PU(p) AS A p-COMPACT GROUP

JESPER M. MØLLER

Abstract. It is shown that the the maximal torus normalizer determines the p-compact group PU(p) and its automorphisms. This leads to homotopy uniqueness results for PU(p) and related p-compact groups and to a novel construction of unstable Adams operations.

1. Introduction

Let X be a p-compact group and N → X the normalizer of a maximal torus of X [9]. We say [22, 7.1] that X is totally N-determined if

• the automorphisms of X are determined by their restrictions to N

• any p-compact group that can accomodate the extended p-compact torus N as a maximal

torus normalizer is isomorphic to X It has been conjectured, see e.g. [19, 5.2] [29, 5.20], that all p-compact groups are totally Ndetermined. Here, we shall consider the p-compact groups, { \atop PU(n)}, arising from the projective unitary groups PU(n) = U(n)/ U(1) by p-completion: B{ \atop PU(n)} = B PU(n)_{p}^{∧}. When 2 ≤ n < p,

{ \atop PU(n)} is totally N-determined for general reasons since the Weyl group order is not divisible by p [22, 7.9]. Thus the first problematic case arises when n = p.

Theorem 1.1 (4.3, 4.4). The p-compact group { \atop PU(p)} is totally N-determined for all primes p > 2.

The general fact [22, 7.2] that for a connected totally N-determined p-compact group X, BX and BN have isomorphic groups of homotopy classes of self-homotopy equivalences allows us to compute the monoid of homotopy classes of self-maps of { \atop PU(p)}.

Corollary 1.2 (4.5). The monoid [B PU(p)_{p}^{∧}, B PU(p)_{p}^{∧}], p > 2, is isomorphic to the monoid (ℤ_{p})^{∗} ∪ {0} consisting of the p-adic units with an adjoined zero element.

Of course, this corollary is a very special case of the much more general and fundamental work of Jackowski–McClure–Oliver [14, 15] for the uniqueness part and Sullivan [32] and Wilkerson [34] for the existence part. So, the novelty of (1.2) does not lie so much in the actual statement but rather in the method of proof.

We use an approach based on the Jackowski–McClure centralizer decomposition [17] rather than the p-stubborn decomposition and regard computations of self-maps and uniqueness questions as two intimately connected issues. The hope is that this aproach may also apply to exotic, non-Lie, p-compact groups. Another characteristic property is that total N-determinacy of { \atop PU(p)} for general reasons imply total N-determinacy of related p-compact groups such as { \atop U(p)}, { \atop SU(p)}, and

{ \atop SU(p, p)}.

The reason that the prime p = 2 is excluded from the statements of (1.1, 1.2) is not due to any inherent complication but simply to the fact that the theory of determinism is at present inadequately developed in that case.

The point of departure is a Jackowski-McClure decomposition of the classifying space B PU(p)_{p}^{∧} as the homotopy colimit defined over a category, a full subcategory of the Quillen category, with only three objects. Section 2 contains a complete description of this three object category. Information about the morphism sets of the category will serve as input for later computations.

Date: June 25, 1997.

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

Key words and phrases. Quillen category, p-compact group, higher limits, determinism, automorphism group.

When using cohomology decompositions to construct maps, certain Wojtkowiak obstruction groups, cohomology groups of the category with coefficients in a center functor, will occur at a crucial point. In our case, the vanishing of the relevant obstruction groups follows as an application of a spectral sequence derived in Section 3 for the cohomology of (certain) categories with only two objects. (By filtering the coefficient functor as in [30], it turns out to be sufficient to be able to handle cohomology computations of categories with just two objects, not three as one might expect.) Section 4 contains the proof, heavily based on the general discussion of determinism for p-compact groups from [22], that { \atop PU(p)} is totally N-determined. With information from Section 5, that discusses determinacy of extensions of totally N-determined p-compact groups, it follows that all (twisted) product combinations of the p-compact groups U(p), SU(p), and PU(p), such as e.g. SU(p, p) and S(U(p) × U(p)) are totally N-determined.

Also A. Viruel and C. Broto have dealt with determinism of PU(p) [5, 6] and I would like to thank them for informing me about their joint work and C. Broto for pointing out to me in [4] the relevance of Aguadé’s long exact sequence [1]. Other valuable information was supplied by F. Cohen, R. Levi, and J.B. Olsson and I am grateful for their help.

2. A full subcategory of the Quillen category

Let p > 2 be an odd prime and let 𝔸(PU(p)) denote the Quillen category of the projective unitary group PU(p). The objects (E, ν) = ν of 𝔸(PU(p)) are conjugacy classes of monomorphisms ν: E → PU(p) from nontrivial elementary abelian p-groups E to PU(p). A morphism f: (E_{1}, ν_{1}) → (E_{2}, ν_{2}) in the Quillen category is a group monomorphism f: E_{1} → E_{2} such that (E_{1}, ν_{1}) = (E_{2}, ν_{2}f). This section contains the detailed examination of the full subcategory 𝔸 of 𝔸(PU(p)) containing the three objects (ℤ/p, λ), (ℤ/p ⊕ ℤ/p, ν), and (Σ(ℤ/p), Σι) defined below.

Let Σ_{p} denote the permutation group on p letters. For any (topological) abelian group A, A^{p} is in a natural way a Σ_{p}-module and we define ΣA by the short exact sequence

0 ⟶ A ⟶^{Δ} A^{p} ⟶ ΣA ⟶ 0

of Σ_{p}-modules where Σ_{p} acts trivially on A and Δ is the diagonal monomorphism. Then Σ is a functor from the category of abelian groups to the category of Σ_{p}-modules. (If A is a module over the field with p elements, 𝔽_{p}, or over the ring of p-adic integers, ℤ_{p}, H^{j}(Σ_{p}; ΣA) ≅ H^{j+1}(Σ_{p}; A) for j > 0 since H^{∗}(Σ_{p}; A^{p}) ≅ H^{∗}(Σ_{p−1}; A) by Shapiro’s lemma).

With this notation, the maximal torus of PU(p) is P T = Σ U(1), the normalizer of P T is N(P T) = Σ U(1)⋊Σ_{p}, and the p-normalizer is N_{p}(P T) = Σ U(1)⋊⟨σ⟩ where ⟨σ⟩ is the cyclic subgroup of Σ_{p} generated by the p-cycle σ = (12 ⋯ p).

First, let ι: ℤ/p → U(1) be an identification between the cyclic p-group ℤ/p and the elements of order p in U(1). Then Σι: Σ(ℤ/p) → Σ U(1) = P T determines an isomorphism between the elementary abelian p-group Σ(ℤ/p) ≅ (ℤ/p)^{p−1} of rank p−1 and the elements of exponent p in the maximal torus P T. We may regard (Σι, Σ(ℤ/p)) as an object of the Quillen category 𝔸(PU(p)).

Next, let ν: ℤ/p ⊕ ℤ/p → PU(p) be the monomorphism that takes the canonical basis elements, e_{1} and e_{2}, of ℤ/p ⊕ ℤ/p to the diagonal matrix ν(e_{1}) = diag(1, ζ, …, ζ^{p−1}), where ζ is a primitive pth root of unity, and ν(e_{2}) = σ.

Finally, let λ: ℤ/p → PU(p) denote the restriction of ν to the first ℤ/p-summand. Define 𝔸 to be the full subcategory of 𝔸(PU(p)) with these three objects.

The image of λ, λ(ℤ/p) = P T^{⟨σ⟩} = Z(N_{p}(P T)), is the σ-invariant subgroup of the maximal torus which is the center of the p-normalizer N_{p}(P T) of the maximal torus. The image of ν is ν(ℤ/p ⊕ ℤ/p) = P T^{⟨σ⟩} × ⟨σ⟩.

In fact λ is the unique rank one object of the Quillen category with disconnected centralizer and ν is the unique object, up to isomorphism, with centralizer of less than maximal rank.

Lemma 2.1. Let (ℤ/p, ε) be a rank one object of 𝔸(PU(p)) with disconnected centralizer C_{PU(p)}(ℤ/p, ε). Then (ℤ/p, ε) = (ℤ/p, λ) as objects of 𝔸(PU(p)).

Proof. If Ž = Ť^{Σ_{p}} denotes the diagonal in the Σ_{p}-module Ť = (ℤ/p^{∞})^{p} there is an induced group action

(2.2) Hom(ℤ/p, Ž) × Σ_{p}\ Hom(ℤ/p, Ť) ⟶ Σ_{p}\ Hom(ℤ/p, Ť) of the abelian group Hom(ℤ/p, Ž) = Rep(ℤ/p, Ž) ≅ ℤ/p on the set of Σ_{p}-orbits Σ_{p}\ Hom(ℤ/p, Ť) = Rep(ℤ/p, U(p)) [24, 3.4.(1)]. The latter set of representations can be identified to the set of partitions (n_{0}, n_{1}, …, n_{p−1}) of p into sums of p nonzero integers and ℤ/p acts by cyclic permutations. Thus the only nontrivial isotropy subgroup occurs at (1, 1, …, 1) corresponding to the representation λ.

Now note that the principal B U(1)-fibration B U(p) ⟶ B PU(p) determines a principal map(Bℤ/p, B U(1))-fibration map(Bℤ/p, B U(p)) ⟶ map(Bℤ/p, B PU(p)). The associated exact homotopy sequence shows that the component group of the centralizer C_{PU(p)}(ℤ/p, ε) is isomorphic to the isotropy subgroup for the action (2.2) of Rep(ℤ/p, U(1)) on ε ∈ Rep(ℤ/p, U(p)). As we have just seen, this isotropy subgroup is nontrivial if and only if ε = λ.

Lemma 2.3. Let E be an elementary abelian p-group and ε: E → PU(p) a monomorphism with centralizer C_{PU(p)}(E, ε) of rank less than p − 1. Then (E, ε) is isomorphic in 𝔸(PU(p)) to (V, ν).

