THE 3-COMPACT GROUP DI(2)
JESPER M. MOLLER
Abstract.The 3-compact group DI(2) is characterized by cohomology algebra*
* with coefficients
in the finite field F3, H*(B DI(2); F3) = F3[x12; x16], being the rank 2 *
*Dickson algebra. We show
that DI(2) is a totally N-determined 3-compact group, compute the endomor*
*phism monoid of
DI(2), and investigate subgroups of DI(2).
1. Introduction
Group number 12 on the Shephard-Todd list [23] of finite irreducible complex *
*reflection groups
is GL (2; F3) or GL (V ) where V is a 2-dimensional F3-vector space. By Clark-E*
*wing [7], GL (V )
is even a 3-adic reflection group, containing and generated by 12 reflections o*
*f order two, and its
mod 3 invariant ring is the Dickson algebra,
DI(2) = F3[V ]GL(V )= F3[x12; x16]
which, with V sitting in grading degree two, is polynomial on a generator of de*
*gree 12 and a
generator of degree 16.
Since Zabrodsky, Smith-Switzer, and Aguade [27, 24, 1] it has been known that*
* DI(2) is the
cohomology algebra of a space.
The question addressed here is: Does there exist a 3-compact group of rank tw*
*o with Weyl
group GL(V )? The answer is presented below in the form of an existence and uni*
*queness theorem.
Although the existence part, as just noted, is not new, the approach, based on *
*the p-compact group
technology as developed by Dwyer and Wilkerson [10, 11, 12], is new (but admitt*
*edly very close
to the one used by Aguade).
Here is an outline of the argument. Let T ~=(Z=31 )2 denote a discrete 3-toru*
*s of rank two and
t the maximal elementary abelian subgroup of T. Assuming the existence of a 3-c*
*ompact group
DI(2) with (discrete) maximal torus T and Weyl group GL(t) = GL(2; F3), let's n*
*ow analyze its
properties.
DI(2) must be connected for no nontrivial 3-group can be the homomorphic imag*
*e of GL(2; F3),
and, since GL(2; F3) contains no elementary abelian 3-groups of order 9, any el*
*ementary abelian
3-subgroup of DI(2) is, up to conjugacy, contained in the maximal torus. Conseq*
*uently [21, 2.9],
the Quillen category of DI(2) has the form
__________________________________________________*
*____________GL(2;F3)=3________________________________________________@
Z=2__99______________________________________________*
*___________________//_tGL(2;F3)dd_____________________________________@
where < t is any nontrivial, proper subgroup, and the morphisms are as indicat*
*ed. According
to Jackowski and McClure [14] B DI(2) is the homotopy colimit of a diagram of t*
*opological spaces
indexed by (the opposite of) the above Quillen category. In order to figure out*
* what the two spaces
in the centralizer diagram for B DI(2) are, we first get a hold on the maximal *
*torus normalizer, N,
for DI(2).
The action of GL (t) on t extends in an essentially unique way to an action o*
*n the ambient
group T and the semi-direct product N = To GL (t) is the only extension (2.4) *
*realizing this
action. Computing centralizers in N , we find that
CN () = ToS3; CN (t) = T;
where S3 is the Weyl group of SU(3), and since these groups are discrete maxima*
*l torus nor-
malizers for totally N-determined 3-compact groups [16], the corresponding cent*
*ralizers in DI(2)
___________
Date: September 2, 1997.
1991 Mathematics Subject Classification. 55R35, 55P15, 55P10.
Key words and phrases. Reflection group, totally N-determined p-compact group*
*, automorphism group.
1
2 J. M. MOLLER
are
CDI(2)() = SU(3); CDI(2)(t) = T
with automorphisms given by the induced automorphisms of the centralizers in N *
*. Thus we
conclude that B DI(2), if it exists, must be the homotopy colimit of a diagram *
*of the form
_______________________________________________________*
*_________________________________op___________________(3)op\GL (2;F3)o@
(Z=2)BoSU(3)p99_____________________________________________*
*_______________________BTGL(2;F3)hh___________________________________@
with (Z=2)op = Z=2 acting on B SU(3) as { 1 }. This is to be understood as a di*
*agram in the
homotopy category of F3-complete topological spaces. After verifying the vanish*
*ing of the Dwyer-
Kan obstructions to lifting to the category of spaces, we define B DI(2) as the*
* F3-completion of
the homotopy colimit of any realization of the above diagram. This leads to the*
* main result of this
note.
Theorem 1.1. (3.9, 3.10, 3.12) There exists a connected, rank two, center-free,*
* simple, totally
N-determined 3-compact group DI(2) with Weyl group GL(2; F3). DI(2) is unique i*
*n the sense that
any other rank two 3-compact group with Weyl group GL (2; F3) in GL (2; Z3) is *
*isomorphic as a
3-compact group to DI(2).
Since DI(2) is totally N-determined in the sense of [16, 7.1], it is easy to *
*compute its endomor-
phism monoid.
Corollary 1.2.[B DI(2); B DI(2)] = Z*3={1} [ {0}.
This follows from (3.11) since by Ishiguro's theorem [20, 5.6] [13, 1.3], B D*
*I(2) is atomic in the
very strong sense that any self-map is either nullhomotopic or an equivalence.
2. Group number 12
The abstract group GL(2; F3) of order 48 = 24.3 is number 12 on the Shephard-*
*Todd list [23] of
finite irreducible complex reflection groups. We shall here investigate this gr*
*oup in its Clark-Ewing
[7] incarnation as a rank two 3-adic reflection group.
