups U and V , the trivial homomor- phism is central [9, Proposition 10.1] meaning that the maps _eBV_// map(BU; BV )0ocBVoBV_ given by evaluation and inclusion of constant maps are each others homotopy in- verses. (If U is a p-compact toral group this already follows from Miller's Sul* *livan conjecture [14] [9, Proposition 5.3, Theorem 6.1] and if V is a p-compact toral* * group from elementary obstruction theory.) Suppose that Bg :BY ! BY and Bh: BS ! BS are maps; not necessarily ra- tional isomorphisms. Then map (BY x BS; BY x BS)BgxBh = map (BY x BS; BY )BgOprBYx map(BY x BS; BS)BhOprBS = map (BY; map(BS; BY )0)cBYOBgx map (BS; map(BY; BS)0)cBSOBh ' map (BY; BY )Bg x map(BS; BS)Bh where the second homeomorphism is provided by adjointness and the homotopy equivalence is induced by composition with evaluation maps. Moreover, composition with the projections Bq1 and Bq2 induce maps Bq2_ ___Bq1 map(BY xBS; BY xBS)BgxBh - -! map(BY xBS; BX2)BfOBq1 --' map (BX1; BX2)Bf ____ where Bq2_is a fibration with fibre BK2 by Lemma 1.2 and Bq1 is a homotopy equivalence by Lemma 1.7. __|_| Corollary 3.5.Let f 2 "Q(X1; X2)be a rational isomorphism with (f) = (g; h) 2 "Q(Y )x "Q(S). Then (1) f is a finite covering homomorphism if and only if g is an isomorphism. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 17 (2) f is an isomorphism if and only if g and h are isomorphisms. Proof.Diagram (7) induces a fibration K2=Z(g)K1 ! Y=gY x S=hS ! X2=fX1 where the fibre has no homotopy in degree > 1, Y=gY is simply connected, and [Lemma 1.6] S=hS ' B(kerh). The corollary follows by inspecting the exact ho- motopy sequence for this fibration. __|_| In short form the content of Theorem 3.3 and Corollary 3.5 is that "Q(X1; X2) "Q(Y )x "Q(S) "Q(X) "Q(Y )x "Q(S) Cov(X1; X2) Out(Y ) x Cov(S) Out(X) Out(Y ) x Out(S) is the set of pairs (g; h) for which Z(g)(K1) K2 and diagram (8) commutes. Her* *e, Out(Y ) Rep(Y; Y ) denotes the group of conjugacy classes of automorphisms of Y , i.e. the group of invertible elements of the monoid "Q(Y ). In the second c* *olumn, where the assumption is that X1 = X = X2, the inclusion respects the monoid or group structure. The lower line represents a direct translation of Baum's expre* *ssion [3, Corollary 6] for the set of finite covering homomorphisms between two local* *ly isomorphic compact connected Lie groups. For instance, "Q(Y x S) ~="Q(Y )x "Q(S)~="Q(P Y x S) where P Y = Y=Z(Y ). Note also that up to isomorphism the only finite covering homomorphism onto X := Y x S=(K; ') are of the form Y x S=(L; ) (idY;h)----!Y x S=(K; ') where L < K, h 2 Cov(S), and h O = '|L. Write X1 X2 if there exists a finite covering homomorphism X1 ! X2. Proposition 3.6.If X1 X2 and X2 X1 then X1 ~=X2. Proof.Let (g1; h1): X1 ! X2and (g2; h2): X2 ! X1be finite covering homomor- phisms where g1; g2 2 Out(Y ) and h1; h2 2 Cov(S) as in Theorem 3.3. Observe that Z(g1) and Z(g2) are isomorphisms and that Z(g1)|K1 takes K1 isomorphically to K2. Injectivity of the p-discrete torus S implies that the commutative diagr* *am K1 _'1__S//O Z(g1)|K~=1|| O |fflfflfflfflO K2 _____S// can be completed by some homomorphism h: S ! S and an application of Naka- yama's lemma [16, Lemma 3.2] shows that h may be chosen to be an isomorphism. Then (g1; h): X1 ! X2 is an isomorphism of p-compact groups. __|_| In contrast, two compact conncted Lie group may cover each other without being isomorphic [3]. 