FINITE CHEVALLEY VERSIONS OF p-COMPACT GROUPS

CARLES BROTO AND JESPER M. MØLLER

Abstract. We describe the spaces of homotopy fixed points of unstable Adams operations acting on p-compact groups and also of unstable Adams operations twisted with a finite order automorphism of the p-compact group. We obtain new exotic p-local finite groups.

Contents

1. Introduction 1 2. p-compact groups 7 3. p-local finite groups 8 4. Recognition of classifying spaces of p-local finite groups 13 5. Homotopy fixed points p-compact groups 21 6. Homotopy fixed points of twisted unstable Adams operations 39 7. General structure of finite Chevalley versions of p-compact groups 44 8. Cohomology rings 49 9. Invariant theory 51 10. Finite Chevalley versions of Aguadé exotic p-compact groups 60 11. Finite Chevalley versions of generalized p-adic Grassmannians 69 References 75

1. Introduction

The main purpose of this paper is the description of the structure of the spaces of homotopy fixed points of unstable Adams operations ψ^{q} acting on p-compact groups and also of unstable Adams operations twisted by automorphisms of p-compact groups τψ^{q}. In the classical case, where τ is an automorphism of a compact connected Lie group G and ψ^{q} an unstable Adams operation of exponent a prime power q, coprime to p, results of Quillen [54] and Friedlander [27, 28], show that the space of homotopy fixed points is, up to p-comple-tion, the classifying space of the finite twisted Chevalley group ^{τ}G(q). Here and throughout, p-completion is understood in the sense of Bousfield-Kan [8]. We will show that in case of exotic p-compact groups we obtain classifying spaces of p-local finite groups, in some cases, new exotic examples.

2000 Mathematics Subject Classification. 55R35, 55P15, 55P10.

Key words and phrases. Chevalley group, p-compact groups, p-local finite groups.

C. Broto is partially supported by MCYT grant BFM2001-2035.

Both authors have been partially supported by the EU grant nr. HPRN-CT-1999-00119.

The concept of p-compact group was introduced by Dwyer and Wilkerson in [22] as a p-local homotopy theoretic analogue of compact Lie group. A p-compact group is a connected p-complete space BX where X = ΩBX is 𝔽_{p}-finite; that is, H^{∗}(X; 𝔽_{p}) is finite. We will usually refer to a p-compact group simply as X. BX is then understood as its classifying space, a concrete loop space structure imposed in the underlying space X. If G is a compact connected Lie group, then the p-completion of its classifying space BG_{p}^{∧} is a p-compact group. A p-compact group that cannot be obtained in this way is called exotic. We postpone till section 2 a more detailed description of the theory of p-compact groups.

The concept of p-local finite group has been recently introduced in [11] as algebraic objects that are modeled on the p-local structure of finite groups and as such they have classifying spaces which are p-complete spaces. In turn, the classifying space of a p-local finite group determines its algebraic structure. We refer to section 3 for the precise definition and main properties of p-local finite groups.

Theorem A. Let p be an odd prime. If X is a 1-connected p-compact group, q is a prime power, coprime to p, and τ is an automorphism of X of finite order coprime to p, then the space of homotopy fixed points of BX by the action of τψ^{q}, denoted B^{τ}X(q), is the classifying space of a p-local finite group.

By analogy with the classical case, we will call p-local finite Chevalley group of type X to any finite p-local finite group X(q) obtained in Theorem A, with classifying space BX(q). If X is obtained as p-completion of a compact Lie group G, BX(q) is homotopy equivalent to the p-completed classifying space of the Chevalley group G(q).

For a prime number p, a prime power q, coprime to p, and a compact connected Lie group G, Friedlander shows a cohomological fibre square that becomes the homotopy pullback diagram

\xymatrix{B^{τ}G(q)_{p}^{∧} \ar[r]^{f} \ar[d]_{f} & BG_{p}^{∧} \ar[d]^{Δ} \\ BG_{p}^{∧} \ar[r]^{(1,τψ^{q})} & BG_{p}^{∧} × BG_{p}^{∧}}

after p-completion, and where ^{τ}G(q) is the twisted Chevalley group over 𝔽_{q} of type G and Δ is the diagonal map. Unstable Adams operations can be defined over p-compact groups (see section 2), hence, following the above pattern, if X is a connected p-compact group and τψ^{q} a twisted Adams operation, then the classifying space B^{τ}X(q) is defined by the homotopy pullback square

\xymatrix{B^{τ}X(q) \ar[r]^{f} \ar[d]_{f} & BX \ar[d]^{Δ} \\ BX \ar[r]^{(1,τψ^{q})} & _{BX × BX.}}

This pullback square provides an alternative definition of the space of homotopy fixed points by the action of τψ^{q} on BX (see section 6 for details).

Our arguments concentrate in the exotic p-compact groups at odd primes, and naturally break into two distinguished steps. One deals with actions of finite groups of order not divisible by p on p-compact groups and the results obtained have an independent interest by their own. The other step deals with the action of unstable Adams operations ψ^{q} where q ≡ 1 mod p and it is the one leading to the new exotic examples of p-local finite groups.

Group actions will be understood in the weak sense of proxy actions; that is, we will say that an action of a group G on a space B is a fibration B → B_{hG} → BG, [22]. The total space B_{hG} is referred to as the homotopy quotient and the space of homotopy fixed points B^{hG} is the space of sections. When we specialize to p-compact groups X, an outer action of G is a homomorphism ρ: G → Out(X), where Out(X) is the group automorphisms of the p-compact group X, in other words, homotopy classes of self-equivalences of BX. It turns out that if G has finite order coprime to p, then an outer action on a connected pcompact group X determines a unique action and the space of homotopy fixed points is again a connected p-compact group:

Theorem B. Let X be a connected p-compact group. If G is a finite group of order coprime to p and ρ: G → Out(X) an outer action, then

(1) ρ lifts to a unique action of G on X.

(2) X^{hG} is a connected p-compact group with H^{∗}(BX^{hG}; ℚ_{p}) ≅ S[QH^{∗}(BX; ℚ_{p})_{G}], the symmetric algebra generated on the coinvariants QH^{∗}(BX; ℚ_{p})_{G}.

(3) (Harper splitting) X^{hG} → X is a p-compact group monomorphism, and

X ≃ X^{hG} × X/X^{hG} ,

thus, in particular, X/X^{hG} is an H-space.

(4) If H^{∗}(BX; 𝔽_{p}) is a polynomial ring, then H^{∗}(BX^{hG}; 𝔽_{p}) is also a polynomial ring.

Here and throughout, H^{∗}(−; ℚ_{p}) stands for H^{∗}(−; ℤ_{p}) ⊗ ℚ, and QH^{∗}(BX; ℚ_{p}) denotes the module of the indecomposables in H^{∗}(BX; ℚ_{p}). This result is proved as a corollary of Theorem 5.2 that establishes, more generally, that if G has order prime to p and acts on a 1-connected p-complete space B then there exists a homotopy equivalence ΩB ⟶^{≃} ΩB^{hG} × Fib(B^{hG} → B).

Some interesting cases to which Theorem B applies are F_{4} at the prime 3 and E_{8} at the prime 5. In the first case, Friedlander’s exceptional isogeny of F_{4} at the prime 3 gives rise to an automorphims of order 2 and the homotopy fixed points p-compact group F_{4}^{hC_{2}} is the pcompact group X_{12} = DI(2) whose cohomology realizes the Dickson algebra H^{∗}(BX_{12}; 𝔽_{3}) ≅ 𝔽_{3}[x_{12}, x_{16}] (subscripts indicate degrees). This case was already considered in our previous work [13]. In the second case, a cyclic group of order 4 generated by the unstable Adams operation ψ^{i}, i = \sqrt{−1}, acts on E_{8}. The homotopy fixed points p-compact group E_{8}^{hC_{4}} is the p-compact group X_{31} corresponding to the reflection group number 31 on the Clark-Ewing list, and its mod 5 cohomology ring is H^{∗}(BX_{31}; 𝔽_{5}) = 𝔽_{5}[x_{16}, x_{24}, x_{40}, x_{48}] (see 5.15).

It turns out that X_{12} = DI(2) and X_{31} are the two exotic p-compact groups originally constructed by Zabrodsky [63], and later included in the Aguadé family [1]. Zabrodsky used invariants by the action of these same automorphisms but only at the level of homotopy groups of BF_{4} and BE_{8}, respectively.

The corresponding splittings are F_{4} ≃ DI(2)×F_{4}/DI(2) at the prime 3, first discovered by Harper [31], and E_{8} ≃ X_{31} × E_{8}/X_{31}, that was obtained by Wilkerson [60]. Other examples appear in 5.3.

Our next Theorem provides the necessary arguments in order to deduce the general case of Theorem A from the two steps.

Theorem C. Let p be an odd prime and X a connected p-compact group, τ an automorphism of X of order prime to p and ψ^{q} an unstable Adams operation, then:

(1) If X is 1-connected and q ≡ 1 mod p, q ≠ 1, then B^{τ}X(q) ≃ BX^{hτ}(q).

(2) If q′ is another p-adic unit such that q ≡ q′ mod p and ν_{p}(1 − q^{r}) = ν_{p}(1 − q′^{r}), where r is the order of q mod p, then BX(q) ≃ BX(q′).

Since we can decompose a p-adic unit q as q = ζq_{0} where ζ is a (p − 1)st-root of unity and q_{0} ≡ 1 mod p, part (1) of the above Theorem will reduce the question of computing BX(q) to the case where q ≡ 1 mod p which turns out to be easier to handle in abstract calculations and concrete examples. The second part of the Theorem tells us that BX(q) does only depend on the order r of q mod p and the p-adic valuation ν_{p}(1 − q^{r}), so we can change the exact value of q at our convenience if we keep those parameters fixed. In particular, Theorem A, could more generally be stated for unstable Adams operations ψ^{q}, with q a p-adic unit of infinite multiplicative order.

Part (2) of Theorem C also explains the often observed fact that finite Chevalley groups G(q) and G(q′) have same cohomology ring or identical p-local structure when q and q′ are prime powers, with q^{r} ≡ q′^{r} ≡ 1 mod p and ν_{p}(1 − q^{r}) = ν_{p}(1 − q′^{r}), for some r, 1 ≤ r ≤ p − 1.

Our second step deals with the action of unstable Adams operations ψ^{q} of exponent q ≡ 1 mod p, q ≠ 1, on connected p-compact groups X. The effect now is opposite in some sense to the case of finite groups of order prime to p. The spaces of homotopy fixed points BX(q) have the same p-rank as the original p-compact groups X, but the maximal tori T^{n} ≃ ((S^{1})^{n})_{p}^{∧} are cut down to finite maximal tori T_{ℓ}^{n} ≅ (ℤ/p^{ℓ})^{n}, ℓ = ν_{p}(1−q) (the p-adic valuation of 1−q), keeping, though, the same Weyl group (see 7.5, 7.6).

We restrict our calculations in this part to p-compact groups for which the mod p cohomology ring H^{∗}(BX; 𝔽_{p}) is a polynomial ring. For simplicity, we will refer to them as polynomial p-compact groups. At odd primes, these include all irreducible exotic examples and will therefore suffice to our purposes.

Theorem D. Let q be a p-adic unit such that q ≡ 1 mod p, q ≠ 1. If X is an irreducible 1-connected polynomial p-compact group, then BX(q) is the classifying space of a p-local finite group.

Proof. The proof is based on the classification theorem for p-compact groups at odd primes [6]. The irreducible 1-connected p-compact groups with polynomial cohomology are

(1) BSU(n)_{p}^{∧} (family 1 in the Clark-Ewing list),

(2) the Quillen generalized Grassmannians (family 2a in the Clark-Ewing list),

(3) the non-modular p-compact groups, and

(4) the Aguadé family (numbers 12, 29, 31, and 34 in the Clark-Ewing list, at primes 3, 5, 5, and 7, respectively).

The different cases are solved in 11.1, 11.4 , 9.7, and 10.3, respectively. □

In cases (1) and (3) we always obtain that BX(q) is the p-completed classifying space of a finite group. The other two families contain the new exotic examples of p-local finite groups.

A complete description of the structure of the p-local finite groups X_{i}(q), i = 12, 29, 31, 34, is obtained in section 10. For X_{12}(q), p = 3, we obtain that if ℓ = ν_{3}(1 + 2^{2n+1}), then BX_{12} ≃ B(^{2}F_{4}(2^{2n+1}))_{3}^{∧} (Example 10.7). For X_{31}(q), p = 5, it turns out that if ℓ = ν_{5}(1 + 2^{4m+2}), then BX_{31}(q) ≃ BE_{8}(2^{2m+1})_{5}^{∧} (Example 10.8). In particular, we can obtain the p-compact groups X_{12} and X_{31} as telescopes of a sequence of p-completed classifying spaces of finite groups (see 10.9):

BX_{12} ≃ hocolim_{n} B(^{2}F_{4}(2^{2n+1}))_{3}^{∧} ,

BX_{31} ≃ hocolim_{m} BE_{8}(2^{2m+1})_{5}^{∧} .

The cases BX_{29}(q) and BX_{34}(q) at primes 5 and 7, respectively, are classifying spaces of exotic p-local finite groups (Example 10.6).

Family 2a in the Clark-Ewing list consists of groups G(m, r, n) with r|m|(p − 1), defined as the pseudoreflection groups in GL_{n}(ℚ_{p}) generated the permutation matrices and the diagonal matrices diag(a_{1}, a_{2}, …, a_{n}) with a_{i}^{m} = 1 and (a_{1}a_{2} … a_{n})^{n/r} = 1. We denote X(m, r, n) the p-compact group of rank n with Weyl group G(m, r, n). We also prove that BX(m, r, n)(q) is the classifying space of an exotic p-local finite group provided n ≥ p and r > 2 (Proposition 11.5).

Proof of Theorem A. We defined B^{τ}X(q) = BX^{h⟨τψ^{q}⟩}, the space of homotopy fixed points of BX by the action of the group generated by τψ^{q}.

If we write q = ζq_{0}, where ζ is a (p − 1)th root of unity and q_{0} ≡ 1 mod p, q_{0} ≠ 1, so that τψ^{q} = τψ^{ζ}ψ^{q_{0}}, then we have

B^{τ}X(q) = BX^{h⟨τψ^{q}⟩} ≃ BX^{h⟨τψ^{ζ}⟩}(q_{0}) ,

according to Theorem C.

X^{h⟨τψ^{ζ}⟩} is a 1-connected p-compact group by Theorem B, hence it splits as a product of irreducible 1-connected p-compact groups

BX^{h⟨τψ^{ζ}⟩} ≃ BX_{1} × ⋯ × BX_{s} ,

and then, also, BX^{h⟨τψ^{ζ}⟩}(q_{0}) ≃ BX_{1}(q_{0}) × ⋯ × BX_{s}(q_{0}). It remains to show that each BX_{i}(q_{0}) is the classifying space of a p-local finite group.

If X_{i} is polynomial, Theorem D applies and BX_{i}(q_{0}) is the classifying space of a p-local finite group.

If X_{i} is the p-completion of a compact Lie group G, then we can find a prime number q′_{0} with q′_{0} ≡ q_{0} ≡ 1 mod p and ν_{p}(1 − q_{0}) = ν_{p}(1 − q′_{0}), and then BX_{i}(q_{0}) ≃ BX_{i}(q′_{0}) by Theorem C (cf. Remark 6.6), and this last is the p-completed classifying space of a finite Chevalley group of type G, by the classical result of Friedlander [28].

By the classification theorem of p-compact groups at odd primes [6] (see section 2), every irreducible, simply-connected p-compact group is either polynomial or the p-completion of a compact Lie group, hence the proof is complete. □

Many authors have been interested in the cohomology rings of finite Chevalley groups at primes different from the defining characteristic. Quillen [54], shows that for an odd prime p and a prime power q coprime to p, if m is the order of q mod p and ℓ = ν_{p}(1 − q^{m}), then

H^{∗}(BGL(n, q); 𝔽_{p}) ≅ P [x_{1}, …, x_{[ \frac{n}{m}]}] ⊗ E[y_{1}, …, y_{[ \frac{n}{m}]}]

where deg(x_{i}) = 2mi and deg(y_{i}) = 2mi − 1.

Fiedorowicz and Priddy, [25, 26] computed the cohomology rings of Chevalley groups of classical type. Kleinerman [33] has computed the cohomology of Chevalley groups of exceptional Lie type at large primes. M. Mimura, M. Tezuka, and S. Tsukuda [38] have recently approached the cohomology rings of finite Chevalley groups at torsion primes, by newly constructing a spectral sequence of Eilenberg-Moore type.

The result that we include here is essentially due to L. Smith, at least part (1) already appears in [57]. We include it here for the convenience of the reader, as it is an important step in our arguments.

Theorem E. Let X be a polynomial p-compact group with

H^{∗}(BX; 𝔽_{p}) ≅ P [x_{1}, …, x_{n}]

and q a p-adic unit with q ≡ 1 mod p, q ≠ 1. Then:

(1) H^{∗}(BX(q); 𝔽_{p}) ≅ P [x_{1}, …, x_{n}] ⊗ E[y_{1}, …, y_{n}] with higher Bockstein relations β_{(ℓ_{i})}(y_{i}) = x_{i}, ℓ_{i} = ν_{p}(1 − q^{d_{i}}), 2d_{i} = deg x_{i}, 2d_{i} − 1 = deg y_{i}, and

(2) the inclusion of the maximal finite torus i: BT_{ℓ}^{n} → BX(q), ℓ = ν_{p}(1 − q), induces a monomorphism i^{∗}: H^{∗}(BX(q); 𝔽_{p}) → H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{W_{X}}.

The inclusion i^{∗}: H^{∗}(BX(q); 𝔽_{p}) → H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{W_{X}} is an isomorphism in many cases. This is checked by direct calculation of the relevant invariant rings. In cases in which X is a connected non-modular p-compact group or a generalized Quillen Grassmannian i^{∗} is an isomorphism (see section 9). It is also an isomorphism in the case of the Aguadé pcompact groups X_{i}(q), i = 29, 31, 34, however, i^{∗} is not an epimorphism in case of X_{12}(q), for which we obtain H^{∗}(BX_{12}(q); 𝔽_{3}) ≅ P [x_{12}, x_{16}] ⊗ E[y_{11}, y_{15}], while H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{W_{X_{12}}} ≅ P [x_{12}, x_{16}] ⊗ E[y_{10}, y_{11}, y_{15}]/(y_{11}y_{15} − x_{16}y_{10}, y_{10}y_{11}, y_{10}y_{15}) (see Example 9.6).

We have restricted our calculations at odd primes, although some of the results are also valid at the prime two. At present the classification of 2-compact groups has not been completed although a plausible conjecture is that the Dwyer-Wilkerson 2-compact group DI(4) is the only irreducible exotic 2-compact group. The finite Chevalley versions of DI(4), named BSol(q), for odd prime powers q, have been first considered by Benson [7] and then by Levi and Oliver [34] who proved that they are classifying spaces of 2-local finite groups and their 2-local structure is in fact a system of fusion relations studied by Solomon [58] and defined over the Sylow 2-subgroup of Spin(7, q).

The paper is organized as follows. In sections 2 and 3 we review the definitions and main results from the theory of p-compact groups and p-local finite groups. In section 4 we further develop some aspects of the theory of p-local finite groups concerning the homotopy characterization of classifying spaces of p-local finite groups. The main results in sections 9 and 10 stating that BX(q) is the classifying space of a p-local finite group if X is a p-compact group in the Aguadé family of a generalized Quillen Grassmannian are based in this homotopy characterization of classifying spaces.

Section 5 deals with what we have called first step. There is a discussion of different ways in which we can understand an action of a group on a p-compact group and it contains the proof of Theorem B. This theorem states that a homotopy fixed point space X^{hG} is again a p-compact group if X was a connected p-compact group and G is a finite group of order prime to p. A different question is to recognize exactly which p-compact group in the classification list is X^{hG}. This problem is considered also in section 5, 5.4 through 5.9. This last contains a criteria for the recognition of the homotopy fixed points p-compact group by action of unstable Adams operations of finite order. This is applied to many examples through the Clark-Ewing list at the end of this section, 5.10 through 5.15.

Section 6 is devoted to the proof of Theorem C, the glue between the first and second steps. It reduces the analysis of the structure of a general B^{τ}X(q) to first analysing a homotopy fixed point p-compact group and then a homotopy fixed point space by the action of an unstable Adams operation ψ^{q} of exponent q ≡ 1 mod p.

The second step starts in sections 7, 8, and 9, where we analyse the general subgroup structure of spaces BX(q), where q ≡ 1 mod p, q ≠ 1, and their cohomological properties. Theorem E is proved in section 8.

Finally sections 10 and 11, are devoted to the more specific properties of the p-compact groups in the Aguadé family and the generalized p-adic Quillen Grassmannians, respectively. With them, we complete the proof of Theorem D.

We would like to thank Bob Oliver and Ran Levi for many helpful discussions. We are particularly indebted with them for the discussions during our stay at the Max-Plank Institute in Bonn in the spring of 2001, about the material presented in section 4. We are also grateful to Bob Oliver for bringing to our attention that the 3-local structure of DI(2) could be related to that of the twisted Chevalley groups of type F_{4}, after our previous work on DI(2) [13]. This was one of our original motivations for the project that led to the present article.

We would also like to thank the Departments of Mathematics of the Københavns Universitet and of the Universitat Autònoma de Barcelona, the Centre de Recerca Matemàtica in Barcelona, and the Max-Planck Institut in Bonn for their hospitality in helping the two authors get together at various stages of the project, and to the European Modern Homotopy Theory Network for helping to finance several of these visits.

2. p-compact groups

A p-compact group is an p-complete pointed space BX such that H^{∗}(X; 𝔽_{p}) is finite where X = ΩBX is the loop space of BX. Throughout the paper, and when no confusion is possible we will simply denote such p-compact group as X. These spaces were introduced by Dwyer and Wilkerson in 1994 as p-local homotopy theoretic versions of compact Lie groups [22]. Examples of p-compact groups include all simply connected p-complete spaces with polynomial 𝔽_{p}-cohomology, and the p-completed classifying spaces of all compact Lie groups G such that π_{0}(G) is a finite p-group. The special p-compact group obtained in this way from a torus is called a p-compact torus.

Theorem 2.1. [22, 9.7] Any p-compact group BX admits a maximal torus BT (X) → BX and a Weyl group W (X). When BX is simply connected, the Weyl group W (X) acts faithfully on the free ℤ_{p}-module L(X) = H_{2}(BT (X); ℤ_{p}) as a reflection group and

H^{∗}(BX; ℤ_{p}) ⊗ ℚ → (H^{∗}(BT (X); ℤ_{p}) ⊗ ℚ)^{W (X)}

is an isomorphism.

The above theorem assigns a ℤ_{p}-reflection group (W, L)(X) to any connected p-compact group BX. It turns out that this assignment is bijective.

Theorem 2.2. [48, 6] Let p be an odd prime. There is a bijective correspondence between connected p-compact groups and ℤ_{p}-reflection groups. We have

Out(X) ≅ N_{GL(L)}(W)/W

for the reflection group (W, L) = (W, L)(X) corresponding to the p-compact group BX.

The group Out(X) consists of the invertible elements of the monoid [BX, BX] of homotopy classes of self-maps of BX. There is an exact sequence of groups

1 → Aut_{ℤ_{p}[W]}(L)/Z(W) → N_{GL(L)}(W)/W → Out_{tr}(W)  (1)

where Out_{tr}(W) = {ϕ ∈ Out(W) | tr(ϕ(w)) = tr(w)} is the group of trace preserving outer automorphisms [48, 3.14–16].

The group Aut_{ℤ_{p}[W]}(L) contains the group ℤ_{p}^{×} of scalar multiplication by p-adic units, thus to any p-adic unit q we can associate a self-equivalence ψ^{q}: BX → BX, that is called the unstable Adams operation of exponent q. These operations are characterized by their effect on the maximal torus: ψ^{q}: BX → BX restricts to a map ψ^{q}: BT_{X} → BT_{X} that induces multiplication by q on H^{2}(BT_{X}; ℤ_{p}).

Classically, unstable Adams operations were first defined by Sullivan on BU(n), for q ∈ ℤ, (p, q) = 1, q > n, as restrictions of Adams operations defined on BU. Then, extended by Wilkerson to all compact Lie groups [60]. In [32] it is shown that p-completed classifying spaces of compact connected Lie groups admit unstable Adams operations ψ^{q} of exponent a p-adic unit q ∈ ℤ_{p}^{∗}. This is extended to p-compact groups for odd primes p in [48].

A p-compact group BX is irreducible if the reflection group (L⊗ℚ_{p}, W) = (L⊗ℚ_{p}, W)(X) is irreducible. In this case, by Schur’s lemma, Aut_{ℤ_{p}[W]}(L) = ℤ_{p}^{×} consists of scalars only so that (1) takes the form

1 → ℤ_{p}^{×}/Z(W) → N_{GL(L)}(W)/W → Out_{tr}(W) .  (2)

The group Out_{tr}(W) turns out to be trivial for most of the simple reflection groups, and in all known examples it consists of elements that lift to finite order elements in Out(X) = N_{GL(L)}(W)/W. In those cases, Out(X) consists only of twisted Adams operations αψ^{q}.

The list of irreducible ℤ_{p}-reflection groups can be derived [48, 11.18] from the Clark-Ewing [16] list of irreducible ℚ_{p}-reflection groups. The simple p-compact groups, corresponding to the irreducible reflection groups, besides the Lie examples, are the non-modular p-compact groups (including the Sullivan spheres), where p does not divide |W |, the Aguadé p-compact groups [1], where p divides |W | exactly once, and the generalized Grasmannians [52] [48, §7] corresponding to the second infinite family in the Clark-Ewing classification table.

We refer to the surveys [42, 51, 18] for more information on p-compact groups.

3. p-local finite groups

The concept of p-local finite group has been introduced in [11] (see also [12]). A p-local finite group is a triple (S, F, L) where S is a finite p-group, F a saturated fusion system over S, and L a centric linking system associated to F. We will state here again all necessary definitions for the convenience of the reader.

A fusion system over a finite group S consists of a set Hom_{F}(P, Q) of monomorphisms for every pair of subgroups P, Q of S, such that it contains at least those monomorphisms induced by conjugation by elements of S and all together form a category where every morphism factors as an isomorphism followed by an inclusion. A fusion system is saturated if it satisfies certain additional axioms formulated by L. Puig (see [11, §1] or the original source [53]). Two subgroups P, P′ of S are called F-conjugate if there is an isomorphism between them in F.

Definition 3.1. Let F be a fusion system over a p-group S.

(1) A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

(2) A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

(3) F is a saturated fusion system if the following two conditions hold:

(i) For each P ≤ S which is fully normalized in F, P is fully centralized in F and Aut_{S}(P) is a Sylow p-subgroup of Aut_{F}(P).

(ii) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φP is fully centralized, and if we set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φP)},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

A subgroup P of S is centric if C_{S}(P′) ≤ P′ for every subgroup P′ ≤ S which is Fconjugate to P. F^{c} denotes the full subcategory whose objects are the centric subgroups of S.

Definition 3.2. Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S, together with a functor

π: L ⟶ F^{c},

and distinguished monomorphisms δ_{P} : P ⟶ Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) π is the identity on objects and surjective on morphisms. More precisely, for each pair of objects P, Q ∈ L, Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

Mor_{L}(P, Q)/Z(P) ⟶^{≅} Hom_{F}(P, Q).

