THE 3-COMPACT GROUP DI(2)

JESPER M. MØLLER

Abstract. The 3-compact group DI(2) is characterized by its cohomology algebra H^{∗}(B DI(2); 𝔽_{3}) = 𝔽_{3}[x_{12}, x_{16}] being the rank 2 mod 3 Dickson algebra. We show that DI(2) is a totally Ndetermined 3-compact group, compute the endomorphism monoid of DI(2), and investigate subgroups of DI(2).

1. Introduction

Group number 12 on the Shephard-Todd list [22]of finite irreducible complex reflection groups is GL(2, 3) or GL(V) where V is a 2-dimensional 𝔽_{3}-vector space. By Clark-Ewing [6], GL(V) is even a 3-adic reflection group, containing and generated by 12 reflections of order two, and its mod 3 invariant ring is the Dickson algebra,

DI(2) = 𝔽_{3}[V]^{GL(V)} = 𝔽_{3}[x_{12}, x_{16}]

which, with V sitting in grading degree two, is polynomial on a generator of degree 12 and a generator of degree 16.

Since Zabrodsky, Smith-Switzer, and Aguadé [25, 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 two with Weyl group GL(V)? The answer is presented below in the form of an existence and uniqueness 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 [9, 10, 11], is new (but admittedly very close to the one used by Aguadé).

Here is an outline of the argument. Let Ť ≅ (ℤ/3^{∞})^{2} denote a discrete 3-torus of rank two and t the maximal elementary abelian subgroup of Ť. Assuming the existence of a 3-compact group DI(2) with (discrete) maximal torus Ť and Weyl group GL(t), let’s now analyze its properties.

DI(2) must be connected for no nontrivial 3-group can be the homomorphic image of GL(2, 3), and, since GL(2, 3) contains no elementary abelian 3-groups of order 9, any elementary abelian 3-subgroup of DI(2) is, up to conjugacy, contained in the maximal torus. Consequently [20, 2.9], the Quillen category of DI(2) has the form

[figure: λ; ℤ/2; GL(2,3)/Σ_{3}; t; GL(t)]

where λ < t is any nontrivial, proper subgroup, and the morphisms are as indicated. According to Jackowski and McClure [13] B DI(2) is the homotopy colimit of a diagram of topological 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, Ň, for DI(2).

The action of GL(t) on t extends in an essentially unique way to an action on the ambient group Ť and the semi-direct product Ň = Ť⋊ GL(t) is the only extension (2.7) realizing this action. Computing centralizers in Ň, we find that

CŇ(λ) = Ť⋊SΣ_{3}, CŇ(t) = Ť,

where SΣ_{3} is the Weyl group of SU(3), and since these groups are discrete maximal torus normalizers for totally N-determined 3-compact groups [17], the corresponding centralizers in DI(2) are

Date: June 25, 1997.

1991 Mathematics Subject Classification. 55R35, 55P15, 55P10.

Key words and phrases. Reflection group, totally N-determined p-compact group, automorphism group.

C_{DI(2)}(λ) = SU(3), C_{DI(2)}(t) = T

with automorphisms given by the the induced automorphisms of the centralizers in Ň. Thus we conclude that B DI(2), if it exists, must be the homotopy colimit of a diagram of the form

[figure: B SU(3); (ℤ/2)^{op}; BT; GL(2,3)^{op}; (Σ_{3})^{op}\ GL(2,3)^{op}]

with (ℤ/2)^{op} = ℤ/2 acting on B SU(3) as {ψ^{±1}}. This is to be understood as a diagram in the homotopy category of 𝔽_{3}-complete topological spaces. After verifying the vanishing of the Dwyer-Kan obstructions to lifting to the category of spaces, we define B DI(2) as the 𝔽_{3}-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, centerfree, simple, totally Ndetermined 3-compact group DI(2) with Weyl group GL(2, 3). DI(2) is unique in the sense that any other rank two 3-compact group with Weyl group GL(2, 3) in GL(2, 3) is isomorphic as a 3-compact group to DI(2).

Since DI(2) is totally N-determined in the sense of [17, 7.1], it is easy to compute its endomorphism monoid.

Corollary 1.2. [B DI(2), B DI(2)] = ℤ_{3}^{∗}/{±1} ∪ {0}.

This follows from (3.11) since by Ishiguro’s theorem [19, 5.6] [12, 1.3], B DI(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, 3) = GL(2, 𝔽_{3}) of order 48 = 2^{3}·3 is number 12 on the Shephard-Todd list [22] of finite irreducible complex reflection groups. We shall here investigate this group in its Clark-Ewing [6] incarnation as a rank two 3-adic reflection group.

Let σ, τ ∈ GL(2, ℤ_{3}), where ℤ_{3} denotes the ring of 3-adic integers, be the matrices

σ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) and τ = ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} )

satisfying the relations τ^{2} = E = σ^{3}, τστ^{−1} = σ^{2}. The two subgroups ⟨σ, τ⟩ and ⟨σ, −τ⟩ of GL(2, ℤ_{3}) are both abstractly isomorphic to the permutation group Σ_{3} but they are not conjugate as subgroups of GL(2, ℤ_{3}). To see this, let L denote the canonical GL(2, ℤ_{3})-modul, i.e. a free ℤ_{3}-module on two generators, and Ť the GL(2, ℤ_{3})-module (L⊗ℚ)/L. Then the fixed point group Ť^{⟨σ,τ⟩} is cyclic of order 3 while Ť^{⟨σ,−τ⟩} is the trivial group.

Let r_{3} : GL(2, ℤ_{3}) → GL(2, 3) denote reduction modulo 3. The images SΣ_{3} = r_{3}(⟨σ, τ⟩) and P Σ_{3} = r_{3}(⟨σ, −τ⟩) are both [8] [23, 10.7.1] abstractly isomorphic to Σ_{3} but again they can be seen to be non-conjugate subgroups of GL(2, 3). We let also σ and τ denote the elements r_{3}(σ) and r_{3}(τ) of GL(2, 3).