18 JESPER MICHAEL MOLLER Definition 4.The local isomorphism system of the connected p-compact group X is the set of isomorphism classes of p-compact groups locally isomorphic to X equipped with the ordering . This definition is a direct copy of the equivalent notion from the category of compact connected Lie groups [3, Definition 8]. For example, let Y be the p-compact group SU(p)^p. The local isomorphism system of Y x S, where S is a p-compact torus of rank 1, has the form Y x S _____U(p)^p//___P/Y/x S and the local isomorphism system of Y x Y has the form P8Y8xOYO qqq OOO qq OOO qqqq OO'' Y x YLL P Y7x7P Y LLL ppp LLL pppp L%% ppp Y x Y= where < Z(Y ) x Z(Y ) ~=Z=p x Z=p is the diagonal subgroup. 4. Centralizers of rational automorphisms The purpose of this section is to investigate centralizers of endomorphisms t* *hat are rational isomorphisms. It is shown that any nontrivial endomorphism of a connected simple p-compact group is a rational isomorphism and in case p divides the Weyl group order even a homotopy equivalence. Let X be a connected p-compact group with maximal torus i: T ! X. The associated Weyl space WT(X) is the group-like topological monoid of self maps of BT over BX [9, Definition 9.2]. Define [9, Definition 9.8] BN(T ) to be the tot* *al space in the fibration BT ! BN(T ) ! BWT(X) ' BWT(X) for which the monodromy action is the inclusion homomorphism BWT(X) = WT(X) ,! aut(BT ) of the Weyl space monoid into the group-like topological monoid aut(BT ) of self homotopy equivalences of BT . Let also BT ! BNp(T ) ! BWp be the restriction of BN(T ) to the classifying space of a p-Sylow subgroup Wp * *of WT(X). The loop space N(T ) = BN(T ) is called the normalizer and Np(T ) = BNp(T ) the p-normalizer of T in X. Note that, unless WT(X) happens to be a p-group, the normalizer N(T ) will not be a p-compact group. The p-normalizer Np(T ), however, is always a p-compact toral group. Sine the action of WT(X) on BT respects the map Bi: BT ! BX , the homomorphism i: T ! X extends to T ! Np(T ) ! N(T ) j!X and [9, Proposition 9.9] Np(T ) jp!X is a monomorphism. RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 19 Suppose f :X ! X is an endomorphism of X. The space of all lifts in the diagram 5BN(T5) k k k k k Bj|| k k |fflffl BN(T ) Bj___BX//__Bf__BX_// is the homotopy fixed point space (X=N(T ))hfj(N(T)). Lemma 4.1. If the endomorphism f :X ! X is a rational isomorphism, then (X=N(T ))hfjN(T)' *. Proof.The universal covering map BT ! BN(T ) induces a covering map WT(X) ! (X=T )hT ! (X=N(T ))hT where the total space is homotopy equivalent [Theorem 2.5] to the homotopi- cally discrete fibre. Thus the base space (X=N(T ))hT and [9, Lemma 10.5] (X=N(T ))hN(T)= (X=N(T ))hThWT(X) are contractible. __|_| In particular there exists a self map BN(f), unique up to vertical homotopy, such that the diagram BN(T ) _BN(f)_BN(T/)/ Bj|| Bj|| |fflffl |fflffl BX ___Bf_____BX// commutes. In view of the proposition below and the results of [11], it seems not totally unlikely that the monoid homomorphism N :"Q(X) ! [BN(T ); BN(T )] is injective. Proposition 4.2.Let X be a connected p-compact group and f0; f1: X ! Xtwo endomorphisms of X. If BN(f0) and BN(f1) are homotopic, then H*Qp(Bf0) = H*Qp(Bf1) and f0 O and f1 O are conjugate for any homomorphism :G ! X from a p-compact toral group G into X. Proof.Since H*Qp(Bj): H*Qp(BX) ! H*Qp(BN(T ))is an isomorphism, the first as- sertion immediately follows. Because ss1(X=N(T )) maps onto ss0(N(T )) we may assume by adjust Bfj by a vertical homotopy to achieve that ss1(BN(fj)) preserves the p-Sylow subgroup Wp = ss1(BNp(T )) < ss1(BN(T )) = WT(X). Let BNp(fj): BNp(T ) ! BNp(T )be the unique based