p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS William G. Dwyer and Clarence W. Wilkerson University of Notre Dame Purdue University x0. Introduction In a recent series of papers, the authors have proposed a homotopy theoretic generalization of the notion of compact Lie groups to "p-compact" groups, and have shown that much of the discrete algebraic data for Lie groups, such as Weyl groups, can be developed in parallel for these "p-compact" groups. Since the We* *yl group is a key ingredient in the classification of compact connected Lie groups* *, it's natural to conjecture that it should play such a role for "p-compact " groups a* *lso. This paper explores the classification in the simplest case, where the Weyl * *group is generated by commuting reflections. The answer (appearing in 0:5A and 0:5B below) is that up to finite central quotients, the "p-compact" groups with abel* *ian Weyl groups are products of the simply connected rank one examples. For odd primes, the rank one examples are just the "Sullivan" spheres, [25] and these h* *ave trivial centers, so the classification at odd primes reduces to earlier joint w* *ork of the authors with H. R. Miller, [8]. As is often the case for Lie groups, the prime 2 plays a special role in our* * analy- sis. In the work with Miller, [8], for odd primes p, cohomological methods base* *d on Lannes' T -functor were used to prove strong uniqueness results for certain cla* *ssify- ing spaces, characterizing them by the isomorphism type of the mod-p cohomology algebra as an algebra over the Steenrod algebra. The key element of that proof was the construction of a function space analogue of the maximal torus. The fun* *c- tion space calculation generalized the observation that in a connected compact * *Lie group G, the centralizer CG (V ) of the elements V of order p in the maximal to* *rus T , for p an odd prime, is just the maximal torus T . But this may fail for p =* * 2_ for example, in G = SU(2), V is the center of SU(2), so its centralizer is all* * of SU(2). A related failure is that even for a Lie group G such that H*(G; Z) has * *no 2-torsion, H*(BG; F2) may fail to all of H*(BT; F2)W(G) , in contrast to the odd prime case. It turns out however, that this problem with SU(2) is essentially the entire* * extent of the failure. Our first result has a relatively simple proof in the case of L* *ie groups. A result similar to 0.1.2 appears in Broto-Henn, [5]: ______________ Both authors were supported in part by the National Science Foundation. Typeset by AM S-T* *EX 1 2 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Theorem 0.1. Let G be a compact connected Lie group. (1) If for an odd prime p, each element of p in G is central, then G is a t* *orus. (2) If each element of order 2 is central in G, then G is a product of a to* *rus by products of SU(2) (quaternionic tori). (3) For an odd prime p, the mod p vector space V of elements of order p in* * a maximal torus T of G has CG (V ) = T . (4) For p = 2 the mod 2 vector space V of elements of order 2 in a maximal torus T , the identity component of CG (V ) is a product of a torus by a quaternionic torus. If H*(G; Z) has no p-torsion, then H*(BG; Fp) H*(BCG (V ); Fp)WG (V ); where WG (V ) = NG (V )=CG (V ). For an odd prime p, CG (V ) = T and WG (V ) = W (G). For p = 2 there is a short exact sequence 1 ! E ! W (G) ! WG (V ) ! 1; where E is an elementary abelian 2-group. E can be identified with the Weyl group of CG (V ). Our goal is to prove that with suitable analogues of the Lie structures used above, Theorem 0.1 holds for the class of connected p-compact groups. Recall from [13] that a p-compact group consists of a pair of spaces (BX; X) together with an homotopy equivalence g : BX ! X such that BX is Fp-complete in the sense of Bousfield-Kan, [3], that H*(X; Fp) is finite, and that the component g* *roup of X is a finite p-group. For such spaces [13] show that X has precise analogues of the usual homomorphisms, maximal torus, Weyl group, and centralizers. For example, the analogue of the centralizer is the loop space on the mapping space Map(BV; BX)Bf . It is a main result of [13] and [10] that these centralizers in* *herit the p-compact group properties. The centralizers of the elements of order p in the torus have special cohomo- logical properties - in the p-torsion free homology case, H*(BCG (V ); Fp) is a sub-Hopf-algebra of H*(BT; Fp),[8] and [27]. For odd primes, Hopf invariant one considerations force (even in the p-compact group case) the centralizer to be t* *he torus. However, for p = 2, these considerations do not force in general the cen* *tral- izer to be a torus. In fact, one original motivation for this work is the foll* *owing characterization of these centralizers: Theorem 0.2. Let (BX; X) be a 2-compact group such that X is connected. Then BX is homotopy equivalent to the the 2-completion of a product of BU(1)'s and BSU(2)'s under either of the following conditions: (1) H*(BX; F2) is a finitely generated polynomial algebra with a Hopf algeb* *ra structure compatible with the action of A2 or (2) each element of order 2 in X is central. In the proof of 0.2, we use one fast corollary of our work on maximal tori a* *nd Weyl groups in [13]: p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 3 Proposition 0.3. If BX is a connected 2-complete space such that H*(BX; F2) is a finitely generated polynomial algebra, and the cohomology is concentrated * *in dimensions divisible by 8, then BX is contractible. Proposition 0.3 also appears in [20] as an application of the [15] extension* * of mod 2 Hopf invariant one type results. The second part of Theorem 0.1 is known for p-compact groups at an odd prime, [8]. The point at the the prime 2 is to identify the the homotopy type o* *f the "centralizer" and to show that it accounts for the "inseparable" part of the no* *rmal extension given by [27]. Corollary 0.4. Let (BX; X) be a 2-compact group such that X is connected and H*(BX; F2) is concentrated in dimensions divisible by 2. Then there is a map f : BG = (BSU(2))r1 x (BU(1))r2 ! BX such that the homotopy fiber of f has finite mod 2 cohomology with a non-zero Euler characteristic. We also have H*(BX; F2) H*(BG; F2)W(X) ; but the action of W (X) need not be faithful on the mod 2 cohomology of BG. If the cohomology is concentrated in dimensions divisible by 4, then r2 = 0, and X has a maximal rank subgroup isomorphic to a product of completed SU(2)'s.That is, X has a quaternionic maximal torus. In this case, the mod 2 reduction of W * *(X), WX (V ), may be considered as a subgroup of ss0Aut((F2 )1 BSU(2)r). This group of components is isomorphic to the symmetric group r wreath product with the subgroup of squares in the 2-adic integer units. However, since WX (V ) is fini* *te, it is conjugate to a subgroup of the r subgroup. The existence of maximal tori and Weyl groups for p-compact groups yields us some primary data for classification. For example, for odd primes, the work of * *[8] easily classifies all connected p-compact groups with abelian Weyl groups, beca* *use these must be generated by reflections of order prime to p, and hence the order* * of the Weyl group is prime to p. Theorem 0.5A. Let (BX; X) be a p-compact group for an odd prime p such that X is connected. Then the following are equivalent: (1) W (X) is abelian. (2) W (X) is a direct product of cyclic groups Z=rZ, where r is a divisor of (p - 1). (3) BX is homotopy equivalent to a product of classifying spaces of "Sulliv* *an" spheres, BS2r-1 . (4) H*(BX; Fp) is isomorphic over the Steenrod algebra to a tensor product of monogenic polynomial algebras, each closed under the Steenrod algebra action. That is, the rank one connected p-compact groups for odd primes are the Sull* *i- van spheres and these have trivial centers. However, for p = 2, the classificat* *ion of connected 2-compact groups with abelian Weyl groups is more difficult. Somewhat surprisingly, we achive the same classification as in the Lie case: 4 