(B) For each F-centric subgroup P ≤ S and each g ∈ P, π sends δ_{P} (g) ∈ Aut_{L}(P) to

c_{g} ∈ Aut_{F}(P). (C) For each f ∈ Mor_{L}(P, Q) and each g ∈ P, the following square commutes in L:

\xymatrix{P \ar[r]^{f} \ar[d]_{δ_{P} (g)} & Q \ar[d]^{δ_{Q}(π(f)(g))} \\ P \ar[r]^{f} & Q.}

The classifying space of the p-local finite group (S, F, L) is defined as the p-completion |L|_{p}^{∧} of the nerve of the category L. The classifying space determines the p-local finite group in the sense that two p-local finite group are isomorphic if and only if they have homotopy equivalent classifying spaces. Actually, the complete structure of a p-local finite group can be recovered from its classifying space by homotopy theoretic methods.

Finite groups are the main source of examples and motivation for p-local finite group theory. If G is a finite group and S a Sylow p-subgroup, the monomorphisms from P to Q induced by conjugation in G, Hom_{G}(P, Q) ≅ N_{G}(P, Q)/C_{G}(P), where N_{G}(P, Q) = { x ∈ G | xP x^{−1} ≤ Q }, form a saturated fusion system over S, F_{S}(G). Furthermore, we define L_{S}^{c}(G) as the category with objects all subgroups of S which are p-centric in G, and morphisms Hom_{L}(P, Q) ≅ N_{G}(P, Q)/C′_{G}(P), where C′_{G}(P) is the p′-complement in C_{G}(P) of the center of P, C_{G}(P) = Z(P) × C′_{G}(P), which is well defined because P is p-centric. L_{S}^{c}(G) is a centric linking system associated to F_{S}(G), and (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} (see [10, 11]). We call exotic those p-local finite groups that are not obtained in this way from any finite group. Examples of exotic p-local finite groups are already shown in [11]. Recently, Levi and Oliver have obtained a family of exotic 2-local finite groups, B Sol(q) [34], based on fusion systems originally described by Solomon [58].

Definition 3.3. (a) For any saturated fusion system F over a p-group S, and any P ≤ S, fully centralized in F, the centralizer fusion system C_{F}(P) over C_{S}(P) is defined by setting

Hom_{C_{F} (P)}(Q, Q′) = {(φ|_{Q}) | φ ∈ Hom_{F}(P Q, P Q′), φ(Q) ≤ Q′, φ|_{P} = Id_{P} }

for all Q, Q′ ≤ C_{S}(P).

(b) For a p-local finite group (S, F, L) and P ≤ S is fully centralized in F, we define the category C_{L}(P) whose objects are C_{F}(P)-centric subgroups Q ≤ C_{S}(P) and where

Mor_{C_{L}(P)}(Q, Q′) = { φ ∈ Hom_{L}(P Q, P Q′) | π(φ)|_{P} = Id_{P} , π(φ)(Q) ≤ Q′ } .

It is proved in [11, §2] that if (S, F, L) is a p-local finite group and P ≤ S is fully centralized in F, then (C_{S}(P), C_{F}(P), C_{L}(P)) is a p-local finite group.

In [34] Levi and Oliver have obtained sufficient conditions for a fusion system to be saturated. We reproduce here their result for the convenience of the reader. We will write C_{F}(x) = C_{F}(⟨x⟩) for x ∈ S.

Proposition 3.4 ([34]). Let F be any fusion system over a p-group S. Then F is saturated if and only if there is a set 𝔛 of elements of of order p in S such that the following conditions hold:

(a) Each x ∈ S of order p is F-conjugate to some element of 𝔛.

(b) If x and y are F-conjugate and y ∈ 𝔛, then there is some ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)) such that ψ(x) = y.

(c) For each x ∈ 𝔛, C_{F}(x) is a saturated fusion system over C_{S}(x).

Let F be a fusion system over a finite p-group S. A subgroup P ≤ S is called radical in F if Out_{F}(P) = Aut_{F}(P)/ Inn(P) is p-reduced, namely, it does not contain non-trivial normal p-subgroups.

Alperin’s fusion theorem for saturated fusion systems [11, A.10] establishes that morphisms in a saturated fusion system are composites of automorphisms of fully normalized, centric, and radical subgroups of the system, or restrictions of those. Hence in order to describe a saturated fusion system F over a finite p-group S it is enough to describe Aut_{F}(Q_{i}) for a set Q_{1}, …, Q_{r} of fully normalized representatives of F-conjugacy classes of centric, radical subgroups of S in F. This motivates the next construction.

If F_{0} is a fusion system over S, Q_{1}, …, Q_{r} are subgroups of S and Δ_{i} is a group of automorphisms Inn(Q_{i}) ≤ Δ_{i} ≤ Aut(Q_{i}), for each i, then we denote by F_{Q_{i}}(Δ_{i}) the fusion system over Q_{i} whose morphisms are restrictions of elements of Δ_{i}, and define

F = ⟨F_{0}; F_{Q_{1}}(Δ_{1}), …, F_{Q_{r}}(Δ_{r})⟩

the fusion system over S whose morphisms are composites of morphisms belonging to any of the generating fusion systems (cf. [11, §9]).

Thus, in particular, if F is a saturated fusion system over a finite p-group S and Q_{1}, …, Q_{r} is a set of fully normalized representatives of F-conjugacy classes of centric, radical subgroups of S in F, then

F = ⟨F_{S}(Aut_{F}(S)); F_{Q_{1}}(Aut_{F}(Q_{1})), …, F_{Q_{r}}(Aut_{F}(Q_{r}))⟩ .

Let G be a finite group and S a Sylow p-subgroup. Let F_{S}(G) the fusion system of G over S. P is centric in F_{S}(G) if and only if it is p-centric in G. A p-subgroup P of G is called p-radical if it is the maximal normal p-subgroup of N_{G}(P), P = O_{p}(N_{G}(P)), or, equivalently, if N_{G}(P)/P is p-reduced. Notice though, that being radical in F_{S}(G) means that Out_{F_{S}(G)}(P) ≅ N_{G}(P)/P C_{G}(P) = Out_{G}(P) is p-reduced.

If P ≤ S is centric and radical in F_{S}(G), then it is p-centric and p-radical in G. Assume that P is not p-radical in G, then there is another p-subgroup Q with P ◁ Q ◁ N_{G}(P) and Q ≠ P. Since P is p-centric, C_{G}(P) = Z(P) × C′_{G}(P), where C′_{G}(P) is a p′-group, hence also C′_{G}(P) ∩ Q = 1, so, therefore P ◁ Q ◁ N_{G}(P)/C′_{G}(P) and Q/P ◁ N_{G}(P)/P C′_{G}(P) = N_{G}(P)/P C_{G}(P), hence Out_{G}(P) is not p-reduced. The converse it is not always true.

We end this section by describing the fusion systems of GL_{p}(q) and SL_{p}(q) over the respective Sylow p-subgroups, where p is a prime number and q is a prime power q ≡ 1 mod p. This will be useful in later sections calculations.

Example 3.5. We will describe the fusion system of GL_{p}(q) over a Sylow p-subgroup, for p a prime and q a prime power such that q ≡ 1 mod p. We can use Alperin and Fong description of p-radical subgroups of general linear groups [4].

The p-primary part of the multiplicative group of units 𝔽_{q}^{∗} is isomorphic to ℤ/p^{ℓ}. Call T_{ℓ}^{p} ≅ (ℤ/p^{ℓ})^{p} the maximal finite torus. S̄ ≅ ℤ/p^{ℓ} ≀ ℤ/p is the Sylow p-subgroup of GL_{p}(q). We can choose S̄ generated by T_{ℓ}^{p}, diagonal matrices of p-power order, and the cycle

C = (\begin{matrix}0 & 0 & … & & 1 \\ 1 & 0 & & & 0 \\ 0 & 1 & & & 0 \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix}) .

The center of GL_{p}(q) is Z_{ℓ} ≅ ℤ/p^{ℓ} embedded diagonally in T_{ℓ}^{p}. Let ζ be a primitive pth root of unity in 𝔽_{q}^{∗}, and define the diagonal matrix B = diag(1, ζ, ζ^{2}, …, ζ^{p−1}) and the subgroup Γ_{ℓ} generated by Z_{ℓ}, together with the matrices B and C. This is a central product of the center Z_{ℓ} and an extraspecial group Γ_{1} of order p^{3} and exponent p, generated by A = diag(ζ, ζ, …, ζ), B, and C.

There is an standard inclusion 𝔽_{q^{p}}^{∗} ⊆ GL_{p}(q), obtained by letting 𝔽_{q^{p}}^{∗} act on 𝔽_{q^{p}} by multiplication and considering 𝔽_{q^{p}} as 𝔽_{q}-vector space. We define U_{ℓ+1} as the image in GL_{p}(q) of the cyclic group ℤ/p^{ℓ+1} ≤ 𝔽_{q^{p}}^{∗}, (q ≡ 1 mod p and ℓ = ν_{p}(1 − q)) of p-power roots of unity in 𝔽_{q^{p}}.

With this notation and according to [4], if R is a p-radical subgroup of GL_{p}(q), q ≡ 1 mod p, ℓ = ν_{p}(1 − q), then R is conjugated to one of the following subgroups:

\begin{matrix}R & N_{GL_{p}(q)}(R) & Out_{GL_{p}(q)}(R) \\ Z_{ℓ} & GL_{p}(q) & 1 \\ T_{ℓ}^{p} & (𝔽_{q}^{∗})^{p} ⋊ Σ_{p} & Σ_{p} \\ S̄ & (𝔽_{q}^{∗})^{p} ⋊ (ℤ/p ⋊ ℤ/p − 1) & ℤ/p − 1 \\ Γ_{ℓ} & (𝔽_{q}^{∗}) · Γ_{ℓ} · SL_{2}(p) & SL_{2}(p) \\ U_{ℓ+1} & 𝔽_{q^{p}}^{∗} ⋊ ℤ/p & ℤ/p. \end{matrix}  (3)

Notice that Z_{ℓ} is clearly non-centric, but the other are all centric in GL_{p}(q).

It is now easy to extract from (3) the centric radical subgroups in the fusion system of GL_{p}(q) over S̄:

\begin{matrix}R & Out_{GL_{p}(q)}(R) \\ T_{ℓ}^{p} & Σ_{p} \\ S̄ & ℤ/p − 1 \\ Γ_{ℓ} & SL_{2}(p). \end{matrix}  (4)

Example 3.6. We proceed now by describing the fusion system of SL_{p}(q) over a Sylow p-subgroup, for p a prime and q a prime power such that q ≡ 1 mod p.

We first show that every p-radical subgroup of SL_{p}(q) is the intersection Q ∩ SL_{p}(q) of a p-radical subgroup Q of GL_{p}(q) with SL_{p}(q). For a given p-radical p-subgroup P of SL_{p}(q) define Q = O_{p}(N_{GL_{p}(q)}(P)). Q ∩ SL_{p}(q) is a normal subgroup of N_{SL_{p}(q)}(P) and since P is the maximal normal p-subgroup of N_{SL_{p}(q)}(P), we have Q ∩ SL_{p}(q) ≤ P. Same argument with N_{GL_{p}(q)}(P) shows that P ≤ Q and therefore Q ∩ SL_{p}(q) ≤ P.

Every element g ∈ GL_{p}(q) normalizes SL_{p}(q), so if g normalizes Q it also normalizes Q ∩ SL_{p}(q) ≤ P, so N_{GL_{p}(q)}(Q) ≤ N_{GL_{p}(q)}(P). But, by definition of Q, this is normal in N_{GL_{p}(q)}(P), hence we actually have N_{GL_{p}(q)}(Q)=N_{GL_{p}(q)}(P). So, therefore, Q=O_{p}(N_{GL_{p}(q)}(Q)) is p-radical.

Fix the Sylow p-subgroup S = S̄ ∩ SL_{p}(q) of SL_{p}(q). Assume that P ≤ S is centric and radical in the fusion system F_{S}(SL_{p}(q)), q ≡ 1 mod p, ℓ = ν_{p}(1 − q). Then P is p-centric and p-radical in SL_{p}(q). In particular P = Q ∩ SL_{p}(q) where Q is p-radical in GL_{p}(q), hence conjugate by an element g ∈ GL_{p}(q) to a p-subgroup in the list (3). Among those intersections, only S = S̄ ∩ SL_{p}(q), T_{ℓ}^{(p−1)} = S ∩ T_{ℓ}^{p}, and Γ_{1} = S ∩ Γ_{ℓ} are also p-centric. Hence the complete list of conjugacy classes of p-centric and p-radical subgroups of SL_{p}(q), is obtained by conjugating these three subgroups by elements g ∈ GL_{p}(q):

P Out_{SL_{p}(q)}(P) Conditions
T_{ℓ}^{(p−1)} Σ_{p} p > 3
S  ℤ/p − 1
Γ_{1}(ξ^{r})  SL_{2}(p)  r = 0 if ℓ = 1, p = 3; ; r = 0, 1, …, p − 1 if ℓ > 1 or ; p > 3,

(5)

where Γ_{1}(ξ^{r}), r = 0, 1, …, (p − 1) is the conjugated subgroup of Γ_{1} by the diagonal matrix diag(ξ^{r}, 1, …, 1), ξ a (q − 1)st root of unity. Notice that for g ∈ GL_{p}(q), gSg^{−1} lies in S if and only it it is exactly S and the same happens with T_{ℓ}^{(p−1)}. In the case of Γ_{1} we just need to check which of the subgroups Γ_{1}(ξ^{r}) are conjugated in SL_{p}(q). In fact, Alperin’s fusion theorem [11, A.10], together with the list of p-radical p-centric subgroups that we have obtained so far, tells us that if two subgroups Γ_{1}(ξ^{r}) and Γ_{1}(ξ^{s}) are conjugated in SL_{p}(q) they are already conjugated in N_{SL_{p}(q)}(S), hence we obtain the list (5) by direct calculation.

4. Recognition of classifying spaces of p-local finite groups

In [11] it is shown that a p-local finite group can be completely recovered from its classifying space by homotopy theoretic methods. Also, a recognition principle for classifying spaces of p-local finite groups is provided in [11, Thm. 7.5]. We will briefly describe these methods and derive an inductive method that will be useful in our situation.

We will first recall how a fusion system F_{(S,f)}(X) or a linking system L_{(S,f)}(X) are attached to a space X equipped with a map f: BS → X, where S is a finite p-group. Then, the basic tool in order to show that these systems define a p-local finite group with classifying space X is [11, Thm. 7.5]. In order to apply the theorem we are generally faced to two main difficulties, namely, to show that the nerve of L_{(S,f)}(X) is homotopy equivalent to X and to show that F_{(S,f)}(X) is a saturated fusion system. We will overcome these difficulties by an inductive method mainly based on the centralizers decomposition of p-local finite groups that we develop in this section.

Definition 4.1. Given spaces X and Y, we say that a map α: X → Y is a homotopy monomorphism at p if the homotopy fibre of α, F, over any connected component of Y, is pquasi-finite; that is, the inclusion F → Map(Bℤ/p, F) as constant maps is a weak homotopy equivalence.

It is not hard to prove that a composition of homotopy monomorphisms at p is again a homotopy monomorphism at p.

Definition 4.2. Let X be a space. A finite p-subgroup of X is a pair (P, f), where P is a finite p-group and f: BP → X a homotopy monomorphism at p. A p-subgroup (S, f) of X is called a Sylow p-subgroup of X if for any other p-subgroup (Q, g) of X, g: BQ → X factors through f: BS → X, up to homotopy.

If (P, f) is a p-subgroup of X, then we denote BC_{X}(P, f) = Map(BP, X)_{f}.

We will need later the next technical lemma.

Lemma 4.3. Assume that X and Y are spaces for which Map(Bℤ/p, X)_{ct} ≃ X and Map(Bℤ/p, Y)_{ct} ≃ Y.

Let f: X → Y be a homotopy monomorphism at p and μ: BP → X a finite p-subgroup of X, then each map in the diagram

\xymatrix{BC_{X}(P, μ) \ar[r]^{ev} \ar[d]_{f_{♯}} & X \ar[d]^{f} \\ BC_{Y} (P, f ∘ μ) \ar[r]_{ev} & Y}

is a homotopy monomorphism at p.

Proof. Let F be the homotopy fibre of the evaluation map

BC_{X}(P, μ) = Map(BP, X)_{μ} ⟶^{ev} X .

There is an induced fibration

Map(Bℤ/p, F) → Map(Bℤ/p, Map(BP, X)_{μ})c̃t → Map(Bℤ/p, X)_{ct}

where c̃t stands for all components mapping down to the component of the constant map in Map(Bℤ/p, X). Since Map(Bℤ/p, X)_{ct} ≃ X, also

Map(Bℤ/p, Map(BP, X)_{μ})c̃t ≃ Map(BP, Map(Bℤ/p, X)_{ct})μ̃ ≃ Map(BP, X)_{μ} ,

and therefore Map(Bℤ/p, F) ≃ F; that is, F is p-quasi finite and ev: C_{X}(P, μ) → X is a homotopy monomorphism at p. Similarly, ev: BC_{Y} (P, f ∘μ) → Y is a homotopy monomorphism at p and then, it is easy to obtain that also f_{♯} is a homotopy monomorphism at p. □

For any space X we denote F_{p}(X) the category in which the objects are finite p-subgroups (P, f) of X, and the morphisms are defined

Mor_{F_{p}(X)}((P, f), (Q, g)) = { φ ∈ Hom(P, Q) | f ≃ g ∘ Bφ } .

Next construction appears already in [10]. We denote L_{p}(X) the category in which objects are p-subgroups (P, f) of X and morphisms are defined as

Mor_{L_{p}(X)}((P, f), (Q, g)) = { (φ, [H]) | φ ∈ Hom(P, Q) and

[H] is the homotopy class of a homotopy from f to g ∘ Bφ } .

We denote L_{p}^{c}(X) the full subcategory whose objects are p-subgroups (P, f) where f is a p-centric map; that is, the induced map f_{♯}: Map(BP, BP)_{Id} → Map(BP, X)_{f} is a mod p homology equivalence.

If (S, f) is a p-subgroup of a space X we can define a fusion system over S, F_{(S,f)}(X), by declaring

Hom_{F_{(S,f)}(X)}(P, Q) = { φ ∈ Hom(P, Q) | f|_{BP} ≃ f|_{BQ} ∘ Bφ }

for all P, Q ≤ S, where f|_{BP} denotes the composition BQ ⟶^{Bi_{P}} BS ⟶^{f} X. Notice that if (S, f) is a Sylow p-subgroup of X, then, as categories, F_{(S,f)}(X) is equivalent to F_{p}(X). Next, we define the category L_{(S,f)}(X) that has objects the subgroups of S and

Mor_{L_{(S,f)}(X)}(P, Q) = { (φ, [H]) | φ ∈ Hom(P, Q) and

[H] is the homotopy class of a homotopy from f|_{BP} to f|_{BQ} ∘ Bφ } ,

and teh full subcategory L_{(S,f)}^{c}(X) whose objects are F_{(S,f)}(X)-centric subgroups P ≤ S. There is also a natural functor π: L_{p}(X) → F_{p}(X), obtained by forgetting the concrete homotopy classes [H] in morphisms sets.

The important question and the aim of the rest of this section consists in finding sufficient conditions on a space X and a p-subgroup (S, f) under which

(S, F_{(S,f)}(X), L_{(S,f)}^{c}(X))

is a p-local finite group and X is its classifying space |L_{(S,f)}^{c}(X)|_{p}^{∧} ≃ X.

One first important case is that of X = |L|_{p}^{∧}, the classifying space itself of a given plocal finite group (S, F, L). In this case there is a natural inclusion δ_{S}: BS → |L|_{p}^{∧} and (S, F_{(S,δ_{S})}(|L|_{p}^{∧}), L_{(S,δ_{S})}^{c}(|L|_{p}^{∧})) is isomorphic to the original (S, F, L). This is how a p-local finite group is completely recovered from its classifying space.

In [11, §7] it is also considered the case of an arbitrary p-complete space X equipped with a p-subgroup (S, f). The main argument establishes sufficient conditions on X and (S, f) under which (S, F_{(S,f)}(X), L_{(S,f)}^{c}(X)) is a p-local finite group. The question of whether or not X is the classifying space is left to directly checking if there is a homotopy equivalence |L_{(S,f)}^{c}(X)|_{p}^{∧} ≃ X.

There seems to be no natural way to construct a map between X and |L_{(S,f)}^{c}(X)| in either direction. This problem was solved in [10] by means of an auxiliary simplicial space M_{•}^{c}(X) that comes equipped with a natural simplicial map

τ_{X}: M_{•}^{c}(X) ⟶ N_{•}(L_{p}^{c}(X))

which induces a homotopy equivalence |τ_{X}|: |M_{•}^{c}(X)| → |L_{p}^{c}(X)|, provided the spaces Map(BP, X)_{α} are aspherical for any p-centric subgroup α: BP → X (see [10, Lemma 4.2] and its proof). The geometric realization |M_{•}^{c}(X)| admits an evaluation map

ev_{X}: |M_{•}^{c}(X)| → X

thus |τ_{X}| and ev_{X} can be used in order to connect geometrically X and |L_{p}^{c}(X)|, or equivalently |L_{(S,f)}^{c}(X)|.

Proposition 4.4. There is a natural map Mf: BS → |M_{•}^{c}(X)| that makes the diagram

\xymatrix{ & BS \ar[dl]_{|δ_{S}|} \ar[d]_{Mf} \ar[dr]^{f} \\ |L_{p}^{c}(X)| & |M_{•}^{c}(X)| \ar[l]^{≃} \ar[r] & X}

commutative up to homotopy.

Proof. Proposition 2.7, Lemma 4.2 and Lemma 4.3 of [10] provide homotopy equivalences

\xymatrix{|L_{p}^{c}(S)| \ar[r]^{≃} & |L_{p}^{c}(BS)| & |M_{•}^{c}(BS)| \ar[l]_{≃} \ar[r]^{≃} & BS}  (6)

hence the Proposition could be proven by showing a map |M_{•}^{c}(BS)| → |M_{•}^{c}(X)| making commutative the necessary diagrams.

M_{•}^{c}(X) is a simplicial space where n-simplices are maps η: Δ(P) → X, where P = (P_{0} →^{φ_{1}} P_{1} →^{φ_{2}} ⋯ →^{φ_{n}} P_{n}) is a sequence of p-subgroups of S and monomorphisms, and Δ(P) can be regarded as the homotopy colimit of the sequence BP_{0} ⟶^{Bφ_{1}} BP_{1} ⟶^{Bφ_{2}} ⋯ ⟶^{Bφ_{n}} BP_{n}, with the condition that the restriction of η to and BP_{i} is a centric p-subgroup of X (see [10, §4] for details).

This last condition is what prevents the obvious construction of a map |M_{•}^{c}(BS)| → |M_{•}^{c}(X)| from being well defined. In fact a subgroup P ≤ S which is centric in S, need not be centric when regarded as a p-subgroups of X as BP ⟶^{Bincl} BS ⟶^{f} X.

We will have to restrict M_{•}^{c}(BS) to the subspace M_{•}^{S}(BS) of simplices η: Δ(P) → BS of M_{•}^{c}(BS) where every group in the sequence Δ(P) is S itself. Accordingly, we call L_{p}^{S}(BS) the full subcategory of L_{p}^{c}(BS) with objects the homotopy equivalences g: BS → BS. With this notation we have a diagram of homotopy equivalences

[figure: |BS|; ≃; ≃; |L_{p}^{S}(BS)|; ≃; |M_{•}^{S}(BS)|; ev_{BS}; ≃; ≃; ≃; BS; |L_{p}^{c}(S)|; ≃; |L_{p}^{c}(BS)|; |M_{•}^{c}(BS)|; ≃; ev_{BS}; ≃; BS]  (7)

where same arguments as in [10] for the sequence (6) are used.

Now, for every equivalence g: BS → BS, the composition BS →^{g} BS →^{f} X defines a centric p-subgroup of X, and then f induces a well defined map of simplicial spaces M_{•}^{S}(BS) → M_{•}^{c}(X), that makes commutative the diagram

\xymatrix{|BS| \ar[r]^{≃} \ar[dr] & |L_{p}^{S}(BS)| \ar[d]^{f_{♯}} & |M_{•}^{S}(BS)| \ar[r]^{ev_{BS}}_{≃} \ar[d]^{f_{♯}} \ar[l]_{≃} & BS \ar[d]^{f} \\  & |L_{p}^{c}(X)| & |M_{•}^{c}(X)| \ar[l]_{≃} \ar[r]^{ev_{X}}_{≃} & X}  (8)

from which the proposition follows. □

The next is a useful result that provides conditions on the space X and a Sylow psubgroup (S, f) under which the fusion system F_{(S,f)}(X) is saturated. An element x ∈ S of order p determines a homomorphism i_{x}: ℤ/p → S an then a map f ∘Bi_{x}: Bℤ/p → X. We write BC_{X}(x) = Map(Bℤ/p, X)_{x}, the connected component that contains the map f ∘ Bi_{x}, and f_{x}: BC_{S}(x) → BC_{X}(x) the map induced by f.

Proposition 4.5. Let X be a space, (S, f) a Sylow p-subgroup of X, and 𝔛 a set of elements of order p in S. Assume that:

(1) Map(Bℤ/p, X)_{ct} ≃ X.

(2) For all x ∈ 𝔛, the natural map f_{x}: BC_{S}(x) → BC_{X}(x) is a Sylow p-subgroup for BC_{X}(x).

(3) For all x ∈ 𝔛, F_{(C_{S}(x),f_{x})}(BC_{X}(x)) is a saturated fusion system over C_{S}(x).

(4) For all x ∈ S of order p, there is φ ∈ Hom_{F_{(S,f)}(X)}(⟨x⟩, S) such that φ(x) ∈ 𝔛.

Then F_{(S,f)}(X) is a saturated fusion system over S and C_{F_{(S,f)}(X)}(x) coincides with F_{(C_{S}(x),f_{x})}(BC_{X}(x)) as fusion systems over C_{S}(x), for all x ∈ 𝔛.

Proof. Write F = F_{(S,f)}(X) for short. Clearly, F is a fusion system over S. Condition (a) of Proposition 3.4 holds by (4); and it remains to show that conditions (b) and (c) of 3.4 hold.

Condition (b) of 3.4: Fix x, y ∈ S of order p such that y ∈ 𝔛, and such that there is ψ_{0} ∈ Hom_{F}(⟨x⟩ , ⟨y⟩) with ψ_{0}(x) = y. We must show that ψ_{0} extends to some ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)).

