NDETERMINED pCOMPACT GROUPS
JESPER M. MfflLLER
Abstract.One of the major problems in the homotopy theory of finite loop *
*spaces is the
classification problem for pcompact groups. It has been proposed to use *
*the maximal torus
normalizer (which at an odd prime essentially means the Weyl group) as th*
*e distinguishing
invariant. We show here that the maximal torus normalizer does indeed cla*
*ssify many pcompact
groups up to isomorphism when p is an odd prime.
Contents
1. Introduction 1
2. Higher limits of center functors *
* 4
2.14. Relation between A(W, t) and the orbit category O(W ) *
* 7
2.17. Centralizers *
* 7
3. Ndeterminism 8
3.1. Ndetermined automorphisms *
* 8
3.5. Ndetermined pcompact groups *
* 9
3.11. Centers and automorphism groups of pcompact groups *
* 13
3.18. Canonical factorizations *
* 14
4. Cohomologically unique pcompact groups *
* 14
5. The pcompact group PGL (n, C) *
* 19
5.17. The action of A(GL (n, C))(T, æ) on CGL(n,C)(T, æ) *
* 26
5.19. Representations of extraspecial pgroups. *
* 26
6. The 3compact group F4 28
7. Polynomial pcompact groups *
* 28
7.10. Construction of modular, centerless, polynomial, simple pcompact grou*
*ps 32
7.13. Automorphisms of X(G(m, r, n)) *
* 34
7.17. Automorphisms of other modular polynomial pcompact groups *
* 35
7.18. Structure of polynomial pcompact groups *
* 35
8. Proofs of Theorem 1.2 and Corollaries 1.31.6 *
* 39
9. Ndeterminism of product pcompact groups *
* 39
10. Maximal rank subgroups of DI2 *
* 43
11. Free Zpmodules and pdiscrete tori *
* 48
12. Shapiro's lemma 57
13. Cellular cohomology of small categories *
* 58
References 64
1. Introduction
This paper addresses the classification problem at odd primes for the pcompa*
*ct groups intro
duced by W.G. Dwyer and C.W. Wilkerson in their seminal paper [30] (surveyed in*
* [65, 53, 26]).
A pcompact group is a connected, pointed, H*Fplocal space BX such that H*(X; *
*Fp) is finite
___________
Date: 15th February 2002.
1991 Mathematics Subject Classification. 55R35, 55P15.
Key words and phrases. Classification of pcompact groups, automorphisms of p*
*compact groups, reflection
subgroup, Quillen category, left derived functors of the inverse limit functor,*
* spaces with polynomial cohomology,
Lie group.
1
2 J.M. MØLLER
where X = BX is the loop space [30, x2]. It is customary, though ambiguous, to*
* refer to BX by
the name, X, for its underlying loop space.
It has been conjectured [53, 72, 26], in analogy with the classification theo*
*rem for compact Lie
groups [25, 83], that pcompact groups are determined by their maximal torus no*
*rmalizers. The
maximal torus normalizer N(X) for the pcompact group X is an extension
(1.1) T (X) ! N(X) ! W (X)
of the maximal torus T (X) by the Weyl group W (X) [30, 9.8], and X is said to *
*be totally N
determined [68, 7.1] if
oX is determined by N(X), and,
othe automorphisms of X are determined by their restrictions to N(X).
We show here that almost all simple pcompact groups are totally Ndetermined a*
*t odd primes.
1.2. Theorem. Let X be a simple pcompact group, where p is an odd prime. Assum*
*e that the
rational Weyl group (r0W (X)) 6= (r0W (E8)) if p = 3 and (r0W (X)) 6= (r0W (Ej)*
*), j = 6, 7, 8, if
p = 5. Then X is totally Ndetermined.
The Weyl group W (X) [30, 9.7] of a connected pcompact group X is a finite g*
*roup of auto
morphisms of the free, finitely generated Zpmodule L(X) = ß1T (X), i.e. W (X) *
* GL (L(X)).
The rational Weyl group, r0W (X), is the image of W (X) in GL (L(X) Zp Qp), an*
*d rpW (X),
the FpWeyl group, the image of W (X) in GL (L(X) Zp Fp). As usual, (r0W (X))*
* stands for
the conjugacy class of the rational Weyl group. The connected pcompact group X*
* is simple if
L(X) Zp Qp is an irreducible r0W (X)module [67, 5.4].
At an odd prime p, the maximal torus normalizer extension (1.1) for a connect*
*ed pcompact
group splits in an essentially unique way [5] and thus N(X) is in fact complete*
*ly determined
by the reflection group (W (X), L(X)). This explains the first part, merely a *
*reformulation of
Theorem 1.2, of the the corollary below; see (4.3) for the precise meaning of t*
*he other statements.
1.3. Corollary.Let X be a simple pcompact groups as in Theorem 1.2. Then X is *
*determined
up to (local) isomorphism by its (rational) Weyl group, and the automorphism gr*
*oup Aut(X) is
isomorphic to NGL(L(X))(W (X))=W (X). Furthermore, if X is centerless or simply*
* connected, then
X is determined by its FpWeyl group, and X is a cohomologically unique pcompa*
*ct group.
For the bigger class of connected (but not necessarily simple) pcompact grou*
*ps Theorem 1.2
takes on a particularly appealing form.
1.4. Corollary.Assume that p > 5. The map
æ oe æ oe
Isomorphism classes of(W,L) Similarity classes of
connected pcompact groups!Zpreflection groups
is a bijection, and Aut(X) is isomorphic to NGL(L)(W )=W for the connected pco*
*mpact group X
corresponding to the reflection group (W, L).
In the general case, for the class of not necessarily connected pcompact gro*
*ups, Theorem 1.2
takes the following form.
1.5. Corollary.Let X be a pcompact group such that all its simple factors sati*
*sfy the assumptions
of Theorem 1.2. Then X is totally Ndetermined and Out(X) ~=Out(N(X)).
The simple factors of the pcompact group X are the simple, centerless pcomp*
*act groups in the
splitting [32, 80] of P X0 = X0=Z(X0), the adjoint form of the identity compone*
*nt of X.
Let me also mention the following partial classification result for connected*
* finite loop spaces
with maximal tori [59, 1.1].
1.6. Corollary.(Cf. [96], [59, 1.6]) Let X be a connected finite loop spaces wi*
*th a maximal torus.
Assume that X has the same Weyl group as the compact, connected simple Lie grou*
*p G and that
no simple factor of G is locally isomorphic to E6, E7, or E8. Then (BX)[1_2] a*
*nd (BG)[1_2] are
homotopy equivalent spaces.
NDETERMINISM 3
In light of the observation by C. Wilkerson [97] that the Weyl group of any c*
*onnected finite
loop space with maximal torus must agree with the Weyl group of a compact conne*
*cted Lie group,
this proves the maximal torus conjecture [49, Conjecture D, p. 68] [99] away fr*
*om the prime 2 in
a number of particular cases.
The proof that the simple pcompact groups of Theorem 1.2 are Ndetermined go*
*es in outline as
follows. Consider some connected pcompact group X with maximal torus normalize*
*r j :N ! X
and assume that the same extended pcompact torus N also can serve as the maxim*
*al torus
normalizer j0:N ! X0 for some other pcompact group X0. Starting with the confi*
*guration
j j0
X oo__N_ ____//_X0
our task is to construct an isomorphism f :X ! X0under N. We observe (3.7) that*
* it suffices to
consider the centerless form of X. According to the Homology Decomposition Theo*
*rem [31, x8],
BX is (the pcompletion of) the homotopy colimit of the A(X)opspace of central*
*izers BCX (E, )
of nontrivial elementary abelian psubgroups :E ! X of X. For any monomorphi*
*sm :E ! X
it is possible to find (nonuniquely) a preferred lift ~: E ! N of such that *
*the morphisms
Cj Cj0
CX (E, )oo_CN_(E, ~)___//_CX0(E, 0)
are again maximal torus normalizers for the centralizers of (E, ) and (E, 0) *
*where 0= j0~ [69].
As the center of X is trivial, the centralizer of (E, ) will have smaller coho*
*mological dimension than
that of X [30, 6.14, 6.15]. Assuming, as part of an inductional argument, that *
*CX (E, ) (which
very well may be nonconnected) is totally Ndetermined, there will therefore b*
*e an isomorphism
f(E, ~): CX (E, ) ! CX0(E,un0)der CN (E, ~). It remains to show that these lo*
*cally defined
isomorphisms f(E, ~) do not depend on the choice of preferred lifts and that th*
*ey combine to yield
a morphism f :X ! X0under N.
Ndeterminism is actually not a property of the pcompact group X itself but *
*rather a property
of the extended pcompact torus N(X): If X is Ndetermined, so is, by the very *
*nature of the
concept, any other pcompact group that admits N(X) for a maximal torus normali*
*zer.
Most of the time the prime p will be assumed to be odd. Some modifications wi*
*ll be needed to
handle the case where p = 2 [60]. Even the formulation of the Nconjecture itse*
*lf will have to be
refined as N(O(2)) = O(2) = N(SO (3)) but O(2) and SO(3) are distinct 2compact*
* groups.
Organization of the paper. In Section 3, I set up the general theory that will *
*be applied in a case
bycase verification of the Nconjecture for the simple, centerless pcompact g*
*roups. We deal with
Afamily, represented by the pcompact groups PGL (n, C) = PSL(n, C), in Sectio*
*n 5, and with
the polynomial case, which includes nearly all remaining compact simple Lie gro*
*ups and all the
exotic (nonLie) simple pcompact groups, in Section 7. The proofs of (1.21.6)*
* are in Section 8.
Sections 2 and 13 contain material dealing with the general problem of computin*
*g cohomology
groups of categories. (There is no claim to originality here as the vanishing *
*result of (2.4) was
proved in [34] and the spectral sequence of (13.2) seems to be that of Lück [54*
*, 17.28] or S_lomi'nska
[87].)
Notation. Write Zp for the ring of padic integers, Qp for the field of padic *
*numbers, and Fp for
the field with p elements. For a pcompact group X, let
oT (X) denote the maximal torus of X [30, 8.9],
oL(X) = ß2(BT (X)) the lattice of X,
oT~(X) = L(X) Z=p1 the pdiscrete maximal torus of X [30, x6],
ot(X) = L(X) Z=p the maximal elementary abelian subgroup of ~T(X),
oW (X) the Weyl group of X [30, 9.6], r0W (X) the rational and rpW (X) the m*
*od p Weyl
group of X (Section 4),
oN(X) the maximal torus normalizer of X [30, 9.8],
oZ(X) the center of X [31, 58],
or(X) the rational rank (of the identity component) of X [30, 5.11],
oAut(X) the group of invertible elements in the monoid End(X) = [BX, *; BX] *
*[68, x3]
of based homotopy classes of based selfmaps of BX, and Out(X) = Aut(X)=ß0(*
*X) the
corresponding group in the unbased category, and,
oA(X) the Quillen category of X.
4 J.M. MØLLER
The objects (E, ) of A(X) are conjugacy classes of monomorphism :E ! X of no*
*ntrivial
elementary abelian pgroups E into X. The morphisms (E0, 0) ! (E1, 1) of A(X)*
* consists of all
group homomorphisms f :E0 ! E1such that (E0, 0) = (E0, 1f). An object (E, ) *
*of A(X) is
toral if :E ! X factors through the maximal torus T (X) ! X. Let
oA(X) t denote the full subcategory of all toral objects, and
oA(X) t the full subcategory of all objects with a morphism to some nontora*
*l object.
The notation for categories is
opcg is the category of pcompact groups,
oGrp is the category of groups,
oAb is the category of abelian groups,
oSp is the category of simplicial sets, and
oTop is the category of topological spaces.
In [pcg], [Grp ], [Sp ] the objects are pcompact groups, groups, topological s*
*paces and the mor
phisms are conjugacy classes of pcompact group morphisms, conjugacy classes of*
* group homomor
phisms, homotopy classes of continuous maps.
Acknowledgments. I would like to thank Kasper Andersen, Jesper Grodal, Dietrich*
* Notbohm, and
Antonio Viruel for several fruitful discussions and for very valuable help at m*
*any points, and the
referee for a long list of constructive comments. This work was partially suppo*
*rted by Centre de
Recerca Matem`atica.
2.Higher limits of center functors
This section contains a vanishing result (2.4) for the derived limits of a ce*
*rtain functor, defined
in purely algebraic terms, which informs on the obstruction theory associated t*
*o the Jackowski
McClure centralizer homology decomposition [45, 31] of BX.
Let W be a finite group and t a nontrivial FpW module which is finite dimen*
*sional as an
Fpvector space. For nontrivial subgroups E0 and E1 of t, put
__
(2.1) W (E0, E1) = {w 2 W  w(E0) E1} and W (E0) = {w 2 W  we = e for all *
*e 2}E0
and note_that the set of orbits for the action of the pointwise stabilizer grou*
*p W (E0) on the
set W (E0, E1) is the_set of group homomorphisms of E0 ! E1 induced_by elements*
* of W . The
stabilizer subgroup W (E0, E0) of E0 will also be written as W (E0).
2.2. Definition.A(W, t) is the category with
objects:nontrivial elementary abelian subgroups E of t, and,
morphisms: group homomorphisms E0 ! E1 induced by elements of W .
For any ZpW module L, Lj:A(W, t) ! Ab, j 0, is the functor that takes the ob*
*ject E t to
the cohomology group*Hj(W (E); L) and the morphism E0 w!E1 in A(W, t) to the h*
*omomorphism
Hj(W (E0); L) resOw!Hj(W (E1); L).
Here is a more detailed explanation_of the functors Lj: Any morphism E0 ! E1 *
*in A(W, t),
represented by an element w 2 W (E0, E1), can be factored E0 ! wE0 E1 into an*
* isomorphism
w :E0 ! wE0 followed by an inclusion. Consider the corresponding group homomorp*
*hisms
W (E0) c(w)!W (E0)w W (E1)
where c(w)w0 = ww0w1 is conjugation by w and W (E0)w = wW (E0)w1 = W (wE0) an*
*d let,
as usual [37, 4.1.1], w*:Hj(W (E0); L) ! Hj(W (E0)w; L)be the isomorphism induc*
*ed by c(w)1
and multiplication by w on L. Define Lj(w) as the composition
* res
Hj(W (E0); L) w!Hj(W (E0)w; L) !Hj(W (E1); L)
of w* followed by the restriction morphism. Since for all w0 2 W (E0) we have t*
*hat W (E0)ww0 =
W (E0)w and cohomology is insensitive to inner conjugation [55, IV.5.6], Lj(ww0*
*) = Lj(w) for all
w0_2 W (E0) and thus this morphism is independent of the choice of representati*
*ve for wW (E0) 2
W (E0, E1)=W (E0) = A(W, t)(E0, E1), cf. [45, 7.6].
NDETERMINISM 5
For instance, for a connected pcompact group X, the functors
(2.3) L(X)2j:A(W (X), t(X)) ! Ab, j = 1, 2,
take the nontrivial elementary abelian psubgroup E of t(X) to H2j(W (X)(E); *
*L(X)).
2.4. Lemma. [34, 8.1] Lj:A(W, t) ! Ab is an acyclic functor in the sense that
( j
limi(A(W, t); Lj)= H (W ; L) i = 0
0 i > 0
for all j 0.
For a pcompact group X, let BCX :A(X)op! pcg and BZCX :A(X) ! Top be the f*
*unctors
that take the object (E, ") of A(X) to
(2.5) BCX (E, ")= map(BE, BX)B"
(2.6) BZCX (E, ")= map(BCX (E, "), BX)Be(")
where Be("): BCX (E, ") ! BXis the evaluation map, and define
(2.7) ßj(BZCX ): A(X) ! Ab, j = 1, 2,
to be the composition of BZCX with the jth homotopy functor. (There is no basep*
*oint problem
here since only abelian pcompact groups are involved.)
2.8. Lemma. Let p be an odd prime and X a connected pcompact group. Assume tha*
*t the identity
component CX (E)0 of the centralizer of any nontrivial elementary abelian psu*
*bgroup E of T (X)
has Ndetermined automorphisms [68, 3.10]. Then there is an equivalence of cate*
*gories
A(W (X), t(X)) ! A(X) t
such that the functors ßj(BZCX ) when restricted to A(X) t correspond to the fu*
*nctors L(X)2j,
j = 1, 2, of ( 2.3).
Proof.Take wW (E0): E0 ! E1in A(W (X), t(X)) to the morphism wE0:(E0, ie0) ! (*
*E1, ie1)
in A(X) t (where ej:Ej ! t(X), j = 0, 1, is the inclusion and i the pcompact g*
*roup morphism
t(X) ! T (X) ! X). This provides a functor
(2.9) A(W (X), t(X)) ! A(X) t
Since the natural map W \[BE, BT (X)] ! [BE, BX], induced by BT (X) ! BX, is in*
*jective for
any elementary abelian pgroup E [67, 3.4] [32, 3.4], this functor is full and *
*as it is also clearly
faithful, (2.9) is an equivalence of categories.
Let now ~N(X) be a discrete approximation [31, 3.12] to the maximal torus nor*
*malizer N(X). For
any elementary abelian psubgroup E of ~T(X), CN~(X)(E) is a discrete approxima*
*tion to CN(X)(E)
and ZCN~(X)(E) is a discrete approximation to ZCN(X)(E) which is isomorphic to *
*ZCX (E) (2.19)
[68, 4.12]. Since the prime p is odd, ~N(X) = ~T(X) o W (X) is a semidirect pr*
*oduct [5, 2.1] and
hence
(2.10) CN~(X)(E) = CT~(X)oW(X)= ~T(X) o W (X)(E), Z(CN~(X)(E)) = ~T(X)W(X)(E*
*),
so that ßjBZCX (E) = ßj((BH0(W (X)(E); ~T(X)))^p) = H2j(W (X)(E); L(X)) = L(X)*
*2j(E).
If the Weyl group element w 2 W (X) takes the elementary abelian psubgroup E*
*0 ~T(X)
into the elementary abelian psubgroup E1 ~T(X), w represents a morphism w :E*
*0 ! E1 in
A(W (X), t(X)). We want_to determine the effect of w on the centralizer center*
*s. Choose a
lift ~w2 N~(X) of w 2 W (X)(E0, E1) W (X) = N~(X)=T~(X). Conjugation by ~w, *
*given by
c(w~)(n) = ~wnw~1, n 2 ~N(X), takes E0 into E1 and conjugation by ~w1, c(w~1*
*), takes CN~(X)(E1)
into CN~(X)(E0) in such a way that the diagram
1)x1
CN~(X)(E0) x E0c(w~oo___ CN~(X)(E1) x E0
e 1xc(w~)
fflffl fflffl
~N(X)oo_______e______CN~(X)(E1) x E1
6 J.M. MØLLER
where e is group multiplication, commutes up to inner automorphism of ~N(X) (as*
* namely c(w~) O
e O (c(w~1) x 1) = e O (1 x c(w~))). Therefore, the diagram of adjoint maps be*
*tween spaces
1)
BCN~(X)(E0)oo_____Bc(w~________BCN~(X)(E1)
' '
fflffl fflffl
map (BE0, BN~(X))B~1oo________ map(BE1, BN~(X))B~2
Bc(w~)
is homotopy commutative. (The vertical maps are equivalences by [40, Lemma 2].)*
* This shows
that the map CX (E1)! CX (E0)induced by the A(X)morphism E0 ! E1 represented b*
*y w lifts __
to the map c(w~1): CN~(X)(E1) ! CN~(X)(E0)between maximal torus normalizers. *
* __
2.11. Corollary.Let p be an odd prime and X a connected pcompact group. Then
(
limi(A(X) t, ßj(BZCX ))= ßj(BZ(X)) i = 0
0 i > 0.
for j = 1, 2. In particular, lim*(A(X) t, ß*(BZCX )) = 0 if and only if X is ce*
*nterless.
Proof.By (2.4, 2.8, 3.12.(2)),
lim0(A(X) t, ßj(BZCX ))= lim0(A(W (X), t(X)), L(X)2j)= H2j(W (X); L(X))
= ßj(BZ(X))
and, similarly, limi(A(X) t, ßj(BZCX )) = 0 for i > 0. *
* ___
Let ßj(BZCX ) t be the subfunctor of ßj(BZCX ) which vanishes on all toral o*
*bjects of A(X)
and has the same value as ßj(BZCX ) on all nontoral objects of A(X). (To see t*
*hat this is indeed
a functor, observe that there can be no morphism from a nontoral object to a t*
*oral object of the
Quillen category.)
2.12. Corollary.Let p be an odd prime and X a connected pcompact group. Then t*
*here is an
exact sequence
0 ! lim0(A(X) t; ßj(BZCX ) t)! lim0(A(X); ßj(BZCX ))! ßj(BZ(X))
! lim1(A(X) t; ßj(BZCX ) t)! lim1(A(X); ßj(BZCX ))! 0
while limi(A(X); ßj(BZCX )) = limi(A(X) t; ßj(BZCX ) t) for i 2. In particula*
*r,
lim*(A(X); ßj(BZCX ) t)~=lim*(A(X) t; ßj(BZCX )~t)=lim*(A(X); ßj(BZCX ))
if and only if X is centerless.
Proof.The quotient functor ß*(BZCX )=ß*(BZCX ) t vanishes on all nontoral obje*
*cts so that,
by (13.12), for all i 0,
limi(A(X); ßj(BZCX )=ßj(BZCX )=t)limi(A(X) t; ßj(BZCX )=ßj(BZCX ) t)
= limi(A(X) t; ßj(BZCX ))
which was computed in (2.11). Combine this with the fact that restriction
lim*(A(X) t; ßj(BZCX ) t) lim*(A(X); ßj(BZCX ) t)
is an isomorphism by (13.12) again. *
* ___
Let St(E) denote the Steinberg representation for GL(E).
2.13. Corollary.Let p be an odd prime, X a connected pcompact group with trivi*
*al center, and
let j be equal to 1 or 2. If Hom A(X)(E,()St(GL (E)), ßj(BZCX (E, ))) = 0 for *
*all nontoral objects
(E, ) of rank j + 1 and j + 2, then
limj(A(X); ßj(BZCX )) = 0 = limj+1(A(X); ßj(BZCX ))
NDETERMINISM 7
Proof.Use (2.12) and Oliver's cochain complex [81] for computing higher limits *
*over A(X). ___
For example, when (X, p) is (F4, 3) or (E8, 5) we have lim*(A(X); ßj(BZCX )) *
*= 0 because the
Quillen category A(X) contains, up to isomorphism, a unique nontoral object (V*
*, ); this V has
order p3, V ~= CX (V, ), and A(X)(V, ) = SL(V ) [41, 7.4, 10.3]. The situati*
*on is much more
complicated for the other members of the Efamily at p = 3 [3].
2.14. Relation between A(W, t) and the orbit category O(W ). Let O0(W ) denote *
*the full
subcategory of the orbit category of W generated by all objects W=G with tG 6= *
*0. There are
obvious functors
__L_//_
A(W, t)oo___O0(W )op
R
given by
` ' ` '
L E0 wW(E0)!E1= W=W (E0) wW(E0)W=W (E1), w(E0) E1,
i wH j ` wW(tH) '
R W=G ! W=H = tG  tH , w1Gw H.
Using that G W (tG) and E tW(E) we see that L and R are adjoint functors in*
* that
A(W, t)(E, R(W=G)) = O0(W )op(L(E), W=G)
for all objects E of A(W, t) and all objects W=G of O0(W )op. Observe also that
__ G __
(2.15) NW (G) W (t ) and W (E) NW (W (E))
for all nontrivial subspaces E t and all subgroups G W . In particular, th*
*e endomorphism
monoid of E is the quotient
__
A(W, t)(E) = W (E)=W (E)
__
of the group W (E) by its normal subgroup W (E). Thus A(W, t) is an EIcategory*
* [54], a category
in which all endomorphisms are isomorphisms.
A collection is a set C of subgroups of W which is closed under conjugation. *
*Let OC(W ) denote
the Corbit category, the full subcategory of O(W ) generated by all objects W=*
*G with G 2 C, and
AC(W, t) the full subcategory of A(W, t) generated by all objects of the form t*
*G for G 2 C.
The collection C is said to be subgroupsharp for the ZpW module L [27, 1.13*
*] if
( j
limi(OC(W )op; Lj)= H (W ; L) i = 0
0 i > 0
where Lj(W=G) = Hj(G; L) as in (2.14).
2.16. Corollary.If the collection C is subgroupsharp for L and tG 6= 0 for all*
* g in C then Lj
restricts to an acyclic functor on AC(W, t) with lim0(AC(W, t); Lj) = Hj(W ; L).
Proof.This is immediate from (13.11) as AC(W, t) = ROC(W ). *
* ___
It is known [45, x5] that the collection C(p) of all psubgroups of W is subgro*
*upsharp for any
ZpW module L and, for general reasons, tP 6= 0 for any pgroup P 2 C(p).
2.17. Centralizers. I close this section with a simplified proof of the followi*
*ng wellknown result
from [74, 3.9] which was used in connection with the mapping spaces of (2.6).
Let P be a ptoral Lie group (i.e. the identity component of P is a torus and*
* ß0(P ) is a finite
pgroup), G a compact Lie group having a finite pgroup as its component group,*
* and CG(P ) the
Lie group centralizer, which also has a finite pgroup as component group [47, *
*A4], of a Lie group
homomorphism f :P ! G. The standard Lie group multiplication homomorphism CG(P *
*)xP ! G
extending f induces a pcompact group morphism C"G(P )x ^P! G^extending ^f:^P! *
*^G. (G^
denotes the pcompact group BG^pobtained by pcompleting the classifying space *
*of the compact
Lie group G.) We shall now see that "CG(P )= CG^(P^) and in particular that [Z(*
*P )= Z(P^), i.e.
that centralizers and centers of ptoral Lie groups can be computed either in t*
*he Lie group category
or in the pcompact group category.
8 J.M. MØLLER
2.18. Lemma. [35, 101, 73] The adjoint
BC"G(P )! map(BP^, BG^)Bf^
of the above standard morphism is a homotopy equivalence. In particular,
BZ[(P )' map(BP^, BP^)B1
where Z(P ) is the Lie group center of P .
*
* S
Proof.The ptoral Lie group P contains [48, 1.1] a dense pdiscrete toral subgr*
*oup ~P= ~Pmwhich
is the union of an ascending sequence of finite pgroups ~Pm. The inclusion of *
*~Pinto P induces
a discrete approximation i: ~P! ^Pto the pcompact toral group ^Pand so we have*
* homotopy
equivalences [30, x6]
map(BP^, BG^)Bf^' map(BP~, BG^)Bf^i' map(BP~m, BG^)B(f^iP~m)
for m large enough. In particular, the above mapping spaces are Fpcomplete [31*
*, 2.5] [30, 6.20].
Furthermore, by DwyerZabrodsky [35, 1.1] and [36, 2.5] or Lannes [52], the can*
*onical map
BCG(P~m) ! map(BP~m, BG^)B(f^iP~m)
is an H*Fpequivalence and here
CG(P~m) ~=CG(P~) ~=CG(P )
when m is large enough and since ~Pis dense in P . *
* ___
Let now G be an extended pcompact torus and ~Gits discrete approximation [31*
*, 3.12].
2.19. Lemma. Let ~: ß ! ~Gbe a homomorphism from a discrete group ß into the ex*
*tended p
discrete torus ~G.
1.The group theoretic centralizer CG~(~) of ~ is a discrete approximation to *
*the extended p
compact torus BCG(~) = map(Bß, BG)B~.
2.The group theoretic center Z(G~) of ~Gis a discrete approximation to the ex*
*tended pcompact
torus BZ(G) = map(BG, BG)B1
Proof.The maps
BCG~(~) ! map(Bß, BG~)B~ ! map(Bß, BG)B~
are H*Fpequivalences: The first map is even a homotopy equivalence [40, Lemma *
*2] and the
fibre of the second map is [62] a K(V, 1), for some rational vector space V , b*
*ecause the fibre of
BG~! BG has this form [31, 3.1]. Taking ~ to be the identity map of ~G, we obta*
*in_a discrete
approximation to Z(G). *
*__
3.Ndeterminism
This section contains comments on and further development of the material in *
*[68] concerning
Ndetermined pcompact groups.
3.1. Ndetermined automorphisms. Let j :N(X) ! X be the maximal torus normalize*
*r for
a pcompact group X. Turn this maximal torus normalizer Bj :BN(X) ! BX into *
*a fibra
tion. Any automorphism f :X ! X of the pcompact group X restricts to an autom*
*orphism
AM (f): N(X) ! N(X) of the maximal torus normalizer, unique up to the action of*
* the Weyl
group W (X0) = ß1(X=N(X)) of the identity component X0 of X, such that the diag*
*ram
B(AM (f))
BN(X) _________//_BN(X)
Bj Bj
fflffl fflffl
BX ______Bf_____//BX
commutes up to based homotopy [68, x3] [1] [101, Theorem C]. The AdamsMahmud h*
*omomor
phism is the resulting homomorphism
(3.2) AM :Aut (X) ! Aut(N(X))=W (X0)
NDETERMINISM 9
of automorphism groups, and X is said to have Ndetermined automorphisms if thi*
*s homomor
phism is injective [68, 3.10].
The following lemma, collecting results from [68, 4.2, 4.3, 4.8] and (9.4), r*
*educes the problem
of determining which pcompact groups have Ndetermined automorphisms to the co*
*nnected and
centerless case. (The simple factors of the pcompact group X are the simple, c*
*enterless pcompact
groups in the splitting [32, 80] of P X0 = X0=Z(X0), the adjoint form of the id*
*entity component
of X.)
3.3. Lemma. Let p be any prime number.
1.The connected pcompact group X has Ndetermined automorphisms if its adjoi*
*nt form P X
does.
2.The pcompact group X has Ndetermined automorphisms if its identity compon*
*ent X0 does.
3.The pcompact group X has Ndetermined automorphisms if all of its simple f*
*actors do.
In the connected, centerless case we use an inductive procedure based on homo*
*logy decomposi
tion [31, 8.1] and preferred lifts [69].
3.4. Proposition.[68, 4.9] Suppose that the pcompact group X is connected and *
*centerless. If
1.CX (L, ) has Ndetermined automorphisms for each rank 1 object (L, ) of A*
*(X).
2.lim1(A(X) ; ß1(BZCX )) = 0 = lim2(A(X) ; ß2(BZCX )).
Then X has Ndetermined automorphisms.
Proof.Let f :X ! X be an automorphism of X such that AM (f): N ! N is conjugate*
* to the
identity map of N. Then (E, f ) = (E, ) for each object (E, ) of A(X) for if *
*~: E ! N is a lift
of :E ! X we have f = fj~ = j O AM (f) O ~ = j O ~ = . Thus composition wit*
*h f determines
an automorphism Cf: CX (E, ) ! CX (E,of)each centralizer in the homology decom*
*position
hocolimBCX ! BX [31, x8]. In particular, when (L, ) is a rank 1 object with p*
*referred lift
~: L ! T ! N [69, 4.10], we obtain a commutative diagram
CN (L,M~)
Cjqqqq MMMCj0M
qqq MMMM
xxqqq ~= M&&
CX (L, )_______Cf_______//_CX (L, )
which implies, using the first assumption, that Cf is conjugate to the identity*
* [68, 3.9]. But then
Cf is conjugate to the identity for all (E, ) 2 Ob(A(X)). To see this, choose *
*any line L < E and
let __:E ! CX (L, L)be the canonical factorization (3.18) of though the cen*
*tralizer of L. Then
note that under the isomorphism CCX(L, L)(E, __(L)) ~=CX (E, ) the isomorphis*
*m Cf induced by
f on X corresponds (3.20) to the isomorphism CCf induced by Cf on CX (L, L).
The second assumption of the lemma assures that there are no further obstruct*
*ions to conju_
gating f to the identity [100] [68, 4.9]. *
* __
3.5. Ndetermined pcompact groups. Let j :N ! X be the maximal torus normalize*
*r for
the pcompact group X. Suppose that N may also serve as the maximal torus norma*
*lizer for some
other pcompact group X0 so that we have two monomorphisms
0
(3.6) X ooj_N _j__//X0
that are both maximal torus normalizers. The pcompact group X is Ndetermined*
* if, in this
situation, there always exists an isomorphism f :X ! X0 under N, i.e. a morphis*
*m f :X ! X0
such that fj and j0 are conjugate. A totally Ndetermined pcompact group is an*
* Ndetermined
pcompact group with Ndetermined automorphisms [68, 7.1].
The following lemma, collecting results from [68, 7.8, 7.10] and (9.6), reduc*
*es the problem of
determining which pcompact groups are Ndetermined to the connected and center*
*less case.
3.7. Lemma. Let p be an odd prime.
1.The connected pcompact group X is Ndetermined if its adjoint form P X is.
2.The pcompact group X is Ndetermined if its identity component X0 is.
3.The pcompact group X is Ndetermined if all of its simple factors are.
10 J.M. MØLLER
Again, in the connected, centerless case we use an inductive procedure.
3.8. Proposition.(Cf. [68, 7.17]) In the situation of ( 3.6), suppose that X is*
* connected and
centerless and that
1.All objects of A(X) of rank 2 have totally Ndetermined centralizers.
2.For each nontoral rank 2 object (V, ) of A(X) there exist a rank 2 object*
* (V, 0) of A(X0)
and an isomorphism f(V, ): CX (V, ) ! CX0(V,su0)ch that j0~ = 0 and
CN (V, ~)M
Cj rrrr MMMCj0M
rrr MMMM
xxrrr ~= &&M
CX (V, )_____f(V,_)_____//CX0(V, 0)
commutes for any of the p + 1 [68, 6.2] special preferred lifts (V, ~) of (*
*V, ).
3.lim2(A(X) ; ß1(BZCX )) = 0 = lim3(A(X) ; ß2(BZCX )).
Then there exists an isomorphism f :X ! X0 under N.
Proof.For each rank 1 object or toral rank 2 object (V, ) of A(X), put 0= j0~*
* where ~: V ! N
is the preferred lift [69, 4.10], and define f(V, ): CX (V, ) ! CX0(V,to0)be *
*the unique isomor
phism under CN (V, ~). Then 0equals the composition
_ f(V, ) res
V ______//CX (V,__)___//CX0(V, _0)___//X0
__
and f(V, )__is the canonical factorization (3.18) 0of 0.
Any nontoral rank 2 object (V, ) has p+1 special preferred lifts (V, ~) ind*
*exed by the set of lines
in V [68, 6.2]. By assumption, neither j0~ nor the isomorphism f(V, ~): CX (V, *
* ) ! CX0(V, j0~)
under CN (V, ~) depend on the choice of ~. Put 0= j0~ and f(V, ) = f(V, ~) wh*
*ere ~ is any of
the p + 1 preferred lifts of .
These morphisms f(V, ) for V  p2 respect morphisms in A(X): Consider for*
* instance a
morphism ff: (V1, 1) ! (V2,fr2)om a rank 1 object to a rank 2 object. Let ~2:*
*V2 ! N be
the special preferred lift of 2 for which ~1 = ~2ff is the preferred lift of *
*1 = 2ff. Since 01=
j0~1 = j0~2ff = 02ff, the group homomorphism ff is an A(X0)morphism (V1, 01)*
* ! (V, 02). Then
02= j0~2 = j0OresNO__~2= j0OresNOCN (ff)__~2= resXOCj0OCN (ff)__~2= resXOf(V1,*
* 1)OCX (ff)__2
as we see from the commutative diagram
CN(ff)_~2________CN0(V1,0~1)Q_______resN___________//_N______*
*______________________________________________________________________@
_____________Cjmmmmmm______QQQQCj0QQ__________________0______*
*_____________________________________________________________________
_______________mmm_________________QQQQQ_________________j______*
*__________
_______________vvmmmm_____________________((Q______________~fflffl=
V2 CX(ff)_2//_CX (V1,__1)____f(V1,_1)_______//_CX0(V1,__01)resX0//_X0
NDETERMINISM 11
__ __
and CX0(ff) 02= f(V1, 1) O CX (ff) 2 as we see from the argument of (10.13). T*
*aking centralizers
of V2 we obtain the commutative diagram
CN (V2, ~2)
rr OO MMMM
rrr ~= MMM
rrr  MMM
rrr  MMMM_
Cj rrrrCCN(V1,~1)(V2, CN (ff)~MMCj0M2)
rr rr LLL MMM
rrr rrr LLL MMM
rrr rrr LLL MMM
xxrrr rrrrr ~= LLLL M&&
CX (V2, 2)_____r__________________________LLL_____//_CX0(V2, 02)
OO rrr f(V2, 2) LLL OO
~  rrr LLL ~
=  rrr LLLL =
 yyrrr __ ~ &&  *
* __
CCX(V1, 1)(V2, CX (ff)_2)_______C_=_______________//_CC 0(V1, 0)(V2, CX0(f*
*f) 02)
f(V1, 1) X 1
 