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Theorem 0.5B. Let (BX; X) be a 2-compact group such that X is connected. Then the following are equivalent (1) W (X) is abelian. (2) W (X) is an elementary abelian 2-group. (3) H*(BX; Z2) Q is a finitely generated polynomial algebra on generators of dimensions 2 or 4. (4) H*(X; F2) is isomorphic over the Steenrod algebra to the cohomology of a product of SU(2)'s, SO(3)'s, and U(1). (5) there exists a Lie group GX isomorphic to a product of U(1)'s and SU(2* *)'s with a central elementary abelian 2-subgroup EX , such that BX B(GX =EX ): The cohomology rings of such BX need not be polynomial. The smallest non- polynomial example occurs as the quotient of a product of three SU(2)'s divided by certain rank 2 subgroup of its center. In the cases that the cohomology is polynomial, the 2-complete homotopy type of the spaces BX is determined u- niquely by the A2 -structure on the mod 2 cohomology. The example of BSO(4) versus BSU(2) x BSO(3) shows that the Steenrod algebra action as well as the abstract ring structure is needed. We show this uniqueness here only for the t* *he "semi-simple" case. Corollary 0.6. Let BY is a simply connected 2-complete space with H*(BY; F2) a finitely generated polynomial algebra and ss1(Y ) finite. Then the following* * are equivalent: (1) W (Y ) is abelian (and hence an elementary 2-group). (2) the indecomposabbles of H*BY are concentrated in dimensions 2, 3, and 4. (3) BY is homotopy equivalent to the 2-completions of product of spaces BH(k), where H(k) (SU(2)k=(Z=2Z): equivalent to BY . Note. Interesting examples of the groups H(k) of Corollary 0.6 arise in the ana* *lysis of centralizers in low rank 2-compact groups. For the Lie group G2, the central* *izer of any element of order 2 is SO(4) which is the product SU(2) x SU(2) modulo the diagonal central Z=2Z subgroup. For the example DI(4) constructed in [12], the centralizer of any rank 2 elementary abelian subgroup is SU(2) x SU(2) x SU(2) modulo the diagonal central Z=2Z subgroup (up to completions). Theorems 0.5X are suggested prototypes of a general classification of connec* *t- ed p-compact groups. The representation of the Weyl group on BT determines potential "simple" factors of X and then one works to decompose X (up to finite covers) as products of these simple factors. The methods in this paper are dire* *ctly cohomologically based. It's clear that this approach has its limits. Work in pr* *ogress of the authors use the results of [13] and [14] in a more subtle way and we bel* *ieve it will lead to a general theory of semi-simplicity for connected p-compact gro* *ups. In the proof of Theorem 0.2, one ingredient needed beyond the cohomological tools of [8] is information about certain fibrations involving classifying spac* *es. We p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 5 can invoke the powerful results of [17] and its generalization in [14] here, al* *though the particular results applied in this note are already proved in [12]: Theorem 0.8. Let G be a connected compact Lie group. Then the monoid of self equivalences of the 2-completion of BG, AutId((F2 )1 BG) which are homo- topic to the identity is homotopy equivalent to the 2-completion of BC(G) and BAutId((F2 )1 BG) B2C(G): This amounts to the assertion that the universal fibration (F2 )1 BG ! BMap*((F2 )1 BG; (F2 )1 BG)Id ! BMap((F2 )1 BG; (F2 )1 BG)Id is fiber homotopy equivalent to the 2-completion of the fibration BG ! BGadjoint! B2C(G): Corollary 0.9. Let G be a compact connected Lie group with a finite center and BG ! E ! B a fibration in which ss1(B; *) acts trivially on BG. Let p be a prime. Then the p-completion of the fibration is f.h.e. to the product fibratio* *n if H2(B; C(G)) Fp is zero. In particular, if the center of G is trivial, the comp* *leted fibration is trivial. With suitable definitions of the center of p-compact group, both 0.8 and 0.9* * are valid for p-compact groups also. These fibration theorems show that properties * *of centers and ideas of semisimplicity for Lie groups are strongly related. Moreo* *ver these concepts are mirrored in the homotopy theoretic properties of classifying spaces. Establishing such properties for p-compact groups is one goal for our future work. Method of proof. : The method of proof of 0.2 is to pass to the 3-connected cov* *er of BX. It's shown that this cover inherits the cohomological coproduct, and hen* *ce has polynomial generators in dimensions 2N for N > 1. We use the coproduct to show that the Weyl group is an elementary abelian 2-group and hence the rational cohomology of BX is generated by dimension 4 and 2. Hence the 3-connected cover of BX is a "fake" product of BSU(2)'s. Using homotopical analogues of the center, we construct a quotient space BY which is a "fake" product of BSO(3)'s. We then apply the diagram methods of [16] and [9] to inductively prove that BY * *is a product of 2-completed BSO(3)'s. Then we analyse the various fibrations that were used to reduce to the 3-connected case. The authors would like to thank D. Benson, J. Harper, H.W. Henn, H. Miller, and R. Oliver for helpful discussions. Notation.. All cohomology is with F2 coefficients unless otherwise stated, and * *will be denoted by H*X. The 2-adic integers are denoted by Z2 and the usual rational numbers by Q. The 2-completion used is the Bousfield-Kan completion and is de- noted by (F2 )1 X. The mod p Steenrod algebra is denoted as Ap . The n-connected cover of the space X is denoted X and the n-th Postnikov approximation to X is denoted by X[n]. The category of unstable A2 -modules is U and that of unstable algebras over A2 is K. 6 WILLIAM G. DWYER AND CLARENCE W. WILKERSON x1. Central Quotients of U(1)r x SU(2)s In this section we present a sketch of Theorem 0.1 for Lie groups (proofs are deferred to the end of the section): Lemma 1.1. Let G be a connected compact Lie group in which each element of order 2 is central. Then the Weyl group of G is an elementary abelian 2-group a* *nd the action of W (G) on the Lie algebra of the maximal torus is the product acti* *on. The next lemma is a special case of the fact that each connected compact Lie group has a finite cover is the product of a torus and a simply connected compa* *ct Lie group: Lemma 1.2. Let G be a connected compact Lie group in which the Weyl group is an elementary abelian 2-group. Then G has a finite covering which is isomorphic as Lie groups to a product of U(1)'s and SU(2)'s. Lemma 1.3. Let G be the product of T = U(1)r and H = SU(2)s. Let C be a finite 2-group which is a subgroup of the center of G. (1) Every element of order 2 is central in G=C if and only if the projectio* *n of C into H is trivial. (2) ss1(G=C) is torsion-free if and only if the intersection of C with H is* * trivial. Before proving Theorem 0.1, we note a few homological properties of the quo- tients studied in this section. These properties will be useful for modeling th* *e finite loop space analogues of Theorem 0.1: Proposition 1.4. Let Ge be the product of T = U(1)r and H = SU(2)s. Let G = eG=C, where C is finite and central in eG. Then (1) G is isomorphic to a quotient in which C is an elementary 2-group. In fact C can be replaced by C=(C2). Here C2 denotes the subgroup of C consisting of squares of elements of C. (2) G is homeomorphic as a topological space to a product of U(1)'s, SU(2)'* *s, and SO(3)'s. The number of SO(3) factors is the 2-rank of C \ (1 x H). (3) If H*BG is a finitely generated polynomial algebra, its indecomposables* * are concentrated in dimensions 2, 3, and 4. (4) H*(G; Z) has torsion of order at most 2. (5) H*(BG; Q) is a finitely generated polynomial algebra on generators of d* *i- mensions two and four. In the odd prime case, the Weyl group of G acts faithfully on the mod p co- homology of BT and hence its structure can be inferred entirely from the mod p cohomology of BG. At p = 2 the action on cohomology is not not always faithful, but we can at least recover the Weyl group up to an extension: Theorem 1.5. Let G be a compact connected Lie group. Let T be a maximal torus for G, and V the set of elements of order 2 in T . Then there is a short * *exact sequence of groups 1 ! W (CG (V )) ! W (G) ! NG (V )=CG (V ) ! 