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THE CLASSIFICATION of P-COMPACT GROUPS for P ODD 97 Annals of Mathematics, 167 (2008), 95–210 The classification of p-compact groups for p odd By K. K. S. Andersen, J. Grodal, J. M. Møller, and A. Viruel* Abstract A p-compact group, as defined by Dwyer and Wilkerson, is a purely ho- motopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classifi- cation similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for p odd. Contents 1. Introduction Relationship to the Lie group case and the conjectural picture for p =2 Organization of the paper Notation Acknowledgements 2. Skeleton of the proof of the main Theorems 1.1 and 1.4 3. Two lemmas used in Section 2 4. The map Φ : Aut(BX) → Aut(BNX ) 5. Automorphisms of maximal torus normalizers 6. Reduction to connected, center-free simple p-compact groups *The first named author was supported by EU grant EEC HPRN-CT-1999-00119. The second named author was supported by NSF grant DMS-0104318, a Clay Liftoff Fellowship, and the Institute for Advanced Study for different parts of the time this research was carried out. The fourth named author was supported by EU grant EEC HPRN-CT-1999-00119, FEDER-MEC grant MTM2007-60016, and by the JA grants FQM-213 and FQM-2863. 96 K. K. S. ANDERSEN, J. GRODAL, J. M. MøLLER, AND A. VIRUEL 7. An integral version of a theorem of Nakajima and realization of p-compact groups 8. Nontoral elementary abelian p-subgroups of simple center-free Lie groups 8.1. Recollection of some results on linear algebraic groups 8.2. The projective unitary groups 8.3. The groups E6(C) and 3E6(C), p =3 8.4. The group E8(C), p =3 8.5. The group 2E7(C), p =3 9. Calculation of the obstruction groups 9.1. The toral part 9.2. The nontoral part for the exceptional groups 9.3. The nontoral part for the projective unitary groups 10. Consequences of the main theorem 11. Appendix: The classification of finite Zp-reflection groups 12. Appendix: Invariant rings of finite Zp-reflection group, p odd (following Notbohm) 13. Appendix: Outer automorphisms of finite Zp-reflection groups References 1. Introduction It has been a central goal in homotopy theory for about half a century to single out the homotopy theoretical properties characterizing compact Lie groups, and obtain a corresponding classification, starting with the work of Hopf [75] and Serre [123, Ch. IV] on H-spaces and loop spaces. Materi- alizing old dreams of Sullivan [134] and Rector [121], Dwyer and Wilker- son, in their seminal paper [56], introduced the notion of a p-compact group, as a p-complete loop space with finite mod p cohomology, and proved that p-compact groups have many Lie-like properties. Even before their introduc- tion it has been the hope [120], and later the conjecture [59], [89], [48], that these objects should admit a classification much like the classification of com- pact connected Lie groups, and the work toward this has been carried out by many authors. The goal of this paper is to complete the proof of the classifica- tion theorem for p an odd prime, showing that there is a one-to-one correspon- dence between connected p-compact groups and finite reflection groups over the p-adic integers Zp. We do this by providing the last—and rather intricate— piece, namely that the p-completions of the exceptional compact connected Lie groups are uniquely determined as p-compact groups by their Weyl groups, seen as Zp-reflection groups. In fact our method of proof gives an essentially self-contained proof of the entire classification theorem for p odd. