THE CLASSIFICATION OF 2COMPACT GROUPS
KASPER K. S. ANDERSEN AND JESPER GRODAL
Abstract.We prove that any connected 2compact group is classified by it*
*s 2adic
root datum, and in particular the exotic 2compact group DI(4), construc*
*ted by Dwyer
Wilkerson, is the only simple 2compact group not arising as the 2compl*
*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 pcompact groups, stating th*
*at, up to isomorphism,
there is a onetoone correspondence between connected pcompact groups *
*and root data
over the padic integers. As a consequence we prove the maximal torus co*
*njecture, giving
a onetoone 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 AndersenGrodalMollerViruel metho*
*ds to incorporate
the theory of root data over the padic 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 JackowskiMcClureOliver in the early 1*
*990s.
1.Introduction
In this paper we prove that any connected 2compact group X is classified, up*
* to isomor
phism, by its root datum DX over the 2adic 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
pcompact groups for all primes p. The classification states that, up to isomor*
*phism, there
is a onetoone correspondence between connected pcompact groups and root data*
* over Zp.
Hence the classification of pcompact 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
2compact groups has the following consequence, which captures most of the clas*
*sification
statement.
Theorem 1.1 (Classification of 2compact groups; splitting version). Let X be a*
* connected
2compact 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 2compact group DI(4)*
* constructed
by DwyerWilkerson [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 pcompact group, introduce*
*d by
DwyerWilkerson [25], can be defined as a pointed, connected, pcomplete 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 pcompact group simply by X. A pcompact group is called connected if the s*
*pace X is
connected, and two pcompact groups are said to be isomorphic if their classify*
*ing spaces
are homotopy equivalent. For more background on pcompact groups, including det*
*ails on
___________
The secondnamed author was partially supported by NSF grant DMS0354633 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 DwyerWilkerson 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 Rroot datum D is a triple (W, L, {Rboe}),*
* where L is a
finitely generated free Rmodule, W AutR(L) is a finite subgroup generated by*
* reflections
(i.e., elements oe such that 1oe 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) = Rbwoew1for 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 onetoone correspondence between Zroot 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 "Rreflection datum*
*"). For
R = Zp, p odd, the notion of an Rroot datum agrees with that of an Rreflectio*
*n group
(W, L); see Section 8. Given two Rroot 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 Zproot data is given as Theorems 8.1 a*
*nd 8.12.
We now explain how to associate a Zproot datum to a connected pcompact grou*
*p. By a
theorem of DwyerWilkerson [25, Thm. 8.13] any pcompact group X has a maximal *
*torus,
which is a map i : BT = (BS1^p)r ! BX satisfying that the homotopy fiber has fi*
*nite
Fpcohomology and nontrivial Euler characteristic. Replacing i by an equivalen*
*t fibration,
we define the Weyl space WX (T ) as the topological monoid of selfmaps 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 Zpreflection 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 pcompact groups). The assignment which to a con*
*nected
pcompact group X associates its Zproot datum DX gives a onetoone correspond*
*ence be
tween connected pcompact groups, up to isomorphism, and Zproot 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 selfhomotopy 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 AdamsMahmud map [8, Lem. 4.1*
*] given
THE CLASSIFICATION OF 2COMPACT GROUPS 3
by lifting a selfequivalence 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 pcompact 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*
* onetoone
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 JackowskiMcClureOliver, DwyerWilkerson, 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 Fpcoefficients, for p "large enough", by AdamsWilkers*
*on [2] in
1980, extending work of ClarkEwing [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 pcompact groups is a spacelevel 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 pcompact 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 Fpequivalent if there exists a space Y 00and a*
* zigzag
Y ! Y 00 Y 0inducing isomorphisms on Fphomology. 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 F2cohomology). If A* is a gr*
*aded
polynomial F2algebra over the Steenrod algebra A2, in finitely many variables,*
* all in degrees
3, then there exists at most one space Y , up to F2equivalence, with H*(Y ; *
*F2) ~=A*, as
graded F2algebras over the Steenrod algebra.
If P *is a finitely generated graded polynomial F2algebra, then there exists*
* at most finitely
many spaces Y up to F2equivalence such that H*(Y ; F2) ~=P *as graded F2algeb*
*ras.
