THE CLASSIFICATION OF 2-COMPACT GROUPS KASPER K. S. ANDERSEN AND JESPER GRODAL Abstract.We prove that any connected 2-compact group is classified by it* *s 2-adic root datum, and in particular the exotic 2-compact group DI(4), construc* *ted by Dwyer- Wilkerson, is the only simple 2-compact group not arising as the 2-compl* *etion of a compact connected Lie group. Combined with our earlier work with Moller and Viru* *el for p odd, this establishes the full classification of p-compact groups, stating th* *at, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups * *and root data over the p-adic integers. As a consequence we prove the maximal torus co* *njecture, giving a one-to-one correspondence between compact Lie groups and finite loop s* *paces admitting a maximal torus. Our proof is a general induction on the dimension of th* *e group, which works for all primes. It refines the Andersen-Grodal-Moller-Viruel metho* *ds to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-* *Wilkerson and the authors, and we show that certain occurring obstructions vanish, by * *relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1* *990s. 1.Introduction In this paper we prove that any connected 2-compact group X is classified, up* * to isomor- phism, by its root datum DX over the 2-adic integers Z2. This, combined with ou* *r previous work with Moller and Viruel [8] for odd primes, finishes the proof of the class* *ification of p-compact groups for all primes p. The classification states that, up to isomor* *phism, there is a one-to-one correspondence between connected p-compact groups and root data* * over Zp. Hence the classification of p-compact groups is completely parallel to the clas* *sification of compact connected Lie groups [9, x4, no. 9], just with Z replaced by Zp. The cl* *assification of 2-compact groups has the following consequence, which captures most of the clas* *sification statement. Theorem 1.1 (Classification of 2-compact groups; splitting version). Let X be a* * connected 2-compact group. Then BX ' BG^2x B DI(4)s, s 0, where G is a compact connecte* *d Lie group, and B DI(4) is the classifying space of the exotic 2-compact group DI(4)* * constructed by Dwyer-Wilkerson [24]. This is the traditional form of the classification conjecture e.g., stated by* * Dwyer in his 1998 ICM address as [19, Conjs. 5.1 and 5.2]. A p-compact group, introduce* *d by Dwyer-Wilkerson [25], can be defined as a pointed, connected, p-complete space * *BX with H*( BX; Fp) finite over Fp, and X is then the pointed loop space BX. The space* * BX is hence the classifying space of the loop space X, justifying the convention o* *f referring to the p-compact group simply by X. A p-compact group is called connected if the s* *pace X is connected, and two p-compact groups are said to be isomorphic if their classify* *ing spaces are homotopy equivalent. For more background on p-compact groups, including det* *ails on ___________ The second-named author was partially supported by NSF grant DMS-0354633 and * *an Alfred P. Sloan Research Fellowship. 1 2 K. ANDERSEN AND J. GRODAL the history of the classification conjecture, we refer to [8] and the reference* *s therein_we also return to it later in this introduction. To make the classification more precise, we now recall the notion of a root d* *atum over Zp. For p = 2 this theory provides a key new input to our proofs, and was devel* *oped in the paper [28] of Dwyer-Wilkerson and in our paper [4] (see also Section 8); we say* * more about this later in the introduction where we give an outline of the proof of Theorem* * 1.2. For a principal ideal domain R, an R-root datum D is a triple (W, L, {Rboe}),* * where L is a finitely generated free R-module, W AutR(L) is a finite subgroup generated by* * reflections (i.e., elements oe such that 1-oe 2 EndR(L) has rank one), and {Rboe} is a coll* *ection of rank one submodules of L, indexed by the reflections oe in W , satisfying the two co* *nditions im(1 - oe) Rboeand w(Rboe) = Rbwoew-1for allw 2 W. The element boe2 L, determined up to a unit in R, is called the coroot correspo* *nding to oe, and together with oe, it determines a root fioe: L ! R via the formula oe(x) = * *x + fioe(x)boe. There is a one-to-one correspondence between Z-root data and classically define* *d root data, by associating (L, L*, { boe}, { fioe}) to (W, L, {Zboe}); see [28, Prop. 2.16]* *. For both R = Z or Zp, one can, instead of {Rboe}, equivalently consider their span, the coroot* * lattice, L0 = +oeRboe L, the definition given in [8, x1] (under the name "R-reflection datum* *"). For R = Zp, p odd, the notion of an R-root datum agrees with that of an R-reflectio* *n group (W, L); see Section 8. Given two R-root data D = (W, L, {Rboe}) and D0= (W 0, L* *0, {Rb0oe}), an isomorphism between D and D0 is an isomorphism ' : L ! L0 such that 'W '-1 = W 0as subgroups of Aut(L0) and '(Rboe) = Rb0'oe'-1for every reflection oe 2 W .* * We let Aut(D) be the automorphism group of D, and we define the outer automorphism gro* *up as Out(D) = Aut(D)=W . A classification of Zp-root data is given as Theorems 8.1 a* *nd 8.12. We now explain how to associate a Zp-root datum to a connected p-compact grou* *p. By a theorem of Dwyer-Wilkerson [25, Thm. 8.13] any p-compact group X has a maximal * *torus, which is a map i : BT = (BS1^p)r ! BX satisfying that the homotopy fiber has fi* *nite Fp-cohomology and non-trivial Euler characteristic. Replacing i by an equivalen* *t fibration, we define the Weyl space WX (T ) as the topological monoid of self-maps BT ! BT* * over i. The Weyl group is defined as WX (T ) = ss0(WX (T )) and the classifying space o* *f the maximal torus normalizer is defined as the Borel construction BNX (T ) = BThWX(T). By d* *efinition, WX acts on LX = ss2(BT ) and if X is connected, this gives a faithful represent* *ation of WX on LX as a finite Zp-reflection group [25, Thm. 9.7(ii)]. There is also an easy* * formula for the coroots boein terms of the maximal torus normalizer NX , for which we refer to * *Section 8.1 (or [28], [4]). We define DX = (WX , LX , {Zpboe}). We are now ready to state the precise version of our main theorem. Theorem 1.2 (Classification of p-compact groups). The assignment which to a con* *nected p-compact group X associates its Zp-root datum DX gives a one-to-one correspond* *ence be- tween connected p-compact groups, up to isomorphism, and Zp-root data, up to is* *omorphism. Furthermore the map : Out(BX) ! Out(DX ) is an isomorphism and more generally B Aut(BX) '-!(B2Z(DX ))hOut(DX) where the action of Out(DX ) on B2Z(DX ) is the canonical one. Here Aut(BX) is the space of self-homotopy equivalences of BX, Out(BX) is its* * com- ponent group, B2Z(DX ) is the double classifying space of the center of the Zp-* *root datum DX (see Proposition 8.4(1)) and is the standard Adams-Mahmud map [8, Lem. 4.1* *] given THE CLASSIFICATION OF 2-COMPACT GROUPS 3 by lifting a self-equivalence BX to BT , see Recollection 8.2. We remark that t* *he existence of the map B Aut(BX) ! (B2Z(DX ))hOut(DX) in the last part of the theorem, requ* *ires the knowledge that the fibration B Aut(BX) ! B Out(BX) splits, which was establ* *ished in [4, Thm. C]. Theorem 1.2 implies that connected p-compact groups are classif* *ied by their maximal torus normalizer, the classification conjecture in [19, Conj. 5.3* *]. For p odd, Theorem 1.2 is [8, Thm. 1.1] (with an improved description of B Aut(BX) by [4, * *Thm. C]). Our proof here is written so that it is independent of the prime p_see the outl* *ine of proof later in this introduction for a further discussion. The main theorem has a number of important corollaries. The "maximal torus co* *njec- ture", gives a purely homotopy theoretic characterization of compact Lie groups* * amongst finite loop spaces: Theorem 1.3 (Maximal torus conjecture). The classifying space functor, which to* * a com- pact Lie group G associates the finite loop space (G, BG, e : G '-! BG) gives a* * one-to-one correspondence between compact Lie groups and finite loop spaces with a maximal* * torus. Furthermore, for G connected, B Aut(BG) ' (B2Z(G))hOut(G). The automorphism statement above is included for completeness, but follows ea* *sily by combining previous work of Jackowski-McClure-Oliver, Dwyer-Wilkerson, and de Si* *ebenthal (cf., [33, Cor. 3.7], [26, Thm. 1.4], and [15, Ch. I, x2, no. 2]). The maximal * *torus conjecture seems to first have made it into print in 1974, where Wilkerson [65] described * *it as a "popular conjecture towards which the author is biased". The "Steenrod problem" from around 1960 (see Steenrod's papers [57, 56]), ask* *s which graded polynomial algebras are realized as the polynomial ring of some space X?* * The problem was solved with Fp-coefficients, for p "large enough", by Adams-Wilkers* *on [2] in 1980, extending work of Clark-Ewing [13], and for all odd p, by Notbohm [51] in* * 1999. The case p = 2 is different from odd primes, for instance since generators can appe* *ar in odd degrees. Theorem 1.4 (Steenrod's problem for F2). Suppose that P *is a graded polynomial* * algebra over F2 in finitely many variables. If H*(Y ; F2) ~= P *for some space Y , the* *n P *is isomorphic, as a graded algebra, to H*(BG; F2) H*(B DI(4); F2) r H*(R P1 ; F2) s H*(C P1 ; F2) t for some r, s, t 0, where G is a compact connected Lie group with finite cent* *er and R P1 and C P1 denotes infinite dimensional real and complex projective space, respec* *tively. In particular if P *has all generators in degree 3 then P *is a tensor product o* *f the following graded algebras: H*(B SU(n); F2)~=F2[x4, x6, . .,.x2n], H*(B Sp(n); F2)~=F2[x4, x8, . .,.x4n], H*(B Spin(7); F2)~=F2[x4, x6, x7, x8], H*(B Spin(8); F2)~=F2[x4, x6, x7, x8, y8], H*(B Spin(9); F2)~=F2[x4, x6, x7, x8, x16], H*(BG2; F2)~=F2[x4, x6, x7], H*(BF4; F2)~=F2[x4, x6, x7, x16, x24], H*(B DI(4); F2)~=F2[x8, x12, x14, x15]. 4 K. ANDERSEN AND J. GRODAL Since the classification of p-compact groups is a space-level statement, it a* *lso gives which graded polynomial algebras over the Steenrod algebra can occur as the cohomolog* *y rings of a space; e.g., the decomposition in Theorem 1.4 where P *is assumed to have * *generators in degrees 3 also holds over the Steenrod algebra. It should also be possible* * to give a more concrete list even without the degree 3 assumption, by finding all polyn* *omial rings which occur as H*(BG; F2) for G a compact connected Lie group with finite cente* *r; for G simple a list can be found in [36, Thm. 5.2], cf. Remark 7.1. In a short compan* *ion paper [6] we show how the theory of p-compact groups in fact allows for a solution to the* * Steenrod problem with coefficients in an arbitrary commutative ring R. We can also determine to which extent the realizing space is unique: Recall t* *hat two spaces Y and Y 0are said to be Fp-equivalent if there exists a space Y 00and a* * zig-zag Y ! Y 00 Y 0inducing isomorphisms on Fp-homology. The statement below also ho* *lds verbatim when p is odd, where the result is due to Notbohm [51, Cors. 1.7 and 1* *.8]_again complications arise for p = 2, e.g., due to the possibility of generators in od* *d degrees. Theorem 1.5 (Uniqueness of spaces with polynomial F2-cohomology). If A* is a gr* *aded polynomial F2-algebra over the Steenrod algebra A2, in finitely many variables,* * all in degrees 3, then there exists at most one space Y , up to F2-equivalence, with H*(Y ; * *F2) ~=A*, as graded F2-algebras over the Steenrod algebra. If P *is a finitely generated graded polynomial F2-algebra, then there exists* * at most finitely many spaces Y up to F2-equivalence such that H*(Y ; F2) ~=P *as graded F2-algeb* *ras. The early uniqueness results on p-compact groups starting with [21], which pr* *edate root data, or even the formal definition of a p-compact group, were formulated in th* *is language_ we give a list of earlier classification results later in the introduction. The* * assumption that all generators are in degrees 3 for the first statement cannot be dropped sin* *ce for instance B(S1 x SU(p2)) and B((S1 x SU(p2))=Cp) have isomorphic Fp-cohomology algebras o* *ver the Steenrod algebra, but are not Fp-equivalent. Also, the same graded polynom* *ial Fp- algebra can of course often have multiple Steenrod algebra structures, the opti* *on left open in the second statement: B SU(2) x B SU(4) and B Sp(2) x B SU(3) have isomorphi* *c Fp- cohomology algebras, but with different Steenrod algebra structures at all prim* *es. Bott's theorem on the cohomology of X=T , the Peter-Weyl Theorem, and Borel's* * char- acterization of when centralizers of elements of order p are connected, given f* *or p odd as Theorems 1.5, 1.6, and 1.9 of the paper [8], also hold verbatim for p = 2 as a * *direct con- sequence of the classification (see Remark 7.3). Likewise [8, Thm. 1.8], givin* *g different formulations of being p-torsion free, holds verbatim except that condition (3) * *should be removed, cf. [8, Rem. 10.10]. We also remark that the classification together w* *ith results of Bott for compact Lie groups, gives that H*( X; Zp) is p-torsion free and concen* *trated in even degrees for all p-compact groups. This result was first proved by Lin and * *Kane, in fact in the more general setting of finite mod p H-spaces, in a series of celebrated* *, but highly technical, papers [38, 39, 40, 35], using completely different arguments. Theorem 1.2 also implies a classification for non-connected p-compact groups,* * though, just as for compact Lie groups, the classification is less calculationally expl* *icit: Any discon- nected p-compact group X fits into a fibration sequence BX1 ! BX ! Bss with X1 connected, and since our main theorem also includes an identification o* *f the classify- ing space of such a fibration B Aut(BX1) with the algebraically defined space (* *B2Z(DX1))hOut(DX1), THE CLASSIFICATION OF 2-COMPACT GROUPS 5 this allows for a description of the moduli space of p-compact groups with comp* *onent group ss and whose identity component has Zp-root datum D. More precisely we have th* *e fol- lowing theorem, which in the case where ss is the trivial group recovers our cl* *assification theorem in the connected case. Theorem 1.6 (Classification of non-connected p-compact groups). Let D be a Zp-r* *oot datum, ss a finite p-group and set B aut(D) = (B2Z(D))hOut(D). The space M = (map (Bss, B aut(D)))hAut(Bss) classifies p-compact groups whose identity component has Zp-root datum isomorph* *ic to D and component group isomorphic to ss, in the following sense: (1) There is a one-to-one correspondence between isomorphism classes of p-co* *mpact groups X with ss0(X) ~=ss and DX1 ~=D, and components of M, given by ass* *ociating to X the component of M given by the classifying map Bss ! B Aut(BX1) -'! B aut(DX1). In particular the set of isomorphism classes of such p-compa* *ct groups identifies with the set of Out(ss)-orbits on [Bss, B aut(D)], which is f* *inite. (2) For each p-compact group X the corresponding component of M has the homo* *topy type of B Aut(BX) via the zig-zag B Aut(BX) '- (map (Bss, B Aut(BX1))C(f))hAut(Bss)'-!