Since x and y are F-conjugate,

[f ∘ Bi_{x}] = [f ∘ Bi_{y}] ∈ [Bℤ/p, X],

and thus Map(Bℤ/p, X)_{x} = Map(Bℤ/p, X)_{y}. Since C_{S}(y) is a Sylow p-subgroup of Map(Bℤ/p, X)_{y} by (2), the natural map BC_{S}(x) → Map(Bℤ/p, X)_{x} factors through BC_{S}(y). In other words, there is some ψ ∈ Hom(C_{S}(x), C_{S}(y)) such that the following square commutes up to homotopy

[figure: BC_{S}(x) × Bℤ/p; f∘B(incl×i_{x}); Bψ×Id; X; BC_{S}(y) × Bℤ/p; f∘B(incl×i_{y}); X.]  (9)

Thus ψ ∈ Hom_{F}(C_{S}(x), C_{S}(y)). If ρ, ρ′ ∈ Hom(C_{S}(x) × ℤ/p, S) denote the homomorphisms ρ(g, t) = gx^{t} and ρ′(g, t) = ψ(g)y^{t}, then f ∘ Bρ ≃ f ∘ Bρ′ by (9), and hence Ker(ρ) = Ker(ρ′) by [11, Proposition 5.4(d)] (and point (1)). And this implies that ψ(x) = y.

Condition (c) of 3.4: Fix some x ∈ 𝔛; we must show that C_{F}(x) is a saturated fusion system. By (3), the fusion system F′ =^{def} F_{(C_{S}(x),f_{x})}(BC_{X}(x)) is saturated, so it suffices to show that these two fusion systems over C_{S}(x) are equal.

To see this, fix P, Q ≤ C_{S}(x), and let φ ∈ Hom(P, Q) be any monomorphism. Set P = P · ⟨x⟩ and Q = Q· ⟨x⟩. Let ρ ∈ Hom(P × ℤ/p, S) and ρ′ ∈ Hom(Q × ℤ/p, S) be defined by ρ(g, t) = gx^{t} and ρ′(g, t) = gx^{t}. Then φ ∈ Hom_{F′}(P, Q) if and only if the following square commutes up to homotopy

[figure: BP × Bℤ/p; f∘Bρ; Bφ×Id; X; BQ × Bℤ/p; f∘Bρ′; X.]  (10)

By (1) and [11, Proposition 5.4(d)], this holds if and only if K =^{def} Ker(ρ) = Ker(ρ′ ∘ (φ × Id)) and the induced maps from B((P × ℤ/p)/K) to X are homotopic. The kernels are equal if and only if φ extends to a monomorphism φ from P to Q which sends x to itself. And in this case, the induced maps on B((P × ℤ/p)/K) are homotopic if and only if f|_{BP} ≃ f|_{BQ} ∘ Bφ, if and only if φ ∈ Hom_{C_{F} (x)}(P, Q).

Now, Proposition 3.4 implies that F_{(S,f)}(X) is a saturated fusion system over S and the argument for condition (c) already contains the proof that C_{F}(x) coincides with F′ = F_{(C_{S}(x),f_{x})}(BC_{X}(x)) as fusion systems over C_{S}(x). □

We derive now another characterization that will be useful in the specific cases in which we are interested or more generally in cases in which there is a good knowledge of elementary abelian p-subgroups of X and of its centralizers.

Theorem 4.6. Let X be a p-complete space and (S, f) a p-subgroup of X. Assume that

(1) Map(Bℤ/p, X)_{ct} ≃ X, and

(2) for each non-trivial element x ∈ S of order p (a) BC_{X}(x) is the classifying space of a p-local finite group, and

(b) if (H, g) is a Sylow p-subgroup for BC_{X}(x), there is a group homomorphism ρ: H → S that makes the diagram

\xymatrix{BH \ar[d]_{g} \ar[r]^{Bρ} & BS \ar[d]^{f} \\ BC_{X}(x) \ar[r]^{ev} & X}

commutative up to homotopy,

then, (S, f) is a Sylow p-subgroup for X and

(S, F_{(S,f)}(X), L_{(S,f)}^{c}(X))

is a p-local finite group.

Furthermore, X ≃ |L_{(S,f)}(X)|_{p}^{∧} if and only if the natural map induced by evaluation

hocolim_{F_{(S,f)}^{e}(X)^{op}} Map(BE, X)_{f|_{BE}} ⟶ X

is a mod p homology equivalence. Here F_{(S,f)}^{e}(X) denotes the full subcategory of F_{(S,f)}(X) consisting of non-trivial fully centralized elementary abelian p-subgroups of S.

Proof. The proof is divided in four steps. First, we prove that (S, f) is a Sylow p-subgroup for X. Next, that the fusion system of X over (S, f), F_{(S,f)}(X) is saturated. In the third step we show that for each F_{(S,f)}(X)-centric subgroup P ≤ S the map f|_{BP} is p-centric.

These two last steps are the hypothesis (a) and (c) of [11, Theorem 7.5]. According to the remarks after the proof of this theorem in [11], this suffices in order to conclude that (S, F_{(S,f)}(X), L_{(S,f)}^{c}(X)) is a p-local finite group. This is the first part of the theorem.

The second part states that X ≃ |L_{(S,f)}(X)|_{p}^{∧} if and only if the natural map induced by evaluation hocolim_{F_{(S,f)}^{e}(X)^{op}} Map(BE, X)_{f|_{BE}} ⟶ X is a mod p homology equivalence. This is proves in step 4. Notice that X ≃ |L_{(S,f)}(X)|_{p}^{∧} is condition (b) in [11, Theorem 7.5]. Hence this second part of the theorem gives a necessary and sufficient condition for X to be the classifying space of the p-local finite group (S, F_{(S,f)}(X), L_{(S,f)}^{c}(X)).

Step 1: (S, f) is a Sylow p-subgroup for X. Let (P, μ) be a finite p-subgroup of X. Choose a central element x of order p in P. It determines a homomorphism i_{x}: ℤ/p → P for which C_{P} (ℤ/p) = P, and a map μ ∘ Bi_{x}: Bℤ/p → X. According to our hypothesis, BC_{X}(x) is the classifying space of a p-local finite group, and if (H, g) is its Sylow p-subgroup, there are homomorphisms ρ: H → S and φ: C_{P} (ℤ/p) → H that make the diagram

\xymatrix{BC_{P} (ℤ/p) \ar[rr]^{Bφ} \ar[d]_{≃}^{ev} \ar[r]^{μ_{♯}} & BC_{X}(ℤ/p, μ ∘ Bi_{x}) \ar[d]^{ev} & BH \ar[d]^{Bρ} \ar[l]_{g} \\ BP \ar[r]^{μ} & X & BS \ar[l]_{f}}

commutative up to homotopy. Hence, ρ ∘ φ: P = C_{P} (ℤ/p) → S provides the factorization of (P, μ) through (S, f).

Step 2: The fusion system of X over (S, f), F_{(S,f)}(X) is saturated. This part of the proof will be based on Proposition 4.5. Define

𝔛 = { x ∈ S | x of order p and f_{x}: BC_{S}(x) → BC_{X}(x)

is a Sylow p-subgroup for BC_{X}(x) } .

Notice now that conditions (1) and (2) of Proposition 4.5 are satisfied by our hypothesis and by definition of the class 𝔛. Condition (3) is easily verified, too. In fact, by hypothesis, for each x ∈ 𝔛, BC_{X}(x) is the classifying space of a p-local finite group and since f_{x}: BC_{S}(x) → BC_{X}(x) is a Sylow p-subgroup for BC_{X}(x), the fusion system F_{(C_{S}(x),f_{x})}(BC_{X}(x)) is saturated.

It remains to verify condition (4); that is, that every element x ∈ S of order p is F_{(S,f)}(X)- conjugate to an element of the class 𝔛.

Assume that x ∈ S has order p. It gives a homomorphism i_{x}: ℤ/p → S and a map f ∘ Bi_{x}: Bℤ/p → X. There is an evaluation map ev: Bℤ/p × BC_{X}(x) → X. Let (H, g) be a Sylow p-subgroup of BC_{X}(x). Since (S, f) is a Sylow p-subgroup of X, there is a homomorphism ρ: ℤ/p × H → S making the diagram

\xymatrix{Bℤ/p × BH \ar[r]^{Bρ} \ar[d]_{1×g} & BS \ar[d]^{f} \\ Bℤ/p × BC_{X}(x) \ar[r]^{ev} & X}

commutative up to homotopy.

Let φ = ρ|_{ℤ/p} the restriction of ρ to the first component ℤ/p. From the above diagram we deduce that φ ∈ Hom_{F_{(S,f)}(X)}(ℤ/p, S). Let y = φ(x).

Then, ρ induces

BH ⟶^{Bρ̃} BC_{S}(y) ⟶^{f_{y}} BC_{X}(y) ⟶^{ev} X

where all maps are homotopy monomorphisms at p. The first one because ρ̃ is a monomorphism, the others by Lemma 4.3.

Now, φ induces a homotopy equivalence BC_{X}(y) ≃ BC_{X}(x), hence also an isomorphism between the respective Sylow p-subgroups. Since (H, g) is a Sylow p-subgroup for C_{X}(x), it follows from the above sequence of maps that (C_{S}(y), f_{y}) is a Sylow p-subgroup for C_{X}(y). Hence y = φ(x) ∈ 𝔛.

Step 3: f|_{BP} is a p-centric map for each F_{(S,f)}(X)-centric subgroup P ≤ S. Suppose that P ≤ S is F_{(S,f)}(X)-centric. Choose a central element x ∈ S or order p. Since P is centric, x ∈ P and we have a sequence of homotopy monomorphisms at p

BP ⟶^{Bincl} BS ⟶^{f_{x}} BC_{X}(x) ⟶^{ev} X .

By hypothesis, BC_{X}(x) is the classifying space of a p-local finite group, and from the above sequence of maps we easily obtain that (S, f_{x}) is a Sylow p-subgroup for BC_{X}(x). Furthermore, P is also F_{S,f_{x}}(BC_{X}(x))-centric, and then f_{x}|_{BP} is a p-centric map. There is a sequence of equivalences

Map(BP, BP)_{Id} ≃ Map(BP, BC_{X}(x))_{f_{x}|_{BP}} ≃ Map(BP × Bℤ/p, X)_{f|_{BP} ∘Bm} ≃ Map(BP, X)_{f|_{BP}}

where m: P × ℤ/p → P denotes multiplication by x, the generator of ℤ/p = ⟨x⟩. The last equivalence is implied by the Zabrodsky’s lemma applied to the fibration Bℤ/p ⟶ BP × Bℤ/p ⟶^{Bm} BP. The above composition shows that f|_{BP} is a p-centric map.

Step 4: X ≃ |L_{(S,f)}(X)|_{p}^{∧} if and only if the natural map

hocolim_{F_{(S,f)}^{e}(X)^{op}} Map(BE, X)_{f|_{BE}} ⟶ X

induced by evaluation is a mod p homology equivalence. Since the categories L_{p}^{c}(X) and L_{(S,f)}^{c}(X) are equivalent, we can write the diagram of of Proposition 4.4 as the homotopy commutative triangle

\xymatrix{ & BS \ar[dl]_{θ} \ar[dr]^{f} \\ |L_{(S,f)}^{c}(X)| \ar[rr]^{h} &  & X .}  (11)

It induces an equivalence of fusion systems over S:

F_{(S,θ)}(|L_{(S,f)}^{c}(X)|) = F_{(S,f)}(X)

and a natural map

η_{P} : Map(BP, |L_{(S,f)}^{c}(X)|)_{θ|_{BP}} ⟶ Map(BP, X)_{f|_{BP}}  (12)

for every P ≤ S. Moreover, the diagram

\xymatrix{Map(BP, |L_{(S,f)}^{c}(X)|)_{θ|_{BP}} \ar[r]^{η_{P}} \ar[d]_{ev} & Map(BP, X)_{f|_{BP}} \ar[d]^{ev} \\ |L_{(S,f)}^{c}(X)| \ar[r]^{h} & X}

is strictly commutative, with vertical maps induced by evaluation at the base point. As a consequence, we obtain a map between the corresponding homotopy colimits together with compatible maps induced by evaluation:

\xymatrix{hocolim_{F_{(S,f)}^{e}(X)^{op}} Map(BE, |L_{(S,f)}^{c}(X)|_{p}^{∧})_{θ|_{BE}} \ar[d]_{ev} \ar[r]^{η} & hocolim_{F_{(S,f)}^{e}(X)^{op}} Map(BE, X)_{f|_{BE}} \ar[d]^{ev} \\ |L_{(S,f)}^{c}(X)|_{p}^{∧} \ar[r]^{h} & X}  (13)

where F_{(S,f)}^{e}(X) is the full subcategory of F_{(S,f)}(X) consisting of non-trivial elementary abelian subgroups of E ≤ S that are fully centralized.

Then problem is then reduced to showing that every map η_{P} in (12) is a homotopy equivalence. In fact, the map η in the diagram (13) would be a homotopy equivalence, too. The left vertical map of (13) is also a homotopy equivalence by [11, 2.6 and 6.3]. The theorem would follow as the right vertical map ev in (13) would be a homotopy equivalence if and only if h is a homotopy equivalence.

We will show that η_{P} in (12) is a homotopy equivalence by induction on the order of the group P. If P = ⟨x⟩, for some x ∈ S of order p, then BC_{X}(x) = Map(BP, X)_{f|_{BP}} is the classifying space of a finite p-local group, by hypothesis. According to Step 2 above, we can assume without loss of generality that x ∈ 𝔛, and so, the induced map f_{x}: BC_{S}(x) → BC_{X}(x) is the inclusion of a Sylow p-subgroup, and the fusion system F_{(C_{S}(x),f_{x})}(BC_{X}(x)) coincides with C_{F_{(S,f)}(X)}(x) by Proposition 4.5.

Now, diagram (11) induces the new homotopy commutative diagram

\xymatrix{ & BC_{S}(x) \ar[dl]_{θ_{♯}} \ar[dr]^{f_{x}} \\ Map(BP, |L_{(S,f)}^{c}(X)|_{p}^{∧})_{θ|_{BP}} \ar[rr]_{η_{P}} &  & BC_{X}(x)}  (14)

where, according to [11, 6.3], the map θ_{♯} is the inclusion of a Sylow p-subgroup of Map(BP, |L_{(S,f)}^{c}(X)|_{p}^{∧})_{θ|_{BP}} which is the classifying space of a centralizer p-local finite group with fusion system C_{F_{(S,f)}(X)}(x). Furthermore, η_{P} induces an equivalence of fusion systems, and therefore a homotopy equivalence.

For an arbitrari non-trivial subgroup P ≤ S, we fix an element x of order p in the center of P. Again, we can assume that x belongs to 𝔛. Then, the map η_{P}, factors as the composition

Map(BP, |L_{(S,f)}^{c}(X)|)_{θ|_{BP}} ⟶ Map(BP × B ⟨x⟩ , |L_{(S,f)}^{c}(X)|)_{θ|_{BP} ∘Bm}
⟶ Map(BP, Map(B ⟨x⟩ , |L_{(S,f)}^{c}(X)|)_{Bincl})_{θ|_{BP}}
⟶ Map(BP, Map(B ⟨x⟩ , X)_{x})_{f|_{BP}}
⟶ Map(BP × B ⟨x⟩ , X))_{f|_{BP} ∘Bm}
⟶ Map(BP, X)_{f|_{BP}}

where all arrows are homotopy equivalences. That concludes the proof that η_{P} in equation (12) is a natural mod p homology equivalence for subgroups P ≤ S. □

Notice also, that, reciprocally, if X is the classifying space of a p-local finite group with Sylow p-subgroup (S, f), then all conditions of Theorem 4.6 are satisfied according to [11, §7].

5. Homotopy fixed points p-compact groups

Let M be a space and G a group. It will be convenient for our purposes, to define an action of G on M as a fibration

\xymatrix{M \ar[r] & M_{hG} \ar[r]^{p} & BG}  (15)

and, accordingly, a G-equivariant map between M and another space M′ supporting an action of G is a map f: M → M′ that extends to a map f_{hG}: M_{hG} → M′_{hG} over BG. Notice that this is not actually a real action G × M → M, but only a proxy action ([22, §10]). It determines a homotopy action of G on M, that is a homomorphism ρ: G → [M, M], which is obtained as the homomorphism induced on fundamental groups by the classifying map φ: BG → B aut(M). Thus, for a fixed homotopy action ρ: G → [M, M], a map φ: BG → B aut(M) with π_{1}(φ) = ρ is interpreted as a lifting of ρ to a proxy action, while a lifting to an actual action would be a homomorphism ρ̄: G → aut(M) whose composition with the projection aut(M) → [M, M] is ρ. The total space M_{hG} of the fibration (15) is the homotopy quotient of the action and the homotopy fixed point set M^{hG} is defined as the space of sections of M_{hG} → M.

Similarly, if X is a loop space with classifying space BX, we will say that an action of the group G on X is a split fibration

\xymatrix{BX \ar[r]^{i} & BX_{hG} \ar[r]^{p} & BG. \ar[l]^{s}}

The section guarantees an induced action of G on X, compatible with the loop structure. The homotopy quotient for this action on X is defined as the pullback space in the diagram

\xymatrix{X_{hG} \ar[d]_{p̄} \ar[r]^{p̄} & BG \ar[d]^{s} \\ BG \ar[r]_{s} & BX_{hG}.}  (16)

This diagram turns out to be a diagram of spaces over BG. The homotopy fibre of p̄ is X, and it has a canonical section s̄ defined by the pullback diagram (16) that we can interpret as the homotopy constant loop

\xymatrix{X \ar[r]^{ī} & X_{hG} \ar[r]^{p̄} & BG. \ar[l]^{s̄}}

The action of G on X depends on the section s: BG → BX_{hG}, and for this action we obtain that the homotopy fixed point space X^{hG} is a loop space with classifying space B(X^{hG}) ≃ (BX)_{s}^{hG}, the connected component of (BX)^{hG} with base point the section s. Furthermore, the evaluation map X^{hG} → X is seen to be the loop map of the evaluation map (BX)_{s}^{hG} → BX, thus we have a sequence of fibrations

\xymatrix{X^{hG} \ar[r]^{ev} & X \ar[r] & X/X^{hG} \ar[r] & (BX)_{s}^{hG} \ar[r]^{ev} & BX}

where we write X/X^{hG} for the homotopy fibre of the evaluation map (BX)_{s}^{hG} → BX.

By analogy with discrete group theory, we will write Out(X) = [BX, BX] and will say that an outer action of G on X is a homomorphism of groups ρ: G → Out(X). Thus, an action of G on BX, classified by a map φ: BG → B aut(BX), gives rise to an outer action, obtained as ρ = π_{1}(φ): G → Out(X). Equivalently, we say that a fibration over BG with fibre BX induces the outer action ρ: G → Out(X) if it is classified by a lifting of Bρ to B aut(BX):

\xymatrix{ & B aut(BX) \ar[d] \\ BG \ar[r]_{Bρ} \ar[ur]^{φ} & B Out(X) .}

As we have explained, the fibration over BG with fibre BX is not yet an action of G on X. An action of G on X inducing the given action on BX is classified by a further lifting

\xymatrix{ & B aut_{∗}(BX) \ar[d] \\ BG \ar[r]_{φ} \ar[ur]^{ψ} & B aut(BX) .}

We can also lift ρ directly to an action of G on X, independently of the given action on BX:

\xymatrix{ & B aut_{∗}(BX) \ar[d] \\ BG \ar[r]_{Bρ} \ar[ur]^{ψ} & B Out(X) .}

The above classifying spaces fit together in a diagram of fibrations

[figure: BX; BX^{ad}; B^{2}Z(X); BX; B aut_{∗}(BX); B aut(BX); B Out(X); B Out(X)]  (17)

that will enable us to compute the obstructions to the different liftings.

Consider the fibration (15) with homotopy action ρ: G → [M, M]. The homotopy action determines an action of G on the group of path-components of M and π_{0}(M)^{G}, or H^{0}(G; π_{0}(M)), denote the set of path-components of M that remain fixed under this action. With base point m ∈ π_{0}(M)^{G} in a G-invariant path-component of M there is a short exact sequence

1 → π_{1}(M, m) → π_{1}(M_{hG}, m) → π_{1}(BG, b) → 1  (18)

of fundamental groups, where b = p(m). If m ∈ π_{0}(M)^{G} happens to be in the image of the evaluation map

π_{0}(M^{hG}) ⟶^{π_{0}(ev)} π_{0}(M)^{G}  (19)

then s(b) = m for some homotopy fixed-point s ∈ M^{hG} and then (18) does have a section, namely π_{1}(s). Since π_{1}(M_{hG}, m) acts on the homotopy groups π_{i}(M, m) of the fibre, also G = π_{1}(BG, b) acts on π_{i}(M, m) through π_{1}(s). We let π_{i}(M, m)^{s_{∗}G}, i ≥ 1, denote the fixed-point group for this action.

Recall that, if the exact sequence (18) splits, then we can identify the set of π_{1}(M, m)-conju-gacy classes of sections π_{1}(BG, b) → π_{1}(M_{hG}, m) with the cohomology group H^{1}(G;π_{1}(M, m)). (We refer to [55] for the definition and properties of group cohomology with non-abelian coefficients.)

Lemma 5.1. Suppose that G is a finite group of order prime to p and that π_{i}(M, m) is a module over the ring ℤ_{(p)} of p-local integers for all i ≥ 2 and all base points m ∈ π_{0}(M)^{G}.

(1) A point m ∈ π_{0}(M)^{G} is in the image of the evaluation map (19) if and only if the exact sequence (18) splits.

(2) If m ∈ π_{0}(M)^{G} is in the image of the evaluation map (19), then there is an exact sequence of pointed sets

∗ → H^{1}(G; π_{1}(M, m)) ⟶ π_{0}(M^{hG}) ⟶^{π_{0}(ev)} π_{0}(M)^{G}

where m is the base point of π_{0}(M)^{G}.

(3) If s ∈ M^{hG} is a homotopy fixed-point with s(b) = m then

π_{i}(M^{hG}, s) ≅ π_{i}(M, m)^{s_{∗}G}

for all i ≥ 1.

Proof. The Postnikov functors P_{r}, defined as nullification with respect to S^{r−1} (see [17]), determine a tower of fibrations

M_{hG} → ⋯ → P_{r}M_{hG} → P_{r−1}M_{hG} → ⋯ → P_{1}M_{hG} → BG

so that M^{hG} is the homotopy inverse limit of a sequence

⋯ → P_{r}M^{hG} → P_{r−1}M^{hG} → ⋯ → P_{1}M^{hG}

of Postnikov homotopy fixed-point spaces.

Note that π_{0}(P_{1}M_{hG}) = π_{0}(M_{hG}) and that each path-component of P_{1}M_{hG} is aspherical with fundamental group π_{1}(P_{1}M_{hG}, m) = π_{1}(M_{hG}, m) for all m ∈ P_{1}M. It is now easy to see that H^{1}(G; π_{1}(M, m)) is indeed the fibre over m ∈ π_{0}(P_{1}M)^{G} = π_{0}(M)^{G} of the evaluation map π_{0}(P_{1}M^{hG}) → π_{0}(M)^{G} and also that π_{1}(P_{1}M^{hG}, s) = π_{1}(M, m)^{s_{∗}G} for any s ∈ P_{1}M^{hG} with s(b) = m, cf. [41, §6]. Obstruction theory implies that π_{0}(M^{hG}) = π_{0}(P_{1}M^{hG}).

Suppose that the homotopy fixed-point space is non-empty and let s ∈ M^{hG} be a homotopy fixed-point. Then the component (M^{hG}, s) containing s is the homotopy inverse limits of the corresponding components

⋯ → (P_{r}M^{hG}, s_{r}) → (P_{r−1}M^{hG}, s_{r−1}) → ⋯ → (P_{1}M^{hG}, s_{1})

of the Postnikov homotopy fixed-point spaces. To finish the proof, observe [41, 3.1] that the fibre of (P_{r}M^{hG}, s_{r}) → (P_{r−1}M^{hG}, s_{−1}r) is the Eilenberg-Mac Lane space K(π_{r}(M, m)^{s_{∗}G}, r).

□

For an alternative formulation, let (M, m) denote the path-component of M containing m ∈ M. If the path component (M, m) ∈ π_{0}(M) is G-invariant, then (M, m) is sub-G-space of M in the sense that the inclusion of (M, m) into M is a G-map; that is, the fibration M → M_{hG} → BG contains a fibration of the form

(M, m) → (M, m)_{hG} → BG

as a sub-fibration over BG. The homotopy fixed-point space

M^{hG} = ⋃_{m∈π_{0}(M)^{G}}(M, m)^{hG}

is a disjoint union of the homotopy fixed-point spaces (M, m)^{hG} where (M, m) runs through the set of G-invariant path-components in M. Since (M, m) by its very definition is a pathconnected G-space the homotopy groups of its homotopy fixed-point spaces are

π_{i}(M, m)^{hG} = \begin{cases}H^{1}(G; π_{1}(M, m)) & i = 0 \\ π_{i}(M, m)^{s_{∗}G} & i > 0 \end{cases}

by the lemma.

Theorem 5.2. Let B be any simply connected p-complete space, G a finite group of finite order prime to p, and

B → B_{hG} → BG

an action of G on B. There exists a homotopy equivalence

ΩB ⟶^{≃} ΩB^{hG} × Fib(B^{hG} → B)

In particular, the fibre Fib(B^{hG} → B) of the evaluation map B^{hG} → B is an H-space.

Proof. By obstruction theory, the space of sections B^{hG} is non-empty. We will show first how to turn this action with a homotopy fixed point into a honest action of G on a space homotopy equivalent to B and with a fixed point. The pullback diagram

\xymatrix{B \ar[d] \ar[r] & EG \ar[d] \ar@{-->}[l] \\ B_{hG} \ar[r] & BG \ar@{-->}[l]}

realizes B → B_{hG} as a regular covering space with G acting on B. Liftings of sections BG → B_{hG} provide G-equivariant maps EG → B. Let B/EG = B∪C(EG) be the homotopy cofibre of any such G-map. Then B → B/EG is a G-equivariant homotopy equivalence and the G-action on B/EG has a fixed point.

Now, we can assume that there is a honest G-action on B with a fixed point. Let ΩB denote the loop space based at any G-fixed point. It suffices to construct a homotopy left inverse for the inclusion ΩB^{hG} → ΩB.

Define tr: ΩB → ΩB to be the map that takes any loop ω to the product ∏ gω of the loops gω where g runs through the elements of G in some fixed order. The image of the induced map tr_{∗}: π_{∗}(ΩB) → π_{∗}(ΩB) on homotopy groups is contained in the fixed group π_{∗}(ΩB)^{G} and the composition π_{∗}(ΩB)^{G} → π_{∗}(ΩB) → π_{∗}(ΩB)^{G} is an isomorphism. This implies that the composition ΩB^{hG} → ΩB → T, where T is the mapping telescope of ΩB ⟶^{tr} ΩB ⟶^{tr} ⋯, is a (weak) homotopy equivalence and we have the left inverse we were looking for. □

Proof of Theorem B. Fix a finite group G of order prime to p, and ρ: G → Out(X) an outer action of G on a connected p-compact group X. Recall that we have a fibration sequence

B^{2}Z(X) → Baut(BX) → BOut(X)

and that the center of X, Z(X) is p-local. By obstruction theory we obtain a unique lifting of ρ to an action φ: BG → B aut(BX). Furthermore, since π_{1}(BX) = 1, Lemma 5.1 implies that π_{0}(BX^{hG}) = ∗; that is, ρ lifts to a unique action of G on X

\xymatrix{BX \ar[r] & BX_{hG} \ar[r] & BG . \ar[l]}  (20)

This is part (1) of the theorem. Now, Theorem 5.2 provides the splitting X ≃ X^{hG} × X/X^{hG}. It follows that X/X^{hG} is an 𝔽_{p}-finite H-space, X^{hG} is a loop space with classifying space BX^{hG} and it is also 𝔽_{p}-finite. Furthermore, BX^{hG} is p-complete because BX is pcomplete [22, 11.13], hence X^{hG} is a connected p-compact group.