The early uniqueness results on pcompact groups starting with [21], which pr*
*edate root
data, or even the formal definition of a pcompact 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 Fpcohomology algebras o*
*ver
the Steenrod algebra, but are not Fpequivalent. 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 PeterWeyl 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 ptorsion 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 ptorsion free and concen*
*trated in
even degrees for all pcompact groups. This result was first proved by Lin and *
*Kane, in fact
in the more general setting of finite mod p Hspaces, 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 nonconnected pcompact groups,*
* though,
just as for compact Lie groups, the classification is less calculationally expl*
*icit: Any discon
nected pcompact 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 2COMPACT GROUPS 5
this allows for a description of the moduli space of pcompact groups with comp*
*onent group
ss and whose identity component has Zproot 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 nonconnected pcompact groups). Let D be a Zpr*
*oot
datum, ss a finite pgroup and set B aut(D) = (B2Z(D))hOut(D). The space
M = (map (Bss, B aut(D)))hAut(Bss)
classifies pcompact groups whose identity component has Zproot datum isomorph*
*ic to D
and component group isomorphic to ss, in the following sense:
(1) There is a onetoone correspondence between isomorphism classes of pco*
*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 pcompa*
*ct groups
identifies with the set of Out(ss)orbits on [Bss, B aut(D)], which is f*
*inite.
(2) For each pcompact group X the corresponding component of M has the homo*
*topy
type of B Aut(BX) via the zigzag
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 pcompact 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 pcompact groups: Homotopy classes of maps BX*
* ! BX0
which induce isomorphism in rational cohomology (the notion of an isogeny for p*
*compact
groups) are in onetoone correspondence with the conjugacy classes of isogenie*
*s between
the associated Zproot data, sending isomorphisms to isomorphisms. Here an isog*
*eny of Zp
root data (W, L, {Zpboe}) ! (W 0, L0, {Zpb0oe}) is a Zplinear 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 selfequivalences of pcompleted*
* classifying
spaces of compact connected Lie groups, the most general of the theorems obtain*
*ed by
JackowskiMcClureOliver 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 2compact 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 Zproot data, Theorem 8.1, shows that *
*all Zproot
data have already been realized as root data of pcompact 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
Z2root datum not coming from a Zroot datum; for p odd see [8].)
The proof that two connected pcompact groups with isomorphic Zproot data ar*
*e iso
morphic, is divided into a prestep and three steps, spanning Sections 26. 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
firstnamed 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 DwyerWilkerson [28] showed how to exten*
*d part of
the classical paper of Tits [61] to pcompact 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 2compact
groups of rank 2 was eliminated in [4, version 2].
In [4], recalled in Recollection 8.2, we show that the "AdamsMahmud" 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 selfhomotopy 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 46).
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, centerfr*
*ee p
compact groups X and X0with isomorphic Zproot 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 2COMPACT GROUPS 7
Step 1 (Section 4): Using that in a connected pcompact 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 pcompact 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 selfequivalences of BCX ( )1.
Now for a general elementary abelian psubgroup : 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 pfusion, 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 pcompact groups (only DI(4*
*) for p = 2)
are easily dealt with, we can assume that BX = BG^pfor some simple, centerfree*
* Lie group.
JackowskiMcClureOliver showed in [32] that BG^pcan be expressed as a homotopy*
* colimit
of certain subgroups P of G, the socalled pradical (also known as pstubborn)*
* subgroups.
For a pradical 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 porbit category of eGwith objects*
* eG=Pefor ePa
pradical 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 casebycase calculations, JackowskiMcClureOliver 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 centerfree simple pcompact groups *
* 9
3. Preliminaries on selfequivalences of nonconnected pcompact 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 Zproot data *
*34
8.1. Root datum, normalizer extension, and root subgroups of a pcompact 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 2compact groups were obtained by *
*DwyerMiller
Wilkerson [21] twenty years ago, in the fundamental cases SU(2) and SO(3). Notb*
*ohm [50]
and MollerNotbohm [46] covered SU(n). Viruel covered G2 [64], ViruelVavpeti~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 inductiononcentralizers approa*
*ch to the
classification, and Antonio Viruel gave us the idea of trying to compare the ce*
*ntralizer and
THE CLASSIFICATION OF 2COMPACT GROUPS 9
pstubborn 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 pcompact 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 2compac*
*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 centerfree simple pcompact groups
In this section we reduce the classification of connected pcompact groups to*
* the case
of simple centerfree groups, in the sense that the classification statement, T*
*heorem 1.2,
holds for a connected pcompact group if it holds for the simple factors occurr*
*ing in the
corresponding adjoint (centerfree) pcompact 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
Zproot 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 pcompact gro*
*up X is
said to be determined by its Zproot datum DX if any connected pcompact 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
pcompact 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 pcompact 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 pcompact groups X, then all connecte*
*d pcompact
groups are determined by their root datum, as is seen by considering products.