Let oe; o 2 GL(2; Z3), where Z3 denotes the ring of 3-adic integers, be the m*
*atrices
oe = 01 -1-1 and o = -01 -10
satisfying the relations o2 = E = oe3; ooeo-1 = oe2. The two subgroups *
* and of
GL(2; Z3) are both abstractly isomorphic to the permutation group 3 but they ar*
*e not conjugate
as subgroups of GL (2; Z3). To see this, let L denote the canonical GL (2; Z3)-*
*module, i.e. a free
Z3-module on two generators, and T the GL(2; Z3)-module (LQ)=L. Then the fixed *
*point group
T is cyclic of order 3 while T is the trivial group.
Let r3: GL(2; Z3) ! GL(2; F3)denote reduction modulo 3. The images S3 = r3() and
P 3 = r3() are both [9] [25, 10.7.1] abstractly isomorphic to 3 but aga*
*in they can be seen
to be non-conjugate subgroups of GL(2; F3). We let also oe and o denote the ele*
*ments r3(oe) and
r3(o) of GL(2; F3).
The subgroup D = = of GL (2; F3) generated by S3 and P 3*
*, or,
alternatively, by S3 and -E, is dihedral of order 12.
Define {G > H} to be the set of conjugacy classes of subgroups of the group G*
* abstractly
isomorphic to the group H.
Lemma 2.1. Conjugacy classes of subgroups of GL(2; Z3) isomorphic to 3; D, or G*
*L(2; F3) are
given by:
1.The map r3:{GL (2; Z3) > H} ! {GL (2; F3) >iH}s a bijection for H = 3; D,*
* and
GL (2; F3).
2.{GL (2; F3) > 3} = {S3; P 3} and {GL (2; F3) > D} = {D}.
Proof.The last part of the lemma is not hard to verify. To transfer these obser*
*vations to GL(2; Z3),
we_compute some obstruction groups._Let V = F23denote the canonical GL(2; F3)-m*
*odule. Then
H*(GL (2; F3); V V *) = 0 = H*(D; V V *) for both groups contain Z=2 as a cen*
*tral subgroup.
Direct computations show that Hi(S3; V V *) = 0 = Hi(P 3; V V *) for i = 1; 2*
*. Therefore
THE 3-COMPACT GROUP DI(2) 3
the obstructions to lifting to GL(2; Z3) and to uniqueness of such lifts vanish*
* [3,_7.1.2].(See [2] for
more information.) *
*|__|
Let now W denote any subgroup of GL(2; Z3) isomorphic to GL(2; F3), for insta*
*nce the explicit
subgroup of GL (2; Z3) from [25, Example 1 p. 201]. The center of W is the subg*
*roup Z(W ) =
{E}. The restriction of r3 to W is [25, 10.7.1] an isomorphism r3|W :W ! GL(2; *
*F3)and if we
(in spite of ambiguous notation) put S3 = (r3|W )-1(S3), P 3 = (r3|W )-1(P 3), *
*and also
D = (r3|W )-1(D), then r3|W is an isomorphism
(2.2) (W ; D; S3; P 3; Z(W )) ! (GL (2; F3); D; S3; P 3; {E})
of groups with subgroups.
The group W acts on L, T, and on the maximal elementary abelian 3-subgroup, t*
*, of T.
Lemma 2.3. The cohomology groups H*(G; L), H*(G; LQ), H*(G; T), and H*(W ; t) a*
*re trivial
for G = W or G = D.
Proof.Since the functor H*(Z(W ); -) vanishes on all three modules, so does H*(*
*G; -) by_the
Lyndon-Hochschild-Serre spectral sequence. *
* |__|
Corollary 2.4.Let N be any extension of G by T realizing T as a G-module. Then *
*N is isomor-
phic to the semi-direct product To G, G = W; D.
Proof.H2(G; T) = 0 by (2.3). *
* |___|
Since T is a characteristic subgroup of N = ToW , there is an exact sequence
1 -!AutW (T ) -!Aut(N ) -!Aut(W );
where AutW (T ) denotes the group of W -equivariant automorphisms of T, and an *
*exact sequence
T ) Aut(N )
1 -!Aut_W(__Z(W-)!____W-!Out(W )
obtained by quotienting out the normal subgroup W / Aut(N ).
Lemma 2.5. [2] In the above exact sequence Out(W ) is cyclic of order two, gene*
*rated by O, and
Aut(N )=W ~=AutW (T )=Z(W ) is isomorphic to Z*3={1}.
Above, O 2 Aut(GL (2; F3)) denotes the automorphism given by O(C) = (detC)C f*
*or all C 2
GL(2; F3). Note that O has order two and that O, swapping the two subgroups S3 *
*and P 3 of
GL(2; F3), is not inner.
The Weyl group WT(X) of a connected p-compact group X with maximal torus T ! *
*X may
naturally be considered as a subgroup (conjugacy class) of Aut(ss1(T )), Aut(T *
*), or, if p is odd,
Aut(t), where T is a discrete maximal torus and t the maximal elementary abelia*
*n subgroup of T.
It is often convenient to speak of the Weyl group as a subgroup (conjugacy cl*
*ass) of Aut(L) =
GL(r; Zp), Aut((Z=p1 )r), or (if p is odd) Aut(V ) = GL(r; Fp), where L is a fr*
*ee Zp-module of rank
r and V elementary abelian of order pr. The following definition is intended to*
* make this usage,
which already appeared in (1.1), precise and to stress the point that Weyl grou*
*ps should be viewed
as subgroups of general linear groups.
Definition 2.6.The connected p-compact group X has Weyl group W in GL(r; Zp) if*
* W -1 =
WT(X) for some maximal torus T -! X and some isomorphism :L ! ss1(T.)