The rational cohomology algebra H^{∗}(BY; ℚ_{p}) is polynomial for any connected p-compact group Y and it follows that the Hurewicz homomorphism induces an isomorphism

QH^{∗}(BY; ℚ_{p}) → π_{∗}(BY)^{∨} ⊗ ℚ

between the indecomposables and the rationalized dual (π^{∨} = Hom_{ℤ_{p}}(π, ℤ_{p})) of the homotopy groups of the simply connected space BY. For the connected fixed-point p-compact group BX^{hG}, in particular, we have

QH^{∗}(BX^{hG}; ℚ_{p}) ≅ π_{∗}(BX^{hG})^{∨} ⊗ ℚ ≅ (π_{∗}(BX)^{∨} ⊗ ℚ)_{G} ≅ (QH^{∗}(BX; ℚ_{p}))_{G}

for π_{∗}(BX^{hG}) = π_{∗}(BX)^{G} as the order of G is prime to p. This proves points (2) and (3). We finish by proving point (4). If X is a polynomial p-compact group

H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(X^{hG}; 𝔽_{p}) ⊗ H^{∗}(X/X^{hG}; 𝔽_{p})

is an exterior algebra, hence H^{∗}(X^{hG}; 𝔽_{p}) is an exterior algebra, too. Therefore, H^{∗}(BX^{hG}; 𝔽_{p}) is a polynomial algebra.. □

Example 5.3. At any odd prime, let C_{2} act on E_{6} through the unstable Adams operation ψ^{−1}. Since the fixed point p-compact group BE_{6}^{hC_{2}} is the p-compact group BF_{4} (5.15), there is a splitting

E_{6} ≃ F_{4} × E_{6}/F_{4}

of homogeneous spaces. This splitting is due to Harris [30]. Also, BP E_{6}^{hC_{2}} ≃ BF_{4}, where P E_{6} is the adjoint form of E_{6}, (5.15), thus there is also a splitting P E_{6} ≃ F_{4} × P E_{6}/F_{4}.

Let p be an odd prime and m a divisor of p − 1 so that the cyclic group C_{m} of order m acts on BSU(mn + s), 0 ≤ s < m, through unstable Adams operations. Since the fixed point p-compact group BSU(mn+s)^{hC_{m}} is (5.12) the generalized Grassmannian BX(m, 1, n) with polynomial cohomology H^{∗}(BX(m, 1, n); 𝔽_{p}) = 𝔽_{p}[x_{m}, …, x_{nm}], |x_{im}| = 2im, there is a splitting

SU(mn + s) ≃ X(m, 1, n) × SU(mn + s)/X(m, 1, n)

of homogeneous spaces. This splitting is originally due to Mimura, Nishida, and Toda [39]), although the recognition of X(m, 1, n) as a loop space is due to Quillen [54] (see also [59, 63, 15]). The case m = 2 is the classical splitting SU(2n) ≃ Sp(n) × SU(2n)/Sp(n). Similar splittings for central quotients of SU(n) can be worked out.

Similarly, at p = 5, let C_{4} act on BE_{8} through unstable Adams operations. Since (5.15) the fixed point p-compact group BE_{8}^{hC_{4}} is the p-compact group BX(G_{31}) corresponding to reflection group number 31 on the Clark-Ewing list, H^{∗}(BX(G_{31}); 𝔽_{p}) = 𝔽_{p}[x_{16}, x_{24}, x_{40}, x_{48}] where subscripts indicate degrees, there is a splitting

E_{8} ≃ X(G_{31}) × E_{8}/X(G_{31})

of homogeneous spaces, that was obtained in [60].

At p = 3, BF_{4} admits an exceptional isogeny of order 2 and the fixed point group BF_{4}^{hC_{2}} is [13] the p-compact group BDI(2) whose cohomology realizes the Dickson algebra 𝔽_{3}[x_{12}, x_{16}]. The corresponding splitting

F_{4} ≃ DI(2) × F_{4}/DI(2)

was first obtained in [31]. Later proofs of this splitting were obtained independently by Wilkerson and by Kono, using Friedlander’s exceptional isogeny of F_{4} localized away from two.

In these last two cases, it was Zabrodsky [63, 4.3], who first recognized the factors DI(2) and X(G_{31}) as loop spaces. Later, Aguadé gave a nice uniform construction of a family of modular p-compact groups included these cases [1].

Our next objective is to obtain a recognition principle for the homotopy fixed point pcompact group BX^{hG}.

Let N → X be the maximal torus normalizer for the p-compact group X. Again, the short exact sequence of topological monoids

BZ(N) = aut(BN)_{1} → aut(BN) → Out(N)

induces a fibration sequence

B^{2}Z(N) → Baut(BN) → BOut(N)

and we may write B^{2}Z(N)_{hOut(N)} = Baut(BN) for the classifying space for BN-fibrations.

Since G is a finite group of order prime to p, we see from this that equivalence classes of BX-fibrations over BG is in one-to-one correspondence with

[BG, B^{2}Z(X)_{h Out(X)}] = [BG, BOut(X)] = Hom(G, Out(X))

and that equivalence classes of BN-fibrations over BG is in one-to-one correspondence with

[BG, B^{2}Z(N)_{h Out(N)}] = [BG, BOut(N)] = Hom(G, Out(N)) .

However, Out(X) ≅ Out(N) and therefore there is a bijective correspondence between BXfibrations over BG and BN-fibrations over BG. We shall now make this correspondence more explicit.

Turn the maximal torus normalizer Bj: BN → BX into a fibration. Write aut(Bj) for the group-like topological monoid of commutative diagrams

\xymatrix{BN \ar[d]_{Bj} \ar[r] & BN \ar[d]^{Bj} \\ BX \ar[r] & BX}

where both horizontal arrows are homotopy equivalences.

Lemma 5.4. Assume that p is odd. The forgetful homomorphisms

\xymatrix{aut(BN) & aut(Bj) \ar[l] \ar[r] & aut(BX)}

are homotopy equivalences.

Proof. The group homomorphisms π_{0} aut(BN) ← π_{0} aut(Bj) → π_{0} aut(BX) are injective because X has N-determined automorphisms [48, 6]. The group homomorphism to the left is surjective because X is N-determined and the one to the right is surjective because any self-homotopy equivalence of BX lifts to a self-homotopy equivalence of BN [46, §3]. The identity components fit into a map of fibrations [23, 11.10]

[figure: aut_{BX}(Bj)_{1}; aut(Bj)_{1}; aut(BX)_{1}; ≃; aut_{BX}(Bj)_{1}; aut(BN)_{1}; Map(BN, BX)_{Bj}]

where the right vertical map, defined by composition with Bj, is a homotopy equivalence [23, 7.5, 1.3] [22, 9.1] [46, 3.4]. The fibre, consisting of the space of maps BN → BN over BX and vertically homotopic to the identity map of BN, is (one component) of the space (X/N)^{hN} which is contractible [44, 5.1]. □

Thus we have bijections

[B, Baut(BN)] = [B, Baut(Bj)] = [B, Baut(BX)]

for any space B and this means BN-fibrations and BX-fibrations over B are in bijective correspondence.

Proposition 5.5. Let X be a connected p-compact group with maximal torus normalizer N → X. If G is a finite group of order prime to p, then any outer action ρ: G → Out(X), lifts to a unique G-action on BX and unique G-action on BN. Moreover, these actions make the map BN → BX G-equivariant; that is, the diagram

[figure: BN; BN_{hG}; BG; BX; BX_{hG}; BG]

is homotopy commutative.

Proof. Let us say that our input is an outer action

ρ: G → Out(X) = W \N_{GL(L)}(W) = Out(N)  (21)

of the finite group G on X and N. The induced map

Bρ ∈ [BG, Baut(Bj)] = Hom(G, W \N_{GL(L)}(W))

corresponds [19] to an iterated fibration

\xymatrix{BN_{hG} \ar[r]^{Bj_{hG}} & BX_{hG} \ar[r] & BG}

over BG. □

Next, we need to lift the action of G on BN and BX to and action on the loop spaces N and X, such that the inclusion N → X is still equivariant.

Again Lemma 5.1 applies to show that the fibration BX → BX_{hG} → BG admits one and only one section; that is, there is a unique lifting of the action on BX to an action on X. However, π_{1}(BN) ≅ W and then Lemma 5.1 does not ensure neither, the existence, nor the uniqueness of a lifting of the action of G on BN to an action of G on N. Instead, it leads to the next description of the possible actions.

Proposition 5.6. If a finite group G of order prime to p acts on BN with outer action ρ: G → W \N_{GL(L)}(W) ≅ Out(N), then there are natural one-to-one correspondences between the sets:

(1) π_{0}(BN^{hG}),

(2) lifts to a G-action on N, and

(3) W -conjugacy classes of lifts in the diagram

\xymatrix{ & N_{GL(L)}(W) \ar[d] \\ G \ar[r]_{ρ} \ar@{.>}[ur] & W \N_{GL(L)}(W) .}

If those sets are non-empty, then they are also in one-to-one correspondence with H^{1}(G; W).

Proof. An action of G on BN is by definition a fibration

BN → BN_{hG} → BG ,  (22)

and according to 5.5 this action of G on BN is uniquely determined by ρ.

The map from π_{0}(BN^{hG}) to the set (2) is immediate because we can identify π_{0}(BN^{hG}) with the vertical homotopy classes of sections of (22), and a sectioned fibration is an action of G on N, by definition. Also, if ψ: BG → B aut_{∗}(BN) is a lift of ρ to an action of G on N, then σ = π_{1}(ψ): G → N_{GL(L)}(W) is an element in the set (3).

Next, we can map π_{0}(BN^{hG}) directly to the set (3). Let φ: BG → B aut(BN) be a classifying map for the fibration (22). Thus, φ extends to a map of fibrations

[figure: BN; BN_{hG}; BG; Bρ; BN; Baut_{∗}(BN); Baut(BN)]

which on the level of fundamental groups [45, 5.2] [5, 3.3] induces a morphism

[figure: W; π_{0}(N_{hG}); G; ρ; W; N_{GL(L)}(W); W \N_{GL(L)}(W)]  (23)

of group extensions. We have seen (Lemma 5.1) that the existence of an action of G on N lifting the action on BN is equivalent to the existence of a section of the exact sequence on the top row of (23), and the diagram shows that this is equivalent to the existence of a lifting of ρ to a homomorphism σ: G → N_{GL(L)}(W). This gives the bijection between π_{0}(BN^{hG}) and the set (3), and shows that all of the three sets are empty if one is empty.

Finally, if they are non empty, then obstruction theory shows that all of them are parametrized by H^{1}(G, W), which coincides with both H^{1}(G; π_{1}(BN^{ad})) (that parametrizes (2), see diagram (17)) and H^{1}(G; π_{1}(BN)) (that parametrizes (1), see Lemma 5.1). □

Proposition 5.7. Let X be a connected p-compact group with Weyl group W and maximal torus normalizer N → X. If G is a finite group of order prime to p and

ρ: G → Out(X) ≅ W \N_{GL(L)}(W)

an outer action, then ρ lifts to a unique action of G on X, and each lift

σ: G → N_{GL(L)}(W)

determines a unique action of G on N such that the inclusion N → X is G-equivariant.

Proof. As we mentioned before, (1) follows directly from Lemma 5.1, and according to Proposition 5.6, the actions of G on N that lift the given outer action are in one-to-one correspondence with lifts of ρ to N_{GL(L)}(W). If we view one of these actions as a sectioned fibration

\xymatrix{BN \ar[r] & BN_{hG} \ar[r] & BG \ar[l]}

it clearly induces an action on X that makes N → X equivariant:

\xymatrix{BN_{hG} \ar[dr] \ar[rr] &  & BX_{hG} \ar[dl] \\  & BG. \ar@{.>}[ul] \ar@{.>}[ur]}

The proposition follows because there is only one action of G on X inducing ρ. □

Proposition 5.8. Let p be an odd prime and G a finite group of order prime to p. Assume that G acts on a connected p-compact group X and

\overline{ρ}: G → N_{GL(L)}(W)

is a lift of the given outer action. If Y is a connected p-compact group that satisfies

(1) W^{\overline{ρ}G} contains a subgroup \overline{W}, complementary to the kernel of W^{\overline{ρ}G} → GL(L^{\overline{ρ}G}), such that (\overline{W} , L(X)^{\overline{ρ}G}) is a reflection group similar to (W (Y), L(Y)), and

(2) QH^{∗}(BY; ℚ_{p}) ≅ QH^{∗}(BX; ℚ_{p})_{G},

then BY = BX^{hG}.

Proof. By the classification theorem for p-compact groups at odd primes [48, 6], it suffices [47, 1.2] to find an map BN(Y) → BX^{hG} that induces an isomorphism on H^{∗}(−; ℚ_{p}) and restricts to monomorphism on the p-normalizer N_{p}(Y), is a p-monomorphism. The homomorphism \overline{ρ} corresponds (5.7) to compatible G-actions BG → BN(X)_{hG} → BX_{hG} on N(X) and X. Taking homotopy fixed points we obtain a commutative diagram of loop space morphisms

\xymatrix{N(X)^{hG} \ar[d] \ar[r] & X^{hG} \ar[d] \\ N(X) \ar[r] & X}

which shows that N(X)^{hG} → X^{hG} is a p-monomorphism. Since the discrete approximation to N(X), N(X)^{hG}, and N(Y) are semi-direct products [5], there is a p-monomorphism N(Y) → N(X)^{hG} for W (Y) is a subgroup W^{\overline{ρ}G} = π_{0}N(X)^{hG} by the first condition. By the second condition, H^{∗}(BY; ℚ_{p}) = H^{∗}(BN(Y); ℚ_{p}) and H^{∗}(BX^{hG}; ℚ_{p}) are abstractly isomorphic graded vector spaces. Therefore, Y and X^{hG} have the same rank [22, 5.9] so that T (Y) → N(X)^{hG} → X^{hG} is a maximal torus and H^{∗}(BX^{hG}; ℚ_{p}) → H^{∗}(BN(Y); ℚ_{p}) is injective [22, 9.7], hence bijective. □

A special case arises when G acts through unstable Adams operations so that the action π_{0}ρ: G → Out(N) → Out(W) is trivial. Then the image of G in Out(N) = W \N_{GL(L)}(W) is contained in the subgroup Z(W)\C_{GL(L)}(W) [48, 3.16] and we have a morphism

[figure: W; π_{0}(N_{hG}); G; Bρ; W; W.C_{GL(L)}(W); Z(W)\C_{GL(L)}(W)]

of group extensions. The possible extensions occurring in the upper line, realizing the trivial action G → Out(W), are classified by H^{2}(G; Z(W)); they are all isomorphic to

W → Z(W)\(D × W) → G

for some central extension Z(W) → D → G [36, IV.§8]. If Z(W) = 1 is trivial, π_{0}(N_{hG}) = G × W and H^{1}(G; W) = Rep(G, W).

Assume that G = C_{r} is a cyclic group of order r, and the outer action of G on X, ρ: C_{r} → Out(X), is given by an Adams operation ρ(λ) = ψ^{λ}, where λ ∈ ℤ_{p}^{×} is a p-adic unit of order r|(p − 1). We can lift ψ^{λ} ∈ Z(W)\C_{GL(L)}(W) to an element ζ ∈ C_{GL(L)}(W), that verifies ζ^{r} ∈ Z(W). If there is a choice of ζ with ζ^{r} = 1, then ρ̄λ = ζ provides a lifting of ρ.

Assume, otherwise, that ζ^{r} has order s in Z(W). Since p is odd, Z(W) has order prime to p, hence s is prime to p. Now, even if there is no lift of the action of C_{r} on X to an action on N, we can reduce the problem by extending the action of C_{r} to an action of C_{sr} on X determined by ρ′(λ) = ψ^{λ} ∈ Z(W)\C_{GL(L)}(W) ⊂ Out(X), that now admits the lift ρ̄′(λ) = ζ. Notice that C_{s} = ⟨λ^{r}⟩ acts trivially on X, so that BX^{hC_{s}} ≃ BX, and then BX^{hC_{sr}} ≅ BX^{hC_{r}}, so we can still determine BX^{hC_{r}} by analysing the equivariant action of C_{sr} on N and X.

Notice also, that if W is irreducible, then C_{GL(L)}(W) consists of diagonal matrices and therefore ζ is an Adams operation.

Corollary 5.9. Let λ ∈ ℤ_{p}^{×} be a p-adic unit of order r|(p − 1). Consider the outer action ρ: C_{r} = ⟨λ⟩ → W \N_{GL(L)}(W) through unstable Adams operations given by ρ(λ) = ψ^{λ}. Then, if ρ admits a lift ρ̄: C_{r} → N_{GL(L)}(W), then all possible lifts are parametrized by H^{1}(C_{r}; W) = Rep(C_{r}, W), the set of conjugacy classes of order r elements w of W, and

(W^{\overline{ρ}C_{r}}, L^{\overline{ρ}C_{r}}) = (C_{W} (w), L^{⟨λw⟩})

for the lift \overline{ρ}(λ) = λw corresponding to w.

Proof. The lifts

\xymatrix{ & W.C_{GL(L)}(W) \ar[d] \\ C_{r} = ⟨λ⟩ \ar[r]_{ρ} \ar@{.>}[ur]^{\overline{ρ}} & Z(W)\C_{GL(L)}(W)}

are given by \overline{ρ}(λ) = wψ^{λ} where w ∈ W is any element of order r. □

We next apply the recognition principle (5.9) in some concrete cases.

5.10. The infinite families. We identify the fixed point p-compact groups for actions through unstable Adams operations on the p-compact groups of the three infinite families in the Clark-Ewing classification table [16].

Proposition 5.11. Suppose that r and m divide p − 1, m > 1, then

(S^{2r−1})^{hC_{m}} = \begin{cases}S^{2r−1} , & m | r \\ ∗ , & otherwise, \end{cases}

for the action through unstable Adams operations of exponent of C_{m} ≤ ℤ_{p}^{∗} on the p-compact group S^{2r−1}.

Proof. Let λ be a primitive mth root of unity, so that C_{m} = ⟨λ⟩ ≤ ℤ_{p}^{∗}. According to Theorem B, (S^{2r−1})^{h⟨λ⟩} is a connected polynomial p-compact group. If m does not divide r, H^{2r}(ψ^{λ}) = λ^{r} is nontrivial, so that the vector space of covariants QH^{∗}(BS^{2r−1}; ℚ_{p})_{⟨λ⟩} vanishes in positive degrees, and the fixed point p-compact group is trivial. If m does divide r, ψ^{λ} acts trivially on S^{2r−1} and the fixed point p-compact group is again S^{2r−1}. □

The next results, 5.12, 5.13, and 5.14, deal with complex and generalized grasmannians. The results of 5.12 and 5.14 were obtained by Castellana [15] using different methods.

Proposition 5.12. Let p be an odd prime. Suppose that m|(p − 1), m > 1, and let C_{m} = ⟨λ⟩ ⊂ ℤ_{p}^{×} be the cyclic group generated by a primitive mth root of unity acting through unstable Adams operations. Then

X(mn + s)^{hC_{m}} = U(mn + s)^{hC_{m}} = \begin{cases}X(m, 1, n) & n > 0 \\ ∗ & n = 0 \end{cases}

for any p-compact group X(mn + s) locally isomorphic to SU(mn + s), 0 ≤ s < m.

Proof. In H^{∗}(BU(mn+s); ℚ_{p})=ℚ_{p}[c_{1}, …, c_{mn+s}] and H^{∗}(BX(mn+s);ℚ_{p})=ℚ_{p}[c_{2}, …, c_{mn+s}] we have

c_{i} is preserved by H^{2i}(ψ^{λ}) ⇔ m|i

and therefore

QH^{∗}(BU(mn + s); ℚ_{p})_{C_{m}} = ℚ_{p}{c_{m}, …, c_{mn}} = QH^{∗}(BX(m, 1, n); ℚ_{p}) = QH^{∗}(BX(mn + s); ℚ_{p})_{C_{m}} .

The Weyl group W = Σ_{mn+s} is the symmetric group in its natural representation on L = ℤ_{p}^{mn+s}. Let e_{1}, …, e_{mn+s} be the canonical basis vectors of L. The permutation

w = (1 ⋯ m)(m + 1 ⋯ 2m) ⋯ (m(n − 1) + 1 ⋯ mn) ∈ Σ_{mn+s}

has order m and

(C_{Σ_{mn+s}}(w), L^{⟨λw⟩})

= (C_{m} ≀ Σ_{n} × Σ_{s}, ℤ_{p}{λe_{1} + λ^{2}e_{2} + ⋯ + λ^{m}e_{m}, …, λe_{m(n−1)+1} + ⋯ + λ^{m}e_{mn}})

contains the reflection group G(m, 1, n) = C_{m} ≀ Σ_{n} as a a subgroup complementary to the kernel, Σ_{s}, of the action of (C_{Σ_{mn+s}}(w) on L^{⟨λw⟩}. This means (5.9) that the fixed point p-compact group U(mn + s)^{hC_{m}} = X(m, 1, n).

From the two short exact sequences of ℤ_{p}Σ_{mn+s}-modules [48, §10]

0 → ℤ_{p} ⟶^{Δ} L → LP U(mn + s) → 0, 0 → LX(mn + s) → LP U(mn + s) → π̌ → 0

where Δ is the diagonal and π̌ a subgroup of π_{1}(P U(mn + s)) = ℤ_{p}/ℤ_{p}(mn + s) (with trivial Σ_{mn+s}-action), we get that

L^{⟨λw⟩} = LP U(mn + s)^{⟨λw⟩} = LX(mn + s)^{⟨λw⟩}

as ℤ_{p}C_{Σ_{mn+s}}(w)-modules. □

Let p be an odd prime and r ≥ 1 and m ≥ 2 natural numbers such that r | m | p − 1. Then the cyclic group C_{m} of order m is contained in the group of units ℤ_{p}^{×} for ℤ_{p}. The ℤ_{p}-reflection group (G(m, r, n), ℤ_{p}^{n}), n ≥ 2, is the group generated by all permutations of the n coordinates and the diagonal matrices in

A(m, r, n) = {diag(a_{1}, …, a_{n}) ∈ C_{m}^{n} | (a_{1} ⋯ a_{n})^{m/r} = 1}

which is an index r subgroup of A(m, 1, n) = C_{m}^{n}. As abstract groups G(m, r, n) = A(m, r, n)⋊ Σ_{n}.

The proof of (5.13) will make use of these facts:

• For arbitrary natural numbers m and n we write m_{n} for m/ gcd(m, n). Then m_{n}n = lcm(m, n) and m_{n}n_{m} = lcm(m, n)/ gcd(m, n).

• C_{lcm(q,m)} = ⟨λ, μ|λ^{q} = 1, μ^{m} = 1, λμ = μλ, λ^{q_{m}} = μ^{m_{q}}⟩.

• Let A(a, t) ∈ GL(ℤ_{p}, t) denote the linear automorphism

A(a, t)(x_{1}, …, x_{t}) = (ax_{t}, x_{1}, …, x_{t−1})

where a ∈ ℤ_{p}^{×} is a unit. The ith power A(a, t)^{i} has characteristic polynomial (x^{t_{i}} − a^{i_{t}})^{t/t_{i}} and A(a, t)^{t} = aE.

• If λ ∈ ℤ_{p}^{×} has order q, then A(λ^{−q_{m}}, q_{m}) also has order q for A(λ^{−q_{m}}, q_{m})^{q_{m}} = λ^{−q_{m}}E has order gcd(q, m). The λ^{−1} eigenspace of A(λ^{−q_{m}}, q_{m}) has rank one and A(λ^{−q_{m}}, q_{m})^{−1} acts on it as multiplication by λ.

• In the exact sequence 1 → A^{⟨g⟩} → C_{A⋊G}(a, g) → C_{G}(g) the image in C_{G}(g) consists of those h ∈ C_{G}(g) that fix a ∈ A/(1 − g)A.

Proposition 5.13. Let X(m, r, n), m ≥ 2, r ≥ 1, n ≥ 2, r | m | p−1, be the simple polynomial p-compact group whose Weyl group is the imprimitive reflection group G(m, r, n). Suppose that the natural number ℓ divides p − 1 and let the cyclic group C_{ℓ} ⊂ C_{lcm(ℓ,m)} ⊂ ℤ_{p}^{×} act on X(m, r, n) through unstable Adams operations. The homotopy fixed point group for this action is

X(m, r, n)^{hC_{ℓ}} = \begin{cases}X(lcm(ℓ, m), r, n/ℓ_{m}) & rℓ | mn \\ X(lcm(ℓ, m), 1, n/ℓ_{m} − 1) & rℓ ∤ mn , ℓ | mn, \\ X(lcm(ℓ, m), 1, [n/ℓ_{m}]) & ℓ ∤ mn \end{cases}

where ℓ_{m} = ℓ/ gcd(ℓ, m) and [n/ℓ_{m}] is the biggest integer ≤ n/ℓ_{m}. (By convention, G(m, r, 1) is cyclic of order m/r and G(m, r, 0) is the trivial group.)

Proof. Let λ ∈ ℤ_{p}^{×} be a primitive ℓth root of unity. In the rational cohomology algebra H^{∗}(BX(m, r, n); ℚ_{p}) ≅ ℚ_{p}[x_{1}, …, x_{n−1}, e] the degrees |x_{i}| = 2im and |e| = 2\frac{m}{r}n so that

x_{i} is preserved by H^{2im}(ψ^{λ}) = λ^{im} ⇔ ℓ | im ⇔ ℓ_{m} | i
e is preserved by H^{2 \frac{m}{r} n}(ψ^{λ}) = λ^{\frac{m}{r} n} ⇔ ℓ | nm/r ⇔ ℓ_{m/r} | n

and thus QH^{∗}(BX(m, r, n); ℚ_{p})_{C_{ℓ}} is isomorphic to the indecomposables of the rational cohomology algebra of the p-compact group on the right hand side of the equation.