Define {G > H} to be the set of conjugacy classes of subgroups of G abstractly isomorphic to H.

Lemma 2.1. [2] The map r_{3} : {GL(2, ℤ_{3}) > Σ_{3}} → {GL(2, 3) > Σ_{3}} is a bijection and {GL(2, 3) > Σ_{3}} = {SΣ_{3}, P Σ_{3}}.

There are at least two distinct semi-direct product decompositions of GL(2, 3),

Q_{8}⋊SΣ_{3} = GL(2, 3) = Q_{8}⋊P Σ_{3} (2.1)

where

Q_{8} = {±E, ±A, ±B, ±AB}

is the Fitting subgroup, the maximal nilpotent normal subgroup, of GL(2, 3), generated by the matrices

A = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and B = ( \begin{matrix}−1 & 1 \\ 1 & 1 \end{matrix} ) .

Since these matrices satisfy the relations A^{2} = −E = B^{2}, BA = −AB, Q_{8} is abstractly isomorphic to the quaternion group, and the formulas

σ^{−1}Aσ = BA  σ^{−1}Bσ = A^{−1}
τ^{−1}Aτ = A^{−1}  τ^{−1}Bτ = BA

specify the conjugation actions of SΣ_{3} and P Σ_{3} on Q_{8}. The proof of the following result uses the semi-direct product decomposition (2.1) in order to ensure the vanishing of the relevant obstruction groups.

Theorem 2.2. [2] Reduction modulo 3, r_{3} : GL(2, ℤ_{3}) → GL(2, 3), admits a section s: GL(2, 3) → GL(2, ℤ_{3}) and any two such sections are conjugate.

Corollary 2.3. The set {GL(2, ℤ_{3}) > GL(2, 3)} consists of a single element.

Proof. Let H_{1} and H_{2} be two subgroups of GL(2, ℤ_{3}), both abstractly isomorphic to GL(2, 3). Then r_{3}H_{1} = GL(2, 3) = r_{3}H_{2}, so H_{1} = s_{1}(GL(2, 3)) and H_{2} = s_{2}(GL(2, 3)) for some sections s_{1} and s_{2}. But any two such sections are conjugate, so H_{1} and H_{2} are conjugate subgroups of GL(2, ℤ_{3}).

Let now W denote any subgroup of GL(2, ℤ_{3}) isomorphic to GL(2, 3), for instance the explicit subgroup of GL(2, ℤ_{3}) from [23, Example 1 p. 201]. The center of W is the subgroup Z(W) = {±E}. The restriction of r_{3} to W is [23, 10.7.1] an isomorphism r_{3}|W: W → GL(2, 3) and if we put SΣ_{3} = (r_{3}|W)^{−1}(SΣ_{3}) and P Σ_{3} = (r_{3}|W)^{−1}(P Σ_{3}) then r_{3}|W is an isomorphism

(W; SΣ_{3}, P Σ_{3}, Z(W)) → (GL(2, 3); SΣ_{3}, P Σ_{3}, {±E}) (2.4)

of groups with subgroups.

The group W acts on L, Ť, and on the maximal elementary abelian 3-subgroup, t, of Ť.

Lemma 2.5. Let G be a group and M a G-module. If G contains a normal subgroup H ◁ G such that H^{∗}(H; M) = 0, then also H^{∗}(G; M) = 0.

Proof. Use the Lyndon-Hochschield-Serre spectral sequence.

Lemma 2.6. The cohomology groups H^{∗}(W; L), H^{∗}(W; L ⊗ ℚ), H^{∗}(W; Ť), and H^{∗}(W; t) are trivial.

Proof. Since the functor H^{∗}(Z(W); −) vanishes on all three modules, so does H^{∗}(W; −).

Corollary 2.7. Let Ň be any extension of W by Ť realizing Ť as a W -module. Then Ň is isomorphic to the semi-direct product Ť⋊W.

Proof. H^{2}(W; Ť) = 0 by (2.6).

Since Ť is a characteristic subgroup of Ň = Ť⋊W, there is an exact sequence

1 ⟶ Aut_{W} (Ť) ⟶ Aut(Ň) ⟶ Aut(W),

where Aut_{W} (Ť) denotes the group of W -equivariant automorphisms of Ť, and an exact sequence

1 ⟶ \frac{Aut_{W} (Ť)}{Z(W)} ⟶ \frac{Aut(Ň)}{W} ⟶ Out(W)

obtained by quotienting out the normal subgroup W ◁ Aut(Ň).

Lemma 2.8. [2] In the above exact sequence Out(W) is cyclic of order two, generated by χ, and Aut(Ň)/W ≅ Aut_{W} (Ť)/Z(W) is isomorphic to ℤ_{3}^{∗}/{±1}.

Above, χ ∈ Aut(GL(2, 3)) denotes the automorphism given by χ(C) = (det C)C for all C ∈ GL(2, 3). Note that χ has order two and that χ, swapping the two subgroups SΣ_{3} and P Σ_{3} of GL(2, 3), is not inner.

The Weyl group W_{T} (X) of a connected p-compact group X with maximal torus T → X may naturally be considered as a subgroup (conjugacy class) of Aut(π_{1}(T)), Aut(Ť), or, if p is odd, Aut(t), where Ť is a discrete maximal torus and t the maximal elementary abelian subgroup of Ť.

It is often convenient to speak of the Weyl group as a subgroup (conjugacy class) of Aut(L) = GL(r, ℤ_{p}), Aut((ℤ/p^{∞})^{r}), or (if p is odd) Aut(V) = GL(r, 𝔽_{p}), where L is a free ℤ_{p}-module of rank r and V an elementary abelian p-group. The following definition is intended to make this usage, which already appeared in (1.1), precise.

Definition 2.9. The connected p-compact group X has Weyl group W in GL(r, ℤ_{p}) if θW θ^{−1} = W_{T} (X) for some maximal torus T ⟶ X and some isomorphism θ: L → π_{1}(T).

