CG be the product of all simple summands whose Weyl group has order a multiple of p. Let R Hom (; Z) be the set of roots of G, and let * *R

R be the subset of roots in G

. For each n 1, let GnC G be the product of all normal subgroups isomorphic to SO(2n + 1). Then, for any admissible epimorphism OE : T1 i T1 , OEW 2Im () if and only if the following two conditions are sati* *sfied (a) Ker(OE) \ G

= 1 and (b) (if p = 2) OE(Gn \ T1 ) Gn \ T1 8n 1; if and only if ! := L(OE)|^p2End (^p) satisfies the two conditions (i) !*(R) ^Zp. R and (ii) !*(R

) (^Zp)* . R

. Proof. We will prove the implications (OEW 2 Im()) =Step=1==)(a,b)Step=2===)(i,ii)Step=3===)(!W 2 Im()): Step 1 Assume that OEW = (f); i. e., f : BG^p-! BG^pis such that f|BT ^p'BOE. We will show that OE satisfies conditions (a) and (b) above. Write G0 = G

for short, set G00= G=G0, and let ff : G0 -! G and fi : G -! * *G00 denote the induced maps. Let T 0= T \G0 and T 00= T=T 0be their maximal tori, a* *nd set OE0= OE|T10 : T10 -! T10. The composite BG0^p-Bff---!BG^p--f--! BG^p--Bfi--!BG00^p is null homotopic, since it is trivial in rational cohomology (cf. [JMO, Theore* *m 3.11]). Hence fO Bff pulls back along the fibration BG0^p-! BG^p-! BG00^p(cf. [BK, VI.6* *.5]) to a map f0 : BG0^p-!BG0^p. By Proposition 1.3, f0 is a homotopy equivalence, a* *nd so Ker (OE0) = Ker(OE)\ G

= 1 by Proposition 1.2(iii). This proves point (a). Now assume p = 2; we must prove condition (b). We have just seen that f rest* *ricts to a self map (and homotopy equivalence) of G<2>. So we can assume that G = G<2* *>; i. e., that G is semisimple and OE2 Aut (T1 ). By Lemma 3.3, OE permutes the simple summands of G. Fix some SO(2n + 1)~=HC * *G, and let H0C G be the simple summand such that OE(T1 \H) = T1 \H0. Since Z(H) = * *1, H must be a direct factor of G (i. e., not just up to a finite covering). Hence* * T1 \H0 is a direct factor of T1 , and so BH0^2is a direct factor of BG^2(i. e., H0 is a d* *irect factor of G up to odd degree covering). The composite BH^2--incl---!BG^2---f--! BG^2--proj---!BH0^2 is a homotopy equivalence; so ss2(BH0^2) ~=ss2(BSO(2n + 1)^2) ~=Z=2. Also, by L* *emma 3.3, H0 has the same Dynkin diagram as H, except possibly for the direction of * *the arrows; and hence (since ss1(H0) 6= 1) must be isomorphic to one of the groups * *SO(2n+1) or P Sp(n). These two are isomorphic if n 2; while if n 3 then ss5(BSO(2n + 1)) ~=ss5(BO) ~=0 and ss5(BP Sp(n)) ~=ss5(BSp) ~=Z=2 19 (cf. [Ml, p.142]). Thus, H0~=H~=SO(2n+1). Alternatively, this last point can be* * proven by showing that any admissible homomorphism between the (2-adic) integral latti* *ces of SO(2n + 1) and P Sp(n) (for n 3) must (after composing with a Weyl group eleme* *nt) be a scalar multiple of the map used in Example 3.1 _ and hence is not an isomo* *rphism. We have now shown that (for any n 1), OE permutes those simple factors isom* *orphic to SO(2n + 1) among themselves. This proves condition (b). Step 2 Let Rs R be the simple roots with respect to some Weyl chamber. Then R