map that covers BN(f), j = 0; 1. Since BN(f0) and BN(f1) are homotopic, BNp(f1) = w OBNp(f0) for some covering translation w of the covering BNp(T ) ! BN(T ). Let f :G ! X be a homomorphism from a p-compact toral group G into X. Since the Euler characteristic of X=Np(T ) is prime to p [9, Proof of 2.3] ther* *e exists by [18, Lemma 3.13] or [8, Proposition 2.14] a lift i :G ! Np(T )such that and jp O i are conjugate. Then Bf0 O B ' Bf0 O Bjp O Bi ' Bjp O BNp(f0) O Bi ' Bjp O w O BNp(f1) O Bi ' Bjp O BNp(f1) O Bi ' Bf1 O Bjp O Bi ' Bf1 O B. __|_| 20 JESPER MICHAEL MOLLER Now follows another observation with regard to the loop map N(f) associated to the rational automorphism f. Lemma 4.3. Let f :X ! X be a rational isomorphism and BN(f) the self-map of BN(T ) that covers Bf [Lemma 4.1]. Then precomposition with BN(f) ______ BN(f) : map(BN(T ); BX)Bj ! map(BN(T ); BX)BfOBj is a homotopy equivalence. Proof.Any lift T (f): T ! Tto the maximal torus is an epimorphism [Theorem 2.5] so the induced map ______ BT (f):BCX (iT ) ! BCX (fiT ) is a homotopy equivalence by [9, Lemma 7.5] or [8, Lemma 10.3]. Note that both BCX (iT ) = map (BT; BX)Bi and BCX (fiT ) = map (BT; BX)BfOBi are WT(X)- spaces in that the homotopy orbit_spaces_BCX (iT )hWT(X) and BCX (fiT )hWT(X) exist. The homotopy equivalence BT (f)induces a homotopy equivalence BCX (iT )hWT(X) ! BCX (fiT )hWT(X) of homotopy fixed point spaces. In particular, the component map(BN(T_);_BX)Bj of the domain is mapped by a homotopy equivalence, identifiable to BN(f) , to t* *he component map (BN(T ); BX)BfOBj of the target. __|_| The analogous statement with the normalizer N(T ) replaced by the p-normalizer Np(T ) also holds. Definition 5.The connected p-compact group X is simple if ss2(BT ) Q is a simple Qp[WT(X)]-module. Since (ss2(BT ) Q)WT(X) ~=ss1(Z(X)) Q ~=ss1(X) Q by [18, Proposition 5.1, Corollary 5.2], any simple p-compact group with nontrivial Weyl group is semisi* *m- ple in the sense that it has finite fundamental group and center. The preeminence of simple p-compact groups is evident from the following theo- rem which in the case of compact Lie groups follows from the classification the* *orem [12, Theorem 2]. Theorem 4.4. Let X be a connected simple p-compact group and f :X ! X a nontrivial endomorphism. Then f is a rational isomorphism. Proof.Suppose f :X ! X is not a rational isomorphism. Since the Euler charac- teristic of X=Np(T ) is prime to p there exists by [18, Lemma 3.13] or [8, Prop* *osition 2.14] an endomorphism Np(f) of the p-normalizer such that BNp(f) covers Bf. We obtain a commutative diagram jp T _____Np(T/)/____X// | | T(f)|| Np(f)|| f|| |fflffl |fflffl |fflffl T _____Np(T/)/jp__X// RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 21 where T (f) is the restriction of Np(f) to the identity component T of Np(T ). * *It follows from admissibility [Theorem 2.4] that the image of H2Qp(BT (f)) is a WT* *(X)- submodule of H2Qp(BT ). Since f is not rational isomorphism and H2Qp(BT ) an irreducible WT(X)-representation, T (f) is trivial. In order to show that f is trivial, it suffices by Corollary 5.6 below to show that f O jp is trivial. Let :Z=pn ! Np(T ) be any homomorphism from a cyclic p-group into the p-normalizer. Since X is connected there exists by [9, Proposi* *tion 5.5, Proposition 8.11] a homomorphism i :Z=pn ! Twith i O i conjugate to jpO . Hence f O jp O = f O i O i = i O T (f) O i is trivial. Since this holds for a* *ny homomorphism , f O jp is trivial by [9, Lemma 7.7]. __|_| Under the additional assumption that