This definition apply equally well to subgroups of Aut((ℤ/p^{∞})^{r}) or (if p is odd) Aut(V) = GL(r, p). For instance, the subgroup SΣ_{3} (P Σ_{3}) of GL(2, 3) is the Weyl group of the 3-compact group SU(3) (PU(3)). Thus SU(3) and PU(3) do not have the same Weyl group in GL(2, 3). Conversely, any connected 3-compact group with Weyl group SΣ_{3} in GL(2, 3) is isomorphic to SU(3), cf. [20, 4.4, 4.7]. If p is odd and W a reflection group in GL(r, p) of order prime to p then there is [8] a unique p-compact group with W as its Weyl group connected in GL(r, p).

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 objects, 0 and 1, and morphism sets

• 𝕀(G, H)(0, 0) = N_{G}(H)/H

• 𝕀(G, H)(1, 1) = G

• 𝕀(G, H)(0, 1) = G/H

• 𝕀(G, H)(1, 0) = ∅ and composition of morphisms given by group multiplication in N_{G}(H) and G and by the right-left actions

G × G/H × N_{G}(H)/H ⟶ G/H

also determined by the group multiplication. Here is a pictorial presentation

[figure: 0; N_{G}(H)/H; G/H; 1; G]

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

• an object C(0) equipped with an action of N_{G}(H)/H ((N_{G}(H)/H)^{op})

• an object C(1) equipped with an action of G (G^{op})

• an N_{G}(H)-equivariant (N_{G}(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 takes the object 0 to G/H and the object 1 to G/{e}. This functor identifies 𝕀(G, H)^{op} with the full subcategory of O(G) containing the orbits G/H and G/{e}.

Example 3.1. Let M be an R[𝕀(G, H)]-module (a functor from 𝕀(G, H) into the category of Rmodules) where R is the field 𝔽_{p}, the ring ℤ_{p}, or the field ℚ_{p}. Assuming that

• p |̸ |G: H|

• S_{p} = Syl_{p}(H) is cyclic of order p

• N_{N_{G}(H)}(S_{p}) = N_{G}(S_{p}) we have ([20, 3.8]) an exact sequence

0 → lim_{𝕀(G,H)}^{0}M → M(H)^{N_{G}(H)/H} → M(G)^{N_{G}(H)}/M(G)^{G} → lim_{𝕀(G,H)}^{1}M → 0

while lim_{𝕀(G,H)}^{j} = 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 M^{H} is an N_{G}(H)/H-module and the inclusion M^{H} ⟶ M is N_{G}(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

lim_{𝕀(G,H)}^{j} F (M) = \begin{cases}M^{G} & j = 0 \\ 0 & j > 0 \end{cases}

for the higher limits over 𝕀(G, H) of the functor F (M).

We specialize now to the pair (G, H) = (W, SΣ_{3}) from (2.4), which by [1] satisfies the conditions of (3.1), and consider the 𝕀(W, SΣ_{3})^{op}-diagram of groups and conjugacy classes of group homomorphisms schematically given by

[figure: Ť⋊SΣ_{3}; (N_{W} (SΣ_{3})/SΣ_{3})^{op}; Ť; W^{op}; (SΣ_{3})^{op}\W^{op}]

(3.3)

where the morphism from left to right is the inclusion, W^{op} acts on Ť through the inversion map w ⟶ w^{−1}, and w ∈ N_{W} (SΣ_{3})^{op} takes (t, g) in the semi-direct product to (w^{−1}t, w^{−1}gw). This diagram is centric in the sense that the inclusion induces group homomorphisms of centralizers

Ť^{SΣ_{3}} = C_{Ť⋊SΣ_{3}}(Ť⋊SΣ_{3}) ⟶ C_{Ť⋊SΣ_{3}}(Ť) ⟵^{≅} CŤ(Ť) = Ť

with the right morphism an isomorphism. Hence there is an induced 𝕀(W, SΣ_{3})-module

[figure: Ť^{SΣ_{3}}; N_{W} (SΣ_{3})/SΣ_{3}; W/SΣ_{3}; Ť; W] (3.4)

of abelian centers. Here, the quotient group N_{W} (SΣ_{3})/SΣ_{3} = (SΣ_{3} × Z(W))/SΣ_{3} ≅ Z(W) is cyclic of order two and it acts nontrivially on Ť^{SΣ_{3}} ≅ ℤ/3.

Now regard the 𝕀(W, SΣ_{3})^{op}-diagram (3.3) as a diagram of maximal torus normalizers and consider the corresponding 𝕀(W, SΣ_{3})^{op}-diagram, BC,

[figure: B SU(3); (N_{W} (SΣ_{3})/SΣ_{3})^{op}; BT; W^{op}; (SΣ_{3})^{op}\W^{op}]

(3.5)

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 [7] and so there is an induced diagram, BZC, of centers

[figure: B(Ť^{SΣ_{3}}); N_{W} (SΣ_{3})/SΣ_{3}; W/SΣ_{3}; BT; W] (3.6)

which may also be viewed as the result of applying the functor B(−)_{3}^{∧} to the 𝕀(W, SΣ_{3})-module ZC of diagram (3.4).

The higher limits of the two 𝕀(W, SΣ_{3})-modules π_{1}(BZC) and π_{2}(BZC) were computed in (3.1).

Lemma 3.7. lim_{𝕀(W,SΣ_{3})}^{∗} π_{1}(BZC) = 0 = lim_{𝕀(W,SΣ_{3})}^{∗} π_{2}(BZC).

The Dwyer–Kan obstruction groups being trivial we know that there exists an 𝕀(W, SΣ_{3})^{op}- diagram in the category of topological spaces inducing diagram (3.5) in the homotopy category. Define B DI(2) to be the 𝔽_{3}-completion of the homotopy colimit of any such diagram. By construction, 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 = 𝔽_{3}, ℤ_{3}, ℚ_{3}.

Proof. This follows from the Bousfield-Kan spectral sequence [5, XII.4.5] and (3.2).