We have rℓ | mn ⇔ ℓ_{m/r} | n, ℓ | mn ⇔ ℓ_{m} | n, and ℓ_{m} | ℓ_{m/r} | ℓ|p − 1. \underline{ℓ_{m/r} | n}: The element

w = diag ( \underbrace{A(λ^{−ℓ_{m}}, ℓ_{m}), …, A(λ^{−ℓ_{m}}, ℓ_{m})}_{n/ℓ_{m}} ) ∈ G(m, r, n)

has order ℓ. Since ((λ^{−ℓ_{m}})^{n/ℓ_{m}})^{m/r} = λ^{−mn/r} = 1 because ℓ|(mn/r) by assumption, w does indeed belong to the index r subgroup G(m, r, n) of G(m, 1, n) = C_{m} ≀ Σ_{n}. Let {e_{1}, …, e_{n}} be the canonical basis for the free ℤ_{p}-module L = ℤ_{p}^{n} on which G(m, r, n) acts. The free ℤ_{p}-module

L^{⟨λw⟩} = ⟨e_{1} + λe_{2} + ⋯ + λ^{ℓ_{m}−1}e_{ℓ_{m}}, …, e_{(n−ℓ_{m})+1} + λe_{(n−ℓ_{m})+2} + ⋯ + λ^{ℓ_{m}−1}e_{n}⟩ ,

has rank n/ℓ_{m}. We shall now compute the centralizer of w. Let ζ be a generator of the cyclic group C_{lcm(ℓ,m)} ⊂ ℤ_{p}^{×} so that C_{m} = ⟨μ⟩ and C_{ℓ} = ⟨λ⟩ with μ = ζ^{ℓ_{m}} and λ = ζ^{m_{ℓ}}. The homomorphisms \xymatrix{A(ℓ, 1, n/ℓ_{m}) \ar[r] & C_{G(m,1,n)}(w) & A(m, 1, n/ℓ_{m}) \ar[l]} defined by

λ_{i} → diag ( \underbrace{E, …, E}_{i−1}, A(λ^{−ℓ_{m}}, ℓ_{m})^{−1}, E, …, E), diag ( \underbrace{E, …, E}_{i−1}, μE, E, …, E) ← μ_{i}

combine to a homomorphism defined on A(lcm(ℓ, m), 1, n/ℓ_{m}) since they agree on their common domain A(gcd(ℓ, m), 1, n/ℓ_{m}) = ⟨μ^{m_{ℓ}}⟩^{n/ℓ_{m}} = ⟨λ^{ℓ_{m}}⟩^{n/ℓ_{m}}, Observe that (λ^{a_{1}}, …, λ^{a_{n/ℓm}}) ∈ A(ℓ, 1, n/ℓ_{m}) lies in the subgroup A(lcm(ℓ, m), r, n/ℓ_{m}) if and only if its image lies in G(m, r, n) and that (μ^{a_{1}}, …, μ^{a_{n/ℓm}}) ∈ A(m, 1, n/ℓ_{m}) lies in the subgroup A(m, r, n/ℓ_{m}) if and only if its image lies in G(m, r, n). Together with the diagonal Δ: Σ_{n/ℓ_{m}} → Σ_{n} given by Δ(σ)((i − 1)ℓ_{m} + j) = (σ(i) − 1)ℓ_{m} + j, 1 ≤ ℓi ≤ n/ℓ_{m}, 1 ≤ j ≤ ℓ_{m}, we obtain a group isomorphism

G(lcm(ℓ, m), 1, n/ℓ_{m}) ⟶^{≅} C_{G(m,1,n)}(w)

that restricts to a group isomorphism G(lcm(ℓ, m), r, n/ℓ_{m}) ≅ C_{G(m,r,n)}(w) between index r subgroups. This isomorphism identifies the pair (C_{G(m,r,n)}(w), L^{⟨λw⟩}) and the imprimitive reflection group (G(lcm(ℓ, m), r, n/ℓ_{m}), ℤ_{p}^{n/ℓ_{m}}). \underline{ℓ_{m/r} ∤ n, ℓ_{m} | n}: It will suffice to consider the case of G(m, m, n) where ℓ ∤ n and ℓ_{m} | n. The element

w = diag ( \underbrace{A(λ^{−ℓ_{m}}, ℓ_{m}), …, A(λ^{−ℓ_{m}}, ℓ_{m})}_{n/ℓ_{m}−1}, A(λ^{−ℓ_{m}}, ℓ_{m})^{1−n/ℓ_{m}}) ∈ G(m, m, n)

has order ℓ. Note that λ^{−1} is not an eigenvalue for A(λ^{−ℓ_{m}}, ℓ_{m})^{1−n/ℓ_{m}} because

A(λ^{−ℓ_{m}}, ℓ_{m})^{1−n/ℓ_{m}} has eigenvalue λ^{−1} ⇔ A(λ^{−ℓ_{m}}, ℓ_{m})^{n/ℓ_{m}−1} has eigenvalue λ

⇔ λ^{(ℓ_{m})_{n/ℓm−1}} = λ^{−ℓ_{m}(n/ℓ_{m}−1)_{ℓm}} ⇔ ℓ | (ℓ_{m})_{n/ℓ_{m}−1} + ℓ_{m}(n/ℓ_{m} − 1)_{ℓ_{m}}

⇔ ℓ | n/gcd(ℓ_{m}, n/ℓ_{m} − 1) ⇒ ℓ | n ⇒ ℓ_{m} | n

which is not the case. Therefore the λ^{−1}-eigenspace

L^{⟨λw⟩} = ⟨e_{1} + λe_{2} + ⋯ + λ^{ℓ_{m}−1}e_{ℓ_{m}}, …, e_{(n−2ℓ_{m})+1} + λe_{(n−2ℓ_{m})+2} + ⋯ + λ^{ℓ_{m}−1}e_{n−ℓ_{m}}⟩

has rank n/ℓ_{m} − 1. The two monomorphisms A(ℓ, 1, n/ℓ_{m} − 1) → C_{G(m,m,n)}(w) and C_{G(m,m,n)}(w) ← A(m, 1, n/ℓ_{m} − 1) given by

λ_{i} → diag ( \underbrace{E, …, E}_{i−1}, A(λ^{−ℓ_{m}}, ℓ_{m})^{−1}, E, …, E, A(λ^{−ℓ_{m}}, ℓ_{m}))

diag ( \underbrace{E, …, E}_{i−1}, μE, E, …, E, μ^{−1}E) ← μ_{i}

agree on their common domain A(gcd(ℓ, m), 1, n/ℓ_{m}−1) and together with the monomorphism

\xymatrix{Σ_{n/ℓ_{m}−1} \ar@{^{(}->}[r]^{Δ} & Σ_{n−ℓ_{m}} \ar@{^{(}->}[r] & Σ_{m}} they define a homomorphism on the group A(lcm(ℓ, m), 1, n/ℓ_{m} − 1) ⋊ Σ_{n/ℓ_{m}−1} such that the composition

A(lcm(ℓ, m), 1, n/ℓ_{m} − 1) ⋊ Σ_{n/ℓ_{m}−1} ↪ C_{G(m,m,n)}(w) ↠ Im (C_{G(m,m,n)}(w) → GL(L^{⟨λw⟩}))

is an isomorphism with image similar to G(lcm(ℓ, m), 1, n/ℓ_{m} − 1). \underline{ℓ_{m} ∤ n}: It will suffice to consider the case of G(m, m, n). The element

w = diag ( \underbrace{A(λ^{−ℓ_{m}}, ℓ_{m}), …, A(λ^{−ℓ_{m}}, ℓ_{m})}_{[n/ℓ_{m}]}, \underbrace{λ^{ℓ_{m}[n/ℓ_{m}]}, 1, …, 1}_{n−ℓ_{m}[n/ℓ_{m}]} ) ∈ G(m, m, n)

has order ℓ. Note that λ^{−1} is not an eigenvalue for λ^{ℓ_{m}[n/ℓ_{m}]} because

λ^{ℓ_{m}[n/ℓ_{m}]} = λ^{−1} ⇔ ℓ | ℓ_{m}[n/ℓ_{m}] + 1 ⇔ ℓ_{m} gcd(ℓ, m) | ℓ_{m}[n/ℓ_{m}] + 1 ⇒ ℓ_{m} | 1

which is not the case as ℓ_{m} > 1. Therefore the λ^{−1} eigenspace L^{⟨λw⟩} has rank [n/ℓ_{m}]. The two monomorphisms \xymatrix{A(ℓ, 1, [n/ℓ_{m}]) \ar[r] & C_{G(m,m,n)}(w) & A(m, 1, [n/ℓ_{m}]) \ar[l]} given by

λ_{i} → diag ( \underbrace{E, …, E}_{(i−1)ℓ_{m}}, A(λ^{−ℓ_{m}}, ℓ_{m})^{−1}, E, …, E, \underbrace{λ^{−ℓ_{m}}, 1, …, 1}_{n−ℓ_{m}[n/ℓ_{m}]} )

diag ( \underbrace{E, …, E}_{(i−1)ℓ_{m}}, μE, E, …, E, \underbrace{μ^{−ℓ_{m}}, 1, …, 1}_{n−ℓ_{m}[n/ℓ_{m}]} ) ← μ_{i}

agree on their common domain A(gcd(ℓ, m), 1, [n/ℓ_{m}]) and together with the inclusion

of permutation groups \xymatrix{Σ_{[n/ℓ_{m}]} \ar@{^{(}->}[r]^{Δ} & Σ_{ℓ_{m}[n/ℓ_{m}]} \ar@{^{(}->}[r] & Σ_{m}}, they define a homomorphism on A(lcm(ℓ, m), 1, [n/ℓ_{m}]) ⋊ Σ_{[n/ℓ_{m}]} such that the composition

A(lcm(ℓ, m), 1, [n/ℓ_{m}]) ⋊ Σ_{[n/ℓ_{m}]} ↪ C_{G(m,m,n)}(w) ↠ Im (C_{G(m,m,n)}(w) → GL(L^{⟨λw⟩}))

is an isomorphism with image similar to the reflection group G(lcm(ℓ, m), 1, [n/ℓ_{m}]). □

The outer automorphism group of X(G(m, r, n)) is isomorphic to A(m, r, n)\ℤ_{p}^{×}A(m, 1, n) except in the cases (m, r, n) ∈ {(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)} [52, §6] [48, 7.14]. The (exotic) homotopy action

ρ: C_{m} = ⟨μ⟩ → Out(X(m, r, n)) ≅ A(m, r, n)\ℤ_{p}^{×}A(m, 1, n)

that takes the generator μ of C_{m} to A(m, r, n)(μ, 1, …, 1) is distinct form the actions through unstable Adams operations of (5.13) when gcd(r, n) > 1 [48, 7.14].

Proposition 5.14. Assume that m≥2, r ≥1, n≥2, and (m, r, n)∉{(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)}. Then the homotopy fixed point group

X(m, r, n)^{hC_{m}} = X(m, 1, n − 1)

for the above exotic homotopy action on X(m, r, n).

Proof. The second assumption of (5.8) is clearly satisfied as the action preserves the generators x_{1}, …, x_{n−1} but does not preserve the generator e. To verify the first assumption, take \overline{ρ}: C_{m} → N_{GL(L)}(G(m, r, n)) = ℤ_{p}^{×}G(m, 1, n) to be the obvious choice \overline{ρ}(μ) = (μ, 1, …, 1). Then

G(m, r, n)^{\overline{ρ}C_{m}} = A(m, r, n) ⋊ Σ_{n−1}, L^{\overline{ρ}C_{m}} = ℤ_{p}^{n−1}

and the composition

\xymatrix{A(m, 1, n − 1) ⋊ Σ_{n−1} \ar@{^{(}->}[r] & G(m, r, n)^{\overline{ρ}C_{m}} \ar@{->>}[r] & Im (G(m, r, n)^{\overline{ρ}C_{m}} → GL(L^{\overline{ρ}C_{m}}))}

where the first morphism is (μ_{2}, …, μ_{m}) → ((μ_{2} ⋯ μ_{n})^{−1}, μ_{2}, …, μ_{n}), Σ_{n−1} ↪ Σ_{n}, identifies the group to the right as the reflection group G(m, 1, n − 1). □

5.15. The sporadic p-compact groups. We identify the fixed point p-compact groups for actions through unstable Adams operations on the p-compact groups corresponding to the 34 sporadic reflection groups of the Clark-Ewing classification table. These p-compact groups are determined by their rational Weyl groups except that the local isomorphism system of G_{35} contains two 3-compact groups E_{6} and P E_{6} [48, 11.18]. However, (E_{6})^{hC_{2}} and (P E_{6})^{hC_{2}} are identical so that in diagram (17) G_{35} can mean either of these two.

36 CARLES BROTO AND JESPER M. MØLLER

The relationship in terms of homotopy fixed point groups displayed in the diagram

\xymatrix{ & C_{30} & G_{32} \ar[dr]^{4} \ar[l]_{5} \ar[r]^{5} & C_{30} & C_{18} \\ G_{16} \ar@{-->}[ur]^{3} \ar[dr]_{4} & G_{37} \ar[l]_{5} \ar@{-->}[ur]^{3} \ar[dr]^{4} &  & G_{10} \ar[d]^{8} & G_{34} \ar[l]_{4} \ar[u]_{9} \ar[d]^{5} \ar[r]^{7} \ar@{-->}[ul]_{5} & C_{42} \\  & C_{20} & G_{31} \ar[l]^{5} \ar@{-->}[ur]^{3} \ar[d]^{8} & C_{24} & C_{30} \\  &  & G_{9} \ar@{-->}[ur]_{3}}  (16)

is justified by (5.9) and the following computer computations:

(1) (G_{37} = W (E_{8}), C_{3}, G_{32}, p ≡ 1 mod 3) There is an element w ∈ G_{37} of order 3 and a

primitive 3rd root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{37}}(w), L_{37}^{⟨λw⟩}) = (G_{32}, L_{32})

meaning that that E_{8}^{hC_{3}} = X(G_{32}).

(2) (G_{37} = W (E_{8}), C_{4}, G_{31}, p ≡ 1 mod 4) There is an element w ∈ G_{37} of order 4 such

that

(C_{G_{37}}(w), L_{37}^{⟨iw⟩}) = (G_{31}, L_{31})

meaning that E_{8}^{hC_{4}} = X(G_{31}).

(3) (G_{37} = W (E_{8}), C_{5}, G_{16}, p ≡ 1 mod 15) There is an element w ∈ G_{37} of order 5 and a

primitive 5th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{37}}(w), L_{37}^{⟨λw⟩}) = (G_{16}, L_{16})

meaning that E_{8}^{hC_{5}} = X(G_{16}).

(4) (G_{34}, C_{4}, G_{10}, p ≡ 1 mod 12). There exists an element w ∈ G_{34} of order 4, a (index 4)

subgroup G of C_{G_{34}}(w), and a primitive 4th root of unity λ ∈ ℤ_{p}^{×} such that

(G, L_{34}^{⟨λw⟩}) = (G_{10}, L_{10})

meaning that X(G_{34})^{hC_{4}} = X(G_{10}).

(5) (G_{32}, C_{4}, G_{10}, p ≡ 1 mod 12) There is an element w ∈ G_{32} of order 4 and a primitive

4th root of unity i ∈ ℤ_{p}^{×} such that

(C_{G_{32}}(w), L_{32}^{⟨iw⟩}) = (G_{10}, L_{10})

which means that X(G_{32})^{hC_{4}} = X(G_{10}).

(6) (G_{32}, C_{30}, C_{5}, p ≡ 1 mod 30) There is an element w ∈ G_{32} of order 5 and a primitive

5th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{32}}(w), L_{32}^{⟨λw⟩}) = (C_{30}, ℤ_{p})

which means that X(G_{32})^{hC_{5}} = S^{59}.

(7) (G_{31}, C_{3}, G_{10}, p ≡ 1 mod 12). There exists an element w ∈ G_{31} of order 3 and a

primitive 3rd root of unity λ ∈ ℤ_{p}^{×} such that (C_{G_{31}}(w), L_{31}^{⟨λw⟩}) = (G_{10}, L_{10}) .

This means that X(G_{31})^{hC_{3}} = X(G_{10}). (The group that the computer finds is G_{10} and not G_{15} (of the same rank and the same degrees) because the elements of order 8 square to central elements [56, p. 281].)

(8) (G_{31}, C_{8}, G_{9}, p ≡ 1 mod 24). There exists an element w ∈ G_{31} of order 8 and a primitive 8th root of unity λ ∈ ℤ_{p}^{×} such that the reflection group

(C_{G_{31}}(w), L_{31}^{⟨λw⟩}) = (G_{9}, L_{9})

which means that X(G_{31})^{hC_{8}} = X(G_{9}).

(9) (G_{10}, C_{8}, C_{24}, p ≡ 1 mod 24) There is an element w ∈ G_{10} of order 8 and a primitive 8th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{10}}(w), L_{10}^{⟨λw⟩}) = (C_{24}, ℤ_{p})

which means that X(G_{10})^{hC_{8}} = S^{47}.

(10) (G_{9}, C_{3}, C_{24}, p ≡ 1 mod 24) There is an element w ∈ G_{9} of order 3 and a primitive 3rd root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{9}}(w), L_{9}^{⟨λw⟩}) = (C_{24}, ℤ_{p})

which means that X(G_{9})^{hC_{3}} = S^{47}.

(11) (G_{34}, C_{9}, C_{18}, p ≡ mod18) There is an element w ∈ G_{34} of order 9 and a primitive 9th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{34}}(w), L_{34}^{⟨λw⟩}) = (C_{18}, ℤ_{p})

which means that X(G_{34})^{hC_{9}} = S^{37}.

The homotopy fixed point p-compact groups shown in

\xymatrix{C_{18} & G_{36} \ar[l]_{18} \ar[dl]_{6} \ar[dr]^{4} &  &  & G_{35} \ar[dl]_{2} \ar[dr]^{3} \ar[r]^{5} & C_{5} \\ G_{26} \ar[u]^{18} \ar[dr]_{12} &  & G_{8} \ar[dl]^{12} \ar[d]^{8} & G_{28} \ar[l]_{4} \ar[dr]_{3} &  & G_{25} \ar[d]^{12} \ar[dl]^{2} \\  & C_{12} & C_{8} &  & G_{5} \ar[r]_{12} & C_{12}}  (17)

are justified by (5.9) and the following computer computations:

(1) (G_{36} = W (E_{7}), C_{6}, G_{26}, p ≡ 1 mod 6) There is an element w ∈ G_{36} of order 6 and a primitive 6th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{36}}(w), L_{36}^{⟨λw⟩}) = (G_{26}, L_{26})

which means that X(G_{36})^{hC_{6}} = X(G_{26}).

(2) (G_{36} = W (E_{7}), C_{4}, G_{8}, p ≡ 1 mod 8). There is an element w ∈ G_{36} of order 4, a subgroup \overline{W} < C_{G_{36}}(w) of index 8, faithfully represented in L_{36}^{⟨iw⟩}, and a primitive 4th root of unity i ∈ ℤ_{p}^{×} such that

(\overline{W} , L_{36}^{⟨iw⟩}) = (G_{8}, L_{8})

which means that X(G_{36})^{hC_{4}} = X(G_{8}). (The reflection group \overline{W} contains elements of order 8 with central square so it is not similar to G_{13} [56, p. 281].)

(3) (G_{36}, C_{14}, C_{14}, p ≡ 1 mod 14) There is an element w ∈ G_{36} of order 14 and a primitive 14th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{36}}(w), L_{36}^{⟨λw⟩}) = (C_{14}, ℤ_{p})

which means that X(G_{36})^{hC_{14}} = S^{27}.

(4) (G_{36}, C_{18}, C_{18}, p ≡ 1 mod 18) There is an element w ∈ G_{36} of order 18 and a primitive 18th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{36}}(w), L_{36}^{⟨λw⟩}) = (C_{18}, ℤ_{p})

which means that X(G_{36})^{hC_{18}} = S^{35}.

(5) (G_{26}, C_{18}, C_{18}, p ≡ 1 mod 18) There is an element w ∈ G_{26} of order 18 and a primitive 18th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{26}}(w), L_{26}^{⟨λw⟩}) = (C_{18}, ℤ_{p})

which means that X(G_{26})^{hC_{18}} = S^{35}.

(6) (G_{8}, C_{12}, C_{12}, p ≡ 1 mod 12) There is an element w ∈ G_{8} of order 12 and a primitive 12th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{8}}(w), L_{8}^{⟨λw⟩}) = (C_{12}, ℤ_{p})

which means that X(G_{8})^{hC_{12}} = S^{23}.

(7) (G_{8}, C_{8}, C_{8}, p ≡ 1 mod 8) There is an element w ∈ G_{8} of order 8 and a primitive 8th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{8}}(w), L_{8}^{⟨λw⟩}) = (C_{8}, ℤ_{p})

which means that X(G_{8})^{hC_{8}} = S^{15}.

(8) (G_{35} = W (E_{6}), C_{2}, G_{28} = W (F_{4}), p ≡ 1 mod 2) There is an element w ∈ G_{35} of order 2

such that

(C_{G_{35}}(w), L_{35}^{⟨−w⟩}) = (G_{28}, L_{28})

which means that E_{6}^{hC_{2}} = F_{4}.

(9) (G_{35}, C_{3}, G_{25}, p ≡ 1 mod 3) There is an element w ∈ G_{35} of order 3 and a primitive 3rd root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{35}}(w), L_{35}^{⟨λw⟩}) = (G_{25}, L_{25})

which means that X(G_{35})^{hC_{3}} = X(G_{25}).

(10) (G_{35}, C_{5}, G_{25}, p ≡ 1 mod 5) There is an element w ∈ G_{35} of order 5 and a primitive 5th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{35}}(w), L_{35}^{⟨λw⟩}) = (C_{5}, ℤ_{p})

which means that X(G_{35})^{hC_{5}} = S^{9}

(11) (G_{35}, C_{4}, G_{8}, p ≡ 1 mod 4) There is an element w ∈ G_{35} of order 4 and a primitive 4th root of unity i ∈ ℤ_{p}^{×} such that

(C_{G_{35}}(w), L_{35}^{⟨iw⟩}) = (G_{8}, L_{8})

which means that X(G_{35})^{hC_{4}} = X(G_{8}).

(12) (G_{25}, C_{2}, G_{5}, p ≡ 1 mod 6) There is an element w ∈ G_{25} of order 2 such that

(C_{G_{25}}(w), L_{25}^{⟨−w⟩}) = (G_{5}, L_{5})

which means that X(G_{25})^{hC_{2}} = X(G_{5}).

(13) (G_{28}, C_{3}, G_{5}, p ≡ 1 mod 6) There is an element w ∈ G_{28} of order 3 and a primitive 3rd root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{28}}(w), L_{28}^{⟨λw⟩}) = (G_{5}, L_{5})

which means that X(G_{28})^{hC_{3}} = X(G_{5}).

(14) (G_{28}, C_{4}, G_{4}, p ≡ 1 mod 4) There is an element w ∈ G_{28} of order 4 and a primitive 4th root of unity i ∈ ℤ_{p}^{×} such that

(C_{G_{28}}(w), L_{28}^{⟨iw⟩}) = (G_{8}, L_{8})

which means that X(G_{28})^{hC_{4}} = X(G_{8}).

(15) (G_{25}, C_{12}, C_{12}, p ≡ 1 mod 12) There is an element w ∈ G_{25} of order 12 and a primitive 12th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{25}}(w), L_{25}^{⟨λw⟩}) = (C_{12}, ℤ_{p})

which means that X(G_{25})^{hC_{12}} = S^{23}.

(16) (G_{55}, C_{12}, C_{12}, p ≡ 1 mod 12) There is an element w ∈ G_{25} of order 12 and a primitive 12th root of unity λ ∈ ℤ_{p}^{×} such that

(C_{G_{5}}(w), L_{5}^{⟨λw⟩}) = (C_{12}, ℤ_{p})

which means that X(G_{5})^{hC_{12}} = S^{23}.

6. Homotopy fixed points of twisted unstable Adams operations

Let X be a p-compact group and set α: X → X a p-compact group automorphism. The homotopy pullback diagram

\xymatrix{BF α(X) \ar[d]_{ι} \ar[r]^{ι} & BX \ar[d]^{Δ} \\ BX \ar[r]^{1×α} & BX × BX}  (18)

serves as the definition of the space BF α(X). If α is homotopic to α′, then one easily checks that BF α(X) ≃ BF α′(X).

The homotopy class of α is an element α ∈ Out(X), and, in turn, this is represented by a loop α: S^{1} → B aut(BX), hence representing an action of ℤ on BX, BX → BX_{hℤ} → S^{1}, in the sense of section 5 (see equation (15)). This fibration can also be obtained as the Borel contruction for the action of the positive integers, ℕ, on BX, determined by Bα: BX → BX, thus BX_{hℤ} ≃ BX ×_{ℕ} ℝ^{+}, hence, the homotopy fixed point space for this action is BX^{hℤ} ≃ Map_{ℕ}(ℝ^{+}, BX). This last can be easily identified with BF α(X).

In the special case where α = τψ^{q} is a twisted unstable Adams operation with q ∈ ℤ_{p}, q ≠ 1, and q ≢ 0 mod p, we have BF τψ^{q}(X) ≃ B^{τ}X(q), or just BX(q), if τ = 1. For q = 1 we trivially obtain BX(1) ≃ Λ(BX), the free loop space.

Assume that α represents an element of finite order r in Out(X), with r prime to p, and X is a connected p-compact group. According to Theorem B, it defines an action of the cyclic group C_{r} on BX. Next proposition shows that the natural map ΛBX^{hC_{r}} → BF α(X) is a homotopy equivalence.

Proposition 6.1. Assume that X is a connected p-compact group. If α: BX → BX represents an element of Out(X) of finite order r, coprime to p, then BF α(X) is homotopy equivalent to the space of free loops on BX^{hC_{r}}, where the action of the cyclic group C_{r} on BX is given by α.

Proof. According to Theorem B, α defines an action of C_{r} on X,

\xymatrix{BX \ar[r]^{i} & BX_{hC_{r}} \ar[r]^{p} & BC_{r} \ar[l]^{s}}

and the space of homotopy fixed points is the homotopy fibre of the induced map Map(BC_{r}, BX_{hC_{r}})_{s} → Map(BC_{r}, BC_{r})_{id}, thus we have an adjoint map BX^{hC_{r}} × BC_{r} → BX_{hC_{r}} that produces a lifting to a map i: BX^{hC_{r}} → BX that makes the triangle

\xymatrix{ & BX \ar[dd]^{α} \\ BX^{hC_{r}} \ar[ur]^{i} \ar[dr]_{i} \\  & BX}

commutative up to homotopy (BX^{hC_{r}} is simply connected by Lemma 5.1). Therefore, we can form a homotopy commutative diagram

[figure: ΛBX^{hC_{r}}; BX^{hC_{r}}; Δ; BX^{hC_{r}}; Δ; BX^{hC_{r}} × BX^{hC_{r}}; BF α(X); BX; Δ; BX; (1,α); BX × BX.]  (19)

We will show that ΛBX^{hC_{r}} → BF α(BX) is a homotopy equivalence. According to Theorem 5.2, BX^{hC_{r}} is the classifying space of a connected p-compact group and by Lemma 5.1 the natural map i: BX^{hC_{r}} → BX induces an identification of the homotopy groups of BX^{hC_{r}} with the invariant elements in the homotopy groups of BX by the action of C_{r}: π_{i}(BX^{hC_{r}}) ≅ π_{i}(BX)^{C_{r}} ↪ π_{i}(BX). There is a long exact sequence for the homotopy groups of BF α(X):

… ⟶ π_{i}(BF α(X)) ⟶ π_{i}(BX) ⟶^{1−α_{∗}} π_{i}(BX) ⟶ π_{i−1}(BF α(X)) ⟶ …

The same construction for the top square of diagram (19) degenerates to

… ⟶ π_{i}(ΛBX^{hC_{r}}) ⟶ π_{i}(BX^{hC_{r}}) ⟶^{0} π_{i}(BX^{hC_{r}}) ⟶ π_{i−1}(ΛBX^{hC_{r}}) ⟶ …

Both long exact sequences together give

\xymatrix{0 \ar[r] & π_{i+1}(BX)^{C_{r}} \ar[r] \ar[d] & π_{i}(ΛBX^{hC_{r}}) \ar[r] \ar[d] & π_{i}(BX)^{C_{r}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Coker\{1 − α_{∗}\} \ar[r] & π_{i}(BF α(X)) \ar[r] & Ker\{1 − α_{∗}\} \ar[r] & 0 .}

Now, Ker{1 − α_{∗}} = π_{i}(BX)^{C_{r}} and Coker{1 − α_{∗}} ≅ π_{i+1}(BX)_{C_{r}}. Since r is coprime to p, and the homotopy groups π_{i}(BX) are ℤ_{(p)}-modules for every i ≥ 2, the composition π_{i+1}(BX)^{C_{r}} → π_{i+1}(BX) → π_{i+1}(BX)_{C_{r}} is an isomorphism. Hence also the middle vertical map π_{i}(ΛBX^{hC_{r}}) → π_{i}(BF α(X)) is an isomorphism. □

Our next result will reduce, in many cases, the question of describing BF α(X) to two separate steps. The computation of homotopy fixed points BX^{hC_{r}}, for elements α of order r coprime to p, and the case in which α = ψ^{q} is an unstable Adams operation of exponent q ≡ 1 mod p, q ≠ 1 (see Theorem 2.2 and formula (2) in section 2). It is one of the two ingredients of Theorem C

Proposition 6.2. Let X be a connected p-compact group. If α is an automorphism of X that factors α = ψ^{q}β with

(1) q ≡ 1 mod p, and (ψ^{q})^{∗}: H^{∗}(X; 𝔽_{p}) → H^{∗}(X; 𝔽_{p}) is the identity, and

(2) β is an automorphism of X that represents an element of finite order r, coprime to p, in Out(X),

then BF α(X) ≃ BX^{hβ}(q).