Construction 2.2 (Quotients of pcompact groups). For explicitness we recall th*
*e quotient
construction for pcompact groups, and describe when a selfhomotopy equivalenc*
*e induces
a homotopy equivalence on quotients, since this will be used in what follows:
Let X be a pcompact group, A an abelian pcompact group and i : BA ! BX a
central homomorphism. By assumption BA is homotopy equivalent to map (BA, BA)1 *
*and
map (BA, BX)iev!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 pointset 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 selfhomotopy equivalence f : BX ! BX such that there e*
*xists
a homotopy equivalence g : BA ! BA making the diagram
g
BA ____//_BA
i i
fflfflffflffl
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 g1. Next no*
*te that
in general, if ' : G ! G0is a map of monoids, h : Y ! Y 0is a map from a Gspac*
*e Y to
a G0space 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
onesided 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 g1, and h the selfmap of map (BA, BX)i given by fi 7! f O f*
*i O g1. 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 pcompact group with center i : BZ ! BX and let
q : BX ! BX=Z denote the quotient map. Then any selfhomotopy 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 2COMPACT 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 nullhomotopic 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 centerfree case). Let X be a connected pcompact*
* 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 pcompact groups with the s*
*ame
Zproot 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 pcompact 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 pcompact torus, and furthermore BTe! BYe*
* is a
maximal torus by the diagram
Ye=Te_____//_BTe_____//BYe
  
  
fflffl fflffli fflffl
Y=T _____//_BT______//BY
 qOi q
  
fflffl fflffl fflffl
*______//P2BY______P2BY.
Any selfhomotopy equivalence f : BY ! BY lifts to a selfhomotopy 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 AdamsMahmud map , recalled in Recollection 8.2, together with Proposition*
* 8.14
THE CLASSIFICATION OF 2COMPACT 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 righthand 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 pgroup and i : BA ! BX0 is central. Proposition 8*
*.4(2)
shows that X0 has Zproot 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
selfhomotopy 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 selfhomotopy equivalence*
* __'of BX
with the property that (__') = (')=A = ff0=A = ff as desired.
3.Preliminaries on selfequivalences of nonconnected pcompact groups
In this short section we prove a fact about detection of selfequivalences of*
* nonconnected
pcompact groups on maximal torus normalizers, which we need in the proof of th*
*e main
theorem, where nonconnected groups occur as centralizers of elementary abelian*
* pgroups
in connected groups.
14 K. ANDERSEN AND J. GRODAL
Proposition 3.1. Let X be a (not necessarily connected) pcompact 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 AdamsMahmud 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 basepoint. Hence the fivelemma 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) pcompact grou*
*p such
~=
that : Out(BX1) ! Out(DX1). Let i : BNp ! BX be the inclusion of a pnormali*
*zer
of a maximal torus, and let ' : BX ! BX be a selfhomotopy 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 AdamsMahmud map, cf. [8, Lem. 4.1], ' lifts *
*to a map
THE CLASSIFICATION OF 2COMPACT 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
psubgroup 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 centerfree si*
*mple p
compact groups with isomorphic Zproot 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 pcompact group Y depe*
*nds
only on DY . We make the following inductive hypothesis:
(?) For all connected pcompact groups Y with cdY < cdX, Y is determined by *
*its
Zproot datum DY and : Out(BY ) ! Out(DY ) is an isomorphism.
Let D be a fixed Zproot 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 psubgroup 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 ssequivariant 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 Gaction, 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 Haction, then eYhas a free G *
*x Haction.
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 2COMPACT 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 Kequivariant. By Propositions 8.4(3)and 8.9, BC^X( )1and BCX^0( 0)1have is*
*omor
phic Zproot 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 Kequivariant so that passing to a quotient of diagram (4.3)with '*
* inserted
produces a leftto right map making the diagram (4.1)homotopy commute.
Consider the AdamsMahmudlike zigzag
(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 Kaction*
*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 fixedpoints, so the maps in (4*
*.4)induce
homotopy equivalences between the fixedpoints. 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 none*
*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
fflffly____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 psubgroup 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 pullback 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 2COMPACT GROUPS 19
Theorem 5.1. Let X and X0 be two connected simple centerfree pcompact groups *
*with
isomorphic Zproot 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 centerfree pcompact group. If *
*: BE !
BX is a nontoral elementary abelian psubgroup of rank two, then CX ( )1 is no*
*ntrivial 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 ac. Step 1a assumes toral, Step 1b assu*
*mes rank
two nontoral, and finally Step 1c considers the general case.
Step 1a: Assume : BE ! BX is a toral elementary abelian psubgroup. 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 lefttoright composit*
*e in the
following diagram
(5.1)
BCN (~VO)OO
ppp  OOOO
ppp ' OOO
ppp  OOOO 0
j Vppppp OOOj 0V
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( 0V)1)=K____//_BCX0( 0V )
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( 0V)(_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( 0V)( 0) BC^X0( 0).