The convention in force above is that H^{∗}(−; ℚ_{3}), where ℚ_{3} is the field of 3-adic numbers, stands for H^{∗}(−; ℤ_{3}) ⊗ ℚ.

The algebra H^{∗}(BT; 𝔽_{3})^{W} = H^{∗}(BT; 𝔽_{3})^{GL(2,3)} is the Dickson algebra DI(2): A polynomial algebra on a generator x_{12} in degree 12 and a generator x_{16} = P^{1}x_{12} in degree 16.

Corollary 3.9. B DI(2) is a connected, centerfree, simple, 3-compact group with maximal torus i: T → DI(2), Weyl group W, and discrete maximal torus normalizer Ň = Ť⋊W.

Proof. By definition and by (3.8) B DI(2) is an 11-connected 𝔽_{3}-complete space with loop space cohomology H^{∗}(ΩB DI(2); 𝔽_{3}) exterior on two generators, in particular, finite. This shows [9] that DI(2) is a 3-compact group.

The 3-compact group morphism i: T → DI(2) is a monomorphism for H^{∗}(BT; 𝔽_{3}) is a finitely generated [3, 1.3.1] [23, 2.3.1] H^{∗}(B DI(2); 𝔽_{3})-module via H^{∗}(Bi; 𝔽_{3}). By construction, W is contained in the Weyl group of DI(2), and since we know from the formula [9, 9.7]

ℚ_{p}[x_{12}, x_{16}] = H^{∗}(B DI(2); ℚ_{3}) = H^{∗}(BT; ℚ_{3})^{W_{T} (DI(2))}

that [23, 5.5.4] the order of the Weyl group of DI(2) is \frac{12}{2} · \frac{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 (2.7) the semi-direct product Ň = Ť⋊W.

The center of DI(2) is isomorphic [10, 7.1] [17, 3.4] to the center of Ň which is trivial. Since also π_{1}(T) ⊗ ℚ is a simple W -module, DI(2) is a simple 3-compact group in the sense of [10, 1.2].

We now addresss uniqueness questions, in particular whether DI(2) is determined by its maximal torus normalizer in the sense of [17, 7.1].

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 monomorphisms of elementary abelian 3-groups to DI(2). Note that there is an obvious functor 𝕀(W, SΣ_{3}) ⟶ 𝔸(DI(2)) that takes 0 to the rank one monomorphism Ť^{SΣ_{3}} ⟶ Ť ⟶ DI(2) and 1 to the rank two monomorphism t ⟶ Ť ⟶ DI(2). Since the cohomology of B DI(2) embeds into the cohomology of BT, this is an equivalence of categories [20, 2.9]. The centralizer C_{DI(2)}(Ť^{SΣ_{3}}) is isomorphic to the N-determined 3-compact group SU(3) for its discrete maximal torus normalizer is CŇ(Ť^{SΣ_{3}}) = Ť⋊SΣ_{3}. Similarly, the centralizer C_{DI(2)}(t) is isomorphic to the N-determined 3-compact group T for its discrete maximal torus normalizer is CŇ(t) = Ť. Thus the 𝔸(DI(2))- module π_{i}(BZC_{DI(2)}) is equivalent to the 𝕀(W, SΣ_{3})-module π_{i}(BZC), i = 1, 2. Now the vanishing lemma (3.7) together with general results about N-determinism [17, 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 equivalences 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(Ň)/W which was computed in (2.8).

DI(2) is the only 3-compact group with GL(2, 3) as Weyl group (in the sense of (2.9)).

Corollary 3.12. Let X be a connected 3-compact group of rank two and suppose that the Weyl group of X is either W in GL(2, ℤ_{3}) or GL(2, 3) in GL(2, 3). 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 GL(2, ℤ_{3}), the subgroup W < GL(2, ℤ_{3}) is the Weyl group of X. Since H^{2}(W; Ť) = 0, the maximal torus normalizer of X can only be the semi-direct product Ň. But this is also the maximal torus normalizer of the N-determined 3-compact group DI(2), so X ≅ DI(2).

Similarly, B DI(2) is the only 𝔽_{3}-complete space realizing the Dickson algebra as its cohomology algebra.

Corollary 3.13. Let X be a 3-compact group such that H^{∗}(BX; 𝔽_{3}) and H^{∗}(B DI(2); 𝔽_{3}) are isomorphic unstable algebras. Then X and DI(2) are isomorphic as 3-compact groups.

Proof. Note that X is connected and that X and DI(2) have isomorphic Quillen categories by Lannes theory [14, 0.4]. In particular, X is connected of rank two with Weyl group GL(2, 3) in GL(2, 3). Hence X and DI(2) are isomorphic by (3.12).

4. G_{2} as a 3-compact group

The subgroup D = ⟨SΣ_{3}, Z⟩ = ⟨σ, −τ⟩ of W generated by SΣ_{3} and the center or, alternatively, by the two elements σ and tau, is dihedral of order 12. SΣ_{3} is normal with quotient D/SΣ_{3} isomorphic to Z which is the center of D. There is a contravariant functor from the index category 𝕀(D, SΣ_{3})

[figure: 0; D/SΣ_{3}; Z; 1; D]

to the homotopy category of topological spaces given by the diagram, BC,

[figure: B SU(3); ℤ^{op}; BT; D^{op}; (SΣ_{3})^{op}\D^{op}]

(4.1)

where the group Z^{op} = ℤ acts on B SU(3) as {ψ^{±1}} and the two morphisms from right to left are the elements of the orbit Bi · D^{op} ⊂ [BT, B SU(3)] for the action of D^{op} on the maximal torus Bi: BT → B SU(3). Since (4.1) is a centric diagram [7] there are associated 𝕀(D, SΣ_{3})-modules

[figure: Z(SU(3)); Z; D/SΣ_{3}; 0; D; 0; Z; D/SΣ_{3}; π_{2}(BT); D]

denoted π_{1}(BZC) and π_{2}(BZC), respectively. Since

lim_{𝕀(D,SΣ_{3})}^{∗}π_{1}(BZC) = 0 (4.2)

because Z ≅ ℤ/2 acts nontrivially on the center Z(SU(3)) ≅ ℤ/3, and

lim_{𝕀(D,SΣ_{3})}^{∗}π_{2}(BZC) = 0 (4.3)

by (3.1), we know from [7, 1.1] that there exists an esentially unique realization of (4.1).