 
fflffl ~= fflffl
CX (V1, 1)________________f(V1,_1)________________//_CX0(V1, 01)
which shows that the isomorphism f(V2, 2): CX (V2, 2) ! CX0(V2,un02)der CN (V*
*2, ~2) is in
duced from the isomorphism f(V1, 1): CX (V1, 1) ! CX0(V1,un01)der CN (V1, ~1)*
*. This implies
naturality as we may enlarge the commutative diagram by the morphisms CX (V2, *
*2) ! CX (V1, 1)
and CX0(V2, 02) ! CX0(V1, 01) induced by ff (3.20).
Also, if ff 2 A(X)(V, ) GL(V ) is a Quillen automorphism of the rank 2 obj*
*ect (V, ), and
~: V ! N a special preferred lift of , then ~ff is again a special preferred l*
*ift of and hence
0ff = j0~ff = j0~ = 0by assumption. Thus A(X)(V, ) A(X0)(V, 0) and as
CX(ff)1 f(V, ) CX(ff)
CX (V, )______//_CX (V,__)~=_//_CX0(V,__0)____//CX0(V, 0)
is an isomorphism under CN (V, ~ff), it equals f(V, ) by assumption. This is n*
*aturality for Quillen
automorphisms of (V, ).
Let now (E, ) be an object of A(X) of any rank > 2. Choose a line L < E. Def*
*ine 0:E ! X0
to be the composite monomorphism
_ res f(L, L) res
E ______//_CX (E,__)___//_CX (L, _L)~=_//CX0(L, ( L)0)___//X0
and define the isomorphism of centralizers f(E, ): CX (E, ) ! CX0(E,to0)be th*
*e isomorphism
Cres Cf(L, L) C__res
CX (E, o)o~=_CCX(L, L)(E, __(L))~=//_CCX0(L, L)(E, f(L, L)~(L))=//_CX0(E,*
* 0)
induced by f(L, L).
To see that this is welldefined, let L1 < E and L2 < E be two distinct rank *
*1 subgroups of
E and let V < E be the subgroup generated by them. Naturality for morphisms fro*
*m a rank 1
object to a rank 2 object gives a commutative diagram
f(L1, L1)
CX (L1, L1)______//_CX0(L1, ( L1)0)
_ nn77 OO OO QQ
(L1)nnnn   QQQQ
nnnnn   QQQQresQQQ
nnn _(V )  f(V, V )  0res Q((Q
E _________//PPPCX (V,__V_)___//CX0(V, ( V_)_)____//X066mm
PPP   mmmmm
_(L2)PPPPPP  mmmmm res
PP'' fflffl fflfflmmm
CX (L2, L2)f(L2,_L2)//_CX0(L2, ( L2)0)
showing that neither 0nor f(E, ) depend on the choice of the rank 1 subgroup *
*of E.
12 J.M. MØLLER
In order to show functoriality of this construction, let ff: (E1, 1) ! (E2,b*
*e2)a morphism in
the category A(X). Choose a rank one subgroup L1 < E1 and put ff(L1) = L2 < E2.*
* Naturality
for the rank 1 case gives a commutative diagram
f(L1, L1) 0
E1 ________//_CX (L1,OO1L1)____//CX0(L1,O(O1L1)_)___//_X0
ff CX(ff) CX0(ff) 
fflffl   
E2 ________//_CX (L2, 1L2)f(L2,/L2)/_CX0(L2,_(_1L2)0)//_X0
which shows that 01= 02ff, thus A(X)((E1, 1), (E2, 2)) A(X0)((E1, 01), (*
*E2, 02)), and implies
commutativity of the diagram
f(E1, 1)
CX (E1,OO1)__~=__//_CX0(E1,OO01)
CX(ff) CX(ff)
 ~= 
CX (E2, 2)f(E2,_2)//_CX0(E2, 02)
which is naturality.
We have now constructed a collection CX (E, ) f(E,)!CX0(E, 0) res!X0,*
* (E, ) 2 Ob(A(X)),
of homotopy A(X)invariant centric [28] monomorphisms from the centralizers of *
*the homology
decomposition of BX [31, 8.1] to BX0. Because the obstruction groups are assume*
*d to vanish,
this collection can [100] [68, x2] be realized by a morphism
Bf :BX ' hocolimBCX ! BX0
such that
f(E, )
CX (E, _)______//CX0(E, 0)
 
 
fflffl fflffl
X ______f______//_X0
commutes for all (E, ) 2 Ob(A(X)). In particular, f is a morphism under the ma*
*ximal torus, for
f is a morphism under the maximal rank monomorphisms [31, x4] X CN (L, ~) ! X*
*0 for some
rank 1 object (L, ) of A(X). Thus f :X ! X0 is in fact an isomorphism [32, 5.6*
*] [69, 3.11] and
since f is the identity on the maximal torus T = N0, also ß0AM (f): W ! W is th*
*e identity map,
for W is faithfully represented as a group of operators on T [30, 9.7]. Thus ß**
*(BAM (f)) is the_
identity automorphism of ß*(BN) and AM (f) is the identity of N [66, 5.2] [5, 3*
*.3]. __
Verification of the third assumption reduces to a computation involving Stein*
*berg representa
tions (2.13). For the verification of the second condition we shall use the fol*
*lowing lemma which
may look rather specialized but in fact applies in all cases considered in this*
* paper.
3.9. Lemma. Let (V, ) be a nontoral rank 2 object of A(X) with special prefer*
*red lift ~: V ! N
and put 0= j0~. Assume that
1.All rank 2 objects of A(X), whose centralizers are isomorphic to CX (V, ),*
* are isomorphic
to (V, ).
2.A(X)(V, ) = SL(V ). (Then also A(X0)(V, 0) = SL(V ).)
3.The isomorphism f(V, ~): CX (V, ) ! CX0(V,un0)der CN (V, ~) is SL(V )opeq*
*uivariant.
Then j0~1 = 0and f(V, ~1) = f(V, ~): CX (V, ) ! CX0(V,fo0)r all special prefe*
*rred lifts (V, ~1)
of (V, ).
Proof.The GL (V )orbit (V, ) . GL(V ) contains p  1 objects, the GL (V )orb*
*it (V, ~) . GL(V )
contains (p  1)(p + 1) objects, and the map j :(V, ~) . GL(V ) ! (V, ) .iGL(V*
*s)(p + 1)to1 [68,
6.2]. By assumption, the orbit (V, ~) . GL(V ) contains all special preferred l*
*ifts whose centralizers
in N are isomorphic to N(CX (V, )). Since X and X0 have the same special prefe*
*rred lifts [68,
7.13], also j0:(V, ~) . GL(V ) ! (V, j0~) .iGL(Vs)(p + 1)to1. Since the orbit*
* (V, j0~) . GL(V )
NDETERMINISM 13
thus contains p  1 objects, the stabilizer subgroup of (V, j0~) must be SL(V )*
* as this in the only
subgroup of GL(V ) of that index. Thus the Quillen automorphism group A(X0)(V, *
* 0) = SL(V ).
Any other special preferred lift of has the form ~ff for an ff in SL(V ) [6*
*8, 6.2], so, clearly,
j0(~ff) = 0ff = 0is independent of the choice of ff. The commutative diagram
CN(ff) CN(ff)
CN (V, ~ff)oo_____________CN_(V, ~)_________________//_CN (V, ~ff)
qq MMMCj0MM 
Cj Cjqqqqq MMMM Cj0
fflffl xxqqq MM&& fflffl
CX (V, )CX(ffCX)(V,oo)_____f(V,~)_____//CX0(V, C0)_//_CX0(V, 0)
X0(ff)
shows that f(V, ~ff) = CX0(ff)f(V, ~)CX (ff)1, since CX (V, ) has Ndetermine*
*d automorphisms_
so that f(V, ~ff) = f(V, ~) by the third assumption. *
* __
The canonical factorizations of and 0 are SL(V )opequivariant (3.19) and *
*they provide a
commutative diagram
(3.10) V II
_ vvvv II__I0
vv IIII
vvv I$$
CX (V, _)__f(V,~)__//CX0(V, 0)
which shows that the restriction of f(V, ~) to V is SL(V )opequivariant. It i*
*s a tautology that
f(V, ~ff) = f(V, ~) for all ff in the Borel subgroup stabilizing ~ so it is in *
*fact only necessary to
check equivariance with respect to one other element (of order p) [91, 3.6.21] *
*of SL(V ).
3.11. Centers and automorphism groups of pcompact groups. The following theorem
collects some useful facts from various sources that will be applied several ti*
*mes in this paper.
3.12. Theorem. Let p be an odd prime and X a connected pcompact group.
1.[4, 5] The semidirect product ~N(X) = ~T(X) o W (X) is a discrete approxim*
*ation [31, 3.12]
to the maximal torus normalizer N(X).
2.[31, x7] The abelian group ~Z(X) given by
0 0 1
H0(W (X); ~T(X)) = H (W (X); L(X)) Q =H (W (X); L(X)) x H (W (X); L(X*
*))
is a discrete approximation to the center [58, 31] of X.
AM
3.[68, 7.2] Aut(X) ~= Out(N(X)) provided X is totally Ndetermined.
The automorphism group of N(X) sits [66, 5.2] in a short exact sequence
(3.13) 0 ! H1(W (X); ~T(X)) ! Aut(N(X)) !Aut(W (X), ~T(X), e(X)) ! 1
where the normal subgroup to the left consists of all automorphisms of N(X) tha*
*t induce the
identity on homotopy groups, and the group to the right consists of all pairs (*
*ff, `) 2 Aut(W (X))x
Aut(T~(X)) such that ` is fflinear and the induced automorphism H2(ff1, `) [9*
*5, 6.7.6] preserves
the extension class e(X) 2 H2(W (X); ~T(X)). The image of W (X0) W (X) = ß0N*
*(X) [58,
3.8] in Aut(N(X)) does not intersect the subgroup H1(W (X); ~T(X)) (as W (X0) i*
*s represented
faithfully in Aut(T~(X)) [30, 9.7]) so there is an induced short exact sequence
(3.14)0 ! H1(W (X); ~T(X)) ! Aut(N(X))=W (X0) !Aut(W (X), ~T(X), e(X))=W (X0) *
*! 1
whose middle term is the target of the AdamsMahmud homomorphism (3.2).
If X is connected and p is odd, the cohomology group to the left is trivial a*
*nd e(X) = 0 [5] so
Aut(N(X)) ~=Aut(W (X), ~T(X), 0) ~=NGL(L(X))(W (X))
is [68, 3.5] [5, 3.3] the group of selfsimilarities of the Zpreflection group*
* (W (X), L(X)) (4.1), and
the target of the AdamsMahmud homomorphism (3.2)
(3.15) Out(N(X)) = Aut(N(X))=W (X) ~=NGL(L(X))(W (X))=W (X)
14 J.M. MØLLER
is [64, x2] the middle term of an exact sequence
(3.16) 1 ! AutZp[W(X)](L(X))=Z(W (X)) ! Out(N(X)) ! Out(W (X))
of automorphism groups. An automorphism of X is exotic if its lift to N(X) [68,*
* 3.7] induces a
nontrivial outer automorphism of W (X).
3.17. Remark. Let p and X be as in (3.12).
1.The formula
ßj(BZ(X)) = H2j(W (X); L(X)), j = 1, 2,
is an alternative version of (3.12.(2)).
2.The endomorphism monoid of X is given by
(
End(X)  {0} ~= NGL(L(X))(W (X))=W (X) = Aut(X) p  W (X)
NGL(L(X) Q)(W (X)) \ End(L(X)) =W (X)p 6  W (X)
provided X is totally Ndetermined and simple [67, 5.4]; use [67, 5.6, 5.6]*
* and [66, 5.2] to see
this. See [46, 47, 48] for the Lie case.
3.18. Canonical factorizations. [30, 8.2] Let :V ! Xbe a monomorphism from an*
* elementary
abelian pgroup to the pcompact group X. The canonical factorization of thro*
*ugh its centralizer
is the central monomorphism___(V ): V ! CX (V,wh)ose adjoint is V x V +!V !*
*X. The
composition V ! CX (V, ) res!X equals . If ff: (V1, 1) ! (V2,is2)a morphi*
*sm in A(X) then
the canonical factorizations are related by a commutative diagram
_1 res
(3.19) V1 _______//_CXO(V1,__1)___//_XO
ff CX(ff) 
fflffl  
V2 ____2__//_CX (V2,__2)res//_X
so that ff: (V1, __1) ! (V2, CXi(ff)__2)s a morphism in A(CX (V1, 1)). The in*
*duced morphisms
CX (ff) and CCX(V1, 1)(ff) can be identified in that the diagram
(3.20) CCX(V1, 1)(V1, __1)Cres~=//_CX4(V1,4 1)
OO jjj OO
 resjjjjjjj 
CCX(V1, 1)(ff)jjjjj CX(ff)
 j C 
CCX(V1, 1)(V2, CX (ff)__2)res~=//_CX (V2, 2)
commutes up to conjugacy.
4. Cohomologically unique pcompact groups
We shall here discuss to what extent Ndetermined pcompact groups are determ*
*ined by their
Weyl groups or their mod p cohomology algebras (4.3). The message intended is t*
*hat cohomolog
ical uniqueness [36, 74, 16, 93] is incidental while Ndeterminism is universal*
*. As the Weyl group
of a connected pcompact group is a reflection subgroup of the automorphism gro*
*up of the lattice
we start out by introducing the category of reflection subgroups.
For a commutative domain R, an element g of GL (r, R) is a reflection if the *
*(r x r) matrix
Ir g has rank at most 1 where Ir is the (r x r) identity matrix. A subgroup W *
*of GL(r, R) is a
reflection subgroup if it is generated by the reflections contained in it.
4.1. Definition.For R = Zp, Qp, Fp, let R  Reflbe the category with
objects:pairs (W, L) where L is a finitely generated free Rmodule and W a fi*
*nite reflection
subgroup of AutR(L), and,
morphisms: pairs (ff, `): (W1, L1) ! (W2,wL2)here ff: W1 ! W2is a group homo*
*morphism
and ` 2 Hom R(L1, L2) an fflinear Rmodule homomorphism.
NDETERMINISM 15
A similarity is an isomorphism in RRefl. Two objects of ZpReflare Rsimilar i*
*f they are taken
to isomorphic objects of R  Reflby the functor rR :Zp Refl! R  Reflinduced b*
*y  ZpR.
G0(W, L) (Gp(W, L)) is the set of similarity classes of objects of Zp  Refltha*
*t are Qpsimilar
(Fpsimilar) to the object (W, L). An object (W, L) of Zp Reflis said to be si*
*mple if L Zp Qp
is a simple QpW module.
A similarity class of objects of R  Reflamounts to an integer r 0 and a co*
*njugacy class (W )
of a reflection subgroup of GL(r, R). The automorphism group AutRRefl(W, L) of*
* an Rreflection
subgroup is isomorphic to the normalizer NGL(L R)(W ) of W in GL(L R) [64, x2*
*] [63].
In Zp  Reflwe shall often write r0 (rp) for the functor rR if R = Qp (R = Fp*
*). (Of course,
if R = Zp, then rR is the identity functor.) By [29, Proof of 5.2], W is a refl*
*ection subgroup of
GL(L) if and only if rpW is a reflection subgroup of GL(L Z=p); also, W and rpW*
* are abstractly
isomorphic groups as the kernel of GL (r, Zp) ! GL (r, Fp) contains no nontriv*
*ial finite order
elements when p is odd [57] [88, 10.7.1]. Two objects, (W1, L1) and (W2, L2), o*
*f Zp  Reflare
Qpsimilar iff there exists a morphism (ff, `): (W1, L1) ! (W2,iL2)n Zp Reflsu*
*ch that r0(ff, `)
is an isomorphism in Qp  Refl, and they are Fpsimilar iff there exists a grou*
*p isomorphism
ff: W1 ! W2and a Zplinear isomorphism ` :L1 ! L2such that (ff, ` ZpFp) is an *
*isomorphism
in Fp Refl. All elements of G0(W, L), which is a finite set according to the J*
*ordanZassenhauss
Theorem [24, 24.2], are represented by centerings of (W, L), i.e. by objects of*
* the form (W, M)
where M is ZpW submodule of L and the index [L : M] is finite. Two centerings,*
* (W, M1) and
(W, M2), are similar if and only if A(M1) = M2 for some A in the normalizer NGL*
*(L Q)(W ) of W
in GL(L Q) [84, 2.12.3].
4.2. Proposition.Let (W, L) be an object of Zp Refl.
1.G0(W, L) = {(W 0) < GL(L)  (r0W 0) = (r0W )}
2.Gp(W, L) = {(W 0) < GL(L)  rpW 0= rpW }
As usual, (W ) stands for the conjugacy class of the subgroup W .
Proof.1. Let A(W ) = {U 2 GL (L Q)  U1W U GL (L)} be the set of automorph*
*isms of
L Q that conjugate the subgroup W GL (L) into (another) subgroup of GL (L).*
* We shall
define surjections
{(W 0) < GL(L)  (r0W 0) = (r0W )} j A(W ) i G0(W, L)
and show that the corresponding equivalence relations on A(W ) are the same. Th*
*e surjection to
the left simply takes U 2 A(W ) to the subgroup conjugacy class (U1W U). The m*
*ap to the right
takes U 2 A(W ) to the similarity class of (W, UL). This is indeed a welldefin*
*ed surjection because
for U 2 GL(L Q) we have
1
U1W U GL(L) , U W U (L) = L , W (UL) = UL
meaning that UL is a ZpW submodule of L Q if and only if U 2 A(W ). The Zp *
*Reflobjects
(W, UL) and (W, V L), U, V 2 A(W ), are similar if and only if V AU1W = W V AU*
*1 for some
isomorphism of the form
1 A V
UL U!~=L !~=L !~=V L
for an A 2 GL(L). In other words, (W, UL) and (W, V L) are similar if and only *
*if U1W U and
V 1W V are conjugate as subgroups of GL(L).
2. The map
{(W 0) < GL(L)  rpW 0= rpW } ! Gp(W, L)
taking (W 0) to (W 0, L) is clearly welldefined and injective. To see that it *
*is also surjective , let
(W1, L1) be an object of Zp  Reflthat admits a similarity (ffp, `p): rp(W1, L1*
*) ! rp(W,iL)n
Fp  Refl. Lift the isomorphism `p to a Zplinear isomorphism ` :L1 ! L. Then (*
*W1, L1) and
(W 0, L), W 0= `W1`1, are similar and rpW 0= `prp(W1)`1p= rpW and thus the su*
*bgroup W 0_
is mapped to the element of Gp(W, L) represented by (W1, L1). *
* __
The Weyl group W (X) of a connected pcompact group X is by birth a finite re*
*flection subgroup
of GL(L(X)) [30, 9.7] and (W (X), L(X)), (r0W (X), L(X) Qp), and (rpW (X), L(*
*X) Fp) are
16 J.M. MØLLER
objects of R  Reflfor R = Zp, Qp, Fp, called the ZpWeyl group (or just the We*
*yl group), the
QpWeyl group, and the FpWeyl group of X, respectively. (As to functoriality w*
*e note that any
toric morphism [70] between connected pcompact groups determines a morphism be*
*tween the
corresponding reflection subgroups.)
4.3. Definition.Let X be a connected pcompact group.
1.X is determined by its RWeyl group if any connected pcompact group Y with*
* the same
RWeyl group as X, i.e. with W (Y ) Rsimilar to W (X), is isomorphic to X.
2.X is a cohomologically unique pcompact group if any connected pcompact gr*
*oup Y with
H*(BY ; Fp) isomorphic to H*(BX; Fp) as an algebra over the mod p Steenrod *
*algebra, is
isomorphic to X.
All pcompact tori are clearly cohomologically unique.
4.4. Corollary.Let p be an odd prime and X an Ndetermined connected pcompact *
*group.
1.X is determined by its ZpWeyl group W (X).
2.If G0(W (X), L(X)) = *, then X is determined by its QpWeyl group r0W (X).
3.If Gp(W (X), L(X)) = *, then X is determined by its FpWeyl group rpW (X)
4.If X is determined by its FpWeyl group, then X is a cohomologically unique*
* pcompact
group.
Proof.At odd primes, the (discrete) maximal torus normalizer of a connected pc*
*ompact group,
which is a split extension (3.12), is determined, up to isomorphism, by the sim*
*ilarity class of the
Weyl group. The next two items are immediate consequences of this, since we are*
* assuming W (X)
recoverable from r0W (X), respectively rpW (X).
The rational rank r(X) as well as the FpWeyl group rpW (X) can be read off f*
*rom H*(BX, Fp)
thanks to Lannes theory [52]. Indeed, r(X) is the maximal r 0 for which there*
* exists a monomor
phism (Z=p)r æ X whose centralizer is a pcompact torus and rpW (X) is (2.8) th*
*e automorphism_
group in the Quillen category of the object t(X) ! X. *
* __
4.5. Lemma. Let W be a finite reflection subgroup of GL(L). Put t = L=pL.
1.[84, (1) p. 248] If t is an irreducible Fp[W ]module, then G0(W, L) = *.
2.[7, 7.1.2] If H1(rpW ; Hom(t, t)) = 0, then Gp(W, L) = *.
Proof.For item 1, let M be a ZpW submodule of L not contained in pL. Since the*
* image of M in
t = L=pL is nontrivial, we get L = M + pL by irreducibility and L = M by Nakay*
*ama's lemma __
[86, 9.2]. The H1condition of item 2 assures that rpW lifts uniquely to GL(r, *
*Zp). __
The sets G0(W, L) and Gp(W, L) are determined in (11.18, 11.25, 11.26) for (W*
*, L) a simple
reflection subgroup and p an odd prime.
For a connected pcompact group X, let SX stand for the universal covering gr*
*oup of X and
P X = X=Z(X) the adjoint form of X [31, 58]. Recall that for (W, L) 2 Ob(Zp  R*
*efl) there are
associated objects (SW, SL), (P W, P L) 2 Ob(Zp Refl) [75] (11.1).
4.6. Lemma. SSX = SX = SP X and P P X = P X = P SX for any connected pcompact *
*group
X.
Proof.Use [58, 4.7, 5.4, 5.5] and that BSX = BX<2>is the 2connected cover of B*
*X. ___
4.7. Proposition.Let p be an odd prime and X an Ndetermined connected pcompac*
*t group.
1.H0(W (X); ~T(X)) = ~Z(X) and H0(W (X); L(X)) = ß1(X).
2.SL(X) = L(SX) and P L(X) = L(P X).
Proof.The formula for the center of X (3.12.(2)) immediately shows that P L(X) *
*= L(P X). By
inspection we see that
(4.8) H0(W (P X); L(P X)) = ß2(BP X)
for any simple pcompact group X. (The formula is known to hold in the Lie case*
* by classical
results. The exotic simple pcompact groups are all centerless and polynomial (*
*7.9) so in this case
NDETERMINISM 17
X = P X and ß2(BX) = 0 because H2(BX; Fp) = 0. Also H0(W (X); L(X)) = 0 by (11.*
*4.3) for
G0(W (X)) = * (11.18) so that L(X) = SL(X).) Therefore,
SL(P X) = kerL(P X) ! H0(W (P X); L(P X)) = kerL(P X) ! ß1(P X)
= L(SP X) = L(SX)
for any simple X. For a generalQconnected X, the Splitting Theorem for centerl*
*ess pcompact
groups [32] tells us that P X = P Xiwhere Xiis simple. Consequently,
Y Y Y
SL(X) = SP L(X) = SL(P X) = SL(P Xi) = L(SXi) = L(SP Xi)
= L(SP X) = L(SX).
From the finite covering ß ! SX x Z(X)0 ! X of [58, 5.4] we obtain a short exac*
*t sequence of
ZpW (X)modules
(4.9) 0 ! SL(X) x H0(W (X); L(X)) ! L(X) ! ß ! 0
and, using H1(W (X); ß) = 0 = H0(W (X); SL(X)) (11.3, 11.4.3), a short exact se*
*quence of Zp
modules
0 ! H0(W (X); L(X)) ! H0(W (X); L(X)) ! ß ! 0
identical to the short exact sequence for computing ß1(X). *
* ___
Recall, that we write X1 X2 if there exists an isogeny X1 i X2 [67, p. 216]*
* in pcg and
(W1, L1) (W2, L2) if there exists an isogeny (W1, L1) ! (W2, L2) in Zp Refl(*
*11.23).
4.10. Corollary.Let p be an odd prime and X1 and X2 two connected pcompact gro*
*ups. Assume
that P X2 is totally Ndetermined.
1.X1 and X2 are locally isomorphic, (W, L)(X1) and (W, L)(X2) are Qpsimilar.
2.X1 X2 , (W, L)(X1) (W, L)(X2).
3.The local isomorphism system [67, 4.7] of X2 is poset isomorphic to G0(W (X*
*2), L(X2)).
Proof.Write (Wi, Li) for (W (Xi), L(Xi)), i = 1, 2. It is clear from the result*
*s of [67, x2x4] that
if X1 and X2 are locally isomorphic (and X1 X2) then (W1, L1) and (W2, L2) ar*
*e Qpsimilar
(and (W1, L1) (W2, L2)). Conversely, suppose that (W1, L1) and (W2, L2) are Q*
*psimilar. Then
(W1, L1) ~=(W2, P~ff(SL2)) for some diagram ~ff:~ß(SL2) oe ß(L1) '!~TH0(W2; L2*
*) of Zpmodules
(11.20). Since SL2 = L(SX2), this means that (W1, L1) is similar to (W (X02)L(*
*X02)) for the
pcompact group
X02= SX2x_Z(X2)0_(ß(L
1), ')
locally isomorphic to X2[67, 2.8]. But X02is totally Ndetermined if P X2is (3.*
*3, 3.7), and therefore
X1 is actually isomorphic to X02(4.4). Moreover, if (W2, P~ff(SL2)) (W2, L2) *
*then (11.21) there
is a commutative diagram
~ß(SL2)oo_o~ß(L1)o'_//_~TH0(W2; L2)
fflffl
~=  
fflffl fflffl fflfflfflffl
~ß(SL2)oo_o~ß(L2)o__//_~TH0(W2; L2)
induced by some automorphism of SL2 and some epimorphism of ~TH0(W2; L2) = ~Z(X*
*2)0 onto
itself with finite kernel. But any automorphism of SL2 = L(SX2) comes from an a*
*utomorphism __
of SX2 (3.12.(3)) and so the above diagram determines [67, 4.3, 4.5] an isogeny*
* X1 i X2. __
4.11. Corollary.Let p be an odd prime. There are fibration sequences
Bß(L(X)) ! BSX x B2H0(W (X); L(X)) ! BX
BX ! B2LH0(W (X); ~T(X)) x BP X ! B2~ß(L(X))
for any Ndetermined connected pcompact group X.
18 J.M. MØLLER
Proof.Write (W, L) for the reflection subgroup (W (X), L(X)) associated to X. T*
*he first of these
fibration sequences will follow if we can show that
(4.12) Bß(L) _______//BSX
 
 
fflffl fflffl
B2H0(W ; L)____//BX
is homotopy fibre square. The top horizontal map corresponds to the monomorphi*
*sm ß(L) æ
~ß(SL) = H0(W ; ~T(SL)) = ~Z(SX) of (11.8.2) and the bottom one corresponds to *
*the monomor
phism ~TH0(W ; L) æ H0(W ; ~T(L)) = ~Z(X) of (11.4.1). There is a fibration
BH0(W ; L) ! Bß(L) ! B2H0(W ; L)
induced from the short exact sequence 0 ! H0(W ; L) ! H0(W ; L) ! ß(L) ! 0 of a*
*belian groups.
But H0(W ; L) and ß1(X) are (4.7) isomorphic abelian groups and thus the left a*
*nd the right
vertical maps in (4.12) have identical fibres.
For the second fibration, it is enough to prove that there exists a homotopy *
*fibre square
(4.13) BX _____//_B2LH0(W ; ~T(L))