(map (Bss, B aut(DX1))C(f))* *hAut(Bss) where C(f) denote the Out(ss)-orbit on [Bss, B Aut(BX1)] of the element * *classifying f : BX ! Bss. Finally, remark that the uniqueness part of the classification Theorem 1.2 ca* *n be reformu- lated as an isomorphism theorem stating that the isomorphisms, up to conjugatio* *n, between two arbitrary connected p-compact groups are exactly the isomorphisms, up to co* *njugation, between their root data. For algebraic groups the isomorphism theorem can be st* *rengthened to an isogeny theorem stating that isogenies of algebraic groups correspond to * *isogenies of root data; see e.g., [60]. In another companion paper [5] we deduce from our cl* *assification theorem that the same is true for p-compact groups: Homotopy classes of maps BX* * ! BX0 which induce isomorphism in rational cohomology (the notion of an isogeny for p* *-compact groups) are in one-to-one correspondence with the conjugacy classes of isogenie* *s between the associated Zp-root data, sending isomorphisms to isomorphisms. Here an isog* *eny of Zp- root data (W, L, {Zpboe}) ! (W 0, L0, {Zpb0oe}) is a Zp-linear monomorphism ' :* * L ! L0with finite cokernel, such that the induced isomorphism ' : Aut(L Zp Qp) ! Aut(L0 Z* *p Qp) sends W isomorphically to W 0and such that '(Zpboe) = Zpb0'(oe), for every refl* *ection oe 2 W where the corresponding factor of W has order divisible by p. As a special case* * this the- orem also contains the description of rational self-equivalences of p-completed* * classifying spaces of compact connected Lie groups, the most general of the theorems obtain* *ed by Jackowski-McClure-Oliver in [33], illuminating their result. Structure of the paper and outline of the proof of the classification. Our proo* *f of the clas- sification of 2-compact groups, written to work for any prime, follows the same* * overall structure as our proof for p odd with Moller and Viruel in [8], but with signif* *icant additions and modifications. Most importantly we draw on the theory of root data [28] [4]* * and have a different way of dealing with the obstruction group problem. We outline our * *strategy below, and also refer the reader to [8, Sec. 1] where we discuss the proof for * *p odd. 6 K. ANDERSEN AND J. GRODAL An inspection of the classification of Zp-root data, Theorem 8.1, shows that * *all Zp-root data have already been realized as root data of p-compact groups by previous wo* *rk, so only uniqueness is an issue. (For p = 2 the root datum of DI(4) [24] is the only ir* *reducible Z2-root datum not coming from a Z-root datum; for p odd see [8].) The proof that two connected p-compact groups with isomorphic Zp-root data ar* *e iso- morphic, is divided into a prestep and three steps, spanning Sections 2-6. Befo* *re describing these steps, we have to recall some necessary results about root data and maxim* *al torus normalizers from [28] and [4]: The first thing to show is that the maximal toru* *s normalizer NX can be explicitly constructed from DX . For p odd this follows by a theore* *m of the first-named author [3] stating that the maximal torus normalizer NX is always s* *plit (i.e., the fibration BT ! BNX ! BWX splits). For p = 2 this is not necessarily the cas* *e, and the situation is more subtle: Recently Dwyer-Wilkerson [28] showed how to exten* *d part of the classical paper of Tits [61] to p-compact groups, in particular reconstruct* *ing NX from DX . Since the automorphisms of NX differ from those of X, one however for cla* *ssifica- tion purposes has to consider an additional piece of data, namely certain "root* * subgroups" {Noe}, which one can define algebraically for each reflection oe. We construct * *these for p- compact groups in [4], see also Section 8, and describe a candidate algebraic m* *odel for the space B Aut(BX). (In the setting of algebraic groups, this "root subgroup" Noe* *will be the maximal torus normalizer of , where Uffis the root subgroup in t* *he sense of algebraic groups corresponding to the root ff dual to the coroot boe; see [4, R* *em. 3.8].) For the connoisseur we note that the reliance in [28] on a classification of connec* *ted 2-compact groups of rank 2 was eliminated in [4, version 2]. In [4], recalled in Recollection 8.2, we show that the "Adams-Mahmud" map, wh* *ich to a homotopy equivalence of BX associates a homotopy equivalence of BNX , factors ~= : Out(BX) ! Out(BN , {BNoe}) -! Out(DX ) where Out(BN , {BNoe}) is the set of homotopy classes of self-homotopy equivale* *nces of BNX permuting the root subgroups BNoe(see Recollection 8.2 for the precise defi* *nition). As a part of our proof we will show that is a isomorphism by induction. The main argument proceeds by induction on the cohomological dimension of X, * *which can be determined from (WX , LX ) alone. (Almost equivalently one could do indu* *ction on the order of WX .) It is divided into a prestep (Section 2) and three steps (Se* *ctions 4-6). Prestep (Section 2): The first step is to reduce to the case of simple, center-* *free groups. For this we use rather general arguments with fibrations and their automorphism* *s, in spirit similar to the arguments in [8]. However the theory of root data and root subg* *roups is needed both for the statements and results for p = 2, and these are incorporate* *d throughout. With this in hand, we can assume that we have two connected simple, center-fr* *ee p- compact groups X and X0with isomorphic Zp-root data DX and DX0. As explained in* * the discussion above, [28, 4] implies that the corresponding maximal torus normaliz* *ers and root subgroups are isomorphic, and we can hence assume that they both equal (BN , {B* *Noe}) embedded via maps j and j0in X and X0, (BN , {BNoe})N j qqqq NNNNj0N qqqq NN xxqqq NNN&& BX ______________________//____________________________* *____________BX0 where the dotted arrow is the one that we want to construct. THE CLASSIFICATION OF 2-COMPACT GROUPS 7 Step 1 (Section 4): Using that in a connected p-compact group every element of * *order p can be conjugated into the maximal torus, uniquely up to conjugation in N , we * *have, for every element : BZ=p ! BX, of order p in X, a diagram of the form (BCN ( ), {BCNR( )oe}) llllll RRRRR lllll RRRRR vvll RR))R BCX ( ) BCX0( ). We can furthermore take covers of this diagram with respect to the fundamental * *group ss1(D) of the root datum, which we indicate by adding a tilde f(.). (This uses * *the formula for the fundamental group of a p-compact group [29], but see also Theorem 8.6.) In * *Section 4 we prove that one can use the induction hypothesis to construct a homotopy equi* *valence between BC^X( )and BC^X0( )under BC^N( ). The tricky point here is that these c* *entralizers need not themselves be connected, so one first has to construct the map on the * *identity component BCX ( )1 and then show that it extends, and this in turn requires tha* *t one has control of the space of self-equivalences of BCX ( )1. Now for a general elementary abelian p-subgroup : BE ! BX of X we can pick * *an element of order p in E, and restriction provides a map BC^X( )! BCX^(Z=p)! BCX^0(Z=p)! BXf0. Step 2 (Section 5): To make sure that these maps are chosen in a compatible way* *, one has to show that this map does not depend on the choice of rank one subgroup of E. In * *Section 5 we show that this lift does not depend on the choices, relying on techniques de* *veloped in [8]. We furthermore show that they combine to form an element [#] 2 lim0 2A(X)[BC^X( ), BXf0] where A(X) is the Quillen category of X, with objects the elementary abelian p-* *subgroups of X and morphisms induced by conjugation in X. Step 3 (Section 6): The construction of the element [#] basically guarantees th* *at X and X0 have the same p-fusion, and the last step, which we carry out in Section 6, dea* *ls with the rigidification question, where our approach differs significantly from [8]. In * *particular, since we work on universal covers throughout, we are able to relate our obstruction g* *roups to groups already calculated in [32]. Since the exotic p-compact groups (only DI(4* *) for p = 2) are easily dealt with, we can assume that BX = BG^pfor some simple, center-free* * Lie group. Jackowski-McClure-Oliver showed in [32] that BG^pcan be expressed as a homotopy* * colimit of certain subgroups P of G, the so-called p-radical (also known as p-stubborn)* * subgroups. For a p-radical subgroup P of G, our element [#] above gives maps BPe^p! BCG ^(pZ(P ))^p! BXf0, where pZ(P ) denotes the subgroup of elements of order at most p in Z(P ). Thes* *e maps combine to form an element in lim0eG=Pe2Or[BeeP^p, BXe0] p(G) where Orp(Ge) is the full subcategory of the p-orbit category of eGwith objects* * eG=Pefor ePa p-radical subgroup of eG. 8 K. ANDERSEN AND J. GRODAL The obstructions to rigidifying this to get a map on the homotopy colimit hocolimeG=Pe2Orp(Ge)(EGexGeeG=Pe)^p! BXe0 lies in obstruction groups which identify with lim*eG=Pe2Orsse*(Z(Pe)^p). p(G) Using extensive case-by-case calculations, Jackowski-McClure-Oliver showed in [* *32] that these obstructions in fact vanish. Hence we have constructed a map BGe^p'-(hocolimeG=Pe2Orp(Ge)(EGexGeeG=Pe)^p)^p! BXf0 which is easily seen to be an equivalence. Passing to a quotient we get the wan* *ted equivalence BG^p! BX0, finishing the proof of uniqueness. The remaining statements of Theor* *em 1.2 also fall out of this approach. Section 7 proves the stated consequences of the classification and the appendix* * Section 8 is used to establish a number of general properties of root data over Zp used thro* *ughout the paper. Here is the contents in table form: Contents 1. Introduction 1 2. Reduction to center-free simple p-compact groups * * 9 3. Preliminaries on self-equivalences of non-connected p-compact groups * * 13 4. First part of the proof of Theorem 1.2: Maps on centralizers * * 15 5. Second part of the proof of Theorem 1.2: The element in lim0 * * 18 6. Third and final part of the proof of Theorem 1.2: Rigidification * * 24 7. Proof of the corollaries of Theorem 1.2 * * 29 8. Appendix: Properties of Zp-root data * *34 8.1. Root datum, normalizer extension, and root subgroups of a p-compact gro* *up 35 8.2. Centers and fundamental groups 36 8.3. Covers and quotients 38 8.4. Automorphisms 42 8.5. Finiteness properties * *43 References 44 Related work and acknowledgments. We refer to the introduction of our paper [8]* * with Moller and Viruel for a detailed discussion of the history of the classificatio* *n for odd primes. The first classification results for 2-compact groups were obtained by * *Dwyer-Miller- Wilkerson [21] twenty years ago, in the fundamental cases SU(2) and SO(3). Notb* *ohm [50] and Moller-Notbohm [46] covered SU(n). Viruel covered G2 [64], Viruel-Vavpeti~c* * covered F4 and Sp(n) [63] [62], Morgenroth and Notbohm handled SO(2n+1) and Spin(2n+1) * *[47] [52], and Notbohm proved the result for DI(4) [53], all using arguments specifi* *c to the case in question. Obviously this paper owes a great debt to our earlier work with Moller and Vi* *ruel [8] for odd primes. Jesper Moller introduced us to the induction-on-centralizers approa* *ch to the classification, and Antonio Viruel gave us the idea of trying to compare the ce* *ntralizer and THE CLASSIFICATION OF 2-COMPACT GROUPS 9 p-stubborn decomposition, a method he had used in the paper [63] in a special c* *ase. We are very grateful to them for sharing their insights. We would furthermore like to * *thank Bill Dwyer and Clarence Wilkerson for helpful correspondence, and for sharing an ear* *ly version of their manuscript [29] on fundamental groups of p-compact groups with us, and* * Haynes Miller for useful questions. The results of this paper were announced in Spring* * 2005 e.g., at the Conference on Topology at the Isle of Skye, June 2005 [7]. Independently of* * our results, Jesper Moller has announced a proof of the classification of connected 2-compac* *t groups (Theorem 1.1) using computer algebra [41]. We benefited from the hospitality of* * Aarhus University, University of Copenhagen, and University of Chicago while writing t* *his paper. 2.Reduction to center-free simple p-compact groups In this section we reduce the classification of connected p-compact groups to* * the case of simple center-free groups, in the sense that the classification statement, T* *heorem 1.2, holds for a connected p-compact group if it holds for the simple factors occurr* *ing in the corresponding adjoint (center-free) p-compact group (see Propositions 2.1 and 2* *.4). We do this by extending the proofs given in [8, Sec. 6] for p odd, to all primes by i* *ncorporating Zp-root data, and root subgroups. Since this additional data requires restructu* *ring of most of the proofs, we present this reduction in some detail. As in [8] we make the following working definition: A connected p-compact gro* *up X is said to be determined by its Zp-root datum DX if any connected p-compact group * *X0 with DX0 ~=DX is isomorphic to X. (Theorem 1.2 will eventually show that this always* * holds.) Proposition 2.1 (Product Reduction). Suppose X = X1x . .x.Xk is a product of si* *mple p-compact groups. (1) If : Out(BXi) ! Out(DXi) is injective for each i, then so is : Out(B* *X) ! Out(DX ). (2) If : Out(BXi) ! Out(DXi) is surjective and Xiis determined by DXi for * *each i, then : Out(BX) ! Out(DX ) is surjective. Proof.The proof of (1)is identical to the proof of [8, Lem. 6.1(2)] (the key fa* *ct is knowing that a map out of a connected p-compact which is trivial when restricted to the* * maximal torus is in fact trivial, which e.g., follows from [42, Thm. 6.1]). The statem* *ent in (2)is a direct consequence of the description of Out(D) in Proposition 8.13 together * *with the assumption that if DXi is isomorphic to DXj then Xiis isomorphic to Xj. The reader might want to note that conversely to Proposition 2.1(2), if : O* *ut(BX) ! Out (DX ) is surjective for all connected p-compact groups X, then all connecte* *d p-compact groups are determined by their root datum, as is seen by considering products. Construction 2.2 (Quotients of p-compact groups). For explicitness we recall th* *e quotient construction for p-compact groups, and describe when a self-homotopy equivalenc* *e induces a homotopy equivalence on quotients, since this will be used in what follows: Let X be a p-compact group, A an abelian p-compact group and i : BA ! BX a central homomorphism. By assumption BA is homotopy equivalent to map (BA, BA)1 * *and map (BA, BX)i-ev!BX is an equivalence. Recall that the quotient BX=A is defined* * as the Borel construction of the composition action of map (BA, BA)1 on map (BA, BX)i,* * cf. [25, Pf. of Prop. 8.3]. This action and the resulting quotient space BX=A only depen* *ds on the (free) homotopy class of i, even on the point-set level, and we have a canonica* *l quotient map q : BX -' map(BA, BX)i! BX=A, well defined up to homotopy. 10 K. ANDERSEN AND J. GRODAL Now suppose we have a self-homotopy equivalence f : BX ! BX such that there e* *xists a homotopy equivalence g : BA ! BA making the diagram g BA ____//_BA i|| |i| fflffl|ffflffl| BX ____//_BX commute up to homotopy. We claim that f naturally induces a map on quotients: First, by using the bar construction model for BA, we can without restriction* * assume that g is induced by a group homomorphism and has a strict inverse g-1. Next no* *te that in general, if ' : G ! G0is a map of monoids, h : Y ! Y 0is a map from a G-spac* *e Y to a G0-space Y 0, which is '-equivariant in the sense that h(g . y) = '(g) . h(y)* *, then there is a canonical induced map on Borel constructions YhG ! Yh0G0under h : Y ! Y 0and * *over B' : BG ! BG0, e.g., by viewing the Borel construction as a homotopy colimit vi* *a the one-sided bar construction. In the above setup take ' = cg, the monoid automorphism of map (BA, BA)1 give* *n by cg(ff) = g O ff O g-1, and h the self-map of map (BA, BX)i given by fi 7! f O f* *i O g-1. Then f induces a map ~f: BX=A ! BX=A, which fits into the homotopy commutative diagr* *am f BX ______//_BX q|| |q| fflffl|_f fflffl| BX=A ____//_BX=A. The quotient construction furthermore behaves naturally with respect to the max* *imal torus: If j : BT ! BX is a maximal torus of X, and h : BT ! BT is a lifting of f : BX * *! BX, then i : BA ! BX factors through j and g : BA ! BA lifts h [26, Lem. 6.5], and * *the above construction produces a homotopy commutative diagram _ BT=A __h_//_BT=A j=A|| |j=A| fflffl|_f fflffl| BX=A ____//_BX=A up to homotopy under the diagram one has before taking quotients. Lemma 2.3. Let X be a connected p-compact group with center i : BZ ! BX and let q : BX ! BX=Z denote the quotient map. Then any self-homotopy equivalence ' : B* *X ! BX fitting into a homotopy commutative diagram BZ II iuuuu IIiI uuu II zzuu ' II$$ BX HH______________//_BXv HHH vvvv q HH$$HH zzvqvvv BX=Z is homotopic to the identity. THE CLASSIFICATION OF 2-COMPACT GROUPS 11 Proof.Let q : BX ! BX=Z denote the quotient map, turned into a fibration. By ch* *anging ' up to homotopy, we can assume that ' is a map strictly over q. By [20] (see * *also [26, Prop. 11.9]) the homotopy class of ' as a map over q corresponds to an ele* *ment ['] 2 ss1(map (BX=Z, B Aut(BZ))f), where f : BX=Z ! B Aut(BZ) is the map classi* *fying the fibration q. Note that the class ['] could a priori depend on how we choose* * ', although this turns out not to be the case. The composite ~f: BX=Z ! B Aut(BZ) ! B Out(BZ) is null-homotopic since ss1(BX* *=Z) = 0, and obviously map(BX=Z, B Out(BZ))0 ev-!B Out(BZ) is a homotopy equivalence.