Denote the bottom lefttoright 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
fflfflgh  fflffl J$$J
____//V_ _____//
BCX^( V )' BCX^0( 0V ) 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 nontoral .
Step 1b: Assume : BE ! BX is a rank two nontoral elementary abelian psubgro*
*up. By
Proposition 5.2 either CX ( )1 is nontrivial or DX ~=DPU(p)^p.
THE CLASSIFICATION OF 2COMPACT 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 fflfflh^ ff(V ) fflffl 
BC^X( ) _______//_BCX ^( ff(V))'//_BCX0(j0~^O (ff1ff(V)))_//_BXf0.
Here all the nonidentity vertical maps are given on the level of mapping space*
*s (i.e., without
the tilde) by f 7! f Off1, which induces a map on the indicated spaces by taki*
*ng homotopy
fibers of the map to B2ss1(D). The lefthand and righthand 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(ff1)' 'g ,Vfor all ff 2*
* WX ( ).
Now, since D ~=DPU(p)^pwe have ss1(D) ~=Z=p, and because E is nontoral, we see*
* that
BCX ( ) ' BE and BC^X( )' BP , where P is the extraspecial 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 pgroup*
* 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 nontrivial, 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
psubgroup j : BU = BZ=p ! BCX ( )1 in the center of a pnormalizer 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 )
nnnn77nOOMMMM'MVM
nnnnn  MMM
nn  M&&M
BU x BEPP____//_BCX (j x V )8BX08
PPP  qqqqq
PPPP  qq'jqq
PP''Pfflfflqq
BCX (j)
Here the lefthand side of the diagram is constructed by taking adjoints of j x*
* and hence
it commutes. The righthand 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 lefttoright composite (for some rank one subgroup V*
* E) or
the bottom lefttoright 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 pnormalizer of a max*
*imal torus in
CX ( ), and hence CX^(j x )contains a pnormalizer 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 psubgroup 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 psubgroup 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)
sss99sOOIII'^IV1I
sssss  III
s  II$$
BC^X( )_____//BCX^( U ) BXf0::
KK uuu
KKK  uu
KKK  uuu
K%%Kfflffl^'uV2u
BCX^( V2)
The lefthand side of this diagram is constructed by adjointness and hence comm*
*utes, and
the righthand side of the diagram commutes up to homotopy by Steps 1a and 1b. *
*Thus the
top lefttoright composite ]' ,V1is homotopic to the bottom lefttoright 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 2COMPACT 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
fflfflgh,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(aeV) MMMMM
fflffl fflffl MMM&&
BC^X( ) _______//_BCX^( V_)____//BXf0
The lefthand side commutes by construction and the righthand 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 casebycase arguments on the level of Zproot data.
Proof of Proposition 5.2.Assume first that (WX , LX ) is an exotic Zpreflectio*
*n group. We
claim that the centralizer of any rank one elementary abelian psubgroup of X i*
*s connected,
which in particular implies that there can be no rank two nontoral elementary *
*abelian
psubgroups: For p > 2 this follows from [8, Thms. 11.1, 12.2(2) and 7.1] combi*
*ned with
DwyerWilkerson'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 2subgroup of rank one up to conjug*
*ation and
that the centralizer of this subgroup is connected.
By the classification of Zproot 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*
*erfree
we may assume that so is G. If ss1(G) has no ptorsion, [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 psubgroup j : BZ=p ! BX of X, and hence X *
*does
not have any rank two nontoral elementary abelian psubgroups.
We can thus furthermore assume that ss1(G) has ptorsion, which implies that *
*DX ~=DG^p
for one of the following G: G = PU (n) for pn; 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 pcompact toral group, since then [23, Cor. 1.4] implies that for any element*
*ary abelian
psubgroup of rank two, CX ( )1 is nontrivial.
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 psubgroup 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 casebycase check of the above cases shows *
*that this
inequality can only hold when G = PU (n), pn. 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 pullback square of pcompact groups
f0
BX0 _____//BY 0
g0 g
fflfflf 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 Fpfinite) and g : BY 0! BY is an epimorphism (i.e., Y=Y 0is the cla*
*ssifying
space of a pcompact group). Then f0 : BX0! BY 0is a centric monomorphism, and *
*g0 is
an epimorphism.