Proof. It is possible to find [21] a factorization ε: E → P Ť⋊⟨σ⟩ af ε through (the discrete approximation to) the p-normalizer of the maximal torus such that the induced homomorphism C_{P Ť⋊⟨σ⟩}(ε) ⟶ C_{PU(p)}(ε) of centralizers is conjugate to (the discrete approximation to) the pnormalizer of a maximal torus. The drop in rank forces the composite E ⟶ P Ť⋊⟨σ⟩ ⟶ ⟨σ⟩ to be surjective; let E_{1} denote the kernel of this epimorphism. Then the monomorphism ε embeds E_{1} into P Ť^{⟨σ⟩} = λ(ℤ/p). Thus E has rank two and ε|E_{1} is isomorphic in 𝔸(PU(p)) to (ℤ/p, λ). Let E_{2} ≅ ℤ/p be a complement to E_{1} and let ε_{2} : E_{2} → C_{P Ť⋊⟨σ⟩} = P Ť⋊⟨σ⟩ be the adjoint of ε. Then ε_{2} is a lift

\xymatrix{ & P Ť⋊⟨σ⟩ \ar[d] \\ E_{2} \ar[r]_{≅} \ar[ur]^{ε_{2}} & ⟨σ⟩}

of an isomorphism E_{2} ⟶^{≅} ⟨σ⟩. However, there is essentially just one such lift for H^{1}(E_{2}; P Ť) = 0 since H^{1}(E_{2}; Ť) = 0 by Shapiro’s lemma and also H^{2}(E_{2}; Ž) = 0. Therefore, ε_{2} is P Ť-conjugate to the canonical lift E_{2} ⟶^{≅} ⟨σ⟩↣P Ť⋊⟨σ⟩ and this shows that (E, ε) is isomorphic in 𝔸(PU(p)) to (V, ν).

We shall now determine the morphisms in the category 𝔸.

Let τ denote the element

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

of the special linear group SL(2, p). The cyclic group of order p generated by τ, ⟨τ⟩, is a Sylow p-subgroup of SL(2, p).

Proposition 2.4. In the category A the morphisms are as follows.

1. The groups of endomorphisms are • 𝔸(Σι) ≅ Σ_{p}

• 𝔸(ν) ≅ SL(2, p)

• 𝔸(λ) ≅ (ℤ/p)^{∗}

2. The above groups act transitively on appropriate sets of incoming morphisms giving bijections • 𝔸(λ, Σι) = Σ_{p}/⟨σ⟩

• 𝔸(λ, ν) = SL(2, p)/⟨τ⟩

• 𝔸(ν, Σι) = ∅

3. The group 𝔸(λ) acts on the above sets of morphisms and • A(λ, Σι)/𝔸(λ) = Σ_{p}/N(⟨σ⟩)

• 𝔸(λ, ν)/𝔸(λ) = SL(2, p)/N(⟨τ⟩) where N(⟨σ⟩) and N(⟨τ⟩) are the normalizers of the cyclic Sylow p-subgroups ⟨σ⟩ < Σ_{p} and ⟨τ⟩ < SL(2, p), respectively.

Before the proof, some remarks of a general nature.

Remark 2.5. Let X be any p-compact group with maximal torus i: T → X and Weyl group W. Let E_{1} and E_{2} be nontrivial elementary abelian p-groups and μ_{1} : E_{1} → T and μ_{2} : E_{2} → T two monomorphisms giving two objects ν_{1} = i ∘ μ_{1} and ν_{2} = i ∘ μ_{2} of the Quillen category 𝔸(X). We want to investigate the set 𝔸(X)(ν_{1}, ν_{2}) of 𝔸(X)-morphisms.

Let μ̌_{1} : E_{1} → Ť and μ̌_{2} : E_{2} → Ť be discrete approximations to μ_{1} and μ_{2}, respectively. Then there are actions

\overline{W} (μ_{2}) × \overline{W} (μ_{1}, μ_{2}) × \overline{W} (μ_{1}) ⟶ \overline{W} (μ_{1}, μ_{2}) (2.6)

of the groups

\overline{W} (μ_{i}) = \{w ∈ W | wμ̌_{i}(E_{i}) = μ̌_{i}(E_{i})\}, i = 1, 2

of Weyl group elements that preserve the image of μ̌_{i} on the set

\overline{W} (μ_{1}, μ_{2}) = \{w ∈ W | wμ̌_{1}(E_{1}) < μ̌_{2}(E_{2})\}

of Weyl group elements that take the image of μ̌_{1} into the image of μ̌_{2}.

Put also

W (μ_{i}) = {w ∈ W | wμ̌_{i} = μ̌_{i}}, i = 1, 2.

Since W (μ_{1}) ⊲ \overline{W} (μ̌_{1}) and W (μ_{2}) ⊲ \overline{W} (μ̌_{2}) are normal subgroups and W (μ_{2})w ⊂ wW (μ_{1}) for all w ∈ W (μ_{1}, μ_{2}), the actions (2.6) induce actions

\overline{W} (μ_{2})/W (μ_{2}) × \overline{W} (μ_{1}, μ_{2})/W (μ_{1}) × \overline{W} (μ_{1})/W (μ_{1}) ⟶ \overline{W} (μ_{1}, μ_{2})/W (μ_{1})

that can be identified to the actions

𝔸(X)(ν_{2}) × 𝔸(X)(ν_{1}, ν_{2}) × 𝔸(X)(ν_{1}) ⟶ 𝔸(X)(ν_{1}, ν_{2}) (2.7)

of the automorphism groups of ν_{1} and ν_{2} on the set of morphisms of ν_{1} into ν_{2}: The identification

\overline{W} (μ_{1}, μ_{2})/W (μ_{1}) = 𝔸(X)(ν_{1}, ν_{2}) (2.8)

associates to the the orbit wW (μ_{1}) the unique monomorphism f: E_{1} → E_{2} such that wμ̌_{1} = μ̌_{2} ∘f. In particular, using [31, 10.7.1] or [8, 4.3] we get

𝔸(X)(τ) = W (2.9)

when X is connected and τ is the inclusion into X of the maximal elementary abelian p-subgroup t of Ť (and p > 2). These assertions are consequences of the fact [24, 3.4] [11, 3.4] that [BE, BT]/W injects into the representation set [BE, BX] for any elementary abelian p-group E.

Let now w̌ be a representative for an element of \overline{W} (μ_{1}, μ_{2}) ⊂ W = Ň/Ť. Conjugation by w,given by c(w)(n) = wnw^{−1}, n ∈ Ň, takes E_{1} into E_{2} and conjugation by w^{−1}, c(w^{−1}), takes CŇ(E_{2}) into CŇ(E_{1}) such that the diagram

\xymatrix{CŇ(E_{1}) × E_{1} \ar[d]_{e} & CŇ(E_{2}) × E_{1} \ar[l]_{c(w^{−1})×1} \ar[d]^{1×c(w)} \\ Ň & CŇ(E_{2}) × E_{2} \ar[l]^{e}}

where e is group multiplication, commutes up to inner automorphism of Ň (as namely c(w) ∘ e ∘ (c(w^{−1}) × 1) = e ∘ (1 × c(w))). Therefore, the diagram of adjoint maps between spaces

\xymatrix{BCŇ(E_{1}) \ar[d]_{≃} & BCŇ(E_{2}) \ar[d]^{≃} \ar[l]_{Bc(w^{−1})} \\ map(BE_{1}, BŇ)_{Bμ_{1}} & map(BE_{2}, BŇ)_{Bμ_{2}} \ar[l]^{\overline{Bc(w)}}}

Does this extend to is homotopy commutative. This shows that the map \overline{c(w)}: C_{X}(E_{2}) → C_{X}(E_{1}) induced by the E < Ň? 𝔸(X)-morphism w is lifts to the map c(w^{−1}): CŇ(E_{2}) → CŇ(E_{1}) between maximal torus nor-

malizers. In particular, the map Z(\overline{c(w)}): Z(C_{X}(E_{1})) → Z(C_{X}(E_{2})) between centers equals the corresponding map Z(c(w^{−1})): Z(CŇ(E_{1})) → Z(CŇ(E_{2})) [22, §3].

.

Remark 2.10. Define 𝔸(W) to be the category whose objects are nontrivial elementary abelian p-subgroups of t < Ť and whose morphisms are group monomorphisms induced by W; i.e.

𝔸(W)(E_{1}, E_{2}) = \overline{W} (E_{1}, E_{2})/W (E_{1}) ⊂ (W/W (E_{1}))^{W (E_{2})} = O(W)(W/W (E_{2}), W/W (E_{1}))

is the set of morphism from E_{1} ≤ t to E_{2} ≤ t. There are functors

O(W)^{op} ⟵ 𝔸(W) ⟶ 𝔸(X)

where O(W) is the orbit category of W. The functor to the Quillen category 𝔸(X) is faithful and full; it is an equivalence of categories if any elementary abelian p-subgroup of X factors through the maximal torus T. Thus the Quillen category contains no more information than the Weyl group representation in t in these cases. The functor to the opposite orbit category, which takes the elementary abelian p-group E ≤ t to W/W (E), is faithful and also full provided

w^{−1}W (E_{2})w ≤ W (E_{1}) ⇒ w(E_{1}) ≤ E_{2}

holds for all w ∈ W and all nontrivial E_{1} ≤ t, E_{2} ≤ t.

Proof of Proposition 2.4. The formulas involving Σι and λ only follow from (2.8, 2.9) as W (λ) = ⟨σ⟩ and \overline{W} (λ) = N(⟨σ⟩).

By a slight abuse of notation, ν(ℤ/p ⊕ ℤ/p) = ⟨λ, σ⟩ where λ = ν(e_{1}) = diag(1, ζ, …, ζ^{p−1}) and σ = ν(e_{2}) is the permutation matrix. The group 𝔸(ν) is the group of automorphisms, λ ⟶ λ^{a}σ^{b}, σ ⟶ λ^{c}σ^{d}, of ⟨λ, σ⟩ induced by inner automorphisms of PU(p). Any such automorphism lifts to an automorphism of the preimage H in U(p) of ⟨λ, σ⟩. The relation λσ = ζσλ which holds in H forces ad − bc ≡ 1 mod p.

To show that 𝔸(ν) is all of SL(⟨λ, σ⟩) it suffices to realize the two generators [33, 6.21]

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )

of SL(2, p) as restrictions of inner automorphisms in PU(p). For this we use the matrices

diag(ζ_{0}, ζ_{1}, …, ζ_{p−1}) and p^{− \frac{1}{2}} A

where ζ_{0} = 1 and ζ_{i} = ζ^{i}ζ_{i−1}, 1 ≤ i ≤ p − 1, and where the first row in the matrix A is

1A = (ζ_{0}, ζ_{1}, …, ζ_{p−1})

and the entries of subsequent rows are given by a_{i,σ(j)} = a_{i−1,j} for 2 ≤ i ≤ p. (For instance, for p = 5, the first generator is realized as conjugation by the diagonal matrix

diag(1, ζ, ζ^{3}, ζ, 1)

and the second generator by

\frac{1}{\sqrt{5}} ( \begin{matrix}1 & ζ & ζ^{3} & ζ & 1 \\ 1 & 1 & ζ & ζ^{3} & ζ \\ ζ & 1 & 1 & ζ & ζ^{3} \\ ζ^{3} & ζ & 1 & 1 & ζ \\ ζ & ζ^{3} & ζ & 1 & 1 \end{matrix} )

where any row is obtained from the one above by a shift to the right.)