\Rs is a ^Qp-basis for L(^p

) (cf. [Ad, Proposition 5.33]). By Lemma 3.3, there is a permutation oe 2 (R) such that for each 2 R, !*() = a . oe for some a 2 (Q^p)*. We may assume (by Lemma 3.3 again) that oe(Rs) = * *Rs. For each 2 R, let v 2 be the element defined in Proposition 3.2(v,vi): v ? Ke* *r () and (v ) = 2. Since ! preserves angles (and in particular orthogonality), we se* *e that !(voe) = a .v for each . Assume now that conditions (a) and (b) hold. If p is odd, then by Propositio* *n 3.2(vi), (Q^p.v ) \ ^p= ^Zp.v for each 2 R. Hence, since !(voe) = a .v and !() , we g* *et a 2 ^Zpfor each , and so !*(R) ^Zp.R. If p = 2, then let R0 R be the set of those roots such that 12v 2 . By 3* *.2(vi) again, the elements 2 R0 are precisely the short roots of summands SO(2n + 1)C * *G; i. e., the short roots in the Gn (for n 1). Also, for each i, OE(Gn\ T1 ) = * *Gn\ T1 by condition (b), and OE|Gn\ T1 is injective by (a). Thus, ! restricts to an a* *dmissible automorphism of the 2-adic integral lattice of Gn, which permutes the simple fa* *ctors by Lemma 3.3. Also, the only admissible automorphisms of the 2-adic integral la* *ttice of SO(2n + 1) are given by multiplication by scalars a 2 (^Z2)*, and so we can * *conclude that oe(R0) = R0. The same argument as for odd p now shows that a 2 ^Z2for all* * ; and so condition (i) also holds in this case. Finally, as noted above, the elements of Rs\ R

form a ^Qp-basis for L(^p

). Hence by (a), Y p - det(! | L(G

\ T1 )) = a ; (* *2) 2Rs\R

and so p-a for 2 R

= W . (Rs\ R

). And this proves condition (ii). Step 3a Assume that G = Sx H, where S is a torus and H is a semisimple Lie g* *roup with trivial center. We show here that for such G, conditions (i) and (ii) suff* *ice to imply that !.W 2Im (). Write G = Sx H1 x : :x:Hm , where the Hm are simple. Let = 0 x 1 x . .x.m and W = W1 x : :x:Wm be the corresponding decompositions of and W . By Lemma 3.3, there is some o 2 m such that !(i^p) = (oi)^pfor all 1 i m (and !(0) = * *0); and Hi and Hoi have the same Dynkin diagram (up to arrow reversal) for each i. * *Also, since !(R