* * We thus have a fibration sequence map(BX=Z, B Aut1(BZ))[f]! map(BX=Z, B Aut(BZ))f ! B Out(BZ) where [f] denotes the components which map to the component of f. Since B Aut1(* *BZ) ' B2Z is a loop space, ss1(map (BX=Z, B Aut1(BZ))g) ~=ss1(map (BX=Z, B Aut1(BZ))0) = [BX=Z, BZ] = 0, for any g : BX=Z ! B Aut1(BZ), using that BX=Z is simply connected and ss2(BX=Z) is finite. Hence ss1(map (BX=Z, B Aut(BZ))f) ! Out(BZ) is injective. But, since* * ['] by assumption maps to the identity in Out(BZ), we conclude that ['] is the identit* *y, and in particular ' is homotopic to the identity as wanted. Proposition 2.4 (Reduction to center-free case). Let X be a connected p-compact* * group with center Z. (1) If X=Z is determined by DX=Z and : Out(BX=Z) ! Out(DX=Z) is surjective, then X is determined by DX . (2) If : Out(BX=Z) ! Out(DX=Z) is injective then so is : Out(BX) ! Out(D* *X ). (3) If : Out (BX=Z) ! Out (DX=Z) is surjective, then so is : Out (BX) ! Out(DX ). Proof.The proof of (1)follows the outline of the corresponding statement for od* *d primes [8, Lem. 6.8(1)], but with the important additional input that we need to keep * *track of the root subgroups: Suppose that X and X0 are connected p-compact groups with the s* *ame Zp-root datum D. By [28, Prop. 1.10] X and X0have isomorphic maximal torus norm* *alizers ND , cf. Section 8. By [4, Thm. 3.1(2)] we can choose monomorphisms j : BND ! * *BX and j0 : BND ! BX0 such that the root subgroups in BND with respect to j and * *j0 agree. Furthermore, the centers of X and X0 agree, and can be viewed as a subgr* *oup Z of ND . Now, ND =Z will be a maximal torus normalizer for both X=Z and X0=Z (s* *ee e.g., [44, Thm. 1.2]) and the root subgroups in BN~D=Z~ with respect to j=Z and* * j0=Z also agree by construction. By our assumptions there hence exists a homotopy eq* *uivalence f : BX=Z ! BX0=Z such that BND =ZL j=Zssss LLLj0=ZL sss LLLL yysss f L%% BX=Z _________________//BX0=Z commutes up to homotopy. We need to see that f is a map over B2Z, since this im* *plies that f induces a homotopy equivalence BX ! BX0 as wanted. This is a short argum* *ent given as the last part of [8, Pf. of Lem. 6.8(1)]. 12 K. ANDERSEN AND J. GRODAL To see (2) consider ' 2 Aut (BX) corresponding to an element in the kernel of* * : Out(BX) ! Out(DX ). By definition, cf. Recollection 8.2, this means that if i :* * BT ! BX is a maximal torus then the diagram BT F ixxxxx FFFiFF __xxx ' F""F BX ____________//_BX commutes up to homotopy. Since the center BZ ! BX factors through i : BT ! BX [* *26, Thm. 1.2], Construction 2.2 combined with our assumption shows that we get a ho* *motopy commutative diagram BZ I uuuu III uuu III zzuu ' I$$I BX HH______________//BXv HHH vvvv qHHH$$H zzvqvvv BX=Z so Lemma 2.3 gives the desired conclusion. We now embark on showing (3), i.e., that : Out(BX) ! Out(DX ) is surjective* *, which requires some preparation. Recall that for any connected p-compact group Y , eY* *is the p- compact group whose classifying space BYeis the fiber of the fibration BY -q!P2* *BY , where P2BY is the second Postnikov section. Let i : BT ! BY be a maximal torus, which* * we can assume to be a fibration, and let BTedenote the fiber of the fibration q O i : * *BT ! P2BY . Since ss2(i) : ss2(BT ) ! ss2(BY ) is surjective (see Proposition 8.5), the lon* *g exact sequence of homotopy groups shows that BTeis a p-compact torus, and furthermore BTe! BYe* * is a maximal torus by the diagram Ye=Te_____//_BTe_____//BYe | | | | | | fflffl| fflffl|i fflffl| Y=T _____//_BT______//BY | |qOi |q | | | fflffl| fflffl| fflffl| *______//P2BY______P2BY. Any self-homotopy equivalence f : BY ! BY lifts to a self-homotopy equivalence * *ef: BYe! BYe, by taking fibers, and it is clear that the assignment f 7! efinduces a hom* *omorphism Out(BY ) ! Out(BYe). ~= For a Y which satisfies ss1(DY ) -! ss1(Y ), Proposition 8.9 shows that DeY~=* *gDY. Hence, the Adams-Mahmud map , recalled in Recollection 8.2, together with Proposition* * 8.14 THE CLASSIFICATION OF 2-COMPACT GROUPS 13 provides the maps in the following diagram (2.1) Out (BY )____//_Out(DY ) | | | | fflffl| fflffl| Out (BYe)____//_Out(gDY). The diagram commutes, since for a given f : BY ! BY , both compositions give a * *map BTe! BTeover ef: BYe! BYe, and hence they give the same element in Out(gDY). ~= * * ' By Proposition 8.10(2), gDX -! D^X=Z, and, chasing through the definitions, B* *Xe -! ~= BX]=Z. By the fundamental group formula, Theorem 8.6, ss1(DX=Z) -! ss1(X=Z). (N* *ote that for the proof of Theorem 1.2 we can assume that X=Z is determined by DX=Z * *making this reference to Theorem 8.6 unnecessary.) Hence applying diagram (2.1)with Y * *= X=Z and using the aforementioned identifications for Y = X=Z produces the diagram Out(BX=Z) ______////_Out(DX=Z) | |~ | |= fflffl| fflffl| Out(BXe )_________//Out(gDX). Here the right-hand vertical map is an isomorphism by Proposition 8.10(2) and C* *orol- lary 8.16, and the top horizontal map is surjective by assumption. Hence : Ou* *t(BXe ) ! Out(DXe) is also surjective. By [45, Thm. 5.4] there is a short exact sequence BA -i!BX0 ! BX, BX0 = BXe x BZ(X)1, where A is a finite p-group and i : BA ! BX0 is central. Proposition 8* *.4(2) shows that X0 has Zp-root datum DX0 = (W, L0, {Zpboe}) x (1, LW , ;) and we hav* *e the identification DX ~=DX0=A as in Proposition 8.14. We are now ready to show that Out(BX) ! Out(DX ) is surjective, by lifting an* * arbitrary element ff 2 Out(DX ) to Out(BX). By Proposition 8.14, ff identifies with an e* *lement ff02 Out(DX0) with ff0(A) = A. Since : Out(BXe ) ! Out(DXe) is surjective it * *follows from Proposition 8.13 that : Out (BX0) ! Out (DX0) is surjective, so we can f* *ind a self-homotopy equivalence ' of BX0 with (') = ff0. Since ff0(A) = A there exis* *ts a lift '0: BA ! BA of ' fitting into a homotopy commutative diagram '0 BA ______//BA i|| i|| fflffl|' fflffl| BX0 ____//_BX0. Finally, since X ~=X0=A, Construction 2.2 now gives a self-homotopy equivalence* * __'of BX with the property that (__') = (')=A = ff0=A = ff as desired. 3.Preliminaries on self-equivalences of non-connected p-compact groups In this short section we prove a fact about detection of self-equivalences of* * non-connected p-compact groups on maximal torus normalizers, which we need in the proof of th* *e main theorem, where non-connected groups occur as centralizers of elementary abelian* * p-groups in connected groups. 14 K. ANDERSEN AND J. GRODAL Proposition 3.1. Let X be a (not necessarily connected) p-compact group with ma* *ximal ~= torus normalizer N and identity component X1. If : Out (BX1) -! Out (DX1) is* * an isomorphism, then : Out(BX) ! Out(BN ) is injective. Proof.Let j : BN ! BX be a normalizer inclusion map, which we can assume is a f* *ibration. Let f : BX ! Bss0(X) = Bss be the canonical fibration and set q = f O j. We first argue that we can make the identification ~= ss0(Aut (q)) -! {' 2 Out(BN ) | '(ker(ss1(q))) = ker(ss1(q))}. Surjectivity is obvious, so we have to see injectivity, where we first observe * *that we can pass to a discrete approximation ~q: BN~ ! Bss, where BN~ and Bss are the standard b* *ar con- struction models. The simplicial maps BN~ ! BN~ are exactly the group homomorph* *isms, so any map ' : ~q! ~qwith BN~ ! BN~ homotopic to the identity is induced by con* *jugation by an element in N~. Hence ' is homotopic to the identity as a map of fibration* *s, proving the claim. Since B Aut(f) -'!B Aut(BX), we have the following diagram, with horizontal m* *aps fibrations, where BN1 denotes the fiber of BN ! Bss: map (Bss, B Aut(BX1))C(f)___//_B Aut(BX)___//_B Aut(Bss) | | || | | || fflffl| fflffl| || map (Bss, B Aut(BN1))C(q)____//_B Aut(q)___//_B Aut(Bss). Here the horizontal fibrations are established in [20] (see also [26, Prop. 11.* *9]) and the vertical maps are induced by Adams-Mahmud maps, cf. [8, Lem. 4.1], so the diagr* *am is homotopy commutativity by the naturality of these maps. To establish the proposition it is enough to verify that ss1(B Aut(BX)) ! ss1* *(B Aut(q)) is injective since we already saw that ss1(B Aut(q)) injects into Out(BN ). By [4,* * Thm. B] the map : B Aut(BX1) ! B Aut(BN1) factors through the covering space Y of B Aut(B* *N1) with respect to the subgroup Out(BN1, {(BN1)oe}), and B Aut(BX1) ! Y has left h* *omo- topy inverse. Since Y ! B Aut(BN1) is a covering, map map(Bss, Y ) ! map(Bss, B* * Aut(BN1)) is likewise a covering map over each component where it is surjective, and henc* *e induces a monomorphism on ss1 on all components of map (Bss, Y ). Since B Aut(BX1) ! Y has a homotopy retract, map (Bss, B Aut(BX1)) ! map (Bss, B Aut(BN1)) also indu* *ces an injection on ss1 for all choices of base-point. Hence the five-lemma and th* *e long- exact sequence in homotopy groups applied to the pair of fibration above guaran* *tees that Out(BX) = ss1(B Aut(BX)) ! ss1(B Aut(q)) is injective as wanted. Proposition 3.2. Suppose that X is a (not necessarily connected) p-compact grou* *p such ~= that : Out(BX1) -! Out(DX1). Let i : BNp ! BX be the inclusion of a p-normali* *zer of a maximal torus, and let ' : BX ! BX be a self-homotopy equivalence. If ' O* * i is homotopic to i then ' is homotopic to the identity map. Proof.Let j : BN ! BX be the inclusion of a maximal torus normalizer, turned in* *to a fibration. By construction of the Adams-Mahmud map, cf. [8, Lem. 4.1], ' lifts * *to a map THE CLASSIFICATION OF 2-COMPACT GROUPS 15 '0: BN ! BN , making the diagram '0 BN ____//_BN j|| |j| fflffl|'fflffl| BX ____//_BX (strictly) commute, and the space of such lifts is contractible. We want to see that '0is homotopic to the identity, since Proposition 3.1 the* *n implies * * ~= that ' is homotopic to the identity, as wanted, using the assumption that : O* *ut(BX1) -! Out(DX1). Lift i : BNp ! BX to a map k : BNp ! BN . By [8, Pf. of Lem. 4.1] the space o* *f such lifts is contractible, so since ' O i is homotopic to i, we conclude that '0O k* * is homotopic to k. Replacing k and '0by discrete approximations, we get the following diagra* *m which commutes up to homotopy BN~pE ~kyyy EE~kE yyy EE __yy ~'0 E""E BN~ ______________//BN~. This is a diagram of K(ss, 1)'s, so after changing the spaces and maps up to ho* *motopy we can assume that all maps are induced by group homomorphisms, and that the diagr* *am commutes strictly. But now ~'0is a group homomorphism which is the identity on * *im(~k), and in particular it is the identity on ~T. Hence ~'0=T~: W ! W is the identity* * on the Weyl group W1 of X1, since W1 acts faithfully on ~T. But W is generated by W1 and th* *e image of im(~k), so we conclude that ~'0=T~: W ! W is the identity as well. Hence ~'0is * *in the image of Der(W ; ~T) ! Aut(N~), cf. [8, Pf. of Prop. 5.2], and the homotopy class of * *~'0is the image of an element in H1(W ; ~T) under the homomorphism H1(W ; ~T) ! Out(N~). Since* * ~'0is the identity on im(~k), this element restricts trivially to H1(Wp; ~T), where W* *p is a Sylow p-subgroup in W . By a transfer argument the element in H1(W ; ~T) is therefore* * also trivial, and ~'0is homotopic to the identity map, as wanted. 4.First part of the proof of Theorem 1.2: Maps on centralizers In this section we carry out the first part of the proof of Theorem 1.2, by c* *onstructing maps on certain centralizers. We have chosen to be quite explicit about when we* * replace spaces by homotopy equivalent spaces, since some of these issues become importa* *nt later on in the proof, where we want to conclude that various constructions really ta* *ke place in certain over- or undercategories. Recall that our setup is as follows: Let X and X0 be connected center-free si* *mple p- compact groups with isomorphic Zp-root data DX and DX0. We want to prove that BX is homotopy equivalent to BX0, by induction on cohomological dimension, where b* *y [27, Lem. 3.8] the cohomological dimension cdY of a connected p-compact group Y depe* *nds only on DY . We make the following inductive hypothesis: (?) For all connected p-compact groups Y with cdY < cdX, Y is determined by * *its Zp-root datum DY and : Out(BY ) ! Out(DY ) is an isomorphism. Let D be a fixed Zp-root datum, isomorphic to both DX and DX0 and let N = ND denote the associated normalizer; see Section 8.1. By [4, Thm. 3.1(2)] we can c* *hoose maps 16 K. ANDERSEN AND J. GRODAL j : BN ! BX and j0: BN ! BX0 making N a maximal torus normalizer in both X and X0 in such a way that the root subgroups BNoeof BN with respect to j : BN ! BX * *and j0: BN ! BX0 agree. By Theorem 8.6, X and X0have canonically isomorphic fundamental groups, which* * both identify with ss1(D) via the inclusions j and j0. Applying the second Postnikov* * section P2 to BX and BX0 hence gives us a diagram j lBN SS j0 llll SSS uulll SS))S BX QQQQ mmBX0 QQ((Q vvmmmm B2ss1(D) where we by changing BN and the maps up to homotopy can assume that all maps are fibrations, and that the diagram commutes strictly. After doing this, we now le* *ave these maps fixed throughout the proof. Suppose that : BV ! BX is a rank one elementary abelian p-subgroup of X, a* *nd let ~ : BV ! BT ! BN denote the factorization of through the maximal torus T* * , which exists by [25, Prop. 5.6], and is unique up to conjugacy in N by [27, Pro* *p. 3.4]. Set 0= j0O ~ for short. Taking centralizers, these maps produce the following diagram (4.1) BCN (~)M j rrrr MMMj0MM rrrr MM yyrr M&&M BCX ( ) BCX0( 0). where we by a slight abuse of notation keep the labeling j and j0. Now the fund* *amental groups of BCX ( ) and BCX0( 0) identify via j and j0 with a certain quotient gr* *oup ss of ss1(BCN (~)), explicitly described in [26, Rem. 2.11] and Proposition 8.4(3). P* *assing to the universal cover of BCX ( ) and BCX0( 0) and the cover of BCN (~) determined by * *the kernel of ss1(BCN (~)) ! ss produces a diagram (4.2) BCN (~)1 j qqqq MMMMj0M0 qqqq MMM xxqq MM&&M BCX ( )1 BCX0( 0)1 where the maps, which are the covers of j and j0, are ss-equivariant with respe* *ct to the natural free action of ss on all three spaces in the diagram. Note that in general if Y is a space with a map f : Y ! BG, with BG the class* *ifying space of a simplicial group G, a specific model for the homotopy fiber eYof f i* *s given by the subspace of Y x EG, consisting of pairs whose images in BG agree. In partic* *ular it has a canonical free G-action, via the action on the second coordinate and the proj* *ection map eY! Y induces a homotopy equivalence eY=G ! Y . We use this model f(.)for the h* *omotopy fiber in what follows. Note that if Y has a free H-action, then eYhas a free G * *x H-action. The spaces in (4.2)all have maps to B2ss1(D) making the obvious diagrams comm* *ute, so we can take homotopy fibers of these maps by pulling back along the map EBss* *1(D) ! THE CLASSIFICATION OF 2-COMPACT GROUPS 17 B2ss1(D), as described above. This produces the diagram (4.3) BC^N(~)1 e ss LLLfj0 0 j ss LL sss LLL yyss LL%% BC^X( )1 BCX^0( 0)1. Note that by construction K = ss x Bss1(D) acts freely on the spaces in (4.3), * *and the maps are K-equivariant. By Propositions 8.4(3)and 8.9, BC^X( )1and BCX^0( 0)1have is* *omor- phic Zp-root data and strictly smaller cohomological dimension than X, so the i* *nductive assumption (?) guarantees that they are homotopy equivalent. Furthermore by con* *struction ejand fj0a0re both maximal torus normalizers and they define the same root subg* *roups in BC^N(~)1, since this is true for j and j0. Therefore, by the above and the i* *nductive hy- pothesis (?) there exists a map ' : BC^X( )1! BCX^0( 0)1, unique up to homotopy* *, making the above diagram (4.3)homotopy commute. We now want to argue that this map can* * be chosen to be K-equivariant so that passing to a quotient of diagram (4.3)with '* * inserted produces a left-to right map making the diagram (4.1)homotopy commute. Consider the Adams-Mahmud-like zig-zag (4.4) : map(BC^X( )1, BCX^0( 0)1)' -' map(je, fj0)0''-!map(BC^N(~)1, BCN^(~)* *1)1 where the first map is a homotopy equivalence by [8, Pf. of Lem. 4.1]. That the* * composite is a homotopy equivalence will follow once we know that the center of CX ( )1 agre* *es with the center of CN (~)1, and this follows from Lemma 8.7 applied to the subgroup C^X(* * )1of eX, where the assumptions are satisfied since ss1(DXe) = 0 by Proposition 8.9 and T* *heorem 8.6. By construction the maps in (4.4)are equivariant with respect to the K-action* *s. Likewise, since the action on the sources in the mapping spaces is already free, taking h* *omotopy fixed- points agrees up to homotopy with taking actual fixed-points, so the maps in (4* *.4)induce homotopy equivalences between the fixed-points. This produces homotopy equivale* *nces map K(BC^X( )1, BCX^0( 0)1)[']'- mapK (je, fj0)0[']'-!mapK(BC^N(~)1, BC^N(~)* *1)[1], where the subscript ['] denotes that we are taking all components of maps non-e* *quivariantly homotopy equivalent to '. We can therefore pick an equivariant map _ 2 mapK (BC* *^X( )1, BCX^0( 0)1)['] corresponding to 1 2 mapK (BC^N(~)1, BC^N(~)1)[1]. Define fh as the composite fh: BC^X( )-' (BC^X( )1)=ss _=ss--!