The set 𝔸(λ, ν) of morphisms from λ to ν is the set of all monomorphisms f: ℤ/p → ℤ/p ⊕ ℤ/p. To see this, observe that ν ∘ f and λ both equal the regular representation for any f. The two formulas involving 𝔸(λ, ν) now easily follow.

Finally, there can be no morphisms from ν to Σι since the automorphism group 𝔸(ν) is not isomorphic to a subgroup of the Weyl group Σ_{p} (or because the image of ν is known to be a maximal elementary abelian p-group [2, 6.4]).

Here is a graphic presentation

[figure: Σι; Σ_{p}; λ; (ℤ/p)^{∗}; Σ_{p}/⟨σ⟩; SL(2,p)/⟨τ⟩; ν; SL(2,p)]

of the (very) small category 𝔸.

We now determine the centralizers and their centers for the three objects of 𝔸 Proposition 2.11. The centralizers of the three objects of the category 𝔸 are

• C_{PU(p)}(Σι) = P T

• C_{PU(p)}(λ) = N_{p}(P T)

• C_{PU(p)}(ν) = ν(ℤ/p ⊕ ℤ/p) with centers

• Z(C_{PU(p)}(Σι)) = P T

• Z(C_{PU(p)}(λ)) = λ(ℤ/p)

• Z(C_{PU(p)}(ν)) = ν(ℤ/p ⊕ ℤ/p)

Proof. The centralizers in the maximal torus normalizer of the elementary abelian p-groups Σι and λ(ℤ/p) = P T^{⟨σ⟩}

C_{P T⋊Σ_{p}}(Σι) = P T and C_{P T⋊Σ_{p}}(λ) = P T ⋊⟨σ⟩

are p-compact toral groups so by N-determinism [22] they equal the centralizers taken in PU(p). Now the computation

C_{PU(p)}(ν) = C_{C_{PU(p)(λ)}}(⟨σ⟩) = C_{P T⋊⟨σ⟩}(σ) = ⟨λ, σ⟩ = ν

shows that ν is a self-centralizing elementary abelian p-group.

Let now BC_{PU(p)}(−) denote the contravariant functor from 𝔸 to topological spaces that takes the object (E, ε) of 𝔸 to the space

BC_{PU(p)}(E, ε) = map(BE, BP U(p))_{Bε}

which is homotopy equivalent to the p-completion of the classifying space of the centralizer C_{PU(p)}(E, ε) of ε(E) in PU(p) [12, 1.1] [7, 2.5]. Our interest in the category 𝔸 comes from the following fundamental approximation theorem of Jackowski and McClure [17, 6.8, 7.7].

Theorem 2.12. The canonical map

hocolim_{𝔸^{op}} BC_{PU(p)}(−) ⟶ BP U(p)

is an H^{∗}𝔽_{p}-equivalence

Thus B PU(p) is the homotopy colimit of the 𝔸-diagram

[figure: BP (T); (Σ_{p})^{op}; ⟨σ⟩^{op}\Σ_{p}^{op}; BN_{p}(T); (ℤ/p)^{∗}; BV; SL(V)^{op}; ⟨τ⟩^{op}\ SL(V)^{op}]

(2.10)

of classifying spaces.

When using this decomposition later on certain obstruction groups will occur.

Let BZC_{PU(p)}(−) denote the covariant functor from 𝔸 to topological spaces that takes the object (E, ε) to

map(BC_{PU(p)}(E, ε), BP U(p))_{Be(ε)} (2.13)

where Be(ε): BC_{PU(p)}(E, ε) → BP U(p) is the evaluation map. Then BZC_{PU(p)}(E, ε) is homotopy equivalent to the p-completion of the classifying space of the center Z(C_{PU(p)}(E, ε)) of the centralizer; this follows from [10, 1.3, 12.1] and (2.16) at the end of this section. The composite functor π_{1}(BZC_{PU(p)}(−)) produces an 𝔸-module of the form

[figure: 0; Σ_{p}; ℤ/p; (ℤ/p)^{∗}; Σ_{p}/⟨σ⟩; SL(V)/⟨τ⟩; V; SL(V)] (2.14)

where V = ℤ/p ⊕ ℤ/p. The composite functor π_{2}(BZC_{PU(p)}(−)) produces the 𝔸-module

[figure: Σℤ_{p}; Σ_{p}; 0; (ℤ/p)^{∗}; Σ_{p}/⟨σ⟩; SL(V)/⟨τ⟩; 0; SL(V)] (2.15) of abelian groups.

In the next section we shall compute the higher limites of these two 𝔸-diagrams of abelian groups.

I close this section with the following well-known result which was used in connection with the mapping spaces of (2.13).

Let P be a p-toral Lie group (i.e. the identity component of P is a torus and π_{0}(P) is a finite p-group), G a compact Lie group having a finite p-group as its component group, and C_{G}(P) the Lie group centralizer, which also has a finite p-group as component group [15, A4], of a Lie group homomorphism f: P → G. The standard Lie group multiplication homomorphism C_{G}(P)×P → G extending f induces a p-compact group morphism { \atop C_{G}(P)}×P̂ → Ĝ extending f̂: P̂ → Ĝ. We shall now see that { \atop C_{G}(P)} = CĜ(P̂) and in particular that { \atop Z(P)} = Z(P̂), i.e. that centralizers and centers of p-toral Lie groups can be computed either in the Lie group category or in the p-compact group category.

Lemma 2.16 (Cf. [26]). The adjoint

B{ \atop C_{G}(P)} → map(BP̂, BĜ)_{Bf̂}

of the above standard morphism is a homotopy equivalence. In particular,

B{ \atop Z(P)} ≃ map(BP̂, BP̂)_{B1}

where Z(P) is the Lie group center of P.

Proof. The p-toral Lie group P contains [16, 1.1] a dense p-discrete toral subgroup P̌ = ⋃ P̌_{m} which is the union of an ascending sequence of finite p-groups P̌_{m}. The inclusion of P̌ into P induces a discrete approximation i: P̌ → P̂ to the p-compact toral group P̂ and so we have homotopy equivalences [9, §6]

map(BP̂, BĜ)_{Bf̂} ≃ map(BP̌, BĜ)_{Bf̂i} ≃ map(BP̌_{m}, BĜ)_{B(f̂i|P̌_{m})}

for m large enough. In particular, the above mapping spaces are 𝔽_{p}-complete [10, 2.5] [9, 6.20]. Furthermore, by Dwyer-Zabrodsky [12, 1.1] and [7, 2.5] or Lannes [18], the canonical map

BC_{G}(P̌_{m}) → map(BP̌_{m}, BĜ)_{B(f̂i|P̌_{m})}

is an H^{∗}𝔽_{p}-equivalence and here

C_{G}(P̌_{m}) ≅ C_{G}(P̌) ≅ C_{G}(P)

when m is large enough and since P̌ is dense in P.

3. The Aguadé spectral sequence

In his construction of modular classifying spaces, J. Aguadé [1] analyzed higher limits taken over small categories with just two objects. This section contains a generalization in the form of a spectral sequence of his long exact sequence.

Suppose that 𝕀 is a finite category with two objects, 0 and 1, of the form

[figure: 0; 𝕀(0); 𝕀(0,1); 1; 𝕀(1)]

where • the morphism monoids 𝕀(0) = 𝕀(0, 0) and 𝕀(1) = 𝕀(1, 1) are groups

• 𝕀(1, 0) = ∅ and let M be any 𝕀(1)-module. Alternatively, we may view M as the functor from 𝕀 to abelian groups with values M(0) = {0} and M(1) = M.

Let M[𝕀(0, 1)] denote the abelian group of all functions u from the morphism set 𝕀(0, 1) to M. The composition functions

𝕀(1) × 𝕀(0, 1) × 𝕀(0) ⟶ 𝕀(0, 1)

make M[𝕀(0, 1)] into a left 𝕀(1)-module by the rule

(a_{1}u)(a) = a_{1}u(a_{1}^{−1}a), a_{1} ∈ 𝕀(1), a ∈ 𝕀(0, 1),

and into a left 𝕀(0)-module by the rule

(a_{0}u)(a) = u(aa_{0}), a_{0} ∈ 𝕀(0), a ∈ 𝕀(0, 1).

These two actions commute, a_{1}(a_{0}u) = a_{0}(a_{1}u), so that in fact the group 𝕀(0) acts on the the 𝕀(1)- module M[𝕀(0, 1)] and on the cohomology groups H^{∗}(𝕀(1); M[𝕀(0, 1)]) of 𝕀(1) with coefficients in the 𝕀(1)-module M[𝕀(0, 1)]. Furthermore, the diagonal map Δ: M → M[𝕀(0, 1)], given by Δ(m)(a) = m for all m ∈ M, a ∈ 𝕀(0, 1), is a homomorphisms of 𝕀(1)-modules with 𝕀(0)-actions (trivial 𝕀(0)- action on M). Therefore, the induced homomorphism Δ_{∗}: H^{∗}(𝕀(1); M) → H^{∗}(𝕀(1); M[𝕀(0, 1)]) factors through the subgroup H^{0}(𝕀(0); H^{∗}(𝕀(1); M[𝕀(0, 1)])) of 𝕀(0)-invariant cohomology classes.

Theorem 3.1. There exists a (cohomological) first quadrant spectral sequence

E_{2}^{pq} ⇒ lim_{𝕀}^{p+q}M

where E_{2}^{pq} = H^{p−1}(𝕀(0); H^{q}(𝕀(1); M[𝕀(0, 1)])) for p ≥ 2 while the exact sequence

0 ⟶ E_{2}^{0∗} ⟶ H^{∗}(𝕀(1); M) ⟶^{Δ_{∗}} H^{0}(𝕀(0); H^{∗}(𝕀(1); M[𝕀(0, 1)])) ⟶ E_{2}^{1∗} ⟶ 0

determines the vertical lines p = 0 and p = 1.