1: p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 7 W (CG (V )) = (Z=2Z)e. In terms of cohomological data, WG (V ) = NG (V )=CG (V ) can be identified with the automorphism group of the fraction field extension c* *or- responding to the ring extension H*BG=ker(f* ) ! H*BT: In this setting, e is the inseparability degree of that extension. In cases tha* *t G has a unique (up to conjugacy) maximal elementary abelian 2-group U, WG (V ) is just the subquotient DWG (U)(V )=IWG (U)(V ) where D is the subgroup of WG (U) that transports V into itself and I is the is* *otropy subgroup of V . We remark that Theorem 1.5 has a valid translation to p-compact groups. From 1.5, we deduce another characterization of the Lie groups for which the Weyl gr* *oup is an elementary 2-group: Lemma 1.6. Let G be a compact connected Lie group. Suppose that T is a maximal torus of G and V the subgroup of T of elements of order 2. Then the Weyl group of G is an elementary abelian 2-group if and only if each g in G that normalizes V also centralizes V . Again, 1.6 can be translated to p-compact groups. Proof of Theorem 0.1. : By 1.1 and 1.2 there is a finite covering of G by a gro* *up eG isomorphic to U(1)r x (SU(2))s. Thus G = eG=C for some finite central subgroup C. By 1.3, the projection of C to H = (SU(2))s must be trivial. But in that cas* *e, Ge=C is abstractly isomorphic to eG. For the cohomology statements, the odd pri* *me case is in [8] ( Borel [2] only covers the prime to Weyl group order case). For* * p = 2, the centralizer is question is U(1)s x SU(2)r and by [27] the extension of rings H*BG ! H*BCG (V ) is separable and Galois with Galois group WG (V ). See Quillen,[22], also for * *this result up to F -isomorphism and localization. Proof of 1.1. : Choose a maximal torus T for G. Let V be the group of elements of order 2 in T . If g 2 NG (T ), then conjugation by g takes V to V by the ide* *ntity map. This implies that the mod 2 reduction of the action of W (G) on the integr* *al lattice of L(T ) is trivial. By 9.1, W (G) must be an elementary abelian 2-gro* *up. The action can be diagonalized over Q. Proof of 1.2. : Consider the adjoint form Gadj of G. It is a product of simple centerless compact Lie groups, Gi. The only Weyl group which is an elementary abelian 2-group and which is the Wey group of a simple Lie group is Z=2Z. Hence each factor of the adjoint form is a SO(3), since the rank one simple groups are just SU(2) and SO(3). But SU(2) has a non-trivial center so the only choice is SO(3). General Lie theory, e.g. [26] gives that G has a finite cover by a produ* *ct of a torus and a simply connected simple factors. Since the adjoint form is a prod* *uct of SO(3)'s the corresponding terms in the finite cover must be SU(2)'s. 8 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Proof of 1.3. : The center of G is isomorphic to the product T x (Z=2Z)r2. Let (t; u) be an element in C such that u is not the identity. Denote by p the proj* *ection to G=C, and by padjthe projection to Gadj = Hadj. Choose v in H such that v2 = u and choose s in T such that s2 = t. Then (p(s; v)2 = p(s2; v2) = p(t; u) = Id,* * so p(s; v) has order 2 in G=C. Passing down to Gadj, padj(s; v) is not the identi* *ty, because the kernel of H ! Hadj has no elements of order 4. But the center of Ha* *dj is trivial by construction, so there must be h 2 H such that padj(t; h) does not commute with padj(s; v) . That is, if the projection of C to H is non-trivial, * *G=C has elements of order 2 not in the center. Conversely, if the projection is tr* *ivial, G=C is abstractly isomorphic to G, since T=C is isomorphic to T for C finite. If C contains an non-trivial element of H, then clearly ss1(G=C) has torsion* *. On the other hand, if ss1(G=C) has torsion, it's of exponent 2. Let ff be a class * *of order 2. Let Gffbe the covering group corresonding to ff. Proof of 1.4. : (1) is clear. (3) and (4) follow directly from (2). For (2), th* *e claim is that (T x H)=C is homeomorphic to T x (H=(C \ (1 x H)). For a general compact connected Lie group G, let G0 be the commutator subgroup. Then G0 is connected and G=G0 is a compact connected abelian Lie group, and hence a torus T . The map G ! T has a section, so G is homeomorphic (but not isomorphic) to G0x T . In our example, for G = (T x H)=E, G0 is isomorphic to H=(E \ H), since it is t* *he image of the commutator subgroup in T x H under the quotient map. Proof of 1.5. From the inclusion V ! T are induced inclusions CG (T ) ! CG (V ) and NG (T ) ! NG (V ). For a possibly disconnected compact Lie group H, define W (H) = NH (TH )=TH . Then if H is maximal rank in G, there is an inclusion of Weyl groups W (H) ! W (G). For such H, there is a natural surjection W (H) ! ss0(H). In this case, we apply these facts to H = NG (V ) or CG (V ). W (CG (V )) = (NG (T ) \ CG (V ))=T: So W (CG (V )) is kernel of W (G) ! WG (V ). Also, W (NG (V )) ! W (G) is an in* *clu- sion, but since W (NG (T )) ! ss0(NG (T )) = W (G) is onto, we have W (NG (V ))* * = W (G). Thus W (G) ! ss0(NG (V )) is onto. WG (V ) is a subgroup of Aut(V ), a f* *inite group. The map NG (V ) ! WG (V ) therefore factors through ss0(NG (V )). Putting these together, we have that W (G) ! WG (V ) is a surjection. Thus the short ex* *act sequence 1 ! W (CG (V )) ! W (G) ! WG (V ) ! 1 is established. The cohomological assertions follow from [22] together with [4]. Proof of 1.6. The condition is just that WG (V ) is the trivial group. From the identifications, WG (V ) is the quotient of W (G) by the elements that reduce t* *o the identity mod 2. Hence WG (V ) = {Id} implies that W (G) is an elementary abelian 2-group. On the other hand, if W (G) is an elementary abelian 2-group, then G has been identified as a quotient of Ge = T x H, where T is a torus and H is a product of SU(2)'s. On this covering group eG, W (G) acts trivially on V . p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 9 x2. Finite loop spaces with (Z=2Z)s as Weyl group Recent work of the authors in [13] shows that for p-compact groups, there ar* *e a maximal torus and a Weyl group with most of the properties that the classical L* *ie group constructs possess. The definitions are modeled on those of [23], adapted* * to the p-compact groups. More precisely, let (BX; X be a p-compact group such that X is connected. Then there exists a maximal torus, i.e. a map f : BT ! BX with many of the usual properties that BT ! BG has in the Lie group case. Here BT can mean an Eilenberg-MacLane space of type either K((Zp )r; 2) or K((Z=p1 Z)r; 1). The latter choice we denote as the discrete torus. The discrete form has the advantage that the elements of order a power of p can visualized in the usual w* *ay as elements in a torus. In this environment, W (X) is just the component group * *of Aut(f : BT ! BX), [23], [13]. Proposition 2.1. Let (BX; X) be a 2-compact group such that BX is simply connected and ss*(BX) Q has rank n. Then any one of the following conditions imply that W (X) is an elementary abelian 2-group contained in GL(n; Z2): (1) The "rational" cohomology H*(BX; Z2) Q has a set of algebra generators over Q2 concentrated in dimensions 2 and 4. (2) The mod 2 cohomology algebra H*BX is a polynomial algebra and its indecomposables are concentrated in dimensions 2, 3, and 4. (3) The mod 2 cohomology algebra H*BX has a coproduct consistent with the action of the mod 2 Steenrod algebra. (4) For each g : RP 1 ! BX, the natural evaluation map e : Map(RP 1; BX)g ! BX is a homotopy equivalence. That is, each element of order 2 in X is cen* *tral in X. Consequently the subgroup of elements of order 2 in the maximal torus T is central in X. (5) If f : BT ! BX is the maximal torus, and i : BV ! BT the inclusion of the elements of order 2 in T , then for each w 2 