Proof. Notice first that BX^{hβ} is again a p-compact group, according to Theorem B, and α restrict to ψ^{q} on BX^{hβ}. We will write BY = BX^{hβ} for simplicity. With this notation we have a homotopy commutative diagram

[figure: BY (q); BY; Δ; BY; (1,ψ^{q}); BY × BY; BF α(X); BX; Δ; BX; (1,α); BX × BX]

where the top and bottom faces are homotopy pullback diagrams, and the front face commutes up to homotopy because α ≃ ψ^{q} ∘ β and β is homotopic to the identity when restricted to BY. Consequently, the homotopy fibres of the vertical maps form another homotopy pullback diagram:

\xymatrix{E \ar[r] \ar[d] & X/Y \ar[d]^{Δ} \\ X/Y \ar[r]^{(1,α)} & X/Y × X/Y}

where E = hofib(BY (q) → BF α(X)), and we still denote by α the self-equivalence of X/Y induced by α: BX → BX. Again, Theorem B implies that X/Y is a connected H-space and then we can also describe E as the homotopy fibre of 1 − α: X/Y → X/Y, and it also implies that the map (ψ^{q})^{∗}: H^{∗}(X/Y; 𝔽_{p}) → H^{∗}(X/Y; 𝔽_{p}) can be read off the map (ψ^{q})^{∗} defined on H^{∗}(X; 𝔽_{p}), which, by hypothesis is the identity. This fact easily implies that (1 − α)^{∗} = (1 − β)^{∗}.

According to Proposition 6.1, the homotopy fibre of 1 − β is contractible, hence (1 − β)^{∗} is an automorphism of H^{∗}(X/Y; 𝔽_{p}). Thus, a spectral sequence argument shows that E is mod p acyclic. Finally, it is easy to see that E is p-complete, hence contractible, and therefore BY (q) ≃ BF α(X). □

Remark 6.3. If X polynomial, the effect of ψ^{q}, q ≡ 1 mod p, on mod p cohomology of X is determined by the effect on H^{∗}(BX, 𝔽_{p}) and this is in turn determined by the effect on H^{∗}(BT_{X}; 𝔽_{p}) which is multiplication by q, hence the identity. For X = F_{4}, E_{6}, E_{7}, E_{8} at the prime 3 or X = E_{8} at the prime 5, we also obtain that ψ^{q}, q ≡ 1 mod p, acts trivially on H^{∗}(X; 𝔽_{p}). The generators for this cohomology algebras either transgress to elements detected in the maximal torus or are linked to such elements by Steenrod operations (cf. [40, Ch7]). In particular, 6.2 applies to all 1-connected p-compact groups, p odd, according to the classification theorem [6].

One further reduction is obtained by extending the action of ℤ on BX to an action of ℤ_{p}. We will se that this is possible for the action of unstable Adams operations ψ^{q} of exponent q ≡ 1 mod p, and in this case we obtain that the homotopy fixed point space BX(q) = BX^{hℤ} depends only on the p-adic valuation ν_{p}(1 − q).

Let α be an element of Out(X) = π_{1}B aut(BX) represented by a loop ω_{α}: S^{1} → B aut(BX) that classifies an action BX → BX_{hℤ} → S^{1}. If the homomorphism π_{1}(ω_{α}): ℤ → Out(X) extends to ℤ_{p}, then we can also extend ω_{α} to a map ω̂_{α}: Ŝ_{p}^{1} → B aut(BX), which is an action of ℤ_{p} on BX that extends the original action determined by α.

Lemma 6.4. Let X be a p-compact group. Assume that the action of ℤ on BX determined by an element α ∈ Out(X) extends to the p-adics, then BF α(X) ≃ BX^{hℤ} ≃ BX^{hℤ_{p}}.

Proof. There is a map of fibrations

[figure: BX; BX; BX_{hℤ}; BX_{hℤ_{p}}; Bℤ; Bℤ_{p}]

where the right fibration is the p-completion of the left one. In fact, ℤ_{p} can only act nilpotently on H^{i}(BX, 𝔽_{p}), which are finite 𝔽_{p}-vector spaces, hence the fibration on the right is preserved by p-completion. Since the base and the fibre are p-completed spaces, so is the total space BX_{hℤ_{p}}. The above diagram is a pullback diagram, so the left fibration is also preserved by p-completion, and then, since the top and bottom horizontal arrows are p-equivalences, so is the middle one.

The functor Map(S^{1}, −) preserves pullback diagrams, thus, we have another pullback diagram

[figure: BX^{hℤ}; BX^{hℤ}; Map(Bℤ, BX_{hℤ})ˆ1; Map(Bℤ, BX_{hℤ_{p}})ˆ1; Map(Bℤ, Bℤ)_{1}; Map(Bℤ, Bℤ_{p})_{1}]

and then, the map Bℤ → Bℤ_{p}, which is a mod p equivalence, induces a diagram

\xymatrix{BX^{hℤ_{p}} \ar[r]_{≃} \ar[d] & BX^{hℤ} \ar[d] \\ Map(Bℤ_{p}, BX_{hℤ_{p}})ˆ1 \ar[r]_{≃} \ar[d] & Map(Bℤ, BX_{hℤ_{p}})ˆ1 \ar[d] \\ Map(Bℤ_{p}, Bℤ_{p})_{1} \ar[r]_{≃} & Map(Bℤ, Bℤ_{p})_{1}}

where the middle and bottom horizontal maps are weak equivalences by [8, II,2.8], hence so is the top horizontal map and the lemma follows. □

Using to the description of Out(X) in section 2 we will see how those extensions are obtained in case of actions of unstable Adams operations ψ^{q} of exponent q ≡ 1 mod p. In order to compare actions of ℤ of ℤ_{p} given by unstable Adams operations, we must analyse the diagram of group homomorphisms

\xymatrix{Hom(ℤ_{p}, ℤ_{p}^{×}) \ar[d] \ar[r]^{res} & Hom(ℤ, ℤ_{p}^{×}) \ar[d] \\ Hom(ℤ_{p}, Out(X)) \ar[r]^{res} & Hom(ℤ, Out(X))}

where the horizontal homomorphisms are given by restriction and the vertical ones by the inclusion of the Adams operations q ∈ ℤ_{p}^{×} ↦ ψ^{q} ∈ Out(X).

Recall that, for an odd prime p, ℤ_{p}^{×} ≅ ℤ/p − 1 × ℤ_{p}, where ℤ/p − 1 correspond to the roots of unity contained in ℤ_{p}^{×} and ℤ_{p} is identified with the subgroup of elements q ∈ ℤ_{p}^{×}, with q ≡ 1 mod p, via the exponential map:

a ∈ ℤ_{p} ↦ exp(pa) ∈ ℤ_{p}^{×}

(exp defined by the usual expansion exp(pa) = 1 + pa + …). Since there are no non-trivial homomorphisms ℤ_{p} → ℤ/p − 1, the group Hom(ℤ_{p}, ℤ_{p}^{×}) can be parametrized by ℤ_{p} in the following way m ∈ ℤ_{p} ↦ ω_{m} ∈ Hom(ℤ_{p}, ℤ_{p}^{×}), defined ω_{m}(a) = exp(pma).

Using the standard identification Hom(ℤ, ℤ_{p}^{×}) ≅ ℤ_{p}^{×} given by evaluation at 1 ∈ ℤ, the restriction map is described by

ℤ_{p} ⟶^{≅} Hom(ℤ_{p}, ℤ_{p}^{×}) ⟶^{res} Hom(ℤ, ℤ_{p}^{×}) ⟶^{≅} ℤ_{p}^{×}
m ↦  ω_{m}  ↦  ω_{m}|_{ℤ}  ↦ ω_{m}(1) = exp(pm)
= 1 + pm + …

It follows that the image of the restriction consists of actions given by unstable Adams operations ψ^{q} with q ≡ 1 mod p, for which we can choose m = \frac{1}{p} log(q).

Now, we can prove the second ingredient of Theorem C.

Proposition 6.5. If q, q′ ∈ ℤ_{p}^{×}, q ≡ q′ mod p, both are of order r mod p, and ν_{p}(1 − q^{r}) = ν_{p}(1 − q′^{r}), then BX(q) ≃ BX(q′), for any connected p-compact group X.

Proof. The proof is divided in two steps. First, we will consider the case q ≡ q′ ≡ 1 mod p (r = 1). In these cases, the actions of ℤ given by ψ^{q} and ψ^{q′}, respectively, extend to actions of the p-adics described by m_{q} = \frac{1}{p} log(q) and m_{q′} = \frac{1}{p} log(q′), respectively. The homotopy fixed points space BX^{hℤ_{p}} depends only of the image of the action ℤ_{p} → Out(X). The image of the two actions are clearly the same if and only if m_{q} and m_{q′} differ by a p-adic unit; that is, if and only if ν_{p}(m_{q}) = ν_{p}(m_{q′}), if and only if ν_{p}(1 − q) = ν_{p}(1 − q′), in which case, we have

BX(q) ≃ BX^{hψ^{q}} ≃ BX^{hℤ_{p}} ≃ BX^{hψ^{q′}} ≃ BX(q′) .

In the general case, we can decompose q = ζ · q_{0} and q′ = ζ · q′_{0}, where ζ is a primitive rth of unity and q_{0} ≡ q′_{0} ≡ 1 mod p, thus

BX(q) ≃ BX^{hζ}(q_{0}) ≃ BX^{hζ}(q′_{0}) ≃ BX(q′) .  □

Remark 6.6. If q is a p-adic unit, we can find a prime number q_{0} such that q ≡ q_{0} mod p and ν_{p}(1 − q^{r}) = ν_{p}(1 − q_{0}^{r}), where r is the order of q mod p, and then,

BX(q) ≃ BX(q_{0})

by Proposition 6.5.

In fact, we can assume that q is an integer, otherwise change it by the sum of enough first terms in its p-adic expansion. Then, by Dirichlet’s theorem there is a prime number q_{0} of the form q_{0} = p^{N}c + q, with N > ν_{p}(1 − q^{r}), satisfying the above conditions.

Proof of Theorem C. Part (1) follows from Proposition 6.2 and Remark 6.3. Part (2) is Proposition 6.5. □

7. General structure of finite Chevalley versions of p-compact groups

In this section we will study the first general properties of finite Chevalley versions BX(q) of p-compact groups X. The main results being the identification of the maximal finite torus, the Weyl group, and the fusion category of elementary abelian p-subgroups.

Proposition 7.1. Let X be a connected p-compact group and α a self homotopy equivalence of X. Then

(1) BF α(X) is connected and p-complete.

(2) ι: BF α(X) → BX is a homotopy monomorphism at p.

(3) For any finite p-group P, Map(BP, BF α(X))_{c} ≃ BF α(X).

Proof. From the definition we obtain a fibration X ⟶ BF α(X) ⟶^{ι} BX where X and BX are p-complete, X is connected and BX is simply-connected. It follows that BF α(X) is connected and p-complete.

For any finite p-group P, Map(BP, BX)_{c} ≃ BX, and Map_{∗}(BP, X) ≃ X for any choice of base point. It then follows that ι: BF α(X) → BX is a homotopy monomorphism at p, and from the induced fibration

Map(BP, X) → Map(BP, BF α(X))_{c} → Map(BP, BX)_{c}

it follows that Map(BP, BF α(X))_{c} ≃ BF α(X). □

Lemma 7.2. Let X be a p-compact group, α a self homotopy equivalence of BX, and (P, ν) an object of F_{p}(BX) fixed by α up to homotopy; that is, ν ≃ α∘ν. If C_{X}(P, ν) is connected, then there is a unique lifting of ν: BP → BX to a homotopy monomorphism g: BP → BF α(X), and

\xymatrix{Map(BP, BF α(X))_{g} \ar[d] \ar[r] & Map(BP, BX)_{ν} \ar[d]^{Δ} \\ Map(BP, BX)_{ν} \ar[r]^{1×α_{♯}} & Map(BP, BX)_{ν} × Map(BP, BX)_{ν}}

is a homotopy pullback diagram.

Proof. Since (18) is a homotopy pullback diagram, there is at least a lifting of ν, g: BP → BF α(X).

The homotopy fibre of Map(BP, BX)_{ν} ⟶^{Δ} Map(BP, BX)_{ν} ×Map(BP, BX)_{ν} is C_{X}(P, ν) = Ω Map(BP, BX)_{ν}, hence pulling back along 1 × α_{♯} we obtain a fibration, up to homotopy,

C_{X}(P, ν) ⟶ Map(BP, BF α(X))ν̂ ⟶^{ι_{♯}} Map(BP, BX)_{ν}

where Map(BP, BF α(X))ν̂ consists of all possible liftings of ν up to homotopy. The base space consists of just one connected component, hence if we assume that the fibre C_{X}(P, ν) is also connected, then the total space must be connected, and therefore any other lifting of ν is homotopic to g. □

The following lemma will help us determine the restriction of α to the centralizers.

Lemma 7.3. Let X be a connected p-compact group and α a self-equivalence of BX. Let T (α) be a given restriction of α to the maximal torus T = T_{X}, and (P, ν) an object of F_{p}(BX).

Suppose that ν: BP → BX admits a factorization μ: BP → BT through the maximal torus j: BT → BX. Then, the object (P, ν) is fixed by α if and only if T (α)μ = wμ for an element w of the Weyl group. If this is the case, the restriction to the maximal torus of the induced self homotopy equivalence α|_{C_{X}(P,ν)} of the centralizer C_{X}(P, ν) is T (α|_{C_{X}(P,ν)}) = w^{−1} ∘ T (α).

Proof. (P, ν) is fixed by α means that ν ≃ α ∘ Bν, and if ν factors as j ∘ μ, that is to say, j ∘ Bμ ≃ α ∘ j ∘ μ ≃ j ∘ T (α) ∘ μ, and according to [49, 4.1], [44, 3.4], this is equivalent to the existence of w, in the Weyl group of X, such that w ∘ μ ≃ BT (α) ∘ μ.

Now assuming the existence of such element w, we read from the commutative diagram

\xymatrix{BT \ar[r]^{T (α)} & BT & BT \ar[l]_{w} \\ Map(BP, BT)_{μ} \ar[u]_{ev}^{≃} \ar[d]^{j_{♯}}_{≃} \ar[r]^{T (α)_{♯}} & Map(BP, BT)_{wμ} \ar[d]^{j_{♯}}_{≃} \ar[u]_{ev}^{≃} & Map(BP, BT)_{μ} \ar[l]_{w_{♯}} \ar[dl]^{j_{♯}}_{≃} \ar[u]_{ev}^{≃} \\ Map(BP, BX)_{ν} \ar[r]^{α_{♯}} & Map(BP, BX)_{ν}}

that the restriction of α|_{C_{X}(P,ν)} = α_{♯} to the maximal torus of C_{X}(V, ν) is w^{−1} ∘ T (α). □

If the centralizer C_{X}(V, ν) is connected, this determines the restriction α|C_{X}(V, ν) (see §2).

Corollary 7.4. Let X be a p-compact group and ν: BV → BX a toral elementary abelian psubgroup such that its centralizer C_{X}(V, ν) is connected. If ψ^{q} is an unstable Adams operation of exponent q ≡ 1 mod p, q ≠ 1, then

(a) there is a unique lift of ν to g: BV → BX(q),

(b) ψ^{q}|_{C_{X}(V,ν)} = ψ^{q} is as well an unstable Adams operation of exponent q, and

(c) the centralizer of (V, g) in X(q) is C_{X(q)}(V, g) ≅ C_{X}(V, ν)(q).

Proof. In particular, when ν: BV → BX is a toral elementary abelian p-group in X and α = ψ^{q} is an Adams operation of exponent q ≡ 1 mod p, then we can write T (ψ^{q}) = ψ^{q}, the qth power map in the maximal torus T = T_{X} and ψ^{q} ∘ μ ≃ μ, where μ: BV → BT is a lift to BT of ν: BV → BX, so, by Lemma 7.3, there is a commutative diagram

\xymatrix{BT \ar[r]^{T (ψ^{q})=ψ^{q}} \ar[d] & BT \ar[d] \\ BC_{X}(V, ν) \ar[r]^{ψ^{q}|_{C_{X} (V,ν)}} \ar[d] & BC_{X}(V, ν) \ar[d] \\ BX \ar[r]^{ψ^{q}} & BX}

this proves (b), namely, ψ^{q}|_{BC_{X}(V,ν)} is, as well, an unstable Adams map ψ^{q}.

Now, (a) and (c) follow from Lemma 7.2. □

We will now restrict our attention to cases with q ≡ 1 mod p, q ≠ 1. According to Proposition 6.2, the general case can be reduced to this one, in the cases that are of interest to us (see Remark 6.3). Hence, essentially, there will be no loss of generality in our assumption.

Proposition 7.5. Let X be a connected p-compact group, p an odd prime, and ψ^{q} an unstable Adams operation of exponent q ∈ ℤ_{p}^{∗}, with q ≡ 1 mod p, q ≠ 1. Then the inclusion ν: Bt_{X} → BX of the subgroup of elements of order p in the maximal torus T_{X} has a unique lift to g: Bt_{X} → BX(q) and its centralizer is

C_{X(q)}(t_{X}, g) = T_{X}(q) .

Proof. Since C_{X}(t_{X}, ν) = T_{X} and ψ^{q}|_{T_{X}} = T (ψ^{q}) = ψ^{q} this follows from 7.3 (see 7.4). □

The subgroup T_{X}(q) ≅ T_{ℓ}^{n} ≅ (ℤ/p^{ℓ})^{n}, where n is the rank of X and ℓ = ν_{p}(q − 1), established in Proposition 7.5 will be referred to as the maximal finite torus of X(q). Then, we define the Weyl group of X(q) as the automorphism group

W_{X(q)}(T_{ℓ}^{n}) = Aut_{F_{p}(BX(q))}(T_{ℓ}^{n})

of T_{ℓ}^{n} in the category F_{p}(BX(q)).

Proposition 7.6. Let X be a connected p-compact group, p an odd prime, and ψ^{q} an unstable Adams operation of exponent q ∈ ℤ_{p}^{∗}, with q ≡ 1 mod p, q ≠ 1. If T_{ℓ}^{n} ≅ (ℤ/p^{ℓ})^{n} is the maximal finite torus of X(q), then its Weyl group is

W_{X(q)}(T_{ℓ}^{n}) ≅ W_{X}

the Weyl group of X, with action on T_{ℓ}^{n} given by the mod p^{ℓ} reduction of the p-adic representation of W_{X}. The extension N_{X(q)}(T_{ℓ}^{n}) = T_{ℓ}^{n} ⋊ W_{X(q)}(T_{ℓ}^{n}) fits in the homotopy commutative diagram

\xymatrix{BN_{X(q)}(T_{ℓ}^{n}) \ar[r] \ar[d] & BN(T_{X}) \ar[d] \\ BX(q) \ar[r] & BX.}

Proof. It follows from the diagram

\xymatrix{Map_{∗}(BT_{ℓ}^{n}, BT_{ℓ}^{n})î \ar[r]^{≃} \ar[d] & Map_{∗}(BT_{ℓ}^{n}, BT_{X})ι̂∘i \ar[d] \\ Map(BT_{ℓ}^{n}, BX(q))_{i} \ar[r]_{ι_{♯}} & Map(BT_{ℓ}^{n}, BX)_{ι∘i},}  (20)

where î is the set of components that map down to the component of the inclusion i: BT_{ℓ}^{n} → BX(q) and similarly, ι̂ ∘ i is the set of components that map down to the component of ι ∘ i. Now, Map(BT_{ℓ}^{n}, BX)_{ι∘i} ≃ BT_{X} and by Lemma 7.2 the upper horizontal arrow in (20) induces a bijection on components, and hence an equivalence of homotopy discrete spaces. □

For X a p-compact group and α a self equivalence, the inclusion ι: BF α(X) → BX induces a functor between the respective fusion categories

ι_{♯}: F_{p}(BF α(X)) ⟶ F_{p}(BX)

and Lemma 7.2 above give some useful information in order to compare the morphism sets. Thus, for instance,

Mor_{F_{p}(BF α(X))}((P, g), (Q, h)) ⟶ Mor_{F_{p}(BX)}((P, ι ∘ g), (Q, ι ∘ h))  (21)

is a bijection provided C_{X}(P, ι ∘ g) is connected. It rarely happens that those centralizers are connected for a general p-group P, but it is not so unusual if we restrict to some particular classes of small groups. For a space Y, we denote F_{p}^{e}(Y) the full subcategory of F_{p}(Y) whose objects are the elementary abelian subgroups of Y.

Corollary 7.7. Let p be an odd prime. If X is a connected polynomial p-compact group and α a self homotopy equivalence, then the functor

ι_{♯}: F_{p}^{e}(BF α(X)) ⟶ F_{p}^{e}(BX)

is both full and faithful.

Proof. If X is a connected polynomial p-compact group, then centralizers of elementary abelian p-subgroups are connected and Lemma 7.2 applies. In fact, if (E, ν) is an elementary abelian p-subgroup of X, then the centralizer C_{X}(E, ν) is also a polynomial p-compact group, hence H^{1}(BC_{X}(E, ν); 𝔽_{p}) = 0 and therefore C_{X}(E, ν) is connected (see [24, 1.3]) and the map (21) is a bijection for every elementary abelian p-subgroups (P, g) and (Q, h) of BF α(X). □

Corollary 7.8. Let p be an odd prime. If X is a connected polynomial p-compact group and ψ^{q} an unstable Adams operation of exponent q ∈ ℤ_{p}^{∗}, with q ≡ 1 mod p, then

ι_{♯}: F_{p}^{e}(BX(q)) ⟶ F_{p}^{e}(BX)

is an equivalence of categories.

Proof. By Corollary 7.7 we only have to check that ι_{♯} induces in this case a bijection between isomorphism classes of objects, and this follows from Proposition 7.5, because in a polynomial p-compact group every elementary abelian subgroup is toral. □

Let X be a polynomial p-compact group with trivial center and q ∈ ℤ_{p}^{∗} a p-adic unit with q ≡ 1 mod p, q ≠ 1. Putting BC_{X(q)}(V, g) = Map(BV, BX(q))_{g} for any object (V, g) of F_{p}^{e}(BX(q)) we get a functor from F_{p}^{e}(BX(q))^{op} to topological spaces. There is natural map

hocolim_{F_{p}^{e}(BX(q))^{op}} BC_{X(q)} → BX(q)  (22)

from the homotopy colimit of this functor. When C_{X}(V, g) is connected, we have

BC_{X(q)}(V, g) ≃ BF (ψ^{q}|_{C_{X}(V,g)})(C_{X}(V, ι ∘ g)) ≃ BC_{X}(V, ι ∘ g)(q)

according to Lemma 7.3 and Remark 7.4.

Let T_{X} be the maximal torus and W_{X} the Weyl group of a p-compact group X, p odd. As usually, we denote by t_{X} the group of all elements of order p in T_{X}, and g: Bt_{X} → X(q) the inclusion. For any nontrivial elementary abelian p-subgroup E ≤ T, write W (E) for the point-wise stabilizer subgroup of E.

Proposition 7.9. Let X be a polynomial p-compact group with trivial center, p odd, and q ∈ ℤ_{p}^{∗} a p-adic unit with q ≡ 1 mod p, q ≠ 1. Assume that

H^{∗}(BX(q); 𝔽_{p}) ≅ H^{∗}(BT_{X}(q); 𝔽_{p})^{W_{X}}

and that

H^{∗}(BC_{X(q)}(E, g|_{BE}); 𝔽_{p}) ≅ H^{∗}(BT_{X}(q); 𝔽_{p})^{W (E)}

for any nontrivial, subgroup E of t_{X}. Then (22) is an 𝔽_{p}-equivalence.

A similar statement holds with F_{p}^{e}(BX(q)) replaced by the full subcategory generated by all objects of the form (t_{X})^{P} where P runs through the subgroups of a Sylow p-subgroup of W.

Proof. This follows from the Bousfield-Kan spectral sequence because the functor

E → H^{∗}(BC_{X(q)}(E, g|_{BE}); 𝔽_{p}) = H^{∗}(BT_{X}(q); 𝔽_{p})^{W (E)}

is exact with limit H^{∗}(BT_{X}(q); 𝔽_{p})^{W_{X}} = H^{∗}(BX(q); 𝔽_{p}), [21, 8.1] [48, 2.16]. □

This result motivates the research in next sections of the cohomology rings H^{∗}(BX(q); 𝔽_{p}) and the invariant rings H^{∗}(BT_{X}(q); 𝔽_{p})^{W_{X}}.

8. Cohomology rings

This section is devoted to the proof of Theorem E. The Eilenberg-Moore spectral sequence is used in order to get a hold of the cohomology rings of the spaces BX(q) of fixed points of unstable Adams operations acting on polynomial p-compact groups BX. We follow the arguments of [57] that already contain the first part of the theorem.

We end this section with an application to the unitary groups BU(n), BSU(n), in which we show that at the prime p, and for a p-adic unit q, the homotopy type of BU(n)(q), or BSU(n)(q), does only depend on the p-adic valuation ν_{p}(1 − q^{m}), where m is the order of q mod p.

Proof of Theorem E. Part (1) is due to L. Smith [57]. We will sketch his arguments here and then will continue with the proof of the second part of the theorem.

There is an Eilenberg-Moore spectral sequence associated to the pullback diagram

\xymatrix{BX(q) \ar[r]^{ι} \ar[d]^{ι} & BX \ar[d]^{Δ} \\ BX \ar[r]^{1×ψ^{q}} & BX × BX.}  (23)

This is a second quadrant spectral sequence with

E_{2}^{s,t} ≅ Tor_{H^{∗}(BX;𝔽_{p})^{⊗2}}^{s,t}(H^{∗}(BX; 𝔽_{p}), H^{∗}(BX; 𝔽_{p})) ⟹ H^{s+t}(BX(q); 𝔽_{p})

converging to a graded ring associated of H^{∗}(BX(q); 𝔽_{p}).