p divides the order of the Weyl group an even stronger statement, which may be seen as a generalization of Ishiguro's theorem [10], holds. Theorem 4.5. Let X be a connected simple p-compact group where p divides the order of the Weyl group WT(X). Then "Q(X) = Out(X). Proof.Let f :X ! X be a rational isomorphism and BN(f): BN(T ) ! BN(T ) the map [Lemma 4.1] that covers Bf :BX ! BX . Then ss1(BN(f)) is an isomor- phism (namely conjugate to WT(T (f)) where BT (f): BT ! BT covers Bf). Thus (some map homotopic to) BN(f) lifts to self map BNp(f) of the covering space BNp(T ) of BN(T ). The discrete approximation Np(f) to Np(f) is a self map 1_____T//____Np/(T/)____Wp//____1// Np(f)|T|| Np(f)|| ~=WT(T(f))| fflffl||fflffl|fflffl||fflffl 1_____T//____Np/(T/)____Wp//____1// of the group extension inducing the Postnikov tower for BNp(T ). Choose an element w 2 Wp of order p (remember that p divides the order of WT(X)) with a lifting g 2 Np (T ). Since the discrete approximation T to T is p-divisible, gp = tp for some t 2 T. Then gt-1 2 Np(T ) has order p and projects onto w; denote also by w this lift gt-1 of w. Note that Np(f)(w) 6= 1. Let Np(T ) be the cyclic group generated by w and B(jp|) the monomorphism B B()!BNp(T ) jp!BX. In the commutative diagram BN__B()_/BNp(T/)B(Np(f))BNp(T_)// NNN NN |Bjp Bjp| B(jp|)NNNN''||fflffl ||fflffl BX _____Bf____BX// the composition of the two top horizontal maps is a monomorphism since Np(f)| is a monomorphism [18, Proposition 3.4]. As also the vertical homomorphism jp is monomorphic, f O jp| is a monomorphism. 22 JESPER MICHAEL MOLLER Since X is connected, the monomorphism jp| factors through i: T ! X (use [9, Proposition 5.5, Proposition 8.11] to see this) and we obtain a homotopy co* *m- mutative diagram BT(f) qBT88_________BT// qq qBkqqq Bi|| |Bi| qq |fflffl |fflffl B B(jp|BX)_//_Bf____BX// of maps of classifying spaces. By assumption ss2(BT ) Q is an irreducible WT(X* *)- module, so ss2(BT (f)) is scalar multiplication by some p-adic integer ; i.e. B* *T (f) = . We must show that is a p-adic unit for then f will be an automorphism [Theorem 2.5]. Assume the converse, i.e. that is divisible by p. Then, since w has order p, BT (f) O Bk is nullhomotopic so also Bf O B(jp|) is nullhomotopic. This contradicts [9, Proposition 5.4] the fact that f O jp| is a monomorphism. _* *_|_| . The above proof is a transcription of that of [11, Proposition 1.3]. We now turn to the case where the prime p does not divide the order of the We* *yl group. The main result [Theorem 1.3] of [8] asserts that the natural map BZ(X) ! map(BX; BX)B1 of the center Z(X) into the the centralizer of the identity map, denoted 1, of * *X is a homotopy equivalence. The following proposition explicitly computes this cent* *er as well as the centralizer of any rational automorphism of X provided the order* * of the Weyl group is prime to p. Proposition 4.6.Let X be a connected p-compact group with Weyl group order prime to p and let f :X ! X be a rational isomorphism. Then precomposition with Bf induces a homotopy equivalence ___Bf BCX (X) = map(BX; BX)B1 --! BCX (fX) = map(BX; BX)Bf where CX (X) and CX (fX) are p-compact tori of rank equal to the rank of the free Z^p-module ss2(BT )WT(X). If, additionally, X is semisimple, then CX (X) a* *nd CX (fX) are trivial p-compact groups. Proof.Composition with the maps of the relation Bf O Bj = Bj O BN(f) induce a commutative diagram __ map (BX;OBX)BfO__Bj'map_(BN(T/);/BX)BfOBjOO ___Bf| _____| | BN(f)'| | | map (BX; BX)B1 ____Bj'map_(BN(T/);/BX)Bj of maps between