Proof. The graded abelian group lim_{𝕀}^{∗}M is [30, Lemma 2] [3, XI.6.2] the cohomology of a cochain complex (C, δ) of the form

0 ⟶ M ⟶^{δ^{0}} M^{𝕀(1)} × M^{𝕀(0,1)} ⟶^{δ^{1}} M^{𝕀(1)×𝕀(1)} × M^{𝕀(1)×𝕀(0,1)} × M^{𝕀(0,1)×𝕀(0)} ⟶ ⋯

with general term

C^{n} = M^{𝕀(1)^{n}} × M^{𝕀(1)^{n−1}×𝕀(0,1)} × M^{𝕀(1)^{n−2}×𝕀(0,1)×𝕀(0)} × ⋯ × M^{𝕀(1)×𝕀(0,1)×𝕀(0)^{n−2}} × M^{𝕀(0,1)×𝕀(0)^{n−1}}

= M^{𝕀(1)^{n}} × ∏_{r+s=n−1} M^{𝕀(1)^{r}×𝕀(0,1)×𝕀(0)^{s}}

and with coboundary map δ^{n}: C^{n} → C^{n+1} given by

δ^{n}(u)(a_{n+1}, a_{n}, …, a_{1}) =

u(a_{n+1}, …, a_{2}) − u(a_{n+1}, …, a_{3}, a_{2}a_{1}) + ⋯ + (−1)^{n+1}M(a_{n+1})u(a_{n}, …, a_{1})

for all u ∈ C^{n} and for all (n + 1)-tuples (a_{n+1}, …, a_{1}) of composable arrows in the category 𝕀. The spectral sequence of the theorem arises from the descending filtration

C = C_{0} ⊃ C_{1} ⊃ ⋯ C_{p} ⊃ C_{p+1} ⊃ ⋯ ⊃ {0}

where C_{p}, p ≥ 1, is the subcomplex

0 ⟶ ⋯ ⟶ 0 ⟶ M^{𝕀(0,1)×𝕀(0)^{p−1}} ⟶ M^{𝕀(1)×𝕀(0,1)×𝕀(0)^{p−1}} × M^{𝕀(0,1)×𝕀(0)^{p}} ⟶ ⋯

that starts at degree p and has general term

C_{p}^{n} = M^{𝕀(1)^{n−p}×𝕀(0,1)×𝕀(0)^{p−1}} × M^{𝕀(1)^{n−p−1}×𝕀(0,1)×𝕀(0)^{p}} × ⋯

⋯ × M^{𝕀(1)×𝕀(0,1)×𝕀(0)^{n−2}} × M^{𝕀(0,1)×𝕀(0)^{n−1}}

for n ≥ p. Thus the quotient complex C_{p}/C_{p+1}, p ≥ 0,

0 ⟶ ⋯ ⟶ 0 ⟶ M^{𝕀(0,1)×𝕀(0)^{p−1}} ⟶^{δ^{p}} M^{𝕀(1)×𝕀(0,1)×𝕀(0)^{p−1}} ⟶ ⋯

starts at degree p and has

(C_{p}/C_{p+1})^{n} = M^{𝕀(1)^{n−p}×𝕀(0,1)×𝕀(0)^{p−1}}

as general term for n ≥ p.

In particular, C_{0}/C_{1} is the cochain complex

0 ⟶ M ⟶ M^{𝕀(1)} ⟶ M^{𝕀(1)×𝕀(1)} ⟶ ⋯ ⟶ M^{𝕀(1)^{n}} ⟶ ⋯

with cohomology

E_{1}^{0q} = H^{q}(C_{0}/C_{1}) = H^{q}(𝕀(1); M).

The subquotient complex C_{1}/C_{2} has the form

0 ⟶ 0 ⟶ M^{𝕀(0,1)} ⟶^{δ^{1}} M^{𝕀(1)×𝕀(0,1)} ⟶^{δ^{2}} ⋯ ⟶ M^{𝕀(1)^{n−1}×𝕀(0,1)} ⟶^{δ^{n}} ⋯

with coboundary map δ^{n}: M^{𝕀(1)^{n−1}×𝕀(0,1)} → M^{𝕀(1)^{n}×𝕀(0,1)} given by

− δ^{n}(u)(a_{n}, …, a_{1}, a) =

u(a_{n}, …, a_{2}, a_{1}a) − u(a_{n}, …, a_{2}a_{1}, a) + ⋯ + (−1)^{n}a_{n}u(a_{n−1}, …, a_{1}, a)

for u ∈ M^{𝕀(1)^{n−1}×𝕀(0,1)} and a_{n}, …, a_{1} ∈ 𝕀(1), a ∈ 𝕀(0, 1). We want to compare this cochain complex and the cochain complex

0 ⟶ M[𝕀(0, 1)] ⟶^{d^{0}} M[𝕀(0, 1)]^{𝕀(1)} ⟶^{d^{1}} M[𝕀(0, 1)]^{𝕀(1)^{2}} ⟶ ⋯

with coboundary map

d^{n−1} : M[𝕀(0, 1)]^{𝕀(1)^{n−1}} → M[𝕀(0, 1)]^{𝕀(1)^{n}}

given by

d^{n−1}(u)(a_{n}, …, a_{1}, a) =

u(a_{n}, …, a_{2}, a) − u(a_{n}, …, a_{2}a_{1}, a) + ⋯ + (−1)^{n}a_{n}u(a_{n−1}, …, a_{1}, a_{n}^{−1}a)

for u ∈ M[𝕀(0, 1)]^{𝕀(1)^{n−1}} = M^{𝕀(1)^{n−1}×𝕀(0,1)} and a_{n}, …, a_{1} ∈ 𝕀(1), a ∈ 𝕀(0, 1). In fact, these two cochain complexes are isomorphic by means of the cochain complex isomorphism

\xymatrix{⋯ \ar[r] & M^{𝕀(1)^{n−1}×𝕀(0,1)} \ar[r]^{δ^{n}} & M^{𝕀(1)^{n}×𝕀(0,1)} \ar[r] & ⋯ \\ ⋯ \ar[r] & M[𝕀(0, 1)]^{𝕀(1)^{n−1}} \ar[u]^{s^{n−1}} \ar[r]^{d^{n−1}} & M[𝕀(0, 1)]^{𝕀(1)^{n}} \ar[u]^{s^{n}} \ar[r] & ⋯}

where the 𝕀(0)-module isomorphism s^{n} is given by

s^{n}(u)(a_{n}, …, a_{1}, a) = (−1)^{n}u(a_{n}, …, a_{1}, a_{n} ⋯ a_{1}a)

for all a_{n}, …, a_{1} ∈ 𝕀(1) and a ∈ 𝕀(0, 1). Commutativity of the above diagram follows from the computation

−δ^{n}(s^{n−1}(u))(a_{n}, …, a_{1}, a)
= s^{n−1}(u)(a_{n}, …, a_{2}, a_{1}a) − s^{n−1}(u)(a_{n}, …, a_{2}a_{1}, a) + ⋯
⋯ + (−1)^{n}a_{n}s^{n−1}(u)(a_{n−1}, …, a_{1}, a)
= (−1)^{n−1}(u(a_{n}, …, a_{2}, a_{n} ⋯ a_{1}a) − u(a_{n}, …, a_{2}a_{1}, a_{n} ⋯ a_{1}a) + ⋯
⋯ + (−1)^{n}a_{n}u(a_{n−1}, …, a_{1}, a_{n} ⋯ a_{1}a))
= (−1)^{n−1}d^{n−1}(u)(a_{n}, …, a_{1}, a_{n} ⋯ a_{1}a)
= −(s^{n}(d^{n−1}(u)))(a_{n}, …, a_{1}, a)

for all a_{n}, …, a_{1} ∈ 𝕀(1) and a ∈ 𝕀(0, 1). Since the lower cochain complex is the standard cochain complex for the cohomology of 𝕀(1) with coefficients in M[𝕀(0, 1)], we get isomorphisms

E_{1}^{1q} = H^{1+q}(C_{1}/C_{2}) ≅ H^{q}(𝕀(1); M[𝕀(0, 1)])

of 𝕀(0)-modules for all q ≥ 0.

In general, for p ≥ 2,

(C_{p}/C_{p+1})^{n} = M^{𝕀(1)^{n−p}×𝕀(0,1)×𝕀(0)^{p−1}} = ((C_{1}/C_{2})^{n−(p−1)})^{𝕀(0)^{p−1}}

and an inspection of the coboundary map reveals that the complex C_{p}/C_{p+1} is just the (diagonal) complex (C_{1}/C_{2})^{𝕀(0)^{p−1}} up to a shift of degrees and consequently

E_{1}^{pq} = H^{p+q}(C_{p}/C_{p+1}) = H^{1+q}(C_{1}/C_{2})^{𝕀(0)^{p−1}} = H^{q}(𝕀(1); M[𝕀(0, 1)])^{𝕀(0)^{p−1}}

for p ≥ 2.

We now determine the differential d_{1} : E_{1}^{pq} → E_{1}^{p+1q} in the spectral sequence.

For p = 0 we have a d_{1}-differential

d_{1} : E_{1}^{0q} = H^{q}(𝕀(1); M) → E_{1}^{1q} = H^{q}(𝕀(1); M[𝕀(0, 1)])

easily seen to be induced by the diagonal homomorphism Δ: M → M[𝕀(0, 1)].

For p ≥ 1, the d_{1}-differential

d_{1} : E_{1}^{pq} = H^{q}(𝕀(1); M[𝕀(0, 1)])^{𝕀(0)^{p−1}} → E_{1}^{p+1q} = H^{q}(𝕀(1); M[𝕀(0, 1)])^{𝕀(0)^{p}}

is induced from the connecting homomorphism of the short exact sequence

0 ⟶ C_{p+1}/C_{p+2} ⟶ C_{p}/C_{p+2} ⟶ C_{p}/C_{p+1} ⟶ 0

of cochain complexes. It follows that d_{1} is induced from the cross term

δ: M^{𝕀(1)^{q}×𝕀(0,1)×𝕀(0)^{p−1}} → M^{𝕀(1)^{q}×𝕀(0,1)×𝕀(0)^{p}}

of the cochain complex (C, δ). Writing out this component of the coboundary map reveals that d_{1} is the usual coboundary map for the cohomology of 𝕀(0) so that

E_{2}^{pq} = H(E_{1}^{pq}, d_{1}) = H^{p−1}(𝕀(0); H^{q}(𝕀(1); M[𝕀(0, 1)]))

for p ≥ 2.

Here are some special cases of this spectral sequence.