Proof.It is clear from the pullback 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 2COMPACT 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, centerfree pcompact 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 Zproot
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 selfmaps, 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 DwyerWilkerson [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 pcompact groups for p odd, has polynomial Fpcohomology 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 selfmaps 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 centerfree compact Lie group G, wit*
*h universal
cover eG. Let Orp(Ge) be the full subcategory of the orbit category with object*
*s the eGsets
eG=Pewith ePa pradical subgroup (i.e, ePis an extension of a torus by a finite*
* pgroup, such
that NGe(Pe)=Peis finite and contains no nontrivial normal psubgroups).
For the pradical 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 eGset map eG=Pe*
*f!eG=Qe
26 K. ANDERSEN AND J. GRODAL
to the map cg1 : BPe ! BQe, where gQe = f(ePe), via the canonical equivalences*
* BPe =
(EPe)=Pe'!EGexGeeG=Pe. (Note that F 0is not welldefined 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 pradical, 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 welldefine*
*d (free)
homotopy class of maps cg1: BCG ^(pZ(P ))! BCG ^(pZ(Q)). Consider the diagram
_____'_____//
BPe^p__________//(BCG ^(pZ(P )))^p BCG^^p(i(pZ(P)))
 QQQQ'i^
cg1 cg1 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 righthand 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 pullback square
e_Ge=Pe
BPe^p____//_BXf0
 
 
fflffl_G=fflfflP
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! ssi1(Z(Pe)^p). Since eGis simple and simpl*
*y connected it
now follows from the fundamental calculations of JackowskiMcClureOliver [32, *
*Thm. 4.1]
THE CLASSIFICATION OF 2COMPACT GROUPS 27
that
limi+1Orssei1(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 pnormalizer 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 JackowskiMcClureOliver, cf. [32, Pf. of*
* Thm. 4.2]:
We are assuming that BX ' BG^p, for some compact connected centerfree 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 righthan*
*d vertical
map, which uses Theorem 8.6 for compact Lie groups.) The lefthand 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 JackowskiMcClureOliver's ca*
*lculation
of higher limits [32, Thm. 4.1]
limieG=Pe2Orssei1(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 AdamsMahmud 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 pcompact group). In [8], with Moller and
Viruel, we established the fundamental group formula for pcompact 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 DwyerWilkerson [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 wellknown case of c*
*ompact Lie
groups. Hence we can assume that X and X0are simple centerfree pcompact group*
*s with
the same Zproot datum D, and that we know the fundamental group formula for X *
*and
Theorem 1.2 for connected pcompact 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*
*onobvious
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 Z2root 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 selfequivalences of nonconnected 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 2COMPACT 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 1.31.6 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 selfequivalences ' 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 pcompact group X^pand that Fpcompletion allows us to identify (WX^p, *
*LX^p) with
(W, L Zp), where L = ss1(T ). In particular (W, L) is a finite Zreflection g*
*roup and all
reflections have order 2. Furthermore, for a fixed Zreflection group (W, L), t*
*here is a bijec
tion between Zroot data with underlying Zreflection group (W, L) and Z2root *
*data with
underlying Z2reflection 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 Zroot datum with underlying Zreflection 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 Zroot data DG ~=D. The Zproot 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 lefthand square in this diagram is homotopy commutative by construction. F*
*or the
righthand 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 fflfflvvmmm 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 EilenbergMac 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 pullbacks 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 Zroot datum of G from BX).
We now give the description of B Aut(BG), also providing a quick description *
*of how the
pcompleted 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 pcomplete 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 nonconnected 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
nonconnected, 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 2COMPACT 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 onetoone 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 onetoone 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 EilenbergMoore 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 F2complete and simply connected.
Write ss2(Y 0) ~=F T , where F is a finitely generated free Z2module and T*
* is a finite
abelian 2group, 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 F2complete. Since H*(Y 00; F2) is polynomial, the Eilenber*
*gMoore spec
tral sequence shows that H*( Y 00; F2) is F2finite, so Y 00is the classifying *
*space of a con
nected 2compact 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 F2algebra A**
* with
given action of the Steenrod algebra A2, can be realized by at most one space Y*
* , up to
F2equivalence, 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 2c*
*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 F2algebra P *, th*
*ere are
only finitely many spaces Y , up to F2equivalence, with H*(Y ; F2) ~= P *as ri*
*ngs. By
the proof of Theorem 1.4, we can assume that Y is 2complete, 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 2compact group X. By Proposi*
*tion 8.18,
Out(DX ) only contains finitely many 2subgroups 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 2compact
groups: The rank of X is bounded above by the Krull dimension of the cohomology*
* ring,
so by the classification of pcompact groups, Theorem 1.2, it is hence enough t*
*o see that
there are only a finite number of Z2root 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 2compact 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 2subgroup : 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 A2algebra.