W (X), w O i = i: That is, the Weyl group action restricts to the trivial action on V . We need one key algebraic lemma. Although related to the congruence sub- group problem, the case needed has an elementary proof. The proof is supplied in appendix A. Lemma 2.2. Let W be a finite subgroup of GL(Z2 ; n) Then the kernel of the mod 2 reduction map to GL(F2; n) is a finite elementary abelian 2-subgroup of W . Lemma 2.3. Let W in GL(N; C) be a reflection group with exactly n reflections, all of order 2. If the order of W is 2n, then W is an elementary 2-group. Note that it is not assumed that n = N. 10 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Lemma 2.4. Let W in GL(N; C) be a reflection group with exactly n reflecting hyperplanes. If W is abelian, then W is isomorphic to a product of n cyclic ref* *lection groups. Recall that f : Y ! Z is said to be central if the natural evaluation map ev* *al : Map(Y; Z)f ! Z is a homotopy equivalence. If this holds for Bf : BY ! BX we abbreviate by saying that Y is central in X or that f is central. We note some properties of this idea of centrality that are immediate in the Lie group case. Lemma 2.5. Let (BX; X) be a 2-compact group such that X is connected. Then (1) If f : BAxBB ! BX has the property that f|*xBB and f|BAx* ! BX are each central, then f is central. (2) The subgroup V of order 2 in a maximal torus T for X are central in X i* *f and only if for any Bg : B(Z=2Z) ! BX, we have Map(BZ=2Z; BX)Bg BX. (3) Suppose f :! X is central in X and a and b in V are represented by Bi : BZ=2Z ! BV and Bj : BZ=2Z ! BV respectively. Then if Bf OBi = Bf O Bj up to homotopy, then, a = b as elements of V . Proof of 2.1:. Case (1): By [13], there is a Weyl group W (X) such that H*(BX; Z2) Q (H*(BT; Z2) Q)W(X) : W (X) is a generalized reflection group generated by re- flections of order 2. Classical formulas from [24] give the number of reflectio* *ns in W (X) as X (ni- 2)=2 where {ni} are the dimensions of the generators of the invariant polynomial alg* *ebra. The only roots of unity in Z2 are plus and minus one, so each reflection has or* *der 2. Hence in this case, the dimension 4 generators each contribute 1 to the sum, and the dimension 2 generators none. Hence the number of reflectionsQis N, the number of dimension 4 generators. On the other hand, |W (X)| = (ni=2) = 2N . By Lemma 2.3, W (X) is an elementary abelian 2-group. Case(2): By Proposition 5.1 such a BX has its rational cohomology generated by dimensions 2 and 4, so case (1) applies. Case (3) If H*BX is a Hopf algebra, each component of the Lannes' T -functor T V(H*BX) is isomorphic to the component of the trivial map, [11], which is H*BX, since H*BX is noetherian. By [18] and [13] this implies that for any f : BV ! BX we have H*Map(BV; BX)f TfV(H*BX) H*BX: That is, condition (3) implies condition (4). Case (4) : We show that condition (4) implies condition (5) Let Bf : BT ! BX be the maximal torus, and let Bi : BV ! BT represent the inclusion of the elements of order 2. Then V is central in X if and only if each Z=2Z is centra* *l in X. In one direction, in [13], it's shown that for connected X, h : BZ=2Z ! BX factors though Bf : BT ! BX. Case (4) by Lemma 2.4 implies that V is central in X. So it remains to show that V central in X implies that the Weyl group is p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 11 an elementary abelian 2-group. By the algebraic lemma (2.2), it's enough to show that the Weyl group acts as the identity on V or BV up to homotopy. Suppose that there exists an element of W (X) which fails to act trivially on BV up to homot* *opy, i.e. as the identity on V . Then there exists an element a 2 V and an eleme* *nt w 2 W (X) [6 such w O a = b 6= a. But by the definition of W (X), the inclusion* * into BX induces homotopic maps representing a and b. By 2.5.2, a = b, contradicting the claim that w O a 6= a. Case (5) : Consider the discrete version of BT , B(Z=21 Z)n. Aut(BT ) acts on this. To get the action W (X), use 0 ! ss2(BT ) ! (Q ss2(BT ) ! (Z=21 Z)n ! 0: Then V can be identified with the image of ((1=2)ss2(BT )) under the projectio* *n. But this is also the mod 2 reduction of the action on ss2(BT ). By 2.2, W (X) i* *s an elementary abelian 2-group. Proof of 2.3. : Proof: Since W is finite, there exists a W -invariant hermitian* * inner product for V = CN . For each reflection wi 2 W choose a unit normal vector vi.* * We need to know that wiwj = wjwi for all possible i and j. If n = 1, this is immed* *iate. If n > 1, for each 1 k n, define Vk to be the span of all the unit normal vec* *tors except vk. Now the action of W on V permutes the reflecting hyperplanes, and hence permutes the normal vectors, up to unit complex numbers. In particular, t* *he action of wk on V must take Vk into itself, because wk(vj) = ak;jvi where i 6=* * k unless k = j. By definition, wk has only a single eigenvector for = -1, so its eigenvalues on Vk must be all 1's. Hence wk(vj) = vj if j 6= k. Hence wkwj = wj* *wk for all j. But k was arbitrary, so any two reflections commute. Thus W is an elementary abelian 2-group. Proof of 2.5. : These appear in [14]. 12 WILLIAM G. DWYER AND CLARENCE W. WILKERSON x3. Reduction to the case H*BX H*BSU(2)r Both 0.2 and 0.5B can be reduced to the H*BSU(2)r case by taking the 3-conne* *cted cover. In the polynomial algebra cases, we can do the calculations directly: Proposition 3.1. Let H*BX be a finitely generated polynomial algebra with a coproduct over A2 . Then the 3-connected cover BX <3> has the same property. Proposition 3.2. Let (BX; X) be a 2-compact group such that X is connected. Suppose that H*BX is a polynomial algebra on generators of dimensions 2, 3, and 4. Then H*BX <3> H*BSU(2)r for some r as algebras over A2 . For the BX of 3.1, from section 2 it follows that the Rector-Weyl group for * *BX < 3 > must be an elementary abelian 2-group and that the "rational" cohomology has generators only in dimension 4. Hence H*BX <3> is polynomial on dimension 4 generators. Proposition 3.3. For BX as in Theorem 0.2, H*BX < 3 > H*BSU(2)r as algebras over the Steenrod algebra. If one wants to use only rational information and not explicit mod 2 cohomol* *ogy information, the spirit of 3.2 and 3.3 is still vaild. However, the proof requ* *ires results of Browder on finite H-spaces, [6]. We do not have a specific classify* *ing space argument. Proposition 3.4. Let (BX; X) be a 2-compact group such that X is connected. Then (BX <3>; BX <3>) is a 2-compact group also. Using further results of Browder, we have Proposition 3.5. Let (BX; X) be a 2-compact group such that X is connected and ss*(BX) Q is concentrated in dimensions 2 and 4. Then H*BX <3> H*BSU(2)r Thus all the classifying spaces that we're seeking to characterize have 3-co* *nnected covers which are fake products of completed BSU(2)'s. x4. Construction of the adjoint form for BX <3> In this section we contruct the analogue of the adjoint form of SU(2)r. Usi* *ng the mod 2 cohomology we recognize the analogue of the center and show that the quotient is well defined. Proposition 4.1. Let (BX; X) be a 2-compact group such that BX is 3-connected and H*BX H*BSU(2)r as algebras over the Steenrod algebra. Then (1) There exists f : BV = (BZ=2Z)r ! BX so that f* is a monomorphism and eval : Map(BV; BX)f ! BX is a homotopy equivalence and (2) The action map BV x Map(BV; BX)f ! Map(BV; BX)f defines a space BY = (F2)1 (EBV xBV Map(BV; BX)f) such that H*BY H*BSO(3)r as algebras in all dimensions and as modules over the Steenrod algebra though dimension three. p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 13 In the next section we'll show that property (2) above forces that H*BY H*BSO(3)r as algebras over the Steenrod algebra. Proof of 4.1. : By Lannes [18], the existence of f is determined by the existen* *ce of a suitable f* . By the isomorphism to H*BSU(2)r, such an algebraic map exists. By further work of Lannes, H*Map(BV; BX)f = TfV(H*BX), using the notation of [10]. Using either the fact that the