Corollary 3.2. Suppose in addition to the assumptions of (3.1) that multiplication by the order |𝕀(0)| of the group 𝕀(0) is an isomorphism on the graded abelian group H^{∗}(𝕀(1); M[𝕀(0, 1)]). Then there is a long exact sequence

⋯ ⟶ H^{n−1}(𝕀(1); M) ⟶^{Δ_{∗}} H^{0}(𝕀(0); H^{n−1}(𝕀(1); M[𝕀(0, 1)])) ⟶ lim_{𝕀}^{n}M ⟶ H^{n}(𝕀(1); M) ⟶ ⋯

of abelian groups.

Proof. The spectral sequence of (3.1) degenerates to a long exact sequence since

E_{2}^{pq} = H^{p−1}(𝕀(0); H^{q}(𝕀(1); M[𝕀(0, 1)])) = 0

for p ≥ 2.

Another special case occurs when the group 𝕀(1) acts transitively on the set 𝕀(0, 1) of incoming morphisms. Pick a morphism a: 0 → 1 and consider the two subgroups

B(1) = {a_{1} ∈ 𝕀(1) | a_{1}a ∈ a𝕀(0)}
C(1) = {a_{1} ∈ 𝕀(1) | a_{1}a = a}

of 𝕀(1). Then we have a commutative diagram

\xymatrix{𝕀(1)/C(1) \ar[d] \ar[r]^{≅} & 𝕀(0, 1) \ar[d] \\ 𝕀(1)/B(1) \ar[r]^{≅} & 𝕀(0, 1)/𝕀(0)}

where the horizontal maps are bijections of 𝕀(1)-sets. By Shapiro’s lemma [13, 4.1.3]

H^{∗}(𝕀(1); M[𝕀(0, 1)]) ≅ H^{∗}(C(1); M)

as the 𝕀(1)-module M[𝕀(0, 1)] ≅ Hom_{ℤ}(ℤ[𝕀(1)/C(1)], M) is isomorphic to the 𝕀(1)-module induced from M viewed as a C(1)-module. Since C(1) is a normal subgroup of B(1), H^{∗}(C(1); M) is a B(1)/C(1)-module and an 𝕀(0)-module via the group epimorphism

𝕀(0) ⟶ B(1)/C(1)

that takes an element a_{0} ∈ 𝕀(0) to the coset a_{1}C(1) given by a_{1}a = aa_{0}. The restriction homomorphism

H^{∗}(𝕀(1); M) ⟶^{res^{∗}} H^{0}(𝕀(0); H^{∗}(C(1); M)) ⊂ H^{∗}(C(1); M)

from the cohomology of 𝕀(1) to the cohomology of C(1) factors through the subgroup of 𝕀(0)- invariant elements.

Corollary 3.3. Assume in addition to the assumptions of (3.1) that the group 𝕀(1) acts transitively on 𝕀(0, 1) and let C(1) < 𝕀(1) be defined as above. Then, in the spectral sequence of (3.1),

E_{2}^{pq} = H^{p−1}(𝕀(0); H^{q}(C(1); M))

for p ≥ 2 while the short exact sequence

0 ⟶ E_{2}^{0∗} ⟶ H^{∗}(𝕀(1); M) ⟶^{res^{∗}} H^{0}(𝕀(0); H^{∗}(C(1); M)) ⟶ E_{2}^{1∗} ⟶ 0

determines the vertical lines p = 0 and p = 1.

An even more special case, but in fact the only one that we shall need, occurs when both the extra conditions of (3.2) and (3.3) are in force, i.e. when multiplication by |𝕀(0)| is an isomorphism on the graded abelian group H^{∗}(𝕀(1); M[𝕀(0, 1)]) and 𝕀(1) acts transitively on 𝕀(0, 1). Then the spectral sequence of (3.1) takes the form of a long exact sequence

(3.4) ⋯ ⟶ H^{n−1}(𝕀(1); M) ⟶^{res^{∗}} H^{0}(𝕀(0); H^{n−1}(C(1); M)) ⟶ lim_{𝕀}^{n}M ⟶ H^{n}(𝕀(1); M) ⟶ ⋯

resembling the original long exact sequence of Aguadé [1]. Here,

H^{0}(𝕀(0); H^{n−1}(C(1); M)) = H^{n−1}(C(1); M)^{B(1)/C(1)} = H^{n−1}(B(1); M)

if M is an R[𝕀(1)]-module and |𝕀(0)| (or |B(1): C(1)|) is invertible in the ring R.

Now follow two lemmas to enhance the applicability of the spectral sequence (3.1).

Corollary 3.5. Let 𝕀 = 𝕀_{1} ∪ 𝕀_{2} be a category with three objects, 0, 1_{1}, and 1_{2}, of the shape

[figure: 1_{1}; 0; 𝕀_{1}; 𝕀_{2}; 1_{2}]

and let M be an 𝕀-module with M(0) = 0, M(1_{1}) = M_{1}, and M(1_{2}) = M_{2}. Then

lim_{𝕀_{1}∪𝕀_{2}}^{∗}M = lim_{𝕀_{1}}^{∗}M_{1} × lim_{𝕀_{2}}^{∗}M_{2}

where the 𝕀(1_{1})-module M_{1} is considered as an 𝕀_{1}-module and the 𝕀(1_{2})-module M_{2} as an 𝕀_{2}-module.

Proof. The cochain complex for computing lim_{𝕀_{1}∪𝕀_{2}}^{∗} M is the product of the cochain complexes for lim_{𝕀_{1}}^{∗} M_{1} and lim_{𝕀_{2}}^{∗} M_{2}.

Of course, (3.5) extends to an arbitrary star shaped finite category with outward pointing arrows.

Corollary 3.6. Let 𝕀 = 𝕀_{1} ∪ 𝕀_{2} be a category with three objects, 0_{1}, 0_{2}, and 1, of the shape

[figure: 0_{1}; 𝕀_{1}; 1; 0_{2}; 𝕀_{2}]

and let M be an 𝕀-module with M(0_{1}) = 0 = M(0_{2}) and M(1) = M. Then there is a Mayer-Vietoris sequence

⋯ → H^{j}(𝕀(1); M) → lim_{𝕀_{1}}^{j}M_{1} × lim_{𝕀_{2}}^{j}M_{2} → lim_{𝕀_{1}∪𝕀_{2}}^{j}M → H^{j+1}(𝕀(1); M) → ⋯

for the higher limits of M over 𝕀.

Proof. The cochain complex C(𝕀; M) for computing lim_{𝕀}^{∗} M is the push out

\xymatrix{C(𝕀(1); M) \ar[r] \ar[d] & C(𝕀_{1}(1); M) \ar[d] \\ C(𝕀_{2}(1); M) \ar[r] & C(𝕀; M)}

over the cochain complex for computing H^{∗}(𝕀(1); M) of the cochain complexes for computing lim_{𝕀_{1}}^{∗} M and lim_{𝕀_{2}}^{∗} M.

We now apply the above results to the particular of the the full subcategory 𝔸 of the Quillen category for PU(p).

Lemma 3.7. Let G be a finite group, H a subgroup of G, S_{p} a p-Sylow subgroup of G, and M a G-module. Assume that S_{p} is cyclic of order p and that multiplication by the index |G: S_{p}| is an automorphism of M. Then

1. \overline{H}^{∗}(G; M) = \overline{H}^{∗}(S_{p}; M)^{N_{G}(S_{p})/S_{p}} = \overline{H}^{∗}(N_{G}(S_{p}); M).

2. \overline{H}^{∗}(H; M) = \overline{H}^{∗}(G; M) if S_{p} < H < G and N_{H}(S_{p}) = N_{G}(S_{p}).

Proof. The monomorphic image of H^{∗}(N_{G}(S_{p}); M) in H^{∗}(S_{p}; M) is the invariant ring H^{∗}(S_{p}; M)^{N_{G}(S_{p})/S_{p}}. Since S_{p} is cyclic of order p, this is also the monomorphic image of H^{∗}(G; M) in H^{∗}(S_{p}; M) [13, 4.2.7].

An R[𝕀(0, 1)]-module, where R is a ring, is a functor from the category 𝕀(0, 1) to the category of R-modules.

Corollary 3.8. Let M be an R[𝕀(0, 1)]-module where R is a ring. Under the assumptions that • 𝕀(1) acts transitively on 𝕀(0, 1)

• |𝕀(0)| is invertible in R

• The Sylow p-subgroup, S_{p}, of 𝕀(1) is cyclic of order p and |𝕀(1): S_{p}| is invertible in R

• S_{p} is contained in B(1) and N_{B(1)}(S_{p}) = N_{𝕀(1)}(S_{p}) there is an exact sequence

0 → lim_{𝕀(0,1)}^{0}M → M(0)^{𝕀(0)} → M^{B(1)}/M^{𝕀(1)} → lim_{𝕀(0,1)}^{1}M → 0

and lim_{𝕀(0,1)}^{j}M = 0 for j > 1.

Proof. Let

⋯ → lim_{𝕀(0,1)}^{j}F_{0}(M) → lim_{𝕀(0,1)}^{j}M → lim_{𝕀(0,1)}^{j}F_{1}(M) → lim_{𝕀(0,1)}^{j+1}F_{0}(M) → ⋯

be the long exact sequence [3, XI.6.1] induced by the short exact sequence of R[𝕀(0, 1)]-modules

0 ⟶ F_{0}(M) ⟶ F (M) ⟶ F_{1}(M) ⟶ 0

where F_{0}(M) is the R[𝕀(0, 1)]-submodule of M with F_{0}(M)(0) = 0, F_{0}(M)(1) = M(1), and F_{1}(M) = M/F_{0}(M). Then

lim_{𝕀(0,1)}^{j}F_{0}(M) = \begin{cases}M(1)^{B(1)}/M(1)^{𝕀(1)} & j = 1 \\ 0 & j ≠ 1 \end{cases}

by the long exact sequence (3.4) and the isomorphism of (3.7.2), and

lim_{𝕀(0,1)}^{j}F_{1}(M) = H^{j}(𝕀(0); M(0)) = \begin{cases}M(0)^{𝕀(0)} & j = 0 \\ 0 & j ≠ 0 \end{cases}

since |𝕀(0)| is invertible in R.

Corollary 3.9. Let M be an R[𝔸]-module where R is either the field 𝔽_{p} or the ring ℤ_{p}. Then there is an exact sequence

0 → lim_{𝔸}^{0}M → M(λ)^{𝔸(λ)} → \frac{M(Σι)^{N(⟨σ⟩)} × M(ν)^{N(⟨τ⟩)}}{M(Σι)^{Σ_{p}} × M(ν)^{SL(V)}} → lim_{𝔸}^{1}M → 0

and lim_{𝔸}^{j}M = 0 for j > 1.