Proof.By Lannes theory [37, Thm. 0.4] homotopy classes of maps from classifying*
* spaces of
elementary abelian 2groups 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 2subgrou*
*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 wellknown 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, Vn01), WSp(n)( ) = ( n, Vn),
WG2( ) = GL 3(F2), WDI(4)( ) = GL 4(F2),
THE CLASSIFICATION OF 2COMPACT 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 ndimensional permutation module for F2[ n] and Vn01is the (n *
* 1)
dimensional submodule consisting of elements with coordinate sum 0. The Weyl g*
*roup
WSp(n)( ) = ( n, Vn) decomposes as ( n, Vn01) 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 F2reflection group, and hence the two notions are ac*
*tually equiv
alent.) By the KrullSchmidt 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, Vn01)*
* arises
from both SU(n)^2and Sp(n)^2. However since H6(B SU(n); F2) ! H6(BVn01; F2) is*
* injec
tive and H6(B Sp(n); F2) ! H6(BVn01; 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 *
*A2action
on H*(BX; F2).
Proof of Theorem 1.6.It is obvious that there is a onetoone correspondence be*
*tween iso
morphism classes of pcompact 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 selfhomotopy equivalences of the fibration p.
By the classification of fibrations (see [20]), equivalence classes of such f*
*ibrations are in
onetoone 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 pcompact 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 PeterWeyl 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 2compact 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
PeterWeyl 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 Zproot data
The purpose of this section is to establish some general results about Zproo*
*t data of
pcompact groups, needed in the proof of the main theorem. The analogous resul*
*ts for
Zroot data and compact Lie groups are often well known; see [9, 16]. We build*
* on the
paper [28] by DwyerWilkerson and our earlier paper [4].
We briefly recall the definition of root data from the introduction: For an i*
*ntegral domain
R, an Rreflection group is a pair (W, L) where L is a finitely generated free *
*Rmodule 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 Rroot
datum to be a triple D = (W, L, {Rboe}) where (W, L) is a finite Rreflection g*
*roup and
for each reflection oe 2 W , Rboeis a rank one submodule of L with im(1  oe) *
* Rboeand
w(Rboe) = Rbwoew1for all w 2 W . If R ! R0is a monomorphism of integral domain*
*s, and D
an Rroot datum, we can define an R0root 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+. .+.oeoe1is the norm element (cf. the proof of*
* Lemma 8.8
below), so giving an Rroot datum with underlying reflection group (W, L) corre*
*sponds to
choosing a cyclic Rsubmodule of H1(; L) for each conjugacy class of reflec*
*tions oe. In
particular for R = Zp, p odd, the notions of a Zpreflection group and a Zproo*
*t datum
agree. If R has characteristic zero, an Rroot datum, or an Rreflection 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 Zproot data, which follows e*
*asily from
the classification of finite Zpreflection groups [8, Thm. 11.1]. This classif*
*ication is again
based on the classification of finite Qpreflection groups [13] [22] which stat*
*es that for a
fixed prime p, isomorphism classes of finite irreducible Qpreflection groups a*
*re in natural
onetoone correspondence with isomorphism classes of finite irreducible Crefl*
*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 Zproot data; splitting version).Any (1)Zpr*
*oot da
tum is isomorphic to a Zproot datum of the form (D1 Z Zp) x D2, where *
*D1 is a
Zroot datum and D2 is a direct product of exotic Zproot data.
(2) There is a onetoone correspondence between isomorphism classes of exot*
*ic Zproot
data and isomorphism classes of exotic Qpreflection groups given by the*
* assignment
D = (W, L, {Zpboe}) _ (W, L Zp Qp). Moreover ss1(D) = 0 for any exotic *
*Zproot
datum D.
THE CLASSIFICATION OF 2COMPACT GROUPS 35
Proof.For any Zproot 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 Zreflection gr*
*oup and
(W2, L2) is a direct product of exotic Zpreflection groups. It follows by def*
*inition that
there are unique Zproot 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 Zproot 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 onetoo*
*ne corre
spondence between exotic Zpreflection groups up to isomorphism and exotic Qpr*
*eflection
groups up to isomorphism. To prove the first part of (2)it thus suffices to sho*
*w that any
exotic Zpreflection group (W, L) can be given a unique Zproot 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 Zproot 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 (1w)(x), w *
*2 W , x 2 L.
Hence the final claim follows from the fact that H0(W ; L) = 0 for any exotic Z*
*preflection
group (W, L) [8, Thm. 11.1].