cohomology is a Hopf algebra over the Stee* *nrod algebra, [11] or by comparison to the classical case, this shows that the evalu* *ation map induces a homotopy equivalence eval : Map(BV; BX)f ! BX, since both spaces are 2-complete and the map in cohomology is an isomorphism. Now choose a model for BV such that it is an abelian topological group with the identity e* *lement as its basepoint. The action of BV on BV induces an action on Map(BV; BX) which restricts to an action on Map(BV; BX)f since BV is connected. So this gives us a well defined way to define a BV action on space homotopy equivalent * *to BX, a proxy action in the language of [13]. Hence the Borel construction is we* *ll defined. The fibration sequence (up to homotopy) BY ! BV ! BX as far as algebras over the Steenrod algebras are concerned looks like the prod* *uct of r copies of the fibration SO(3) ! BZ=2Z ! BSU(2) Therefore the Eilenberg-Moore Spectral Sequence for it collapses at E2, and H*BY H*SO(3)r as algebras over the Steenrod algebra. Applying the Borel transgressio* *n, we see that H*BY is a polynomial algebra on generators of dimensions 2 and 3, and that Sq1 is an isomorphism between dimensions 2 and 3. We can not however easily deduce the action of Sq2 on the dimension 3 generators. If only one or t* *wo dimension 3 generators are present, the expected action of Sq2 is almost immedi* *at. However, the higher rank cases lead to an interesting linear algebra problem in characteristic 2, which is solved in section 5. 14 WILLIAM G. DWYER AND CLARENCE W. WILKERSON x5. A Diagonalization Criteria In this section we prove a useful criteria for showing that a matrix with en* *tries in a field of characteristic p is diagonalizable over the field with q elements* *. The criteria is suggested by the information implied by the Steenrod algebra action* * in the case p = 2 in analysing fake products of BSO(3)'s. Let Vq be a vector space of dimension n over some finite field Fq of charact* *eristic p. Suppose that K is an algebraically closed field which contains Fq . We form * *the vector space V over K by tensoring up with K over Fq . This gives a preferred choice of Fq structure for V . It specifies a Frobenius operator on the elemen* *ts of V , OE : V ! V which is Fq linear, by choosing a Fq basis {vi} for Vq and de* *fining OE(v) = ffqivi if v = ffivi for ffi2 K. This does not depend on the particular choice of Fq basis for Vq. A subspace W of V is said to be defined over Fq * * if OE(W ) W . Notice that as a map of abelian groups, OE is one to one and onto. Proposition 5.1. For M 2 EndK (V ), with matrix {mij} with respect to the basis {vi}, there exists B 2 GLn(Fq ) such that BMB-1 is diagonal with entries in K if and only if OE(M) = (M)q where OE(M) is the matrix with entries {mqij} Notice that 5:1 is equivalent to showing that M is semi-simple, with eigensp* *aces defined over Fq . Proof of 5.1. : We must show that V decomposes as an M-module into eige nspaces which are defined over Fq . Suppose that 2 K is an eigenvalue for M, and denote the generalized eigenspace of K-vectors annihilated by some power of (M - ) as V . Now if (M - )N v = 0 then OE((M - )N v) = (Mq - q)N OE(v) = (M - )qN OE(v) = 0 so OE(v) 2 V . That is, OE(V ) V : Thus, without loss of generality, we can assume then that is the only eigenval* *ue for M. Suppose that V is not spanned by eigenvectors. Then there exists N such that (M - )N+1 = 0, but (M - )N v 6= 0 for some v 2 V . Since OE is a monomorphism on V , for such a v 0 6= OE((M - )N v) = (M - )Nq OE(v): This is impossible unless N = 0: That is, every vector in V is an eigenvector. In particular, there is a basis of V of eigenvectors defined over Fq . Corollary 5.2. Let R be the graded polynomial ring over Fq in variables {xi}, where dim(xi) = m for all i. Let M be an n x n matrix with entries which are homogeneous linear forms in the {xi}. If OE(M) = Mq p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 15 then there exists an automorphism B of R such that BMB-1 is diagonal with entries in the Fq -vector space spanned by the {xi}. Proof of 5.2. : Choose an algebraic closure K of Fq , and let {ff : xi ! ffi 2 * *K} be a point. Denote by M(ff) the matrix obtained by evaluating the entries of M at the point ff. Then for each such ff, by 5.1, there exists B(ff) 2 GLn(Fq ) such* * that BffM(ff)B-1ff is diagonal. Choose an ff such that its components are linearly independent ov* *er Fq . Then consider M0 = BffMB-1ff: Any off-diagonal term of M0 is zero when evaulated at ff, by the choice of Bff. Since Bffhas Fq entries, and the terms of M are linear forms, the terms of M0(* *ff) are Fq -linear combinations of the components of ff. By the choice of ff, since M0ij(ff) = 0; i 6= j, each must be the trivial linear combination. That is M0ij* *= 0 2 R for i 6= j. That is, by choosing a sufficently generic ff, we see that BffMB-1ff is diagonal. The diagonal entries are clearly in R and since R is a free commut* *ative algebra, the action of Bffon the vector space Rm extends to an automorphism of the algebra R. 16 WILLIAM G. DWYER AND CLARENCE W. WILKERSON x6. Classification of Unstable Polynomial Algebras with Generators in Dimensions 2,3, and 4 Proposition 6.1. Let R* be a finitely generated unstable algebra which is poly- nomial on generators of dimensions 2,3, or 4. Then (1) Sq1|R3 = 0: (2) Sq1 : R2 ! R3 is a surjection. (3) There is a choice of algebra generators {zi} generators of dimension 4 * *such that Sq1zi = 0 for all i. (4) The Sq1-homology of R* is a finitely generated polynomial algebra on generators in dimensions 2 and 4. Thus far this has been formal work in unstable algebras over A2 . For the to* *po- logical results, we need information on the Bockstein spectral sequence. But if H*BX = R*, we see that the Bockstein spectral sequence collapses after taking the Sq1-homology, since that answer is concentrated in even dimensions. In fact, it's polynomial with representatives for algebra generators {xi} for i > N, {x2* *i} for i N, and the choice of generators {zk} as constructed above with Sq1zk = 0. Thus in the topological case with R* = H*BX, we have (H*(BX; Z2)=torsion) F2 = i>N F2 [xi] jN F2[x2j] k=1;lF2 [zk] and the rational cohomology is generated by classes of dimensions 2 and 4. Corollary 6.2. Let R* be an unstable algebra over A2 which is polynomial as an algebra and generated by elements in degrees 2 and 3. Suppose further that dimF2R2 = N and Sq1 : R2 ! R3 is an isomorphism. Then R* N H*BSO(3) as algebras over A2 . That is, there is a choice of generators {xi} in dimensio* *n 2 so that setting yi = Sq1xi for each i yields Sq2yi = xiyi for each i. In fact 6.2 is a special case of 6.3 , for which more substantial calculatio* *ns are required: Corollary 6.3. Let R* be an unstable algebra over A2 which is polynomial as an algebra and generated by elements in degrees 2, 3, and 4. Suppose further that dimF2R2 = N and Sq1 : R2 ! R3 is an isomorphism. Then R* H*BH(k), for various k, where H(k) is the Lie group SU(2)k=diag(Z=2). H*BH(k) = F2 [x2; y3; z1; : :;:zk-1 ], where dim zi = 4. The A2 action is determined by (1) Sq1x2 = y3 (2) Sq1y3 = 0 (3) Sq2y3 = x2y3 (4) Sq1zi = 0 (5) Sq2zi = x2zi In this labelling scheme, H0 = SU(2), H1 = SO(3), and H2 = SO(4). Proof of 6.1. : In the topological case with R* = H*BX, one can use the methods of [6] to analyse the Bockstein spectral sequence for X to obtain most of this information. However, we give a direct algebraic argument based on a careful analysis of the Steenrod algebra action in low dimensions. Choose an algebra ba* *sis p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 17 {xi} in dimension 2, {yj} in dimension 3, and {zk} in dimension 4. Let "Y denote the column vector with entries Yj = yj, for 1 j N. For dimensional reasons Sq2"Y = M(x)"Y where M(x) is an N x N matrix with entries from R2. Notice that M(x) must have rank N as a map from R3 ! R5, since Sq1Sq2"Y = OE("Y) where OE("Y) is the vector with entries the square of the original ent ries