 
 
fflffl fflffl
BP X ________//_B2~ß(L)
with an abelian topological group in the lower right corner. The top horizontal*
* map corresponds to
the epimorphism H0(W ; L) i LH0(W ; ~T(L)) of (11.4.4) and the bottom one to th*
*e epimorphism
H0(W ; P L) = ß(P L) i ~ß(L) of (11.8.2). There is a fibration
BH0(W ; ~T(L))^p! B2LH0(W ; ~T(L)) ! B2~ß(L)
obtained by applying the Fibre Lemma [14, II.5.1] to the fibration BH0(W ; ~T(L*
*)) ! B2~ß(L) with
BH0(W ; ~T(L)) as fibre reflecting the defining short exact sequence for ~ß(L).*
* But H0(W ; ~T(L))
is (3.12.(2)) a discrete approximation to the center of X and thus the left and*
* the right_vertical
maps in (4.13) have identical fibres. *
* __
The Ndetermined connected pcompact group BX is, in other words, the quotien*
*t pcompact
group of BSX x B2H0(W (X); L(X)) corresponding to the subgroup ß(L(X)) (11.8.(4*
*)) of the
center ~ß(SL(X)) x ~TH0(W (X); L(X)) (4.7, 11.8.(1)), or, the covering pcompac*
*t group [17] [58,
3.3] of B2LH0(W (X); ~T(X)) x BP X corresponding to the quotient group ~ß(L(X))*
* (11.8.(3)) of
the fundamental group LH0(W (X); ~T(X)) x ß(P L(X)) (4.7, 11.8.(1)).
According to [74, 8.1], any "pconvenient and simply connected or pseudo simp*
*ly connected"
compact connected Lie group satisfies (4.5.(2)). For our purposes, however, the*
* following corollary
will suffice.
4.14. Corollary.Let p be an odd prime and let X be the pcompact group represen*
*ted by
othe product subgroup U(n1) x . .x.U(nk), n1+ . .+.nk = n, ni 0, of U(n), o*
*r,
othe intersection with SU(n) of such a subgroup of U(n), or,
othe image in PU(n) = U(n)=U(1) of such a subgroup of U(n).
Then Gp(W (X), L(X)) = *.
Proof.Write t = t(U(n)), t0 = t(SU (n)), and t1 = t(PU (n)) (the dual to t0). I*
*t suffices (4.5.2) to
show that H1(W ; ) = 0 where W is a subgroup of the form n1x . .x. nk of W (U*
*(n)) = n
and the blank is any of the Fp nmodules Hom (t, t), Hom (t0, t0) or Hom (t1, t*
*1).
Let i be 1 or 2. From the fact that Hi( n; Fp) = 0 for all n when p is odd [5*
*0, 12.2.2], we
inductively deduce that also Hi(W ; Fp) = 0. But then also
Hi(W ; t0) ~=Hi(W ; t) ~=Hi(W ; t1)
H1(W ; Hom(t0, t0)) ~=H1(W ; Hom(t, t0)), H1(W ; Hom(t1, t1)) ~=H1(W ; Ho*
*m(t, t1))
NDETERMINISM 19
as we see from the exact sequences in cohomology induced by the short exact seq*
*uences
0 ! t0 ! t +!Fp ! 0,0 ! Fp ! t ! t1 ! 0,
0 ! t0 ! Hom (t, t0) ! Hom (t0, t0)0!!0,Hom(t1, t1) ! Hom (t, t1) ! t1 ! 0
of Fp nmodules.
Since the representation t = Ind nn1(Fp) is induced from the trivial 1dimen*
*sional representa
tion, its restriction to W ,
Y
resWn(t) = resWnInd nn1(Fp) = IndW\x n1(Fp),
x2W\ n= n1
is a product of representations induced from trivial 1dimensional representati*
*ons. But W \x n1,
the intersection of W with aQconjugate of n1 = 1x n1, is again a subgroup *
*of W type, so
it follows that Hi(W ; t) = Hi(W \ x n1; Fp) = 0. Furthermore, Hom (t, ) = *
*Ind nn1() so
that, by the same argument, H1(W ; Hom(t, )) = 0 where the blank can be t, t0 *
*or t1. ___
5.The pcompact group PGL (n, C)
In this section we show Ndeterminism for the Afamily of pcompact groups wh*
*ere p is an odd
prime. See Broto and Viruel [16, 15] for an alternative proof and [68, 7.19] f*
*or a prototype of
Theorem 5.1.
5.1. Theorem. The pcompact group PGL (n, C) is totally Ndetermined for all n *
* 1 and all
odd primes p.
As a consequence (5.3) of this theorem also GL(n, C), for instance, is totall*
*y Ndetermined so
that we may conclude from (3.12.(3), 3.16) that
Aut(GL (n, C)) = AutZpW(GL(n,C))(L(GL (n, C))) = AutZp[ n](Znp)
when n > 2.
5.2. Corollary.Let X be a pcompact group whose QpWeyl group r0W (X) is in Cla*
*rkEwing
family 1 and assume that p is odd. Then:
1.X is totally Ndetermined.
2.X is determined by its ZpWeyl group.
3.For n > 2
(
End (X) ~= Zp n < p
Zxp[ {0} n p
while End(SL(2, C)) = Zp=Zx.
4.If ß1(X) = 1, or Z(X) = 1, or n < p3, then X is determined by its FpWeyl g*
*roup and X is
a cohomologically unique pcompact group.
Proof.This is immediate from (3.3, 3.7) and (3.17.(2), 4.4, 11.18). In connecti*
*on with the appli
cation of (3.17.(2)), observe that the outer automorphism of the symmetric grou*
*p 6 [91, 2.2.18,
2.2.20] can not be lifted to an automorphism of N(X) because all such automorph*
*isms_take_reflec
tions to reflections. *
* __
5.3. Corollary.Let X be the pcompact group represented by
othe product subgroup GL(n1, C) x . .x.GL(nk, C), n1+ . .+.nk = n, ni 0, of*
* GL(n, C),
othe intersection of such a subgroup with SL(n, C), or,
othe image of such a subgroup in PGL (n, C).
Then X is totally Ndetermined, X is determined by its RWeyl group for R = Zp,*
* Fp, and X is
a cohomologically unique pcompact group (p odd).
Proof.That X is totally Ndetermined follows from (5.1) together with (3.3, 3.7*
*). Apply_(4.14,
4.4) for the other properties of X. *
* __
20 J.M. MØLLER
We shall prove (5.1) by inductively verifying that BPGL (n, C) satisfies the *
*sufficient criteria (3.4,
3.8, 3.9) for total Ndeterminism. For this process, it is crucial (2.12) to ha*
*ve information about
the nontoral elementary abelian psubgroups of PGL (n, C) = GL(n, C)=Cx and th*
*eir centralizers.
Thus we shall start out by identifying the nontoral elementary abelian psubgr*
*oups of PGL (n, C),
their Quillen automorphism groups, and their centralizers.
Nontoral elementary abelian psubgroups of PGL (n, C) can be constructed fro*
*m extraspecial
pgroups contained in GL (n, C) as follows: Let P be an extraspecial psubgro*
*up (this means
[P, P ] = Z(P ) is of order p [85, 5.3]) and E an elementary abelian psubgroup*
* of GL(n, C) such
that
Z(P ) Z(GL (n, C)), [P, E] = {1} = P \ E,
where Z(P ) is the center of P and Z(GL (n, C)) = Cx the center of GL (n, C). T*
*hen T = P E
is a nonabelian subgroup of GL(n, C) that maps onto a nontrivial nontoral el*
*ementary abelian
psubgroup, V , of PGL (n, C) with kernel Z(P ). (V is nontoral because the pr*
*eimage of toral
subgroup of PGL (n, C) is toral in GL(n, C).)
In fact, all nontrivial nontoral elementary abelian psubgroups of PGL (n, *
*C) have this form.
5.4. Lemma. [41, 3.1] Let V be a nontrivial nontoral elementary abelian psub*
*group of PGL (n, C).
Then
op divides n, and,
othere is an inclusion morphism of short exact sequences of groups
1____//_Z(P_)_____//_T_________//_V_______//1
 æ æ
  
fflffl fflffl fflffl
1_____//_Cx____//_GL(n, C)__//_PGL(n, C)__//_1
where T = P E is the direct product of an extraspecial pgroup P GL (n*
*, C) and an
elementary abelian pgroup E GL(n, C) such that P \ E = {1} = [P, E]. The*
* extraspecial
pgroup P can be chosen to have exponent p.
Proof.If n is not divisible by p, then all elementary abelian psubgroups of PG*
*L (n, C) are toral.
Assume now that p divides n. As H2(V ; Z=p) maps onto Hom (H2(V ); Cx) = H2(V, *
*Cx), there is
a subgroup R GL(n, C) that maps onto V with a kernel that is cyclic of order *
*p and central in
GL(n, C). If R were abelian, then R and V would be toral subgroups.
The commutator subgroup [R, R] is cyclic of order p for it is nontrivial and*
* contained in the
kernel of the epimorphism R ! V . Thus V = R=[R, R]. Let P be a normal subgr*
*oup of R
such that P=[R, R] is complementary to Z(R)=[R, R]. Then R = P Z(R) and P is ex*
*traspecial as
Z(P ) = P \ Z(R) = [R, R] = [P Z(R), P Z(R)] = [P, P ].
The commutative diagram
P x Z(R)__________//_GL(n, C)
 
 
fflffl fflffl
P=[R, R] x Z(R)=[R,_R]__//PGL(n, C)
has an adjoint diagram
Z(R) ___________//CGL(n,C)(P_)_____//_GL(n, C)
  
  
fflffl fflffl fflffl
Z(R)=[R, R]____//_CPGL(n,C)(P=[R,_R])0//_PGL(n, C)
where the horizontal maps are inclusions and the two rightmost vertical maps ar*
*e epimorphisms
with kernel Cx. The centralizer of P in GL(n, C) is a product of general linear*
* groups [82, Propo
sition 4] and Z(R) is included here as an abelian, hence toral, subgroup. There*
*fore, Z(R)=[R, R]
is included as a toral subgroup in (the identity component) of the centralizer *
*of P=[R, R] in
PGL (n, C). It follows that Z(R)=[R, R] is the isomorphic image of an elementar*
*y abelian pgroup
E CGL(n,C)(P ) GL(n, C).
NDETERMINISM 21
By construction, [P, E] = {1} = E \Z(P ) = E \P , so P and E permute, T = P E*
* is a subgroup
of GL(n, C) that maps onto Vpwith_kernel Z(P ). For any extraspecial pgroup P*
* GL(n, C)
of exponent p2 with Z(P) = p1 there is an extraspecial pgroup P+ GL(n, C) *
*of exponent p
that has the same center, the same centralizer, and the same image in PGL (n, C*
*) as P_(5.19).
Therefore we can assume P = P+ has exponent p. *
* __
The commutatorpsubgroup_and the center of the covering group T = P E of V are [*
*T, T ] = [P, P ] =
Z(P ) = p1 and Z(T ) = Z(P )E [T, T ].
By taking commutators in T we get an alternating bilinear form
(5.5) f :V x V ! [T, T ]
on V = T=[T, T ], i.e. f(u1, u2) = [__u1, __u2] where __u2 T is a lift of u 2 V*
* . This bilinear form may
be degenerate in that
V ?= Z(T )=[T, T ] = E
and we obtain a nondegenerate alternating bilinear form
__
(5.6) f:V=V ?x V=V ?! [T, T ]
by factoring out V ? or, equivalently, by restricting to the subspace P=[P, P ]*
* ~=V=V ?~=T=Z(T )
of V .
Define
Isom(V, f)= {ff 2 Aut(V )  f(ff(u1), ff(u2)) = f(u1, u2)}
__ ? __ __
Aut(f)= {(A, a) 2 Aut(V=V ) x Aut(Z(T ))  fO (A x A) = a O f}
to be the group of all isometries of (V, f) and, respectively, the group of all*
* pairs of automorphisms
(A, a) 2 Aut(V=V ?) x Aut(Z(T )) that make
_f "
V=V ?x V=V ?_____//[T,ØT_]_//Z(T )
AxA  a
fflffl " fflffl
V=V ?x V=V ?___f_//[T,ØT_]_//Z(T )
commutative.
Any outer automorphism ff of T induces an an automorphism_a(ff) of Z(T ) and *
*an automorphism
A(ff) of T=Z(T ) = V=V ?such that (A(ff), a(ff)) 2 Aut(f).
5.7. Lemma. For odd p there is a short exact sequence
__
1 ! Hom (V=V ?, V ?) ! Out(T ) (A,a)!Aut(f) ! 1
for the outer automorphism group of T .
Proof.The 2cocycle for the extension Z(T ) ! T ! T=Z(T ) = V=V ?is c where
__u __ ____
1. u2= c(u1, u2)u1u2, u1, u2 2 T=Z(T ),
where __u2 T is a lift of u 2 T=Z(T ). Since
__ __ __ __ __ ____ 1 __ __ ____ 1 __ __ ____ 1
f(u1, u2)c(u2, u1) = [u1, u2] u2. u1 u2u1 = u1. u2 u2u1 = u1. u2 u1u2
= c(u1, u2)
__
for all u1, u2 2 T=Z(T ), the 2cochain_fmeasures the failure of the 2cocycle *
*c in being symmetric.
If the pair (A, a) is in Aut(f), then the 2cochain d = (A*c)1(a*c) is symme*
*tric, for
__ __ 1 1
f(Au1, Au2) = af(u1, u2) , c(Au1, Au2) ac(u1, u2) = c(Au2, Au1) ac(u2*
*, u1),
and hence 2d = ffiq where q is the associated quadratic form, q(u) = d(u, u), v*
*iewed as a 1cocycle.
Thus (A, a) can be lifted to an automorphism of T .
The kernel of the map ff ! (A(ff), a(ff)) is easily determined as follows. Th*
*ere is a surjection
__
Hom(T=Z(T ), Z(T )) i ker Out(T ) ! Aut(f)
22 J.M. MØLLER
taking the homomorphism ': T=Z(T ) ! Z(Tt)o the automorphism t ! '(t)t of T . *
*This au
tomorphism is inner precisely when '(t) = [u, t] for some u 2 T . Since any ho*
*momorphisms
T ! [T, T ] is of this form, the kernel is isomorphic to Hom(T=Z(T ), Z(T ))= H*
*om(T=Z(T_), [T, T ]) ~=
Hom (T=Z(T ), Z(T )=[T, T ]) ~=Hom(V=V ?, V ?). *
* __
*
* __
For example, if P is an extraspecial pgroup then Out(P ) is isomorphic to t*
*he group Aut(f)
(when p is odd).
We shall next determine the Quillen automorphism groups of the subgroups T *
*GL(n, C) and
V PGL (n, C) of (5.4).
5.8. Definition.For a homomorphism æ: H ! Gof a (finite) group H into a Lie gro*
*up G, define
the Quillen automorphism group A(G)(H, æ) as
A(G)(H, æ) = {ff 2 Out(H)  (æff) = (æ)}
where (æ) denotes the representation (æ) 2 Rep(H, G) = Hom (H, G)=G represented*
* by the homo
morphism æ.
If the target of æ is G = GL(n, C), in particular, then
A(G)(H, æ) = {ff 2 Out(H)  tr(æff) = tr(æ)}
by complex representation theory.
5.9. Lemma. Let T = P E and V = T=[T, T ] be as in ( 5.4) and assume that T has*
* exponent p.
Then the homomorphism
A(GL (n, C))(T, æ) ! A(PGL (n, C))(V, æ)
is surjective.
Proof.Suppose that BCx normalizes V in PGL (n, C) for some B 2 GL(n, C). Then T*
* B T Cx.
But if g 2 T and gB = zh for some z 2 Cxpand_some h 2 T , then z must have orde*
*r p since g and__
h have order p. Thus z is an element of p 1= [T, T ] T . Consequently, T B= T*
* . __
In the situation of (5.4), consider first the special case where E is trivial*
* and T = P is an extra
special pgroup whose center is central in GL(n, C). The extraspecial pgroup *
*P has P :[P, P ] =
p2d characters of degree 1 and p  1 irreducible characters of degree pd (descr*
*ibed in (5.19)).
These irreducible representations of degree pd are faithful and they are [43, V*
*.16.14] in bijective
correspondence with the nontrivial linear forms ~: Z(P ) ! Cx; the representat*
*ion corresponding
to ~ is the representation ~P induced from any extension of ~ to a linear form *
*on a maximal normal
abelian subgroup of P . Thus the representation æ of P has the form
X X
æ = ~P + Ø
for some nontrivial linear forms ~ on Z(P ) and some homomorphisms Ø: V ! Cx .*
* Since æ
is faithful at least one ~ must appear, and, since æ takes the center of P into*
* the center of
GL(n, C), no Øs can occur and exactly one ~ occurs. (Observe that for nonident*
*ity g 2 Z(P ),
~P(g) 6= 1 = Ø(g).) Thus in fact
æ = m~P, pdm = n,
for some nontrivial homomorphism ~: Z(P ) ! Cx. From the formula [43, V.16.14]*
* [44, 7.5]
( d
træ(g) = p m~(g) g 2 Z(P )
0 g 62 Z(P )
we see that the Quillen automorphism groups
A(GL (n, C))(P, æ) = {ff 2 Out(P )  a(ff) = 1}, A(PGL (n, C))(V, æ) = *
*Sp(V )
consist of those outer automorphisms of P that restrict to the identity on the *
*center Z(P ) and (5.7,
5.9) of all isometries of the nondegenerate space (V, f), respectively. (Note *
*also that V = P=P \Cx
is unique up to isomorphism as an object of A(PGL (n, C)).)
In general, T = P E is the direct product of an extraspecial pgroup with an*
* elementary abelian
pgroup E. Since the restriction of æ to P is of the form æP = m~P, as we hav*
*e just seen,
representation theory for products of groups [43, V.10.3] [44, 8.1] tells us th*
*at the representation
NDETERMINISM 23
æ = ~P]Ø is the outer tensor product of ~P with a faithful mdimensional repres*
*entation Ø of
E = V ?. From the formula (
d~(g)Ø(e) g 2 Z(P )
træ(g, e) = p
0 g 62 Z(P )
we see that the Quillen automorphism groups
A(GL (n, C))(T,=æ){ff 2 Out(T )  a(ff) 2 A(GL (n, C))(Z(T ), ~]Ø)}
A(PGL (n, C))(V,=æ){ff 2 Isom(V, f)  ffV ?2 A(PGL (n, C))(V ?, Ø)}
consist of those outer automorphisms of T that restrict to Quillen automorphism*
*s of the m
dimensional representation ~]Ø of Z(T ) = Z(P )E and of those isometries of the*
* inner product
space (V, f) whose restrictions to V ?leave the representation Ø invariant, res*
*pectively.
Define A(T ) Out(T ) and A(V, f) Aut(V ) to be the groups
A(T )= {ff 2 Out(T )  a(ff) = 1}
A(V, f)= {ff 2 Isom(V, f)  ff is the identity}on V ?
consisting of those outer automorphisms that restrict to the identity on Z(T ),*
* respectively of all
isometries of (V, f) that restrict to the identity on E = V ?. Then A(T ) is a*
* subgroup of the
Quillen automorphism group A(GL (n, C))(T, æ) (and equal to this Quillen automo*
*rphism group
if T is extraspecial). It follows from (5.7) that A(T ), of order Sp(V=V ?)*
*Hom (V=V ?, V ?), is
isomorphic to A(V, f). This proves the following lemma.
5.10. Lemma. The Quillen automorphism group A(GL (n, C))(T, æ) contains A(T ) a*
*nd the Quillen
automorphism group A(PGL (n, C))(V, æ) contains A(V, f). If T is extraspecial,*
* the Quillen au
tomorphism group A(PGL (n, C))(V, æ) = Sp(V ).
The final step consists in identifying the centralizers and their centers for*
* the subgroups T
GL(n, C) and V PGL (n, C) of (5.4). The information we need is obtained in *
*(5.12) as an
application of the more general, and elementary, (5.11).
5.11. Lemma. Let T be any subgroup of GL (n, C), æ: T ! GL(n, C)the inclusion, *
*and Z a
central subgroup of GL(n, C).
1.There is a short exact sequence of Lie groups
1 ! CGL(n,C)(T )=Z ! CGL(n,C)=Z(T ) @!Hom(T, Z)(æ)! 1
where the group to the right is the isotropy subgroup for the action of Hom*
* (T, Z) on (æ) 2
Rep(T, GL(n, C)) and @(BZ)(g) = [B, g].
2.The connected component of CGL(n,C)=Z(T ) is
CGL(n,C)=Z(T )0 = CGL(n,C)(T )=Z
and the group of components ß0 CGL(n,C)=Z(T ) is isomorphic to
Hom (T, Z)(æ)= {OE: T ! Z 9B 2 GL(n, C)8g 2 T :OE(g) = [B, g]}
Proof.The exact sequence of the first point is a consequence of the short exact*
* sequence
1 ! CGL(n,C)(T ) ! {B 2 GL(n, C)[B, T ] Z} @!Hom(T, Z)(æ)! 1
because CGL(n,C)=Z(T ) is the quotient of the middle group by the central subgr*
*oup Z. The second
point follows from the first because the centralizer of T in GL(n, C), a produc*
*t of general_linear
groups [82, Proposition 4], is connected. *
* __
5.12. Lemma. Let T and V be as in ( 5.4).
1.If T = P is extraspecial, then
CPGL(pdm,C)(V ) = V x PGL(m, C), Z CPGL(pdm,C)(V ) = V,
where the Quillen automorphism ff 2 A(V, f) = Sp(V ) acts as ff1 x 1 and f*
*f, respectively.
2.If V ?has rank one, then the component group of Z CPGL(pdm,C)(V ) is isomo*
*rphic to V=V ?
or to V .
3.ß1Z CPGL(pdm,C)(V ) is a finitely generated free abelian group with trivia*
*l A(V, f)action.
24 J.M. MØLLER
Proof.In the special case where T = P is extraspecial, all elements OE of Hom *
*(P, Cx) are of the
form OE(g) = [h, g] for some h 2 P . Thus all OE preserve the representation (æ*
*) and it follows from
(5.11) that the natural homomorphism
(5.13) V x PGL(m, C) = V x CGL(pdm,C)(P )=Cx ! CPGL(pdm,C)(V )
is an isomorphism. Use (5.17) to get the action of the Quillen automorphism gro*
*up.
For the second item of the lemma, suppose that T = P E where E = V ? is oned*
*imensional.
Then
CPGL(pdm,C)(V ) = CPGL(pdm,C)(P E) = CCPGL(pdm,C)(P)(V ?)
= CP=[P,P]xPGL(m,C)(V ?) = P=[P, P ] x CPGL(m,C)*
*(V ?)
and consequently,
ZCPGL(pdm,C)(V ) = P=[P, P ] x ZCPGL(m,C)(V ?).
Here (5.14), the second factor is either connected, in which case
ß0CPGL(pdm,C)(V ) = ß0ZCPGL(m,C)(V ) = P=[P, P ] = V=V ?,
or disconnected, in which case the center of Z(CPGL(pdm,C)(V )) = V . *
* __
Use (5.17) for the final item of the lemma. *
* __
5.14. Lemma. For any elementary abelian pgroup E PGL (n, C) of rank one, eit*
*her the cen
tralizer CPGL(n,C)(E) as well as its center ZCPGL(n,C)(E) are both connected or*
* ZCPGL(n,C)(E) =
E.
Proof.There is (5.11) an exact sequence
1 ! Cx ! CGL(n,C)(E) ! CPGL(n,C)(E) ! Hom (E, Cx)(Ø)! 1
where Ø: E ! GL(n, C)is a lift. The group to the right is either trivial or cyc*
*lic of order p. If it
is trivial, then
CPGL(n,C)(E) = CGL(n,C)(E)= GL(1, C), Z CPGL(n,C)(E) = Z CGL(n,C)(E) = GL(*
*1, C)
are both connected Lie groups [58, 4.6]. Otherwise, n = rp and Ø = ræ is a dire*
*ct sum of a number
of copies of the regular representation æ of E. Then the centralizer
CPGL(n,C)(E) = GL(r, C)p= GL(1, C) o
where oe has order p and acts on GL(r, C)p by permuting the factors cyclically.*
* Thus the center of
the centralizer,
i p j 1
Z CPGL(n,C)(E) = GL(1, C) = GL(1, C) = H (; GL(1, C))
is cyclic of order p. *
* ___
The information collected so far suffices to establish the vanishing of some *
*of the higher limits
for the functors ßj(BZCPGL(n,C)): A(PGL (n, C)) ! Ab(2.7). We shall make use of*
* the following
lemma which, together with its application in the proof of (5.16), is due to J.*
* Grodal.
5.15. Lemma. Let A be a subgroup and P a parabolic subgroup of G = GL (n, Fp) s*
*uch that
U A P where P = UL is the Levi decomposition [23, x69A]. Then
Hom Fp[A](St(G), M) = Hom Fp[A=U](St(L), M)
for any Fp[A]module M which is trivial as an Fp[U]module and finitedimension*
*al as an Fpvector
space.
~=
Proof.The standard Fp[A]module isomorphism Hom (St(G), Fp) M ! Hom(St(G), *
*M)) re
stricts to an isomorphism
~=
Hom(St(G), Fp)U M ! Hom Fp[U](St(G), M)
NDETERMINISM 25
of Fp[A=U]modules. Since Steinberg modules are selfdual and St(G)U = St(L) [8*
*9, 18, 42] we
have
Hom(St(G), Fp)U ~=St(G)U ~=St(L) ~=Hom(St(L), Fp)
as Fp[P=U]modules. Thus Hom Fp[U](St(G), M) ~=Hom (St(L), Fp) M ~= Hom(St(L*
*), M) as
Fp[A=U]modules and consequently
Hom Fp[A](St(G), M) ~=HomFp[U](St(G), M)A=U ~=Hom(St(L), M)A=U
= Hom Fp[A=U](St(L), M)
as vector spaces. *
* ___
5.16. Lemma. limi(A(PGL (n, C)), ßj(BZCPGL(n,C))) = 0 for j = 1, 2 and i = j, j*
* + 1.
Proof.It suffices (2.13, 5.10, 5.12) to show that the following homomorphism gr*
*oups are trivial:
oHom Sp(V()St(V ), V ) where dimFpV = 2,
oHom A(V,f)(St(V ), V ) and HomA(V,f)(St(V ), V=V ?) where dimFpV = 3 and f *
*is a nontrivial
alternating bilinear form on V ,
oHom A(V,f)(St(V ), Zp) where dimFpV is 3 or 4, f is a nontrivial alternati*
*ng bilinear form
on V , and Zp carries the trivial A(V, f)action.
Note that Zp can be replaced by Fp as target module since the Steinberg module *
*is finitely
generated. The first of these groups is clearly trivial as Sp(V ) = SL(V ) cont*
*ains 1 which acts
trivially on the Steinberg module but has no nontrivial fixed points in V . Fo*
*r the remaining cases,
we apply (5.15). For us, n is 3 or 4, and the group A consists of the matrices
` '
Ik *
0 SL(2)
where Ik is a (k x k)identity matrix, k = 1, 2. Take P and U = Op(P ) to be th*
*e subgroups of
G = GL(n, Fp) consisting of matrices of the form
` ' ` '
GL(k) * Ik *
0 GL(2) , respectively,0 I2
so that L = GL(k) x GL(2). Then
Hom Fp[A](St(G), Fp)= Hom Fp[SL(2)](St(GL (k)) St(GL (2)), Fp)
= Hom Fp[SL(2)](St(GL (2)), HomFp(St(GL (k), Fp)))
= HomFp[SL(2)](St(GL (2)), Fp)
and, for n = 3,
Hom Fp[A](St(G), V=V ?) = Hom Fp[SL(2)](St(GL (2)), V=V ?)
Using that St(GL (2)) is an irreducible Fp[SL(2)]module we see that both these*
* groups are trivial._
Since V ?is a trivial Fp[A]module, also Hom Fp[A](St(G), V ) must be trivial. *
* __
Proof of Theorem 5.1.PGL(n, C) is nonmodular, hence totally Ndetermined [68, *
*3.11, 7.4],
for n < p. We may therefore, inductively, assume that all elementary abelian p*
*subgroups of
PGL (n, C) have totally Ndetermined centralizers (3.3, 3.7, 5.12) [82, Proposi*
*tion 4]. But then also
PGL (n, C) itself has Ndetermined automorphisms according to (3.4, 5.16) and i*
*s Ndetermined
according to (3.8, 5.16) provided we can verify the conditions of (3.9) when n *
*= pm is divisible by
p. It only remains to consider the third condition as the first two have been v*
*erified in (5.4, 5.10).
Let j0:N(PGL (n, C)) ! Xbe the maximal torus normalizer for some pcompact gr*
*oup X. Let
(V, ) denote the nontoral rank 2 object of A(PGL (n, C)), ~: V ! N(PGL (n, C)*
*)a preferred lift
of :V ! PGL (n, C), and put 0= j0~. The object (V, 0) of A(X) does not depe*
*nd on the choice
26 J.M. MØLLER
of ~ (3.9). We must show that the diagram
CPGL(n,C)(V,O_)f(V,~)//_CXO(V,OO0)
CPGL(n,C)(ff) CX(ff)
 