This definition apply equally well to subgroups of Aut((Z=p1 )r) or (if p is *
*odd) Aut(V ) =
GL(r; Fp). For instance, the subgroup S3 (P 3) of GL(2; F3) is the Weyl group o*
*f the 3-compact
group SU(3) (PU (3)). Thus SU(3) and PU (3) do not have the same Weyl group in*
* GL (2; F3).
Conversely, any connected 3-compact group with Weyl group S3 in GL (2; F3) is i*
*somorphic to
SU(3), cf. [6] [21, 4.4, 4.7]. If p is odd and W a reflection group in GL(r; Zp*
*) of order prime to p
then there is [9] a unique connected p-compact group with W as its Weyl group i*
*n GL(r; Zp).
4 J. M. MOLLER
3.Construction of DI(2)
We shall construct the 3-compact group DI(2) as the homotopy colimit over an *
*index category
with just two objects.
For any group G with subgroup H let I(G; H) denote the category with two obje*
*cts, 0 and 1,
and morphism sets
oI(G; H)(0; 0) = NG(H)=H
oI(G; H)(1; 1) = G
oI(G; H)(0; 1) = G=H
oI(G; H)(1; 0) = ;
and composition of morphisms given by group multiplication in NG(H) and G and b*
*y the right-left
actions
G x G=H x NG(H)=H -!G=H
also determined by the group multiplication. Here is a pictorial presentation
____________________________________________*
*____________________G=H_______________________________________________@
NG(H)=H __:0:_______________________________________*
*____________________________//_1Gdd___________________________________@
of the index category I(G; H).
A functor C :I(G; H) ! C(C :I(G; H)op! C) into some category C consists of
oan object C(0) equipped with an action of NG(H)=H ((NG(H)=H)op)
oan object C(1) equipped with an action of G (Gop)
oan NG(H)-equivariant (NG(H)op-equivariant) morphism C(0) ! C(1) (C(0) C(1*
*)).
There is for instance a functor I(G; H)op-! O(G) into the orbit category that t*
*akes the object 0
to G=H and the object 1 to G={e}. This functor identifies I(G; H)op with a full*
* subcategory of
O(G).
Example 3.1. Let M be an R[I(G; H)]-module (a functor from I(G; H) into the cat*
*egory of R-
modules) where R is a field or the ring Zp of p-adic integers. Assuming that
op - |G: H|
oSp = Sylp(H) is cyclic of order p
oNNG(H)(Sp) = NG(Sp)
we have [21, 3.8] an exact sequence
0 ! lim0I(G;H)M ! M(0)NG(H)=H! M(1)NG(H)=M(1)G ! lim1I(G;H)M ! 0
while limjI(G;H)= 0 for j > 1.
Also fixed point functors are naturally functors defined on I(G; H).
Example 3.2. Let M be an R[G]-module. Then MH is an NG(H)=H-module and the incl*
*usion
MH -! M is NG(H)-equivariant. Let F (M) be the I(G; H)-module defined by these *
*data. With
the same assumptions on G, H, and R as in (3.1) we have
( G
limjI(G;H)F (M)= M j = 0
0 j > 0
for the higher limits over I(G; H) of the functor F (M).
We specialize now to the pair (G; H) = (W; S3) from (2.2), which by [1] satis*
*fies the con-
ditions of (3.1), and consider the I(W; S3)op-diagram of groups and conjugacy c*
*lasses of group
homomorphisms schematically given by
_________________________________________________*
*______________________________________________________________________@
(3.3) (NW (S3)=S3)op__ToS3<<________________________________________*
*___________TWop_______________________________oo_
________________________________________________*
*________________________________________________bb____________________@
where the morphism from left to right is the inclusion, W opacts on T through t*
*he inversion map
w -! w-1, and w 2 NW (S3)op takes (t; g) in the semi-direct product to (w-1t; w*
*-1gw). This
diagram is centric in the sense that the inclusion induces group homomorphisms *
*of centralizers
T S3 = CTo S3 (T oS3) -!CTo S3 (T ) ~=-CT(T ) = T
THE 3-COMPACT GROUP DI(2) 5
with the right morphism an isomorphism. Hence there is an induced I(W; S3)-modu*
*le
_________________________________________________*
*_______________________________________________________W=S3___________@
(3.4) NW (S3)=S3 ___T9S39_________________________________________*
*__________//_TW____________________________________
_________________________________________________*
*___________________________cc_________________________________________@
of abelian centers. Here, the quotient group NW (S3)=S3 = (S3 x Z(W ))=S3 ~=Z(W*
* ) is
cyclic of order two and it acts non-trivially on T S3~=Z=3.
Now regard the I(W; S3)op-diagram (3.3) as a diagram of maximal torus normali*
*zers and
consider the corresponding I(W; S3)op-diagram, BC,
_________________________________________________*
*______________________________________________________________________@
(3.5) (NW (S3)=S3)op_B_SU(3)<<______________________________________*
*________________BTWop________________________________oo_
________________________________________________*
*____________________________________^^________________________________@
of 3-compact groups and conjugacy classes of morphisms. (Here, B SU(3) is short*
* for (B SU(3))^3.)
This is a central diagram in the sense of Dwyer-Kan [8] and so there is an indu*
*ced diagram, BZC,
of centers
_________________________________________________*
*______________________________________________________________________@
(3.6) (NW (S3)=S3)_B(T;S3);_______________________________________*
*_____________________//_BTW""____________________________________
________________________________________________*
*______________________________________________________________________@
which may also be viewed as the result of applying the functor B(-)^3to the I(W*
*; S3)-module
ZC of diagram (3.4).