8.1. Root datum, normalizer extension, and root subgroups of a pcompact
group. For any connected pcompact group X with maximal torus T , the Weyl group
WX acts naturally on LX = ss1(T ) as a finite Zpreflection group [25, Thm. 9.7*
*(ii)]. For p
odd, H1(; L) = 0 for any reflection oe, so the finite Zpreflection group (*
*WX , LX ) gives
rise to a unique Zproot datum DX . The construction of root data for connected*
* 2compact
groups, in the present form, is due to DwyerWilkerson [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 2compact 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 1oe!LX )if (8.1)splits,
Z2boe= 1+oe
ker(LX ! LX ) otherwise.
The root datum of X is then the Z2root 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 Zproot 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 pcompact group X with Zproot 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 fiberwise Fpcompletion [10, Ch. I, x8] of BN~D and li*
*kewise
introduce the (nondiscrete) root subgroups BNX,oeand BND,oeby fiberwise Fpco*
*mpletion
of the corresponding discrete versions BN~X,oeand BN~D,oe.
Recollection 8.2 (The AdamsMahmud 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~
 
 
fflfflf 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], fiberwise Fpcompletion [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 Zproot datum,*
* we
define a subdatum of D to be a Zproot 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 Fpfinite and X and Y have the same r*
*ank.
Proposition 8.3. Let X and Y be connected pcompact 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 Zproot 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 Z2root data of X and Y , cf. [28, Pf. of Lem. 9.16].
For a Zproot 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 2COMPACT 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 Zpsublattice 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*
*wyerWilkerson
[26, x7] on how to compute the center and centralizers of toral subgroups of X.
Proposition 8.4. Let X be a connected pcompact group with Zproot 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 Zproot 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
Zproot 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 Zproot datum. Recall that the coroot lattice L0*
* L is the
Zplattice 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 pcompact 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 2compact 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
2compact group of rank 1 it follows (cf. [28, p. 13691370]) that X(oe) ~= G^2*
*for G =
SU(2) x (S1)r1, SO (3) x (S1)r1 or U(2) x (S1)r2 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 DwyerWilkerson [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 (DwyerWilkerson [29] and Remark 6.2). Let X be a connected pcompa*
*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 Zroot datum D0, and D2 is a direct product of exotic Zproot 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 *
*Zroot
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 DwyerWilkerson [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 pcompact group.
Lemma 8.7. Let f : BX ! BY be a monomorphism of maximal rank between connected
pcompact groups X and Y . If ss1(DY ) = 0 then BZ(X) ' BZ(NX ).
THE CLASSIFICATION OF 2COMPACT GROUPS 39
Proof.For p odd, the conclusion holds for any connected pcompact 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 Zproot 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 + . .+.oeoe1is 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= oeboe6= 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 Zproot datum and let L0 be the coroot lattice. *
*If L0is a
Zplattice 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 Zproot 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 pcompact group with Zproot 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 Zproo*
*t datum,
and A ~Z(D) a subgroup of the discrete_center._We_define a quotient of D to b*
*e a Zproot
datum of the form D=A = (W, ~T=A, {hoe}) where hoedenotes the image of hoein ~T*
*=A; the
fact that this is a Zproot datum is part of the following result.
Proposition 8.10. (1)If D = (W, ~T, {hoe}) is a Zproot datum and A Z~(D),*
* then
___
D=A = (W, ~T=A, {hoe}) is a Zproot datum. Moreover ]D=A ~=eD and Z~(D=*
*A) ~=
~Z(D)=A.
(2) Let X be a connected pcompact group with Zproot datum DX and A ! X a
central monomorphism. If ~Adenotes the discrete approximation to A, then*
* the Zp
root datum DX=A of the pcompact 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*
*plattice
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 Zplattices, 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(boe1)fioe1so (W, L*, {Zpfioe1}) is a Zproot datum (the dua*
*l of D). Applying
Lemma 8.8 to this Zproot 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 Zproot datum. In pa*
*rticular we
obtain ]D=A= (W, L0, {Zpboe}) = eDby definition.
THE CLASSIFICATION OF 2COMPACT 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 Zproot
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 Zproot datum with Z~(D) = 0 or ss1(D) = 0 splits as a di*
*rect
product D ~=D1 x . .x.Dn of irreducible Zproot data Di.
Proof.The case ~Z(D) = 0 is essentially proved by DwyerWilkerson [27, Pf. of T*
*hm. 1.5],
for completeness we briefly sketch the argument: Let (W, L) be the Zpreflecti*
*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 pointwise 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 Zproot datum structure Dion (Wi, Li) su*
*ch that we
get a product decomposition D ~=D1 x . .x.Dn into irreducible Zproot 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 Zproot data; structure version).Let (1)D =*
* (W, L, {Zpboe})
be a Zproot datum with coroot lattice L0, and let D0 = (W, L0 LW , {Z*
*pboe}) =
eDx Dtriv, where Dtriv= (1, LW , ;) is a trivial Zproot 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 Zproot data Di with ss1(Di) = 0.