(BCX^0( 0)1)=ss '-!BC^X0( 0) ' _=K ' 0 and similarly define h : BCX ( ) '- (BC^X( )1)=K ---!'(BCX^0( 0)1)=K -! BCX0( * * ). 18 K. ANDERSEN AND J. GRODAL By construction the maps fh and h fit into following homotopy commutative di* *agram BC^N(~) ej uuu JJJfj0J0 uuuu || JJJ zzuu |fflffl J%%J BC^X( )___ BCN (~) B0==^CX0( 0) __________________j_____________________K0KKKjtt | __________________________________________________* *__________________________________KKKttt| | tt______e_________________________________________* *_________________________________________________________________KKKt| fflffl|y____h_________________________________________* *_____________________________________________________________________________* *_________________________________________________________________%%Kyttfflffl| BCX ( ) ' BC ( 0) 0 _____________________==______________________X ___________________________________________________* *__________________ _________________________________________________* *_____________________________ _____h_________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________________________________________ ' and are in fact uniquely determined up to homotopy by this property by Proposit* *ion 3.1 and the inductive assumption (?). Let f' : BC^X( )! BXf0be the composite of fh with the evaluation BC^X0( 0)! B* *Xf0 and similarly define ' : BCX ( ) ! BX0 as the composite of h with the evaluat* *ion BCX0( 0) ! BX0. We define ' and f' when : BE ! BX is an elementary abelian p-subgroup of r* *ank greater that one, by restricting to a rank one subgroup V E and using adjoint* *ness as follows: As before |V factors through T , uniquely as a map to N , and we let * *~ : BV ! BN denote the resulting map to N . Restriction produces a map h |V 0 0 ' ,V: BCX ( ) ! BCX ( |V ) ---!'BCX0(j ~) ! BX . Similarly, letting BC^X( )denote the homotopy fiber of the map BCX ( ) ! BX ! B* *2ss1(D) as in the rank one case, we define g' ,Vas hg | g' ,V: BC^X( )! BCX^( |V )---V!'BCX^0(j0~)! BXf0. By construction ' ,Vand g' ,Vfit together in that the diagram g' ,V BC^X( )_____//BXf0 | || | | fflffl|' ,Vfflffl| BCX ( )_____//BX0 commutes up to homotopy, and is in fact a homotopy pull-back square by construc* *tion. This concludes the construction of the maps on centralizers which we will use i* *n the next sections to construct our equivalence BX ! BX0. We will in particular prove tha* *t ' ,V and g' ,Vare independent of the choice of the rank one subgroup V E, after wh* *ich we will drop the subscript V . 5. Second part of the proof of Theorem 1.2: The element in lim0 In this section we prove that the maps 'g ,Vconstructed in the previous secti* *on are independent of the choice of rank one subgroup V E and give coherent maps int* *o BXf0. More specifically we prove the following. THE CLASSIFICATION OF 2-COMPACT GROUPS 19 Theorem 5.1. Let X and X0 be two connected simple center-free p-compact groups * *with isomorphic Zp-root data, and assume the inductive hypothesis (?). Then the maps g' ,V: BC^X( )! BXf0 constructed in Section 4 are independent of the choice of V and together form a* *n element in lim0 2A(X)[BC^X( ), BXf0]. Here A(X) is the Quillen category of X with objects the elementary abelian p-* *subgroups : BE ! BX of X and morphisms given by conjugation (i.e., the morphisms from (* * : BE ! BX) to ( 0 : BE0 ! BX) are the linear maps ' : E ! E0 such that is freely homotopic to 0O B'). We need the following proposition, whose proof we postpone to after the rest * *of the proof of Theorem 5.1. Proposition 5.2. Let X be a connected simple center-free p-compact group. If * *: BE ! BX is a non-toral elementary abelian p-subgroup of rank two, then CX ( )1 is no* *n-trivial or DX ~=DPU(p)^p. Proof of Theorem 5.1.We divide the proof into two steps. Step 1 verifies the in* *dependence of the choice of V , and the shorter Step 2 then uses this to construct the ele* *ment in lim0. Step 1: The maps g' ,Vand ' ,Vare independent of the choice of rank one subgrou* *p V : We divide this step into three substeps a-c. Step 1a assumes toral, Step 1b assu* *mes rank two non-toral, and finally Step 1c considers the general case. Step 1a: Assume : BE ! BX is a toral elementary abelian p-subgroup. By assump* *tion factors through BT to give a map ~ : BE ! BN , unique up to conjugation in N * *, and as in the rank one case we let 0= j0~. We want to say that the map g' ,Vdoes not * *depend on V , basically since it is a map suitably under BC^N(~), and hence uniquely d* *etermined, independently of V . This will follow by adjointness, analogously to [8, Pf. o* *f Thm. 2.2], although a bit of care has to be taken, since we have to verify that this happe* *ns over B2ss1(D) in order to be able to pass to the cover f(.), as we now explain. Recall that by construction the map h |V is the bottom left-to-right composit* *e in the following diagram (5.1) BCN (~|VO)OO ppp | OOOO ppp |' OOO ppp | OOOO 0 j |Vppppp OOOj 0|V pp (BC ^(~| ))=K OOOO ppp N V 1O OOO ppp oooo OOOO OOO ppp ooo OOO OOO ppp ooo OOO OOO xxppp' wwo _=K '' ' O'' BCX ( |V )oo____(BCX^( |V )1)=K__'__//_(BCX^0( 0|V)1)=K____//_BCX0( 0|V ) where we notice that all subdiagrams commute up to homotopy over BK. 20 K. ANDERSEN AND J. GRODAL Since BE maps into these spaces via and ~, adjointness produces the followi* *ng diagram (5.2) BCN (~) oo OO PPPPP oooo '|| PPPP oooo | PPPP 0 joooooo PPPjP0P ooo BCCNn(~|V)(~)QQ PPPPP oooo nnnn QQQQ PPPP oooo nnnnn QQQQQ PPPP wwoooo' vvnn ' C(_=K) Q(( ' PP(( BCX ( )oo___BCCX( |V)( )oo_______Z ____'____//_BCCX0( 0|V)(_0)//_BCX0( 0) where Z = map(BE, (BCX^( |V )1)=K) and C(_=K) is the map induced by _=K on map- ping spaces. Since the diagram (5.2)homotopy commutes as a diagram over B2ss1(D) we get a * *ho- motopy commutative diagram by passing to homotopy fibers: (5.3) BC^N(~)OOO qqq | OOOO qqq '| OOO qqq | OOOO ejqqqqq OOOfj0 0 qq BC ^ (~) OOOO qqq CN (~|V)O OOO qqq ppp OOO OOO qqq ppp OOO OOO qqq ppp OOO OOO xxqqq wwpp ^ ''O ''O oo'__ oo__'_____ __C(_=K)_//_ __'__// BC^X( ) BCCX^( |V)( ) eZ ' BCCX^0( 0|V)( 0) BC^X0( 0). Denote the bottom left-to-right homotopy equivalence in (5.3)by C (hg |V), just* *ified by the fact that by construction the following diagram homotopy commutes C (hg |V) (5.4) BC^X( ) __'___//_BC^X0( 0) J | | JJJJ | | JJJ fflffl|gh | fflffl| J$$J ____//V_ _____// BCX^( |V )' BCX^0( 0|V ) BXf0. Diagram (5.3), together with the inductive assumption (?) and Proposition 3.1 s* *hows that the homotopy class of C (hg |V) does not depend on V . Hence by diagram (5.4)th* *e same is true for g' ,Vwhich is what we wanted. (The key point in the above argument is * *that we can choose ~ once and for all, such that ~|V is a factorization of |V through * *BT for every V E.) Note that the construction of C (hg |V) in (5.3)does not depend on being to* *ral, as long ' ,V 0 0 as 0in that case is defined as BE -!BCX ( ) ---! BX (instead of j ~), which ma* *kes sense in this more general setting_the top part of diagram (5.3)is only needed to con* *clude the independence of V . In Step 1b below we will also use the notation C (hg |V) fo* *r non-toral . Step 1b: Assume : BE ! BX is a rank two non-toral elementary abelian p-subgro* *up. By Proposition 5.2 either CX ( )1 is non-trivial or DX ~=DPU(p)^p. THE CLASSIFICATION OF 2-COMPACT GROUPS 21 Assume first that DX ~=DPU(p)^p. Since uniqueness for this group is well know* *n, both for p odd and p = 2 by [21], [43], [11] and [8], the statement of course follows fo* *r this reason. But one can also argue directly, using a slight modification of the proof of [8* *, Lem. 3.2] which we quickly sketch: For ff 2 WX ( ) we have the following diagram hg |V BC^X( ) ________//_BCX^( |V_)__'_______//BCX^0(j0~)__________//BXf0 | | | |||| | | | || fflffl| fflffl|h^ |ff(V ) |fflffl || BC^X( ) _______//_BCX ^( |ff(V))'//_BCX0(j0~^O (ff-1|ff(V)))_//_BXf0. Here all the non-identity vertical maps are given on the level of mapping space* *s (i.e., without the tilde) by f 7! f Off-1, which induces a map on the indicated spaces by taki* *ng homotopy fibers of the map to B2ss1(D). The left-hand and right-hand squares obviously c* *ommute and the middle square commutes up to homotopy by our inductive assumption (?) a* *nd Proposition 3.1. We thus conclude that '^ ,ff(VO)BC^X(ff-1)' 'g ,Vfor all ff 2* * WX ( ). Now, since D ~=DPU(p)^pwe have ss1(D) ~=Z=p, and because E is non-toral, we see* * that BCX ( ) ' BE and BC^X( )' BP , where P is the extra-special group p1+2+of order* * p3 and exponent p if p is odd and Q8 if p = 2. Now [8, Prop. 3.1] shows that WX ( * *) contains SL(E), and the same is true for X0. The proof of [8, Prop. 3.1] furthermore sh* *ows that g' ,VO B^CX (ff)' g' ,Vfor ff 2 SL(E). (Apply the argument there to the p-group* * P instead of E.) Since SL(E) acts transitively on the rank one subgroups of E, combining * *the above gives that ]' ,V'0g' ,Vfor any rank one subgroup V 0 E as desired. We can therefore assume that CX ( )1 is non-trivial, and the proof in this ca* *se is an adaptation of [8, Pf. of Lem. 3.3] to our new setting: Choose a rank one elemen* *tary abelian p-subgroup j : BU = BZ=p ! BCX ( )1 in the center of a p-normalizer of a maxima* *l torus in CX ( ). Let j x : BU x BE ! BX be the map defined by adjointness, and for * *any rank one subgroup V of E, consider the map j x |V : BU x BV ! BX obtained by restri* *ction. By construction j x |V is the adjoint of the composite BU -j!BCX ( )1 res--!BC* *X ( |V )1, so j x |V : BU x BV ! BX factors through a maximal torus in X by [25, Prop. 5.* *6]. It is furthermore straightforward to check that j x is a monomorphism (compar* *e [8, Pf. of Lem. 3.3]). Now consider the following diagram BCX ( |V ) nnnn77nOO|MMMM'M|VM nnnnn | MMM nn | M&&M BU x BEPP____//_BCX (j x |V )8BX08 PPP | qqqqq PPPP | qq'jqq PP''Pfflffl|qq BCX (j) Here the left-hand side of the diagram is constructed by taking adjoints of j x* * and hence it commutes. The right-hand side homotopy commutes by Step 1a (using the induc* *tive assumption (?)), since j x |V is toral of rank two. We can hence without ambig* *uity define (j x )0as either the top left-to-right composite (for some rank one subgroup V* * E) or the bottom left-to-right composite. Denote by 0 the restriction of (j x )0to * *BE. 22 K. ANDERSEN AND J. GRODAL By construction of the map Cjx (hg |V) as the bottom composite in (5.3), the * *diagram Cjx (hg |V) (5.5) BCX^(j x )____'_____//BCX0(^(j x )0) | | | | fflffl| C (hg | fflffl| __________)V___// BC^X( ) ' BC^X0( 0) commutes. Furthermore, since j x |V is toral, diagram (5.3), applied with ~ eq* *ual to a factorization of j x |V through BT , shows that the top horizontal map in (5.5* *)agrees with Cjx (fhj), and in particular it is independent of V , again using Proposit* *ion 3.1 and our inductive assumption (?). We claim that this forces the same to be true for the bottom horizontal map i* *n (5.5): By our choice of j, the centralizer CX (j x ) contains a p-normalizer of a max* *imal torus in CX ( ), and hence CX^(j x )contains a p-normalizer of a maximal torus in ^CX (* *.)Propo- sition 3.2 and our inductive assumption (?) therefore shows that the bottom map* * in (5.5) C (hg |V) is independent of V , so g' ,V: BC^X( )-----! BC^X0( 0)! BXf0is also independe* *nt of V as wanted. Step 1c: Assume : BE ! BX is an elementary abelian p-subgroup of rank 3. Th* *e fact that g' ,Vis independent of V when E has rank two implies the statement in gene* *ral: Let : BE ! BX be an elementary abelian p-subgroup of rank at least three, and sup* *pose that V1 6= V2 are two rank one subgroups of E. Setting U = V1 V2 we get the f* *ollowing diagram BCX^( |V1) sss99sOO|III'^I|V1I sssss | III s | II$$ BC^X( )_____//BCX^( |U ) BXf0:: KK uuu KKK | uu KKK | uuu K%%Kfflffl|^'u|V2u BCX^( |V2) The left-hand side of this diagram is constructed by adjointness and hence comm* *utes, and the right-hand side of the diagram commutes up to homotopy by Steps 1a and 1b. * *Thus the top left-to-right composite ]' ,V1is homotopic to the bottom left-to-right comp* *osite ]' ,V2, i.e., the map g' ,Vis independent of the choice of rank one subgroup V as claim* *ed. Step 2: An element in lim0. With the above preparations in place it is easy to* * see, as in [8, Pf. of Thm. 2.2], that the maps f' : BC^X( )! BXf0fit together to form an e* *lement in lim0 2A(X)[BC^X( ), BXf0]. In order not to cheat the reader of the finale, we repeat the short argument fr* *om [8, Pf. of Thm. 2.2]: For any morphism ae : ( : BE ! BX) ! (, : BF ! BX) in A(X) THE CLASSIFICATION OF 2-COMPACT GROUPS 23 we need to verify that BC^X(ae) BC^X(,)______________//BC^X( ) GGG vvv GGG vvv f',GG##G --vf'vv BXf0 commutes up to homotopy. If F has rank one, then ae is an isomorphism, and we * *let ~ : BF ! BT ! BN be a factorization of , through BT . In this case the claim fo* *llows since fh, (5.6) BC^X(,)___'____//_BCX^0(j0~) MMMM BC^X(ae')|| BC^X0('ae)||MMMMM fflffl|gh,ae fflffl| MM&&M BC^X(,ae)__'___//_BCX^0(j0~ae)___//_BXf0 commutes up to homotopy, since we can view the diagram as taking place under BC* *^N(~)-'! BC^N(~ae), up to homotopy, using the inductive assumption (?) and Proposition 3* *.1 as in Step 1a. The case where F has arbitrary rank follows from the independence of t* *he choice of rank one subgroup V , established in Step 1, together with the rank one case* *: For a rank one subgroup V of E set V 0= ae(V ) and consider the diagram BC^X(,)_______//_BCX^(,|VM0) MMM BC^X(ae)|| BC^X(ae|V)|| MMMMM fflffl| fflffl| MMM&& BC^X( ) _______//_BCX^( |V_)____//BXf0 The left-hand side commutes by construction and the right-hand side commutes si* *nce the diagram (5.6)commutes, proving the claim. This constructs an element [#] 2 lim0 2A(X)[BC^X( ), BXf0] as wanted. We now give the proof of Proposition 5.2, used in the proof of Theorem 5.1, w* *hich we postponed. The proof uses case-by-case arguments on the level of Zp-root data. Proof of Proposition 5.2.Assume first that (WX , LX ) is an exotic Zp-reflectio* *n group. We claim that the centralizer of any rank one elementary abelian p-subgroup of X i* *s connected, which in particular implies that there can be no rank two non-toral elementary * *abelian p-subgroups: For p > 2 this follows from [8, Thms. 11.1, 12.2(2) and 7.1] combi* *ned with Dwyer-Wilkerson's formula for the Weyl group of a centralizer [26, Thm. 7.6]. F* *or p = 2 we have DX ~=DDI(4)and hence WX ~=Z=2xGL 3(F2) where the central Z=2 factor acts b* *y -1 on LX and GL 3(F2) acts via the natural representation on LX F2, cf. [8, Pf o* *f Thm. 11.1] or [28, Rem. 7.2]. In particular it follows directly (cf. [28, Pf. of Prop. 9.1* *2, DI(4) case]) that X contains a unique elementary abelian 2-subgroup of rank one up to conjug* *ation and that the centralizer of this subgroup is connected. By the classification of Zp-root data, Theorem 8.1(1), we may thus assume tha* *t DX is of the form DG^pfor some simple compact connected Lie group G, and since X is cent* *er-free we may assume that so is G. If ss1(G) has no p-torsion, [59, Thm. 2.27] implies* * that the 24 K. ANDERSEN AND J. GRODAL centralizer in G of any element of order p is connected. By the formula for the* * Weyl group of a centralizer [26, Thm. 7.6] (cf. Proposition 8.4(3)), it then follows that CX * *(j) is connected for any rank one elementary abelian p-subgroup j : BZ=p ! BX of X, and hence X * *does not have any rank two non-toral elementary abelian p-subgroups. We can thus furthermore assume that ss1(G) has p-torsion, which implies that * *DX ~=DG^p for one of the following G: G = PU (n) for p|n; G = SO(2n+1), n 2 for p = 2; * *G = PSp(n), n 3 for p = 2; G = PSO (2n), n 4 for p = 2; G = P E6 for p = 3 and G = P E7* * for p = 2. We want to see that if j : BZ=p ! BX has rank one and DX 6~=DPU(p)^p, then CX (* *j) is not a p-compact toral group, since then [23, Cor. 1.4] implies that for any element* *ary abelian p-subgroup of rank two, CX ( )1 is non-trivial. By [26, Thm. 7.6] it is enough to see this for the corresponding Lie group G.