The higher limits of the two I(W; S3)-modules ss1(BZC) and ss2(BZC) were comp*
*uted in (3.1).
Lemma 3.7. lim*I(W;S3)ss1(BZC) = 0 = lim*I(W;S3)ss2(BZC).
The Dwyer-Kan obstruction groups being trivial we know that there exists an I*
*(W; S3)op-
diagram in the category of topological spaces inducing diagram (3.5) in the hom*
*otopy category.
Define B DI(2) to be the F3-completion of the homotopy colimit of any such diag*
*ram. By construc-
tion, B DI(2) comes equipped with a map Bi: BT ! B DI(2)which is equivariant up*
* to homotopy
with respect to the given action of W on BT and the trivial action on B DI(2). *
*On cohomology,
(Bi)* takes H*(B DI(2)) into the invariant ring H*(BT )W .
Proposition 3.8.The homomorphism
(Bi)*:H*(B DI(2); R) ! H*(BT ; R)W
is an isomorphism for R = F3; Z3; Q3.
Proof.This follows from the Bousfield-Kan spectral sequence [5, XII.4.5] and (3*
*.2). |___|
The convention in force above is that H*(-; Q3), where Q3 is the field of 3-a*
*dic numbers, stands
for H*(-; Z3) Q.
The algebra H*(BT ; F3)W = H*(BT ; F3)GL(2;3)is the Dickson algebra DI(2): A*
* polynomial
algebra on a generator x12in degree 12 and a generator x16= P 1x12in degree 16.
Corollary 3.9.B DI(2) is a connected, center-free, simple, 3-compact group with*
* maximal torus
i: T ! DI(2), Weyl group W in GL(2; Z3), and discrete maximal torus normalizer *
*N = ToW .
Proof.By definition and by (3.8) B DI(2) is an 11-connected F3-complete space w*
*ith loop space
cohomology H*(B DI(2); F3) exterior on two generators, in particular, finite. *
*This shows [10]
that DI(2) is a 3-compact group.
The 3-compact group morphism i: T ! DI(2)is a monomorphism for H*(BT ; F3) is*
* a finitely
generated [3, 1.3.1] [25, 2.3.1] H*(B DI(2); F3)-module via H*(Bi; F3). By con*
*struction, W is
contained in the Weyl group of DI(2), and since we know from the formula [10, 9*
*.7]
Q3[x12; x16] = H*(B DI(2); Q3) = H*(BT ; Q3)WT(DI(2))
that [25, 5.5.4] the order of the Weyl group of DI(2) is 12_2. 16_2= 48, which *
*also is the order of W ,
the Weyl group of DI(2) is precisely W .
The (discrete approximation to) the normalizer of the maximal torus of DI(2) *
*is by (2.4) the
semi-direct product N = ToW .
The center of DI(2) is isomorphic [11, 7.1] [16, 3.4] to the center of N whic*
*h is trivial. Since_also_
ss1(T ) Q is a simple W -module, DI(2) is a simple 3-compact group in the sens*
*e of [11, 1.2]. |__|
We now address uniqueness questions, in particular whether DI(2) is determine*
*d by its maximal
torus normalizer in the sense of [16, 7.1].
6 J. M. MOLLER
Theorem 3.10. The 3-compact group DI(2) is totally N-determined.
Proof.Let A(DI(2)) stand for the Quillen category of conjugacy classes of monom*
*orphisms of ele-
mentary abelian 3-groups to DI(2). Note that there is an obvious functor I(W; S*
*3) -!A(DI(2))
that takes 0 to the rank one monomorphism T S3 -! T -! DI(2) and 1 to the rank*
* two
monomorphism t -! T -! DI(2). Since the cohomology of B DI(2) embeds into the *
*cohomol-
ogy of BT , this is an equivalence of categories [21, 2.9]. The centralizer CD*
*I(2)(T S3) is iso-
morphic to the N-determined 3-compact group SU(3) for its discrete maximal toru*
*s normalizer
is CN (T S3) = To S3. Similarly, the centralizer CDI(2)(t) is isomorphic to th*
*e N-determined
3-compact group T for its discrete maximal torus normalizer is CN (t) = T. Thus*
* the A(DI(2))-
module ssi(BZCDI(2)) is equivalent to the I(W; S3)-module ssi(BZC), i = 1; 2. N*
*ow the vanishing
lemma (3.7) together with general results about N-determinism [16, 4.9, 7.15, 7*
*.17]_imply_that
DI(2) is totally N-determined. *
* |__|
Corollary 3.11.The group Aut(DI(2)) of homotopy classes of self-homotopy equiva*
*lences of
B DI(2) is isomorphic to Z*3={1}.
Proof.Since DI(2) is totally N-determined, Aut(DI(2)) is isomorphic to Aut(N )=*
*W which_was
computed in (2.5). *
* |__|
DI(2) is the only 3-compact group with GL(2; F3) as Weyl group (in the sense *
*of (2.6)).
Corollary 3.12.Let X be a connected 3-compact group of rank two and suppose tha*
*t the Weyl
group of X is either W in GL(2; Z3) or GL(2; F3) in GL(2; F3). Then X and DI(2)*
* are isomorphic
as 3-compact groups.
Proof.In either case, since there is essentially just one copy of GL(2; 3) in G*
*L(2; Z3), the subgroup
W < GL(2; Z3) is the Weyl group of X. Since H2(W ; T) = 0, the maximal torus no*
*rmalizer of
X can only be the semi-direct product N . But this is also the maximal torus no*
*rmalizer_of the
N-determined 3-compact group DI(2), so X ~=DI(2). *
* |__|
Similarly, B DI(2) is the only F3-complete space realizing the Dickson algebr*
*a as its cohomology
algebra.