CPGL(n,C)(V, _)f(V,~)//_CX (V, 0)
commutes for all ff 2 Sp(V ) = SL(V ) [43, II.9.12]. This will be the case if a*
*pplication of the identity
component functor ()0 and the component group functor ß0() gives commutative *
*diagrams [66,
5.3] [64, 3.4, 3.10]. The first of these derived diagrams commutes because SL(V*
* )op, generated by el
ements of order p [91, 3.6.21], acts trivially on CPGL(n,C)(V, )0 = PGL (m, C)*
* = CX (V, 0)0 whose
automorphism group (5.2) Aut(PGL (m, C)) ~=Zxp(or Zxp=Zx if m = 2) contains no *
*elements of
order p. The above diagram also commutes on the level of ß0 for ß0(f(V, ~)) is *
*SL(V_)opequivariant
by (3.10, 5.12). *
* __
5.17. The action of A(GL (n, C))(T, æ) on CGL(n,C)(T, æ). Write (æ) = ~1(æ1) + *
*. .+.~t(æt)
as a direct sum of inequivalent irreducible characters (æ1), . .,.(æt) with mul*
*tiplicities ~1, . .,.~t,
respectively. Then
Y Y
(5.18) CGL(n,C)(T, æ) = GL (~i, C), Z CGL(n,C)(T, æ) = Z GL (~i, *
*C)
æi2S(æ) æi2S(æ)
where S(æ) = {æ1, . .,.æt} is the set of irreducible characters occuring in æ. *
* Let (S(æ)) be
the group of all permutation of S(æ) and for a given integer valued function fi*
* on S(æ) write
(S(æ))fifor the subgroup of permutations that preserve fi. We shall need the d*
*egree function d,
recording the degree of æi, and the multiplicity function ~(æ), recording the m*
*ultiplicity ~iof æiin
æ. The elements of NGL(n,C)(T, æ) are Clinear automorphisms of Cn that are ff*
*linear for some
automorphism ff 2 Aut(T ) and there is a homomorphism
A(GL (n, C))(T, æ) = NGL(n,C)(T, æ)=T CGL(n,C)(T, æ) ! (S(æ))d\ (S(æ*
*))~(æ)
since ff permutes the irreducible representations (æi) in a degree and multipli*
*city preserving way.
Now,
NGL(n,C)(T, æ) NGL(n,C)(CGL(n,C)(T, æ)) NGL(n,C)(ZCGL(n,C)(T, æ))
Y
GL(di~i, C) o (S(æ))d~(æ)
and since the first factor of the semidirect product acts trivially on ZCGL(n,*
*C)(T, æ), the action
homomorphism
A(GL (n, C))(T, æ) ! Out(CGL(n,C)(T, æ)) ! Aut(ZCGL(n,C)(T, æ))
factors through (S(æ))d\ (S(æ))~(æ) (S(æ))d~(æ). Observe, in particular, th*
*at the subgroup
A(T ) of A(GL (n, C))(T, æ) acts trivially on CGL(n,C)(T ) because any outer au*
*tomorphism ff 2
A(T ) restricts to the identity on Z(T ) so that it preserves all the irreducib*
*le components Øi~P of
the representation æ.
5.19. Representations of extraspecial pgroups. We construct explicitly the fa*
*ithful irre
ducible representations of the extraspecial pgroups.
Let E be an elementary abelian pgroup of rank d 1 and C[E] its complex gro*
*up algebra, or
rather, its underlying Edimensional complex vector space. Then there is a co*
*mmutative diagram
as in (5.4)
"æ
P ]Ø____//GL(C[E])
 
 
fflfflØ " fflffl
V _æ__//_PGL(C[E])
NDETERMINISM 27
where V = E^ x E is the product of E and its dual E^ = Hom (E, Cx), and P = P,*
* P+ is the
subgroup of GL(C[E]),
P = , P+ = ,
generated by
Ri(v) = i(v)v, Tu(v) = u + v, i 2 E^, u, v 2 E,
where ! is a primitive p2th root of unity. Since the commutator
[!Ri, Tu] = [Ri, Tu] = i(u), i 2 E^, u 2 V,
is scalar multiplication with the complex number i(u), the group P (P+) is ext*
*raspecial of
order p1+2dand exponent p2 (p). A trace computation reveals that the p  1 fait*
*hful irreducible
representations of P are obtained from the inclusion æ by composing with the au*
*tomorphisms
(!)Ri ! (!)Rii, Tu ! Tu, 0 < i < p, of P . Observe that P and P+ have the same*
* center, the
same centralizer in GL(C[E]), and the same image in PGL (C[E]). The same is tru*
*e for P and
P+ considered as subgroups of GL(C[E] m ) by means of the representation mæ.
An object (V, ) of A(X) is said to be doversize if
codimker(ß0(~): V ! ß0N(X) = W (X)) d
for all preferred lifts ~: V ! N(X) of :V ! X and d 0 is the biggest such n*
*atural number.
Thus the 0oversize objects are the toral objects. It may be worthwhile to note*
* that the Afamily
provides examples of highly oversized elementary abelian subgroups.
5.20. Proposition.Let T and V be as in Lemma 5.4. If T = P E where P has order *
*p1+2dthen
(V, æ) is a doversize object of the Quillen category A(PGL (n, C)).
Proof.We shall first consider the case where T = P = p1+2d+is extraspecial and*
* æ is one of the
irreducible and faithful representations that we just considered. Note that P i*
*s contained in the
maximal torus normalizer N GL (n, C) as
x ff
N GL (n, C) = Ri, Toe i 2 map(V, C ), oe 2 (V )
is generated by all the operators Ri(v) = i(v)v, Toe(v) = oe(v) for all functio*
*ns i from V into
Cx and all permutations oe of the elements of V . Similarly, V is contained in *
*the maximal torus
normalizer N PGL (n, C) = N GL (n, C) =Cx of PGL (n, C). The centralizers
CN(GL(n,C))(P ) = Cx, CN(PGL(n,C))(V ) = V
so that the inclusion of V into the maximal torus normalizer is a preferred lif*
*t of the inclusion
of V into PGL (n, C). For this preferred lift the intersection of V with the m*
*aximal torus has
codimension d. But the intersection of V with any maximal torus of PGL (n, C) i*
*s covered by the
intersection of P with a maximal torus of GL(n, C) and such a subgroup has orde*
*r at most p1+d
which is the order of a maximal normal abelian subgroup of the extraspecial p*
*group P . Thus
(V, æ) is a doversize rank 2d object of the Quillen category of PGL (n, C).
When æ = m~P,
CN(GL(n,C))(P ) = N(GL (n, C)) \ CGL(n,C)(P ) = N(GL (n, C)) \ GL(m, C)
= N(GL (m, C))
so that the centralizer of V in N(PGL (n, C)) is V x N(PGL (m, C)). Again, the *
*inclusion of V
into N(PGL (n, C)) is a preferred lift of the inclusion of V into PGL (n, C) an*
*d we conclude, as
above, that (V, æ) is a doversize rank 2d object of A(PGL (n, C)).
In general, T = P E is the direct product of an extraspecial pgroup and an *
*elementary abelian
pgroup. But still the inclusion of V = P=[P, P ] x E into the maximal torus *
*normalizer is a
preferred lift because its adjoint
E ! CN(PGL(n,C))(P=[P, P ]) ! CPGL(n,C)(P=[P, P ])
is a preferred lift as E is toral. For this preferred lift, the intersection of*
* V with the maximal torus
has codimension d and, as above, this is actually the minimum. Thus (V, æ) is a*
* doversize_rank
> 2d object of the Quillen category. *
* __
28 J.M. MØLLER
6. The 3compact group F4
We consider the 3compact group (BF4)^3obtained by completing the classifying*
* space BF4 for
the exceptional Lie group F4of rank 4.
6.1. Theorem. [92] The following hold for the 3compact group F4:
1.F4 is totally Ndetermined.
2.F4 is determined by its RWeyl group for R = Zp, Qp, Fp.
3.F4 is a cohomologically unique pcompact group.
4.End (F4)  {0} = Aut(F4) = NGL(L(F4))(W (F4))=W (F4) is an abelian group is*
*omorphic to
Zx3=Zx x C2 where the group C2 of order 2 is generated by an exotic automor*
*phism.
Proof.The information provided by Griess [41, 7.4] about elementary abelian ps*
*ubgroups of the
Lie group F4 shows that the 3compact group F4 satisfies the conditions of (3.8*
*); see (3.9, 3.10)
and the remark below (2.13). Combined with (4.4, 11.18, 11.25) this proves the *
*first three items.
Direct computation shows that the normalizer
x ff
NGL(4,Z3)(W (F4)) = Z3, ", W (F4)
where
0 1
1 1 0 0
p ___ B 1 1 0 0 C
2" = B@ 0 0 1 1 CA
0 0 1 1
for the Weyl group of F4 in GL(4, Z3) as described e.g. in [12]. The final item*
* of the_theorem is
now a consequence of (3.17.(2)). *
* __
Note that (6.1) yields a new proof of the existence of Friedlander's exceptio*
*nal isogeny [39].
7.Polynomial pcompact groups
All connected Fplocal spaces with polynomial mod p cohomology are pcompact *
*groups. We
study these polynomial pcompact groups in this section. See also D. Notbohm [7*
*2, 76, 79] for
further information and for references to the literature about this classical s*
*ubject.
For any connected pcompact group X, the image of H*(BX; Fp) in H*(BT (X); Fp*
*) is contained
in the invariant ring H*(BT (X); Fp)W(X) for the action of the Weyl group on th*
*e cohomology of
the maximal torus. Much work, summarized in the following lemma, has been done *
*to tell when
H*(BX; Fp) actually equals this invariant ring.
7.1. Lemma. Let p be an odd prime and X a pcompact group. The following condi*
*tions are
equivalent:
1.H*(BX; Fp) is a polynomial algebra.
2.H*(BX; Fp) = H*(BT (X); Fp)W(X).
3.H*(BX; Fp) H*(BT (X); Fp).
4.H*(BX; Fp) is concentrated in even degrees.
5.H*(BX; Zp) is concentrated in even degrees and degreewise free.
6.H*(BX; Zp) is polynomial on even degree generators.
7.H*(BX; Zp) = H*(BT (X); Zp)W(X).
8.H*(BX; Zp) H*(BT (X); Zp).
If X satisfies these equivalent conditions,Qthen the rational rank r of X [30, *
*5.1] equals the Krull
dimension of H*(BX; Fp), and W  = diwhere 2di, 1 i r, are the degrees o*
*f the polynomial
generators [88, 5.3.5, 5.5.4].
Proof.1. ) 2. is [29, 2.11] and 2. ) 3. ) 4. ) 5. is elementary. 5. ) 1., 6., *
*7., 8.: As was
noted in [59, 4.2], H*( BX; Zp) is degreewise free so that Borel's argument [8*
*] [88, 10.7.5] shows
that H*(BX; Fp) and H*(BX; Zp) are polynomial. But then H*(BX; R) is the invari*
*ant ring for __
R = Zp, Fp by [29, 2.11] again. The implications 8. ) 5., 7. ) 5., 6. ) 5. are *
*elementary. __
NDETERMINISM 29
7.2. Definition.A pcompact group X is polynomial if its cohomology ring H*(BX;*
* Fp) is a
polynomial Fpalgebra. A Zpreflection group (W, ~T) is polynomial if its invar*
*iant ring H*(T~, Fp)W
is a polynomial Fpalgebra.
7.3. Example.A pcompact group X is nonmodular if p does not divide the order *
*of W (X). A
Zpreflection group (W, ~T) is nonmodular if p does not divide the order of W *
*. Any nonmodular
pcompact group is connected [58, 3.8] and its Weyl group is a nonmodular Zpr*
*eflection group.
The ShephardTodd theorem [7, 7.2.1] says that any nonmodular Zpreflection gr*
*oup (W, ~T) is
polynomial, and, clearly, H0(W ; ~T) = 0 if (W, ~T) is also simple. Any nonmo*
*dular pcompact
group X is polynomial [68, 3.12], totally Ndetermined [68, 3.11, 7.7], and det*
*ermined by its R
Weyl group for R = Zp, Fp (4.5, 4.4); if X is also simple, then X is centerless*
* (3.12.(2)) and
determined by its QpWeyl group (11.18, 4.4).
The Weyl group of any polynomial pcompact group is a polynomial Zpreflectio*
*n group but
not all polynomial Zpreflection groups are Weyl groups of polynomial pcompact*
* groups (7.4). If
the ring of invariants H*(T~, Fp)W for some ZpW torus ~Tis polynomial then W i*
*s a Zpreflection
group [29, Proof of 5.2].
7.4. Remark. Borel [9, 2.5] shows that for a simple compact Lie group G and p a*
*n odd prime,
the Bousfield Fplocalization (BG)^pof BG [13] is a nonpolynomial pcompact gr*
*oup BG^precisely
when
oG = SU(r + 1)=Z where Z is a nontrivial central psubgroup, or,
oG = F4, PE6, E6, E7, E8 and p = 3, or,
oG = E8 and p = 5.
Kemper and Malle [51] show that a simple Zpreflection group (W, ~T) is nonpol*
*ynomial precisely
when it is the Weyl group of one of the Lie pcompact groups on Borel's list  *
*with the exception
that (W, ~T)(PU (3)) at p = 3 is polynomial because we are in dimension 2 [71, *
*5.1]. Combining
this with (7.27), we see that the invariant ring H*(T~; Zp)W with Zpcoefficien*
*ts is nonpolynomial
precisely when (W, ~T) = (W, ~T)(G^) is the Weyl group of one of the Lie pcomp*
*act groups ^Gon
Borel's list. Thus the polynomial Zpreflection group (W, ~T)(PU (3)) at p = 3*
* is not the Weyl
group of a polynomial pcompact group for then also the invariant ring with Zp*
*coefficients would
be polynomial (7.1). (Combine the method of (7.24) with the results of [71, x4*
*] [51, x5] to see
that (W, ~T)(SU (r + 1)=Z) is nonpolynomial when pr + 1, n 3, and Z is a no*
*ntrivial central
pgroup.)
7.5. Lemma. Let p be an odd prime. Let i: ~T! Xbe a loop space homomorphism fro*
*m a Zp
torus ~Tto a polynomial pcompact group X. If H*(Bi; Fp) induces an isomorphism
H*(BX; Fp) ~=H*(T~; Fp)W
to the ring of invariants for some finite group W of automorphisms of ~T, then *
*i: ~T! X is a
(pdiscrete) maximal torus for X and W and W (X) are Fpsimilar Zpreflection g*
*roups.
Proof.Since H*(Bi; Fp) makes H*(T~; Fp) a finitely generated H*(BX; Fp)module *
*[88, 2.3.1],
i: ~T! Xis a monomorphism [30, 9.11]. Moreover, ~Tand T (X) have the same rank,*
* the Krull
dimension of H*(BX; Fp), so that i: ~T! Xis indeed a maximal torus. Also, W and*
* W (X) have
the same order given by the degrees of the polynomial generators. By Lannes th*
*eory [52], the
homomorphism t ! ~T! X is W equivariant up homotopy because it is so on mod p *
*cohomology.
This means that rpW is contained in the Quillen automorphism group A(X)(t) of t*
* ! X which is __
rpW (X) (2.8). But these two groups have the same order, so they must be identi*
*cal. __
If X is polynomial, then X is connected and, by Lannes theory [52], any monom*
*orphism of
a nontrivial elementary abelian pgroup into X factors through the maximal tor*
*us and hence
(2.8) the Quillen category A(X) is equivalent to A(W (X), t(X)). About the cent*
*ric [28] functor
BCX :A(W (X), t(X)) ! [pcg](2.5) we know that, for an odd prime p,
1.H*(BCX ; Zp) = H*(T~(X); Zp)0,
2.ßj(BZCX ) = L(X)2j, j = 1, 2.
30 J.M. MØLLER
The formula in item (1) is a consequence of [33, 1.2] and (7.1, 2.10) showing t*
*hat polynomiality
is preserved under taking centralizers of elementary abelian subgroups. The for*
*mula in item (2)
follows from (2.8). (Recall that H*(T~(X); Zp)0 is (2.2) the functor given by H*
**(T~(X); Zp)0(E) =
H*(T~(X); Zp)W(X)(E)= H*(BT (X); Zp)W(X)(E)and, similarly (2.3), L(X)2jis the *
*functor given
by L(X)2j(E) = H2j(W (X)(E); L(X)) for all nontrivial subgroups E of t(X).)
Combined with the acyclicity result of (2.4) this leads to a very simple proo*
*f of the homology
decomposition for polynomial pcompact groups.
7.6. Proposition.[45, 31] Let p be an odd prime. For any polynomial pcompact g*
*roup X, the
evaluation map
hocolimA(W(X),t(X))opBCX ! BX
is an H*Zpequivalence. Alternatively, the full subcategory ( 2.14) AC(p)(W (X)*
*, t(X))op based on
the collection C(p) of all psubgroups of W , can be used for index category.
Proof.By (2.4, 2.16) and one of the formulas above, the E2page of the Bousfiel*
*dKan spectral
sequence for the cohomology of a homotopy colimit collapses onto the vertical a*
*xis and_therefore
the evaluation map is a H*Zpequivalence. *
* __
In particular, if X is a polynomial pcompact group and p does not divide ord*
*er of the Weyl group
(i.e. X is a nonmodular pcompact group (7.3)) then BX is H*Zpequivalent to t*
*he homotopy
colimit of a diagram of the form
__________________________________________*
*_________________
BT (X)ee_W(X)op_________________________________*
*__________________________________
i.e. to BN(X); this is the case treated by ClarkEwing [20]. If p divides the o*
*rder of the Weyl
group exactly once, then BX is H*Zpequivalent to the homotopy colimit of a dia*
*gram of the form
___________________________________________________*
*______________________________________op__________________________W(X)@
__W(X)(tS)op=W(X)(tBCXS(tS))op88____________________________________*
*_____________________________BTW(X)(X)ee______________________________@
with just two nodes; this is the cases treated by Aguad'e [2]. In general, BX i*
*s H*Zpequivalent to
the homotopy colimit of a diagram with nodes in onetoone correspondence with *
*the subgroups
of the Sylow psubgroup of W (X). (The objects tP, for P a subgroup of SylpW (X*
*), generate a
skeletal subcategory of AC(p)(W (X), t(X)).)
The decomposition (7.6) is usually only helpful when X is centerless. (Any si*
*mple pcompact
group X for which r0W (X) is not in family 1 of the ClarkEwing list and not eq*
*ual to r0W (E6)
if p = 3, is centerless (3.12.(2), 11.18)).
Conversely, given a finite group W of automorphisms of a Zptorus ~Tsuch that*
* H0(W ; ~T) = 0
and the ring of invariants H*(T~; Fp)W is polynomial, does there exist a polyn*
*omial pcompact
group X(W ) with Zpreflection subgroup (W, ~T) and with mod p cohomology isomo*
*rphic to this
invariant ring? Note that if X(W ) exists, then the Quillen category A(X(W )) *
*= A(W, t), the
maximal torus normalizer ~N(X(W )) = ~To W , and the functor ~NO CX(W), giving *
*the maximal
torus normalizers of the centralizers, is the functor ~N:A(W, t)op! [Grp ]given*
* by
` 1 1 '
N~(E0 wW(E0)!E1) = ~To W (E0) (w,c(w))~To W (E1)
according to the considerations of the proof of (2.8). This means that if BX(E)*
* denotes the value
of BCX(W) on E t then there must exist homotopy commutative diagrams
~N(wW(E0))
(7.7) BN~(E0)oo_________ BN~(E1)
Bj(E0) Bj(E1)
fflffl fflffl
BX(E0) ooX(wW(E0))_BX(E1)
where the vertical arrows are (discrete) maximal torus normalizers.
7.8. Theorem. (Generalized ClarkEwing construction) Let p be an odd prime and *
*(W, ~T) a
polynomial Zpreflection group with H0(W ; ~T) = 0. Suppose that there exist a *
*centric functor [28]
NDETERMINISM 31
BX :A(W, t)op! [pcg]and a natural transformation Bj :BN~ ! BX such that, for e*
*ach non
trivial subgroup E of t, BX(E) is a polynomial pcompact group and Bj(E): BN~(E*
*) ! BX(E)
is a pdiscrete maximal torus normalizer. Then BX determines an essentially un*
*ique functor
BX :A(W, t)op! Top, and H*(BX(W ); Fp) ~=H*(T~; Fp)W as unstable algebras where
^
BX(W ) = hocolimA(W,t)opBX p
is the Fplocalization of the homotopy colimit. X(W ) is a centerless polynomia*
*l pcompact group
whose Weyl group is Fpsimilar to W . If all values of the functor BX are total*
*ly Ndetermined
pcompact groups, then also X(W ) is totally Ndetermined.
Alternatively, the full subcategory ( 2.14) AC(p)(W, t) based on the collecti*
*on C(p) of all p
subgroups of W , can be used for index category.
Proof.For any nontrivial subgroup E of t, the pcompact group BX(E) has pdisc*
*rete center
~Z(X(E)) = Z(N~(E)) = ~T W(E)meaning (3.17.(1)) that (ßjBZX)(E) = H2j(W (E); L*
*(T~)) =
L(T~)2j(E) for j = 1, 2. Since these functors are acyclic (2.4), [28, 1.1] tel*
*ls us that BX lifts, es
sentially uniquely, to a functor taking values in the category of topological s*
*paces. Let BX(W ) be
the (Fplocalization of the) homotopy colimit. The polynomial pcompact group B*
*X(E) has coho
mology H*(BX(E); R) = H*(T~; R)W(E) = H*(T~; R)0(E), R = Fp, Zp. Since this fun*
*ctor is acyclic
(2.4), the BousfieldKan spectral sequence for the cohomology of a homotopy col*
*imit [14, XII.4.5]
collapses onto the vertical axis giving the cohomology of BX(W ) and so H*(BX(W*
* ); Fp) =
H*(T~; Fp)W . As this invariant ring is assumed to be polynomial, X(W ) is inde*
*ed a polynomial
pcompact group. The pcompact group morphism T (GL (n, C)) = CGL(n,C)(t) ! X(W*
* ) is a max
imal torus and rp(W ) = rpW (X(W )) by (7.5). According to [68, 4.9] and (2.11,*
* 3.8), X(W ) is
totally Ndetermined provided all values of the functor BX are totally Ndeterm*
*ined pcompact
groups.
We may replace the index category A(W, t) by any of its full subcategories I *
*as long as
lim1+j(I; L(T~)2j) = 0 = lim2+j(I; L(T~)2j), j = 1, 2, and H*(BT~; Zp)0 is ac*
*yclic on I with
lim0equal to the invariant ring. For instance, I = AC(p)(W, t), where C(p) is t*
*he collection_of all
psubgroups of W is a possibility (2.16). *
* __
In particular, if p divides the order of W exactly once, we may use the full *
*subcategory
A(W, t){t, tS} = I(W, W (tS)) (13.10) generated by the two objects t and tS whe*
*re S = SylpW is
a Sylow psubgroup of W .
The QpWeyl group r0W (X) (4.3) of a connected pcompact group X is a reflect*
*ion subgroup of
Aut(L(X) Qp) [30, 9.7]. If X is simple in the sense that this Weyl group is an *
*irreducible reflection
group then r0W (X) must occur in the ClarkEwing classification table [20]. Th*
*e irreducible
reflection groups of this table are divided into four infinite families, denote*
*d 1, 2a, 2b, and 3, and
34 sporadic reflection groups G4, . .,.G37.
7.9. Theorem. Let p be an odd prime and X a simple pcompact group with Weyl re*
*flection group
(W (X), L(X)). Assume that
or0W (X) is not in family 1,
oif p = 3, then (r0W (X)) 6= (r0W (F4)), (r0W (E6)), (r0W (E7)), (r0W (E8)),*
* and,
oif p = 5, then (r0W (X)) 6= (r0W (E8)).
Then the following hold:
1.X is a centerless, simply connected, totally Ndetermined, polynomial pcom*
*pact group.
2.X is determined by its RWeyl group for R = Zp, Qp, Fp.
3.X is a cohomologically unique pcompact group.
4.End (X) is given by ( 3.17.( 2)).
Proof.A glance at the ClarkEwing classification table [20] (as presented e.g. *
*in [5, Table 1])
reveals that X is either a nonmodular pcompact group, which certainly has the*
* stated properties
(7.3), or one of the modular pcompact groups treated in (7.10) in which case w*
*e apply_(7.8,_5.3)
together with (4.4, 11.18, 11.25.3). *
* __
32 J.M. MØLLER
7.10. Construction of modular, centerless, polynomial, simple pcompact groups.*
* We
apply (7.8) to construct polynomial pcompact groups X(G) where G GL(r, Qp) i*
*s either
oin family 2a,
or0W (G2) at p = 3 from family 2b,
oone of the groups of Aguad'e [2, Table 1], or,
or0W (E6) at p = 5.
There is no ambiguity in pretending that G be a subgroup of Aut(T~) = GL(r, Zp)*
* since G0(G) = *
in each of these cases (11.18). The rings of invariants H*(T~; R)G, R = Fp, Zp,*
* are polynomial
rings (7.4), and from [5, 3.4] we have that H0(G; ~T) = 0. Thus it suffices to *
*find a functor BX
that satisfies the conditions of (7.8).
Family 2a. (Cf. [76]) Let p be an odd prime and r 1, m 2, n 2 natural n*
*umbers
such that rmp  1. Let Cm Zxpbe the order m cyclic subgroup of the padic u*
*nits. Define
G(m, r, n) = A(m, r, n) n as the subgroup of GL (n, Zp) = Aut(T~), ~T= ~T(U(n))*
*, generated by
the group W (U(n)) = n of monomial matrices and the abelian group A(m, r, n) o*
*f diagonal
matrices with entries in Cm and determinant in the index r subgroup of Cm . (*
*For instance,
G(2, 1, n) = W (SO (2n+1)) and G(2, 2, n) = W (SO (2n)).) The subgroup n norma*
*lizes A(m, r, n)
and G(m, r, n) = A(m, r, n) o n is in fact the semidirect product of the two *
*groups. The ring of
invariants [71, 2.4] [88, x7.4, Example 1]
H*(T~; Zp)G(m,r,n)= Zp[y1, . .,.yn1, e], yi = 2im, e = 2m_rn,
is generated by e = (x1. .x.n)m_rtogether with the n  1 first elementary symme*
*tric polyno
mials yi = oei(xm1, . .,.xmn), 1 i n  1, in the mth powers of the coordina*
*te functions
xi:H2(T~; Zp) ! Zp, 1 i n, which are considered as having degree 2.
Define AC(p)(G, t) where G = G(m, r, n) or G = n, t = t(U(n)), to be the ful*
*l subcategory
of A(G, t) generated by all objects of the form E = tP for P Sylp n = SylpG(m*
*, r, n) a
subgroup of a Sylow psubgroup of n (which is also a Sylow psubgroup of G(m, *
*r, n)). These two
small categories have by definition the same set of objects E, with the same po*
*intwise stabilizer
subgroups G(m, r, n)(E) = n(E), and for the morphism sets (2.1) we note that
__ (E ) __
G (m, r, n)(E0, E1) = A(m, r, n) n 1 x n(E0, E1)
meaning that any morphism (a, oe): E0 !iE1n AC(p)(G(m, r, n), t)(E0, E1) factor*
*s uniquely as a
morphism oe :E0 ! E1in AC(p)( n, t)(E0, E1) followed by multiplication a: E1 ! *
*E1by a diagonal
matrix a 2 A(m, r, n) n(E1)= A(m, r, n)0(E1). To see this, it is convenient to *
*observe that that
all objects E = tP of AC(p)(G, t) are of the special form
E = {(x1, . .,.xn) 2 Fnp xi= xj iff i and j are n(E)equivalent}
for some partition n(E) of n = {1, . .,.n} into disjoint subsets. (Thus AC(p)(G*
*(m, r, n), t) can be
viewed as the Grothendieck construction on the functor A(m, r, n)0 from AC(p)( *
*n, t) to categories
with one object.)
We now define the functor BX :AC(p)(G(m, r, n), t)op! [pcg]which shall serve *
*as input for the
generalized ClarkEwing construction (7.8). On objects E = tP t = t(U(n)) we *
*are forced to put
BX(E) = BCU(n)(E) for the pointwise stabilizer group G(m, r, n)(E) = n(E) = W*
* (CU(n)(E))
and CU(n)(E), a product of unitary groups [82, Proposition 4], is determined by*
* its ZpWeyl group
(5.3). For each morphism E0 oe!E1 a!E1 in AC(p)(G(m, r, n), t) we are require*
*d (7.7) to fill in
the commutative diagram
1)) (a1,1)
T~o n(E0)o(1,c(oeo~To_ n(E1)oo______T~o_ n(E1)
  
  
fflffl fflffl fflffl
CU(n)(E0)oo````` ` CU(n)(E1)oo````` `CU(n)(E1)
of pcompact groups with discrete maximal torus normalizers. To the left we may*
* put the value
BCU(n)(oe): BCU(n)(E0) BCU(n)(E1) on E0 oe!E1of the functor BCU(n):AC(p)( n,*
* t) ! [pcg].
NDETERMINISM 33
To the right there is just one possibility, denoted _a1, for CU(n)(E1) have N*
*determined auto
morphisms (5.3). This prompts us to declare
oe a BCU(n)(oe) B_a1
BX E0 !E1 !~=E1 = BCU(n)(E0)  BCU(n)(E1) ~=BCU(n)(E1)
However, for this to be a valid_definition of a functor we need to verify that *
*the relation ø O a =
øaø1 O ø, a 2 n(E0), ø 2 n(E0, E1), which holds in AC(p)(G(m, r, n), t), als*
*o holds in [pcg],
i.e. that the diagram
a1
CU(n)(E0)oo______CU(n)(E0)OOOO
CU(n)(ø) CU(n)(ø)
 