of * *"Y. First we prove that Sq1"Y = 0. Consider Sq1Sq2"Y = Sq3"Y = OE("Y) = Sq1(M(x))"Y + M(x)(Sq1"Y). Each entry of Sq1(M(x)) is a linear form in the y-variables. Hence the remaining term in the equation , M(x)Sq1("Y), is on the one hand a function of only the y-variables, and on the other hand, a linear combination of terms of the form xiyj. Hence it is zero. Since M(x) has rank N, in fact Sq1"Y = 0. Next we show that M(x) satisfies the equation M(x)M(x) = M(OE(x)): Consider Sq2Sq2"Y = Sq3Sq1"Y = 0 from above. Expanding, we obtain Sq2Sq2"Y = Sq2(M(x)"Y) = M(OE(x))"Y + Sq1(M)Sq1"Y + M(x)M(x)"Y = 0: By 5.1 and 5.2 of the previous section, there is another choice of the y-genera* *tors so that with respect to this basis, M(x) is diagonal. That is, there exists a * *basis {yj} for R3 and elements {xi} of R2 so that Sq2yi = xiyi for 1 i N: Notice that since Sq1Sq2yi = Sq3yi = y2i= (Sq1xi)yi, we have Sq1xi = yi. Extend the chosen {xi} to a basis of R2 in such a way that Sq1xi = 0 if i > N. The final step is to show that there is a choice of the dimension 4 generators which are killed by Sq1. Suppose z 2 R4 is not a polynomial in the x-variables. Then if X Sq1z = aijxiyj; define X w = aijxiyj: i>N Then X Sq1( aijxixj) = w: i>N 18 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Hence X Sq1(z - w) = aijxiyj: iN But apply Sq1 again: X X 0 = aijSq1(xi)yj = aijyiyj iN iN Hence aij = 0 for all 1 i; j N. So z - w represents the same indecomposable as z and has Sq1 zero. Proof 6.3. This requires showing that the dimension four generators {zi} can be chosen to have Sq2zi = xzi for some x 2 R2. Again we use the matrix notation. Let Z denote the column vector with i-th entry the generator zi. For dimensiona reasons, Sq2Z = A(x)Z + F (y) + G(x); where A is a square matrix with entries linear forms in the x-variables, where * *F and G are column vectors with entries respectively quadratic and cubic homogeneous forms in the y- and x-variables. From 6.1, we can assume that Sq1Z = 0: Now apply the Adem relation Sq2Sq2Z = Sq3Sq1Z = 0 to obtain 0 = Sq2(A(x)Z) + Sq2F + Sq2G = A(x2)Z + A(x)(A(x)Z + F + G) + Sq2F + Sq2G = = A(x2)Z + A(x)2Z + A(X)F + A(x)G + Sq2F + Sq2G Hence as in 6.1, A(x) = A(x2), and we can change the choice of z-variables so that A(x) is diagonal. That is, let z be a typical generator. We can assume that Sq2z = xz + f(y) + g(x): Let Sq(0;1)denote the Milnor primitive Sq2Sq1 + Sq1Sq2: Then Sq(0;1)Sq2z = Sq2Sq3z + Sq1Sq2Sq2z = Sq5z + Sq4Sq1z + Sq1Sq3Sq1z = 0; using the Adem relations. Note that (1) Sq(0;1)xi = xiyi (2) Sq(0;1)yi = y2i Hence 0 = (Sq(0;1)x)z + xSq0;1z + Sq(0;1)f + Sq(0;1)g = (xSq1x)z + x(Sq1Sq2z) + Sq(0;1)(f + g) = (xSq1x)z + x((Sq1x)z + Sq1(f + g)) + Sq(0;1)(f + g) That is, (xSq1x)z = x((Sq1x)z: p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 19 That is, x is either one of the canonical generators xi or x = 0. We also dedu* *ce that x(Sq1(f + g)) + Sq(0;1)(f + g) = 0 Now Sq1f = 0, so xSq1g = Sq(0;1)(f) + Sq(0;1)(g): But Sq(0;1)(f) is purely a polynomial in the y-variables, Sq(0;1)(f) = 0. If X f = aijyiyj; ij then X Sq(0;1)(f) = aijyiyj(yi+ yj) = 0 ij Thus aij= 0 if i < j. That is f is a square, in fact it's expressible uniquely * *in the form (Sq1u)2 = f. If x = 0, then z0 = z + u2 will be a suitable replacement for z. If x 6= 0, we need an expression for g similar in spirit to that for f. With* *out loss of generality, we can assume that x = x1. Write X g = bijkxixjxk: ijk X xSq1g = x1 bijk(yixjxk + xiyjxk + xixjyk) ijk X = Sq(0;1)(g) = bijkxixjxk(yi+ yj + yk) ijk Comparing coefficients on the two sides, we see that b123 = 0 unless i = 1; j = k. That is, g = xv2 for a two class v. Finally we must show t* *hat v = u, where f = (Sq1u)2. But from Sq2Sq2z = 0 = Sq2(f + g) + x(f + g); we have Sq2(xv2 + (Sq1u)2) = x(xv2 + (Sq1u)2) = x2v2 + x(Sq1v)2 so since x 6= 0, we have (Sq1u)2 = (Sq1v)2: 20 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Taking square roots is unique in characteristic 2, and we had assumed that Sq1 * *was an isomorphism from dimension 2 to dimension 3, so we conclude that u = v. Thus f + g = xu2 + f = xu2 + (Sq1u)2: We can adjust our choice of z by z0 = z + u2 and verify that Sq2z0 = xz0, and Sq1z0 = 0. The mod 2 Weyl group for H(k), k > 0, can be identified with the subgroup of GLk+1 (F2 ) which pointwise fixes the span of the first k - 1 basis vectors.* * We establish the identification of the unstable algebras of 6.3 with the tensor pr* *oduct of the cohomologies of BH(k)'s by choosing the canonical basis pairs in dimensi* *ons two and three, xi; yj with the property that yi = Sq1xi and Sq2yi = xiyi. This uniquely defines these up to permutation. Next we pick the zk as above. Each su* *ch zk belongs to a particular xi, in the sense that Sq2zk = zkxi. Several differen* *t zk's can use the same xi. The decomposition is then by eigenspaces. A given xi has an eigenspace of four dimensional classes {z} for which Sq2z = xiz. This contribut* *es an H(k) factor, where k is the dimension over F2 of that eigenspace. The SU(2) factors correspond to the zero eigenvalue. p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 21 x7. Uniqueness for the adjoint and simply connected forms Proposition 7.1. Let (BX; X) be a 2-compact group such that X is connected. (1) If H*BX H*BSO(3)r as algebras over A2 , then BX is homotopy equiv- alent to (F2 )1 (BSO(3)r). (2) If H*BX H*BSU(2)r as algebras over A2 , then BX is homotopy equiv- alent to (F2 )1 (BSU(2)r). The main ingredient is the [9] version of the [16] diagram description of BG* *. We quote from [9] and [13]: Definition 7.2. Let BX be a simply connected p-complete space such that H*(BX; Fp) is finite. Let ABX be the category with objects {(V; f)} where f : BV ! BX such that V is a finite dimensional vector space over Fp and H*(BV; Fp) is a finitely generated H*(BX; Fp) module via f* . The morphisms are : (V; f) ! (U; g) induced from an injection S : V ! U such that g O B(S ) =* * f up to homotopy. We can also define a purely algebraic version. Let R* be an un- stable algebra over the Steenrod algebra which is notherian. AR has objects (V;* * f* ) corresponding to maps over the Steenrod algebra f* : R* ! H*(BV; Fp) which are finite. The morphisms are : (V; f* ) ! (U; g*) induced from an injection S : V ! U such that B(S )* O g* = f* . If BX = BG, where G is a compact Lie group, then the category ABX is equivalent to AH*(BG;Fp) , by work of [22] and [18]. If H*(BX; Fp) is finite th* *en Dwyer-Wilkerson in [9] and [13] generalize [16] to obtain a diagram description* * of BX: Theorem 7.3. Let BX be a simply connected p-complete space with H*(BX; Fp) finite. Define on the category ABX a functor F , with F ((V; f)) = Map(BV; BX)* *f. To each morphism : (V; f) ! (U; g), F assigns the induced map B(S )* : Map(BU; BX)g ! Map(BV; BX)f. Then the p-completion of the homotopy direct limit of this functor over AopBXis homotopy equivalent to BX. Similiarly* *, the functor : (V; f* ) ! TfV(H*(BX; Fp)) has its inverse limit isomorphic over the Steenrod algebra to H*(BX; Fp) and the higher derived functors of are zero. The category AY can be defined more generally for a space Y with noetherian mod p cohomology. However, Theorem 7.3 even in the algebraic case is not true in general. One example is the wedge of two copies of RP 1. In this case the inver* *se limit is the cohomology of the disjoint union of a pair of RP 1's. One hypothes* *is that suffices for the algebraic version is that R* be a ring of invariants. In our construction, we need a modified smaller version of the category ABX . Let J be an ideal in H*(BX; Fp) which is closed under the Steenrod algebra acti* *on. Define the full subcategory ABX (J) to include only those objects (V; f) such t* *hat J 2 ker(f* ). Proposition 7.4. Suppose for BX in 7.2, H*BX R*1 R*2 as algebras over the Steenrod algebra, where R*1is a ring of invariants and R*2* *is noetherian. Take J to be the ideal generated by positive dimensional elements in 22 WILLIAM G. DWYER AND CLARENCE W. WILKERSON R2 