Define BG_{2} to be the homotopy colimit of any realization of (4.1). By construction, there is map Bi: BT → BG_{2} which is homotopy invariant under the action of D^{op} on BT.

Proposition 4.4. The homomorphism The homomorphism

(Bi)^{∗}: H^{∗}(BG_{2}; R) → H^{∗}(BT; R)^{D}

is an isomorphism for R = 𝔽_{3}, ℤ_{3}, ℚ_{3}.

Proof. The 𝕀(D, SΣ_{3})-module

[figure: H^{∗}(BT)^{SΣ_{3}}; D/SΣ_{3}; Z; H^{∗}(BT); D]

is the result of applying the cohomology functor to (4.1). Hence the cohomology of the homotopy colimit is

H^{∗}(BG_{2}) = lim_{𝕀(D,SΣ_{3})}^{0}F (H^{∗}(BT)) = H^{∗}(BT)^{D}

by the Bousfield-Kan spectral sequence [5, XII.4.5] and (3.2).

The invariant ring

H^{∗}(BT; 𝔽_{3}) = 𝔽_{3}[x_{4}, x_{12}]

is [23, 7.4 Example 3] polynomial on a generator x_{4} in degree 4 and a generator X_{12} = P^{2}x_{4} in degree 12.

Corollary 4.5. BG_{2} is a connected, centerfree, simple, 3-compact group with maximal torus i: T → G_{2}, Weyl group D, and discrete maximal torus normalizer Ň = Ť⋊D.

Proof. The proof is similar to that of (3.9) since also H^{2}(D; Ť) = 0 and Z(Ť⋊D) = Ť^{D} = 0.

Theorem 4.6. The 3-compact group G_{2} is totally N-determined.

Proof. Noting that the p-compact groups in the centralizer diagram (4.1) for BG_{2} are totally N-determined [17, 20], the theorem follows from the vanishing results (4.2, 4.3) and the general N-determinism theory [17, 4.9, 7.15, 7.17]. (Note that in [17, 4.9, 7.17] it is harmless to replace 𝔸(X) by any subcategory 𝔸 for which hocolim_{𝔸^{op}} BC_{X}(ν) is H^{∗}𝔽_{p}-equivalent to BX.)

Remark 4.7. In diagram (4.1) we are using a full subcategory of the Quillen category as index category. The centralizer decomposition based on the the Quillen category 𝔸(G_{2}) = 𝕀(D, SΣ_{3}) ∪ 𝕀(D, ⟨−τ⟩) is a diagram of the form

[figure: B SU(3); Z^{op}; BT; D^{op}; (SΣ_{3})^{op}\D^{op}; ⟨−τ⟩op\D^{op}; B U(2); Z^{op}]

where Z^{op} = Z ≅ ℤ/2 acts on B SU(3) and B U(2) as {ψ^{±1}}.

Corollary 4.8. The group Aut(G_{2}) of homotopy classes of self-homotopy equivalences of BG_{2} is isomorphic to ℤ_{3}^{∗}/{±1}.

Proof. Since G_{2} is totally N-determined, Aut(G_{2}) is [17, 7.2] isomorphic to the group Aut(Ť⋊D)/D.

Corollary 4.9. Let Y be a connected 3-compact group with Weyl group D in GL(2, 3). Then Y and G_{2} are isomorphic as 3-compact groups.

Proof. Since H^{2}(D; Ť) = 0 the maximal torus normalizer of Y must be isomorphic to the semidirect product Ť⋊D and then Y must be isomorphic to G_{2} because G_{2} is totally N-determined.

Corollary 4.10. [21] Let Y be a 3-compact group such that H^{∗}(BX; 𝔽_{3}) and H^{∗}(BG_{2}; 𝔽_{3}) are isomorphic as unstable algebras. Then Y and G_{2} are isomorphic as 3-compact groups.

Proof. Since H^{1}(BX; 𝔽_{3}) = 0, X is connected, and since X and G_{2} have isomorphic cohomology algebras, X and G_{2} have isomorphc Quillen categories [14]. In particular [20, 2.9], X has Weyl group D in GL(2, 3).

5. Subgroups of DI(2)

This section contains some general theory for monomorphisms of between p-compact groups and it is shown that DI(2) contains essentially unique copies of each of the 3-compact groups SU(2) × SU(2), U(2), Spin(5), SU(3), PU(3), and G_{2}.

Let X_{1} and X_{2} be two connected p-compact groups of the same rank. Let j_{1} : N_{1} → X_{1} and j_{2} : N_{2} → X_{2} be normalizers of maximal tori i_{1} : T_{1} → X_{1} and i_{2} : T_{2} → X_{2}.

Consider the homomorphism [16, 3.11]

(5.1) N: Mono(X_{1}, X_{2}) → Mono(N_{1}, N_{2})

that to any conjugacy class of a monomorphism f: X_{1} → X_{2} associates the unique conjugacy class N(f): N_{1} → N_{2} such that

\xymatrix{N_{1} \ar[r]^{N(f)} \ar[d]_{j_{1}} & N_{2} \ar[d]^{j_{2}} \\ X_{1} \ar[r]_{f} & X_{2}}

commutes up to conjugacy. Here, Mono(X_{1}, X_{2}) ⊂ [BX_{1}, BX_{2}] denotes the set of conjugacy classes of monomorphisms of X_{1} into X_{2} and Mono(N_{1}, N_{2}) denotes the set of conjugacy classes of maps BN_{1} → BN_{2} inducing monomorphisms on π_{1} and isomorphisms on π_{2}. Note that if Ň_{1} → N_{1} and Ň_{2} → N_{2} are discrete approximations then

[BŇ_{1}, BŇ_{2}] = [BN_{1}, BN_{2}]

so that

Mono(N_{1}, N_{2}) = Mono(Ň_{1}, Ň_{2})/Ň_{2}

consists of conjugacy classes of monomorphisms of Ň_{1} into Ň_{2}. For any monomorphism f ∈ Mono(X_{1}, X_{2}), we let Ň(f) ∈ Mono(Ň_{1}, Ň_{2}), determined up to conjugacy, denote any discrete approximation to N(f).