* * So, let V G be a rank one elementary abelian p-subgroup of G. If CG (V )1 is a toru* *s then WCG(V )1= 1 and hence WCG(V )~=ss0(CG (V )). By [9, x5, Ex. 3(b)] or [58, Thm. * *9.1(a)] ss0(CG (V )) is isomorphic to a subgroup of ss1(G), so |WCG(V )| divides |ss1(G* *)|. In particular, if x is a generator of V then the number of elements in a maximal torus which a* *re conjugate to x in G is at least |WG |=|ss1(G)| (since two elements in a maximal torus are* * conjugate in G if and only if they are conjugate by a Weyl group element). In particular |WG | pn - 1 _______, |ss1(G)| where n is the rank of G. A direct case-by-case check of the above cases shows * *that this inequality can only hold when G = PU (n), p|n. In this case, a generator of V m* *ust have the form x = [diag(~1, . .,.~n)]. For n > p, some of the ~'s must agree, so CG * *(V )1 is not a torus in this case. This proves the claim. 6.Third and final part of the proof of Theorem 1.2: Rigidification In this section we finish the proof of Theorem 1.2 by showing that our elemen* *t in lim0 from Theorem 5.1 rigidifies to produce a homotopy equivalence BX ! BX0. We firs* *t need a lemma: Lemma 6.1. Suppose that we have a homotopy pull-back square of p-compact groups f0 BX0 _____//BY 0 |g0| |g| fflffl|f fflffl| BX _____//BY where f : BX ! BY is a centric monomorphism (i.e., map (BX, BX)1 '-!map(BX, BY * *)f and Y=X is Fp-finite) and g : BY 0! BY is an epimorphism (i.e., Y=Y 0is the cla* *ssifying space of a p-compact group). Then f0 : BX0! BY 0is a centric monomorphism, and * *g0 is an epimorphism. Proof.It is clear from the pull-back square that f0 is a monomorphism and that * *g0 is an epimorphism. To see that f0 is centric, observe that we have a map of fibrations (Y=Y 0)hX0____//map(BX0, BX0)[g0]____//map(BX0, BX)g0 || | | || | | || 0 fflffl| fflffl| (Y=Y 0)hX_____//map(BX0, BY 0)[fOg0]//_map(BX0, BY )fOg0 THE CLASSIFICATION OF 2-COMPACT GROUPS 25 where e.g., the subscript [g0] denotes the components of the mapping space mapp* *ing to the component of g0. Since the wanted map map (BX0, BX0)1 ! map (BX0, BY 0)f0is* * the restriction of the map of total spaces to a component, it is hence enough to se* *e that the map between the base spaces is an equivalence, which follow since we have equiv* *alences map (BX, BX)1 __'___//map(BX0, BX)g0 |'| || fflffl| fflffl| map (BX, BY )f__'__//map(BX0, BY )fOg0. Here the vertical equivalence follows from the centricity of f and the horizont* *al equivalences follows from [18, Prop. 3.5] combined with [26, Prop. 10.1] (in [18, Prop. 3.5]* * taking E = BX0, B = BX, and X = BX and BY respectively). Proof of Theorem 1.2.By the results of Section 2 (Propositions 2.1 and 2.4) we * *are reduced to the case where we consider two simple, center-free p-compact groups BX and B* *X0 with the same root datum D. Furthermore, in Theorem 5.1 of the previous section we c* *onstructed an element [#] 2 lim0 2A(X)[BC^X( ), BXf0]. We want to use this element to cons* *truct the map from BX to BX0, and show that it is an equivalence. By the classification o* *f Zp-root data, Theorem 8.1(1), D is either exotic or D ~=DG^pfor a compact connected Lie* * group G and we handle these two cases separately. Suppose that D is exotic, and notice that in this case ss1(D) = 0 by Theorem * *8.1(2). If p is odd we are exactly in the situation covered by rather easy arguments in* * [8] (see [8, Pf. of Thm. 2.2] and [8, Prop. 9.5]). For p = 2 uniqueness of B DI(4), as well * *as the state- ments about self-maps, are already well known by the work of Notbohm [53], but * *we never- theless quickly remark how a proof also falls out of the current setup, noticin* *g that the argu- ments from [8] from this point on, in the special case of DI(4), carries over v* *erbatim: We have D ~=DDI(4)and since ss1(D) = 0, the functor 7! ssi(map (BC^X( ), BXf0) ) iden* *tifies with 7! ssi(BZ(CX ( ))), as explained in detail in [8, Pf. of Thm. 2.2]. Since we * *are considering X = DI(4) we know by Dwyer-Wilkerson [24, Prop. 8.1] that limjA(X)ssi(BZ(CX ( )* *)) = 0 for all i, j 0. (The proof of this is a Mackey functor argument, and relies o* *n the regular structure of the Quillen category of DI(4), due to the fact that its classifyin* *g space, like the exotic p-compact groups for p odd, has polynomial Fp-cohomology ring.) The cent* *ralizer decomposition theorem [26, Thm. 8.1] now produces a map BX ! BX0, which by stan- dard arguments given in [8, Pf. of Thm. 2.2] is seen to be an equivalence. The * *statement about self-maps also follows as in [8, Pf. of Thm. 2.2], using Out(BN , {BNoe})* * instead of Out(BN ). Now, suppose that D ~=DG^p, for a simple center-free compact Lie group G, wit* *h universal cover eG. Let Orp(Ge) be the full subcategory of the orbit category with object* *s the eG-sets eG=Pewith ePa p-radical subgroup (i.e, ePis an extension of a torus by a finite* * p-group, such that NGe(Pe)=Peis finite and contains no non-trivial normal p-subgroups). For the p-radical homology decomposition [32, Thm. 4] of the compact Lie grou* *p eG, one considers the functor F : Orp(Ge) ! Spaces given by eG=Pe7! EGexGeeG=Pe, where * *Spaces denotes the category of topological spaces. Viewed as a functor to the homotopy* * category of spaces, Ho(Spaces ), this functor is isomorphic to the functor F 0: Orp(Ge) * *! Ho(Spaces ) given on objects by eG=Pe7! BPeand on morphisms by sending the eG-set map eG=Pe* *-f!eG=Qe 26 K. ANDERSEN AND J. GRODAL to the map cg-1 : BPe ! BQe, where gQe = f(ePe), via the canonical equivalences* * BPe = (EPe)=Pe-'!EGexGeeG=Pe. (Note that F 0is not well-defined as a functor to Space* *s, since the element g is just an arbitrary coset representative for the coset gQe.) We* * can hence in what follows replace EGexGeeG=Pe by BPe in this way, whenever we are working* * in the homotopy category. Since Pe and Qe are p-radical, the same is true for their images P and Q in G* *, and CG (P ) = Z(P ) and likewise for Q (see [32, Prop. 1.6(i) and Lem. 1.5(ii)]). * *Hence there is a well defined induced morphism cg : pZ(Q) ! pZ(P ) as well as a well-define* *d (free) homotopy class of maps cg-1: BCG ^(pZ(P ))! BCG ^(pZ(Q)). Consider the diagram _____'_____// BPe^p__________//(BCG ^(pZ(P )))^p BCG^^p(i(pZ(P))) | QQQQ'i^ cg-1|| |cg-1| BCG^(cg)| QQQ(pZ(P))QQ p | QQ fflffl| fflffl| fflffl| QQQQ((Q _____'_____// ___________// BQe^p__________//(BCG ^(pZ(Q)))^p BCG^^p(i(pZ(Q)))'^ BXf0 i(pZ(Q)) where iV : BV ! BG^pdenotes the map induced by the inclusion of a subgroup V * *G, and where the horizontal maps in the middle square are given by lifting the sta* *ndard homotopy equivalence given by adjointness to the covers. The first two squares* * are ho- motopy commutative by construction, and the right-hand triangle commutes since * *[#] 2 lim0 2A(X)[BC^X( ), BXf0]. Hence this diagram produces an element [i] 2 lim0eG=* *Pe2Or[BeeP^p, BXf0]. ' * * p(G) i(pZ(P)) Denote the composition BPe^p! BCG ^(pZ(P ))^p! BCG^^p(ipZ(P))------!BXf0by e_Ge* *=Pe (i.e., the coordinate of [i] corresponding to eG=Pe), and let _G=P : BP^p! BX0 * *be the map constructed analogously, using 'i(pZ(P))instead. We want to lift [i] to a map hocolimeG=Pe2Orp(Ge)(EGexGeeG=Pe)^p! BXf0. By [10, Prop. XII.4.1 and XI.7.1] (see also [66, Prop. 3] or [34, Prop. 1.4]) t* *he obstructions to doing this lie in limi+1eG=Pe2Orssei(map (BPe^p, BXf0) e ), i 1. p(G) _Ge=Pe By construction, e_Ge=Peand _G=P fit into a homotopy pull-back square e_Ge=Pe BPe^p____//_BXf0 | | | | fflffl|_G=fflffl|P BP^p ____//_BX0 . By [12, Lem. 4.9(1)] the map _G=P is centric, so Lemma 6.1 implies that e_Ge=Pe* *is centric as well. Hence by centricity and naturality, the functor eG=Pe7! ssi(map (BPe^p* *, BXf0)_ee ) * * G=Pe identifies with the functor eG=Pe7! ssi-1(Z(Pe)^p). Since eGis simple and simpl* *y connected it now follows from the fundamental calculations of Jackowski-McClure-Oliver [32, * *Thm. 4.1] THE CLASSIFICATION OF 2-COMPACT GROUPS 27 that limi+1Orssei-1(Z(Pe)^p) = 0, fori 1. p(G) Hence by the homology decomposition theorem [32, Thm. 4] we get a map BGe^p'-(hocolimeG=Pe2Orp(Ge)(EGexGeeG=Pe)^p)^p! BXf0 which by construction is a map under BNfp, the p-normalizer of a maximal torus * *in BGe^p. Dividing out by Z(D) as explained in Construction 2.2 produces the homotopy com* *mutative diagram BNp G xxxx GGG xx GGG --xx GG## BG^p______________//BX0. It is now a short argument, given in detail in [8, Pf. of Thm. 2.2], to see tha* *t BX = BG^p! BX0 is a homotopy equivalence as wanted. We want to show that Out(BX) ! Out(DX ) is an isomorphism: To see surjectivit* *y, note that if ff 2 Out(DX ), then by [4, Thm. B] and [8, Prop. 5.1], ff corresponds t* *o a unique map ff02 Out(BN , {BNoe}). Hence if j : BN ! BX is a maximal torus normalizer, * *then repeating the above argument with respect to the two maps j : BN ! BX and j O f* *f0 : BN ! BX gives a map BX ! BX realizing ff 2 Out(DX ). Finally we show injectivit* *y, essentially repeating the argument of Jackowski-McClure-Oliver, cf. [32, Pf. of* * Thm. 4.2]: We are assuming that BX ' BG^p, for some compact connected center-free Lie grou* *p G. As in the proof of Proposition 2.4(3), we have the following commutative diagram Out (BG^p)____//_Out(DG^p) | | | | fflffl| fflffl| Out (BGe^p)___//_Out(DGe^p). (Compare diagram (2.1), and note that we use that gDG^p= DfG^pfor the right-han* *d vertical map, which uses Theorem 8.6 for compact Lie groups.) The left-hand vertical map* * in the above diagram is injective since factoring out by the center (via the quotient * *construction recalled in Construction 2.2) provides a left inverse (in fact an actual invers* *e, though we do not need this here). So we just have to see that Out(BGe^p) ! Out(DGe^p) is inj* *ective. By [4, Thm. B], this map factors through Out(BNe, {BNeoe}) and Out(BNe, {BNeoe}) ! Out* *(DGe^p) is an isomorphism, where eNdenotes the maximal torus normalizer in eG^p. By the* * homology decomposition theorem [32, Thm. 4] and obstruction theory [32, Thm. 3.9] (cf. [* *66, Prop. 3]), injectivity of Out(BGe^p) ! Out(BNe) follows from Jackowski-McClure-Oliver's ca* *lculation of higher limits [32, Thm. 4.1] limieG=Pe2Orssei-1(Z(Pe)^p) = 0, i 1. p(G) We conclude that Out(BG^p) ! Out(DG^p) is also injective as claimed. Finally, the last statement in Theorem 1.2 about the homotopy type of B Aut(B* *X) fol- lows by combining [4] with what we have proved so far: The Adams-Mahmud map fac* *tors as : B Aut(BX) ! B Aut(BN , {BNoe}) ! B Aut(BN ), where B Aut(BN , {BNoe}) is the 28 K. ANDERSEN AND J. GRODAL covering of B Aut(BN ) with respect to the subgroup Out(BN , {BNoe}) of the fun* *damental group. Furthermore [4, x5] explains how killing elements in ss2(B Aut(BN , {BNo* *e})) con- structs a space denoted B aut(DX ), whose universal cover is B2Z(DX ), where BZ* *(DX ) = (BZ~(DX ))^p. It it furthermore shown there that the fibration B aut(DX ) ! B O* *ut(DX ) is split, i.e., B aut(DX ) has the homotopy type of (B2Z(DX ))hOut(DX). Compositio* *n gives a map B Aut(BX) ! B aut(DX ), which is an isomorphism on ssi for i > 1 by constru* *ction, and an isomorphism on ss1 by what we have shown in the first part of the theore* *m. This concludes the proof of the main Theorem 1.2. Remark 6.2 (The fundamental group of a p-compact group). In [8], with Moller and Viruel, we established the fundamental group formula for p-compact groups, Theo* *rem 8.6, for p odd, as a consequence of the classification. In this remark we sketch ho* *w one by expanding the proof of Theorem 1.2 somewhat can avoid the reliance on Theorem 8* *.6, hence proving Theorem 8.6 also for p = 2 this way_this was the strategy which w* *e had originally envisioned before Dwyer-Wilkerson [29] provided an alternative direc* *t proof of Theorem 8.6 as we were writing this paper. The fundamental group formula was not used in the reduction to simple center-* *free groups, except for a reference in the proof of Proposition 2.4(3), where it, fo* *r the purpose of inductively proving Theorem 1.2, was only needed in the well-known case of c* *ompact Lie groups. Hence we can assume that X and X0are simple center-free p-compact group* *s with the same Zp-root datum D, and that we know the fundamental group formula for X * *and Theorem 1.2 for connected p-compact groups of lower cohomological dimension tha* *n X. In the last section we constructed an element [#] 2 lim0 2A(X)[BC^X( ), BXf0]. By * *not passing to a universal cover, and hence not using the fundamental group formula, a simplif* *ied version of the same argument gives an element [~#] 2 lim0 2A(X)[BCX ( ), BX0 ]. The only n* *on-obvious change is that since the center formula in Lemma 8.7 does not hold in the prese* *nce of direct factors isomorphic to DSO(2n+1)^2in the Z2-root datum, one has to take the targ* *et of in (4.4)to be BZ(DCX( )1) (obtained as a quotient of BZ(CN (~)1); cf. [4, Lem. 5.1* *]) to obtain a homotopy equivalence. With this change the rest of Section 4 proceeds as bef* *ore, but ignoring everything on the level of covers, and one constructs a map like befor* *e without any choices. Section 5 has to be modified in the following way: Instead of hav* *ing maps being under the maximal torus normalizer (which we now do not a priori know), w* *e utilize instead that potentially different maps agree in M = (map (Bss, B aut(D)))hAut(Bss), where B aut(D) = (B2Z(D))hOut(D), which means that they are homotopy equivalent* *, by the description of self-equivalences of non-connected groups, cf., Theorem 1.6,* * which we know by induction. From the element in lim0one can easily get an isomorphism be* *tween fundamental groups: Consider the diagram i HZp*(BNp)UU jiiiiiiii | UUUUUUj0UU iiii | UUUUUU ttttiiiiii~= fflffl| [~#UU****U] HZp*(BX) oo_______colim 2A(X)HZp*(BCX ( ))________//HZp*(BX0) where the vertical map is given by choosing a central rank one subgroup ae of t* *he p- normalizer Np and considering the corresponding inclusion Np ! CX (jOae). (Here* * HZpn(Y ) = THE CLASSIFICATION OF 2-COMPACT GROUPS 29 limHn(Y ; Z=pn).) Note that the maps j and j0 are surjective by a transfer argu* *ment, and that the indicated isomorphism follows by the centralizer homology decompositio* *n theorem [26, Thm. 8.1]. This shows that the kernel of j is contained in the kernel of j* *0. However, since we could reverse the role of X and X0in all the previous arguments (we ha* *ve not used any special model for X), we conclude by symmetry that the kernel of j equals t* *he kernel of j0, and in particular the kernels of the maps j : ss1(D) ! ss1(X) and j0: ss* *1(D) ! ss1(X0) agree, and they are surjective by Proposition 8.5. If D is exotic then ss1(D) =* * 0 by The- orem 8.1(2), so there is nothing to prove. If D is not exotic, then by Theorem* * 8.1(1), D = DG Z Zp for a compact connected Lie group G and so we can assume BX ' BG^p. ~= Hence j : ss1(D) -! ss1(X) by Lie theory (cf., [9, x4, no. 6, Prop. 11]), so th* *e same holds for X0. 7.Proof of the corollaries of Theorem 1.2 In this section we prove the Theorems 1.3-1.6 from the introduction. The firs* *t theorem is the maximal torus conjecture: Proof of Theorem 1.3.The proof in the connected case is an extension of [8, Pf.