CU(n)(E1)oo_1_1CU(n)(E1)_
_øa ø
commutes in [pcg]. This is not difficult as _øa1ø1and the isomorphism induce*
*d by _a1 on
CU(n)(E1) have the same effect on the maximal torus normalizer, so are identica*
*l. Thus the above
definition indeed makes BX into a functor. BX is clearly a centric functor beca*
*use BCU(n)is and
we conclude from (7.8) that there exists a centerless, polynomial, totally Nde*
*termined pcompact
group XG(m, r, n) with reflection subgroup Fpsimilar, and hence even Zpsimila*
*r (11.25), to
(G(m, r, n), ~T).
For future reference, we now compute the centralizer of an arbitrary nontriv*
*ial subgroup E of
t = t(XG(m, r, n)). Suppose that E has rank r > 0. Choose an (n x r)matrix B w*
*hose columns
form a basis for E. Declare i and j to be equivalent if the ith and jth rows in*
* B are Cm multiplies
of each other, 1 i, j n. Let n(E) denote the partition of n = {1, 2, . .,.n*
*} into equivalence
classes. If there is a zerorow in B, call the corresponding equivalence class *
*the nullclass. Suppose
that the nullclass contains u0 0 elements and that there are s 1 more clas*
*ses containing
u1, . .,.us elements, respectively.
The following lemma, describing the pointwise stabilizer subgroup G(m, r, n)*
*(E), implies that
the equivalence relation n(E) does not depend on the choice of basis.
7.11. Lemma. The pointwise stabilizer G(m, r, n)(E) of E is isomorphic to the *
*subgroup
G(m, r, u0) x u1x . . .us
where the reflection subgroup ( uj, Zujp) is similar to (W, L)(U(uj)), 1 j *
*s.
Proof.The element (a, oe) 2 A(m, r, n) o n stabilizes E pointwise if and only*
* if aiBoe(i)= Bi
where Bi, i = 1, . .,.n are the rows of the matrix B. This implies that the per*
*mutation oe of the
n rows of B must respect the Cm equivalence classes. Therefore the group homom*
*orphism
G(m, r, u0) x u1x . .!.utG(m, r, n)(E)
(b, ø), oe1, . .,.oes)! (b, a1(oe1), . .,.aj(oej)), øoe1. *
*.o.es
where aj(oej)iBoej(i)=QBi, 1 j s, is an isomorphism. Observe in this conne*
*ction that the
product aj(oej)i = 1; indeed, for fixed j, the product over all aj(oej)i, whe*
*re i runs through
the elements of a cycle in the decomposition of oej, equals 1. Conjugate this a*
*ction of uj by the
diagonal matrix consisting of the first nonzero entries in the rows of B to ob*
*tain_the standard
permutation action. *
* __
If we take existence for granted, referring to [76], then the above lemma and*
* (3.8) would suffice
to show inductively that XG(m, r, n) is totally Ndetermined.
The 3compact group G2. Take BX to be the functor on the 2object category A(G,*
* t){tS, t}op=
I(G, W (SU (3)))op, G = W (G2) = W (SU (3)) x Z(G), Z(G) = { 1}, indicated by t*
*he diagram
_____________________________________________________*
*______________________________________________________________________@
Z(G)op__BSU9(3)9__________________________________________*
*________________________BTGopbb_______________________________________@
34 J.M. MØLLER
where Z(G) acts on BSU (3) via the unstable Adams operations _ 1.
The Aguad'e groups. These are the reflection groups
(G12, p = 3), (G29, p = 5), (G31, p = 5), (G34, p = 7), (G36, p = 5), (G36*
*, p = 7), (G37, p = 7)
where the index refers to their numbering on the ClarkEwing list [20]. Since *
*p divides the
order of the Weyl group only once, it suffices to specify the functor BX on the*
* full subcategory
A(G, t){tS, t}op= I(G, G(tS))op= I(G, W (SU (r + 1))op (13.7.6) where r denotes*
* the rank. Take
BX to be the functor indicated by the diagram
______________________________________________________*
*______________________________________________________________________@
(7.12) Z(G)op_BSU9(r9+_1)_________________________________________*
*_________________________BTGopbb______________________________________@
where Z(G), which is cyclic of order 2, 4, 4, 6, 2, 2, 2, acts on BSU (r + 1) v*
*ia unstable Adams
operations. See [4] for more details. (This follows Aguad'e's original construc*
*tion very closely.)
The 5compact group E6. Take BX to be the functor on A(G, t){tS, t}op = I(G, G*
*(tS))op,
G = W (E6), indicated by the diagram
________________________________________________________*
*______________________________________________________________________@
Cop2_BU(5)9x9BU(1)_________________________________________*
*__________________________BTGopbb_____________________________________@
where C2 acts on U(5) x U(1) in some way.
7.13. Automorphisms of X(G(m, r, n)). [21, (2.13)][76, x7] Assume first that A(*
*m, r, n) is a
characteristic subgroup of G(m, r, n). Then
NGL(n,Zp)(G(m, r, n)) = ZxpG(m, 1, n)
because this normalizer is contained in the normalizer of A(m, r, n) which equa*
*ls Zxpo n by the
argument of [82, Lemma 3], and, on the other hand, a diagonal matrix diag(u1, .*
* .,.un) 2 (Zxp)n
normalizes G(m, r, n) if and only if it lies in ZxpA(m, 1, n). Thus (3.12.(3)),
Out(X(G(m, r, n))) ~=ZxpG(m, 1, n)=G(m, r, n) ~=ZxpA(m, 1, n)=A(m, r, *
*n)
is an abelian group and the exact sequence (3.16) has the form
(7.14) 1 ! Zxp=ZG(m, r, n) ! ZxpG(m, 1, n)=G(m, r, n) ! C(r,n)! 1
where ZG(m, r, n), the center of G(m, r, n), is cyclic of order m_r(r, n) and C*
*(r,n)denotes a cyclic
group of order the greatest common divisor (r, n) of r and n.
Choose a primitive (p  1)th root of unity i 2 Zxp, choose integers s and t w*
*ith (r, n) = sr + tn,
and put " = diag(i p1_m, 1, . .,.1) 2 ZxpA(m, 1, n). Then Ä (m, r, n) projects*
* onto a generator of
the cyclic group C(r,n)and the element
p1_tD(r,n) p1_r_E D (r,n) p1_r_E 2 x
i m i , i m(r,n)2 = i , i m(r,n) H (C(r,n); Zp=ZG(m, r,*
* n))
classifies extension (7.14) because i p1_mtA(m, r, n) = "(r,n)A(m, r, n). Cons*
*equently,
p1_tD (r,n) p1_r_E p  1
(7.14) splits, i m 2 i , i m(r,n), _____mt 2 Z((r,n),p1_mr_(r,n))
` f'ifi fi
, (r, n), p__1_mr_(r,fn)ififip_m1t , (r, n)fifi*
*p__1minZ(_r_(r,n))
where we at the final stage observe that _r__(r,n)and t are relatively prime si*
*nce 1 = s__r_(r,n)+ t__n_(r,n).
For instance,i(7.14)jsplits whenever (r, n) = (r, n2) for then (r, n) and _r__(*
*r,n)are relatively prime so
p1 p1
that (r, n), p1_mr_(r,n)= (r, n), ___mclearly divides ___mt. More generally,
______________(p__1)(r,_n)__________niijoj
max x2Z p  1, p1_mt + x (r, n), p1_mr_(r,n)
is the smallest possible order of an exotic automorphism of X(G(m, r, n)) proje*
*cting onto a gener
ator of C(r,n).
7.15. Lemma. [76, x6] A(m, r, n) is a characteristic subgroup of G(m, r, n) if *
*and only if (m, r, n) 62
{(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)}.
NDETERMINISM 35
Proof.For n > 4, A(m, r, n) is the Fitting subgroup of G(m, r, n). (Consult e.g*
*. [85] for general
group theoretic information.) For 2 n 4, Fit(G(m, r, n)) = A(m, r, n) o F *
*where F is a
subgroup of Fit( n) which is elementary abelian of order n. If A(m, r, n) is no*
*t characteristic in
G(m, r, n), it is not characteristic in Fit(G(m, r, n)) and then (7.16)
n = 2:m_rm2m_r(r, 2) so that (m, r) = (2, 1), (2, 2), (4, 2) or (4, 4),
n = 3:m_rm23m_r(r, 3) so that (m, r) = (3, 3),
n = 4:m_rm34mm_r(r, 2) so that (m, r) = (2, 1) or (2, 2).
Among these options, (2, 2, 2) is an illegal choice of parameters, A(4, 4, 2) ~*
*=C4 is the unique cyclic
subgroup of order 4 in G(4, 4, 2) ~=D8, A(2, 1, 4) is the unique elementary abe*
*lian subgroup of
order 16 of G(2, 1, 4), and in the remaining four cases it can be verified that*
* A(m, r,_n) is not
characteristic in G(m, r, n). *
* __
7.16. Lemma. Let A o W be the semidirect product for the action of a finite gr*
*oup W on a
finite abelian group A. If A is not characteristic in A o W , then A divides *
*AoeCW (oe) for some
nontrivial element oe 2 W .
Proof.If A is not characteristic, some automorphism of the semidirect product *
*takes an element
of A to an element (a, oe) where oe 2 W is nontrivial. As automorphisms preser*
*ve centralizers up
to isomorphism, we know that A divides CAoW (a, oe). The exact sequence
1 ! Aoe! CAoW (a, oe) ! CW (oe)
shows that CAoW (a, oe) divides AoeCW (oe). *
* ___
Since G(2, 1, 2) is conjugate to G(4, 4, 2) [21, 2.5], its normalizer was fou*
*nd above. In the
remaining three cases, there are exact sequences (3.16)
1 ! Zxp=ZG(4, 2, 2) ! NGL(2,Zp)(G(4, 2, 2))=G(4, 2, 2) ! 3 ! 1
1 ! Zxp! NGL(3,Zp)(G(3, 3, 3))=G(3, 3, 3) ! A4 ! 1
1 ! Zxp=ZG(2, 2, 4) ! ZxpW (F4)=G(2, 2, 4) ! 3 ! 1
describing the automorphism groups of XG(4, 2, 2), XG(3, 3, 3), and XG(2, 2, 4).
7.17. Automorphisms of other modular polynomial pcompact groups. If W is one o*
*f the
Aguad'e reflection groups or W = W (G2) and p = 3, then (3.17.(2))
End(X(W ))  {0} = Out(X(W )) = Zxp=Z(W )
for NGL(r,Zp)(W ) = ZxpW according to [4, 5.7].
The 3compact group BX(G12) is also denoted BDI2 for, since G12 GL(2, Z3) ma*
*ps isomor
phically onto GL(2, F3) [11, p. 272] [88, 10.7.1], the mod 3 cohomology algebra
H*(BX(G12); F3) = H*(BT~; F3)GL(2,3)= F3[x12, x16]
is the rank 2 mod 3 Dickson algebra [88, 8.1.5]: A polynomial algebra on a gen*
*erator x12 in
degree 12 and a generator x16 = P 1x12 in degree 16. DI2has the potential of c*
*ontaining all
other connected 3compact groups of rank 2 as this is certainly true on the lev*
*el of Weyl groups.
Section 10 elaborates on this aspect of DI2.
7.18. Structure of polynomial pcompact groups. (Cf. [79]) We start by noting t*
*hat polyno
miality of a connected pcompact group is determined by the universal covering *
*pcompact group
and the fundamental group.
7.19. Lemma. Let X be connected pcompact group with universal covering pcompa*
*ct group SX
[58, 3.3] and fundamental group ß1(X). Then
1.X is polynomial, SX is polynomial and ß1(X) is a free Zpmodule.
2.If X is polynomial, then H*(BX; R) ! H*(BSX; R) is surjective and the kerne*
*l is the ideal
generated by the degree 2 cohomology classes, R = Fp, Zp, Qp.
3.If X is polynomial, then H*(BX; R) ~=H*(ß2(BX), 2; R) H*(BSX; R) as grade*
*d algebras,
R = Fp, Zp, Qp.
36 J.M. MØLLER
Proof.If X is polynomial, H*(BX; Zp) is (7.1) concentrated in even degrees and *
*is degreewise
free so that, in particular, the second homology group H2(BX; Zp) = ß2(BX) is a*
* free Zpmodule.
The Serre spectral sequence for the Postnikov fibration K(ß1(X), 1) ! BSX ! BX *
*collapses at
the E3page to yield
E3 = R H*(ß2(BX),2;R)H*(BX; R) = H*(BSX; R).
Conversely, if SX is polynomial and ß1(X) is free, then the Serre spectral sequ*
*ence for the fibra
tion BSX ! BX ! K(ß2(BX), 2) collapses at the E2page for degree reasons and sh*
*ows that __
H*(BX; Fp) is concentrated in even degrees and is degreewise free. *
* __
Q
Let Y = Yibe a product of finitely many simple, simplyQconnected pcompact *
*groups Yi, ß a
(finite) subgroup of the pdiscrete center ~Z(Y ) = ~Z(Yi) of Y , and ': ß ! *
*~Sa homomorphism
into the discrete approximation ~Sto a pcompact torus S. Define [67, x2] the p*
*compact group
X = Y x S=(ß, ')
by the short exact sequence ß (incl,')!Y x S ! X. Any connected pcompact g*
*roup has this form
with Y = SX and S = Z(X)0 [58, 5.4].
7.20. Corollary.X = Y x S=(ß, ') is polynomial if and only if ': ß ! ~SisQa mon*
*omorphism
and each simple factor Yi in the universal covering pcompact group Y = Yi eq*
*uals SU(n) for
some n or is one of the pcompact groups from ( 7.9).
Proof.We use criterion (7.19.1). Elementary homological algebra performed on th*
*e short exact
sequence 0 ! ß1(S) ! ß1(X) ! ß ! 0 shows that ß1(X) is a free Zpmodule if and *
*only
if ': ß ! ~Sis injective. (OneQmay also use the functor ~Tof x11 to see this.)*
* The universal
covering pcompact group Y = Yi is polynomial iff each simple factor Yi is po*
*lynomial (7.28).
According to (7.9) and Borel [9, 2.5], Yi is polynomial iff Yi = SU(n) for some*
* n or r0W (Yi) 6=_
r0W (F4), r0W (E6), r0W (E7), r0W (E8) if p = 3 and r0W (Yi) 6= r0W (E8) if p =*
* 5. __
In greaterQdetail, any polynomial pcompact group X is of the form X = X1 x X*
*2 where
X1 = SU (ni) x S =(ß, ') is a polynomial pcompact group whose universal cov*
*ering is a
product of special unitary pcompact groups and X2 is a product of some of the *
*simple, simply
connected, centerless, polynomial pcompact groups of (7.9).
7.21. Corollary.All polynomial pcompact groups are totally Ndetermined.
Q
Proof.Since all simple factors of P X = P Y = P Yiare totally Ndetermined (7*
*.20, 5.2,_7.9), X
is totally Ndetermined (3.3, 3.7). *
* __
7.22. Corollary.Let Y be a simply connected, polynomial pcompact group. Then
1.Y is determined by its FpWeyl group, and,
2.BY is cohomologically unique among Fplocal spaces.
Proof.Since Y is totally Ndetermined (7.21), it suffices (4.4, 4.5) to show th*
*at the cohomology
groupQH1(W (Y ); Hom(t(Y ), t(Y ))) isQtrivial.Q But that is proved by Notbohm *
*in [79, 6.2]: Let
Y = Yi be as in (7.20). Let S = Si W (Yi) be the product subgroup wit*
*h factors
Si = W (Yi) in case Yi = SU(n) and Si = SylpW (Yi) in case Yi is one of the pc*
*ompact groups
from (7.9). Then W (Y ): S is prime to p. The natural homomorphism CY (t(YQ*
*)S) ! Y is a
monomorphism of maximal rank [31, 4.3], and, by inspection, CY (t(Y )S) = CYi*
*(t(Yi)Si) is
isomorphic to a product of SU(n)s and U(n)s. Thus H1(W (t(Y )S); Hom(t(Y ), t(Y*
* ))) = 0 by [74,
8.2]. But then also the cohomology group H1(W (Y ); Hom(t(Y ), t(Y ))) = 0 by a*
* transfer argument_
because W (CY (t(Y )S)) = W (t(Y )S) S has index prime to p in W (Y ). *
* __
7.23. Corollary.Any polynomial pcompact group is determined up to local isomor*
*phism by its
mod p cohomology algebra considered as an unstable algebra over the Steenrod al*
*gebra.
Proof.The mod p cohomological dimension as well as H*(BSX; Fp) (7.19), and henc*
*e (7.22)
BSX, can be read off from H*(BX; Fp) if this is a polynomial algebra. But this*
* information_
is the local isomorphism class of X [67, 2.6]. *
* __
NDETERMINISM 37
Two locally isomorphic pcompact groups, X1 = Y x S=(ß1, '1) and X2 = Y x S=(*
*ß2, '2) are
isomorphic iff there exist automorphisms g 2 Out(Y ) and h 2 Out(S) = Aut(S~) s*
*uch that the
diagram
~Z(Y )oooß1o_//'1//_~S