mapping spaces. Since p is prime to the order of the Weyl group of X, H*(BX; Fp) ~=H*(BT ; Fp)WT(X) ~=H*(BN(T ); Fp) RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 23 by [7, Theorems_2.10-2.11]. Thus Bj :BN(T ) ! BX is_an_H*Fp-equivalence and the maps Bj are homotopy_equivalences. Since also BN(f) is a homotopy equiv- alence by Lemma 4.3, Bf is a homotopy equivalence and so it only remains to compute map (BN(T ); BX)Bj which is a path-component of the space of sections of the fibration BCX (T ) ! BCX (T )hWT(X) ! BWT(X) with fibre BCX (T ) ' BT . Since p is prime to |WT(X)| so that H>0(WT(X); ss2(BT )) = 0 [17, Theorem 6.3] shows that map (BN(T ); BX)Bj = K(ss2(BCX (T ))WT(X); 2) where the action map (BT; BX)Bi x WT(X) ! map (BT; BX)Bi is composition with elements from the Weyl space. Choose a based map w :BT ! BT realizing some w 2 WT(X) and a based map w-1 :BT ! BT which is a homotopy inverse to w. Then the homotopy commutative diagram of homotopy equivalences eBT Bi_ BT oo'__map (BT; BT )B1_'__/map(BT;/BX)Bi w-1|| || |__w| |fflffl' |fflffl ' |fflffl BT oeBTomap_(BT; BT )B1_Bi__map(BT;/BX)Bi/ where the middle vertical map is conjugation by w and eBT evaluation at the base point, shows that the above right action of w 2 WT(X) on map (BT; BX)Bi identifies to the standard left action of w on BT . Thus Z(X) ~=CX (fX) is always a p-compact torus in this nonmodular case. If X is semisimple, the center Z(X) is also finite [18, Theorem 5.3]; hence trivia* *l. __|_| Combining Theorem 4.5 covering the modular case and Proposition 4.6 covering the nonmodular case, we obtain Corollary 4.7.Suppose that f :X ! X is a nontrivial endomorphism of a con- nected simple p-compact group X with nontrivial Weyl group. Then pre- and post- composition with Bf induce maps Bf_ ___Bf map (BX; BX)B1 --! map (BX; BX)Bf -- map (BX; BX)B1 that both are homotopy equivalences. Proof.If the prime p divides |WT(X)|, Bf is a homotopy equivalence by Theo- rem 4.5 and if not, both centralizers are contractible by Proposition 4.6. __|* *_| In the Lie group case the above corollary is contained in [12, Theorem 3]. 24 JESPER MICHAEL MOLLER 5. A triviality criterion The purpose of this section is to prove that if a p-compact group homomorphism vanishes on all elements, then it is trivial. A homomorphism f :X ! Y between p-compact groups is said to vanish on all elements if Z=pn -! X -f!Y is trivial for any n 1 and any homomorphism :Z=pn ! X . (A homomorphism is trivial if the corresponding map of classifying spaces is nullhomotopic.) Theorem 5.1. Let f :X ! Y be a homomorphism that vanishes on all elements. Then f is trivial. The proof uses an inductive principle introduced in [8, x9] and codified in t* *he concept of a saturated class. A class C of p-compact groups is said to be satur* *ated [8, Definition 9.1] if (1) C is closed under isomorphisms. (2) The trivial p-compact group belongs to C. (3) If the identity component X0 of X is in C, then X 2 C. (4) If X is connected and X=Z(X) 2 C, then X 2 C. (5) If X is connected and has trivial center, and Y 2 C for all p-compact gr* *oups Y with cdFp(Y )< cdFp(X), then X 2 C. The point is that the only saturated class is [8, Theorem 9.2] the class of a* *ll p-compact groups. Let now C be the class of all p-compact groups X with the property that any homomorphism defined on X that vanishes on all elements is trivial. The strategy is to show that C is saturated. Obviously, C is closed under isomorphisms. Lemma 5.2. Any p-compact toral group belongs to C. Proof.Suppose that X is a p-compact toral group and f :X ! Y a homomorphism that vanishes on all elements. Then kerf = X so f is trivial by [9, Lemma 7.7].