Corollary 3.13.Let X be a 3-compact group such that H*(BX; F3) and H*(B DI(2); *
*F3) are
isomorphic unstable algebras. Then X and DI(2) are isomorphic as 3-compact grou*
*ps.
Proof.Note that X is connected and that X and DI(2) have isomorphic Quillen cat*
*egories by
Lannes theory [15, 0.4]. In particular, X is connected of rank two with Weyl gr*
*oup GL (2;_3) in
GL(2; 3). Hence X and DI(2) are isomorphic by (3.12). *
* |__|
4.G2 as a 3-compact group
We present a construction of BG2 at the prime p = 3 very similar to the const*
*ruction of DI(2)
of the previous section.
Recall that D, which is dihedral of order 12, is the subgroup D = *
* ~=S3x Z(W )
of W generated by S3 and the center. S3 is normal in D with quotient D=S3 isom*
*orphic
to Z(W ) which also is the center of D. There is a contravariant functor from t*
*he index category
I(D; S3)
_______D=S3______________________________________*
*______________________________________________________________________@
Z(W)__:0___________//:_______________________________*
*_____________________________________1Ddd_____________________________@
to the homotopy category of topological spaces given by the diagram, BC,
____""______________________________________________*
*______________________________________________________________________@
(4.1) Z(W)op__B_SU(3)___________________________________________*
*___________________BTDop_________________________________oo_
____________________________________________________*
*______________________________________________________________________@
where the group Zop= Z acts on B SU(3) as { 1 } and the two morphisms from righ*
*t to left are
the elements of the orbit Bi . Dop [BT; B SU(3)] for the action of Dop on the *
*maximal torus
Bi: BT ! B SU(3). Since (4.1) is a centric diagram [8] there are associated I(D*
*; S3)-modules
_________________________________________________________________*
*_______________________________________________________________D=S3___@
Z(W)__Z(SU<(3))<_____________________________________________________*
*_//_0dd__________________________________0::__________________________@
________________________________________________________________*
*______________________________________________________________________@
THE 3-COMPACT GROUP DI(2) 7
denoted ss1(BZC) and ss2(BZC), respectively. Since
(4.2) lim*I(D;S3)ss1(BZC) = 0
because Z ~=Z=2 acts non-trivially on the center Z(SU (3)) ~=Z=3, and
(4.3) lim*I(D;S3)ss2(BZC) = 0
by (3.1), we know from [8, 1.1] that there exists an essentially unique realiza*
*tion of (4.1).
Define BG2 to be the homotopy colimit of any realization of (4.1). By constru*
*ction, there is
map Bi: BT ! BG2 which is homotopy invariant under the action of Dop on BT .
Proposition 4.4.The homomorphism The homomorphism
(Bi)*:H*(BG2; R) ! H*(BT ; R)D
is an isomorphism for R = F3; Z3; Q3.
Proof.The I(D; S3)-module
_____________D=S3_____________________________________*
*______________________________________________________________________@
Z(W)__H*(BT;)S3_________//_;______________________________*
*__________________________HD(BTb)b____________________________________@
_____________________________________________________*
*______________________________________________________________________@
is the result of applying the cohomology functor to (4.1). Hence the cohomology*
* of the homotopy
colimit is
H*(BG2) = lim0I(D;S3)F (H*(BT )) = H*(BT )D
by the Bousfield-Kan spectral sequence [5, XII.4.5] and (3.2). *
* |___|
The invariant ring
H*(BT ; F3)D = F3[x4; x12]
is [25, 7.4 Example 3] polynomial on a generator x4 in degree 4 and a generator*
* x12in degree 12.
Corollary 4.5.BG2 is a connected, center-free, simple, 3-compact group with max*
*imal torus
i: T ! G2, Weyl group D in GL(2; Z3), and discrete maximal torus normalizer N =*
* ToD.
Proof.The proof is similar to that of (3.9) since also H2(D; T) = 0 and Z(T oD)*
* = T D= 0. |___|
Theorem 4.6. The 3-compact group G2 is totally N-determined.
Proof.Noting that the p-compact groups in the centralizer diagram (4.1) for BG2*
* are totally
N-determined [16, 21], the theorem follows from the vanishing results (4.2, 4.3*
*) and the general
N-determinism theory [16, 4.9, 7.15, 7.17]. (Note that in [16, 4.9, 7.17] it is*
* harmless_to replace
A(X) by any subcategory A for which hocolimAopBCX () is H*Fp-equivalent to BX.)*
* |__|
Remark 4.7. In diagram (4.1) we are using a full subcategory of the Quillen cat*
*egory as index
category. The centralizer decomposition based on the complete Quillen category *
*[21, 2.5] A(G2) =
I(D; S3) [ I(D; <-o >) is a diagram of the form
_________________________________________________*
*__________
ZopB_SU(3)99________________________________________*
*___________________________ddI
I(S3)op\DopIIII
II
I __________________________________*
*_____________
uBT hhDop_____________________________*
*_____________________________
uuu
_____zz<-o>op\Dopuuuuu___________________________*
*____________________
Zop_B_U(2)99________________________________________*
*_________________________________
where Zop= Z ~=Z=2 acts on B SU(3) and B U(2) as { 1 }.
Corollary 4.8.The group Out(G2) of homotopy classes of self-homotopy equivalenc*
*es of BG2 is
isomorphic to Z*3={1}.