Definition 5.2. The monomorphism f ∈ Mono(X_{1}, X_{2}) is N-determined if N^{−1}(N(f)) ⊂ Mono(X_{1}, X_{2}) consists of f alone.

Let W_{1} = π_{0}(N_{1}) and W_{2} = π_{0}(N_{2}) denote the Weyl groups.

Example 5.3. If p |̸ |W_{1}|, then all monomorphisms are N-determined. Indeed, it is not difficult to see that (5.1) is bijective in this case.

In case X_{1} = X_{2}, the map (5.1) is the homomorphism N: Out(X_{1}) → Out(N_{1}) previously encountered. We say that X_{1} has N-determined monomorphisms if this map is injective; if X_{1} is totally N-determined N is a bijection. Note that (5.1) is equivariant in the sense that the diagram

\xymatrix{Mono(X_{1}, X_{2}) × Out(X_{1}) \ar[d]_{N×N} \ar[r] & Mono(X_{1}, X_{2}) \ar[d]^{N} \\ Mono(N_{1}, N_{2}) × Out(N_{1}) \ar[r] & Mono(N_{1}, N_{2})}

commutes.

Proposition 5.4. Let i: X_{1} → X_{2} be a monomorphism. If • i is N-determined

• X_{1} is totally N-determined

• {Ň_{2} > Ň_{1}} is a one-point set then the action Mono(X_{1}, X_{2}) × Out(X_{1}) → Mono(X_{1}, X_{2}) is transitive and all monomorphisms of X_{1} into X_{2} are N-determined.

Proof. Note first that for any α ∈ Out(X_{1}), iα is an N-determined monomorphism. Suppose namely that N(f) = N(iα) = N(i)N(α) for some monomorphism f: X_{1} → X_{2}. Then N(fα^{−1}) = N(f)N(α)^{−1} = N(i), so fα^{−1} = i or f = iα.

Let now f: X_{1} → X_{2} be any monomorphism and Ň(f): Ň_{1} → Ň_{2} a representative for the conjugacy class N(f). Since Ň_{2} contains but a single copy of Ň_{1} up to conjugacy and X_{1} is totally N-determined, Ň(f) = Ň(i)Ň(α) for some automorphism α of X_{1}. Then Ň(f) = Ň(iα) and f = iα.

The third condition is satisfied in case Ň_{1} = Ť_{1}⋊W_{1}, Ň_{2} = Ť_{2}⋊W_{2} are semi-direct products and the set {W_{1} > W_{2}} is a one-point set.

Proposition 5.5. Let i: X_{1} → X_{2} be a monomorphism. Then the Euler characteristic

χ(X_{2}/iX_{1}) = |W_{2} : W_{1}|

and, provided that i induces an epimorphism π_{0}(Z(X_{1})) → π_{0}(C_{X_{2}}(X_{1})), the Weyl group W_{X_{2}}(X_{1}) is the isotropy subgroup for the action of Out(X_{1}) on i ∈ Mono(X_{1}, X_{2}).

Proof. The first part of the proposition is [16, 3.11]. For the second part we use the fibration induced by Bi

(X_{2}/X_{1})^{hX_{1}} → ∐_{iα≃i} map(BX_{1}, BX_{1}) → map(BX_{1}, BX_{2})_{Bi}

where the components of the total space are indexed by the isotropy sybgroup Out(X_{1})_{i} for the action on i ∈ Mono(X_{1}, X_{2}). The fibre is the Weyl space. The assumptions of the proposition assure that the inclusion of the fibre into the total space is a bijection on π_{0}.

The requirement on the monomorphism i is met for instance if i is centric in which case the Weyl space is homotopically discrete.

Proposition 5.6. Let i: X_{1} → X_{2} be an N-determined monomorphism inducing an epimorphism π_{0}(Z(X_{1})) → π_{0}(C_{X_{2}}(X_{1})). Then

W_{X_{2}}(X_{1}) = N_{W_{2}}(W_{1})/W_{1}

provided X_{1} is totally N-determined.

Proof. The assumptions imply that the Weyl group W_{X_{2}}(X_{1}) is isomorphic to the stabilizer subgroup Out(X_{1})_{i} which again is isomorphic to the stabilizer subgroup Out(N_{1})_{N(i)} for the action

Mono(Ň_{1}, Ň_{2})/Ň_{2} × Aut(Ň_{1})/Ň_{1} → Mono(Ň_{1}, Ň_{2})/Ň_{2}

of Out(N_{1}) on N(i) ∈ Mono(N_{1}, N_{2}). Now apply (5.7).

Lemma 5.7. The stabilizer subgroup Out(N_{1})_{N(i)} for a monomorphism N(i) ∈ Mono(N_{1}, N_{2}) is isomorphic to N_{W_{2}}(W_{1})/W_{1}.

Proof. Note that there is an epimorphism

NŇ_{2}(Ň_{1})/Ň_{1}↠ (Aut(Ň_{1})/Ň_{1})_{N(i)} = Out(N_{1})_{N(i)}

given by conjugation by elements of Ň_{2} normalizing Ň_{1}. This bijection is actually also injective, hence bijective, for if conjugation by, say, n_{2} ∈ NŇ_{2}(Ň_{1}) agrees with conjugation by some element n_{1} ∈ Ň_{1}, then n_{1} and n_{2} have the same image in W_{2}, so that n_{2} belongs to Ň_{1}. This follows because the Weyl groups of the connected p-compact groups X_{1} and X_{2} are faithfully represented in their maximal tori. Consequently

W_{X_{2}}(X_{1}) ≅ Out(X_{1})_{i} ≅ Out(N_{1})_{N(i)} ≅ NŇ_{2}(Ň_{1})/Ň_{1}

and this last group is isomorphic to the Weyl group W_{X_{2}}(X_{1}) by the projection Ň_{2}↠W_{2}.