Z~(g~=) ~= ~=h
fflffl fflfflfflffl
~Z(Y )oooß2o_//'2//_~S
commutes [67, 4.3, 4.5].
The next example shows that there are locally isomorphic but nonisomorphic p*
*olynomial p
compact groups with isomorphic mod p cohomology algebras.
7.24. Lemma. Let X = Y x S=(ß, ') be a polynomial pcompact group. If ß pZ~(*
*Y ), then
W (X) and W (Y x S) are Fpsimilar so that H*(BX; Fp) and H*(BY x BS; Fp) are i*
*somorphic
unstable polynomial algebras.
Proof.We may assume that ß, and hence S, is nontrivial as otherwise there is n*
*othing to prove.
From the short exact sequence 0 ! ß ! ~T(Y ) x ~T(S) ! ~T(X) ! 0 we get the exa*
*ct sequence
0 ! H1(ß; Fp) ! H2(T~(X); Fp) ! H2(T~(Y ) x ~T(S); Fp) ! H2(ß; Fp) !*
* 0
of FpW (X)modules.
The map induced by ': ß ! ~S, Ext(T~(S), Fp) = H2(T~(S); Fp) ! H2(ß; Fp) = Ex*
*t(ß, Fp),
is surjective since ' is injective. This implies that the FpW (X)module homom*
*orphism onto
H2(ß; Fp) has a right inverse.
We next show that the FpW (X)module homomorphism out of H1(ß; Fp) has a left*
* inverse.
Write K for the kernel of the map onto H2(ß; Fp) and apply H0(W (X); ) to the *
*short exact
sequence 0 ! H1(ß) ! H2(T~(X)) ! K ! 0 to get the exact sequence
H1(W (X); H2(T~(X))) ! H1(W (X); K) ! H1(ß) ! H0(W (X); H2(T~(X)))
where H1(W (X); K) ~=H1(W (X); H2(T~(Y ) x ~T(S))) ~=H1(W (X); H2(T~(Y ))) sinc*
*e the homol
ogy groups H1(W (X); H2(ß)), H2(W (X); H2(ß)), and H1(W (X); H2(T~(S))) are tri*
*vial [5, 3.2] [74,
3.1]. It suffices to show that the map into H0(W (X); H2(T~(X))) is injective o*
*r, by exactness, that
the map out of H1(W (X); H2(T~(X))) is surjective. The maps ~Z(Y ) ! ~Z(X) = ~Z*
*(Y )xS=(ß, ') !
~Z(Y )=ß induce dual maps Hom (Z~(Y )=ß, Z=p1 ) ! Hom (Z~(X), Z=p1 ) ! Hom (Z~(*
*Y ), Z=p1 )
whose composition is surjective under the assumption of the lemma that ß pZ~(*
*Y ). Thus
the second of these maps, which by (11.17) can be identified to the first map i*
*n the above exact
sequence, must be an epimorphism, too.
We now conclude that
1 2
H2(T~(X); Fp)= H1(ß; Fp) + cokerH (ß; Fp) ! H (T~(X); Fp)
2 2
= H2(ß; Fp) + kerH (T~(Y ) x ~T(S); Fp) ! H (ß; Fp)
= H2(T~(Y ) x ~T(S); Fp)
as FpW (X)modules and therefore the two rings of invariants,
H*(BX; Fp) ~=H*(T~(X); Fp)W(X) ~=H*(T~(Y ) x ~T(S); Fp)W(X) ~=H*(BY x BS;*
* Fp),
are isomorphic unstable algebras over the mod p Steenrod algebra. *
* ___
7.25. Example.[98] [74, 9.6] The pcompact groups
Ui= SU(p ) x U(1)=(Z=pi, incl), 0 i , 3,
from the local isomorphism system of U(p ) (11.28) are distinct, polynomial (7.*
*19) pcompact
groups but (7.24) the Weyl groups W (Ui) are Fpsimilar and the unstable algebr*
*as H*(BUi; Fp)
are isomorphic for 0 i < (and distinct from H*(BU(p ); Fp)). Thus the polyn*
*omial pcompact
group SU(p ) x U(1) is not determined by its FpWeyl group, not even by its mod*
* p cohomology
algebra.
An unstable graded algebra over the mod p Steenrod algebra is
38 J.M. MØLLER
opolynomial if its underlying graded algebra over Fp is polynomial on finite*
*ly many generators,
otopologically realizable if it is isomorphic to the mod p cohomology of a t*
*opological space.
Steenrod's problem [90] asks for the determination of all topologically realiza*
*ble polynomial alge
bras over the the mod p Steenrod algebra. A complete solution was found by D. N*
*otbohm [76, 79]
but it may still bePworthwhile to record also the following form of the answer.
Write P (H, t) = (dimkHi)tifor the Poincar'e series of the graded algebra H*
* over the field k.
7.26. Theorem. A polynomial unstable algebra over the mod p Steenrod algebra is*
* topologically
realizable if and only if it is isomorphic to H*(BX; Fp) = H*(T~(X); Fp)W(X) fo*
*r some polynomial
pcompact group X (as described in ( 7.20)). Moreover, the following conditions*
* are equivalent
1.(W, ~T) is the Weyl group of a polynomial pcompact group,
2.(W, ST~) is polynomial and H0(W ; L(T~)) is a free Zpmodule,
3.(W, ~T) is polynomial and H0(W ; L(T~)) is a free Zpmodule,
4.(W, ~T) is polynomial and H1(W ; ~T) = 0,
5.(W, ~T) is polynomial and P (H*(T~; Fp)W , t) = P (H*(T~; Qp)W , t),
6.H*(T~; Zp)W is polynomial,
for any Zpreflection group (W, ~T).
Proof.If BX is an Fplocal space and H*(BX; Fp) is polynomial, then X is a poly*
*nomial p
compact group and H*(BX; Zp) = H*(T~(X); Zp)W(X) is a polynomial ring (7.1). T*
*his proves
1) 6, and 6, 5by (7.27); we proceed to show 5) 4) 3) 2) 1.
If item 5 holds, then H1(W ; Hj(T~; Zp)) = 0 for all degrees j 0 (7.27). Fo*
*r j = 1, in particular,
H1(W ; L(T~)_) = 0 which, for general reasons (11.11.1, 11.8.89), is equivalen*
*t to H1(W ; ~T) = 0
or to H0(W ; L(T~)) being a free Zpmodule. From the (split) short exact sequen*
*ce 0 ! ST~! ~T!
H0(W ; ~T) ! 0 of ZpW tori (11.8.10) we get an epimorphism H2(T~; Fp) i H2(ST~*
*; Fp) of FpW 
modules and therefore [71, 4.1] (W, ST~) is polynomial. In the splitting (11.15*
*) of (W, ST~) into a
product of simple Zpreflection groups (Wi, ~Ti) with H0(Wi; L(T~i)) = 0, each *
*factor is polynomial
(7.28), i.e. not similar to the Weyl groups of F4, E68at p = 3 and E8at p = 5 *
*(7.4). Thus all simple
factors of (W, ST~) are Weyl groups of simple, simply connected, polynomial pc*
*ompact groups,
(W, ST~) is the Weyl group of the product Y of these, and (W, ~T), where ~T= (S*
*T~x ~S)=(ß, '), is
the Weyl group of the pcompact group X = (Y x S)=(ß, '), BS = (BS~)^p, which i*
*s_polynomial_
by (7.20). *
*__
The map
æ oe æ oe
Isomorphism classes of (W,T~) Similarity classes of polynomial
polynomial pcompact groups! Zpreflection groups with H1 = 0
is surjective by (7.26.(4)) and injective by (7.21).
7.27. Lemma. Let (W, ~T) be a Zpreflection group. Then H*(T~; Zp)W is polynom*
*ial if and only
if (W, ~T) is polynomial and P (H*(T~; Fp)W , t) = P (H*(T~; Qp)W , t). If thi*
*s is the case, then
H*(T~; Zp)W Z=p ~=H*(T~; Fp)W and H1(W ; H*(T~; Zp)) = 0.
Proof.The Poincar'e series condition ensures that the monomorphism of H*(T~; Zp*
*)W Z=p to
H*(T~; Fp)W is an isomorphism. Now, if the Poincar'e series condition is satisf*
*ied, and H*(T~; Zp)W
Z=p = H*(T~; Fp)W is polynomial, then there exist homogeneous elements x1, . .x*
*.r2 H*(T~; Zp)W ,
where r is the rank of ~T, that reduced mod p become polynomial generators for *
*H*(T~; Fp)W . Thus
the ring homomorphism Zp[x1, . .,.xr] ! H*(T~; Zp)W becomes an isomorphism mod *
*p and hence
it is an isomorphism by Nakayama's lemma. Conversely, if H*(T~; Zp)W is polyno*
*mial, then [78,
2.3, 2.4] [77] the polynomiality condition from [11, Ch 5, x5, Exercice 5] [88,*
* 5.5.4, 5.5.5] can be
used to show that H*(T~; Fp)W is polynomial. The last assertion of the lemma f*
*ollows from the
exact sequence
. ...p!H*(T~; Zp)W i H*(T~; Fp)W 0!H1(W ; H*(T~; Zp)) .p!H1(W ; H*(T~;*
* Zp)) ! . . .
where H1(W ; Hj(T~; Zp)) is a finite Zpmodule for fixed degree j. *
* ___
NDETERMINISM 39
7.28. Lemma. Let A and B be finitely generated graded algebras over a field k. *
*If A k B is a
graded polynomial ring over k on homogeneous generators of positive degree, the*
*n both factors A
and B are polynomial.
Proof.Since A kB is free over A (a kbasis for B provides an Abasis for A kB*
*) and the global
dimension of the polynomial algebra A k B is finite by Hilbert's syzygy theore*
*m [7, 4.2.3], the
global dimension of A is also finite. Thus A is polynomial by Serre's converse_*
*[7,_6.2.3] to Hilbert's
theorem. _*
*_
8.Proofs of Theorem 1.2 and Corollaries 1.31.6
This small section contains the proofs of the results stated in the introduct*
*ion.
Proofs of Theorem 1.2, Corollary 1.3, and CorollaryT1.5.he classification of Th*
*eorem 1.2 is the
content of (5.2, 6.1, 7.9). To obtain Corollary 1.3 and Corollary 1.5, combine *
*this with_(4.4, 4.10,
11.18, 11.25) and (3.3, 3.7, 3.12.(3)). *
* __
Proof of Corollary 1.4.Two connected pcompact groups with similar Zpreflectio*
*n groups have
isomorphic maximal torus normalizers (3.12.(1)), so are isomorphic (1.2). Thus *
*the map (W, L) is
injective.
To prove that the map (W, L) is surjective, let (W, L) be any Zpreflection g*
*roup. Then L sits
as the kernel of a short exact sequence (11.5)
0 ! L ! LH0(W ; ~T) x P L ! ~ß(L) ! 0
of ZpW modules. Choose a pcompact torus S and a centerless pcompact group P *
*X such that
S x P X realizes the Zpreflection group in the middle. This is possible since*
* P L is a product
(11.15) of simple Zpreflection groups, each of which is realizable (7.10). The*
* Zpreflection group
(W, L) is now realized by a covering pcompact group of S x P X (4.10). *
* __
The expression for the automorphism group of X is (3.12.(3), 3.15). *
* __
Proof of Corollary 1.6.Observe that that the maximal torus normalizers [59, 1.3*
*] for X and G
become homotopy equivalent after fibrewise completion away from the prime 2. T*
*his is because
the maximal torus normalizers of the associated pcompact groups split [4] when*
* p is odd, cf. [59,
Proof of Proposition 5.5]. Thus there exists a space (BN)[1_2] and rational equ*
*ivalences
(BX)[1_2] (BN)[1_2] ! (BG)[1_2]
that pcomplete to maximal torus normalizers for the pcompact groups (BX)^pand*
* (BG)^pat each
odd prime p. In this situation, Ndeterminism of the pcompact group (BG)^pand *
*the Arithmetic
Square [14, VI.8.1], ensure the existence of a homotopy equivalence (BG)[1_2] '*
* (BX)[1_2]_of spaces
localized away from 2. *
* __
Within the framework of this paper, it easily can be shown (see remark below *
*(2.13)) that also
the simple pcompact group (E8, p = 5) [94] is totally Ndetermined, determined*
* by its RWeyl
group for R = Zp, Qp, Fp, and is a cohomologically unique pcompact group (4.4,*
* 11.18, 11.25).
However, more information is needed for the other members of the Efamily [6]. *
*If this program
goes through we can remove the exceptions from Theorem 1.2, and it will then fo*
*llow that any
finite family Yi of connected, simple, nonabelian, pairwise nonisomorphic p*
*compact groups
is similarity free and any connected pcompact group is completely reducible in*
* the (provisional)
sense of [64, 3.4, 3.10] when p is odd. Thus for instance [66, 5.2] will apply *
*to all (nonconnected)
pcompact groups G and [64, p. 381] will contain a description of the set "Q (X*
*1, X2) of rational
isomorphisms between any two locally isomorphic pcompact groups, X1 and X2, fo*
*r p odd.
9.Ndeterminism of product pcompact groups
We show in this section that determinacy behaves well with respect to formati*
*on of (finite)
products of pcompact groups.
First two lemmas of a general nature. A pcompact group morphism f :X ! Y is *
*said to be
trivial if Bf :BX ! BY is nullhomotopic.
40 J.M. MØLLER
9.1. Lemma. Let X and Y be pcompact groups and X ! Z(Y ) a pcompact group mor*
*phism
into the center of Y . If the composite morphism X ! Z(Y ) ! Y is trivial, then*
* X ! Z(Y ) is
trivial.
Proof.Turn the center Bz :BZ(Y ) ! BY into a fibration (with fibre Y=Z(Y )) and*
* map BX into
it to obtain the fibration
map(BX, Y=Z(Y )) ! map(BX, BZ(Y ))Bz1(B0)! map(BX, BY )B0
where the total space consists of all maps BX ! BZ(Y ) that composed with Bz be*
*come null
homotopic. With the help of the Sullivan Conjecture for pcompact groups [31, 9*
*.3], the fibre of
this fibration identifies to Y=Z(Y ) and the base to BY . Thus the total space*
* identifies_to the
connected space BZ(Y ) = map(BX, BZ(Y ))B0. This shows the lemma. *
* __
9.2. Lemma. Let f :X ! Ybe a pcompact group morphism and jp:Np(X) ! X the pno*
*rmalizer
[30, 9.8] of the maximal torus of X. Then
1.[32, 5.6] f is a monomorphism, fjp is a monomorphism
2.[67, 6.6] f is trivial, fjp is trivial
If X is connected, this remains true with the pnormalizer replaced by the maxi*
*mal torus.
Proof.Suppose that the restriction fjp of f to the pnormalizer is a monomorphi*
*sm. Then [30,
9.11] H*(BNp(X)) is a finitely generated H*(BY )module via H*(Bfjp). Since H**
*(BX) is a
H*(BY )submodule of H*(BNp(X)) thanks to the transfer homomorphism [31, 9.13] *
*and H*(BY )
a noetherian graded ring [30, 2.4], H*(BX) is a finitely generated H*(BY )modu*
*le via H*(Bf).
The converse follows from the fact that the composition of two monomorphisms is*
* a monomorphism.
If X is connected, any monomorphism of Z=p to X factors through the maximal t*
*orus monomor
phism i: T (X) ! X[30, 4.7, 5.6] [31, 3.11]. This implies that if ~Np(X) ! Np(X*
*) ! X ! Y has
a nontrivial kernel, the same is true for ~T(X) ! T (X) ! X ! Y [30, x7]; here*
*, N~p(X) and
~T(X) are discrete approximations [30, 6.4]. In other words, if T (X) ! X ! Y i*
*s injective, so is
Np(X) ! X ! Y [30, 7.3] [31, 3.5]. *
* __
The second part of the lemma is [67, 6.6, 6.7]. *
* __
We now address Ndeterminism of automorphisms of product pcompact groups. Le*
*t X1 and
X2 be pcompact groups and
ooß1_ _ß2_//_
X1 __'1_//X1x X2o'2oX2_
the natural projections and inclusions.
9.3. Lemma. Let f :X1x X2 ! X1 be a pcompact group morphism such that f'1:X1 !*
* X1 is
an isomorphism and f'2:X2 ! X1 is trivial. Then f is conjugate to f'1ß1.
Proof.We want to show that the adjoint of Bf, BX2 ! map(BX1, BX1)B(f'1), which *
*maps into
a space homotopy equivalent [31, 1.3] to BZ(X1), is nullhomotopic. But this fo*
*llows immediately
from (9.1) since composition with the evaluation monomorphism to BX1 gives the *
*nullhomotopic_
map Bf O B'2. *
*__
9.4. Proposition.Let X1 and X2 be two connected pcompact groups with Ndetermi*
*ned auto
morphisms. Then also the product pcompact group X1x X2 has Ndetermined automo*
*rphisms.
Proof.Let f be an automorphism of X1 x X2 under the product N1 x N2 of the two *
*maximal
torus normalizers. The morphism ß1f'1:X1 ! X1 is an isomorphism for [58, 3.7] [*
*31, 4.7] it is
a rational equivalence [30, 9.7] and a monomorphism (9.2). As also ß1f'2:X2 ! X*
*1 is trivial by
(9.2), it follows from (9.3) that ß1f is conjugate to ß1f'1ß1. Similarly, ß2f i*
*s conjugate to ß2f'2ß2
and thus f is conjugate to the product morphism f1x f2 where f1 = ß1f'1 and f2 *
*= ß2f'2. Thus
N(f) = N1(f1) x N2(f2) and, since X1 and X2 have Ndetermined automorphisms, it*
* follows_that
f1 and f2 are conjugate to identity morphisms. *
* __
Next, we address Ndeterminism of products. This is based on a slight reformu*
*lation of the
Splitting Theorem [32, 6.1] [80].
NDETERMINISM 41
9.5. Theorem. Assume that p is odd. Let X be a connected pcompact~group and i:*
* T ! Xa
maximal torus with normalizer j :N ! X. For any decomposition N =!N1 x N2 of N*
* into a
product of two extended pcompact tori, N1 and N2, there exist pcompact groups*
*, X1 and X2, and
an isomorphism s: X ! X1x X2 such that
~=
N ____//_N1x N2
j j1xj2
fflffl~= fflffl
X _s__//_X1x X2
commutes up to conjugacy where j1:N1 ! X1 and j2:N2 ! X2 are normalizers of max*
*imal tori.
Proof.Write Ni, i = 1, 2, as an extension Ti ! Ni ! Wi of a pcompact torus Ti *
*and a finite
group Wi. Then the Weyl group W = ß0(N) of X is isomorphic to W1x W2 and Wiacts*
* [32, 6.3]
as a reflection group on ß1(Ti) Q. According to [32, 6.1], the splitting ß1(T*
* ) ~=ß1(T1) x ß1(T2)
as a W ~=W1x W2module can be realized by a pcompact group splitting s: X ! X1*
*x X2. This
means that if N(s): N ! N01x N02, where j0i:N0i! Xiis the normalizer of the max*
*imal torus
Ti! Xi, i = 1, 2, is the lift [67, 5.1] of s, then the discrete approximation [*
*31, 3.12] ~N(s) to N(s)
determines an isomorphism
0_______//~T________//~N________//W_______//_1
~T(s) N~(s) W(s)
fflffl fflffl fflffl
0_____//~T1x ~T2_//_N01x N02_//_W1x W2____//_1
of short exact sequences where ~T(s) is the given splitting ~T~=~T1xT~2and W (s*
*) the given splitting
W ~=W1x W2. Relative to the given splitting ~N~=~N1x ~N2, the middle isomorphis*
*m ~N(s) takes
~N1x ~N2isomorphically to ~N01x ~N02. The composite
0
~N1'1!~N1x ~N2~N(s)!~N01x ~N02ß2!~N02,
where '1 is the injection and ß02the projection, can be factored through a homo*
*morphism W1 !
~T2as it restricts to the trivial morphism ~T1! T~2. Since p is assumed to be *
*odd, any such
homomorphism is trivial for the reflection group W1 is generated by elements of*
* order prime to p.
This implies that ~N(s): ~N1x ~N2! ~N01xi~N02s the product of two isomorphisms,*
* ~N1! ~N01and
~N2! ~N02. Let ji, i = 1, 2, be the composite of this isomorphism ~Ni! ~N0iwith*
* ji:N~0i! Xi. ___
The assumption that p should be odd is presumably not essential.
9.6. Proposition.The product of two connected Ndetermined pcompact groups is *
*Ndetermined
when p > 2.
Proof.This is immediate from the commutative diagram
N1x N2
r HHHTTTT
rrr HH TTTTT
rrr 0HHH TTTTT
xxrrr j H$$ ~=T))
X1x X2 X0__s__//X01x X02
where X0 is any pcompact group with maximal torus normalizer j0 and s the spli*
*tting iso
morphism from (9.5). For if X1 and X2 are Ndetermined, we get isomorphisms f1*
*:X1 ! X01
and f2:X2 ! X02under N1 and N2, respectively, and s1 O (f1 x f2) is then an is*
*omorphism_
X1x X2 ! X0 under N1x N2. _*
*_
The next step is to generalize (9.4) and (9.6) to possibly nontrivial extensi*
*ons.
9.7. Theorem. Let Y ! G ! X be a short exact sequence of connected pcompact gr*
*oups.
1.If the adjoint forms P X and P Y have Ndetermined automorphisms, so does G.
2.If the adjoint forms P X and P Y are Ndetermined and p > 2, so is G.
42 J.M. MØLLER
Since a connected pcompact group has Ndetermined automorphisms or is Ndete*
*rmined pro
vided this holds for its adjoint form [68, 4.8, 7.10], the proof of the above t*
*heorem is an immediate
consequence of (9.4, 9.6) and the lemma below.
9.8. Lemma. Let Y ! G ! X be an extension of connected pcompact groups. Then
1.G is locally isomorphic [67, 2.7] to X x Y , and,
2.the adjoint form P G is isomorphic to P X x P Y .
Proof.Let SX denote the universal covering pcompact group and S = Z(X)0 the id*
*entity com
ponent of the center of X. Let Y ! E1 ! SX xS be the extension obtained by pull*
*ing back along
the isogeny [58, 5.4] SX x S ! X. Since [66, 3.2, 3.3, 3.4] the projection of S*
*X x S to S induces a
~=
bijection Ext(S, Y ) ! Ext(SX x S, Y ) of equivalence classes of extensions, t*
*he pcompact group
E1, which is locally isomorphic to G, is isomorphic to SX x E2 for some extensi*
*on Y ! E2 ! S
of the pcompact torus S by Y . By [66, 2.6], E2 is locally isomorphic to S x Y*
* and hence G is
locally isomorphic to SX x S x Y , which is locally isomorphic to X x Y .
Any connected pcompact group has the same adjoint form as its universal cove*
*ring pcompact_
group (4.6). Hence P G ~=P (SX x SY ) ~=P X x P Y . *
* __
9.9. Example.Since [66, 3.3, 3.4]
Ext(PU (p), SU(p)) = [BPU (p), B2Z(SU (p))] = H2(BPU (p); Z=p) = Hom (Z=p, Z*
*=p) = Z=p
there are p equivalence classes of extensions of PU(p) by SU(p) in the category*
* of pcompact groups.
However, since the local isomorphism system of the pcompact group SU(p) x SU(p*
*) [67, p. 217]
SU (p) x PU(p)X
ffff33f XXX,,X
SU (p) x SU(p)X 3PU(p)3x PU(p)
XXXX++X ffffff
SU(p)o PU(p)
consists of very few pcompact groups, we see from (9.10) that the middle pcom*
*pact group in any
of these extensions must be isomorphic to either the direct product SU(p) x PU(*
*p) or the semi
direct product SU(p) o PU(p) for the conjugation action of PU (p) on SU(p). All*
* the pcompact
groups locally isomorphic to SU(p) x SU(p) are totally Ndetermined. The automo*
*rphism groups
are, for instance,
Out(SU (p) x PU(p)) = Z*px Z*p
Out(SU (p) o PU(p)) ~={(u, v) 2 Z*px Z*p u v mod p} o Z=2
where Z=2 permutes the coordinates. Formulas like these follow from (5.1) in co*
*mbination with
[67, 4.3] and [64, 3.5].
9.10. Lemma. For any short exact sequence Y '!G ß!X of connected pcompact gr*
*oups there
exists a corresponding short exact sequence Z(Y ) ! Z(G) ! Z(X) of centers. In *
*particular, Y
and G have isomorphic centers if X is centerless.
Proof.Let z :Z(Y ) ! Ybe the center of Y . In the commutative diagram
map(BZ(Y ), BY )Bz___//_map(BZ(Y ), BG)B('z)//_map(BZ(Y ), BX)B0
'   '
fflffl fflffl fflffl
BY __________________//_BG__________________//BX
the horizontal lines are fibration sequences and the vertical arrows are evalua*
*tion maps. Note that
the middle arrow is a homotopy equivalence since the outer two arrows are homot*
*opy equivalences.
This shows that 'z :Z(Y ) ! Gis central and by naturality we obtain [30, 8.3] a*
* short exact
sequence
P Y ! G=Z(Y ) ! X
NDETERMINISM 43
of pcompact groups. This extension is equivalent to the trivial extension [66*
*, 3.4] since P Y is
centerless [58, 4.7] [31, 6.3]. Thus G=Z(Y ) ~=P Y x X and
Z(G)=Z(Y ) ~=Z(G=Z(Y )) ~=Z(P Y x X) ~=Z(X)
by [58, 4.6.(4)]. *
* ___
10.Maximal rank subgroups of DI2
This section contains some general theory for monomorphisms of between pcomp*
*act groups and
it is shown that DI2contains essentially unique copies of each of the 3compact*
* groups SU(2) x
SU(2), U(2), Spin(5), SU(3), PU(3), and G2.
Recall from the previous section that BDI2 is the homotopy colimit (at p = 3)*
* of a diagram of
the form
________________________________________________________*
*______________________________________________________________W(SU(3))@
(10.1) (Z=2)op_BSU9(3)9______________________________________________*
*_____________________BTW(SUo(3))pee___________________________________@
where Z=2 acts on BSU (3) as {_ 1} and W , the Weyl group of DI2, is the subgro*
*up of GL(2, Zp)
W = W (DI2) = (3))= E
generated by the Weyl groups of SU(3) and PU(3) or, alternatively, by the matri*
*ces
` ' ` '
oe = 01 11 and ø = 01 10
together with scalar multiplication with 1. The semidirect product
~N(DI2) = ~To W
where ~T= ~T(SU (3)) is (3.12) the discrete approximation to the maximal torus *
*normalizer N(DI2)
for DI2.
We start our investigation of maximal rank subgroups of DI2with some general *
*remarks.
Let X1 and X2 be two connected pcompact groups of the same rank. Let j1:N1 !*
* X1 and
j2:N2 ! X2 be normalizers of maximal tori i1:T1 ! X1and i2:T2 ! X2.
Consider the map [69, 3.11]
(10.2) N :Mono (X1, X2) ! Mono(N1, N2)
that to any conjugacy class of a monomorphism f :X1 ! X2associates the unique c*
*onjugacy class
N(f): N1 ! N2such that
N(f)
N1_____//N2
j1 j2
fflfflfflffl
X1__f__//X2
commutes up to conjugacy. Here, Mono(X1, X2) [BX1, BX2] denotes the set of co*
*njugacy classes
of monomorphisms of X1 into X2 and Mono(N1, N2) denotes the set of conjugacy cl*
*asses of maps
BN1 ! BN2 inducing monomorphisms on ß1 and isomorphisms on ß2. Note that if ~N1*
*! N1 and
~N2! N2 are discrete approximations then
[BN~1, BN~2] = [BN1, BN2]
so that
Mono (N1, N2) = Mono(N~1, ~N2)=N~2
consists of conjugacy classes of monomorphisms of N~1into N~2. For any monomor*
*phism f 2
Mono(X1, X2), we let N~(f) 2 Mono (N~1, ~N2), determined up to conjugacy, denot*
*e any discrete
approximation to N(f).
10.3. Definition.The monomorphism f 2 Mono(X1, X2) is Ndetermined if the subse*
*t N1(N(f))
of Mono(X1, X2) consists of f alone.
Let W1 = ß0(N1) and W2 = ß0(N2) denote the Weyl groups.
44 J.M. MØLLER
10.4. Example.If p  W1, then all monomorphisms are Ndetermined. Indeed, it *
*is not difficult
to see that (10.2) is bijective in this case.
In case X1 = X2, the map (10.2) is the homomorphism N :Out(X1) ! Out(N1) prev*
*iously
encountered. We say that X1 has Ndetermined monomorphisms if this map is injec*
*tive; if X1 is
totally Ndetermined N is a bijection. Note that (10.2) is equivariant in the s*
*ense that there is a
commutative diagram
Mono(X1, X2) x Out(X1)___//_Mono(X1, X2)
NxN  N
fflffl fflffl
Mono (N1, N2) x Out(N1)__//_Mono(N1, N2)
relating group actions on sets of monomorphisms.
Let G and H be groups. Write {G > H} for the set of conjugacy classes of subg*
*roups abstractly
isomorphic to H of G.
10.5. Proposition.Let i: X1 ! X2be a monomorphism between the two pcompact gro*
*ups X1
and X2 of the same rank. Then the Euler characteristic Ø(X2=iX1) = W2:W1 and *
*if
oi is Ndetermined
oX1 is totally Ndetermined
o{N~2> ~N1} is a onepoint set
then the action Mono(X1, X2) x Out(X1) ! Mono(X1, X2) is transitive and all mon*
*omorphisms
of X1 into X2 are Ndetermined.
Proof.The first part is [69, 3.11]. For the second part, note first that for an*
*y ff 2 Out(X1), iff
is an Ndetermined monomorphism. Suppose namely that N(f) = N(iff) = N(i)N(ff) *
*for some
monomorphism f :X1 ! X2. Then N(fff1) = N(f)N(ff)1 = N(i), so fff1 = i and t*
*herefore
f = iff.
Let now f :X1 ! X2be any monomorphism and ~N(f): ~N1! ~N2a representative for*
* the con
jugacy class N(f). Since ~N2contains but a single copy of ~N1up to conjugacy an*
*d X1 is totally
Ndetermined, N~(f) = N~(i)N~(ff) for some automorphism ff of X1. Then N~(f) =*
* N~(iff)_and
f = iff. *
*__
The third condition is satisfied in case N~1= ~T1oW1, N~2= ~T2oW2 are semidi*
*rect products
and the set {W1 > W2} is a onepoint set.
10.6. Definition.For a monomorphism f :Y ! X of pcompact groups, let WX (f) or*
* WX (Y ),
the Weyl group of f, denote the component group of the Weyl space WX (Y ) [32, *
*4.1, 4.3].
10.7. Proposition.Let f :Y ! X be a monomorphism of pcompact groups.
1.If the homomorphism ß0(Z(Y )) ! ß0(CX (Y )) induced by f is surjective, the*
*n the Weyl group
WX (Y ) is the isotropy subgroup Out(Y )f for the action of Out(Y ) on f 2 *
*Mono(Y, X).
2.If f is centric [28], then there is a short exact sequence of loop spaces [*
*30, 3.2] Y ! NX (Y ) !
WX (Y ) where NX (Y ) is the normalizer of f [32, 4.4].
Proof.The monomorphism f determines a fibration
a Bf_
WX (Y ) ! map(BY, BY )Bff! map(BY, BX)Bf
fOff'f
where the components of the total space are indexed by the isotropy subgroup Ou*
*t(Y )f and the
fibre is the Weyl space. The assumptions of the proposition assure that the inc*
*lusion of the fibre
into the total space is a bijection on ß0. If we make the additional assumption*
* that f be centric,
the Weyl space becomes homotopically discrete and the exact sequence of the pro*
*position_is the
one from [32, 4.6] *
* __
10.8. Lemma. Suppose that i: X1 ! X2is a monomorphism and let N(i) 2 Mono(N1, N*
*2) be
the induced monomorphism of normalizers. Then the stabilizer subgroup Out(N1)N(*
*i)of N(i) is
isomorphic to the quotient group NW2(W1)=W1.
NDETERMINISM 45
Proof.Note that there is an epimorphism
NN~2(N~1)=N~1i Aut (N~1)=N~1N(i)= Out(N1)N(i)
given by conjugation by elements of N~2normalizing N~1. This homomorphism is a*
*ctually also
injective, hence bijective, for if conjugation by, say, n2 2 NN~2(N~1) agrees w*
*ith conjugation by
some element n1 2 N~1, then n1 and n2 have the same image in W2, so that n2 bel*
*ongs to N~1.
This follows because the Weyl groups of the connected pcompact groups X1 and X*
*2 are faithfully
represented in their maximal tori. Consequently
Out(N1)N(i)~=NN~2(N~1)=N~1
and this last group is isomorphic to the quotient group NW2(W1)=W1 by the proje*
*ction_~N2iW2.
__
10.9. Proposition.Let i: X1 ! X2 be an Ndetermined monomorphism between the tw*
*o p
compact groups X1 and X2 inducing an epimorphism ß0(Z(X1)) ! ß0(CX2(X1)). Then
WX2(X1) = NW2(W1)=W1
provided X1 is totally Ndetermined.
Proof.The assumptions imply that the Weyl group WX2(X1) is isomorphic to the st*
*abilizer sub
group Out(X1)iwhich again is isomorphic to the stabilizer subgroup Out(N1)N(i)f*
*or the action
Mono(N~1, ~N2)=N~2x Aut(N~1)=N~1! Mono(N~1, ~N2)=N~2
of Out(N1) on N(i) 2 Mono(N1, N2). Now apply (10.8). *
* ___
10.10. Example.By (10.4), the inclusion ~To<ø>æ T~oW is realizable by an Ndet*
*ermined
monomorphism i: U(2) ! DI2. The monomorphism i is centric (because BU(2) = BThZ*
*=2and
the centralizer CU(2)(T ) ~=T ~=CDI2(T )) so (10.5, 10.7, 10.9)
Ø(DI2=U(2)) = 24 and WDI2(U(2)) ~=Z(W )
and Out(U(2)) acts transitively on Mono(U(2), DI2) since {T~oW > ~To<ø>} is a *
*onepoint set.
10.11. Example.Similarly, {W > Z=2xZ=2} is a onepoint set, so there is an esse*
*ntially unique
monomorphism i: SU(2) x SU(2) ! DI2realizing the inclusion of ~To(Z=2xZ=2) into*
* ~ToW . The
monomorphism i is Ndetermined, centric, and
Ø(DI2=SU (2) x SU(2)) = 12 and WDI2(SU (2) x SU(2)) ~=Z(W )
and Out(SU (2) x SU(2)) acts transitively on Mono(SU (2) x SU(2), DI2).
10.12. Example.Also {W > D8} = {D8}, where D8 is the dihedral group of order 8.*
* It follows
that there exists a unique monomorphism i: Spin(5) ! DI2realizing the inclusion*
* ~ToD8æ ~ToW .
This monomorphism is centric (because BSpin(5) and B(T~oD8) are H*F3equivalent*
*), so
Ø(DI2=Spin(5)) = 6 and WDI2(Spin(5)) ~=Z(W )
and Out(Spin(5)) acts transitively on Mono(Spin(5), DI2).
In a situation where a pair of monomorphisms G ! X1 and G ! X2 are given, let*
* us write
mapBG (BX1, BX2)
for the space of maps BX1 ! BX2 under BG up to homotopy.
10.13. Lemma. Let z :Z ! X1be a central monomorphism and i: X1 ! X2any monomorp*
*hism
inducing an isomorphism X1 ~=CX1(z) ! CX2(f O z). Then f induces a homotopy equ*
*ivalence
mapBZ(BX1, BX1) ! mapBZ (BX1, BX2)
of mapping spaces.
46 J.M. MØLLER
Proof.The spaces BCX1(z) = map(BZ, BX1)Bz and BCX2(f O z) = map(BZ, BX2)B(fOz)a*
*re
X1=Zspaces and BCf(Z): BCX1(z) ! BCX2(f O z)is an X1=Zmap inducing a map
mapBZ (BX1, BX1) = BCX1(z)h(X1=Z)! BCX2(f O z)h(X1=Z)= mapBZ (BX1, BX2)
of homotopy fixed point spaces. If Cf(z) is an isomorphism, then this map is a *
*homotopy_equiva
lence. *
*__
This happens for instance for V ! CX (V ) ! X so that
map BV(BCX (V ), BCX (V )) ' mapBV (BCX (V ), BX)
for any connected pcompact group X, any elementary abelian pgroup V , and any*
* monomorphism
V ! X.
10.14. Example.Let i: SU(3) ! DI2denote the monomorphism arising in the constru*
*ction
(10.1) of BDI2 as a homotopy colimit. By (10.13), Bi induces a homotopy equival*
*ence
mapBZ=3(BSU (3), BSU (3)) ! mapBZ=3(BSU (3), BDI2)
where Z=3 ! SU(3) is the center, and thus a bijection
Out+(SU (3)) ! Mono(SU (3), DI2),
where Out+(SU (3)) consists of the unstable Adams operations _u indexed by unit*
*s u 2 Z*3with
u 1 mod 3. We obtain a commutative diagram
~=
Out+(SU (3))________________//Mono(SU (3), DI2)
fflffl
N ~= N
fflffl fflffl
Out(T~oW (SU (3)))=W (SU (3))//_Mono(T~oW (SU (3)), ~ToW )=W
and using (10.8) we see that the kernel of the composition going down and then *
*right is trivial.
Thus i is Ndetermined and [28, 4.2] centric. Consequently,
Ø(DI2=SU (3)) = 8 and WDI2(SU (3)) ~=Z(W ),
Out(SU (3)) acts transitively on Mono(SU (3), DI2), and all monomorphisms of SU*
*(3) into DI2are
Ndetermined.
10.15. Example.Similarly, the monomorphism i: SU(3) ! G2arising in the construc*
*tion (7.12)
of BG2 as a homotopy colimit is Ndetermined and centric. Also, Out(SU (3)) act*
*s transitively on
Mono(SU (3), G2) with stabilizer subgroup WG2(SU (3)) = { E}, and Ø(G2=SU (3)) *
*= 2.
10.16. Example.The inclusions of the maximal torus and of SU(3) into DI2constit*
*ute a homo
topy coherent set of maps out of the centralizer diagram (7.10) for BG2 into BD*
*I2. Observing that
both maps are centric one sees first that the Wojtkowiak obstruction groups van*
*ish according to
(13.7) and next that the resulting map BG2 ! BDI2 is a centric monomorphism rea*
*lizing the inclu
sion ~ToW (G2) ! ~ToW (G2) of maximal torus normalizers. As also {W > W (G2)} =*
* {W (G2)},
we conclude that
Ø(DI2=G2) = 4 and WDI2(G2) ~={1}
and that Out(G2) acts transitively on Mono(G2, DI2).
10.17. Example.BPU (3) is the homotopy colimit of a diagram of the form
_______________________________________________________________*
*______________________________________________________________op______@
SL(V )op_BV<<________________________________________________________*
*_________________Sop//_BN3[op[______________________BTW(PU(3))bb______@
___________________________________________*
*__________________3_\SL(V_)_______________________S3 \W(PU(3))
__________________________________________*
*___________
Z=2
where S3 is a Sylow 3subgroup of SL(V ) and N3 is the 3normalizer of the maxi*
*mal torus. There
is a canonical map BN3 ! BDI2 because N3 is also the 3normalizer of the maxima*
*l torus of DI2.
This map BN3 ! BDI2 is centric and it respects the maps of the above diagram up*
* to homotopy.
NDETERMINISM 47
The obstructions to extending BN3 ! BDI2 to a map BPU (3) ! BDI2 lie in the hig*
*her limits of
the A(PU (3))module
________________________________________________________________*
*______________________________________________________
SL(V )ß*(BT9(DI2)))9_________________________________________________*
*___________________ß*(BZ3)ZZ______________________oo_//_ß*(BTW)(PU(3))@
SL(V )=S3 ______________________________________*
*______________W(PU(3))=S3_____________________________________
______________________________________*
*_________________________
Z=2
which vanish completely (13.7).(We are here implicitly using computations of ma*
*pping spaces like
map(BV, BDI2)Bi = BT (DI2).) Thus there exists a unique homotopy class Bi: BPU *
*(3) ! BDI2
extending the inclusion of the 3normalizer. Also, the restriction of i to the *
*3normalizer of the
maximal torus is a monomorphism, so i itself is a monomorphism (9.2), and i is *
*centric because the
BousfieldKan spectral sequence [14, XI.7.1] for map(BPU (3), BDI2)Bi shows tha*
*t this mapping
space is weakly contractible. As also {T~oW > ~ToW (PU (3))} is a onepoint se*
*t and PU (3) is
totally Ndetermined (5.1), (10.5, 10.7, 10.9) show that
Ø (DI2=PU (3))= 8 and WDI2(PU (3)) = Z(W )
and that the group Out(PU (3)) acts transitively on the set Mono(PU (3), DI2) o*
*f conjugacy classes
of monomorphisms.
In view of [10], which says that any connected, closed subgroup of maximal ra*
*nk of a compact
connected Lie group is the normalizer of its center, this example is somewhat s*
*urprising.
There is no monomorphism of PU(3) into G2 for A(PU (3))(V ) = SL(V ) (5.10) i*
*s too big to be
a subgroup of A(G2)(V ) = 3x Z=2 (7.10). Indeed, no nontrivial compact, connec*
*ted Lie group
admits a proper, centerless subgroup of maximal rank [10].
The next example describes the normalizers of the elementary abelian subgroup*
*s of DI2. Strictly
speaking, these normalizers are not 3compact groups, but rather extended 3com*
*pact groups, in
that their component groups are not 3groups.
We start with a general observation.
10.18. Proposition.Let :V ! X be a monomorphism of an elementary abelian pgr*
*oup V into
a pcompact group X.
1.There is a short exact sequence of groups
1 ! ß0(CX ( )=V ) ! WX ( ) ! A(X)( ) ! 1
where CX ( )=V is the standard quotient [30, 8.3].
2.There is a short exact sequence of loop spaces
CX ( ) ! NX ( ) ! A(X)( )
where NX ( ) is the normalizer of [32, 4.4].
Proof.Assuming B :BV ! BX to be a fibration, consider the induced fibration
a B_
WX ( ) ! map (BV, BV )Bf ! map(BV, BX)B
f2A(X)( )
where the fibre is the Weyl space [32, 4.1] of and the components, each one h*
*omotopy equivalent
to BV , of the total space are indexed by the automorphism group of in the Qu*
*illen category.
The homotopy exact sequences of this fibration and of its subfibration
CX ( )=V ! BV ! BCX ( )
give the exact sequence of groups and show that B(CX ( )=V ) is the regular cov*
*ering space of
BWX ( ) corresponding to the normal subgroup ß0(CX ( )=V ) . WX ( ). Thus there*
* is a pullback
diagram
BCX ( )_______//BNX ( )
 
 
fflffl fflffl
B(CX ( )=V )___//_BWX ( ),
where the horizontal maps are regular covering spaces. *
* ___
48 J.M. MØLLER
10.19. Example.For any monomorphism ~: Z=3 ! DI2there is (10.18) a short exact *
*sequence
of loop spaces
SU(3) ! NDI2(~) ! Z(W )
where Z(W ) ~=Z=2 acts on SU(3) as {_ 1}. Thus
NDI2(~) = SU(3)o Z(W )
where B(SU (3)o Z(W )) denotes the total space of the unique [66, 3.3, 3,7] BSU*
* (3)fibration over
BZ(W ) realizing the given monodromy action. (It is not essential in [66, x3] t*
*hat the component
group ß0(X) be a pgroup.) Since the homotopy fixed point space BZ(SU (3))hZ(W)*
*is contractible,
the inclusion ~ToW (SU (3))æ SU (3) extends uniquely to a short exact sequence *
*morphism
T~oW (SU (3))___//_NT~oW(~)__//_Z(W )
fflffl  
  