Proof.Since G2 is totally N-determined, Out(G2) is [16, 7.2] isomorphic*
*_ to the group
Aut(T oD)=D. |_*
*_|
Corollary 4.9.Let Y be a connected 3-compact group with Weyl group D in GL(2; Z*
*3) or r3(D)
in GL(2; F3). Then Y and G2 are isomorphic as 3-compact groups.
8 J. M. MOLLER
Proof.Since H2(D; T) = 0 the maximal torus normalizer of Y must be isomorphic t*
*o the semi- __
direct product ToD and then Y must be isomorphic to G2 because G2 is totally N-*
*determined. |__|
Corollary 4.10.[22] Let Y be a 3-compact group such that H*(BX; F3) and H*(BG2;*
* F3) are
isomorphic as unstable algebras. Then Y and G2 are isomorphic as 3-compact grou*
*ps.
Proof.Since H1(BX; F3) = 0, X is connected, and since X and G2 have isomorphic *
*cohomology
algebras, X and G2 have isomorphic Quillen categories [15]. In particular [21, *
*2.9],_X has Weyl
group r3(D) in GL(2; F3). *
* |__|
5.Subgroups of DI(2)
This section contains some general theory for monomorphisms of between p-comp*
*act groups
and it is shown that DI(2) contains essentially unique copies of each of the 3-*
*compact groups
SU(2) x SU(2), U(2), Spin(5), SU(3), PU(3), and G2.
Let X1 and X2 be two connected p-compact 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 homomorphism [17, 3.11]
(5.1) 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|
fflffl|fflffl|
X1__f__//X2
commutes up to conjugacy. Here, Mono(X1; X2) [BX1; BX2] denotes the set of con*
*jugacy 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 N*
*1 ! N1 and
N2 ! N2 are discrete approximations then
[BN1 ; BN2 ] = [BN1; BN2]
so that
Mono (N1; N2) = Mono(N1 ; N2)=N2
consists of conjugacy classes of monomorphisms of N1 into N2. For any monomorp*
*hism f 2
Mono(X1; X2), we let N (f) 2 Mono (N1 ; N2), determined up to conjugacy, denote*
* any discrete
approximation to N(f).
Definition 5.2.The monomorphism f 2 Mono(X1; X2) is N-determined if N-1(N(f))
Mono(X1; X2) consists of f alone.
Let W1 = ss0(N1) and W2 = ss0(N2) denote the Weyl groups.
Example 5.3. If p - |W1|, then all monomorphisms are N-determined. Indeed, it i*
*s not difficult
to see that (5.1) is bijective in this case.
In case X1 = X2, the map (5.1) is the homomorphism N :Out(X1) ! Out(N1) previ*
*ously
encountered. We say that X1 has N-determined monomorphisms if this map is injec*
*tive; if X1 is
totally N-determined N is a bijection. Note that (5.1) is equivariant in the se*
*nse 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.
Proposition 5.4.Let i: X1 ! X2be a monomorphism between the two p-compact group*
*s X1 and
X2 of the same rank. Then the Euler characteristic O(X2=iX1) = |W2:W1| and if
THE 3-COMPACT GROUP DI(2) 9
oi is N-determined
oX1 is totally N-determined
o{N2 > N1} is a one-point set
then the action Mono(X1; X2) x Out(X1) ! Mono(X1; X2) is transitive and all mon*
*omorphisms
of X1 into X2 are N-determined.
Proof.The first part is [17, 3.11]. For the second part, note first that for an*
*y ff 2 Out(X1), iff
is an N-determined monomorphism. Suppose namely that N(f) = N(iff) = N(i)N(ff) *
*for some
monomorphism f :X1 ! X2. Then N(fff-1) = N(f)N(ff)-1 = N(i), so fff-1 = i or f *
*= iff.
Let now f :X1 ! X2be any monomorphism and N(f): N1! N2 a representative for t*
*he con-
jugacy class N(f). Since N2 contains but a single copy of N1 up to conjugacy an*
*d X1 is totally
N-determined, 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 N1 = T1oW1, N2 = T2oW2 are semi-dire*
*ct products
and the set {W1 > W2} is a one-point set.
Definition 5.5.For a monomorphism f :Y ! X of p-compact groups, let WX (f) or W*
*X (Y ), the
Weyl group of f, denote the component group of the Weyl space WX (Y ) [12, 4.1,*
* 4.3].
Proposition 5.6.Let f :Y ! X be a monomorphism of p-compact groups.
1.If the homomorphism ss0(Z(Y )) ! ss0(CX (Y )) induced by f is surjective, t*
*hen 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 [8], then there is a short exact sequence of loop spaces [1*
*0, 3.2] Y ! NX (Y ) !
WX (Y ) where NX (Y ) is the normalizer of f [12, 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 assumptio*
*n that f be centric,
the Weyl space becomes homotopically discrete and the exact sequence of the pro*
*position_is the
one from [12, 4.6] *
* |__|
Lemma 5.7. Let i: X1 ! X2 be a monomorphism and N(i) 2 Mono(N1; N2) the induc*
*ed
monomorphism of normalizers. Then the stabilizer subgroup Out(N1)N(i)of N(i) is*
* isomorphic to
NW2(W1)=W1.