* * __|_| Lemma 5.3. Let X be a p-compact group with identity component X0 and let f :X ! Y be a homomorphism that vanishes on all elements. If X0 2 C, then f is trivial. Proof.Since X0 2 C and f vanishes on all elements, the composition X0 ! X ! Y is trivial and f admits [Corollary 1.8] a factorization X ! ss0(X) ! Y through * *the the finite p-group ss0(X) of components. I claim that ss0(X) ! Y vanishes on all elements; hence [Lemma 5.3] is trivial. To see this, let Z=pn ! ss0(X) be any group homomorphism. Recall that the p- normalizer admits a monomorphism jp: Np(T ) ! Xwith ss0(jp) an epimorphism; see [18, Corollary 3.9] or [8, Remark 2.11]. Choose an m n and a homomorphism Z=pm ! ss0(Np(T )) making Z=pm _____/ss0(Np(T/)) | | modpn| ss0(jp)| fflffl||fflffl|fflffl Z=pn_______ss0(X)// RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 25 commutative. Here, the top horizontal homomorphism lifts to the p-discrete ap- proximation Np (T ) as in the proof of Theorem 4.5. Compose this lift with the monomorphism jp into X to obtain the commutative diagram Z=pm _______X//__f___Y//<< | | zzz modpn| | zz fflffl||fflffl|fflfflzz Z=pn _____ss0(X)// of p-compact group homomorphisms. The composition of the two horizontal ho- momorphisms at the top is trivial since f vanishes on all elements. Hence also Z=pn ! ss0(X) ! Y is trivial by [9, Lemma 7.7]. __|_| For any p-compact group X, put P X = X=Z(X). Lemma 5.4. Let X be a connected p-compact group and f :X ! Y a homomor- phism that vanishes on all elements. If P X 2 C, then f is trivial. Proof.Since the center Z(X) 2 C by Lemma 5.3, f factors [Lemma 1.7] through a homomorphism P X ! Y defined on P X. I claim that that this factorization vanishes on all elements; hence is trivial. Let Z=pn ! P X be a homomorphism. Since P X is connected, the ori- entable fibration BZ(X) ! BX ! BP (X) is classified by some k-invariant k :BP (X) ! B2Z(X) . The obstruction to lifting Z=pn ! P X to X is the re- striction of k to Z=pn corresponding, [20] or [15, x1], to a cohomology class in H2(Z=pn; Z(X)). Assume the order of this obstruction is pm (any element of this cohomology group has finite order). Then it is possible to find a homomorphism Z=pm+n ! X such that Z=pm+n ______X//__f__Y//>> | "" modpn| | "" fflffl||fflffl||fflffl"" Z=pn ______P/X/ commutes up to conjugacy. Since f vanishes on all elements, the composition of * *the two horizontal homomorphisms at the top is trivial. Hence also Z=pn ! P X ! Y is trivial. __|_| Lemma 5.5. Let X be a connected p-compact group with trivial center and f :X ! Y a homomorphism that vanishes on all elements. If all p-compact groups Y with cdFp(Y )< cdFp(X)are in C, then f is trivial. Proof.The category AX has as objects all pairs (V; ) where V is a nontrivial elementary abelian p-group and :V ! X a conjugacy class of monomorphisms V ! X. The morphisms of AX are injections over X. For any object (V; ) of AX , let e(V; ): BCX (V ) ! BX be the evaluation monomorphism. Naturality of these maps makes them combine to a map e: hocolimBCX (V ) ! BX 26 JESPER MICHAEL MOLLER where the homotopy colimit is taken over (the opposite of) AX . The map e is an H*Fp-equivalence [8, Theorem 8.1] so we have homotopy equivalences _e map(BX; BY ) -!'map(hocolimBCX (V ); BY ) ' holimmap(BCX (V ); BY ) for any p-compact group Y . Consider now a homomorphism f :X ! Y that vanishes on all elements. For any object (V; ) of AX also f O