) (^Zp)*.R

(by condition (ii)), the arrow on a double connector * *can be reversed only if p 6= 2, and the arrow on a triple connector can be reversed on* *ly if p 6= 3. Recall that Z(Hi) = 1 for all i 1. Thus, for each i, either Hi and Hoi are isomorphic, or one of them is isomorphic to SO(2n + 1) and the other to P Sp(n)* * for some n 3. And by the remark on reversing arrows in the Dynkin diagram, this la* *st case can occur only if p 6= 2. By a result of Friedlander [Fr], BSO(2n+1)^p'BSp* *(n)^pfor 20 any n and any odd p. So we can compose ! by h(Bff) for some appropriate ff2 Aut* * (G), to arrange that ! sends each simple factor to itself. Q m We can now write ! = i=0 !i, where !i : i^pae i^pfor each i; and where OE0 2 ([BS^p; BS^p]Q ) by Theorem 1.1. We will be done upon showing that !i 2 ([BHi^p; BHi^p]Q ) for each i. In particular, we can simplify the notation, and* * assume that G = Hi is simple. We have seen that !* permutes the roots and simple roots of G up to scalar m* *ulti- ple, and hence induces an automorphism of the Dynkin diagram of G; possibly rev* *ersing arrows. The only simple groups whose Dynkin diagrams have arrow reversing autom* *or- phisms are B2 (= SO(5)~=P Sp(2)), G2, and F4. Also, as noted above, such arrow reversing can occur only if p 6= 2 and G~=B2 or F4; or if p 6= 3 and Hi~=G2. I* *n all of these cases, self maps BG^p-! BG^phave been constructed by Friedlander (in [Fr]* * again), to realize the arrow reversing automorphisms. So if necessary we can compose wi* *th one of these maps, to arrange that ! acts on the Dynkin diagram preserving arrows. Since Z(G) = 1, any arrow preserving automorphism of the Dynkin diagram can be realized by some automorphism ff2 Aut (G) (cf. [Bb2, p.42, Corollaire]). S* *o upon replacing ! by L(ff|T ) O ! for some ff, we are reduced to the case where !* ac* *ts on the Dynkin diagrams via the identity, and sends each root to some scalar multiple o* *f itself. In particular, since the Weyl group W is generated by reflections in the kernel* *s of the roots (3.2(i)), !2 End () is W -equivariant; and is multiplication by some k 2 * *^Zpsince L(T1 ) = ^Qp is irreducible as a W -representation (cf. [Bb1, p.82]). Also, by* * (ii), k 2 (^Zp)* if p|||W |. For such k, unstable Adams operations k : BG^p-! BG^p, have been constructed by Sullivan [Su] (when G = SU(n)) or Wilkerson [Wi] (in general). And since the restriction of k to BT ^pis induced by the k-th power map on T , we see that (* * k) = !. Step 3b Now let G be arbitrary, and fix some admissible map !2 End (^p) such that !*(R) (^Zp.R) and !*(R

) ((^Zp)*.R