Fp. Then H*(BX; Fp) InvLimABX (J)TfV(H*(BX; Fp)) and the higher derived functors are zero. Topologically, we have that BX hocolimABX (J)Map(BV; BX)f after p-completions. Our application is to BY , where H*BY H*BSO(3)r. Choose R1 to be one tensor factor isomorphic over the Steenrod algebra to H*BSO(3), and R2 to be the remaining factors. Proposition 7.5. Suppose that H*BY H*BSO(3)r over the Steenrod algebra. Then there exists f : BY ! (F2 )1 (BSO(3)) such that f* is an injection on mod 2 cohomology. Proof of 7.1. : For r = 1, this is the result of [7]. Suppose that 7.1 is know* *n for r - 1 factors. By 7.5, there is a fibration fib(f) ! BY ! (F2 )1 (BSO(3)) The Eilenberg-Moore spectral sequence collapses and shows that the mod 2 coho- mology of the fiber is an example of the rank r - 1 case. By induction fib(f) i* *s h.e to (F2 )1 (BSO(3)r-1 ). The classification of fibrations quoted in the introduc* *tion then shows that the fibration is f.h.e. to the trivial (product) fibration. Proof of 7.5. : Assume that 7.1 is known for r - 1 factors. Consider the diagram given by F : ABX (J) ! T OP . There are up to isomorphism just two cohomology algebras occuring as H*Map(BV; BX)Bf : (1) for dimV = 1, H*BSO(3)r-1 H*BO(2) and (2) for dimV = 2, H*BSO(3)r-1 H*(RP 1)2. Using 7.5 and 7.1 for r - 1 factors, we will show that only two homotopy typ* *es occur: Denote by BX1 a space with cohomology of type (1) and by BX2 that of type (2). We first prove that (1) BX1 (F2 )1 BSO(3)r-1 x BO(2) and that (2) BX2 (F2 )1 BSO(3)r-1 x (RP 1 x RP 1): The 1-connected cover BX1 <1> has h : BX1 <1> ! K(Z2 ; 2) with H*fib(h) H*BSO(3)r-1 . By the induction hypothesis, fib(h) (F2 )1 BSO(3)r-1 : p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 23 By the fibration theorem, Theorem 0.8, the fibration fib(h) ! BX1 <1> ! K(Z2 ; 2) is f.h.e to the product fibration. Thus the fibration BX1 <1> ! BX1 ! RP 1 is classified by a map into BAut(BX1 <1> ). The fundamental group of this spa* *ce is non-trivial, but the classifying map lifts to BAut1(BX1 <1> ), which is K(* *Z2 ; 2). There are two such maps, but the nontrivial one is the right choice. Thus BX1 is the desired product of completions of BSO(3)r-1 and BO(2). A similar but easier analysis establishes BX2. In particular, the composite functor F [3]into T OP that assigns to (V; f) * *the 3-rd Postnikov approximation to Map(BV; BX)Bf takes, up to homotopy, only the values (1) K(Z=2Z; 2)r-1 x (F2 )1 BO(2) and (2) K(Z=2Z; 2)r-1 x BO(1)2 Because of the choice of J, the mod 2 cohomology of the diagram is naturally* *OB isomorphic to that give by the functor : AH*(BX) (J) ! K, given by Lannes' T -functor. But this latter diagram is naturally equivalent to TfV(H*BX) T0(R2* *), where T0 is the component of the trivial map. Hence by 7.3, the inverse limit * *is just R1 R2 = H*BX, and this is the cohomology of the homotopy direct limit. Now apply the 3-rd Posnikov approximation to the topological diagram. For the spaces of type (1), we obtain K(Z=2Z; 2)r-1 x(F2 )1 BO(2) and for the type * *(2) spaces, K(Z=2Z; 2)r-1 x BO(1)2 Hence the homotopy colimit has the cohomology of K(Z=2Z; 2)r-1 x BSO(3). Call this space M and choose a map h : M ! K(Z=2Z; 2)r-1 which induces an isomorphism onto the corresponding factor of the cohomology of M. Then there is a fibration fib(h) ! M ! K(Z=2Z; 2)r-1 : The Eilenberg-Moore spectral sequence collapses, and we have that H*fib(h) H*BSO(3) as algebras over the Steenrod algebra. By the case for r = 1, fib(h) is homotopy equivalent to (F2 )1 BSO(3). Thus Theorem 0.8 of the introduction applies and the fibration is f.h.e. to the product fibration. In particular, th* *ere is a map BX ! M ! (F2 )1 BSO(3) which is non-trivial on mod 2 cohomology. There is a variant of the above in which BX is replaced by a BY where BY is obtained by killing all but one of the non-zero elements in H2. For the right c* *hoices, this gives a BY such that H*BY is isomorphic to that of BSU(2)r-1 x BSO(3). We should point out that from the view of mod 2 cohomology, all maps BZ=2Z ! BSU(2)r are central, and each can be permuted into any another by an automor- phism of H*BSU(2)r over the Steenrod algebra. Topologically however, there can 24 WILLIAM G. DWYER AND CLARENCE W. WILKERSON be non-equivalent quotients at the classifying space level. For example, two po* *ssi- ble quotients of SU(2) x SU(2) by a central Z=2Z give SU(2) x SO(3) and SO(4) respectively. These two have non-equivalent classifying spaces. This means th* *at if one starts with a space with the mod 2 cohomology of a products of BSU(2)'s, mod 2 cohomology doesn't allow us to immediately choose a central Z=2Z so that the quotient BY=B(Z=2Z) has the cohomology of BSO(3) x BSU(2)r-1 . Proof of 7.4. : The main point is that if R2 is noetherian, then the component * *of the Lannes' T -functor corresponding to the trival map is naturally isomorphic * *to R2 itself, [10], and thus the functor TfV(R1 R2) is equivalent to TfV(R1) R2. p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 25 x8. Homotopical Uniqueness In [[]8], a large number of p-compact groups are shown to satisfy a rather strong uniqueness property: the Ap -isomorphism type of H*(BX; Fp determines the homotopy type of BX. [21] extends this type of result to many Lie groups. While it is plausible to ask for this type of uniqueness for the examples of th* *is note also, given the possible complexity of the cohomology algebras, we choose * *to answer it only in the fundamental group is finite and the cohomology algebra is polynomial. Theorem 7.1. Let X and Y be semi-simple 2-compact groups with abelian Weyl groups and polynomial cohomology at the classifying space level. Then H*BX A2 H*BY if and only if BX is homotopy equivalent to BY . A more reasonable general approach to uniqueness in terms of discrete algebr* *aic data would be to suggest that the normalizer of the maximal torus contains suff* *icient information to determine BX. This is known to be true for BG, but under the proviso that one is testing against other BH. Theorem 7.2. Let X and Y be 2-compact groups with abelian Weyl groups. Let TX and TY be discrete approximations to the maximal tori of X and Y respective* *ly. Then N(TX ) is isomorphic to N(TY ) if and only if BX is homotopy equivalent to BY . Finally, there is a background complication which was ignored in the earlier section. One is often computing quotients in different categories. The follow* *ing comparison is proved in [14]. Theorem 7.3. Let G be a compact Lie group such that the group of components is a p-group. Let A be a subgroup of the center of G, and i : A ! G the inclusi* *on homomophism. Suppose that (BX; X) is a p-compact group and g : BX ! Fp1 (BG) a homotopy equivalence. Then if f : BA ! BX is any map such that g O f Fp1 (Bi), then Fp1 B(G=A) Fp1 (EBA xBA Map(BA; BX)f): x9. Mod p reductions of invariants Let R be a commutative ring and J an ideal of R. If W is a finite subgroup of automorphisms of R such that W J = J, there is an induced action of W on the quotient ring R=J. The long exact cohomology sequence for the coefficient seque* *nce 0 ! J ! R ! R=J ! 