For simplicity, we will write P [x_{i}] = P [x_{1}, …, x_{n}] ≅ H^{∗}(BX; 𝔽_{p}). The Koszul complex

E(x_{i}) = P [x_{i}] ⊗ P [x_{i}] ⊗ E[sx_{1}, … sx_{n}]

with bideg(sx_{i}) = (−1, 2d_{i}) and d(sx_{i}) = x_{i} ⊗ 1 − 1 ⊗ x_{i}, is a free resolution of P [x_{i}] as (P [x_{i}] ⊗ P [x_{i}])-module, with module structure given by the multiplication m = Δ^{∗}. Then, Tor_{P [x_{i}]⊗P [x_{i}]}^{∗∗}(P [x_{i}], P [x_{i}]) is the homology of the complex

P [x_{i}] ⊗_{P [x_{i}]⊗P [x_{i}]} E(x_{i}) ≅ P [x_{i}] ⊗ E[sx_{1}, … sx_{n}]

where now the action of P [x_{i}] ⊗ P [x_{i}] on the left hand side term P [x_{i}] in given by the algebra map (1 × ψ^{q})^{∗}, hence one obtains the expression d(sx_{i}) = x_{i} − q^{d_{i}}x_{i} for the differential, but since q ≡ 1 mod p, we actually have d(sx_{i}) = 0 for all i = 1, …, n. This yields

E_{2}^{∗∗} ≅ Tor_{P [x_{i}]⊗P [x_{i}]}^{∗∗}(P [x_{i}], P [x_{i}]) ≅ P [x_{1}, …, x_{n}] ⊗ E[sx_{1}, …, sx_{n}]

and, since the algebra generators appear in filtration degrees 0 and −1, the spectral sequence collapses at the E_{2}-page and then we can find elements y_{i} in H^{∗}(BX(q); 𝔽_{p}) representing sx_{i} in the graded associated ring, with

H^{∗}(BX(q); 𝔽_{p}) ≅ P [x_{1}, …, x_{n}] ⊗ E[y_{1}, …, y_{n}] .

Let T_{X} be the maximal torus of X and W_{X} the Weyl group. Since X is polynomial, the mod p cohomology ring of BX coincides with the invariants by the action of the Weyl group on the mod p cohomology of BT_{X}, H^{∗}(BT_{X}; 𝔽_{p})^{W_{X}} ≅ H^{∗}(BX; 𝔽_{p}) ≅ P [x_{1}, …, x_{n}].

According to 7.4, 7.5, the classifying space of maximal finite torus of X(q) is BT (q) ≅ BT_{ℓ}^{n} and it is obtained from a pullback diagram

\xymatrix{BT_{ℓ}^{n} \ar[r]^{ι} \ar[d]^{ι} & BT \ar[d]^{Δ} \\ BT \ar[r]^{1×ψ^{q}} & BT × BT .}  (24)

Furthermore, the Weyl group is W_{X} (7.6) hence, the restriction map

i^{∗}: H^{∗}(BX(q); 𝔽_{p}) → H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})

has image in the invariant subring by the action of the Weyl group, W_{X}. It remains to show that this restriction map is injective.

The pullback diagram (24) yields another Eilenberg-Moore spectral sequence:

\overline{E}_{2}^{s,t} ≅ Tor_{H^{∗}(BT;𝔽_{p})^{⊗2}}^{s,t}(H^{∗}(BT; 𝔽_{p}), H^{∗}(BT; 𝔽_{p})) ⟹ H^{s+t}(BT_{ℓ}^{n}; 𝔽_{p}) .

We will pay special attention to the map between the two spectral sequences i^{∗}: E_{r}^{∗∗} → \overline{E}_{r}^{∗∗} induced by the natural map from diagram (24) to diagram (23) given by inclusion of the maximal torus. In order to describe the induced map at the level of E_{2}-pages, we need some elementary algebraic considerations.

Again for simplicity, we will write P [t_{i}] = P [t_{1}, …, t_{n}] ≅ H^{∗}(BT_{X}; 𝔽_{p}). The kernel of the multiplication m: P [t_{i}] ⊗ P [t_{i}] → P [t_{i}] is a Borel ideal

Ker m = (t_{1} ⊗ 1 − 1 ⊗ t_{1}, …, t_{n} ⊗ 1 − 1 ⊗ t_{n})

and then we can define derivations

∂_{i}: P [t_{i}] → P [t_{i}]

for i = 1, …, n, in the following way. For any homogeneous polynomial f ∈ P [t_{i}], f ⊗ 1 − 1 ⊗ f ∈ Ker m, hence we can find an expression f ⊗ 1 − 1 ⊗ f = ∑_{i} c_{i}(f)(t_{i} ⊗ 1 − 1 ⊗ t_{i}), with coefficients c_{i}(f) ∈ P [t_{i}] ⊗ P [t_{i}], and then define ∂_{i}(f) = m(c_{i}(f)) ∈ P [t_{i}]. A routine calculation shows:

(1) ∂_{i} is well defined and does not depend on the choice of coefficients c_{1}(f), …, c_{n}(f),

(2) ∂_{i} is a derivation of P [t_{i}], and

(3) ∂_{i}(t_{i}) = 1 and ∂_{i}(t_{j}) = 0 if j ≠ i.

These properties show that these are the partial derivatives:

∂_{i}(f) = \frac{∂f}{∂t_{i}} .

After these considerations we can easily describe the map between the respective E_{2}-pages and show that it is injective. In order to compute the \overline{E}_{2}^{∗∗}, we define now the Koszul complex

E(t_{i}) = P [t_{i}] ⊗ P [t_{i}] ⊗ E[st_{1}, … st_{n}]

with bideg(st_{i}) = (−1, 2) and d(st_{i}) = t_{i} ⊗ 1 − 1 ⊗ t_{i}. As before, we obtain that

\overline{E}_{2}^{∗∗} ≅ Tor_{P [t_{i}]⊗P [t_{i}]}^{∗∗}(P [t_{i}], P [t_{i}]) ≅ P [t_{i}] ⊗_{P [t_{i}]⊗P [t_{i}]} E(t_{i}) ≅ P [t_{i}] ⊗ E[st_{1}, … st_{n}]  (25)

since the differential in this complex turns out to be trivial, again, because q ≡ 1 mod p. Also as before, the algebra generators of \overline{E}_{2}^{∗∗} appear in filtration degree 0 and −1 and therefore the spectral sequence \overline{E}_{r}^{∗∗} collapses at the E_{2}-page.

Now, the inclusion i^{∗}: P [x_{i}] → P [t_{i}] extends to a map of Koszul complexes

i^{∗}: E(x_{i}) → E(t_{i})

which is a P [x_{i}] ⊗ P [x_{i}]-module map defined by

i^{∗}(sx_{i}) = ∑_{j} c_{i}(x_{i}) ⊗ st_{j}

on generators. Then the induced map

i^{∗}: Tor_{P [x_{i}]⊗P [x_{i}]}^{∗∗}(P [x_{i}], P [x_{i}]) ≅ P [x_{i}] ⊗ E[sx_{1}, … sx_{n}] ⟶ Tor_{P [t_{i}]⊗P [t_{i}]}^{∗∗}(P [t_{i}], P [t_{i}]) ≅ P [t_{i}] ⊗ E[st_{1}, … st_{n}]

is determined by

i^{∗}(sx_{i}) = ∑_{j} ∂_{j}(x_{i}) ⊗ st_{j} = ∑_{j} \frac{∂x_{i}}{∂t_{j}} ⊗ st_{j} .

Now, i^{∗} is injective because the jacobian determinant is non-trivial,

J = det (\frac{∂x_{i}}{∂t_{i}} ) ≠ 0,

by [61]. Since both spectral sequences collapse at the E_{2}-page, it follows that i^{∗}: H^{∗}(BX(q); 𝔽_{p}) → H^{∗}(BT_{ℓ}^{n}; 𝔽_{p}) is also injective. □

Remark 8.1. The argument with the Eilenberg-Moore spectral sequence used in the proof of part (1) of the above Theorem applies more generally to the case of any unstable Adams operation ψ^{q} of arbitrary exponent q ∈ ℤ_{p}^{∗} acting on a polynomial p-compact group (see [57]). Under these more general hypothesis we obtain that if H^{∗}(BX) ≅ P [x_{1}, …, x_{n}] then the cohomology of BX(q) is

H^{∗}(BX(q); 𝔽_{p}) ≅ P [x_{i_{1}}, …, x_{i_{k}}] ⊗ E[y_{i_{1}}, …, y_{i_{k}}]

where the polynomial generators x_{i_{j}} correspond to those x_{i} with degree 2d_{i} = deg x_{i} where m|d_{i}, if m is the order of q mod p, and 2d_{i} − 1 = deg y_{i}.

Notice that we can write q = ζq′ where ζ is an m-root of one in ℤ_{p} and q′ ≡ 1 mod p. Hence ψ^{q} = ψ^{q′} ∘ ψ^{ζ}, and ψ^{ζ} has finite order m as automorphism of the p-compact group X. It follows from 6.2, 6.3, that BY (q′) ≃ BX(q) if BY = BX^{hψ^{ζ}}. Moreover, by Theorem B, Y = X^{hψ^{ζ}} is again a polynomial p-compact group. According to Theorem E the cohomology of BY must be

H^{∗}(BY; 𝔽_{p}) ≅ P [x_{i_{1}}, …, x_{i_{k}}] .

9. Invariant theory

Let X be a polynomial p-compact group of rank n and let q be a p-adic unit, q ≡ 1 mod p, q ≠ 1, and ℓ = ν_{p}(1 − q). In Theorem E(2) we obtained a monomorphism i^{∗}: H^{∗}(BX(q); 𝔽_{p}) ↪ H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{W_{X}}, where T_{ℓ}^{n} is the maximal finite torus of BX(q) and W_{X} the Weyl group (see 7.5, 7.6). Whether or not i^{∗} is an isomorphism, H^{∗}(BX(q); 𝔽_{p}) ≅ H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{W_{X}}, is now a question of invariant theory and this is the subject of this section.

Continuing with the notation of the precedent section we write V = t_{X} for the elements of order p in the maximal finite torus and identify the dual vector space with the two dimensional primitive elements in the cohomology of BT_{ℓ}^{n}, V^{∗} ≅ P H^{2}(BT_{ℓ}^{n}; 𝔽_{p}). The Bockstein operations provide a vector space isomorphism P H^{2}(BT_{ℓ}^{n}; 𝔽_{p}) ≅ H^{1}(BT_{ℓ}^{n}; 𝔽_{p}), that we will denote as d: V^{∗} → dV^{∗}, of degree (−1). If P (V^{∗}) is the symmetric algebra on V^{∗} and E(dV^{∗}) the exterior algebra on dV^{∗}, we can describe the algebra structure of H^{∗}(BT_{ℓ}^{n}; 𝔽_{p}) as

K(V^{∗}) = P (V^{∗}) ⊗ E(dV^{∗}) = P [x_{1}, …, x_{n}] ⊗ E[dx_{1}, …, dx_{n}] ,

and d extends to an algebra derivation on K(V^{∗}). Moreover, any subgroup G ≤ GL(V) of linear substitutions acts on K(V^{∗}) in a natural way that commutes with the derivation d, hence K(V^{∗})^{G} is still a differential algebra.

Assume that P (V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] is a polynomial algebra; in particular, G is a pseudoreflection group. Then dρ_{1}, …, dρ_{n} are also invariant under the action of G. The purpose of the next theorem is to establish the cases in which {ρ_{1}, …, ρ_{n}, dρ_{1}, …, dρ_{n}} is a free system of generators for K(V^{∗})^{G}.

If we write dρ_{i} = ∑_{j=1}^{n} a_{ij}dx_{j}, the jacobian J = det(a_{ij}) ∈ P (V^{∗}) is invariant relative to det^{−1}; that is, for any g ∈ G, g · J = det(g)^{−1}J. The relative invariants form a free module over P (V^{∗})^{G} on one generator P (V^{∗})_{det^{−1}}^{G} = f_{det}^{−1} · P (V^{∗})^{G}, for an element f_{det}^{−1} ∈ P (V^{∗}) which is unique up to an invertible of 𝔽_{p} (see [14]). It follows that f_{det}^{−1} divides J.

Theorem 9.1 ([9]). Let V be a vector space of dimension n over a field of characteristic p ≠ 2. Assume that G ≤ GL(V) is a group of linear substitutions such that P (V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] is a polynomial algebra, then

K(V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] ⊗ E[dρ_{1}, …, dρ_{n}]

if and only if f_{det}^{−1} has degree deg f_{det}^{−1} = ∑_{i=1}^{n}(deg ρ_{i} − 2).

Proof. Since P (V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] is a polynomial ring of invariants, the Jacobian is nonzero, J ≠0 (see [61]), and this implies that the homomorphism P [ρ_{1}, …,ρ_{n}]⊗E[dρ_{1}, …,dρ_{n}]→ K(V^{∗}) defined from the free anticommutative algebra to the subalgebra of K(V^{∗})^{G} by mapping the variable ρ_{i} to the polynomial ρ_{i} of P (V^{∗})^{G} and dρ_{i} to the differential of ρ_{i} in K(V^{∗}) is injective.

If I = (i_{1}, …, i_{k}) is an ordered sequence of integers 1 ≤ i_{1} < ⋯ < i_{k} ≤ n, we write dρ_{I} = dρ_{i_{1}}dρ_{i_{2}} … dρ_{i_{k}} and also dx_{I} = dx_{i_{1}}dx_{i_{2}} … dx_{i_{k}}. Let F P (V^{∗}) be the graded field of fractions of P (V^{∗}). Then, F K(V^{∗}) = F P (V^{∗}) ⊗_{P (V^{∗})} K(V^{∗}) is a vector space over F P (V^{∗}) spanned by {dx_{I}}_{I}. And then, {dρ_{I}}_{I} is also a base of F K(V^{∗}).

Assume that deg f_{det}^{−1} = ∑_{i=1}^{n}(deg ρ_{i} − 2). This is the degree of the Jacobian J, hence J = f_{det}^{−1}, up to an invertible of 𝔽_{p}. Let w ∈ K(V^{∗})^{G} be an arbitrary element. We can write w = ∑_{I} w_{I}dρ_{I}, with w_{I} ∈ F P (V^{∗}) and then we will show that actually, for each index I, w_{I} ∈ P (V^{∗}). We choose I_{0} of minimal length such that w_{I_{0}} ≠ 0. Let I′_{0} be the complementary sequence, then

w dρ_{I′_{0}} = w_{I_{0}}dρ_{I_{0}}dρ_{I′_{0}} = ±w_{I_{0}}dρ_{1} … dρ_{n} = ±w_{I_{0}}Jdx_{1} … dx_{n}

is an element of K(V^{∗})^{G}, and, since dx_{1} … dx_{n} is invariant relative to det, w_{I_{0}}J ∈P (V^{∗})_{det^{−1}}^{G} = f_{det}^{−1}P (V^{∗})^{G}. So, our assumption implies that w_{I_{0}} ∈ P (V^{∗})^{G}. Now we can repeat the argument with w − w_{I_{0}}dρ_{I_{0}} ∈ K(V^{∗})^{G}. It follows that each w_{I} belongs to P (V^{∗})^{G} and then w ∈ P [ρ_{1}, …, ρ_{n}] ⊗ E[dρ_{1}, …, dρ_{n}].

Assume otherwise that deg f_{det}^{−1} ≠ ∑_{i=1}^{n}(deg ρ_{i} − 2); that is, J = ιf_{det}^{−1} for some element ι ∈ P (V^{∗})^{G} of positive degree, then

w = \frac{dρ_{1} … dρ_{n}}{ι} = f_{det}^{−1}dx_{1} … dx_{n}

is an element of K(V^{∗})^{G} which does not belong to P [ρ_{1}, …, ρ_{n}] ⊗ E[dρ_{1}, …, dρ_{n}]. □

Example 9.2 (G a non-modular group [3]). If G ≤ GL(V) is a pseudoreflection group of order not divisible by p, then it is known that P (V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] is a polynomial algebra and also that the degree of f_{det}^{−1} is twice the number of pseudoreflections of G. On the other hand, the number of pseudoreflection is G is ∑_{i=1}^{n}(\frac{deg ρ_{i}}{2} − 1). Hence deg f_{det}^{−1} = ∑_{i=1}^{n}(deg ρ_{i} − 2) and then Theorem 9.1 implies

K(V^{∗})^{G} = P [ρ_{1}, …, ρ_{n}] ⊗ E[dρ_{1}, …, dρ_{n}] .

For a group G ≤ GL(V) we denote [x] = {gx | g ∈ G} the orbit of an element x ∈ V^{∗}. The coefficients c_{i} of the polynomial ∏_{y∈[x]}(X − y) = X^{m} + c_{1}X^{m−1} + ⋯ + c_{m−1}X + c_{m} are the Chern classes of the orbit [x] and belong to P (V^{∗})^{G}. The element c_{m} = ∏_{y∈[x]} y is also called the Euler element of [x]. If we choose just one element z_{L} ∈ L ∩ [x] for each 1-dimensional vector subspace L of V^{∗} that intersects the orbit [x] non trivially, E[x] = ∏ z_{L} is the pre-Euler element of the orbit [x], defined up to a non-zero escalar. This is a relative invariant respect a linear character χ of G that we can associate to the orbit [x] by the equation g(E[x]) = χ(g) · E[x], for all g ∈ G. (See [9, 14].)

Example 9.3 (Family 1 in the Clark-Ewing list: Σ_{n+1}). The symmetric group Σ_{n+1} acts on the integral lattice of SU(n + 1) that we can describe as V = ℤ{(t̂_{1} − t̂_{n+1}), (t̂_{2} − t̂_{n+1}), …, (t̂_{n} − t̂_{n+1})} where Σ_{n+1} permutes the letters t̂_{1}, … t̂_{n+1}. Dually, V^{∗} is generated by classes t_{1}, t_{2}, …, t_{n}, and Σ_{n+1} permutes t_{1}, t_{2}, …, t_{n}, t_{n+1} with the relation t_{1} + t_{2} + ⋯ + t_{n} + t_{n+1} = 0.

The orbit of t_{1} is [t_{1}] = {t_{1}, t_{2}, …, t_{n}, t_{n+1}}, and the Chern classes of this orbit obtained as the coefficients of the polynomial ∏_{i=1}^{n+1}(X − t_{i}) are, up to a sign, the generators c_{i} of the invariant ring P (V^{∗})^{Σ_{n+1}} = P [c_{2}, …, c_{n+1}].

The orbit of t_{1} − t_{2} is

[t_{1} − t_{2}] = {(t_{i} − t_{j}) | 1 ≤ i, j ≤ n + 1, i ≠ j}
= {±(t_{i} − t_{j}) | 1 ≤ i ⪇ j ≤ n + 1}
= {±(t_{i} − t_{j}) | 1 ≤ i ⪇ j ≤ n} ∪ {±(t_{1} + ⋯ + 2t_{i} + ⋯ + t_{n}) | 1 ≤ i ≤ n} ,

thus the pre-Euler element associated to this orbit is

E = E[t_{1} − t_{2}] = ∏_{1≤i⪇j≤n}(t_{i} − t_{j}) ∏_{1≤i≤n}(t_{1} + ⋯ + 2t_{i} + ⋯ + t_{n})

except in the particular case n = 2 at p = 3, in which case E[t_{1} − t_{2}] = (t_{1} − t_{2}). We can check that the linear character associated to the pre-Euler element is precisely the determinant (det = det^{−1} in this case) and also that the degree of E, n^{2} + n, coincides with the degree of the jacobian J in all cases except n = 2 at p = 3. Thus for (n, p) ≠ (2, 3), we have

K(V^{∗})^{Σ_{n+1}} = P [c_{2}, …, c_{n+1}] ⊗ E[dc_{2}, …, dc_{n+1}] .

The particular case n = 2 at the prime 3 will be considered in next Example 9.4.

Example 9.4 (Σ_{3} at the prime 3). The integral lattice of SU(3) is π_{2}(T_{SU(3)}) = ℤ{(t̂_{1} − t̂_{3}), (t̂_{2} − t̂_{3})} with the action of Σ_{3} that permutes t̂_{1}, t̂_{2}, and t̂_{3}. If Σ_{3} is generated by the 3-cycle σ and the transposition τ, the representation afforded by π_{2}(T_{SU(3)}) is determined by

σ ↦ (\begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix}) , τ ↦ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

The dual action in mod 3 cohomology V^{∗} = H^{2}(BT_{SU(3)}; 𝔽_{3}) = 𝔽_{3}{t_{1}, t_{2}} gives P (V^{∗})^{Σ_{3}} ≅ P [x_{4}, x_{6}], where x_{4} = t_{1}^{2} + t_{1}t_{2} + t_{2}^{2} and x_{6} = t_{1}t_{2}(t_{1} + t_{2}). This is the particular case of Example 9.3 with n = 2 at the prime 3.

The action extends to K(V^{∗})) = P [t_{1}, t_{2}] ⊗ E[dt_{1}, dt_{2}] where we obtain invariant elements

y_{3} = dx_{4} = (t_{2} − t_{1})dt_{1} + (t_{1} − t_{2})dt_{2}
y_{5} = dx_{6} = (t_{2}^{2} − t_{1}t_{2})dt_{1} + (t_{1}^{2} − t_{1}t_{2})dt_{2}

and

y_{4} = (t_{2} − t_{1})dt_{1}dt_{2}

so that

y_{3}y_{5} = (t_{1}^{2} + t_{1}t_{2} + t_{2}^{2})(t_{2} − t_{1})dt_{1}dt_{2} = x_{4}y_{4} .

These elements together with the polynomial invariants generate the invariant ring K(V^{∗})^{Σ_{3}}:

K(V^{∗})^{Σ_{3}} ≅ \frac{P [x_{4}, x_{6}] ⊗ E[y_{3}, y_{4}, y_{5}]}{(y_{3}y_{5} − x_{4}y_{4}, y_{3}y_{4}, y_{4}y_{5})} .  (26)

The proof follows the method of Theorem 9.1. In this particular case 1, dt_{1}, dt_{2}, dt_{1}dt_{2} is a basis of K(V^{∗}) as a free P (V^{∗})-module, while 1, y_{3}, y_{5}, y_{3}y_{5} or 1, y_{3}, y_{4}, y_{5} are basis of F K(V^{∗}) as graded F P (V^{∗}) vector spaces. Assume that w is any element of K(V^{∗})^{Σ_{3}}.

We can write w = w_{0} + w_{1}y_{3} + w_{2}y_{4} + w_{3}y_{5}. First, multiply this equality by y_{4}: wy_{4} ∈ K(V^{∗})^{Σ_{3}} and wy_{4} = w_{0}y_{4} = w_{0}(t_{2} − t_{1})dt_{1}dt_{2}. Then, w_{0}(t_{2} − t_{1}) ∈ P (V^{∗})_{det^{−1}}^{Σ_{3}} = (t_{2} − t_{1})P (V^{∗})^{Σ_{3}}, hence w_{0} ∈ P (V^{∗})^{Σ_{3}}.

Next w′ = w − w_{0} = w_{1}y_{3} + w_{2}y_{4} + w_{3}y_{5} ∈ K(V^{∗})^{Σ_{3}}. We now multiply this equality by x_{4}^{−1}y_{5} ∈ K(V^{∗})^{Σ_{3}}: w′x_{4}^{−1}y_{5} ∈ K(V^{∗})^{Σ_{3}} and w′x_{4}^{−1}y_{5} = w_{1}x_{4}^{−1}y_{3}y_{5} = w_{1}y_{4}, and then again the equality w_{1}y_{4} = w_{1}(t_{2} − t_{1})dt_{1}dt_{2} ∈ K(V^{∗})^{Σ_{3}} implies that w_{1} ∈ P (V^{∗})^{Σ_{3}}.

Similarly, we obtain that also w_{2}, w_{3} ∈ P (V^{∗})^{Σ_{3}}, hence w belongs to the ring generated by x_{4}, x_{6}, y_{3}, y_{4}, y_{5}. This proves the isomorphism (26).

Example 9.5 (Family 2a in the Clark-Ewing list: G = G(m, r, n), r|m|p−1, [9]). G(m, r, n) is the subgroup of GL_{n}(ℤ_{p}) generated by the permutation matrices and the diagonal matrices diag(θ_{1}, …, θ_{n}), where θ_{i}^{m} = 1 and (θ_{1} … θ_{n})^{\frac{m}{r}} = 1. In particular, G(m, 1, n) is isomorphic to the semidirect product (ℤ/m)^{n} ⋊ Σ_{n}. In this case we clearly have P (V^{∗})^{G(m,1,n)} = P [ρ_{1}, …, ρ_{n}], where 1 + ρ_{1} + ⋯ + ρ_{n} = ∏_{i=1}^{n}(1 + x_{i}^{m}), if we write P (V^{∗}) = P [x_{1}, …, x_{n}]. Now, ρ_{n} = (x_{1} … x_{n})^{m} is the Euler element associated to the orbit of x_{1}, [x_{1}]. The pre-Euler element is E_{1} = E[x_{1}] = x_{1} … x_{n}. It carries an associated linear character χ_{1}, defined by χ_{1}(diag(θ_{1}, …, θ_{n})) = θ_{1} … θ_{n} and χ_{1}(σ) = 1 if σ ∈ Σ_{n} is a permutation matrix. Notice that

G(m, r, n) = Ker χ1^{\frac{m}{r}} and

P (V^{∗})^{G(m,r,n)} = P [ρ_{1}, …, ρ_{n−1}, E1^{\frac{m}{r}}] .

The orbit of (x_{1} − x_{2}) is [x_{1} − x_{2}] = { θ_{1}x_{i} − θ_{2}x_{j} | θ_{1}^{m} = θ_{2}^{m} = 1 , i < j } and its pre-Euler element is E_{2} = ∏_{i<j}(x_{i}^{m} − x_{j}^{m}). In this case the associated character is χ_{2} defined by χ_{2}(diag(θ_{1}, …, θ_{n})) = 1 and χ_{2}(σ) = sg(σ) is the sign of the permutation. We clearly have det = χ_{1}χ_{2} and then det^{−1} = χ_{1}^{\frac{m}{r} −1}χ_{2}. It follows that f_{det}^{−1} = E_{1}^{\frac{m}{r} −1}E_{2}. Counting degrees, we

obtain ∑_{i=1}^{n−1}(deg ρ_{i}−2)+deg(E1^{\frac{m}{r}})−2 = ∑_{i=1}^{n−1}(2im−2)+2n\frac{m}{r} −2 = n(n−1)m+2n(\frac{m}{r} −1) =

deg f_{det}^{−1}. Hence, Theorem 9.1 implies

K(V^{∗})^{G(m,r,n)} = P [ρ_{1}, …, ρ_{n−1}, E1^{\frac{m}{r}}] ⊗ E[dρ_{1}, …, dρ_{n−1}, d(E1^{\frac{m}{r}})] .

Example 9.6 (G_{12}, G_{29}, G_{31}, and G_{34} in the Clark-Ewing list at modular primes). The groups G_{12} (rank 2, p=3), G_{29} (rank 4, p=5), G_{31} (rank 4, p=5), and G_{34} (rank 6, p=7), of the Clark-Ewing list have polynomial invariants [1, 2, 62].

We obtain by direct calculation that the generator of the det^{−1}-relative invariants f_{det}^{−1} has the same degree as the corresponding jacobian in cases G_{29}, G_{31}, and G_{34}, and then Theorem 9.1 applies.