). We will show that ! extends to* * a Q-equivalence BG^p-! BG^p. Let n be the_exponent of the center of the semisimple part of G, and let ss * *= {z2 Z(G) : zn_= 1}. Set G = G=ss: a quotient_group which satisfies the condition in Step 2* *. Let be the integral lattice in G ; then __ = {x 2 Q : R(x) Z; nx 2 }: __ by Proposition 3.2(iv). For any x 2 ^p, n . !x = !(nx) 2 ^pand R(!x) = (!*R)(x) (^Zp. R)(x) ^Zp __ __ __ __ __ (using (i)). So !( ^p) ^p; and ! extends to a map f : BG ^p-! BG ^pby Step 2. __ Identify ^p=^p~=ssp (the Sylow p-subgroup of ss), and let !02 Aut(ssp) be t* *he map induced by !. The composite __ _f __ BG^p----! BG ^p----! BG ^p----! K(ssp; 2) 21 is nullhomotopic, since H2(BG; ssp)~=Hom (ss1(G); ssp)Hom(ss1(T ); ssp) (Prop. 3.2(i* *ii)) ~= Hom (; ssp) ~=Hom (^p; ssp): __ __ So f pulls back along the fibration BG^p-! BG ^p-! K(ssp; 2) to a map f : BG^p-* *! BG^p; and f extends the original admissible map BOE. An inspection of the proof of Theorem 3.4 shows, at least when G is semisimp* *le with trivial center, that [BG^p; BG^p]Q is generated by products of unstable Adams o* *perations on the separate simple factors of G, by automorphisms of G, and by the "excepti* *onal isogenies" of Friedlander. This is the generalization to connected groups of th* *e theorem of Hubbuck [Hu], which says that for simple G, [BG,BG] is generated by automorp* *hisms and unstable Adams operations. The following description of the self homotopy equivalences of BG^pis now ea* *sy. Corollary 3.5. If p is odd, then for any compact connected Lie group G, any adm* *issible map !2 Aut (^p) extends to a homotopy equivalence f : BG^p-! BG^p. In other wor* *ds, h : [BG^p; BG^p]h ----! NAut(T1 )(W )=W ~=NAut(^p)(W )=W is an isomorphism of groups in this case. If p = 2, then h is onto if and only* * if G contains no direct factor of the form Sp(n)x SO(2n+1) (for some n 1). And if G* * does contain such a factor, then Im (h) is the subgroup of all elements which send f* *actors SO(2n + 1) to factors of the same type. Proof. Recall that [BG^p; BG^p]h = -1 NAut(T1 )(W )=W (Proposition 1.4). Us* *ing this, Theorem 3.4 implies that h is onto (and hence an isomorphism) if p is odd* *, or if p = 2 and G has no direct factor SO(2n + 1). If p = 2 and G does contain a factor SO(2n + 1), then it can only be sent to* * another direct factor which is either isomorphic_to SO(2n + 1), or which has the same i* *ntegral lattice (2-adically) and root system R (restricted to this summand). And a chec* *k of the root systems shows that the only other possibility is for it to be sent to a di* *rect factor Sp(n). Thus, if h is not onto, then G must contain a direct factor SO(2n + 1)x * *Sp(n); and Im(h) is the group of all admissible maps which send factors SO(2n+1) to fa* *ctors of the same type. Finally, Example 3.1 shows that h is never onto when G has a direct factor S* *O(2n+ 1)x Sp(n). Using Sullivan's arithmetic pullback square for completions and localization* *s of sim- ply connected spaces, these results can now be converted to results about globa* *l self maps of BG. Theorem 3.6. There is a monomorphism : [BG; BG]Q ae AdmEpi (T; T ) ~=[NAut(Q) (W ) \ End()]=W 22 such that for any Q-equivalence f : BG -! BG, (f) = OEW for some OE : T i T with f|BT 'BOE. For each prime p, let G