Proof. Filter the R[𝔸]-module as in the proof of (3.8) and use (3.5).

It is now easy to compute the higher limits of diagrams (2.14) and (2.15).

Corollary 3.10. The higher limits for diagram (2.14) are trivial:

lim_{𝔸}^{∗}π_{1}BZC_{PU(p)} = 0.

Corollary 3.11. The higher limits for diagram (2.15) are trivial:

lim_{𝔸}^{∗}π_{2}BZC_{PU(p)} = 0.

The final lemma, which will be used later, is an easy consequence of (3.7).

Lemma 3.12. With G and S_{p} as in (3.7),

H^{∗}(A⋊G; 𝔽_{p}) = H^{∗}(A⋊S_{p}; 𝔽_{p})^{N_{G}(S_{p})/S_{p}} = H^{∗}(A⋊N_{G}(S_{p}); 𝔽_{p})

for any G-module A.

Proof. The first identity follows from the Lyndon-Hochschield-Serre spectral sequence applied to the group extension

1 → A⋊S_{p} → A⋊N_{G}(S_{p}) → N_{G}(S_{p})/S_{p} → 1

where the order of the quotient group is prime to p. For the second identity we use the two spectral sequences

H^{∗}(N_{G}(S_{p}); H^{∗}(A; 𝔽_{p})) ⇒ H^{∗}(A⋊N_{G}(S_{p}); 𝔽_{p})
H^{∗}(G; H^{∗}(A; 𝔽_{p})) ⇒ H^{∗}(A⋊G; 𝔽_{p})

with isomorphic E_{2}-pages.

4. Determinism

We show that PU(p) viewed as a p-compact group is totally N-determined in the sense of [22, 7.1].

Let X be any p-compact group with maximal torus i: T → X and maximal torus normalizer j: N → X. Consider the group homomorphism [22, 3.2]

N: Out(X) → Out(N) (4.1)

obtained by restricting automorphisms of X to the maximal torus N. (Here, Out(X) (Out(N)) stands for the group of invertible elements of the monoid [BX, BX] ([BN, BN])). A connected p-compact group X is said to have N-determined automorphisms if (4.1) is injective. (See [22, 3.10] for the general definition.) All connected p-compact groups with Weyl group order prime to p have N-determined automorphisms.

We say that X is N-determined [22, 6.1] if any diagram of loop spaces of the form

(4.2)  \xymatrix{ & N \ar[dl]_{j} \ar[dr]^{j′} \\ X &  & X′}

where X′ is any p-compact group and j′: N → X′ a monomorphism conjugate to a maximal torus normalizer can be completed by an isomorphism f: X → X′ under N (up to conjugation).

Let from now on PU(p) stand for the p-compact group BP U(p)_{p}^{∧} and let j: N(P T) → PU(p) denote the maximal torus normalizer for PU(p). (We continue to use the same notation as for the Lie group PU(p) since there really is no conflict.) Then (4.1) takes the form of a a group homomorphism N: Out(PU(p)) → Out(N(P T)).

In the following we shall use repeatedly the preferred lifts introduced in [21]. These are defined as follows. Let ν: V → X be a monomorphism of a nontrivial elementary abelian p-group V into the p-compact group X. Then μ: V → N is a preferred lift of ν if the first two of the points

• j ∘ μ is conjugate to ν

• C_{j}: C_{N}(μ) → C_{X}(ν) is conjugate to the normalizer of a maximal torus of C_{X}(ν)

• The homomorphisms V ⟶^{π_{0}(μ)} π_{0}(N) ⟶ Aut(π_{1}(T)) has a nontrivial kernel. are satisfied; if also the third point is satisfied, then μ is a special preferred lift. The main result of [21] is that any ν admits a preferred lift and even a special preferred lift if X is assumed connected. If the monomorphism ν: V → X factors through the maximal torus then this factorization is a preferred lift of ν, in fact, up to conjugation in N, the only special preferred lift of ν.

Theorem 4.3. PU(p) has N-determined automorphisms.

Proof. Let f be any automorphism of PU(p) under N(P T), i.e. making the diagram

\xymatrix{ & N(P T) \ar[dl]_{j} \ar[dr]^{j} \\ PU(p) \ar[rr]_{f} &  & PU(p)}

commutative up to conjugation. We must show that f is conjugate to the identity morphism.

Let (E, ε) be any of the three objects in the category 𝔸. Choose a special preferred lift \overline{ε}: E → N(P T) of ε. The automorphism f preserves ε for f ∘ ε = f ∘ j ∘ \overline{ε} = j ∘ \overline{ε} = ε and we get an induced diagram

\xymatrix{ & C_{N(P T)}(\overline{ε}) \ar[dl]_{C_{j}}^{≅} \ar[dr]^{C_{j}}_{≅} \\ C_{PU(p)}(ε) \ar[rr]_{C_{f}} &  & C_{PU(p)}(ε)}

of centralizers. Since the centralizer of ε is a p-compact toral group, C_{j} will actually be an isomorphism of p-compact toral groups in this case and hence C_{f} must be conjugate to the identity morphism of C_{PU(p)}(ε).

Since the Wojtkowiak obstruction groups [35] lim_{𝔸}^{i}π_{i}BZC_{PU(p)}, i = 1, 2, associated to the Jackowski-McClure decomposition (2.10) are trivial (3.10, 3.11), this is enough to show that f is conjugate to the identity morphism of PU(p).

Theorem 4.4. PU(p) is N-determined.

Proof. The proof consists in checking that the proof of [22, 6.15], which is formulated for the full Quillen category, also applies to the full subcategory 𝔸 of (2.12).

We show first that diagram (4.2) can be completed at the level of cohomology. Note that all centralizers of nontrivial elementary abelian p-groups in PU(p) are totally N-determined as the Weyl groups of their identity components have orders prime to p [22, 3.9, 4.2, 6.7]. Note also that (V, ν) is, up to isomorphism the only object of the Quillen category 𝔸(PU(p)) which does not factor through the maximal torus and that the automorphism group 𝔸(V, ν) = SL(V) acts transitively on the set of lines in V (2.3, 2.4). It thus follows from [22, 7.15] that H^{∗}(Bj)H^{∗}(BP U(p); 𝔽_{p}) = H^{∗}(Bj′)H^{∗}(BX; 𝔽_{p}) in H^{∗}(BN(P T); 𝔽_{p}). As also H^{∗}(Bj) and H^{∗}(Bj′) are injective morphisms of unstable algebra over the Steenrod algebra, we get an isomorphism Φ: H^{∗}(BX; 𝔽_{p}) → H^{∗}(BP U(p); 𝔽_{p}) of unstable algebras over the Steenrod algebra inducing, via Lannes theory [18], an isomorphism Φ^{∗}: 𝔸(PU(p)) → 𝔸(X) of Quillen categories.

The monomorphisms λ: ℤ/p → PU(p) and Σι: Σℤ/p → PU(p) have uniqely determined special preferred lifts, \overline{λ} and Σ\overline{ι}. Put λ′ = j ∘ \overline{λ} and Σι′ = j′ ∘ Σ\overline{ι}. Again, since p-compact toral groups are N-determined [22, 6.5], there exists an esssentially unique isomorphism f(λ): C_{PU(p)} → C_{X}(λ′) under C_{N(P T)}(\overline{λ}) and an isomorphism f(Σι): C_{PU(p)}(Σι) → C_{X}(Σι′) under C_{N(P T)}(Σ\overline{ι}).

The monomorphism ν: V → PU(p), V = ℤ/p ⊕ ℤ/p, which does not factor through the maximal torus, admits p + 1 distinct special preferred lifts [22, 5.2] indexed by the rank one subgroups of V. Let μ: V → N(P T) be any of these p + 1 special preferred lifts. (As defined in Section 2, ν: V → PU(p) factors through the maximal torus normalizer N(P T) and this factorization is a special preferred lift since C_{N(P T)}(V) = V = C_{PU(p)}(V)∩N(P T). Any other special preferred lift of ν is obtained by composing this preferred lift with an element of the automorphism group 𝔸(ν).) Note that the conjugacy class ν′ = Φ^{∗}(ν) is independent of the choice of preferred lift μ. Since p-compact toral groups are N-determined [22, 7.9], there exists an isomorphism f(ν): C_{PU(p)}(ν) → C_{X}(ν′) under C_{N(P T)}(μ) We must verify that f(ν) is independent of the choice of μ. This will follow, cf. the proof of [22, 6.12], if we can show that f(ν) is equivariant up to homotopy with respect to the actions of 𝔸(PU(p))(ν) = 𝔸(X)(ν′) on domain and target. To show equivariance, consider the commutative diagram

\xymatrix{ & map(BV, BV)_{B1} \ar[d]_{\underline{Bμ}}^{≃} \ar[ddl]_{\underline{Bν}} \ar[ddr]^{\underline{Bν′}} \\  & map(BV, BN(P T))_{Bμ} \ar[dl]_{\underline{Bj}}^{≃} \ar[dr]^{\underline{Bj′}}_{≃} \\ map(BV, BP U(p))_{Bν} \ar[rr]_{Bf(ν)} &  & map(BV, BX)_{Bν′}}

where all maps are homotopy equivalences. Equip the identity component map(BV, BV)_{B1} ≃ BV with the GL(V)-action given by conjugation. Since \underline{Bj} ∘ \underline{Bμ} and \underline{Bj′} ∘ \underline{Bμ} are equivariant isomorphisms, also Bf(ν) is equivariant up to homotopy.

As in [22, 6.12], one shows that for any morphism a: (E_{1}, ε_{1}) → (E_{2}, ε_{2}) in the category 𝔸, the diagram

\xymatrix{C_{PU(p)}(ε_{2}) \ar[d]^{≅}_{f(ε_{2})} \ar[r]^{C_{PU(p)}(a)} & C_{PU(p)}(ε_{1}) \ar[d]_{≅}^{f(ε_{1})} \\ C_{X}(ε_{2}′) \ar[r]_{C_{X}(a)} & C_{X}(ε_{1}′)}

commutes up to conjugacy. Thus the maps

C_{PU(p)}(ε) ⟶^{f(ε)} C_{X}(ε′) ⟶^{e(ε′)} X, (E, ε) ∈ Ob(𝔸),

will represent an element in lim_{𝔸}^{0}[BC_{PU(p)}(ε), BX]. The homotopy equivalences

map(BC_{PU(p)}(ε), BX)_{B(f(ε)∘e(ε′))} ⟵ map(BC_{PU(p)}(ε), BC_{PU(p)}(ε))_{B1}

⟶ map(BC_{PU(p)}(ε), BP U(p))_{Be(ε)}

that are compatible with the morphisms of 𝔸, show that the Wojtkowiak obstruction groups [35] are lim_{𝔸}^{i+1}π_{i}BZC_{PU(p)}, i = 1, 2, and these were shown to be trivial in (3.10, 3.11).