* * of Thm. 1.10], where a partial result excluding the case p = 2 was given: Assume that (X, BX, * *e) is a con- nected finite loop space with a maximal torus i : BT ! BX. Let W denote the se* *t of conjugacy classes of self-equivalences ' of BT such that i' is conjugate to i. * *It is straight- forward to see (consult e.g., [8, Pf. of Thm. 1.10]) that BT^p! BX^pis a maxima* *l torus for the p-compact group X^pand that Fp-completion allows us to identify (WX^p, * *LX^p) with (W, L Zp), where L = ss1(T ). In particular (W, L) is a finite Z-reflection g* *roup and all reflections have order 2. Furthermore, for a fixed Z-reflection group (W, L), t* *here is a bijec- tion between Z-root data with underlying Z-reflection group (W, L) and Z2-root * *data with underlying Z2-reflection group (W, L Z2) given by the assignments D 7! D Z Z* *2 and (W, L Z2, {Z2boe}) 7! (W, L, {L \ Z2boe}), as is seen by examining the defini* *tions. Let D be the Z-root datum with underlying Z-reflection group (W, L) corresponding to * *DX^2. By the classification of compact connected Lie groups, cf. [9, x4, no. 9, Pro* *p. 16], there is a (unique) compact connected Lie group G with maximal torus i0: T ! G induci* *ng an isomorphism of Z-root data DG ~=D. The Zp-root data of X^pand G^pare isomorphic* * at all primes, since the root data at odd primes are determined by (W, L Zp). Th* *eorem 1.2 hence implies that for each p, we have a homotopy equivalence 'p : BX^p! BG^psu* *ch that BT^pF i^pxxx FFi0^pF xx FFF --xxx 'p F"" BX^p _____________//_BG^p commutes. As in [8, Pf. of Thm. 1.10] we see that H*(BX; Q) ! H*(BT ; Q)W is * *an isomorphism and since the same is true for BG we also have a map BXQ ! BGQ under BT . We have the following diagram Q Q (7.1) pBX^p ____//_( pBX^p)Qoo_BXQ '|| |'| |'| Q fflffl| Qfflffl| fflffl| pBG^p ____//_( pBG^p)Qoo_BGQ. 30 K. ANDERSEN AND J. GRODAL The left-hand square in this diagram is homotopy commutative by construction. F* *or the right-hand side note that, since all maps in (7.1)are under BT , the following * *diagram commutes Q H*( pBX^p; Q)oo___________________ H*(BX; Q) | hhQQQQQ nnn77n | | QQQQ nnnn | | QQQQ nnnn ~= | |~=| H*(BT ; Q)W ~=|| | mm PPP | | mmmm PPP~=P | | mmmm PPP | Q fflffl|vvmmm P''P fflffl| H*( pBG^p; Q)oo__________________ H*(BG; Q). This implies commutativity on the level of homotopy groups, and since the invol* *ved spaces are all products of rational Eilenberg-Mac Lane spaces (since they have homotop* *y groups only in even degrees) this implies that the diagram (7.1)homotopy commutes. By * *changing the maps up to homotopy we can hence arrange that the diagram strictly commutes* *, and taking homotopy pull-backs produces a homotopy equivalence BX ! BG, by the Sull* *ivan arithmetic square[10, VI.8.1], as wanted. This proves that every connected fini* *te loop space BX with a maximal torus is homotopy equivalent to BG for some compact connected* * Lie group G, and in the course of the analysis, we furthermore saw that G is unique* * (since we can read off the Z-root datum of G from BX). We now give the description of B Aut(BG), also providing a quick description * *of how the p-completed results are used. By [26, Thm. 1.4] B2Z(G) ' B Aut1(BG). (This is a* * conse- quence of that B2Z(G^p) ' B Aut1(BG^p).) By [33, Cor. 3.7] Out(G) ~=Out(BG). (S* *ince (G=T )hT is homotopically discrete with components the Weyl group, as in [8, Le* *m. 4.1] and [49, Thm. 2.1], any map f : BG ! BG gives rise to a map ' : BT ! BT over f, uni* *que up to Weyl group conjugation and since '^p2 Aut(DG^p) for all p one sees that ' 2 * *Aut(DG ); now ' determines the collection {'^p} which by p-complete results determines {f* *^p}, which again determines f by the arithmetic square.) Finally by a theorem of de Sieben* *thal [15, Ch. I, x2, no. 2] (see also [9, x4, no. 10]), the short exact sequence 1 ! G=Z(* *G) ! Aut(G) ! Out(G) ! 1 is split, so the fibration B Aut(BG) ! B Out(BG) has a section. (See* * also [4, x6].) Taken together these facts establish that B Aut(BG) ' (B2Z(G))hOut(G)as c* *laimed. The non-connected case follows easily from the connected, using our knowledge* * of self- maps of classifying spaces of compact connected Lie groups: Suppose that X is p* *otentially non-connected, let X1 be the identity component, and ss the component group. No* *te that homotopy classes of spaces Y such that P1(Y ) is homotopy equivalent to Bss and* * the fiber Y <1> is homotopy equivalent to BX1 can alternatively be described as homotopy * *classes of fibrations Z ! Y ! K, with Z is homotopic to BX1 and K is homotopic to Bss. * *(A homotopy equivalence of fibrations means a compatible triple of homotopy equiva* *lences between fibers, total spaces, and base spaces.) By the first part of the proof we know that BX1 ' BG1 for a unique compact co* *nnected Lie group G1. Homotopy classes of fibrations with base homotopy equivalent to B* *ss and fiber homotopy equivalent to BX1 are classified by Out(ss)-orbits on the set of* * free homotopy classes [Bss, B Aut(BG1)]. By the results in the connected case the space B Aut* *(BG1) sits in a split fibration B2Z(G1) ! B Aut(BG1) ! B Out(G1). THE CLASSIFICATION OF 2-COMPACT GROUPS 31 Hence Out(ss)-orbits on the set [Bss, B Aut(BG1)] correspond to (Out (ss) x Out* *(G1))-orbits on the set a H2ff(ss; Z(G1)). ff2Hom(ss,Out(G1)) This agrees with the classification of isomorphism classes of group extensions * *of the form 1 ! H ! ? ! K ! 1, where H is isomorphic to G1 and K is isomorphic to ss. (An isomorphism of group extensions is here a compatible triple of isomorphisms.) * *Since the identity component H is necessarily a characteristic subgroup, isomorphism clas* *ses of group extensions as above are in one-to-one correspondence with isomorphism classes o* *f compact Lie groups with identity component isomorphic to G1 and component group isomorp* *hic to ss. These equivalences put together give that there is a one-to-one corresponde* *nce between homotopy classes of spaces Y such that P1(Y ) ' Bss and Y <1> ' BG1 and isomorp* *hism classes of compact Lie groups with identity component isomorphic to G1 and comp* *onent group isomorphic to ss. Hence our BX is homotopy equivalent to BG for a unique * *compact Lie group G, completing the proof of the theorem. Proof of Theorem 1.4.Let Y be a space such that H*(Y ; F2) is a graded polynomi* *al algebra of finite type. Let V = H1(Y ; F2) (dual to H1(Y ; F2)) and let Y 0denote the* * fiber of the classifying map Y ! BV . Clearly Y 0is connected and since ss1(BV ) is a * *finite 2- group it follows from [17] that the Eilenberg-Moore spectral sequence for the f* *ibration Y 0! Y ! BV converges strongly to H*(Y 0; F2). The map H*(BV ; F2) ! H*(Y ; F2)* * is an isomorphism in degree 1 and hence injective since H*(Y ; F2) is a polynomial* * algebra, so H*(Y ; F2) is free over H*(BV ; F2). Hence the spectral sequence collapses and * *we get an isomorphism of rings (but not necessarily of algebras over the Steenrod algebra* *) H*(Y ; F2) ~= H*(Y 0; F2) H*(BV ; F2). In particular H1(Y 0; F2) = 0 so by [10, Prop. VII.3* *.2] Y 0is F2- good and ss1(Y 0^2) = 0. So to prove the theorem, we can without restriction as* *sume that Y 0 is F2-complete and simply connected. Write ss2(Y 0) ~=F T , where F is a finitely generated free Z2-module and T* * is a finite abelian 2-group, and let Y 00be the fiber of the map Y 0! B2F . The induced hom* *omorphism H2(Y 0; F2) ! H2(B2F ; F2) is an epimorphism so H*(B2F ; F2) ! H*(Y 0; F2) is i* *njective. As above we obtain an isomorphism H*(Y 0; F2) ~=H*(Y 00; F2) H*(B2F ; F2) as * *rings. By construction Y 00is simply connected, ss2(Y 00) is finite, and by the fibr* *e lemma [10, Lem. II.5.1] Y 00is F2-complete. Since H*(Y 00; F2) is polynomial, the Eilenber* *g-Moore spec- tral sequence shows that H*( Y 00; F2) is F2-finite, so Y 00is the classifying * *space of a con- nected 2-compact group. The first part of Theorem 1.4 now follows from the clas* *sification Theorem 1.1. The second part follows from this using the calculation of the mod* * 2 coho- mology of the simple simply connected Lie groups, cf. [36, Thm. 5.2]. Remark 7.1. In addition to the list in Theorem 1.4, the only polynomial rings a* *rising as H*(BG; F2) for a simple compact connected Lie group G are F2[x2, x3, . .,.xn* *] for G = SO (n), n 5, and F2[x2, x3, x8, x12, . .,.x8n+4] for G = PSp (2n + 1), n 0,* * cf. [36, Thm. 5.2]. It is conceivable that any graded polynomial algebra of finite type * *which is the mod 2 cohomology ring of a space, is a tensor product of these factors and the * *ones listed in Theorem 1.4. Proof of Theorem 1.5.The first statement claims that a polynomial F2-algebra A** * with given action of the Steenrod algebra A2, can be realized by at most one space Y* * , up to F2-equivalence, if A* has all generators in degree 3. For this notice that, a* *s in the proof of Theorem 1.4, the assumptions assure that Y ^2' BX for a simply connected 2-c* *ompact 32 K. ANDERSEN AND J. GRODAL group X. Using the classification Theorem 1.1, the statement can now easily be * *checked as done in Proposition 7.2 below. We now prove the second statement, that for any polynomial F2-algebra P *, th* *ere are only finitely many spaces Y , up to F2-equivalence, with H*(Y ; F2) ~= P *as ri* *ngs. By the proof of Theorem 1.4, we can assume that Y is 2-complete, and any such Y fi* *ts in a fibration sequence Y 0! Y ! BV where V = H1(Y ; F2). It also follows that H*(Y * *0; F2) is a polynomial ring, uniquely determined as an algebra over the Steenrod algeb* *ra from H*(Y ; F2). In particular Y 0' BX for a connected 2-compact group X. By Proposi* *tion 8.18, Out(DX ) only contains finitely many 2-subgroups up to conjugation. Hence the d* *escription of B Aut(BX) in Theorem 1.2 implies that [BV, B Aut(BX)] is finite, so it is en* *ough to see that there are only a finite number of possibilities for Y 0given H*(Y 0; F* *2) as an algebra over the Steenrod algebra. This again follows easily from the classification o* *f 2-compact groups: The rank of X is bounded above by the Krull dimension of the cohomology* * ring, so by the classification of p-compact groups, Theorem 1.2, it is hence enough t* *o see that there are only a finite number of Z2-root data with rank less than a fixed rank* *. This is the result of Proposition 8.17. We now give a proof of the auxiliary uniqueness result referred to in the pro* *of of Theo- rem 1.5. Proposition 7.2. Suppose X is a 2-compact group of the form BX ' BG^2x BDI(4)s,* * for a simply connected compact Lie group G and s 0, such that H*(BX; F2) is a pol* *ynomial algebra. Then X has a unique maximal elementary abelian 2-subgroup : BE ! BX,* * and the Weyl group (W ( ), E) together with the homomorphism H6(BX; F2) ! H6(BE; F2) is an invariant of H*(BX; F2) as an algebra over the Steenrod algebra A2, which* * uniquely determines BX up to homotopy equivalence. In particular BX is uniquely determin* *ed up to homotopy equivalence by H*(BX; F2) as an A2-algebra. Proof.By Lannes theory [37, Thm. 0.4] homotopy classes of maps from classifying* * spaces of elementary abelian 2-groups to BX are determined by H*(BX; F2) as an algebra ov* *er the Steenrod algebra. Furthermore, the fact that H*(BX; F2) is assumed to be a poly* *nomial algebra, guarantees that there is only one maximal elementary abelian 2-subgrou* *p : BE ! BX, up to conjugation, by [54, Cor. 10.7] together with the fact that this is t* *rue for B DI(4). (This can also be deduced using the unstable algebra techniques of [2] and [31]* *, or simply by inspecting the calculations of Griess [30] below.) Let (W ( ), E) denote its* * Weyl group, which also only depends on H*(BX; F2), and we view this as a pair with W ( ) a * *subgroup of GL (E). By [36, Thm. 5.2], G is a direct product of the groups SU (n), Sp(n)* *, Spin(7), Spin(8), Spin(9), G2 and F4. In these cases, if E is the maximal elementary ab* *elian 2- subgroup of G, WG^2( ) = NG (E)=CG (E) whose structure is well-known in these c* *ases, e.g., by computations of Griess [30, x5, Thm. 6.1 and Thm. 7.3]. (See also [63, Prop.* * 3.2] for details on the cases Spin(8) Spin(9) F4.) Since WDI(4)( ) = GL 4(F2) by con* *struction [24], it follows that the group (W ( ), E) is a direct product of the following* * (with matrix groups acting on columns): WSU(n)( ) = ( n, Vn0-1), WSp(n)( ) = ( n, Vn), WG2( ) = GL 3(F2), WDI(4)( ) = GL 4(F2), THE CLASSIFICATION OF 2-COMPACT GROUPS 33 2 3 2 1 0 | * * * 3 __1__|*__*__*__ 6 | 7 6 0 | 7 6__0__1_|_*__*__*__7 WSpin(7)( ) = 64 0 | 7, WSpin(8)( ) = 66 0 0 | 77, |GL 3(F2) 5 4 0 0 | GL (F ) 5 0 || 0 0 || 3 2 | 2 3 2 3 1 * | * * * |* * * 66_0__1_||*__*__*__77 66GL_2(F2)_||**__*__77 WSpin(9)( ) = 66 0 0 | 77, WF4( ) = 66 0 0 | 77. 4 0 0 ||GL 3(F2)5 4 0 0 ||GL3(F2) 5 0 0 || 0 0 || Here Vn is the n-dimensional permutation module for F2[ n] and Vn0-1is the (n -* * 1)- dimensional submodule consisting of elements with coordinate sum 0. The Weyl g* *roup WSp(n)( ) = ( n, Vn) decomposes as ( n, Vn0-1) x (1, L), L = Vn n ~=F2, when n * *is odd, n 3. However, after this decomposition, all the listed pairs satisfy that V i* *s indecompos- able as an F2[W ]-module. (Note that this is a priori stronger than just saying* * that (W, V ) does not split as a product; however, it follows from unstable algebra techniqu* *es [48, Secs. 5 and 7] that (W, V ) is an F2-reflection group, and hence the two notions are ac* *tually equiv- alent.) By the Krull-Schmidt theorem [14, 6.12(ii)] this decomposition V = V1 * * . . .Vm as F2[W ]-modules is unique up to permutation, and since Wiis characterized as * *the point- wise stabilizer of V1 . . .bVi . . .Vm , we get that the decomposition of (W* *, V ) as a product is unique as well, up to permutation. Thus the structure of (W ( ), * *E) as a product of finite indecomposable groups, is an invariant which almost character* *izes BX up to homotopy equivalence, except that for n odd, n 3, the group ( n, Vn0-1)* * arises from both SU(n)^2and Sp(n)^2. However since H6(B SU(n); F2) ! H6(BVn0-1; F2) is* * injec- tive and H6(B Sp(n); F2) ! H6(BVn0-1; F2) is trivial, we conclude that (W ( ), * *E) together with the homomorphism H6(BX; F2) ! H6(BE; F2) characterizes BX up to homotopy equivalence. This proves the proposition since both data are determined by the * *A2-action on H*(BX; F2). Proof of Theorem 1.6.It is obvious that there is a one-to-one correspondence be* *tween iso- morphism classes of p-compact groups with identity component isomorphic to X1 a* *nd com- ponent group isomorphic to ss, and equivalence classes of fibration sequences F* * ! E p-!B with F homotopy equivalent to BX1 and B homotopy equivalent to Bss. It is like* *wise obvious that in this case B Aut(E) is homotopy equivalent to B Aut(p), where Au* *t(p) is the space of self-homotopy equivalences of the fibration p. By the classification of fibrations (see [20]), equivalence classes of such f* *ibrations are in one-to-one correspondence with Out(B)-orbits on [B, B Aut(F )], and the space B* * Aut(p) equals (map (B, B Aut(F ))C(p))hAut(B), where C(p) denotes the Out(B)-orbit on * *[B, B Aut(F )] of the element classifying p : E ! B. The above considerations completely reduces the proof of the theorem, to our * *classi- fication theorem for connected p-compact groups, Theorem 1.2, except for the fi* *niteness statement. For this note that Proposition 8.18 implies that [Bss, B Out(D)] is * *finite so the finiteness of [Bss, B aut(D)] follows. Remark 7.3. As stated in the introduction, our classification also shows that B* *ott's theo- rem on the cohomology of X=T , the Peter-Weyl theorem, as well as Borel's chara* *cterization of when centralizer of elements of order p are connected, stated as Theorems 1.* *5, 1.6 and 34 K. ANDERSEN AND J. GRODAL 1.9 in [8], hold verbatim for 2-compact groups. To prove these results it suffi* *ces by The- orem 1.1 to check them for DI(4), since they are well known for compact Lie gro* *ups. For DI(4) one argues as follows: Bott's theorem [8, Thm. 1.5] follow from [24, Thm.* * 1.8(2)], the Peter-Weyl theorem [8, Thm. 1.6] is a result of Ziemia'nski [67], and it is tri* *vial to check that [8, Thm. 1.9] hold. 8. Appendix: Properties of Zp-root data The purpose of this section is to establish some general results about Zp-roo* *t data of p-compact groups, needed in the proof of the main theorem. The analogous resul* *ts for Z-root data and compact Lie groups are often well known; see [9, 16]. We build* * on the paper [28] by Dwyer-Wilkerson and our earlier paper [4]. We briefly recall the definition of root data from the introduction: For an i* *ntegral domain R, an R-reflection group is a pair (W, L) where L is a finitely generated free * *R-module and W is a subgroup of AutR(L) generated by reflections (i.e. elements oe 2 AutR(L)* * such that 1 - oe 2 End R(L) has rank one). If R is a principal ideal domain, we define a* *n R-root datum to be a triple D = (W, L, {Rboe}) where (W, L) is a finite R-reflection g* *roup and for each reflection oe 2 W , Rboeis a rank one submodule of L with im(1 - oe) * * Rboeand w(Rboe) = Rbwoew-1for all w 2 W . If R ! R0is a monomorphism of integral domain* *s, and D an R-root datum, we can define an R0-root datum by D R R0= (W, L R R0, {Rboe R * *R0}). The element boe, defined up to a unit in R, is called the coroot associated t* *o oe. By definition boedetermines a unique linear map fioe: L ! R called the associated * *root such that oe(x) = x + fioe(x)boefor x 2 L. Define the coroot lattice L0 L as the * *sublattice spanned by the coroots boeand the fundamental group of D by ss1(D) = L=L0. In g* *eneral Rboe ker(N), where N = 1+oe+. .