Proof.Note that there is an epimorphism
NN2(N1 )=N1 i Aut (N1 )=N1N(i)= Out(N1)N(i)
given by conjugation by elements of N2 normalizing N1. This bijection is actual*
*ly also injective,
hence bijective, for if conjugation by, say, n2 2 NN2(N1 ) agrees with conjugat*
*ion by some element
n1 2 N1, then n1 and n2 have the same image in W2, so that n2 belongs to N1. T*
*his follows
because the Weyl groups of the connected p-compact groups X1 and X2 are faithfu*
*lly represented
in their maximal tori. Consequently
Out(N1)N(i)~=NN2(N1 )=N1
and this last group is isomorphic to the quotient group NW2(W1)=W1 by the proje*
*ction_N2i W2.
|__|
Proposition 5.8.Let i: X1 ! X2be an N-determined monomorphism between the two 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 N-determined.
10 J. M. MOLLER
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(N1 ; N2)=N2 x Aut(N1 )=N1 ! Mono(N1 ; N2)=N2
of Out(N1) on N(i) 2 Mono(N1; N2). Now apply (5.7). *
* |___|
Example 5.9. By (5.3), the inclusion To<-o >aeT oW is realizable by an *
* N-determined
monomorphism i: U(2) ! DI(2). The monomorphism i is centric (because B U(2) = *
*BThZ=2
and CU(2)(T ) ~=T ~=CDI(2)(T )) so (5.4, 5.6, 5.8)
O(DI(2)= U(2)) = 24 and WDI(2)(U (2)) ~=Z(W )
and Out(U (2)) acts transitively on Mono(U (2); DI(2)) since {T oW > To<-o >} i*
*s a one-point set.
Example 5.10. Similarly, {W > Z=2 x Z=2} is a one-point set, so there is an ess*
*entially unique
monomorphism i: SU(2) x SU(2) ! DI(2)realizing the inclusion of To (Z=2 x Z=2) *
*into To W .
The monomorphism i is N-determined, centric, and
O(DI(2)= SU(2) x SU(2)) = 12 and WDI(2)(SU (2) x SU(2)) ~=Z(W )
and Out(SU (2) x SU(2)) acts transitively on Mono(SU (2) x SU(2); DI(2)).
Example 5.11. Also {W > D8} = {D8}, where D8is the dihedral group of order 8. I*
*t follows that
there exists a unique monomorphism i: Spin(5) ! DI(2)realizing the inclusion To*
* D8aeTo W .
This monomorphism is centric (because B Spin(5) and B(T oD8) are H*F3-equivalen*
*t), so
O(DI(2)= Spin(5)) = 6 and WDI(2)(Spin(5)) ~=Z(W )
and Out(Spin(5)) acts transitively on Mono(Spin(5); DI(2)).
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.
Lemma 5.12. Let z :Z ! X1be a central monomorphism and i: X1 ! X2any monomorphi*
*sm
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.
Proof.The spaces BCX1(z) = map(BZ; BX1)Bz and BCX2(f O z) = map(BZ; BX2)B(fOz)a*
*re
X1=Z-spaces and BCf(Z): BCX1(z) ! BCX2(f O z)is an X1=Z-map 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 p-compact group X, any elementary abelian p-group V , and any*
* monomorphism
V ! X.
Example 5.13. Let i: SU(3) ! DI(2)denote the monomorphism arising in the constr*
*uction (3.5)
of B DI(2) as a homotopy colimit. By (5.12), Bi induces a homotopy equivalence
map BZ=3(B SU(3); B SU(3)) ! mapBZ=3(B SU(3); B DI(2))
where Z=3 ! SU(3) is the center, and thus a bijection
Out+(SU (3)) ! Mono(SU (3); DI(2));
THE 3-COMPACT GROUP DI(2) 11
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); DI(2))
fflffl
N ~=|| N||
fflffl| fflffl|
Out(T oS3)=S3 ____//_Mono(T oS3; ToW )=W
and using (5.7) we see that the kernel of the composition going down and then r*
*ight is trivial.
Thus i is N-determined and [8, 4.2] centric. Consequently,
O(DI(2)= SU(3)) = 8 and WDI(2)(SU (3)) ~=Z(W );
Out(SU (3)) acts transitively on Mono(SU (3); DI(2)), and all monomorphisms of *
*SU(3) into DI(2)
are N-determined.
Example 5.14. Similarly, the monomorphism i: SU(3) ! G2arising in the construct*
*ion (4.1) of
BG2 as a homotopy colimit is N-determined and centric. Also, Out(SU (3)) acts t*
*ransitively on
Mono(SU (3); G2) with stabilizer subgroup WG2(SU (3)) = {E}, and O(G2= SU(3)) =*
* 2.
Example 5.15. The inclusions of the maximal torus and of SU(3) into DI(2) const*
*itute a homo-
topy coherent set of maps out of the centralizer diagram (4.1) for BG2 into B D*
*I(2). Observing
that both maps are centric one sees first that the Wojtkowiak obstruction group*
*s vanish according
to (4.2, 4.3) and next that the resulting map BG2 ! B DI(2) is a centric monomo*
*rphism realizing
the inclusion To D ! ToD of maximal torus normalizers. As also {W > D} = {D}, w*
*e conclude
that
O(DI(2)=G2) = 4 and WDI(2)(G2) ~={1}
and that Out(G2) acts transitively on Mono(G2; DI(2)).