The resulting p-compact group homomorphism f: PU(p) → X is a monomorphism since (5.2) it is a morphism under the p-normalizer C_{N(P T)}(\overline{λ}) = N_{p}(P T). Therefore, see the last lines of the proof of [22, 6.15], f is also a morphism under N(P T); in particular a rational equivalence, and hence [25, 3.7] [10, 4.7] an isomorphism of p-compact groups.

Since PU(p) is totally N-determined, the homomorphism (4.1) is an isomorphism: Out(PU(p)) ≅ Out(N(P T)). We shall now use this to reprove a special case of [14, 15].

Corollary 4.5. [BP U(p)_{p}^{∧}, BP U(p)_{p}^{∧}] = ℤ_{p}^{∗} ∪ {0}.

Proof. By [24, 5.5, 5.6] any essential self-map of BP U(p)_{p}^{∧} is a self homotopy equivalence and by [22, 3.5, 7.2]

Out(N(P T)) = Sim(Σ_{p}, Σℤ_{p})/Σ_{p}

where Sim(Σ_{p}, Σℤ_{p}) denotes the group of all pairs (χ, ϕ) ∈ Aut(Σ_{p})×Aut(Σℤ_{p}), for which ϕ(gx) = χ(g)ϕ(x) for all g ∈ Σ_{p}, x ∈ Σℤ_{p}. For such a pair

ϕ((Σℤ_{p} ⊗ ℚ)^{g}) = (Σℤ_{p} ⊗ ℚ)^{χ(g)}, g ∈ Σ_{p},

and it follows that χ sends transpositions to transpositions, i.e. that χ is an inner automorphism. (Alternatively, use that any automorphism of the symmetric group Σ_{p} is inner [33, 2.18].) Consequently,

Out(N(P T)) = Aut_{Σ_{p}}(Σℤ_{p}) = ℤ_{p}^{∗}

consists, by Schur’s lemma, entirely of scalar multiplications since Σℤ_{p} ⊗ ℚ is an irreducible Σ_{p}- representation.

For general reasons [22, 4.8, 7.10], the p-compact groups U(p) and SU(p) are totally Ndetermined (as are product combinations of these p-compact groups (Section 5)) since PU(p) is. Consequently,

[BSU(p)_{p}^{∧}, BSU(p)_{p}^{∧}] = ℤ_{p}^{∗} ∪ {0}

also holds [24, 4.3] for the universal covering p-compact group SU(p) of PU(p). Similarly,

Out(U(p)) = Sim(Σ_{p}, ℤ_{p}^{p})/Σ_{p} = Aut_{Σ_{p}}(ℤ_{p}^{p})

describes the automorphisms of the p-compact group U(p).

Cohomological uniqueness [8, 27] occurs when the maximal torus normalizer can be read off from the mod p cohomology algebra. As an illustration of this phenomenon, we now show cohomological uniqueness of the p-compact groups U(p), SU(p), and PU(p).

Lemma 4.6. ([13, 4.2.5]) The cohomology algebra of the symmetric group Σ_{p},

H^{∗}(Σ_{p}; ℤ_{p}) = ℤ_{p}[x]/(px)

is a polynomial algebra on a generator x of degree |x| = 2(p − 1) modulo the principal ideal containing px.

Theorem 4.7. ([27, 5, 6]) Let Y be one of the p-compact groups U(p), SU(p), or PU(p) and let X be any connected p-compact group. If H^{∗}(BX, 𝔽_{p}) and H^{∗}(BY, 𝔽_{p}) are isomorphic unstable algebras over the Steenrod algebra, then X and Y are isomorphic p-compact groups.

Proof. It suffices to show that X and Y have isomorphic maximal torus normalizers.

Let Φ: H^{∗}(BX) → H^{∗}(BY) be an isomorphism of algebras over the Steenrod algebra. By Lannes theory [18], Φ induces an isomorphism Φ^{∗}: 𝔸(Y) → 𝔸(X) and a natural isomorphism H^{∗}(BC_{Y} (−)) ≅ H^{∗}(BC_{X}(−)) of functors.

Let τ be a maximal elementary abelian p-group in Y such that C_{Y} (τ) → Y is a maximal torus. Then C_{X}(Φ^{∗}τ) → X is also a maximal torus of X and Φ^{∗}: 𝔸(Y)(τ) → 𝔸(X)(Φ^{∗}τ) is an isomorphism of Weyl groups (2.9) that extends to an isomorphism of Weyl representations.

Group cohomology computations such as

H^{2}(Σ_{p}; P Ť) = H^{3}(Σ_{p}; Σℤ_{p}) = H^{4}(Σ_{p}; ℤ_{p}) = \begin{cases}ℤ/p & p = 3 \\ 0 & p > 3 \end{cases}

show that unless Y = PU(p) and p = 3, the semi-direct product is the only extension realizing the Weyl group representation. Thus X and Y have isomorphic maximal torus normalizers (semi-direct products) in these cases.

But if Y = PU(3), then H^{2}(Σ_{3}; P Ť) ≅ H^{2}(ℤ/3; P Ť) ≅ ℤ/3 so there is more than one extension of the Weyl group Σ_{3} by the 3-discrete maximal torus P Ť of PU(p). The discrete approximation Ň to the maximal torus normalizer of X is one of these extensions and it order to show that it is the semi-direct extension we consider its restriction to the Sylow 3-subgroup ℤ/3 of Σ_{3}. This extension is represented by the normalizer CŇ(Φ^{∗}λ) where λ = (P Ť)^{ℤ/3} is the object of 𝔸(PU(3)) considered in (2.1). Since λ is a preferred lift, CŇ(Φ^{∗}λ) is the discrete approximation to the pcompact toral group C_{X}(Φ^{∗}λ) which can be seen to be a semi-direct product by using that it has the same cohomology as the semi-direct product C_{PU(p)}(λ). Thus Ň = P Ť⋊Σ_{3} and X ≅ PU(3) since PU(3) is N-determined.

5. Determinism of product p-compact groups

We show in this section that determinacy behaves well with respect to formation of (finite) products of p-compact groups.

First two lemmas of a general nature. A p-compact group morphism f: X → Y is said to be trivial if Bf: BX → BY is nullhomotopic.

Lemma 5.1. Let X and Y be p-compact groups and X ⟶ Z(Y) a p-compact group morphism 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))_{Bz^{−1}(B0)} ⟶ map(BX, BY)_{B0}

where the total space consists of all maps BX ⟶ BZ(Y) that composed with Bz become nullhomotopic. With the help of the Sullivan Conjecture for p-compact groups [10, 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.

Lemma 5.2. Let f: X → Y be a p-compact group morphism and j_{p}: N_{p} → X the p-normalizer [9, 9.8] of the maximal torus of X. Then

1. f is a monomorphism ⇔ fj_{p} is a monomorphism

2. f is trivial ⇔ fj_{p} is trivial

Proof. Suppose that the restriction fj_{p} of f to the p-normalizer is a monomorphism. Then [9, 9.11] H^{∗}(BN_{p}) is a finitely generated H^{∗}(BY)-module via H^{∗}(Bfj_{p}). Since H^{∗}(BX) is a H^{∗}(BY)- submodule of H^{∗}(BN_{p}) thanks to the transfer homomorphism [10, 9.13] and H^{∗}(BY) a noetherian graded ring [9, 2.4], H^{∗}(BX) is a finitely generated H^{∗}(BY)-module via H^{∗}(Bf). The converse follows from the fact that the composition of two monomorphisms is a monomorphism.

The second part of the lemma is [24, 6.7].

We now address N-determinacy of automorphisms of product p-compact groups. Let X_{1} and X_{2} be p-compact groups and

\xymatrix{X_{1} \ar[r]_{ι_{1}} & X_{1} × X_{2} \ar[l]_{π_{1}} \ar[r]^{π_{2}} & X_{2} \ar[l]^{ι_{2}}}

the natural projections and inclusions.

Lemma 5.3. Let f: X_{1} × X_{2} → X_{1} be a p-compact group morphism such that fι_{1} : X_{1} → X_{1} is an isomorphism and fι_{2} : X_{2} → X_{1} is trivial. Then f is conjugate to fι_{1}π_{1}.

Proof. We want to show that the adjoint of Bf, BX_{2} ⟶ map(BX_{1}, BX_{1})_{B(fι_{1})}, which maps into a space homotopy equivalent [10, 1.3] to BZ(X_{1}), is nullhomotopic. But this follows immmediately from (5.1) since composition with the evaluation monomorphism to BX_{1} gives the nullhomotopic map Bf ∘ Bι_{2}.

Theorem 5.4. Let X_{1} and X_{2} be connected p-compact groups with N-determined automorphisms. Then also the product p-compact group X_{1} × X_{2} has N-determined automorphisms.

Proof. Let f be an automorphism of X_{1} × X_{2} under the product N_{1} × N_{2} of the two maximal torus normalizers. The morphism π_{1}fι_{1} : X_{1} → X_{1} is an isomorphism for [25, 3.7] [10, 4.7] it is a rational equivalence [9, 9.7] and a monomorphism (5.2). As also π_{1}fι_{2} : X_{2} → X_{1} is trivial by (5.2), it follows from (5.3) that π_{1}f is conjugate to π_{1}fι_{1}π_{1}. Similarly, π_{2}f is conjugate to π_{2}fι_{2}π_{2} and thus f is conjugate to the product morphism f_{1} × f_{2} where f_{1} = π_{1}fι_{1} and f_{2} = π_{2}fι_{2}. Thus N(f) = N_{1}(f_{1}) × N_{2}(f_{2}) and, since X_{1} and X_{2} have N-determined automorphisms, it follows that f_{1} and f_{2} are conjugate to identity morphisms.

Next, we adress N-determinacy of products. This is based on a slight reformulation of the Splitting Theorem.