CG be the product of all simple summands whose Weyl group has order a multiple of p, and set

= \ L(G

\T ). For each n 1, let HnC G be the product of all normal subgroups isomorphic to SO(2n + 1). Then, f* *or any OE2 AdmEpi (T; T ), OEW 2Im () if and only if (a) Ker(OE) \ G

= 1 for all p|||W |, and (b) OE(Gn \ T ) = Gn \ T for all n 1. Proof. For any f : BG -! BG, f|BT ' BOE for some OE : T -! T by Notbohm's theor* *em [No1]; and ! = L(OE)| satisfies conditions (a,b) since it satisfies them after * *p-completion for each p (Theorem 3.4). Thus, there is a well defined homomorphism as above,* * and OEW 2Im () only if OE satisfies the three given conditions. If (f) = (f0), then (f) = (f0) for each p, and so f^p'f0^pfor each p by the injectivity of (Theorem 2.5). And by [JMO, Theorem 3.1], this implies that f' * *f0. Now fix some admissible map OE : T i T which satisfies conditions (a) and (b* *). For each prime p, Theorem 3.4 applies to show that BOE extends to a Q-equivalence f* *p : BG^p-! BG^p. And then by [JMO, Theorem 3.1] (applied with fT = B(inclOOE)), the* *re exists f : BG -! BG such that f^p'fp for each p, and hence such that f|BT 'BOE. Theorem 3.1 in [JMO] was used here to show both uniqueness and existence of * *maps f : BG -! BG. Its proof is based on the homotopy pullback square of mapping spa* *ces Q map (BG; BG) ----! pmap (BG; BG^p) ?? ? y ?y Q map (BG; BGQ ) ----! map (BG; ( p BG^p)Q ); which is induced by Sullivan'sQarithmetic pullback square for BG [BK, VI.8.1]. * *It also uses the fact that BGQ and ( p BG^p)Q are both products of Eilenberg-Maclane s* *paces. As a final application of these results, we get the following (disappointing* *) result about the global self homotopy equivalences of BG. Corollary 3.7. For any compact connected Lie group G, any homotopy equivalence f : BG -! BG is homotopic to Bff for some ff2 Aut (G). Proof. Assume f|BT 'BOE, where OE2 AdmEpi (T; T ). Then OE2 Aut (T ), since f* * is a homotopy equivalence. Set ! = L(OE)|2 Aut (). By Theorem 3.4, !* sends each root of G to an integral multiple of some other root; and since the simple roots are* * linearly independent those integers must be 1. In other words, !* permutes the roots; an* *d so ! is an automorphism of the root system with integral lattice. It follows that OE* * = ff|T for some ff2 Aut (G) (cf. [Bb2, p.41, Prop. 17]); and f' Bff since is injective (T* *heorem 3.6). 23 References [Ad] J. F. Adams, Lectures on Lie groups, Benjamin (1969) [AM] J. F. Adams and Z. Mahmud, Maps between classifying spaces, Inventiones* * math. 35_(1976), 1-41 [Br] A. Borel, Topics in the homology theory of fiber bundles, Lecture Notes* * in Math. 36_, Springer-Verlag (1967) [Bt] R. Bott, On torsion in Lie groups, Proc. Nat. Acad. Sci. 40_(1954), 586* *-588 [Bb1] N. Bourbaki, Groupes et algebres de Lie, Chapitres 4-6, Hermann (1968) [Bb2] N. Bourbaki, Groupes et algebres de Lie, Chapitre 9, Hermann (1982) [Bf] A. Bousfield, Homotopy spectral sequences and obstructions, Israel J. M* *ath. 66_ (1989), 54-104 [BK] A. Bousfield and D. Kan, Homotopy limits, completions and localizations* *, Lecture Notes in Math. 304_, Springer-Verlag (1972) [DW] W. Dwyer and C. Wilkerson, A new finite loop space at the prime two, Jo* *urnal A.M.S. 6_(1993), 37-64 [DW2] W. Dwyer and C. Wilkerson, The center of a p-compact group (preprint) [DZ] W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic t* *opology, Barcelona, 1976, Lecture Notes in Math. 1298_, Springer-Verlag (1987), * *106-119 [Fe] M. Feshbach, The Segal conjecture for compact Lie groups, Topology 26_(* *1987), 1-20 [Fr] E. Friedlander, Exceptional isogenies and the classifying spaces of sim* *ple Lie groups, Annals of Math. 101_(1975), 510-520 [Hu] J. Hubbuck, Homotopy representations of Lie groups, New developments in* * topology, London Math. Soc. Lecture Notes 11_, Cambridge Univ. Press (1974), 33-41 [Is] K. Ishiguro, Unstable Adams operations on classifying spaces, Math. Pro* *c. Camb. Phil. Soc. 102_(1987), 71-75 [JMO] S. Jackowski, J. McClure, and B. Oliver, Homotopy classification of sel* *f-maps of BG via G-action, Annals of Math. 135_(1992), 183-270 [Ml] J. Milnor, Morse theory, Princeton Univ. Press (1969) [Ms] G. Mislin, The homotopy classification of self-maps of infinite quatern* *ionic projective space, Quarterly J. Math. Oxford 38_(1987), 245-257 [MZ] D. Montgomery & L. Zippin, Topological transformation groups, Interscie* *nce (1955) [Mo] J. M. Moller, The normalizer of the Weyl group, Math. Ann. 294_(1992), * *59-80 [No1] D. Notbohm, Maps between classifying spaces, Math. Z. 207_(1991), 153-1* *68 [No2] D. Notbohm, Maps between classifying spaces and applications (G"ottinge* *n preprint, 1991) [Su] D. Sullivan, Geometric topology, Part I: Localization, periodicity and * *Galois symme- try, Mimeographed notes, M.I.T. (1970) [Wi] C. W. Wilkerson, Self-maps of classifying spaces, Localization in group* * theory and 24 homotopy theory, Lecture Notes in Math. 418_, Springer-Verlag (1974), 150* *-157 [Wo] Z. Wojtkowiak, On maps from holim F to Z, Algebraic topology, Barcelona, * *1986, Lecture Notes in Math. 1298_, Springer-Verlag (1987), 227-236 25