e(V; ): CX (V ) ! Yvanishes on all elements. The assumption that X has trivial center implies [18, Theorem 4.4] that the monomor- phism :V ! X is not central and hence [9, Proposition 6.14, Remark 6.15] that cdFp(CX (V ))< cdFp(X)so that CX (V ) 2 C by assumption. Hence f O e(V; ) is trivial. Let R Rep(X; Y ) denote the set of those homomorphisms X ! Y that van- ish on all elements and let map (BX; BY )R be the corresponding union of path- components of map (BX; BY ). By the above, map (BX; BY )R is homotopy equiv- alent to a union of components of the homotopy inverse limit holimmap(BCX (V ); BY )0 of spaces of nullhomotopic maps. Since trivial homomorphisms are cen- tral [9, Proposition 10.1], evaluation provides a homotopy equivalences map(BCX (V ); BY )0 ! BY compatible with the morphisms of AX . Thus the above homotopy inverse limit is [6, Lemma XI.5.6] homotopy equivalent to BY . In particular, map(BX; BY )R is connected, meaning that the set R contains only the trivial homomorphism 0 2 Rep(X; Y ). __|_| Together with the induction principle, Lemma 5.3-5.5 prove Theorem 5.1. Two easy corollaries can be obtained by combining Theorem 5.1 with the facts that any element of an arbitrary p-compact group is conjugate to an element of * *the p-normalizer and any element of a connected p-compact group is conjugate to an element of the maximal torus. Corollary 5.6.Let X be a p-compact group with maximal torus T and p- normalizer Np(T ) and let f :X ! Y be any p-compact group homomorphism. If f|Np(T ) is trivial, then f is trivial. Proof.Since X=Np(T ) has Euler characteristic prime to p, any homomorphism Z=pn ! X factors through Np(T ) by [18, Lemma 3.13] or [8, Proposition 2.14]. As f|Np(T ) is trivial, this shows that f|Z=pn is trivial and hence f itself is tr* *ivial by Theorem 5.1. __|_| Corollary 5.7.Let X be a connected p-compact group and T ! X a maximal torus. Then the following are equivalent (1) f is trivial (2) f|T_is trivial (3) H*Qp(Bf) = 0 for any homomorphism f :X ! Y . RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS 27 Proof.Let i: T ! X and j :S ! Y be maximal tori and choose [Theorem 2.4] a homomorphism ': T ! S such that f O i and j O ' are conjugate. Let '*: U ! V be the induced linear map of U := H2Qp(BS) into V := H2Qp(BT ). As in the proof of Theorem 2.5, '*(U) is_a_WT(X)-submodule_and '*(S[U]WS(Y )) = S['*(U)]WT(X). This equation shows that H *Qp(B') O H*Qp(Bj) = 0 if and only if __* H Qp(B') = 0 from which the equivalence of (2) and (3) follows. Assuming (2), let :Z=pn ! X be any homomorphism from a cyclic p-group into X. Choosing a homomorphism i :Z=pn ! T with i O i conjugate to shows that Bf O B ' Bf O Bi O Bi ' *. Hence f is trivial by Theorem 5.1. __|_| Together with the Sullivan conjecture for p-compact groups [8, Theorem 9.3], corollaries 5.6-5.7 represent a generalization of [12, Theorem 3.11]. References 1.J.F. 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(to appear* *). 17._____, Spaces of sections of Eilenberg-MacLane fibrations, Pacific J. Math. * *130 (1987), 171-186. 18.J.M. Moller and D. Notbohm, Centers and finite coverings of finite loop spac* *es, Preprint. 19.D. Notbohm, Maps between classifying spaces and applications, J. Pure Appl. * *Algebra (to appear). 20._____, On the functor `Classifying Space' for compact Lie groups, Preprint, * *1991. 21.Z. Wojtkowiak, Maps between p-completed classifying spaces II, Proc. Royal S* *oc. Edinburgh 118A (1991), 133-137. Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn O E-mail address: moller@math.ku.dk