Theorem 5.5 (Splitting Theorem [11, 6.1], [28]). Assume that p is odd. Let X be a connected pcompact group and i: T → X a maximal torus with normalizer j: N → X. For any decomposition N ⟶^{≅} N_{1}×N_{2} of N into a product of two extended p-compact tori, N_{1} and N_{2}, there exist p-compact groups, X_{1} and X_{2}, and an isomorphism s: X → X_{1} × X_{2} such that

\xymatrix{N \ar[d]_{j} \ar[r]^{≅} & N_{1} × N_{2} \ar[d]^{j_{1}×j_{2}} \\ X \ar[r]_{s}^{≅} & X_{1} × X_{2}}

commutes up to conjugacy where j_{1} : N_{1} → X_{1} and j_{2} : N_{2} → X_{2} are normalizers of maximal tori.

Proof. Write N_{i}, i = 1, 2, as an extension T_{i} ⟶ N_{i} ⟶ W_{i} of a p-compact torus T_{i} and a finite group W_{i}. Then the Weyl group W = π_{0}(N) of X is isomorphic to W_{1} × W_{2} and W_{i} acts [11, 6.3] as a reflection group on π_{1}(T_{i}) ⊗ ℚ. According to [11, 6.1], the splitting π_{1}(T) ≅ π_{1}(T_{1}) × π_{1}(T_{2}) as a W ≅ W_{1} × W_{2}-module can be realized by a p-compact group splitting s: X → X_{1} × X_{2}. This means that if N(s): N → N′_{1} × N′_{2}, where j′_{i}: N′_{i} → X_{i} is the normalizer of the maximal torus T_{i} ⟶ X_{i}, i = 1, 2, is the lift [24, 5.1] of s, then the discrete approximation [10, 3.12] Ň(s) to N(s) determines an isomorphism

\xymatrix{0 \ar[r] & Ť \ar[r] \ar[d]_{Ť(s)} & Ň \ar[r] \ar[d]^{Ň(s)} & W \ar[d]^{W (s)} \ar[r] & 1 \\ 0 \ar[r] & Ť_{1} × Ť_{2} \ar[r] & N′_{1} × N′_{2} \ar[r] & W_{1} × W_{2} \ar[r] & 1}

of short exact sequences where Ť(s) is the given splitting Ť ≅ Ť_{1} ×Ť_{2} and W (s) the given splitting W ≅ W_{1} × W_{2}. Relative to the given splitting Ň ≅ Ň_{1} × Ň_{2}, the middle isomorphism Ň(s) takes Ň_{1} × Ň_{2} isomorphically to Ň′_{1} × Ň′_{2}. The composite

Ň_{1} ⟶^{ι_{1}} Ň_{1} × Ň_{2} ⟶^{Ň(s)} Ň′_{1} × Ň′_{2} ⟶^{π′_{2}} Ň′_{2},

where ι_{1} is the injection and π′_{2} the projection, can be factored through a homomorphism W_{1} ⟶ Ť_{2} as it restricts to the trivial morphism Ť_{1} ⟶ Ť_{2}. Since p is assumed to be odd, any such homomorphism is trivial for the reflection group W_{1} is generated by elements of order prime to p. This implies that Ň(s): Ň_{1} × Ň_{2} → Ň′_{1} × Ň′_{2} is the product of two isomorphisms, Ň_{1} ⟶ Ň′_{1} and Ň_{2} ⟶ Ň′_{2}. Let j_{i}, i = 1, 2, be the composite of this isomorphism Ň_{i} ⟶ Ň′_{i} with j_{i}: Ň′_{i} → X_{i}.

The assumption that p should be odd is presumably not essential.

Theorem 5.6. The product of two connected N-determined p-compact groups is N-determined when p > 2.

Proof. This is immediate from the commutative diagram

\xymatrix{ & N_{1} × N_{2} \ar[dl] \ar[dr]_{j′} \ar[drr] \\ X_{1} × X_{2} &  & X′ \ar[r]_{s}^{≅} & X′_{1} × X′_{2}}

where X′ is any p-compact group with maximal torus normalizer j′ and s the splitting isomorphism from (5.5). For if X_{1} and X_{2} are N-determined, we get isomorphisms f_{1} : X_{1} → X′_{1} and f_{2} : X_{2} → X′_{2} under N_{1} and N_{2}, respectively, and s^{−1} ∘ (f_{1} × f_{2}) is then an isomorphism X_{1} × X_{2} ⟶ X′ under N_{1} × N_{2}.

The next step is to generalize (5.4) and (5.6) to possibly nontrivial extensions.

Theorem 5.7. Let Y ⟶ G ⟶ X be a short exact sequence of connected p-compact groups. 1. If the adjoint forms P X and P Y have N-determined automorphisms, so does G.

2. If the adjoint forms P X and P Y are N-determined and p > 2, so is G.

Since a connected p-compact group has N-determined automorphisms or is N-determined provided this holds for its adjoint form [22, 4.8, 7.10], the proof of the above theorem is an immediate consequence of (5.4, 5.6) and the lemma below.

Lemma 5.8. In any extension Y ⟶ G ⟶ X of connected p-compact groups, the middle p-compact group, G, is locally isomorphic [24, 2.7] to the product, X × Y, of the two outer p-compact groups and the adjoint form P G is isomorphic to P X × P Y.

Proof. Let X⟨1⟩ denote the universal covering p-compact group and S = Z(X)_{0} the identity component of the center of X. Let Y ⟶ G_{1} ⟶ X⟨1⟩ × S be the extension obtained by pulling back along the isogeny [25, 5.4] X⟨1⟩ × S ⟶ X. Since [23, 3.2, 3.3, 3.4] the projection of X⟨1⟩ × S to S induces a bijection Ext(S, Y) ⟶^{≅} Ext(X⟨1⟩ × S, Y) of equivalence classes of extensions, the pcompact group G_{1}, which is locally isomorphic to G, is isomorphic to X⟨1⟩×G_{2} for some extension Y ⟶ G_{2} ⟶ S of the p-compact torus S by Y. By [23, 2.6], G_{2} is locally isomorphic to S × Y and hence G is locally isomorphic to X⟨1⟩ × S × Y which is locally isomorphic to X × Y.

Any connected p-compact group has the same adjoint form as its universal covering p-compact group. Hence P G ≅ P (X⟨1⟩ × Y ⟨1⟩) ≅ P X × P Y.

Example 5.9. Since

Ext(PU(p), SU(p)) = [B PU(p), B^{2}Z(SU(p))] = H^{2}(B PU(p); ℤ/p)

= Hom(ℤ/p, ℤ/p) = ℤ/p

there are p equivalence classes of extensions of PU(p) by SU(p). However, since the local isomorphism system of SU(p) × SU(p) [24, p. 217]

\xymatrix{ & SU(p) × PU(p) \ar@{-->}[dr] \\ SU(p) × SU(p) \ar@{-->}[ur] \ar@{-->}[dr] &  & PU(p) × PU(p) \\  & SU(p)⋊ PU(p) \ar@{-->}[ur]}

consists of very few p-compact groups, we see from (5.10) that the middle p-compact group in any of these extensions must be isomorphic to either the direct product SU(p) × PU(p) or the semi-direct product SU(p, p) = SU(p)⋊ PU(p) for the conjugation action of PU(p) on SU(p). All the p-compact groups locally isomorphic to SU(p) × SU(p) are totally N-determined. The automorphism groups are, for instance,

Out(SU(p) × PU(p)) = ℤ_{p}^{∗} × ℤ_{p}^{∗}
Out(SU(p, p)) ≅ {(u, v) ∈ ℤ_{p}^{∗} × ℤ_{p}^{∗} | u ≡ v mod p}⋊ℤ/2

where ℤ/2 permutes the coordinates. Formulas like these follow from (4.5) in combination with [24, 4.3] and [20, 3.5].

Similarly,

Ext(PU(p), U(p)) = [B PU(p), B^{2}Z(U(p))] = H^{3}(B PU(p); ℤ_{p})

= Ext(H_{2}(B PU(p); ℤ_{p}), ℤ_{p}) = Ext(ℤ/p, ℤ_{p}) = ℤ/p

shows that there are p equivalence classes of extensions of PU(p) by U(p). However, the local isomorphism system

\xymatrix{ & SU(p) × PU(p) × S \ar@{-->}[dr] \ar[dddr] \\  &  & U(p)⋊ PU(p) \\ SU(p) × SU(p) × S \ar[uur] \ar[r] \ar[ddr] & S(U(p) × U(p)) \ar@{-->}[ur] \ar@{-->}[dr] \\  &  & U(p) × PU(p) \\  & SU(p) × PU(p) × S \ar@{-->}[ur]}

of SU(p)×SU(p)×S, where S is a rank one p-compact torus, and (5.10) show that the corresponding middle p-compact group must be isomorphic to either the direct or the semi-direct product of U(p) and PU(p). All p-compact groups locally isomorphic to SU(p)×SU(p)×S are totally N-determined.

The automorphism group of S(U(p) × U(p)) is

Out(S(U(p) × U(p))) ≅ {(u, v, w) ∈ ℤ_{p}^{∗} × ℤ_{p}^{∗} × ℤ_{p}^{∗} | u ≡ v ≡ w mod p}⋊ℤ/2

where ℤ/2 commutes two of the coordinates.

Lemma 5.10. For any short exact sequence Y ⟶^{ι} G ⟶^{π} X of connected p-compact groups 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) → Y be the center of Y. In the commutative diagram

\xymatrix{map(BZ(Y), BY)_{Bz} \ar[d]_{≃} \ar[r] & map(BZ(Y), BG)_{B(ιz)} \ar[d] \ar[r] & map(BZ(Y), BX)_{B0} \ar[d]^{≃} \\ BY \ar[r] & BG \ar[r] & BX}

the horizontal lines are fibration sequences and the vertical arrows are evaluation maps. Note that the middle arrow is a homotopy equivalence since the outer two arrows are homotopy equivalences. This shows that ιz: Z(Y) → G is central and by naturality we obtain [9, 8.3] a short exact sequence

P Y ⟶ G/Z(Y) ⟶ X

of p-compact groups. This extension is equivalent to the trivial extension [23, 3.4] since P Y is centerless [25, 4.7] [10, 6.3]. Thus G/Z(Y) ≅ P Y × X and

Z(G)/Z(Y) ≅ Z(G/Z(Y)) ≅ Z(P Y × X) ≅ Z(X)

by [25, 4.6.(4)].

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Matematisk Institut, Universitetsparken 5, DK–2100 København Ø

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