+.oe|oe|-1is the norm element (cf. the proof of* * Lemma 8.8 below), so giving an R-root datum with underlying reflection group (W, L) corre* *sponds to choosing a cyclic R-submodule of H1(; L) for each conjugacy class of reflec* *tions oe. In particular for R = Zp, p odd, the notions of a Zp-reflection group and a Zp-roo* *t datum agree. If R has characteristic zero, an R-root datum, or an R-reflection group* *, is called irreducible if the representation W ! GL (L R K) is irreducible, where K denot* *es the quotient field of R, and it is said to be exotic if furthermore the values of t* *he character of this representation are not all contained in Q. We are now ready to state the classification of Zp-root data, which follows e* *asily from the classification of finite Zp-reflection groups [8, Thm. 11.1]. This classif* *ication is again based on the classification of finite Qp-reflection groups [13] [22] which stat* *es that for a fixed prime p, isomorphism classes of finite irreducible Qp-reflection groups a* *re in natural one-to-one correspondence with isomorphism classes of finite irreducible C-refl* *ection groups (W, V ) [55] for which the values of the character of W ! GL (V ) are embeddabl* *e in Qp; see e.g., [3, Table 1] for an explicit list of groups and primes. Theorem 8.1 (The classification of Zp-root data; splitting version).Any (1)Zp-r* *oot da- tum is isomorphic to a Zp-root datum of the form (D1 Z Zp) x D2, where * *D1 is a Z-root datum and D2 is a direct product of exotic Zp-root data. (2) There is a one-to-one correspondence between isomorphism classes of exot* *ic Zp-root data and isomorphism classes of exotic Qp-reflection groups given by the* * assignment D = (W, L, {Zpboe}) _ (W, L Zp Qp). Moreover ss1(D) = 0 for any exotic * *Zp-root datum D. THE CLASSIFICATION OF 2-COMPACT GROUPS 35 Proof.For any Zp-root datum D = (W, L, {Zpboe}), [8, Thm. 11.1] gives a splitti* *ng (W, L) ~= (W1, L1 Z Zp) x (W2, L2) of (W, L), where (W1, L1) is a finite Z-reflection gr* *oup and (W2, L2) is a direct product of exotic Zp-reflection groups. It follows by def* *inition that there are unique Zp-root data D0and D2 with underlying reflection groups (W1, L* *1 Z Zp) and (W2, L2) such that D ~=D0x D2, and by the same argument D2 splits as a dire* *ct product of exotic Zp-root data. Furthermore, writing D0 = (W1, L1 Z Zp, {Zpbo* *e}) it is clear that D0~=D1 Z Zp where D1 = (W1, L1, {L1 \ Zpboe}). This proves (1). By [8, Thm. 11.1], the assignment (W, L) _ (W, L ZpQp) establishes a one-to-o* *ne corre- spondence between exotic Zp-reflection groups up to isomorphism and exotic Qp-r* *eflection groups up to isomorphism. To prove the first part of (2)it thus suffices to sho* *w that any exotic Zp-reflection group (W, L) can be given a unique Zp-root datum structure* *. For p > 2 this holds since H1(; L) = 0, cf. the discussion in the beginning of this s* *ection. For p = 2, (W, L) ~=(WDI(4), LDI(4)) where the claim follows by direct inspection (cf. [28* *, Rem. 7.2]). For any Zp-root datum D = (W, L, {Zpboe}), the formula oe(x) = x + fioe(x)boe* *shows that the coroot lattice L0 contains the lattice spanned by the elements (1-w)(x), w * *2 W , x 2 L. Hence the final claim follows from the fact that H0(W ; L) = 0 for any exotic Z* *p-reflection group (W, L) [8, Thm. 11.1]. 8.1. Root datum, normalizer extension, and root subgroups of a p-compact group. For any connected p-compact group X with maximal torus T , the Weyl group WX acts naturally on LX = ss1(T ) as a finite Zp-reflection group [25, Thm. 9.7* *(ii)]. For p odd, H1(; L) = 0 for any reflection oe, so the finite Zp-reflection group (* *WX , LX ) gives rise to a unique Zp-root datum DX . The construction of root data for connected* * 2-compact groups, in the present form, is due to Dwyer-Wilkerson [28, x9]: Let T~be the * *discrete approximation to T , N~X the discrete approximation to the maximal torus normal* *izer NX and oe 2 WX a reflection. Define ~T +(oe) = ~T and let ~T0+(oe) denote its * *maximal divisi- ble subgroup. Then X(oe) = CX (T~+0(oe)) is a connected 2-compact group with We* *yl group and N~(oe) = CN~X(T~+0(oe)) is a discrete approximation to its maximal toru* *s normalizer. Furthermore, let ~T0-(oe) denote the maximal divisible subgroup of ~T -(oe) = k* *er(T~-1+oe-!~T) and define the root subgroup N~X,oeby N~X,oe= {x 2 ~N(oe) | 9 y 2 ~T0-(oe) : x is conjugate to y in X(oe)}. Then there is a short exact sequence (8.1) 1 ! ~T0-(oe) ! ~NX,oe! ! 1, and we define ( im(LX -1-oe-!LX )if (8.1)splits, Z2boe= 1+oe ker(LX --! LX ) otherwise. The root datum of X is then the Z2-root datum DX = (WX , LX , {Z2boe}); see [28* *, x6 and 9] and [4] for a further discussion. Conversely, the maximal torus normalizer and the root subgroups of a connecte* *d p- compact group can be reconstructed from its root datum: For a Zp-root datum D = (W, L, {Zpboe}) there is [28, Def. 6.15] [4, x3] an algebraically defined exten* *sion 1 ! ~T! ~ND! W ! 1 36 K. ANDERSEN AND J. GRODAL called the normalizer extension with a subextension 1 ! ~T0-(oe) ! N~D,oe! * *! 1 for each reflection oe 2 W . For a connected p-compact group X with Zp-root datum D* *X there is an isomorphism of extensions [28, Prop. 1.10] [4, Thm. 3.1(2)] 1 ____//_~T__//~ND___//_W___//_1 || ~| |||| || =| || || fflffl||| 1 ____//_~T__//~NX___//_W___//_1 sending the root subgroups ~ND,oeto ~NX,oefor all reflections oe, and any such * *isomorphism is unique up to conjugation by an element in ~T. We define BND as the fiber-wise Fp-completion [10, Ch. I, x8] of BN~D and li* *kewise introduce the (non-discrete) root subgroups BNX,oeand BND,oeby fiber-wise Fp-co* *mpletion of the corresponding discrete versions BN~X,oeand BN~D,oe. Recollection 8.2 (The Adams-Mahmud map). By [8, Lem. 4.1] we have an "Adams- Mahmud" homomorphism : Out(BX) ! Out(N~), given by associating to f : BX ! BX the homomorphism (f) : ~N! ~N, unique up to conjugation, such that the diagram B (f) BN~ _____//BN~ | | | | fflffl|f fflffl| BX _____//BX commutes up to homotopy. By [4, Thm. B], factors through Out(N~, {N~oe}) = {['] 2 Out(N~) | '(N~oe) * *= N~'(oe)}, which is isomorphic to Out(DX ) via restriction to ~T. We again denote this ma* *p by . Likewise, as e.g., explained in [8, Prop. 5.1], fiber-wise Fp-completion [10, C* *h. I, x8] induces ~= a natural isomorphism Out(N~) -! Out(BN ), and we can hence equivalently view * *(f) as an element in Out(BN ), and we will not notationally distinguish between the tw* *o cases. We denote the subgroup of Out(BN ) corresponding to Out(N~, {N~oe}) by Out(BN , {B* *Noe}). 8.2. Centers and fundamental groups. If D = (W, L, {Zpboe}) is a Zp-root datum,* * we define a subdatum of D to be a Zp-root datum of the form (W 0, L, {Zpboe}oe2 0)* * where (W 0, L) is a reflection subgroup of (W, L) and 0is the set of reflections in W 0. For * *the next result, recall [26, Def. 4.1] that a homomorphism f : BX ! BY is called a monomorphism* * of maximal rank if the homotopy fiber Y=X is Fp-finite and X and Y have the same r* *ank. Proposition 8.3. Let X and Y be connected p-compact groups. If f : BX ! BY i* *s a monomorphism of maximal rank then DX naturally identifies with a subdatum of DY* * . Proof.By definition there is a maximal torus i : BT ! BX such that f O i : BT !* * BY is a maximal torus for Y . Thus we can identify LX = LY = ss1(T ) and by [26, Lem.* * 4.4] we have an induced monomorphism WX ! WY . This proves the result for p odd, since * *in that case the Zp-root data DX and DY are uniquely determined by their underlying ref* *lection groups (WX , LX ) and (WY , LY ). In the case p = 2 the result follows from the* * construction of the Z2-root data of X and Y , cf. [28, Pf. of Lem. 9.16]. For a Zp-root datum D = (W, L, {Zpboe}), we let ~T= L Zp Z=p1 be the associ* *ated discrete torus and for a reflection oe 2 W , we define hoe= boe 1_22 T~. Clea* *rly hoeis THE CLASSIFICATION OF 2-COMPACT GROUPS 37 independent of the choice of boeand conversely hoedetermines Zpboe, cf. [28, x2* * and x6]. So instead of (W, L, {Zpboe}) we might as well work with (W, ~T, {hoe}); we will u* *se these two viewpoints interchangeably without further mention. Also note that hoe= 1 for * *p odd. When oe 2 W is a reflection, we define the singular set S(oe) by D E S(oe) = ~T0+(oe), hoe= ker(fioe ZpZ=p1 : ~T! Z=p1 ), * * T cf. [26, Def. 7.3] and [4, Pf. of Prop. 4.2]. Define the discrete center ~Z(D) * *of D as oeS(oe), where the intersection is taken over all reflections oe 2 W . In other words, * *letting M0 denote the root lattice, i.e., the Zp-sublattice of L* spanned by the roots fio* *e, we have the identification i j (8.2) Z~(D) = ker ~T= Hom Zp(L*, Z=p1 ) i Hom Zp(M0, Z=p1.) The following proposition translates into the present language the results of D* *wyer-Wilkerson [26, x7] on how to compute the center and centralizers of toral subgroups of X. Proposition 8.4. Let X be a connected p-compact group with Zp-root datum DX = (* *W, L, {Zpboe}). (1) The center BZ(X) of X is canonically homotopy equivalent to the center B* *Z(D) = (BZ~(D))^pof D. (2) The identity component Z(X)1 of the center has Zp-root datum (1, LW , ;). (3) For A ~T, let W (A) be the pointwise stabilizer of A in W , A the set* * of reflections oe with A S(oe), and W (A)1 the subgroup of W generated by A. Then t* *he centralizer CX (A) has Weyl group W (A) and its identity component CX (A* *)1 has Zp-root datum equal to the subdatum DA = (W (A)1, L, {Zpboe}oe2 A) of D. Proof.The first part is [26, Thm. 7.5]. The second part follows easily from thi* *s since the maximal divisible subgroup of ~Z(D) equals " " ~T0+(oe) = L+ (oe) ZpZ=p1 = LW ZpZ=p1 . oe oe Part (3)follows by combining [26, Thm. 7.6] and Proposition 8.3 since BCX (A)1 * *! BX is a monomorphism of maximal rank [26, Prop. 4.3]. Let D = (W, L, {Zpboe}) be a Zp-root datum. Recall that the coroot lattice L0* * L is the Zp-lattice spanned by the coroots boeand that the fundamental group ss1(D) is t* *he quotient L=L0. Define HZpn(X) = limkHn(X; Z=pk). The following proposition, which refi* *nes [8, Prop. 10.2], constructs a canonical epimorphism ss1(DX ) ! ss1(X) which will be* * shown to be an isomorphism in Theorem 8.6 below. Proposition 8.5. Let X be a connected p-compact group with maximal torus T and * *Zp- root datum DX = (W, L, {Zpboe}). Then the homomorphism L = HZp2(BT ) ! HZp2(BX)* * ~= ss1(X) factors through ss1(DX ) and the induced homomorphism ss1(DX ) ! ss1(X) * *is surjec- tive with finite kernel. Proof.For p odd we have im(1 - oe) = Zpboefor all oe, so L=L0 ~=H0(W ; L) and t* *he result follows from [8, Prop. 10.2]. Assume now that p = 2 and let oe 2 W be a reflection. To see the first part w* *e have to show that the homomorphism L = HZ22(BT ) ! HZ22(BX) ~=ss1(X) vanishes on the co* *roots boe. This follows from the construction of the root datum of X: X(oe) = CX (T~+* *0(oe)) is a 38 K. ANDERSEN AND J. GRODAL connected 2-compact group and Proposition 8.4(3)shows that DX(oe)equals the sub* *datum (, L, {Z2boe}) of DX . The commutative diagram L = HZ22(BT )___//_HZ22(BX(oe)) ~=ss1(X(oe)) || | || | || fflffl| L = HZ22(BT )______//HZ22(BX) ~=ss1(X) shows that it suffices to prove the claim for X(oe). However since X(oe) is a * *connected 2-compact group of rank 1 it follows (cf. [28, p. 1369-1370]) that X(oe) ~= G^2* *for G = SU(2) x (S1)r-1, SO (3) x (S1)r-1 or U(2) x (S1)r-2 where r is the rank of X. T* *he well- known formula for the fundamental group of a compact connected Lie group in ter* *ms of its root datum, cf. [9, x4, no. 6, Prop. 11] or [1, Thm. 5.47], now established the* * first part of the proposition. Since im(1 - oe) Z2boeby definition, L ! L=L0 = ss1(DX ) factors through H0* *(W ; L), so the final claim now follows from [8, Prop. 10.2]. We will also be using the following formula for the fundamental group of a p-* *compact group, proved by Dwyer-Wilkerson [29] by a transfer argument as this paper was * *being written (see also [8, Rem. 10.3]). The formula was previously known for p odd,* * by our classification [8], and we sketch in Remark 6.2 how one can bypass the use of t* *his formula also in the classification for p = 2 by a more cumbersome argument which we had* * originally envisioned using in this paper; in particular providing an independent proof. Theorem 8.6 (Dwyer-Wilkerson [29] and Remark 6.2). Let X be a connected p-compa* *ct ~= group. Then ss1(DX ) -! ss1(X) induced by the maximal torus T ! X. Proof.By Theorem 8.1(1)we may write DX = D1x D2, where D1 is of the form D0 Z Zp for a Z-root datum D0, and D2 is a direct product of exotic Zp-root data. By [2* *7, Thm. 1.4] this induces a splitting BX ' BX1 x BX2 with DXi ~=Di. We have to show that the kernel of L = ss1(T ) ! ss1(X) equals the coroot lattice L0; by the above it su* *ffices to treat the case where DX is exotic and the case where DX is of the form D0 Z Zp for a * *Z-root datum D0. In the first case, Theorem 8.1(2) shows that ss1(DX ) = 0 so the result follo* *ws from Proposition 8.5. In the second case we have DX ~=DG^pfor some compact connected Lie group G. B* *y the result of Dwyer-Wilkerson [29, Thm. 1.1] the kernel of L = ss1(T ) ! ss1(X) equ* *als the kernel of HZp2(BTX ) ! HZp2(BNX ). Since the maximal torus normalizer may be reconstr* *ucted from the root datum by [28, Prop. 1.10] for p = 2 and [3, Thm. 1.2] for p odd, * *we may identify the homomorphism HZp2(BTX ) ! HZp2(BNX ) with the homomorphism H2(BTG ; Z) ! H2(BNG ; Z) tensored by Zp. The result now follows from the corresponding resu* *lt for compact Lie groups, cf. [9, x4, no. 6, Prop. 11] or [1, Thm. 5.47]. 8.3. Covers and quotients. We now start to address how the root datum behaves u* *pon taking covers and quotients of a p-compact group. Lemma 8.7. Let f : BX ! BY be a monomorphism of maximal rank between connected p-compact groups X and Y . If ss1(DY ) = 0 then BZ(X) ' BZ(NX ). THE CLASSIFICATION OF 2-COMPACT GROUPS 39 Proof.For p odd, the conclusion holds for any connected p-compact group X [26, * *Rem. 7.7], so we may suppose p = 2. Since ss1(DY ) = 0, DY does not have any direct factor* *s isomorphic to DSO(2n+1)^2, so [4, x5] implies that the singular set SY (oe) with respect t* *o Y equals ~T(+oe) for any reflection oe 2 WY . By Proposition 8.3, DX identifies with a s* *ubdatum of DY and hence SX (oe) = ~T +(oe) for all reflections oe 2 WX . The claim now fo* *llows from Proposition 8.4(1). Lemma 8.8. Let D = (W, L, {Zpboe}) be a Zp-root datum with coroot lattice L0. * *Then L0 \ LW = 0 and L0 LW has finite index in L. In particular W acts faithfully* * on L0. Proof.The Qp[W ]-module V = L Zp Qp decomposes as V = V W U where UW = 0. Writing boe= x + y with x 2 V W and y 2 U we have oe(boe) = x + oe(y) and hence oe(boe) - boe2 U. Also oe(boe) 6= boe: If N = 1 + oe + . .