Example 5.16. According to [14], B PU(3) is the homotopy colimit of a diagram o*
*f the form
_________________________________________________*
*______________________________________________________________________@
(NP3 ()=)^3>)op;;_______________________________*
*______________________________BVWopVV_________________________________@
________________________________________________*
*______________________________________________________________________@
where V is elementary abelian of order 32, S3 = Syl3(SL(V )), and (NP3 ()=<*
*oe>)op~=opacts
on To by conjugation [21, 2.9]. The obvious map
(5.17) B(T o) ! B(T oW ) ! B DI(2)
is invariant up to homotopy under this action. The Shapiro lemma [18, 4.5] and*
* the identities
CTo (T ) = T = CTo W(T ) show that (5.17) in centric. Since Mono(V; DI(2)) *
*= {*}, there is up
to conjugation a unique monomorphism BV ! B DI(2), which must respect any group*
* action on
BV , and the centralizer of this monomorphism is the maximal torus T of DI(2). *
*These two maps
define a homotopy coherent set of maps out of the centralizer diagram (5.2) for*
* B PU(3). The
Wojtkowiak obstruction groups [26] for piecing them together to an actual map o*
*ut of B PU(3)
are lim1and lim2of the diagram
______________________________________________________*
*______________________________________________________________________@
Z=2___T<SL(V_)=S3//_<________________________________*
*______________________0SL(Vd)d_____________________________________
______________________________________________________*
*______________________________________________________________________@
where Z=2 acts non-trivially on T ~=Z=3, and lim2and lim3of the diagram
________SL(V_)=S3_____________________________________*
*______________________________________________________________________@
Z=2____<0___________//<__________________________________*
*___________________________LSL(Ve)e___________________________________@
______________________________________________________*
*______________________________________________________________________@
where L = ss2(BT ). Since these obstructions groups vanish, indeed the entire l*
*im*= 0 in both
cases [21, 3.4, 3,9], there exists a unique homotopy class Bi: B PU(3) ! B DI(2*
*)extending the two
given maps. Also, the restriction of i to the 3-normalizer of the maximal torus*
* is a monomorphism,
so i itself is a monomorphism [21, 5.2], and i is centric because the Bousfield*
*-Kan spectral sequence
[5, XI.7.1] for map(B PU(3); B DI(2))Bi shows that this mapping space is weakly*
* contractible. As
also {T oW > To P 3} is a one-point set and PU(3) is totally N-determined [21],*
* (5.4, 5.6, 5.8)
show that
O (DI(2)= PU(3))= 8 and WDI(2)(PU (3)) = Z(W )
12 J. M. MOLLER
and that the group Out(PU (3)) acts transitively on the set Mono(PU (3); DI(2))*
* of conjugacy classes
of monomorphisms.
There is no monomorphism of PU (3) into G2 for A(PU (3))(V ) = SL(V ) is too *
*big to be a
subgroup of A(G2)(V ) = D. Indeed, no nontrivial compact, connected Lie group a*
*dmits a proper,
center-free subgroup of maximal rank [4].
The next example describes the normalizers of the elementary abelian subgroup*
*s of DI(2).
Strictly speaking, these normalizers are not 3-compact groups, but rather exten*
*ded 3-compact
groups, in that their component groups are not 3-groups.
We start with a general observation.
Proposition 5.18.Let :V ! X be a monomorphism of an elementary abelian p-grou*
*p V into
a p-compact 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 [10, 8.3].
2.There is a short exact sequence of loop spaces
CX () ! NX () ! A(X)()
where NX () is the normalizer of [12, 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 [12, 4.1] of and the components, each one ho*
*motopy equivalent
to BV , of the total space are indexed by the automorphism group of in the Qui*
*llen category.
The homotopy exact sequences of this fibration and of its sub-fibration
CX ()=V ! BV ! BCX ()
give the exact sequence of groups and show that B(CX ()=V ) is the regular cove*
*ring space of
BWX () corresponding to the normal subgroup ss0(CX ()=V ) . WX (). Thus there i*
*s a pull-back
diagram
BCX () _______//BNX ()
| |
| |
fflffl| fflffl|
B(CX ()=V )____//_BWX ();
where the horizontal maps are regular covering spaces. *
* |___|
Example 5.19. For any monomorphism : Z=3 ! DI(2)there is (5.18) a short exact s*
*equence of
loop spaces
SU (3) ! NDI(2)() ! Z(W )
where Z(W ) ~=Z=2 acts on SU(3) as { 1 }. Thus
NDI(2)() = SU(3)o Z(W )
where B(SU (3)o Z(W )) denotes the total space of the unique [19, 3.3, 3,7] B S*
*U(3)-fibration over
BZ(W ) realizing the given monodromy action. (It is not essential in [19, x3] t*
*hat the component
group ss0(X) be a p-group.) Since the homotopy fixed point space BZ(SU (3))hZ(W*
*)is contractible,
the inclusion To S3aeSU (3) extends uniquely to a short exact sequence morphism
ToS3 _____//NTo W()____//_Z(W )
fflffl
| | ||
| | ||
fflffl| fflffl| ||
SU (3)_____//_NDI(2)()___//Z(W )
__
where NTo W() = ToW () = To(S3x Z(W )).
THE 3-COMPACT GROUP DI(2) 13
For any monomorphism :(Z=3)2 ! DI(2)there is a short exact sequence of loop *
*spaces
T ! NDI(2)() ! W
so NDI(2)() is an extended p-compact torus with To W as discrete approximation *
*[11, 3.12].
Example 5.20. The normalizers of the 3-compact subgroups of DI(2) are (5.6.2)
NDI(2)(G2) = G2 and NDI(2)(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 To*
* W of NX (T )
and the normalizer in DI(2) of X. For X = PU(3), for instance, the picture is
To P 3 ____//_NTo W(T oP 3)___//Z(W )
fflffl
| | ||
| | ||
fflffl| fflffl| ||
PU(3)_______//_NDI(2)(PU_(3))//_Z(W )
where NTo W(T oP 3) = To (P 3 x Z(W )). It seems likely that this is another i*
*nstance of
N-determinism.
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