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 97 We start by very briefly introducing p-compact groups and some objects associated to them, necessary to state the classification theorem—we will later in the introduction return to the history behind the various steps of the proof. We refer the reader to [56] for more details on p-compact groups and also recommend the overview articles [48], [89], and [95]. We point out that it is the technical advances on homotopy fixed points by Miller [94], Lannes [88], and others which make this theory possible. A space X with a loop space structure, for short a loop space, is a triple (X, BX, e) where BX is a pointed connected space, called the classifying space of X, and e : X → ΩBX is a homotopy equivalence. A p-compact group is a ∗ loop space with the two additional properties that H (X; Fp) is finite dimen- sional over Fp (to be thought of as ‘compactness’) and that BX is Fp-local [21], [56, §11] (or, in this context, equivalently Fp-complete [22, Def. I.5.1]). Often we refer to a loop space simply as X. When working with a loop space we shall only be concerned with its classifying space BX, since this determines the rest of the structure—indeed, we could instead have defined a p-compact group to be a space BX with the above properties. The loop space (Gˆp,BGˆp,e), corresponding to a pair (G, p) (where p is a prime, G a compact Lie group with component group a finite p-group, and (·)ˆp denotes Fp-completion [22, Def. I.4.2], [56, §11]) is a p-compact group. (Note however that a compact Lie group G is not uniquely determined by BGˆp, since we are only focusing on the structure ‘visible at the prime p’; e.g., B SO(2n +1)ˆp B Sp(n)ˆp if p =2,as originally proved by Friedlander [66]; see Theorem 11.5 for a complete analy- sis.) A morphism X → Y between loop spaces is a pointed map of spaces BX → BY . We say that two morphisms are conjugate if the corresponding maps of classifying spaces are freely homotopic. A morphism X → Y is called an isomorphism (or equivalence) if it has an inverse up to conjugation, or in other words if BX → BY is a homotopy equivalence. If X and Y are p- compact groups, we call a morphism a monomorphism if the homotopy fiber Y/X of the map BX → BY is Fp-finite. The loop space corresponding to the Fp-completed classifying space BT = r (BU(1) )ˆp is called a p-compact torus of rank r.Amaximal torus in X is a monomorphism i : T → X such that the homotopy fiber of BT → BX has nonzero Euler characteristic. (We define the Euler characteristic as the alter- nating sum of the Fp-dimensions of the Fp-homology groups.) Fundamental to the theory of p-compact groups is the theorem of Dwyer-Wilkerson [56, Thm. 8.13] that, analogously to the classical situation, any p-compact group admits a maximal torus. It is unique in the sense that for any other maximal torus i : T → X, there exists an isomorphism ϕ : T → T such that iϕ and i are conjugate. Note the slight difference from the classical formulation due to the fact that a maximal torus is defined to be a map and not a subgroup. 98 K. K. S. ANDERSEN, J. GRODAL, J. M. MøLLER, AND A. VIRUEL Fix a p-compact group X with maximal torus i : T → X of rank r. Replace the map Bi : BT → BX by an equivalent fibration, and define the Weyl space WX (T ) as the topological monoid of self-maps BT → BT over BX. The Weyl group is defined as WX (T )=π0(WX (T )) [56, Def. 9.6]. By [56, Prop. 9.5] WX (T ) is a finite group of order χ(X/T ). Furthermore, by [56, Pf. of Thm. 9.7], if X is connected then WX (T ) identifies with the set of conjugacy classes of self-equivalences ϕ of T such that i and iϕ are conjugate. In other words, the canonical homomorphism WX (T ) → Aut(π1(T )) is injective, so we can view WX (T ) as a subgroup of GLr(Zp), and this subgroup is independent of T up to conjugation in GLr(Zp). We will therefore suppress T from the notation. Now, by [56, Thm. 9.7] this exhibits (WX ,π1(T )) as a finite reflection group over Zp. Finite reflection groups over Zp have been classified for p odd by Notbohm [107] extending the classification over Qp by Clark-Ewing [34] and Dwyer-Miller-Wilkerson [52] (which again builds on the classification over C by Shephard-Todd [126]); we recall this classification in Section 11 and extend Notbohm’s result to all primes. Recall that a finite Zp-reflection group is a pair (W, L) where L is a finitely generated free Zp-module, and W is a finite subgroup of Aut(L) generated by elements α such that 1−α has rank one. We say that two finite Zp-reflection groups (W, L) and (W ,L) are isomorphic,if −1 we can find a Zp-linear isomorphism ϕ : L → L such that the group ϕW ϕ equals W .
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