+.oe|oe|-1is the norm* * element, then fioe(x)N(boe) = N(oe - 1)(x) = 0 for all x 2 L. Since oe 6= 1 we have fioe6= 0 * *so Nboe= 0. Thus oe(boe) 6= boesince otherwise Nboe= |oe|boe6= 0. This proves that (oe - 1)* *(boe) = rboewith r 6= 0 so boe2 U.PThus L0 UPand L0 \ LW = 0 as desired. Since |W |x = w2W wx + w2W (x - wx) 2 LW + L0, for any x 2 L we see th* *at L0 LW has finite index in L, and in particular W acts faithfully on L0. Let D = (W, L, {Zpboe}) be a Zp-root datum and let L0 be the coroot lattice. * *If L0is a Zp-lattice with L0 L0 L, the formula oe(x) = x + fioe(x)boeshows that L0is W* * -invariant. By Lemma 8.8, W acts faithfully on L0 and hence also on L0, so (W, L0, {Zpboe})* * is a Zp- root datum. We define a cover of D to be any Zp-root datum of this form. In par* *ticular the universal cover eD of D is defined by eD = (W, L0, {Zpboe}). Note that by * *definition, ss1(De) = 0. For the reduction in Section 2 we need the following result which * *does not rely on the fundamental group formula Theorem 8.6. Proposition 8.9. Let X be a connected p-compact group with Zp-root datum DX and* * let H be a subgroup of ss1(X). Let Y ! X be the cover of X corresponding to H. Then* * DY is the cover of DX corresponding to the kernel of the composition LX ! ss1(DX ) ! * *ss1(X) ! ss1(X)=H. Proof.By construction BY is the fiber of BX ! B2(ss1(X)=H). Let BT ! BX be a maximal torus of X and let BNX ! BX the maximal torus normalizer. Now consider the following diagram obtained by pulling the fibration BY ! BX ! B2(ss1(X)=H) * *back along BT ! BNX ! BX (8.3) BT 0_____________//BN 0____________//BY | | | | | | fflffl| fflffl| fflffl| BT ____________//_BNX____________//_BX | | | | | | fflffl| fflffl| fflffl| B2(ss1(X)=H) ______B2(ss1(X)=H)_____B2(ss1(X)=H). Thus BT 0! BY is a maximal torus and BN 0! BY is a maximal torus normalizer by * *[44, Thm. 1.2]. The above diagram shows that the Weyl group of Y identifies with the* * Weyl 40 K. ANDERSEN AND J. GRODAL group W of X. For a reflection oe 2 W we have the diagram BT~0_____//BN~0(oe)__//BY (oe) | | | | | | fflffl| fflffl| fflffl| BT~ ____//_BN~X(oe)__//_BX(oe), where X(oe) = CX (T~+0(oe)), N~X(oe) = CN~X(T~+0(oe)) and similarly for Y (oe) * *and N~0(oe). It now follows by definition (cf. [28, x9]) that the image of h0oe2 ~Te0quals hoe2* * ~T. Diagram (8.3)produces the short exact sequence 0 ! LY ! LX ! ss1(X)=H ! 0 so Zpb0oe LY maps to Zpboe LX . This shows the claim. We next introduce quotients of root data. Let D = (W, ~T, {hoe}) be a Zp-roo* *t datum, and A ~Z(D) a subgroup of the discrete_center._We_define a quotient of D to b* *e a Zp-root datum of the form D=A = (W, ~T=A, {hoe}) where hoedenotes the image of hoein ~T* *=A; the fact that this is a Zp-root datum is part of the following result. Proposition 8.10. (1)If D = (W, ~T, {hoe}) is a Zp-root datum and A Z~(D),* * then ___ D=A = (W, ~T=A, {hoe}) is a Zp-root datum. Moreover ]D=A ~=eD and Z~(D=* *A) ~= ~Z(D)=A. (2) Let X be a connected p-compact group with Zp-root datum DX and A ! X a central monomorphism. If ~Adenotes the discrete approximation to A, then* * the Zp- root datum DX=A of the p-compact group X=A identifies with the quotient * *datum DX =A~. In particular ^DX=A~=gDX. Proof.Write D = (W, L, {Zpboe}) where L = Hom Zp(Z=p1 , ~T) is the associated Z* *p-lattice and boe2 L the associated coroots. By (8.2)the sequence of discrete tori ~T! ~* *T=A ! T~=Z~(D) corresponds to the sequence L ! LT~=A! M0 *of Zp-lattices, where M0 is* * the root lattice spanned by the fioe. Note that (W, L*) is a reflection group via * *the action oe(ff) = ff + ff(boe-1)fioe-1so (W, L*, {Zpfioe-1}) is a Zp-root datum (the dua* *l of D). Applying Lemma 8.8 to this Zp-root datumishowsjthat M0 (L*)W has finite index in L*, * *so the * dual homomorphism L ! M0* (L*)W is injective. The last summand is isomorph* *ic to Hom Zp(H0(W ; L), Zp)* which again identifies with H0(W ; L) modulo its torsion* * subgroup. The formula oe(x) - x = fioe(x)boeshows that the image of the composition L0 ! * *L ! H0(W ; L) is torsion, so by the above the composition L0 ! L ! LT~=A! M0* is in* *jective. In particular W acts faithfully on LT~=Aby Lemma 8.8. The homomorphism fioe ZpZ=p1* * : ~T! Z=p1 factors through ~T=A, and we get a corresponding homomorphism fi0oe: LT~=* *A! Zp fi0oe such that the composition L ! LT~=A-! Zp agrees with fioe. We now claim that o* *e(x) = x+fi0oe(x)boefor x 2 LT~=A. To see this note that the image of L in LT~=Ahas fi* *nite index since we have the exact sequence L ! LT~=A! Ext1Zp(Z=p1 , A) with Ext1Zp(Z=p1 , A) fi* *nite. Thus the claim follows from the above by using the corresponding formula oe(x) = x +* * fioe(x)boe for x 2 L. This proves that D=A = (W, LT~=A, {Zpboe}) is a Zp-root datum. In pa* *rticular we obtain ]D=A= (W, L0, {Zpboe}) = eDby definition. THE CLASSIFICATION OF 2-COMPACT GROUPS 41 To see the claim about Z~(D=A), note that by the above fioe Zp Z=p1 : T~! Z=* *p1 fi0oe ZpZ=p1 identifies with the composition ~T! ~T=A --------! Z=p1 . Hence the singular se* *ts for D and D=A satisfies SD (oe)=A = SD=A(oe) and the claim follows. To see part (2), let i : BT ! BX be a maximal torus and let f : BA ! BX be the central monomorphism. Then f factors through BT by [25, Prop. 8.11] to give a c* *entral monomorphism g : BA ! BT . Moreover i factors through the maximal torus normali* *zer N of X and we obtain the diagram BT _______//BN_______//BX | | | | | | fflffl| fflffl| fflffl| BT=A ____//_BN =A____//BX=A cf. Construction 2.2. It follows that T=A is a maximal torus in X=A and N =A i* *s the maximal torus normalizer. The Weyl groups of X and X=A identifies naturally, c* *f. [45, Thm. 4.6]. By construction the elements h0oe2 ~T=A~corresponding to DX=A are th* *e images of the elements hoe2 ~Tcorresponding to X (cf. the proof of Proposition 8.9). T* *his shows that DX=A ~=DX =A~as desired. As a special case of the quotient construction, we define the adjoint Dad of * *a Zp-root datum D = (W, L, {Zpboe}) by Dad = D=Z~(D). Note that by Proposition 8.10(1)we * *have ~Z(Dad) = 0 and that it follows from the proof that Dad = (W, M0*, {Zpboe}), wh* *ere M0 is the root lattice, i.e., the sublattice of L* spanned by the roots fioe. Proposition 8.11. Any Zp-root datum with Z~(D) = 0 or ss1(D) = 0 splits as a di* *rect product D ~=D1 x . .x.Dn of irreducible Zp-root data Di. Proof.The case ~Z(D) = 0 is essentially proved by Dwyer-Wilkerson [27, Pf. of T* *hm. 1.5], for completeness we briefly sketch the argument: Let (W, L) be the Zp-reflecti* *on group associated to D and write L ZpQp = V1 . . .Vn be the decomposition of L ZpQp* * into T + irreducible Qp[W ]-modules. Define Li = L \ Vi. Since Z~(D) = 0 we have oe~T0* *(oe) = 0 as well, and it follows [27, Pf. of Thm. 1.5] that the homomorphism L1 x . .x.L* *n ! L is an isomorphism. Letting Widenote the point-wise stabilizer of L1 . . .bLi . .* * .Ln we hence (cf. [27, Prop. 7.1]) get a product decomposition (W, L) ~=(W1, L1) x . .* *x.(Wn, Ln). It is now clear that there is a unique Zp-root datum structure Dion (Wi, Li) su* *ch that we get a product decomposition D ~=D1 x . .x.Dn into irreducible Zp-root data. The case where ss1(D) = 0 is easily reduced to the first case using the previ* *ous results: By Proposition 8.10(1), Z~(Dad) = 0 so we can write Dad ~=D1 x . .x.Dn where th* *e Di are irreducible. Proposition 8.10(1) now shows that D = eD~= gDad~=fD1x . .x.f* *Dnas claimed. Theorem 8.12 (The classification of Zp-root data; structure version).Let (1)D =* * (W, L, {Zpboe}) be a Zp-root datum with coroot lattice L0, and let D0 = (W, L0 LW , {Z* *pboe}) = eDx Dtriv, where Dtriv= (1, LW , ;) is a trivial Zp-root datum. Then D ~* *=D0=A for a finite central subgroup A ~Z(D0) and there is a splitting eD~=D1 x .* * .x.Dn of eDinto irreducible Zp-root data Di with ss1(Di) = 0. (2) For p > 2, the assignment D = (W, L, {Zpboe}) _ (W, L Zp Qp) is a one-t* *o- one correspondence between isomorphism classes of irreducible Zp-root da* *ta D with 42 K. ANDERSEN AND J. GRODAL ss1(D) = 0 and isomorphism classes of irreducible Qp-reflection groups. F* *or p = 2, the assignment is surjective and the preimage of every element consists o* *f a single element, except for (WSp(n), LSp(n) Q2) ~=(WSpin(2n+1), LSpin(2n+1) Q2)* * whose preimage consists of DSp(n) Z Z2 and DSpin(2n+1) Z Z2 which are non-isomo* *rphic for n 3. Proof.The fact that D0 is a Zp-root datum follows from Lemma 8.8. The short ex* *act sequence 0 ! L0 LW ! L ! F ! 0 where F is finite with trivial W -action prod* *uces a short exact sequence 1 ! A ! ~T!0~T! 1 between the associated discrete tori. * *Since the roots fi0oefor D0 are given by L0 LW ! L -fioe!Zp, it follows that A S* *D0(oe) = ker((L0 LW ) Zp Z=p1 ! Z=p1 ) for any reflection oe. Hence A ~Z(D0) is cen* *tral and D ~=D0=A. The last part of (1)follows from Proposition 8.11. We now prove (2). For any prime p, Theorem 8.1(2)shows that the assignment in* * (2) gives a one-to-one correspondence between isomorphism classes of exotic Zp-root* * data and isomorphism classes of exotic Qp-reflection groups, and that ss1(D) = 0 for all* * exotic Zp-root data D. Hence it suffices by Theorem 8.1(1)to show the claim for Zp-root data o* *f the form D1 Z Zp for a Z-root datum D1. In this case it is clear that if ss1(D1 Z Zp) = 0, then we can find a Z-root* * datum D01 with ss1(D01) = 0 and D01 Z Zp ~=D1 Z Zp (simply choose D01to be the universal * *cover of D1; this is defined for Z-root data in the same way as for Zp-root data). Hence* * it suffices to study the assignment D = (W, L, {Zpboe}) _ (W, L Z Q) from irreducible Z-ro* *ot data with ss1(D) = 0 to irreducible Q-reflection groups. It is well-known (cf. [9, x* *4]) that this assignment is surjective and that it only fails to be injective in that the Z-r* *oot data DSp(n) and DSpin(2n+1)which are non-isomorphic for n 3 maps to the same Q-reflection* * group. This proves part (2)since for n 3, the Zp-root data DSp(n) Z Zp and DSpin(2n+* *1) Z Zp are non-isomorphic for p = 2 and isomorphic for p = 2. 8.4. Automorphisms. Recall that an isomorphism between two Zp-root data D = (W,* * L, {Zpboe}) and D0= (W 0, L0, {Zpb0oe0}) is an isomorphism ' : L ! L0with the property that* * 'W '-1 = W 0as subgroups of Aut(L0) and '(Zpboe) = Zpb0'oe'-1for every reflection oe 2 W* * . We denote the automorphism group of D by Aut(D); clearly W is a normal subgroup of Aut(D)* *, and we define the outer automorphism group Out(D) by Out(D) = Aut(D)=W . Recall that a Zp-root datum D = (W, L, {Zpboe}) is called irreducible if L Z* *p Qp is an irreducible Qp[W ]-module. The following proposition is a restatement of [8, Pr* *op. 5.4]. Proposition 8.13. Suppose Di = (Wi, Li, {Zpboe}oe2 i), i = 0, . .,.k, is a coll* *ection of pairwise non-isomorphic irreducibleQZp-root data. Assume that W0 = 1 but that W* *i is non- trivial for i 1. Let D = ki=0Dmii, mi 1 denote a product of these Zp-roo* *t data. Then _ ! Yk ~ GL m0(Zp) x Out(Di) o mi =-!Out(D). i=1 Proof.This follows directly by combining [8, Prop. 5.4] with [4, Rem. 4.5]. The following two results are needed for the reduction in Section 2. Proposition 8.14. Let D = (W, L, {Zpboe}) be a Zp-root datum with coroot lattic* *e L0. Let D0= (W, L0 LW , {Zpboe}) = eDxDtriv, where Dtriv= (1, LW , ;) is a trivial Zp-r* *oot datum. THE CLASSIFICATION OF 2-COMPACT GROUPS 43 Then D ~=D0=A for a finite subgroup A ~Z(D0) and the restriction Aut(D) -! Aut(D0) = Aut(De) x Aut(Dtriv) is an isomorphism onto the subgroup {' 2 Aut(D0) | '(A) = A}. In particular Out* *(D) identifies with a subgroup of finite index in Out(D0). Remark 8.15. For a Zp-root datum D = (W, L, {Zpboe}) with ss1(D) = 0 we have Zp* *boe= ker(L N-!L), where N = 1 + oe + . .+.oe|oe|-1is the norm element: By Theorem 8.* *1(1)either D ~=D1 Z Zp for a Z-root datum D1 with ss1(D1) = 0 (cf. the proof of Theorem 8* *.12(2)) or D is exotic. In the first case the result is well known [9, x4], and in the * *second case the claim follows since H1(; L) = 0 for all reflections oe (cf. the proof of Th* *eorem 8.1(2)). In particular we obtain Out(D) = NGL(L)(W )=W , and this group is explicitly compu* *ted for all irreducible (W, L) in [8, Thm. 13.1]. Proof of Proposition 8.14.The fact that D ~= D0=A for a finite subgroup A Z~(* *D0) is part of Theorem 8.12(1), and the identification Aut(D0) = Aut(De) x Aut(Dtriv) * *comes from Proposition 8.13. Clearly both L0 and LW are invariant under Aut(D) so w* *e get a restriction homomorphism Aut(D) -! Aut(D0) which is injective since L0 LW * *has finite index in L by Lemma 8.8. If ' 2 Aut(D), then ' corresponds to an automor* *phism ' : ~T! ~Tand the restriction of ' to L0 LW corresponds to a lift e': ~T 0! * *~T 0which sends A to itself. Conversely, if ' 2 Aut(D0) with '(A) = A, then ' clearly in* *duces an automorphism of D. The last claim follows from the fact that the orbit of A und* *er Out(D0) is finite since ~Z(D0) has only finitely many subgroups isomorphic to A. Corollary 8.16. For any Zp-root datum D there is a canonical isomorphism ~= Aut(De) ! Aut(Dad). Proof.Write D = (W, L, {Zpboe}), and let M0 L* denote the root lattice, i.e.,* * the lattice spanned by the roots fioe. Then eD = (W, L0, {Zpboe}) and the roots for eD are* * given by the composition L0 ! L -fioe!Zp. Hence we can identify the root lattice for eD* * with M0 and hence eD=Z~(De) = (De)ad ~=(W, M0*, {Zpboe}) = Dad. The result now follows* * from Proposition 8.14. 8.5. Finiteness properties. In this final subsection we prove that there are on* *ly finitely many Zp-root data of a given rank, and that for a fixed Zp-root datum D, Out(D)* * only contains finitely many finite subgroups up to conjugation. These results are us* *ed for the proof of the finiteness statements in Theorems 1.5 and 1.6 in Section 7. Proposition 8.17. For any prime p there is, up to isomorphism, only finitely ma* *ny Zp-root data of a fixed rank. Proof.By the classification of finite Qp-reflection groups [13] [22], there are* * only finitely many finite Qp-reflection groups of a fixed rank. For each finite Qp-reflection* * group (W, V ), there are, up to equivalence, only finitely many representations W ! GL (V ) wh* *ich gives rise to the reflection group (W, V ). For each of these representations, the lo* *cal version of the Jordan-Zassenhaus theorem [14, Thm. 24.7] shows that, up to isomorphism, th* *ere are only finitely many Zp[W ]-lattices L with L ZpQp ~=V as Qp[W ]-modules. In part* *icular we conclude that, up to isomorphism, there are only finitely many finite Zp-reflec* *tion groups of fixed rank. Finally, choosing a Zp-root datum for a finite Zp-reflection gr* *oup (W, L) 44 K. ANDERSEN AND J. GRODAL corresponds to choosing a cyclic subgroup of the finite group H1(; L) for e* *ach conjugacy class of reflections oe. Hence any finite Zp-reflection group gives rise to onl* *y finitely many Zp-root data. This proves the result. Proposition 8.18. Let D be a Zp-root datum. Then Out(D) contains only finitely * *many conjugacy classes of finite subgroups. Proof.Let (W, L) be the finite Zp-reflection group underlying D, and let n = ra* *nkL. Note first that the order of a finite subgroup of GL (L) is bounded above: If G GL* * n(Zp) has finite order, then it is easily seen that the composition G ! GL n(Zp) ! GL n(F* *p) is injective for p > 2 and has kernel of order at most 2n2 for p = 2 (cf. [8, Lem. 11.3]). H* *ence G has order at most |GLn(Fp)|for p > 2 and 2n2. |GLn(F2)|for p = 2. In particular the* *re is an upper bound on the order of finite subgroups of NGL(L)(W ). Since Out(D) is con* *tained in NGL(L)(W )=W and W is finite, it follows that there is also an upper bound on t* *he order of finite subgroups of Out(D). Fix a finite group G. By the above it suffices to show that the set Rep(G, O* *ut(D)) (i.e., the set of homomorphisms G ! Out(D) modulo conjugation in Out(D)) is fin* *ite. By Theorem 8.12(1), we can write D ~=D0=A where D0 = eDx Dtrivand Dtrivis a trivial Zp-root datum, and this identifies Out(D) with a subgroup of finite index in Ou* *t(D0), cf. Proposition 8.14. Hence it is enough to prove that Rep(G, Out(D0)) is finite si* *nce this will imply that Rep(G, Out(D)) is finite. By Theorem 8.12(1)and Proposition 8.13, Out(D0) is isomorphic to a direct pro* *duct of GL m0(Zp) and groups of the form Out(Di) o mi where the Diare irreducible Zp-r* *oot data. It thus suffices to know that Rep(G, GLm (Zp)) and Rep(G, Out(D) o m ) are fin* *ite for any m and any irreducible Zp-root datum D. The first claim follows directly from t* *he local version of the Jordan-Zassenhaus theorem [14, Thm. 24.7]. To see the second claim, let W denote the Weyl group of D and note that since* * D is irreducible, Schur's lemma implies that the image of the central homomorphism Z* *xp! Out (D) equals the kernel of the canonical homomorphism Out (D) ! Out (W ). Si* *nce Zxp~= Zp x C where C is finite (C = Z=2 for p = 2 and C = Z=(p - 1) for p > 2) * *and Out (W ) is finite, it follows that Out(D) is a finite (central) extension of Z* *p. 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