(2) For p > 2, the assignment D = (W, L, {Zpboe}) _ (W, L Zp Qp) is a onet*
*o
one correspondence between isomorphism classes of irreducible Zproot da*
*ta D with
42 K. ANDERSEN AND J. GRODAL
ss1(D) = 0 and isomorphism classes of irreducible Qpreflection 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 nonisomo*
*rphic
for n 3.
Proof.The fact that D0 is a Zproot 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 onetoone correspondence between isomorphism classes of exotic Zproot*
* data and
isomorphism classes of exotic Qpreflection groups, and that ss1(D) = 0 for all*
* exotic Zproot
data D. Hence it suffices by Theorem 8.1(1)to show the claim for Zproot data o*
*f the form
D1 Z Zp for a Zroot datum D1.
In this case it is clear that if ss1(D1 Z Zp) = 0, then we can find a Zroot*
* 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 Zroot data in the same way as for Zproot data). Hence*
* it suffices
to study the assignment D = (W, L, {Zpboe}) _ (W, L Z Q) from irreducible Zro*
*ot data
with ss1(D) = 0 to irreducible Qreflection groups. It is wellknown (cf. [9, x*
*4]) that this
assignment is surjective and that it only fails to be injective in that the Zr*
*oot data DSp(n)
and DSpin(2n+1)which are nonisomorphic for n 3 maps to the same Qreflection*
* group.
This proves part (2)since for n 3, the Zproot data DSp(n) Z Zp and DSpin(2n+*
*1) Z Zp
are nonisomorphic for p = 2 and isomorphic for p = 2.
8.4. Automorphisms. Recall that an isomorphism between two Zproot 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 Zproot 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 nonisomorphic irreducibleQZproot 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 Zproo*
*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 Zproot datum with coroot lattic*
*e L0. Let
D0= (W, L0 LW , {Zpboe}) = eDxDtriv, where Dtriv= (1, LW , ;) is a trivial Zpr*
*oot datum.
THE CLASSIFICATION OF 2COMPACT 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 Zproot datum D = (W, L, {Zpboe}) with ss1(D) = 0 we have Zp*
*boe=
ker(L N!L), where N = 1 + oe + . .+.oeoe1is the norm element: By Theorem 8.*
*1(1)either
D ~=D1 Z Zp for a Zroot 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 Zproot 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 Zproot data of a given rank, and that for a fixed Zproot 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 Zproot
data of a fixed rank.
Proof.By the classification of finite Qpreflection groups [13] [22], there are*
* only finitely
many finite Qpreflection groups of a fixed rank. For each finite Qpreflection*
* 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 JordanZassenhaus 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 Zpreflec*
*tion groups
of fixed rank. Finally, choosing a Zproot datum for a finite Zpreflection 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 Zpreflection group gives rise to onl*
*y finitely many
Zproot data. This proves the result.
Proposition 8.18. Let D be a Zproot datum. Then Out(D) contains only finitely *
*many
conjugacy classes of finite subgroups.
Proof.Let (W, L) be the finite Zpreflection 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
Zproot 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 Zpr*
*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 Zproot datum D. The first claim follows directly from t*
*he local
version of the JordanZassenhaus 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. In particular
E = Out(D) o m fits into an extension of the form
(8.4) 1 ! (Zp)m ! E ss!Q ! 1,
where Q is finite. Any homomorphism ' : G ! E gives a homomorphism ss O ' : G !*
* Q
by composition, and for a fixed homomorphism ff : G ! Q, the set of homomorphis*
*ms ' :
G ! E with ssO' = ff equals the set of splittings of the pullback 1 ! (Zp)m ! E*
*0! G ! 1
of (8.4)along ff : G ! Q. The set of such splittings, modulo conjugation by ele*
*ments in
(Zp)m , is in onetoone correspondence with H1(G; (Zp)m ), which is finite for*
* all actions of
G on (Zp)m . Hence the set of homomorphisms ' : G ! E with ss O ' = ff is finit*
*e modulo
conjugation in (Zp)m E. Since Q is finite, there are only finitely many homom*
*orphisms
ff : G ! Q, so we conclude that Rep(G, E) is finite as claimed.
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Institute of Mathematical Sciences, Aarhus University
Email address: kksa@math.ku.dk
Dept. of Mathematics, University of Chicago, and Dept. of Math. Sci., Univ. o*
*f Copenhagen
Email address: jg@math.ku.dk