fflffl fflffl 
SU (3)________//_NDI2(~)___//_Z(W )
__
where NT~oW(~) = ~ToW(~) = ~To(W (SU (3)) x Z(W )).
For any monomorphism :(Z=3)2 ! DI2there is a short exact sequence of loop s*
*paces
T ! NDI2( ) ! W
so NDI2( ) is an extended pcompact torus with ~ToW as discrete approximation [*
*31, 3.12].
10.20. Example.The normalizers of the 3compact subgroups of DI2are (10.7.2)
NDI2(G2) = G2 and NDI2(X) = Xo Z(W )
for X = U(2), SU(2)xSU (2), Spin(5), SU(3), PU(3) where Z(W ) acts on X as {_ 1*
*}. In each case
there is a unique short exact sequence morphism connecting the normalizer in ~T*
*oW of NX (T )
and the normalizer in DI2of X. For X = PU(3), for instance, the picture is
T~oW (PU (3))___//_NT~oW(T~oW (PU (3)))_//Z(W )
fflffl
  
  
fflffl fflffl 
PU (3)____________//NDI2(PU (3))_____//_Z(W )
where NT~oW(T~oW (PU (3))) = T~o(W (PU (3)) x Z(W )). It seems likely that thi*
*s is another
instance of Ndeterminism.
11.Free Zpmodules and pdiscrete tori
Nearly all material of this section is present, in one form or another, in [7*
*5].
A Zpmodule which is isomorphic to Zrpfor some finite r will be called a Zpl*
*attice and a
Zpmodule which is isomorphic to (Z=p1 )r = (Qp=Zp)r for some finite r will be *
*called a Zptorus.
Let ~Tand L denote the endofunctors of the category Ab of abelian groups gi*
*ven by ~T=
Z=p1  and L = Hom (Z=p1 , ). Then Hom Ab(T~(A), B) = Hom Ab(A, L(B)) so (T~*
*, L) is a
pair of adjoint functors. The left adjoint functor ~Tis right exact, ~Tvanishes*
* on finite Zpmodules,
turns Zplattices into Zptori, and its left derived functor ~T1= Tor(Z=p1 , )*
* preserves finite
Zpmodules and vanishes on Zplattices. The right adjoint functor L is left exa*
*ct, L vanishes on
finite Zpmodules, turns Zptori into Zplattices, and its right derived functo*
*r L1 = Ext(Z=p1 , )
preserves finite Zpmodules and vanishes on Zptori. In symbols:
~T0 ! S ! L ! H ! 0 = 0 ! ~T1(L) ! ~T1(H) ! ~T(S) ! ~T(L) ! ~T(H) ! 0
L 0 ! H ! ~T! ~P! 0 = 0 ! L(H) ! L(T~) ! L(P~) ! L1(H) ! L1(T~) ! 0
where S is a Zplattice, ~Pis a Zptorus, and L, H, and ~Tare Zpmodules. In fa*
*ct the pair (T~, L)
provides adjoint equivalences [14, p. 181] between the full subcategories of (t*
*he underlying abelian
groups of) Zplattices and (the underlying abelian groups of) Zptori.
A ZpW module whose underlying Zpmodule is a Zplattice will be called a ZpW*
* lattice and
ZpW module whose underlying Zpmodule is a Zptorus will be called a ZpW toru*
*s.
NDETERMINISM 49
11.1. Definition.[75, 1.1.4, 1.1.5] For a ZpW lattice L and a Zptorus ~T, put
SL = kerL ! H0(W ; L) P L = L(P ~T(L))
0
P ~T= cokerH (W ; ~T) ! ~T ST~= ~T(SL(T~))
In plain language, SL is simply the ZpW submodule of L generated by the unio*
*n of the subsets
(1  w)L, w 2 W , and ~T(P L) is the quotient of ~T(L) by the invariants ~T(L)W*
*for the W action.
We have short exact sequences
(11.2) 0 ! SL ! L ! H0(W ; L) ! 0, 0 ! H0(W ; ~T(L)) ! ~T(L) ! ~T(P L) ! 0
defining SL and P L. (SL could perhaps be called the root lattice and P L the w*
*eight lattice of
L.)
It simplifies matters a great deal to assume that W is generated by elements *
*of order prime to
p (as are Zpreflection subgroups for odd primes p).
11.3. Lemma. Suppose that W is generated by elements of order prime to p. Then *
*H1(W ; H) =
0 = H1(W ; H) for any Zpmodule H with trivial W action.
Proof.Observe that the abelianization H1(W ; Z) is a finite abelian group gener*
*ated by elements_
of order prime to p and apply universal coefficients. *
* __
11.4. Lemma. Let L be a ZpW lattice.
1.The Zpmodule homomorphism H0(W ; L) ! L is split injective and the Zpmodu*
*le homo
morphisms ~TH0(W ; L) ! H0(W ; ~T(L)), H0(W ; L) ! H0(W ; L) are injective.
2.coker H0(W ; L) ! H0(W ; L) is finite.
3.H0(W ; SL) = 0 = H0(W ; ~T(SL)) and H0(W ; SL) is finite. If W is generated*
* by elements
of order prime to p, then H0(W ; SL) = 0.
4.The Zpmodule homomorphism ~T(L) ! H0(W ; ~T(L)) is split surjective and th*
*e Zpmodule
homomorphisms H0(W ; L) ! LH0(W ; ~T(L)), H0(W ; ~T(L)) ! H0(W ; ~T(L)) are*
* surjective.
5.ker H0(W ; ~T(L)) ! H0(W ; ~T(L)) is finite.
6.H0(W ; ~T(P L)) = 0 = H0(W ; P L) and H0(W ; ~T(P L)) is finite. If W is ge*
*nerated by ele
ments of order prime to p, then H0(W ; ~T(P L)) = 0.
7.H0(W ; L) ~=LH0(W ; ~T(L)) and H0(W ; ~T(L)) ~=~TH0(W ; L).
8.H0(W ; L) = 0 , H0(W ; L) is finite, H0(W ; ~T(L)) = 0 , H0(W ; ~T(L)) is f*
*inite.
9.H0(W ; L) = 0 , H0(W ; ~T(L)) = 0 = H1(W ; ~T(L)) , H0(W ; L Zp Z=p) = 0.
10.H0(W ; ~T(L)) = 0 , H0(W ; L) = 0 = H1(W ; L) , H0(W ; Hom(Z=p, ~T(L))) = 0.
Proof.The inclusion H0(W ; L) æ L has a right inverse because its cokernel is a*
* torsionfree,
hence free, Zpmodule. Then also ~TH0(W ; L) ! H0(W ; ~T(L)) ~T(L) is injecti*
*ve by functoriality.
Since the first homology group H1(W ; L=H0(W ; L)) is finite, the long exact co*
*efficient sequence
in homology shows that H0(W ; L) ! H0(W ; L) is injective. The QpW module L *
*Qp contains
H0(W ; L Qp) as a direct summand, so H0(W ; L Qp) H0(W ; L Qp), and it *
*contains
H0(W ; L Qp) as a direct summand, so H0(W ; L Qp) H0(W ; L Qp). Thus t*
*he vector
spaces H0(W ; L) Qp ~=H0(W ; L Qp) and H0(W ; L) Qp ~=H0(W ; L Qp) have*
* the same
dimension. This shows that the cokernel of the monomorphism H0(W ; L) æ H0(W ; *
*L) is finite.
Apply the left exact functor H0(W ; ) to (11.2) and, using item 1, conclude th*
*at H0(W ; SL) = 0.
Apply the right exact functor to (11.2) and conclude that H0(W ; SL) is finite *
*(and, using (11.3),
trivial if W is generated by elements of order prime to p). This proves this f*
*irst three items
and the next three ones are proved in a dual fashion. For item 7, take the shor*
*t exact sequence
0 ! Zp ! Qp ! Z=p1 ! 0 of Zpmodules. Apply H0(W ; ) O (L ) and H0(W ; L) O*
*  to it
and compare the results
H0(W ; L)___//_H0(WO;OL Qp)_//_H0(WO;O~T(L))//_0
 ~  
 =  
  
H0(W ; L)___//_H0(W ; L) _Qp__//_~TH0(W ;_L)_//_0
50 J.M. MØLLER
to see that ~TH0(W ; L) ~=H0(W ; ~T(L)). Dually, compare the values of H0(W ; *
*) O Hom(, ~T(L))
and Hom (, H0(W ; ~T(L))) applied to the same short exact sequence and conclud*
*e that H0(W ; L)
and LH0(W ; ~T(L)) are isomorphic. Combine these isomorphisms with items 2 and *
*5 to obtain
item 8. To get the formulas of items 10 and 9, simply apply the right exact fun*
*ctor H0(W ; ) to
the short exact sequence 0 ! L .p!L ! L Z=p ! 0 and the left exact functor H*
*0(W ; ) to the
short exact sequence 0 ! Hom (Z=p, ~T) ! ~T.p!~T! 0 where L Z=p = Hom (Z=p, *
*~T). ___
From the commutative diagrams with exact rows
0 ____//_SL_________//_L________//_H0(WO;_L)_//_0OhhP
  PP OO
  P P 
 OO PP OO

0 ____//_SL___//_SL x H0(W ;_L)_//_H0(W ;_L)_//_0
0_____//H0(W ; ~T(L))________//~T(L)_________//_~T(P_L)_//_0
k k
 k kk  
fflfflfflffluukkk fflfflfflffl 
0_____//H0(W ; ~T(L))_//H0(W ; ~T(L)) x ~T(P_L)//_~T(P_L)//_0
the Snake Lemma produces exact sequences of ZpW modules
(11.5)0 ! SL x H0(W ; L) ! L ! ß(L) ! 0
0 ! ß(L) ! ~T(SL) x ~TH0(W ; L) ! ~T(L) !*
* 0
0 ! ~ß(L) ! ~T(L) ! H0(W ; ~T(L)) x ~T(P L) ! 0
0 ! L ! LH0(W ; ~T(L)) x P L ! ~ß(L) !*
* 0
where ß(L) and ~ß(L) are the finite groups defined by the short exact sequences
(11.6) 0 ! H0(W ; L) ! H0(W ; L) ! ß(L) ! 0
(11.7) 0 ! ~ß(L) ! H0(W ; ~T(L)) ! H0(W ; ~T(L)) ! 0
of abelian groups. We have thus constructed functors
~s:ZpW  mod ! (ZpW  mod )o o!o s :ZpW  mod ! (ZpW  mod )o!o o
'(L) 0 ~'(L)
~s(L)= ~T(SL) oe ß(L) ! ~TH (W ; L)s(L)= P L i ~ß(L)  LH0(W ; ~T(L))
from the category of ZpW modules into the category of pushout (pullback) dia*
*grams of ZpW 
modules. Since we can recover L from the value of these functors in that colim*
*~s(L) = ~T(L)
and L = lims(L), the classification of ZpW modules has been reduced to the cla*
*ssification of
ZpW modules L with ß(L) = 0 or ~ß(L) = 0.
11.8. Lemma. Let L be a ZpW lattice and assume that W is generated by elements*
* of order
prime to p.
1.ß(P L) = H0(W ; P L), ~ß(SL) = H0(W ; ~T(SL), ~ß(P L) = 0 = ß(SL), and ß(P *
*L) ~=~ß(SL).
2.ß(L) ! H0(W ; ~T(SL)) = ~ß(SL) is injective and ß(P L) = H0(W ; P L) ! ~ß(L*
*) is surjective.
3.0 ! H0(W ; L) ! LH0(W ; ~T(L)) x ß(P L) ! ~ß(L) ! 0 is an exact sequence.
4.0 ! ß(L) ! ~ß(SL) x ~TH0(W ; L) ! H0(W ; ~T(L)) ! 0 is an exact sequence.
5.SSL = SL = SP L and P P L = P L = P SL.
6.ß(L) = 0 , SL x H0(W ; L) = L and ~ß(L) = 0 , L = LH0(W ; ~T(L)) x P L.
7.If H0(W ; L) = 0, then ß(L) = H0(W ; L), ~ß(L) = H0(W ; ~T(L)), and there i*
*s a short exact
sequence 0 ! ß(L) ! ~ß(SL) ! ~ß(L) ! 0.
8.'(L) 2 Hom (ß(L), ~TH0(W ; L)) ~=Ext(ß(L), H0(W ; L)) classifies the above *
*abelian extension
( 11.6) and ~'(L) 2 Hom (LH0(W ; ~T(L)), ~ß(L)) ~=Ext(H0(W ; ~T(L)), ~ß(L))*
* classifies ( 11.7).
9.H0(W ; ~T(L)) ~=coker'(L), H1(W ; ~T(L)) ~=ker'(L), H0(W ; L) ~=ker~'(L), H*
*1(W ; L) ~=
coker~'(L).
10.0 ! H0(W ; L) ! L ! P L ! H1(W ; L) ! 0 and 0 ! H1(W ; ~T(L)) ! ~T(SL) ! ~T*
*(L) !
H0(W ; ~T(L)) ! 0 are exact sequences.
NDETERMINISM 51
Proof.Item 1 is true because H0(W ; P L) = 0 = H0(W ; ~T(P L)) by (11.4.6). For*
* item 2, note that
there is a commutative diagram
(ff,fi)
0____//_ß(L)PP_//~T(SL) x ~TH0(W_;_L)//_~T(L)_//_077n
PPPP  nnnnn
PPPP  nnn+n
PP''Pfflfflnnn
T~(L) x ~T(L)
for some homomorphisms ff and fi. For all elements x of ß(L), ff(x) + fi(x) = *
*0 in ~T(L). If
ff(x) = 0 in ~T(SL), then also ff(x) = 0 in ~T(L) so fi(x) = 0 in ~T(L). But th*
*is means that the
monomorphism (ff, fi) takes x to 0, so x = 0. Thus ff is a monomorphism. Appl*
*y the functor
H0(W ; ) to a short exact sequence from (11.5) to obtain the commutative diagr*
*am
0______//L_________//LH0(W ; ~T(L)) x_P_L_//~ß(L)_//0
  
  
fflffl fflffl 
0___//H0(W ; L)//_LH0(W ; ~T(L)) x H0(W ;_P/L)/_~ß(L)//_0
with a bottom row that is exact according to (11.3). Conclude that SP L = SL. U*
*sing the exact
sequence of item 3 and (11.4.7) we see that there is short exact sequence
0 ! kerH0(W ; L) ! LT~H0(W ; L) ! ß(P L) ! ~ß(L) ! 0
for any ZpW module L. Applied to SL, this gives ß(P L) ~=~ß(SL).
For items 9 and 10 apply the left exact functor H0(W ; ) to one of the short*
* exact sequences
from (11.5) and get the commutative diagram
0________//L_______//LH0(WO;O~T(L))Ox_P_L_______//~ß(L)__________//0OOO
  
  
OO OO ~'(L) OO
0_____//H0(W ;_L)____//LH0(W ; ~T(L))____//ker(~ß(L) ! H1(W ;_L))//_0
using H0(W ; P L) = 0 = H1(W ; P L) (11.4.10). Now apply the Snake Lemma. *
* ___
The group W acts on the dual Zplattice L_ = Hom (L, Zp) according to the rul*
*e (w . ')(x) =
'(w1x), w 2 W , ' 2 L_, x 2 L. The W equivariant duality pairing
(11.9) ~T(L) x L_ ! Z=p1
obtained from the identification L_ = Hom (L, L(Z=p1 ) = Hom (T~(L), Z=p1 ) ind*
*uces pairings
(11.10) H*(W ; ~T(L)) x H*(W ; L_) ! Z=p1 , H*(W ; ~T(L)) x H*(W ; L_) ! Z=*
*p1
relating homology and cohomology groups. (A duality pairing of Zpmodules is a*
* bilinear map
A x B ! C of Zpmodules such that the adjoint homomorphisms A ! Hom Zp(B, C) an*
*d B !
Hom Zp(A, C) are isomorphisms.)
11.11. Lemma. Let L be a ZpW lattice and L_ its dual. Assume that W is generat*
*ed by elements
of order prime to p.
1.The bilinear maps ( 11.10) are duality pairings.
2.S(L_) = (P L)_.
Proof.It is immediate that
H*(W ; L_) = H*(W ; Hom(T~(L), Z=p1 )) ~=Hom(H*(W ; ~T(L)), Z=p1 )
for Hom (, Z=p1 ) is an exact functor. But then also
H*(W ; ~T(L)) ~=Hom(H*(W ; L_), Z=p1 )
52 J.M. MØLLER
because A ~=Hom (Hom (A, Z=p1 ), Z=p1 ) for any Zptorus, Zplattice, or finite*
* Zpmodule A.
Apply the exact functor Hom (, Z=p1 ) to the short exact sequence 0 ! S(L_) ! *
*L_ !
H0(W ; L_) ! 0 to get the short exact sequence
0 ! H0(W ; ~T(L)) ! ~T(L) ! ~T(S(L_)_) ! 0
and conclude that P L = S(L_)_. *
* ___
Suppose that the group W = W1 x . .W.nis the direct product of finitely many *
*of its normal
subgroups W1, . .,.Wn. For j = 1, . .,.n, let
Y
Wj?= Wi
i6=j
T
denote the product of all these subgroups but Wj. Then W = Wj x Wj?and`Wj = i6*
*=jWi?.
Observe that H0(Wi?; L) is a ZpWimodule and also that the direct sum H0(Wi?;*
* L) is a ZpW 
module with a natural ZpW module homomorphism to L given by addition.
11.12. Lemma. [32, 1.5] If H0(W ; ~T(L)) = 0 =`H0(W ; L) for a ZpW lattice L, *
*then there is a
ZpW lattice U and a short exact sequence 0 ! H0(Wi?; L) ! L ! U ! 0 of ZpW *
*lattices.
Each summand H0(Wi?; L) is a ZpWilattice and
oH0(Wi; ~TH0(Wi?; L)) = 0 provided H0(W ; ~T(L)) = 0,
oH0(Wi; H0(Wi?; L)) = 0 provided H0(W ; L) = 0 and each factor group Wi is g*
*enerated by
elements of order prime to p.
Proof.This amounts to showing that the addition maps
a a
H0(Wi?; L) ! L, ~TH0(Wi?; L) ! ~T(L)
are injective. P
Suppose that (xi),Pwith xi 2 H0(Wi?; L), satisfies xi = 0. Then, for an arb*
*itrarily chosen
indexTj, xj =  i6=jxi. The left hand side is fixed by Wj?and the right hand s*
*ide is fixed by
i6=jWi?= Wj. Thus xj is fixed by Wj?x Wj = W , so that xj 2 H0(W ; L). But H0(*
*W ; L) = 0
by (11.4.8). For the other addition map, recall from (11.4.1) that ~TH0(Wi?; L)*
* is contained in
H0(Wi?; ~T(L)) and proceed as above. The computation
H0(Wi; ~T(H0(Wi?; L))) H0(Wi; H0(Wi?; ~T(L))) = H0(Wix Wi?; ~T(L))
= H0(W ; ~T(L))
shows that H0(Wi; ~T(H0(Wi?; L))) = 0 if H0(W ; ~T(L)) = 0. If Wi is generated *
*by elements of
order prime to p, then H1(Wi; ß(H0(Wi?; L)) = 0 so that
H0(Wi; H0(Wi?; L)) H0(Wi; H0(Wi?; L)) = H0(Wix Wi?; L) = H0(W ; L)
proving the final assertion of the lemma. *
* ___
We now specialize to reflection subgroups. If W Aut(L) is a group of automo*
*rphisms of the
Zplattice L, any w 2 W restricts to automorphism Sw of SL and projects to to a*
*n automorphism
P w of P L. If Sw is the identity on SL, then w is the identity on ~T(L) = coli*
*m~s(L) so w is the
identity. If w is a reflection on L, then Sw is a reflection on SL because SL=S*
*L~=L=L. This
means that if W is a reflection subgroup of Aut(L) then also SW (P W ) is a ref*
*lection subgroup
of Aut(SL) (Aut(P L). Thus the Sconstruction and the P construction (11.1) ar*
*e endofunctors
of the category Zp Reflof Zpreflection subgroups (4.1).
We wish to classify the elements of the category Zp  Reflup to similarity. *
*The preceding
general discussion implies the following first reduction of this classification*
* problem.
11.13. Lemma. Let (W1, L1) and (W2, L2) be two objects of Zp Refl. Then the fo*
*llowing three
statements are equivalent:
1.(W1, L1) and (W2, L2) are similar.
NDETERMINISM 53
2.The diagram
~T(SL1)oo_oß(L1)o____//~TH0(W1; L1)
T~(`)~= ~= ~=~T(_*)
fflffl fflffl fflffl
~T(SL2)oo_oß(L2)o____//~TH0(W2; L2)
commutes for some similarity (ff, `): (SW1, SL1) ! (SW2,,SL2)some isomorphi*
*sm between
ß(L1) and ß(L2), and some isomorphism _ :H0(W1; L1) ! H0(W2; L2).
3.The diagram
P L1___////_~ß(L1)LH0(W1;o~T(L1))o_
`~= ~= ~=(T~_)*
fflffl fflffl fflffl
P L2___////_~ß(L2)LH0(W2;o~T(L2))o_
commutes for some similarity (ff, `): (P W1, P L1) ! (P W2,,PsL2)ome isomor*
*phism be
tween ~ß(L1) and ~ß(L2), and some isomorphism _ :H0(W1; ~T(L1)) ! H0(W2; ~T*
*(L2)).
The classification of similarity classes of objects (W, L) of Zp  Reflhas no*
*w been reduced to
the case where ß(L) = 0 or ~ß(L) = 0. Fortunately, this is very easy.
11.14. Theorem. [75] Let (W1, L1) and (W2, L2) be two objects of Zp  Reflwhere*
* p is odd.
Assume that ß(L1) = 0 = ß(L2) or ~ß(L1) = 0 = ~ß(L2) , i = 1, 2. Then (W1, L1) *
*and (W2, L2) are
similar if they are Qpsimilar.
Proof.Assume that (W1, L1) and (W2, L2) are Qpsimilar objects of Zp  Reflwith*
* ~ß(L1) =
0 = ~ß(L2). Since Li = P Lix LH0(Wi, ~T(Li)), i = 1, 2 (11.8.6), it suffices (1*
*1.13) to show that
(P W1, P L1) and (P W2, P L2) are similar. As the splitting constructed in (11.*
*15) below depends
on rational information only, it suffices to prove the theorem under the additi*
*onal hypothesis that
(Wi, Li) be simple. This is done in (11.18) below by going through the ClarkEw*
*ing_classification
table [20]. *
*__
11.15. Lemma. [32, 75] Let (W, L) be an object of ZpRefl where p is odd. If H0*
*(W ; ~T(L)) = 0
(or H0(W ; L) = 0), then
Y
(W, L) = (Wi, Li)
splits as a product of simple objects of Zp Reflwith H0(Wi; ~T(Li)) = 0 (or H0*
*(Wi, Li) = 0) for
all i.
Proof.We shall only consider the case where L = P is a ZpW lattice with H0(W ;*
* ~T(P )) = 0. As
W is a finite reflection`subgroup of Aut(P ) and H0(W ; P ) = 0 (11.4.8), the Q*
*pW module P ZpQp
splits as a direct sum Mi~=P Zp Qp of finitely many irreducible QpW modules*
* M1, . .,.Mn.
Each of these irreducible summands occurs with multiplicity one and carries a n*
*ontrivial W action
[32, p. 280]. Define Wi to be the subgroup of W thatQpointwise fixes j6=iMj so*
* that the action
of Wi is concentrated on the summand Mi. Then W = Wi is is the direct produc*
*t of these
normal subgroups [32, 6.3] and, according to (11.12), P is isomorphic to the di*
*rect sum of the
ZpW lattices H0(Wi?; P ). Observe that each summand H0(Wi?; L) is a ZpWilatti*
*ce and
oWiis a reflection subgroup of AutZp(H0(Wi?; L)),
o(Wi, H0(Wi?; L)) is simple,
oP H0(Wi?; L) = H0(Wi?; L).
Indeed, the first item is implicit in the proof of [32, 6.3], the second item i*
*s clear because the
rationalization H0(Wi?; L) Zp Qp = H0(Wi?; L Zp Qp) = Mi by construction, and*
* the third_
item is contained in (11.12). *
* __
11.16. Lemma. Let (W, L) be a Zpreflection group.
1.(W, L) and (W, L_) are Qpsimilar and ~ß(L) ~=ß(L_).
2.(W, SL) and (W, S(L_)) are Zpsimilar.
3.(W, P L) and (W, P (L_)) are Zpsimilar.
54 J.M. MØLLER
4.(W, SL) and (W, (P L)_) are Zpsimilar.
Proof.For item 2, first note that S(L_) = SP (L_) = S((SL)_) by (11.11.2). But *
*SL is (11.15)
a product of simpleQZpreflection groups (Wi,QLi) with H0(Wi; Li) = 0. So (SL)_*
* is isomorphic
to the product (Wi, L_i) and S((SL)_) = (Wi, S(L_i)). By inspection (of re*
*flection group
family 1 and W (E6) at p = 3), we see that (Wi, Li) and (Wi, S(L_i)) are Zpsim*
*ilar. Thus SL and
S(L_) are Zpsimilar. Moreover, the isomorphisms H0(W ; L_) ~=Hom (H0(W ; ~T(L*
*)), Z=p1 ) ~=
Hom (T~H0(W ; L), Z=p1 ) ~=Hom (H0(W ; L), Zp) from (11.11.1) show that the lat*
*tices H0(W ; L_)
and H0(W ; L) have the same rank. Therefore (W, L) and (W, L_) are Qpsimilar (*
*11.5). Finally,_
(W, SL) ~=(W, P (L_)_) ~=(W, (P L)_) by (11.11.2) again. *
* __
11.17. Lemma. Let (W, L) be a Zpreflection group. Then there are natural group*
* isomorphisms
H0(W ; L_ Z=p) ~=Ext(H0(W ; ~T(L)), Z=p) and H1(W ; L_ Z=p) ~=Hom(H0(W ; ~T*
*(L)), Z=p).
Proof.Using (11.11), we get H0(W ; L_ Z=p) = H0(W ; L_) Z=p = Hom (H0(W ; ~T(L)*
*), Z=p1 )
Z=p = Ext(H0(W ; ~T(L)), Z=p). In the universal coefficient exact sequence
0 ! H1(W ; L_) Z=p ! H1(W ; L_ Z=p) ! Tor(H0(W ; L_), Z=p) ! 0
the term to right identifies to Hom (H0(W ; ~T(L)), Z=p) and the term to the le*
*ft is trivial_because
H1(W ; L_) = Hom (H1(W ; ~T(L)), Z=p1 ) and H1(W ; ~T(L)) = 0 [5, 3.3]. *
* __
Recall that G0(W, L) stands for the set of similarity classes of reflection s*
*ubgroups that are
Qpsimilar to (W, L) (4.1).
Write PpiSU(r + 1) for the quotient SU(r + 1)=Cpiof SU(r + 1) by the central *
*subgroup Cpiof
order pifor 0 i p(r + 1) where p(r + 1) is the highest power of p that di*
*vides r + 1.
11.18. Lemma. Let (W, L) be a simple object of Zp Refl. Then G0(W, L) = * exce*
*pt that
1.G0(W (SU (r + 1))) = {W (PpiSU(r + 1))  0 i p(r + 1)} contains p(r +*
* 1) + 1 elements.
2.G0(W (E6)) = {W (E6), W (PE 6)} contains two elements if p = 3.
Proof.The reflection subgroup rp(W, L) = (W, L Z=p) is irreducible, and hence*
* G0(W ) = *
(4.5.(1)), unless (W, L) is in ClarkEwing family 1 or p = 3 and r0W is r0W (E6*
*) or r0W (G2) [4,
6.2]. All the Lie cases are covered by G. Maxwell [56, Table I]. (See also [22]*
* or [84, 5.1] for the
Afamily.) __
__
We learn from (11.18) that two simple objects, (W1, L1) and (W2, L2), of Zp *
*Reflare similar
if they are Qpsimilar and either ß(L1) ~=ß(L2) or ~ß(L1) ~=~ß(L2). Combined wi*
*th the splitting
result (11.12), this proves (11.14).
Let (W, L) be an object of Zp  Refl. We shall next describe G0(W, L) as a pa*
*rtially ordered
set. For given diagrams
(11.19)ff: ß(P L) = H0(W ; P L) i ~ß LH0(W ; ~T(L))
~ff:~ß(SL) = H0(W ; ~T(SL)) oe ß ! ~TH0(W*
* ; L)
of Zpmodules, put
0
(11.20)Sff(P L) = limP L ! ~ß LH (W ; ~T(L))
~T(P~ff(SL)) = colim~T(SL) ß ! ~TH0(W ; *
*L)
so that ~ß(Sff(P L)) = ~ßand ß(P~ff(SL)) = ß. There are defining short exact se*
*quences
0 ! Sff(P L) ! LH0(W ; ~T(L)) x P L ! ~ß! 0
0 ! SL x H0(W ; L) ! P~ff(SL) ! ß ! 0
of ZpW modules. We have previously (11.5) seen that
Sß(PL)i~ß(L) H0(W;L)(P L) = L = P~ß(SL)oeß(L)!T~H0(W;L)(SL).
NDETERMINISM 55
By naturality, W is a reflection subgroup of Aut(Sff(P L)) and of Aut(P~ff(SL))*
*. Also by naturality,
there are morphisms
Sff(P L) ! Sß(PL)i0 H0(W;L)(P L) = P L x H0(W ; L)
H0(W ; L) x SL = P~ß(SL)oe0!T~H0(W;L)(SL) ! P~f*
*f(SL)
showing that (W, Sff(P L)) and (W, P~ff(SL)) are Qpsimilar to (W, L). Convers*
*ely, any element
of G0(W, L) will have this form because if (W1, L1) and (W2, L2) are Qpsimilar*
* then (SW, SL1)
and (SW, SL2) ((P W, P L1) and (P W, P L2)) are Zpsimilar by (11.14) and clear*
*ly H0(W1; L1) and
H0(W2, L2) are isomorphic Zplattices.
Declare two diagrams of the form considered in (11.19) to be equivalent if th*
*ey can be connected
by an automorphism in AutZpRefl(W, P L) (or AutZpRefl(W, SL)) (4.1) and an au*
*tomorphism
in Aut(H0(W ; L)) as in (11.13).
11.21. Lemma. For any object (W, L) of Zp  Refl, p odd, there is a bijection b*
*etween the
following three sets:
1.G0(W, L).
2.Equivalence classes of diagrams ß(P L) i ~ß LH0(W ; ~T(L)) of Zpmodules.
3.Equivalence classes of diagrams ~ß(SL) oe ß ! ~TH0(W ; L) of Zpmodules.
Since ~ß(SL) ~=ß(P L) is a finite group (11.4.2), G0(W, L) is a finite set. O*
*ur next aim is to
introduce an ordering relation on G0(W, L).
11.22. Lemma. For a ZpRefl morphism (ff, `): (W1, L1) ! (W2,tL2)he following t*
*hree state
ments are equivalent:
1.r0(ff, `): r0(W1, L1) ! r0(W2,iL2)s a similarity in Qp  Refland W2 acts tr*
*ivially on
coker`.
2.S(ff, `): S(W1, L1) ! S(W2,iL2)s a similarity in Zp  Refland the induced m*
*orphism of
Zptori ~T((ff, `)*): ~TH0(W1; L1) ! ~TH0(W2;aL2)n epimorphism with finite *
*cokernel.
3.P (ff, `): P (W1, L1) ! P (W2,iL2)s a similarity in Zp  Refland the induce*
*d morphism of
Zplattices (ff, `)*:H0(W1; L1) ! H0(W2;aL2)monomorphism with finite kernel.
Proof.Assume that L1 ! L2 is injective with finite cokernel H. Then there is a*
* short exact
sequence
0 ! kerH0(W1; L1) ! H0(W2; L2) ! cokerSL1 ! SL2 ! kerH ! H0(W2; H) ! 0
provided by the Snake Lemma. If the middle term is trivial, then H = H0(W2; H).*
* If W2 acts
trivially on H, then the kernel to the left is trivial because H1(W2; H) = 0 by*
* (11.3), and the
kernel to the right is trivial because H = H0(W2; H). The proof for P L1 ! P L2*
*_is completely
dual. _*
*_
11.23. Definition.An isogeny is a ZpReflmorphism (ff, `): (W1, L1) ! (W2,tL2)h*
*at satisfies
one of the three equivalent conditions of ( 11.22).
Write (W1, L1) (W2, L2) if there exists an isogeny (W1, L1) ! (W2, L2).
11.24. Lemma. If (W1, L1) (W2, L2) (W1, L1) then (W1, L1) and (W2, L2) are *
*similar ob
jects of Zp Refl.
Proof.An isogeny (W1, L1) ! (W2, L2) induces a commutative diagram
~T(SL1)oo_oß(L1)o___//_~TH0(W1; L1)
Ø
~= ~= Ø
fflffl fflffl fflfflØ
~T(SL2)oo_oß(L2)o___//_~TH0(W2; L2)
which can be completed [61] by a vertical isomorphism to the right. *
* ___
56 J.M. MØLLER
Thus the relation induces a partial ordering relation on the set of similar*
*ity classes of objects
of Zp Refl; in particular on the set G0(W, L). For any object (W, L),
(W, SL x H0(W ; L)) (W, L) (W, LH0(W ; ~T(L)) x P L)
by (11.5) and actually
G0(W, L) = {(W 0, L0)  (W 0, L0) (W, LH0(W ; ~T(L)) x P L)}
= {(W 0, L0)  (W, SL x H0(W ; L)) (W *
*0, L0)}
is the set of similarity classes of objects above LH0(W ; ~T(L)) x P L or below*
* SL x H0(W ; L).
I close this section with a few remarks about the set Gp(W, L) (4.1).
11.25. Lemma. Let (W, L) be an object of Zp Refl.
1.If (W, L) is simple, then Gp(W, L) G0(W, L).
2.G0(W, L) \ Gp(P W, P L) = * = G0(W, L) \ Gp(SW, SL) if H0(W ; L) = 0.
3.If (W, L) is simple, then Gp(W, L) = * unless (W, L) is similar to (W (X), *
*L(X)) for X =
PpiSU(r + 1)), 0 < i < p(r + 1).
Proof.Gp(W ) G0(W ) when W is simple because any two abstractly isomorphic gr*
*oups from the
ClarkEwing list happen to have the same rank r and to be conjugate as subgroup*
*s of GL(r, Qp)
[4, 2.6]. When H0(W ; L) = 0, P L is the unique object of G0(W, L) with ~ß= 0 *
*(11.21);_this_
condition can (11.8.6) be read off from L Z=p. *
* __
11.26. Example.Put (W, L) = (W, L)(PU (r + 1)) so that ß(L) is cyclic of order *
*p where this
is the highest power of p that divides r + 1. The + 1 elements of G0(W, L) ar*
*e represented by
the centerings (W, Li), 0 i , where Li L is the inverse image of the orde*
*r pi subgroup of
ß(L) (11.21) [84, 5.1]. Thus ß(Li) is cyclic of order piand L = L . Assume now *
*that 0 < i < so
that both ß(Li) and ~ß(Li) are nontrivial cyclic pgroups. As pointed out to m*
*e by D. Notbohm,
tensoring the commutative diagram of ZpW modules with exact rows and columns
0 0
 