0 shows that RW ! (R=J)W may fail to be a surjection. One topological case of interest has R = H*(BT; Zp), J = pR, and W a finite subgroup of GL(n; Zp). Somewhat surprisingly, the action of W is faithfully passed to the reduction, w* *ith a minor reservation for p = 2. The following has an elementary proof which the reader might like to reproduce: 26 WILLIAM G. DWYER AND CLARENCE W. WILKERSON Lemma 9.1. Let A 2 GL(n; Zp) be an element of finite order. If p > 2, then A is not in the kernel of the mod p reduction map GL(n; Zp) ! GL(n; Fp). If p = 2, and W is a finite subgroup of GL(n; Z2), the intersection of W with the kernel * *of the reduction map is an elementary abelian 2-group. A corollary of the p-odd result is the observation that if (BX; X) is a conn* *ected p-compact group such that H*(X; Zp) is torsion free, then in fact H*(BT; Fp)W(X) H*(BX; Fp), [27], [8]. This fails in specific cases for p = 2. However, there* * is a common generalization Theorem 9.2. Let (BX; X) be a connected p-compact group such that H*(X; Zp) is torsion free. Then H*(BX; Zp) H*(BTX ; Zp)W(X) If p > 2 this is true with Fp coefficients. If p = 2, H*(BX; F2) H*(BCX (2TX ); F2)W(X) but the reduced action of W (X) is not effective if TX 6= CX (2TX ). Here 2TX denotes the elements of order 2 in a discrete maximal torus for X. The proof of 9.2 uses two different ways of computing fraction field extensi* *on degrees. First, if W ! Aut(L) is a monomorphism of a finite group into the group of field automorphisms of a given field L, the Galois theory gives that the deg* *ree of the field extension LW ! L is just the order of W , |W |. On the other hand, if R ! S is a finite extension of connected graded domains of finite type over a field K, then [1] provides a calculation of the degree of* * the corresponding fraction field extension: Lemma 9.3. For S and R as described above, the degree of the fraction field extension is limt%1(P (t; S)=P (t; R)) The easiest example of these two principles occurs for a connected compact L* *ie group G. Then a result of Borel identifies H*(BG; Q) and H*(BT; Q)W(G) . So on the one hand the Galois theory gives the fraction field extension degree as |W * *(G)|. On the other hand, if (2n1; : :;:2nr) are the degrees of the generatorsQof the * *rational cohomology, then the Poincare series arguement shows that |W (G)| = (ni). The* *re are of course many other paths to this conclusion. The advantage of the Poincare series argument is that it works in the possible absence of Gaolis extensions. * * We can generalize the Borel result somewhat: Lemma 9.4. Suppose (BX; X) is a connected p-compact group. Let f : BY ! BX be the inclusion of a maximal rank connected sub-p-compact group. Then H*(f; Zp) Q has fraction field degree |W (X)|=|W (Y )|: If the Zp-adic cohomol* *ogy of each is torsion free, the same result holds for the mod p fraction field deg* *ree. Proof of 9.2: Let BY = Map(B(2T ); BX)i = BCX (2T ). Then both H*(BT; F ) and H*(BY; F ) are f.g. modules over H*(BX; F ), for F = Fp or F = Zp. Denote S = H*(BT; Zp), R = H*(BX; Zp), and D = H*(BCX (pT ); Zp) . If M is a graded p-COMPACT GROUPS WITH ABELIAN WEYL GROUPS 27 type free Zp -module of finite type, denote by P (t; M Q) the Poincare series * *for M Q, and similarly for P (t; M Fp). By the torsion free hypothesis on X, P (t; R Fp) = P (t; R Q): A similar statement is true for P (t; S Fp ). Now the analogue of a theorem of Borel for connected p-compact groups , [13] asserts that R Q (S Q)W(X) . Hence the degree of the fraction field extension corresponding to R ! S is by Galois theory exactly |W (X)|. Now R SW(X) by the definition of W (X). But the fraction field degree of SW(X) ! S is also |W (X)| so the degree of R ! SW* *(X) is one. But S is integral over R and hence so is SW(X) . Since R is integrally * *closed in its fraction field, we have R = SW(X) . We now perform a similar analysis for the mod p case. Again R Fp is a finit* *ely generated polynomial algebra, from the torsion free hypothesis on X. Let E denote the kernel of the map W (G) ! Aut(pT ). For odd p, E is trivial and for p = 2, E is an elementary abelian 2-group, by lemma 9.1. Consider first the odd prime case. Then we have the sequence of monomorphisms R Fp ! SW(X) Fp ! (S Fp)W(X) ! S Fp We wish to show that the leftmost two are isomorphisms. We do this by calcu- lating degrees of fraction field extensions. From the characteristic zero calcu* *lation, R Fp ! S Fp has degree |W (X)|. By Galois theory, (S Fp)W(X) ! S Fp has degree |W (G)=|E| = |W (G)|. Hence the degree of R Fp ! (S Fp)W(X) is |W (G)|=|W (G)| = 1. But S Fp is integral over R Fp , so (S Fp )W(X) is also. Since R Fp is integrally closed in its field of fractions, it must i* *nclude (S Fp)W(X) . That is, R = H*(BT; Fp)W(X) . For the p = 2 case, if |E| = 1, the above proof would suffice. If E 6= 1, t* *hen however, R F2 6= H*(BT; F2)W(X) . For example, if X = SU(2), this occurs. In this case, we have already shown that CX (2T ) is a product of tori and S* *U(2)'s. Let r be the number of SU(2) factors. Then W (CX (2T ) is an elementary abelian 2-group of rank r. Hence by 9.3, the degree of the extension of fraction fields corresponding to H*(BX; F2) ! H*(BCX (2T ); F2) is |W (X)|=|W (CX (2T )|. By [14], the Weyl group of CX (2T ) is also the subgroup of W (X) which pointwise * *fixes 2T . But this is just the E defined above. Now W (X)=E acts on 2T and hence on H*(BCX (2T ); F2). We need to show that in fact H*(BX; F2) H*(BCX (2T ); F2)W(X)=E Consider R F2 ! (D F2)W(X)=E ! D F2 The composite degree is |W (X)|=|E|. The degree of the middle to rightmost is t* *he order of W (G)=E, since this acts effectively on D F2. Hence by the integrali* *ty and integral closure arguments of above, H*(BX; F2) H*(BCX (2T ); F2)W(X)=E : 28 WILLIAM G. DWYER AND CLARENCE W. WILKERSON References [[1]]J. F. Adams and C. W. Wilkerson, Finite H-spaces and algebras over the Ste* *enrod algebra, Ann. of Math. 111 (1980), 95-143. [[2]]A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces * *homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207. [[3]]A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localization* *s, Lecture Notes in Mathematics vol. 304, Springer-Verlag, Berlin, 1972. [[4]]N. Bourbaki, Commutative Algebra. [[5]]C. Broto and H.-W. Henn, Some remarks on central elementary abelian p-subg* *roups and cohomology of classifying spaces, Quart. J. Math. Oxford (2)44 (1993), 155-* *163. [[6]]W. Browder, On Differential Hopf Algebras, Transactions of the A.M.S. 107 * *(1963), 153-178. [[7]]W. G. Dwyer, H. R. Miller and C. W. Wilkerson, The homotopic uniqueness of* * BS3, Algebraic Topology, Barcelona 1986, Lecture Notes in Math. 1298, Springer, Berlin, 19* *87, pp. 90-105. [[8]]_____, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 2* *9-45. [[9]]W. G. Dwyer and C. W. Wilkerson, A Cohomological Decomposition Theorem, To* *pology 31 (1992), 433-443. [[10]]____, Smith theory and the functor T , Commentarii Math. Helv. (1991). [[11]]____, Spaces of null homotopic maps, ????. [[12]]____, A New Finite H-space at the Prime 2, Journal of the A.M.S 6 (1993),* * 33-63. [[13]]____, Homotopy fixed point methods for Lie groups and finite loop spaces,* * Annals of Math. (to appear). [[14]]____, The center of a p-compact group (1993), 1-44. [[15]]D. L. Goncalves, Mod 2 homotopy associative H-spaces, Springer Lecture No* *tes in Mathe- matics 657 (1978), 198-216. [[16]]S. Jackowski and J. E. McClure, A cohomology decomposition theorem, Topol* *ogy (1992), ???. [[17]]S. Jackowski, J. E. McClure and R. Oliver, Homotopy classification of sel* *f-maps of BG via G-actions, Annals of Math. 135 (1992), 183-226. [[18]]J. Lannes, Sur la cohomologie modulo p des p-groupes Abelienselementaires* *, Homotopy The- ory, Proc. Durham Symp. 1985, edited by E. Rees and J.D.S. Jones, Cambridge* * Univ. Press, Cambridge, 1987. [[19]]____, Sur les espaces fonctionnels dont la source est le classifiant d'un* * p-groupe abelien elementaire, GET FINAL REFERENCE. [[20]]J. P. Lin, Cup products and finite loop spaces, Topology and its Applicat* *ions 45 (1992), 73-84. [[21]]D. Notbohm, Uniqueness of classifying spaces, Goettingen preprint. [[22]]D. G. Quillen, The spectrum of an equivariant cohomology ring: I, Annals * *of Math. 94 (1971), 549-572. [[23]]D. L. Rector, Subgroups of finite dimensional topological groups, J. Pure* * and Applied Alg. 1 (1971), 253-273. [[24]]Shepherd and Todd, Finite reflection groups, Canadian J. Math. (1954). [[25]]D. Sullivan, Geometric Topology, Part I: Localization, Periodicity and Ga* *lois Symmetry, Lecture Notes, MIT, 1971. [[26]]T. tom Dieck, Lie groups, GET IT RIGHT. [[27]]C. W. Wilkerson, Rings of invariants and inseparable forms of algebras ov* *er the Steenrod Algebra, Proc. Camb. Phil. Soc. (to appear). University of Notre Dame, Notre Dame, Indiana 46556 Purdue University, W. Lafayette, IN 47907