Example 5.8. By (5.3), the inclusion Ť⋊⟨−τ⟩ → Ť⋊W is realizable by an N-determined monomorphism i: U(2) → DI(2). The monomorphism i is centric (because B U(2) = BT_{hℤ/2} and C_{U(2)}(T) ≅ T ≅ C_{DI(2)}(T)) and by (5.4, 5.5, 5.6), since {Ť⋊W > Ť⋊⟨−τ⟩} is a one-point set,

χ(DI(2)/ U(2)) = 24 and W_{DI(2)}(U(2)) ≅ {±E}

and Out(U(2)) acts transitively on Mono(U(2), DI(2)).

Example 5.9. Similarly, {W > ℤ/2 × ℤ/2} is a one-point set, so there is an essentially unique monomorphism i: SU(2) × SU(2) → DI(2) realizing the inclusion of Ť⋊(ℤ/2 × ℤ/2) into Ť⋊W. The monomorphism i is N-determined, centric, and

χ(DI(2)/ SU(2) × SU(2)) = 12 and W_{DI(2)}(SU(2) × SU(2)) ≅ {±E}

and Out(SU(2) × SU(2)) acts transitively on Mono(SU(2) × SU(2), DI(2)).

Example 5.10. Also {W > D_{8}} = {D_{8}}, where D_{8} is the dihedral group of order 8. It follows that there exists a unique monomorphism i: Spin(5) → DI(2) realizing the inclusion Ť⋊D_{8} → Ť⋊W. This monomorphism is centric (because B Spin(5) and B(Ť⋊D_{8}) are H^{∗}F_{3}-equivalent), so

χ(DI(2)/ Spin(5)) = 6 and W_{DI(2)}(Spin(5)) ≅ {±E}

and Out(Spin(5)) acts transitively on Mono(Spin(5), DI(2)).

In a sitaution where a pair of monomorphisms G → X_{1} and G → X_{2} are given, let us write

map^{BG}(BX_{1}, BX_{2})

for the space of maps BX_{1} → BX_{2} under BG up to homotopy.

Lemma 5.11. Let z: Z → X_{1} be a central monomorphism and i: X_{1} → X_{2} any monomorphism inducing an isomorphism X_{1} ≅ C_{X_{1}}(z) → C_{X_{2}}(f ∘ z). Then f induces a homotopy equivalence

map^{BZ}(BX_{1}, BX_{1}) → map^{BZ}(BX_{1}, BX_{2})

of mapping spaces.

Proof. The spaces BC_{X_{1}}(z) = map(BZ, BX_{1})_{Bz} and BC_{X_{2}}(f ∘ z) = map(BZ, BX_{2})_{B(f∘z)} are X_{1}/Z-spaces and BC_{f}(Z): BC_{X_{1}}(z) → BC_{X_{2}}(f ∘ z) is an X_{1}/Z-map inducing a map

map^{BZ}(BX_{1}, BX_{1}) = BC_{X_{1}}(z)^{h(X_{1}/Z)} → BC_{X_{2}}(f ∘ z)^{h(X_{1}/Z)} = map^{BZ}(BX_{1}, BX_{2})

of homotopy fixed point spaces. If C_{f}(z) is an isomorphism, then this map is a homotopy equivalence.

This happens for instance for V → C_{X}(V) → X so that

map^{BV} (BC_{X}(V), BC_{X}(V)) ≃ map^{BV} (BC_{X}(V), BX)

for any connected p-compact group X, any elementary abelian p-group V, and any monomorphism V → X.

Example 5.12. Let i: SU(3) → DI(2) denote the monomorphism arising in the construction of B DI(2) as a homotopy colimit. By (5.11), Bi induces a homotopy equivalence

map^{Bℤ/3}(B SU(3), B SU(3)) → map^{Bℤ/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)),

where Out^{+}(SU(3)) consists of the unstable Adams operations ψ^{u} indexed by units u ∈ ℤ_{3}^{∗} with u ≡ 1 mod 3. We obtain a commutative diagram

\xymatrix{Out^{+}(SU(3)) \ar[r]^{≅} \ar[d]_{N}^{≅} & Mono(SU(3), DI(2)) \ar[d]^{N} \\ Out^{+}(Ť⋊SΣ_{3})/SΣ_{3} \ar[r] & Mono(Ť⋊SΣ_{3}, Ť⋊W)/W}

with bijections as indicated. It follows from (5.7) that the kernel of the lower horizontal map consists of {±1} alone. Thus i is N-determined and also centric by [7]. Consequently,

χ(DI(2)/ SU(3)) = 8 and W_{DI(2)}(SU(3)) ≅ {±E},

Out(SU(3)) acts transitively on Mono(SU(3), DI(2)), and all monomorphisms of SU(3) into DI(2) are N-determined.

Example 5.13. Similarly, the monomorphism i: SU(3) → G_{2} arising in the construction of BG_{2} as a homotopy colimit is N-determined and centric. Also Out(SU(3)) acts transitively on Mono(SU(3), G_{2}) with stabilizer subgroup W_{G_{2}}(SU(3)) = {±E}, and χ(G_{2}/ SU(3)) = 2.