 
fflffl fflffl
0 ____//_L0____//Li____//_ß(Li)__//_0
  
  
 fflffl fflffl
0 ____//_L0____//L_____//_ß(L_)__//_0
 
 
fflffl fflffl
~ß(Li)_____~ß(Li)__//_0
 
 
fflffl fflffl
0 0
with Z=p results in the commutative diagram
Tor(~ß(Li), Z=p)__Tor(~ß(Li), Z=p)
 ~
 =
fflffl fflffl
0___//Tor(ß(Li), Z=p)//_L0 Z=p___//Li Z=p______//ß(Li) Z=p__//_0
with a split epimorphism to the right. We conclude that
0
Li Z=p ~=cokerH (W ; L0 Z=p) ! L0 Z=p H0(W ; Li Z=p), 0 < i < ,
as FpW modules. (These modules are irreducible [38] and it is no coincidence [*
*84, 3.3] that Li Z=p
have the same irreducible constituents, namely
0
cokerH (W ; L0 Z=p) ! L0 Z=p ~=kerL Z=p ! H0(W ; L Z=p) andZ=p,
for all i.) This shows that Gp(W (PpiSU(r + 1))) consists of  2 elements for*
* 0 < i < .
NDETERMINISM 57
11.27. Example.(Cf. (9.9)) For (W, L) = (W (X), L(X)), X = SU(p) x SU(p), the s*
*et G0(W, L)
consists of four elements corresponding to the four subgrouporbits under the a*
*ction of the auto
morphism group AutZpRefl(W, P L) = Zxpx Zxp o Z=2 on H0(W ; P L) = Z=p x Z=p.
11.28. Example.G0(W (X), L(X)) for X = U(p ) is the poset {(i, j) 2 Z x Z  0 *
* j i }
with lexicographic ordering. The point (i, j) corresponds to the diagram
j
Z=p Z=pi.p!Z=p1
where Z=pi is the subgroup of order pi of H0(W (X); ~T(SL(X))) = Z=p Z=p1 . *
*U(p ) corre
sponds to ( , 0) in this formalism. If i1 i2 and j1 j2, then the commutativ*
*e diagram
j1
Z=p oo_oZ=pi1o.p//_Z=p1
 fflffl  j2j1
  .p
 fflffl.fflfflfflfflpj2
Z=p oo_oZ=pi2o_//Z=p1
shows that (i1, j1) (i2, j2).
12. Shapiro's lemma
The main purpose of this section is to introduce some notation to be used in *
*Section 13.
For any set S and any abelian group M we put
M[S] = Z[S] Z M, M = Hom Z(Z[S], M)
where Z[S] stands for the free abelian group with basis S. M[] is a covariant*
* and M< >a
contravariant functor from the category of sets to the category Ab of abelian *
*groups. (M[]
(M< >) is the left (right) adjoint of the forgetful functor from abelian group*
*s to sets.) M can
also be considered as the abelian group of all functions u: S ! M. In case S is*
* a left Gset and
M a left Gmodule for some group G, the rules
g(s m) = gs gm, (gu)(s) = gu(g1s), g 2 G, s 2 S, m 2 M, u: S !*
* M,
make M[S] and M into left Gmodules. A special case occurs when S is the lef*
*t Gset G=H of
left cosets of a subgroup H of G.
12.1. Lemma. M[G=H] is isomorphic to the induced module IndGH(M) and Mis *
*isomor
phic to the coinduced module CoindGH(M).
Proof.Let T be a set of left coset representatives for G=H.
The set T is a basis for the free right ZHmodule ZG. The induced module Ind*
*GH(M) =
ZG ZH M is [95, 6.3.4] the sum over T  copies t M of M with Gaction g(t m) =*
* s hm where
gt = sh, s 2 T , h 2 H. The module M[G=H] = Z[G=H] Z M is the sum over T  co*
*pies t M of
M with Gaction g(t m) = s gm. The Zlinear isomorphism M[G=H] ! IndGH(M) t*
*hat takes
t m to t t1m is Glinear as it takes g(t m) = y gm to y y1gm = y ht1m = g(t *
*t1m).
The set T 1= {t1  t 2 T } is a basis for the free left ZHmodule ZG. The *
*coinduced
module CoindGH(M) is [95, 6.3.4] the product over T  of copies ßtM of M, wher*
*e ßtm: ZG ! M
is the Hmap sending t1 to m 2 M and z1 to 0 for all z 6= t in T . The Gact*
*ion is given
by g(ßtm) = ßy(hm). The module M is the product over T  of copies ætM o*
*f M, where
ætm: G=H ! M is the set map sending tH to m and zH to 0 for all z 6= t in T . *
*The Gaction is
given by g(ætm) = æy(gm). The Zlinear isomorphism CoindGH(M) ! Mthat tak*
*es ßtm to __
æt(tm) is Glinear as it takes g(ßtm) = ßy(hm) to æy(yhm) = æy(gtm) = gæt(tm). *
* __
Let ShG denote the homotopy colimit of S viewed as a functor from the categor*
*y G to the
category of sets. (ShG is the nerve of the small groupoid that has S for object*
* set and {g 2 G 
gs1 = s2} as the set of morphisms s1 ! s2.) The next lemma is just a reformulat*
*ion of Shapiro's
lemma.
12.2. Lemma. There are natural isomorphisms
H*(G; M[S]) ~=H*(ShG; M), H*(G; M) ~=H*(ShG; M)
58 J.M. MØLLER
Proof.Let X be a set of representatives for the Gorbits in S and G(x) the isot*
*ropy subgroup at
x 2 X. Then there are a homotopy equivalence
a
BG(x) ! ShG
x2X
and isomorphisms of Gmodules
a a Y Y
M[S] ~= M[G=G(x)] ~= IndGG(x)(M), M ~= M ~= CoindGG(x)(M)
`
induced by the isomorphism S ~= G=G(x) of Gsets. These isomorphisms combine, *
*with the_help
of Shapiro's lemma, to the isomorphisms of the lemma. *
* __
In other words,
a Y
(12.3) H*(G; M[S]) ~= H*(G(x); M), H*(G; M) ~= H*(G(x); M)
where x 2 S runs through a set of representatives for the orbit set S=G.
13.Cellular cohomology of small categories
The following is a general discussion of the derived functors of the inverse *
*limit.
Let I be a small category such that
oI has only finitely many objects,
oany endomorphism is an isomorphism
oany isomorphism is an automorphism
meaning that I is a special kind of very small ordered category [81] or EIcate*
*gory [54]. I could
for instance be a skeletal subcategory of the Quillen category of a pcompact g*
*roup.
Write S(i, j) for the set of morphisms from the object i to the object j and *
*I(i) for the group of
morphisms i ! i. Under the above assumptions, the set Ob(I) of objects of I has*
* the structure of a
partially ordered set, a poset, where i j if there is a morphism from i to j.*
* Let K(I) = Cx(Ob (I))
denote the ordered simplicial complex associated to Ob(I). The vertex set of K(*
*I) is the poset
Ob(I) and the psimplices, p > 0, is the set of all strictly increasing sequenc*
*es (i0. .i.p) of elements
of Ob(I) (where i < j if i j and i 6= j). The ordered simplicial complex K(I)*
* is ddimensional if
there exists a string i0 ! . .!.id of d morphisms between distinct objects but *
*no such string of
d + 1 morphisms. K(I) is again a poset with ordering given by inclusion.
For any psimplex (i0. .i.p) 2 K(I)p, put
I(i0. .i.p) = I(ip) x . .I.(i0) and S(i0. .i.p) = S(ip1, ip) x . *
*.x.S(i0, i1)
with the convention that for p = 0, S(i0) is understood to be a point. Form the*
* homotopy orbit
space
ShI(i0. .i.p) = S(i0. .i.p)hI(i0...ip)
for the action of the group I(i0. .i.p) on the set S(i0. .i.p) given by
(ap, . .,.a0) . (ap1p, . .,.a01) = (apap1pa1p1, . .,.a1a01a1*
*0)
for all aj 2 I(ij) and aj1j2 I(ij1, ij). This homotopy orbit space constructi*
*on provides a functor
ShI:K(I)op! Sp
from the opposite poset of K(I) to the category Sp of simplicial sets. For any *
*inclusion oe oe0of
simplices, the map ShI(oe) ShI(oe0) is induced by the obvious projection I(oe*
*) I(oe0) and the
map S(oe) S(oe0) given by composition or omission of morphisms in the usual w*
*ay.
Let now M :I ! Ab be a functor. Consider the functor HqM :K(I) ! Ab that t*
*akes the
psimplex (i0. .i.p) 2 K(I)p to the abelian group
Hq(ShI(i0. .i.p); M(ip)) = Hq(I(i0. .i.p); M(ip) )
Define cohomology of K(I) with coefficients in HqM, H*(K(I); HqM), as the cohom*
*ology of the
cochain complex (C(K(I); HqM), ffi):
Y ffip1 Y
(13.1) . .!. HqM(i0. .i.p1) ! HqM(i0. .i.p) ! . . .
(i0...ip1)2K(I)p1 (i0...ip)2K(I)p
NDETERMINISM 59
with differential
Xp
ffip1(U)(i0. .i.p) = (1)jOEj*æ*jU(i0. .b.ij.i.p.)
j=0
for all cochains U 2 Cp1(K(I); HqM) and all psimplices (i0. .i.p) of the simp*
*licial complex K(I).
Here, æj:I(ip) x . .I.(i0) ! I(ip) x . .d.I(ij)xi.s.I.(i0)the projection and th*
*e homomorphisms
8
>>M(ip) if j = 0
< j D E
M(ip) OE>M(ip) I(i0, . .,.bij,i.f.,.ip)0 < j < p
>:
M(ip1) if j = 0
are given by
8
>u(ap1p, . .,.ajj+1aj1j, . .,.a01)if 0 <*
* j < p
:M(ap1p)u(ap2,p1, . .,.a01)if j = p.
It will become clear later that ffiffi = 0, i.e. that (C(K(I); HqM), ffi) is in*
*deed a cochain complex.
Let lim*(I; ) denote the right derived functors of the inverse limit functor*
* lim:AbI ! Ab .
13.2. Theorem. [54] [87] There is a first quadrant cohomological spectral seque*
*nce Epqrwith
Epq1= Cp(K(I); HqM) andEpq2= Hp(K(I); HqM)
converging to limp+q(I; M)
This spectral sequence is associated to a descending filtration on the cochai*
*n complex C(I; M)
that has lim*(I; M) for cohomology groups:
Let be the category of totally ordered finite sets and weakly order preserv*
*ing maps. The
cosimplicial replacement functor [14, XI.5]
Q * I
:Ab ! Ab
Q*
takes the abelian Igroup M the cosimplicial abelian group M that in codegre*
*e n is the abelian
group of twisted ncochains of I with coefficients in M, i.e.
Q * n Y
M = M(in)
i0!...!in2N(I)n
consists of all functions U from N(I)n with values U(i0 ! i1 ! . .!.in) in M(in*
*). (As usual, the
nerve, N(I), of I is the singular set of I: The simplicial set that in degree 0*
* is the set of objects of
I and in degree n > 0 is the set of all sequences i0 ! i1 ! . .!.in of n compos*
*able morphisms
in I.) The coface maps
dj(U)(i0 ! . .!.in+1) = U(i0 ! . .!.bij! . .!.in+1), 0 j n,
dn+1(U)(i0 ! . .!.in+1) = M(in ! in+1)U(i0 ! . .!.in)
are the obvious ones.PDefine C(I; M) to be the underlying cochain complex whose*
* differentialQis the
alternating sum (1)idi. The ith cohomotopy group of the cosimplicial abelian*
* group *M,
Q * i
ßi M = H (C(I; M)), i 0,
is defined [14, X.7.1] as the ith cohomology group of its underlying cochain co*
*mplex C(I; M).
13.3. Lemma. [14, XI.6.2] [81, Lemma 2] The functors
Q* i
AbI ! Ab !ß Ab, i 0,
form a universal cohomological ffifunctor [95, 2.1.1] with limin degree 0.
60 J.M. MØLLER
In other words, lim*= ß* O * and limi(I; M) = Hi(C(I; M)) is the ith cohomol*
*ogy group of
the cochain complex C(I; M) of I with (twisted) coefficients M.
Define l to be the function on N(I) that is 0 on N(I)0; on N(I)1, l(i ! i) = *
*0 while l(i ! j) = 1
if i and j are nonisomorphic; and in general
n1X
l(i0 ! i1 ! . .!.in) = l(ii! ii+1),
i=0
the function l counts the number of strict inequalities in the string i0 i1 *
* . . .in. This makes
the nerve into a filtered simplicial set
; = F0N(I) F1N(I) . . .FpN(I) Fp+1N(I) . . .N(I)
where
FpN(I) = {i0 ! i1 ! . .!.in 2 N(I)  l(i0 ! i1 ! . .!.in) < p}
is the set of all strings of composable morphisms where less than p of the morp*
*hisms have non
isomorphic domain and codomain. Since I has only finitely many equivalence clas*
*ses of objects,
the filtration is finite: Fd+1N(I) = N(I) if K(I) has dimension d.
The filtration we are going to use is the induced descending filtration on th*
*e cochain complex
C(I; M),
(13.4) C(I; M) = F0C(I; M) F1C(I; M) . . .FpC(I; M) Fp+1C(I; M) . . .{*
*0}
where
FpC(I; M) = {U 2 C(I; M)  UFp(N(I)) = 0}
consists of all cochains that vanish on FpN(I). This filtration is finite: Fd+1*
*C(I; M) = {0} if K(I)
is ddimensional.
Proof of Theorem 13.2.Suppose that K(I) is ddimensional. Then the E1page of *
*the spectral
sequence associated [95, 5.4.1] to the filtration (13.4) satisfies Epq1= 0 when*
*ever p > d and
Ed*1= H*(FdC(I; M)[d]) where FdC(I; M)[d] is the translated cochain complex [*
*95, 1.2.8] that
in degree n equals FdC(I; M)d+n. Note that
M
FdC(I; M)[d] = C(i0. .i.d; M)
i0...id2K(I)d
splits as a direct product over the dsimplices in K(I) of the cochain complexe*
*s C(i0. .i.d; M)
given by
* +
a
C(i0. .i.d; M)n = M(id) I(id)rdx I(id1, id) x . .x.I(i0, i1) x *
*I(i0)r0
r0+...+rd=n
with a differential that is the restriction of the differential on C(I; M). Th*
*e claim is that the
cohomology of C(i0. .i.d; M) equals H*M(i0. .i.d) as defined above (13.1). The *
*standard cochain
complex for computing this cohomology group is
(13.5) Hom I(id)x...xI(i0)(B*(I(id)) . .x.B*(I(i0)), M(id) i1))
where B*(I(id)) . .x.B*(I(i0)) as the tensor product of unnormalized bar reso*
*lutions has
@(ado . . .a0o) = ado . . .@a0o+ (1)r0ado . . .@a1o a0o+ . . .
+ (1)r0+...+rd1@ado ad1o . . .a*
*0o
as its differential. Here, ajo= ajrj . . .aj1where ajk2 I(aj) and
rj1X
@ajo= ajrj . . .aj2+ . .a.jij+1ajij . .+.(1)rjajrjajrj1 . . .a1
ij=1
as usual [95, 6.5.1]. In fact, there is an isomorphism, oe, of cochain complexe*
*s from the standard
cochain complex (13.5) to C(i0. .i.d; M) given by
oe(U)(ado, ad1d, . .,.a1o, a01, a0o) = (1)r1+r3+...U(ado, adoad1d, . *
*.,.a1o, a1oa01, a0o)
NDETERMINISM 61
where ajoaj1j= ajrj. .a.j1aj1jand the sign is (1) raised to the power that i*
*s the sum over
all odd j of rj = ajo. I leave it to the reader to check that this isomorphis*
*m oe indeed commutes
with the differentials. The conclusion is that
Y
Edq1= Hq(FdC(I; M)[d]) ~= HqM(i0. .i.d)
(i0...id)2K(I)d
is isomorphic to the degree d group of the simplicial cochain complex C(K(I); H*
*qM) (13.1).
This same pattern repeats itself at all stages of the filtration as
Y
FpC(I; M)p+q= Fp+1C(I; M)p+q C(i0. .i.p; M)p+q
(i0...ip)2K(I)p
and, in fact, there is an isomorphism
Y
FpC(I; M)[p]=Fp+1C(I; M)[p] ~= C(i0. .i.p; M)
(i0...ip)2K(I)p
of cochain complexes. So, by the above computation,
Y
Epq1~= HqM(i0. .i.p)
(i0...ip)2K(I)p
is isomorphic to Cp(K(I); HqM).
It remains to compute the d1differential. Again, it will be sufficient to c*
*onsider the differ
ential dd1q1:Ed1q1! Edq1as similar arguments apply in general. Consider a coh*
*omology class
[U(i0. .i.d1)] in Hq(id1. .i.0; M) represented by the qcocycle
a
U(i0. .i.d1): I(id1)rd1x . .x.I(i0)r0! M(id1) i1)
rd1+...r0=q
and extend this to an element of Fd1C(I; M)q+d1by mapping the other (q + d  *
*1)simplices of
N(I) to 0. The image dd1q1[U(i0. .i.d1)] is represented by (oe1ffioe)U(i0. .*
*i.d1) where ffi is the
zigzaghomomorphism of the short exact sequence
0 ! FdC(I; M) ! Fd1C(I; M) ! FdC(I; M)=Fd1C(I; M) ! 0
of cochain complexes. This means that dd1q1[U(i0. .i.d1)] vanishes on all (q*
* + d)simplices of
N(I) except on the ones of the form
aij1i0j ai0jij
(13.6) i0 ! . .!.ij1 ! i0j!ij ! . .!.id1
for some object i0jof I, where it has the value
(1)q+jU(ad1o, ad2d1, . .,.ajo, aij1i0jai0jij, aj1o, . .,.a01*
*, a0o)
assuming, for simplicity, that 0 < j < d  1. We must compare this to the homom*
*orphism
B*(I(id1)) . . .B*(I(ij)) B*(I(i0j)) B*(I(ij1)) . . .B*(I(i0)) !
*
* j
B*(I(id1)) . . .B*(I(i0)) U(i0...id1)!M(id1) i1)OE!
0 0 ff
M(id1) I(id2, id1) x . .x.I(ij, ij) x I(ij1, ij) x . *
*.x.I(i0, i1)
where the first homomorphism takes ad1o . . .ajo a0jo aj1o . . .a0oto ad*
*1o . . .
ajo 1o aj1o . . .a0o. Assuming U(i0. .i.d1) to be normalized [95, 6.5.5], *
*this agrees with __
the value of (oe1ffioe)U(i0. .i.d1) on the (q + d)simplex (13.6) except that*
* the sign is missing. __
There is also a dual spectral sequence
E2pq= Hp(K(I); HqM) ) colimp+q(I; M)
where HqM(i0. .i.p) = Hq(ShI(i0. .i.p); M(i0)).
62 J.M. MØLLER
13.7. Example.1. [95, 3.5.12] If the category I is a poset S, the spectral sequ*
*ence (13.2) for a
functor M :S ! Ab degenerates to a cochain complex
Y Y
. .!. M(sp1) ! M(sp) ! . . .
(s0...sp1)2K(S)p1 (s0...sp)2K(S)p
with cohomology lim*(S; M).
2. If the category I is a group G and M :Gop! Ab a Gmodule, then the spectral*
* sequence 13.2
collapses onto the vertical axis in the sense that E0j1= Hj(G; M) and Eij1= 0 f*
*or i > 0.
3. Suppose that I is a category
S(0,1)
0___________//1
with two objects and no nonidentity automorphisms. The limn(I; M) = 0 for n > *
*1 and there is
an exact sequence
0 ! lim0(I; M)! M(0) x M(1) ffi!M(1) ! lim1(I; M)! 0
where ffi(m0, m1)(a) = m1 M(a)(m0) for all morphisms a 2 S(0, 1).
4. For a category I with two objects, 0 and 1, there is a long exact sequence
. .!.Hj(I(0); M(0)) Hj(I(1); M(1)) d1!Hj(E(0, 1); M(1)) ! limj+1(I; M*
*)! . . .
where we are assuming that I(1)xI(0) acts transitively on S(0, 1) with stabiliz*
*er subgroup E(0, 1).
5. With I = A(W, t){E, t}, the full subcategory of A(W, t) containing t and one*
* of its nontrivial
subspaces E 6= t, we get a long exact sequence
__ j d1 j __
. .!.Hj(W (E)=W (E); M(E)) x H (W ; M(t)) ! H (W (E); M(t))
! limj+1(I; M)! . . .
*
* __
where the homomorphism d1 is induced from M(E) ! M(t) and from the inclusion of*
* W (E) W .
In case E = tS 6= t is the fixedpoint space for the action of the Sylow psu*
*bgroup S of W and
M(tS) and M(t) are Z(p)modules, we conclude that there is an exact sequence
__ S S __ S
(13.8) 0 ! lim0(I; M)! M(tS)W (t )=W(tx)M(t)W ! M(t)W (t )! lim1(I; M)! 0
and
j(__W(tS); M(t))
(13.9) limj+1(I; M)= H_____________Hj(W,; M(t))j 1.
This quotient group vanishes if S has order p for N__W(tS)(S) = NW (S) (2.15) a*
*nd both cohomol
ogy groups equal Hj(S; M(t))NW (S)as these are the stable elements [19, 9.1, 10*
*.1] in this case.
Assuming, furthermore, that M(tS) = MW(tS)and M(t) = M for some Z(p)[W ]module*
* M, we
rediscover the formula
( W
limj(I; M)= M if j = 0
0 if j > 0
from [2].
6. Let H be a subgroup of the group G and I(G, H) = O(G)op{G=H, G={e}} the full*
* subcategory
_________G=H_____________________________________*
*______________________________________________________________________@
(13.10) NG(H)=H __9G=H9_________//_______________________________*
*____________________________________G={e}Gee__________________________@
of O(G)opcontaining the two objects G={e} and G=H. The limits of any functor M *
*:I(G, H) ! Ab
fit into a long exact sequence
. .!.Hj(NG(H)=H; M(G=H)) Hj(G; M(G={e})) d1!Hj(NG(H); M(G={e}))
! limj+1(I(G, H); M)! . . .
NDETERMINISM 63
where the homomorphism d1 is induced from M(H): M(G=H) ! M(G={e}) and from the*
* inclu
sion of NG(H) into G.
7. Let I be a category with three objects, 0, 11, and 12, of the shape
____________________________________*
*______________
I1o711gg__________________________________*
*_____________________________7o
________________________________________________*
*________________oooo
___:0:__________________________________________*
*________________________OOOO
I2O''O12__________________________________*
*______________________________
gg__________________________________*
*_______________
and let M be an Imodule with M(0) = 0, M(11) = M1, and M(12) = M2. Then
lim*(I; M) = lim*(I1; M1) x lim*(I2; M2)
where I1 is the full subcategory generated by the objects 0 and 11, I2 the full*
* subcategory generated
by the objects 0 and 12, and the I(11)module M1 is considered as an I1module *
*and the I(12)
module M2 as an I2module. Of course, this extends to an arbitrary star shaped *
*finite category
with outward pointing arrows.
8. Let I be a category with three objects, 01, 02, and 1, of the shape
_______________________________________________*
*___
__7017_________________________________________*
*____________________I1OOO
OO''O1__________________________________*
*___________________________________
o77dd__________________________________*
*______________________________o
_______________________________________________*
*___oooo
__7027_________________________________________*
*____________________I2
and let M be an Imodule with M(01) = 0 = M(02) and M(1) = M. Then there is a M*
*ayer
Vietoris sequence
. .!.Hj(I(1); M) ! limj(I1; M1) x limj(I2; M2) ! limj(I; M) ! Hj+1(I(1); M)*
* ! . . .
where I1 is the full subcategory generated by the objects 01 and 1 and I2 the f*
*ull subcategory
generated by the objects 02 and 1.
In the next lemma, R(A) means the full subcategory containing all objects of *
*the form Ra for
a 2 Ob(A).
13.11. Lemma. Let
__L__//
IooR__J
be an adjunction between two small categories, I(i, Rj) = J(Li, j) for all i 2 *
*Ob(I), j 2 Ob(J), and
A a full subcategory of J. Then
1.lim*(J; L*M) ~=lim*(I; M).
2.lim*(A; M) ~=lim*(R(A); L*M) ~=lim*(LR(A); M)
for any functor M :J ! Ab .
Proof.Since any left adjoint functor is left cofinal, the first assertion is a *
*consequence of the
Cofinality Theorem [14, XI.9.2, XI.7.2].
For the proof of the second assertion, where we may assume that Ob(J) = Ob(A)*
* [ Ob(LRA),
we consider the inclusion functors
A ,! J  LRA
The inclusion of LRA into J is left cofinal for the universal arrow LRa ! a is *
*a terminal object in
the over category LRA # a for all a 2 Ob(J). For the other inclusion, consider *
*the Grothendieck
spectral sequence
limp(J; limq(a # A; M)) ) limp+q(A; M)
If a in an object of A, the identity of a is an initial object in the under cat*
*egory a # A. Otherwise,
note that the restrictions
__L__//
Ra # RAoo___LRa # A
R
64 J.M. MØLLER
are adjoint functors so that
(
limq(LRa # A; M) ~=limq(Ra # RA; L*M)= MLRa q = 0
0 q > 0
because the identity of Ra is an initial object in the under category Ra # RA. *
*We conclude that
lim*(A; M) ~=lim*(J; M) ~=lim*(LRA; M). Finally, observe that there is an indu*
*ced adjoint
ness between RA and LRA so that also the two groups lim*(RA; L*M) and lim*(LRA;*
*_M)_are
isomorphic. *
*__
13.12. Proposition.Let J be a full subcategory of I. If M vanishes on all objec*
*ts outside J and
if any object of I with a morphism to an object of J is in J, then lim*(I; M) ~*
*=lim*(J; M).
Proof.The cochain projection map
Y Y
M(in) ! M(jn)
i0!...!in2N(I)n j0!...!jn2N(J)n
is an isomorphism. *
*___
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