Example 5.14. The inclusions of the maximal torus and of SU(3) into DI(2) constitute a homotopy coherent set of maps out of the centralizer diagram (4.1) for BG_{2} into B DI(2). Observing that both maps are centric one sees first that the Wojtkowiak obstruction groups vanish according to (4.2, 4.3) and next that the resulting map BG_{2} → B DI(2) is a centric monomorphism realizing the inclusion Ť⋊D → Ť⋊D of maximal torus normalizers. As also {W > D} = {D}, we conclude that

χ(DI(2)/G_{2}) = 4 and W_{DI(2)}(G_{2}) ≅ {1}

and that Out(G_{2}) acts transitively on Mono(G_{2}, DI(2)).

Example 5.15. According to [13] B PU(3) is the homotopy colimit of a diagram of the form

[figure: B(Ť⋊⟨τ⟩)_{3}^{∧}; (N_{P Σ_{3}} (⟨σ⟩)/⟨σ⟩)^{op}; BV; W^{op}; S_{3}^{op}\ SL(V)^{op}]

where V is an elementary abelian 3-group of order 3^{2}, S_{3} = Syl_{3}(SL(V)), and (N_{P Σ_{3}}(⟨σ⟩)/⟨σ⟩ ≅ ⟨τ⟩ acts on Ť⋊⟨τ⟩ by conjugation [20, 2.9]. The obvious map

B(Ť⋊⟨σ⟩) → B(Ť⋊W) → B DI(2) (5.16)

is invariant up to homotopy under this action. Use the Shapiro lemma [15, 4.5] and the identities C_{Ť⋊⟨σ⟩}(Ť) = Ť = C_{Ť⋊W} (Ť) to show that (5.16) 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 diagram for B PU(3). The Wojtkowiak obstruction groups for piecing them together to an actual map out of B PU(3) are lim^{1} and lim^{2} of the diagram

[figure: Ť^{⟨σ⟩}; SL(V)/S_{3}; ℤ/2; 0; SL(V)]

where ℤ/2 acts nontrivially on Ť^{⟨σ⟩} ≅ ℤ/3, and lim^{2} and lim^{3} of the diagram

[figure: 0; SL(V)/S_{3}; ℤ/2; L; SL(V)]

where L = π_{2}(BT). Since these obstructions groups vanish, indeed the entire lim^{∗} = 0 in both cases [20, 3.4, 3,9], there exists a unique homotopy class Bi: B PU(3) → B DI(2) extending the two given maps. Since the restriction of i to the 3-normalizer of the maximal torus is a monomorphism, i itself is a monomorphism [20, 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 {Ť⋊W > Ť⋊P Σ_{3}} is a one-point set and PU(3) is totally N-determined [20], (5.4, 5.5, 5.6) show that

χ (DI(2)/ PU(3)) = 8 and W_{DI(2)}(PU(3)) = {±E}

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 G_{2} for 𝔸(PU(3))(V) = SL(V) is too big to be a subgroup of 𝔸(G_{2})(V) = D. Indeed, no nontrivial compact, connected Lie group admits a proper, centerfree subgroup of maximal rank [4].

The final example describes the normalizers of the elementary abelian subgroups of DI(2). Strictly speaking, these normalizers are not 3-compact groups, but rather extended 3-compact groups, in that their component groups are not 3-groups.

We start with a general observation.

Proposition 5.17. Let ν: V → X be a monomorphism of an elementary abelian p-group V into a p-compact group X. Assume that the standard map [9, 8.3] V → π_{0}(C_{X}(ν)) is surjective. Then there is a short exact sequence

C_{X}(ν) → N_{X}(ν) → 𝔸(X)(ν)

of loop spaces where N_{X}(ν) is the normaliser of ν [11, 4.4].

Proof. The Weyl space [11, 4.1] of ν, W_{X}(ν), is the fibre of a fibration

W_{X}(ν) → ∐_{f∈𝔸(X)(ν)} map(BV, BV)_{Bf} ⟶^{\underline{Bν}} map(BV, BX)_{Bν}

where each component of the total space is homotopy equivalent to BV. Since V surjects onto π_{0}(C_{X}(V)), the exact homotopy sequence shows that the Weyl group W_{X}(ν) = π_{0}(W_{X}(ν)) is isomorphic to the group 𝔸(X)(ν) of self-maps in the Quillen category amd that the identity component W_{X}(ν)_{1} of the Weyl space is homotopy equivalent to the homogeneous space C_{X}(ν)/V. This implies that we have a pull-back diagram

[figure: BV; BC_{X}(ν); B(W_{X}(ν)_{1}); BV; BN_{X}(ν); B(W_{X}(ν))]

where the bottom fibration sequence is from [11, 4.6] and the top one from [9, 8.3]. A diagram chase of the associated long exact homotopy sequences reveals s short exact sequence

1 ⟶ π_{0}(C_{X}(ν)) ⟶ π_{0}(N_{X}(ν)) ⟶ 𝔸(X)(ν) ⟶ 1

as well as isomorphisms π_{i}(C_{X}(ν)) ≅ π_{i}(N_{X}(ν)) for i ≥ 1. thus BC_{X}(ν) is the regulat covering space of BN_{X}(ν) coresponding to the normal subgroup π_{0}(C_{X}(ν)) ◁ π_{0}(N_{X}(ν)).

Example 5.18. For any monomorphism λ: ℤ/3 → DI(2) there is a short exact sequence of loop spaces

SU(3) → N_{DI(2)}(λ) → ℤ/2

where ℤ/2 acts on SU(3) as {ψ^{±1}}. Thus

N_{DI(2)}(λ) = SU(3)⋊ℤ/2

where B(SU(3)⋊ℤ/2) denotes the total space of the unique [18, 3.3, 3,7] B SU(3)-fibration over Bℤ/2 realizing the given monodromy action. (It is not essential in [18, §3] that the component group π_{0}(X) be a p-group.)

For any monomorphism ν: (ℤ/3)^{2} → DI(2) there is a short exact sequence of loop spaces

T → N_{DI(2)}(ν) → W

so N_{DI(2)}(ν) is an extended p-compact torus with Ť⋊W as discrete approximation [10, 3.12].

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Matematisk Institut, Universitetsparken 5, DK–2100 København Ø

E-mail address: moller@math.ku.dk