The case G_{12} = GL(2, 3) is special. Notice that all those groups contain a copy of the symmetric group of the same rank affording the representation of Example 9.3. G_{12} contains Σ_{3} as described in Example 9.4. The invariant ring K(V^{∗})^{GL(2,3)} was computed by Mui [50] (alternatively, use the arguments in Example 9.4):

K(V^{∗})^{GL(2,3)} ≅ \frac{P [x_{12}, x_{16}] ⊗ E[y_{10}, y_{11}, y_{15}]}{(y_{11}y_{15} − x_{16}y_{10}, y_{10}y_{11}, y_{10}y_{15})}

where

x_{12} = \begin{matrix}| & \begin{matrix}t_{1} & t_{2} \\ t_{1}^{9} & t_{2}^{9} \end{matrix} & | \\ | & \begin{matrix}t_{1} & t_{2} \\ t_{1}^{3} & t_{2}^{3} \end{matrix} & | \end{matrix} , x_{16} = P^{1}(x_{12}) = \begin{matrix}| & \begin{matrix}t_{1}^{3} & t_{2}^{3} \\ t_{1}^{9} & t_{2}^{9} \end{matrix} & | \\ | & \begin{matrix}t_{1} & t_{2} \\ t_{1}^{3} & t_{2}^{3} \end{matrix} & | \end{matrix} ,

y_{11} = dx_{12}, y_{15} = dx_{16}, and y_{10} is defined by the relation y_{11}y_{15} = x_{16}y_{10}.

We can easlily obtain the description of the inclusion

K(V^{∗})^{GL(2,3)} ↪ K(V^{∗})^{Σ_{3}}

as

R: \frac{P [x_{12}, x_{16}] ⊗ E[y_{10}, y_{11}, y_{15}]}{(y_{11}y_{15} − x_{16}y_{10}, y_{10}y_{11}, y_{10}y_{15})} ⟶ \frac{P [x_{4}, x_{6}] ⊗ E[y_{3}, y_{4}, y_{5}]}{(y_{3}y_{5} − x_{4}y_{4}, y_{3}y_{4}, y_{4}y_{5})}

mapping

x_{12} ↦ x_{4}^{3} + x_{6}^{2} ,
x_{16} ↦ x_{6}^{2}x_{4} ,
y_{15} ↦ x_{6}^{2}y_{3} − x_{4}x_{6}y_{5} ,
y_{11} ↦ −x_{6}y_{5} , and
y_{10} ↦ x_{6}y_{4} .

Let X be a connected non-modular p-compact group; that is, a p-compact group for which p does not divide the Weyl group order. Models for those p-compact groups were constructed by Clark-Ewing [16]. If W_{X} is the Weyl group of X, the action of W_{X} on the maximal torus T_{X} is determined by the induced representation ρ: W_{X} → GL_{n}(ℤ_{p}), where n is the rank of X. This representation gives W_{X} the structure of a pseudo-reflection group, thus product of irreducibles listed in [16]. It turns out that BX ≃ (BT_{hW_{X}})_{p}^{∧}, where the action of W_{X} on BT is given by ρ [20]. Our next result is a similar description of X(q), for q ≡ 1 mod p.

Theorem 9.7. Let X be a connected non-modular p-compact group and q ≡ 1 mod p, q ≠ 1, then

BX(q) ≃ ((BT_{X}(q))_{hW_{X}})_{p}^{∧} ≃ B(T_{ℓ}^{n} ⋊ W_{X})_{p}^{∧}

with T_{ℓ}^{n} ≅ (ℤ/p^{ℓ})^{n}, where n is the rank of X and ℓ = ν_{p}(q − 1).

Proof. In Proposition 7.5 we have obtained a map BT_{ℓ}^{n} ⟶^{≃} Map(BV, BX(q))_{Bi} → BX(q) and according to Proposition 7.6 we have a factorization

\xymatrix{BT_{ℓ}^{n} ≃ Map(BV, BX(q))_{Bi} \ar[r] & (Map(BV, BX(q))_{Bi})_{hW_{X}} \ar[r] & BX(q) .}

The induced maps in cohomology are

\xymatrix{H^{∗}(BX(q); 𝔽_{p}) \ar[r] & H^{∗}((BT_{ℓ}^{n})_{hW_{X}}) \ar[r]^{≅} & H^{∗}(BT_{ℓ}^{n})^{W_{X}}}

where the second arrow is an isomorphism because the order of W_{X} is coprime to p and the composition is a monomorphism by Theorem E.

According to theorems E and 9.1, H^{∗}(BX(q); 𝔽_{p}) and H^{∗}(BT_{ℓ}^{n})^{W_{X}} has the same Poincare series, hence H^{∗}(BX(q); 𝔽_{p}) ≅ H^{∗}((BT_{ℓ}^{n})_{hW_{X}}) and the result follows. □

Example 9.8 (BSU(2) at odd primes). The Weyl group of SU(2) is ℤ/2 that acts on the maximal torus S^{1} ⊂ ℂ by sign change, that is, as ψ^{−1}. In particular, it is non-modular, and Theorem 9.7 applies. All spaces will be considered completed at p.

Let ψ^{q} be an Adams map of exponent q ∈ ℤ_{p}^{∗}, q ≠ 1. For q ≡ 1 mod p, define ℓ = ν_{p}(1 − q), and then BSU(2)(q) has maximal finite torus ℤ/p^{ℓ} and Weyl group ℤ/2, acting by sign change, so

BSU(2)(q) ≃ (Bℤ/p^{ℓ})_{hℤ/2}

is an equivalence at the prime p. In case q ≡ −1 mod p, we can write ψ^{q} = ψ^{−1} ∘ ψ^{−q}, with ψ^{−1} in the Weyl group and −q ≡ 1 mod p, in which case we should define ℓ = ν_{p}(1 + q), and the above equivalence holds.

Notice that if q ≢ ±1 mod p, then we can write ψ^{q} = ψ^{ζ} ∘ψ^{q′}, where ζ is a (p−1)th root of 1, different than ±1, and q′ ≡ 1 mod p. Then, by Proposition 6.2, BSU(2)(q) ≃ BSU(2)^{hψ^{ζ}}(q′), and according to Proposition 5.8 (see 5.11), BSU(2)^{hψ^{ζ}} is trivial, hence BSU(2)(q)_{p}^{∧} is also trivial.

For q a prime power, coprime to p, SU(2)(q) is equivalent at p to the finite Chevalley group SU_{2}(q). This agrees with the above calculations, for in any case ℓ = ν_{p}(1 − q^{2}).

Example 9.9 (Sullivan spheres S^{2m−1}, m | p − 1). This generalizes the previous example. When m divides p − 1, the cyclic group C_{m} of order m acts on ℤ/p^{∞}. The Sullivan sphere BS^{2m−1} is the p-completion of the classifying space for the semi-direct product ℤ/p^{∞} ⋊ C_{m} for this action. The mod p cohomology of BS^{2m−1} is polynomial on a single generator in degree 2m.

The maximal torus of S^{2m−1} is S^{1} and the Weyl group is cyclic C_{m}, identified with the mth roots of 1 in ℤ_{p}. For q ∈ ℤ_{p}^{∗}, we obtain

S^{2m−1}(q) = \begin{cases}∗ & ℓ = 0 \\ S^{2m−1}(1 + p^{ℓ}) & 0 < ℓ < ∞ \\ S^{2m−1}(1) & ℓ = ∞, \end{cases}

where ℓ = ν_{p}(1 − q^{m}), S^{2m−1}(1 + p^{ℓ}) ≃ B(ℤ/p^{ℓ} ⋊ C_{m})_{p}^{∧}, and S^{2m−1}(1) ≃ ΛS^{2m−1}. In fact, if we write q = ζ · q′, with q′ ≡ 1 mod p and ζ a root of 1 in ℤ_{p}, then

S^{2m−1}(q) ≅ (S^{2m−1})^{h⟨ζ⟩}(q′)

by Proposition 6.2. Since q^{m} ≡ 1 mod p if and only if ζ ∈ C_{m} is an mth root of unity, we have,

(i) for 0 = ν_{p}(1 − q^{m}), ζ ∉ C_{m} and (S^{2m−1})^{h⟨ζ⟩} is contractible (see 5.11), (ii) for 0 < ℓ = ν_{p}(1 − q^{m}) < ∞, ζ ∈ C_{m}, hence (S^{2m−1})^{h⟨ζ⟩} ≅ S^{2m−1} (see 5.11), and

therefore S^{2m−1}(q) ≅ S^{2m−1}(q′). Moreover, ℓ = ν_{p}(1 − q′) and Theorem 9.7 implies that BS^{2m−1}(q′) ≃ B(ℤ/p^{ℓ}⋊C_{m})_{p}^{∧}. Notice that this result does only depend on ℓ = ν_{p}(1−q^{m}), hence also BS^{2m−1}(1 + p^{ℓ}) ≃ B(ℤ/p^{ℓ} ⋊ C_{m})_{p}^{∧},

(iii) and finally, if ν_{p}(1 − q^{m}) = ∞, we have 1 = q^{m}: q = ζ ∈ C_{m} is itself an mth root of 1, so q′ = 1, and BS^{2m−1}(q) ≅ BS^{2m−1}(1) ≃ ΛBS^{2m−1}.

Example 9.10 (SU(3)(q) at the prime 3). Fix q a 3-adic integer with 0 < ℓ = ν_{3}(1−q) < ∞. According to Theorem E

H^{∗}(SU(3)(q); 𝔽_{3}) ≅ P [x_{4}, x_{6}] ⊗ E[y_{3}, y_{5}] ,

with β_{(ℓ)}(y_{3}) = x_{4} and β_{(ℓ)}(y_{5}) = x_{6}.

According to propositions 7.5 and 7.6, T_{ℓ}^{2} ≅ (ℤ/3^{ℓ})^{2} is the maximal finite torus of SU(3)(q) with Weyl group Σ_{3}. Now, the invariant ring

H^{∗}(T_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}} ≅ \frac{P [x_{4}, x_{6}] ⊗ E[y_{3}, y_{4}, y_{5}]}{(y_{3}y_{5} − x_{4}y_{4}, y_{3}y_{4}, y_{4}y_{5})}

computed in Example 9.4, turns out to differ from H^{∗}(SU(3)(q); 𝔽_{3}). The natural map H^{∗}(SU(3)(q); 𝔽_{3}) ↪ H^{∗}(T_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}} (see Theorem E) has cokernel isomorphic to P [x_{6}]y_{4}.

Some interesting examples involve the group SU(3) or the spaces BSU(3)(q) and in this cases Propostion 7.9 will not apply. Instead, we need to develope ad hoc techniques in order to obtain mod p homology decompositions of such spaces.

Given a finite group G and subgroups H_{1}, H_{2}, …, H_{k} ≤ G, we define a finite category 𝕀(k) with objects {0; 1, 2, …, k}, where G is the group of automorphisms of 0 and for each i > 0, H_{i}\G = Hom_{𝕀(k)}(i, 0) as G-sets and Aut_{𝕀(k)}(i) = N_{G}(H_{i})/H_{i}. We will write 𝕀_{i} for the full subcategory with objects 0 and i. Those categories appear in the context of the Aguadé p-compact groups and other compact Lie groups, as Quillen categories of elementary abelian subgroups (see [1, 48]). Next result is essentially contained in [1, 48].

Proposition 9.11. Let M be a given diagram of ℤ_{p}-modules index by the category 𝕀(k). Assume that

(a) Restriction gives an isomorphism H^{j}(G; A) ≅ H^{j}(H_{1}; A), for any ℤ_{(p)}G-module A and j ≥ 1.

(b) p ∤ |N_{G}(H_{i})| and M_{i} = M_{0}^{H_{i}}, for every i ≥ 2.

Then, there is an exact sequence

0 → \underleftarrow{lim}^{0}_{𝕀(k)} M → M_{1}^{N_{G}(H_{1})/H_{1}} ⊕ M_{0}^{G} → M_{0}^{N_{G}(H_{1})} → \underleftarrow{lim}^{1}_{𝕀(k)} M → 0 ,

and \underleftarrow{lim}^{j}_{𝕀(k)} M = 0 if j ≥ 2.

Proof. We consider a star-shaped category 𝕀(k) with k + 1 objects {0, 1, 2, …, k}. There is an exact sequence of the form [48]

0 →lim^{0} M → M_{0}^{G} × ∏_{i>0} M_{i}^{N_{G}(H_{i})/H_{i}} → ∏_{i>0} M_{0}^{N_{G}(H_{i})}
→lim^{1} M → H^{1}(G; M_{0}) × ∏_{i>0} H^{1}(N_{G}(H_{i})/H_{i}; M_{i}) → ∏_{i>0} H^{1}(N_{G}(H_{i}); M_{0})
→lim^{2} M → H^{2}(G; M_{0}) × ∏_{i>0} H^{2}(N_{G}(H_{i})/H_{i}; M_{i}) → ∏_{i>0} H^{2}(N_{G}(H_{i}); M_{0})
→lim^{3} M → ⋯

Under condition (b) this exact sequence reduces to the exact sequence

0 →lim^{0} M → M_{0}^{G} × M_{1}^{N_{G}(H_{1})/H_{1}} → M_{0}^{N_{G}(H_{1})}
→lim^{1} M → H^{1}(G; M_{0}) × H^{1}(N_{G}(H_{1})/H_{1}; M_{1}) → H^{1}(N_{G}(H_{1}); M_{0})
→lim^{2} M → H^{2}(G; M_{0}) × H^{2}(N_{G}(H_{1})/H_{1}; M_{1}) → H^{2}(N_{G}(H_{1}); M_{0})
→lim^{3} M → ⋯

Condition (a) implies that H_{1} and G have the same Sylow p-subgroup. Hence p ∤ |N_{G}(H_{1})/H_{1}| and so therefore H^{∗}(N_{G}(H_{1}); A) ≅ H^{∗}(H_{1}; A)^{N_{G}(H_{1})/H_{1}}. Now, in the diagram given by restrictions H^{j}(G; A) → H^{j}(N_{G}(H_{1}); A) → H^{j}(H_{1}; A), j ≥ 1, the composition is an isomorphism and the second arrow is a monomorphism, hence both arrows are isomorphisms:

H^{j}(G; A) ≅ H^{j}(N_{G}(H_{1}); A) ≅ H^{j}(H_{1}; A) , j ≥ 1 ,

and the Proposition follows. □

Example 9.12 (G_{2} at the prime 3). The exceptional Lie group G_{2} has rank two and the Weyl group is dihedral D_{12}, listed in family 2b for m = 6 in the Clark-Ewing list. The category F_{3}^{e}(G_{2}) of non-trivial elementary abelian 3-subgroups of G_{2} is equivalent to the category 𝕀(2), with G = D_{12}, the Weyl group of G_{2}, H_{1} = σ_{3}, and H_{2} = Σ_{2}. The centralizer diagram for elementary abelian 3-subgroups is equivalent to

[figure: BSU_{3}; ℤ/2; BT^{2}; (D)^{op}; (Σ_{3})^{op}\(D)^{op}; (Σ_{2})^{op}\(D)^{op}; BU_{2}; ℤ/2]

and it contains SU(3) as one of the centralizers, hence Propostion 7.9 does not apply to BG_{2}(q) at the prime 3 (see Example 9.10). We will see how Proposition 9.11 instead, implies that the centralizers diagram for elementary abelian 3-subgroups (diagram (27) below) is in fact a sharp homology decomposition for BG_{2}(q) at the prime 3.

The cohomology of BG_{2} at the prime 3 is H^{∗}(BG_{2}; 𝔽_{3}) ≅ H^{∗}(BT^{2}; 𝔽_{3})_{12}^{D} ≅ P [x_{4}, x_{12}]. Fix q a 3-adic integer with 0 < ℓ = ν_{3}(1 − q) < ∞. According to Theorem E

H^{∗}(SU(3)(q); 𝔽_{3}) ≅ P [x_{4}, x_{12}] ⊗ E[y_{3}, y_{11}] ,

with β_{(ℓ)}(y_{3}) = x_{4} and β_{(ℓ)}(y_{11}) = x_{12}.

On the other hand, according to Corollary 7.8 the categories of non-trivial elementary abelian 3-subgroups of G_{2} and G_{2}(q) coincide: F_{3}^{e}(G_{2}(q)) ≅ F_{3}^{e}(G_{2}), and furthermore, for every object (E, ν) of F_{p}^{e}(G_{2}), BC_{G_{2}(q)}(E, ν) ≃ BC_{G_{2}}(E, ν)(q), thus the centralizer diagram of elementary abelian subgroups of G_{2}(q) is equivalent to

[figure: BSU_{3}(q); ℤ/2; BT_{ℓ}^{2}; (D)^{op}; (Σ_{3})^{op}\(D)^{op}; (Σ_{2})^{op}\(D)^{op}; BU_{2}(q); ℤ/2]  (27)

and there is a natural map hocolim_{F_{p}^{e}(G_{2}(q))^{op}} BC_{G_{2}(q)} → BG_{2}(q). We will see by direct computation that this map is in fact a sharp homology decomposition.

Notice that H^{∗}(BU(2)(q); 𝔽_{3}) ≅ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{2}}, and then, by Proposition 9.11, there is an exact sequence

0 → \underleftarrow{lim}^{0}_{F_{p}^{e}(G_{2}(q))} H^{∗}(BC_{G_{2}(q)}; 𝔽_{3}) → H^{∗}(BSU(3)(q); 𝔽_{3})^{ℤ/2} ⊕ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{D_{12}}

→ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}×ℤ/2} → \underleftarrow{lim}^{1}_{F_{p}^{e}(G_{2}(q))} H^{∗}(BC_{G_{2}(q)}; 𝔽_{3}) → 0 ,

and \underleftarrow{lim}^{i}_{F_{p}^{e}(G_{2}(q))} BC_{G_{2}(q)} = 0 if i ≥ 2, where Σ_{3} × ℤ/2 ≅ N_{D_{12}}(Σ_{3}) = D_{12}. It clearly follows that

\underleftarrow{lim}^{0}_{F_{p}^{e}(G_{2}(q))} H^{∗}(BC_{G_{2}(q)}; 𝔽_{3}) ≅ H^{∗}(BSU(3)(q); 𝔽_{3})^{ℤ/2} ≅ P [x_{4}, x_{12}] ⊗ E[y_{3}, y_{11}]

with x_{12} = x_{6}^{2} and y_{11} = x_{6}y_{5} in H^{∗}(BSU(3)(q); 𝔽_{3}).

The Bousfield-Kan spectral sequence

\underleftarrow{lim}^{i}_{F_{p}^{e}(G_{2}(q))} H^{j}(BC_{G_{2}(q)}; 𝔽_{3}) ⟹ H^{i+j}( hocolim_{F_{p}^{e}(G_{2}(q))^{op}} BC_{G_{2}(q)}; 𝔽_{3})

collapses to the isomorphism

H^{∗}( hocolim_{F_{p}^{e}(G_{2}(q))^{op}} BC_{G_{2}(q)}; 𝔽_{3}) ≅ \underleftarrow{lim}^{0}_{F_{p}^{e}(G_{2}(q))} H^{∗}(BC_{G_{2}(q)}; 𝔽_{3}) ≅ P [x_{4}, x_{12}] ⊗ E[y_{3}, y_{11}] ;

in other words, hocolim_{F_{p}^{e}(G_{2}(q))^{op}} BC_{G_{2}(q)} → BG_{2}(q) is a sharp homology decomposition at the prime 3 and

H^{∗}(G_{2}(q); 𝔽_{3}) ≅ \underleftarrow{lim}^{0}_{F_{p}^{e}(G_{2}(q))} H^{∗}(BC_{G_{2}(q)}; 𝔽_{3}) ≅ P [x_{4}, x_{12}] ⊗ E[y_{3}, y_{11}] .

Example 9.13 (G_{2} at odd primes). We will now complete the description of G_{2}(q) at odd primes. In the previous example we have describe it at the prime 3. At primes bigger than 3, G_{2} turns out to be a connected non-modular p-compact group. Recall that the exceptional Lie group G_{2} has rank two and the Weyl group is dihedral D_{12}, listed in family 2b for m = 6 in the Clark-Ewing list. Let p be an odd prime and q a p-adic unit. We will distinguish three cases:

(1) q^{2} ≡ 1 mod p, q^{2} ≠ 1: In this case F_{p}^{e}(G_{2})(q) = F_{p}^{e}(G_{2}), in particular the p-rank of BG_{2}(q) is two again, and its cohomology can be derived from Theorem E.

(2) q^{2} ≢ 1 mod p, q^{6} ≡ 1 mod p, q^{6} ≠ 1: The element of order 3 in W (G_{2}) has a 1-dimensional eigenspace of eigenvalue 2 in L(G_{2}). Thus ℤ/3 × ℤ/2 = ℤ/6 acts on this eigenspace. We get an embedding Ň(S^{11}) → Ň(G_{2}) and hence a monomorphism S^{11} → G_{2} inducing BS^{11} ≃ (BG_{2})^{hψ^{ζ}}, where ζ is a 3rd primitive root of 1. Then, if we write q = ζq′, with q′ ≡ ±1 mod p, we have BG_{2}(q) ≃ (BG_{2})^{hψ^{ζ}}(q′) ≃ BS^{11}(q′) = BS^{11}(q), this last equality because ±ζ belongs to the Weyl group of S^{11}. Now the p-rank of BG_{2}(q) ≃ BS^{11}(q) is one and the cohomology ring follows from Theorem E.

(3) q^{6} ≢ 1 mod p: In this case q = ζq′ with q′ ≡ 1 mod p and ζ is a primitive root of one whose order does not divide 6. It follows from Proposition 5.8 that (BG_{2})^{hψ^{ζ}} ≃ ∗, hence BG_{2}(q) is as well contractible.

In case q^{2} = 1, G_{2}(q) is the free loop space ΛG_{2}, while for q^{2} ≠ 1 and q^{6} = 1, we have G_{2}(q) ≃ ΛS^{11}.

This result provides the geometric explanation of Kleinermann’s computation of cohomology rings of finite Chevalley groups of type G_{2} (see [33]).

10. Finite Chevalley versions of Aguadé exotic p-compact groups

In [1], Aguadé constructed the exotic p-compact groups X_{i}, i = 12, 29, 31, 34, with Weyl groups the groups G_{12} (rank 2, p = 3), G_{29} (rank 4, p = 5), G_{31} (rank 4, p = 5), and G_{34} (rank 6, p = 7), on the Sheppard-Todd and Clark-Ewing lists, respectively. All four of them are obtained as the homotopy colimit of a diagram that we proceed by describing.

Write G_{i} to denote one of the groups G_{12}, G_{29}, G_{31}, or G_{34}, and Z its center, namely, Z ≅ ℤ/2 for G_{12}, Z ≅ ℤ/4 for G_{29}, Z ≅ ℤ/4 for G_{31}, Z ≅ ℤ/6 for G_{34}, in all cases represented by diagonal matrices with entries p − 1 roots of unity. In all four cases we also fix a subgroup isomorphic to Σ_{p}. Then, the index category is the opposite category of 𝕀(1), with two objects 0 and 1 and

Aut_{𝕀(1)}(0) = G_{i} ,
Aut_{𝕀(1)}(1) = N_{G_{i}}(Σ_{p})/Σ_{p} ≅ Z ,
Mor_{𝕀(1)}(1, 0) = Σ_{p}\G_{i} , and
Mor_{𝕀(1)}(0, 1) = ∅ .

The functor assigns BT^{p−1} to 0 and BSU_{p} to 1, up to homotopy, where the center of G_{i}, Z, acts on BSU_{p} via unstable Adams operations. The diagram is described in the following picture

[figure: BSU_{p}; Z; BT^{p−1}; (G_{i})^{op}; (Σ_{p})^{op}\(G_{i})^{op}] .

Each X_{i} is a p-compact group with maximal torus T_{X_{i}} = T^{p−1} and Weyl group W_{X_{i}} = G_{i}. The respective cohomology rings coincide with the invariant rings H^{∗}(BX_{i}; 𝔽_{p}) ≅ H^{∗}(BT_{X_{i}}; 𝔽_{p})^{G_{i}}, and these are the polynomial rings ([1, 2, 62]):

H^{∗}(BX_{12}; 𝔽_{3}) ≅ P [x_{12}, x_{16}] ,
H^{∗}(BX_{29}; 𝔽_{5}) ≅ P [x_{8}, x_{16}, x_{24}, x_{40}] ,
H^{∗}(BX_{31}; 𝔽_{5}) ≅ P [x_{16}, x_{24}, x_{40}, x_{48}] ,
H^{∗}(BX_{34}; 𝔽_{7}) ≅ P [x_{12}, x_{24}, x_{36}, x_{48}, x_{60}, x_{84}] .

Throughout this section we fix an unstable Adams operation ψ^{q} of exponent q ∈ ℤ_{p}^{∗} with q ≡ 1 mod p, q ≠ 1. We will describe the p-local structure of the spaces BX_{i}(q), that have been defined by the homotopy pullback diagram

\xymatrix{BX_{i}(q) \ar[r]^{ι} \ar[d]_{ι} & BX_{i} \ar[d]^{Δ} \\ BX_{i} \ar[r]^{(1,ψ^{q})} & BX_{i} × BX_{i}}

and will show that they are classifying spaces of p-local finite groups. In particular, cases i = 29, 34 provide new exotic examples of p-local finite groups.

The first results on the p-local structure of BX_{i}(q) are given by propositions 7.5 and 7.6. Set ℓ = ν_{p}(1 − q). The maximal elementary abelian p-subgroup of X_{i}, (t_{X_{i}}, ν), factors as a p-subgroup (t_{X_{i}}, g) of X_{i}(q), and the centralizer of this group

C_{X_{i}(q)}(t_{X_{i}}, g) ≃ T_{ℓ}^{p−1} ≅ (ℤ/p^{ℓ})^{p−1}

is the maximal finite torus of X_{i}(q). All elementary abelian p-subgroups of X_{i}(q) factor through this one. Moreover, the Weyl group is W_{X_{i}(q)}(T_{ℓ}^{p−1}) = G_{i}, and the normalizer N_{X_{i}(q)}(T_{ℓ}^{p−1}) = T_{ℓ}^{p−1} ⋊ G_{i} sits in the maximal torus normalizer of X_{i}(q), making homotopy commutative the diagram

\xymatrix{BN_{X_{i}(q)}(T_{ℓ}^{p−1}) \ar[r] \ar[d] & BN_{X_{i}}(T^{p−1}) \ar[d] \\ BX_{i}(q) \ar[r] & BX_{i}.}

Now, we fix the Sylow p-subgroup S = (ℤ/p^{ℓ})^{(p−1)} ⋊ ℤ/p of N_{X_{i}(q)}(T_{ℓ}^{p−1}), generated by T_{ℓ}^{p−1}and a p-cycle of Σ_{p} ≤ G_{i}. We will denote by f: BS → BX_{i}(q) the homotopy monomorphism obtained as the composition BS → BN_{X_{i}(q)}(T_{ℓ}^{p−1}) → BX_{i}(q). Then (S, f) is a p-subgroup of BX_{i}(q), and it will play the role of a Sylow p-subgroup.

Since X_{i}, i = 12, 29, 31, 34, are polynomial p-compact groups, according to Corollary 7.8, ι: BX_{i}(q) → BX_{i} induces an equivalence of categories

ι_{♯}: F_{p}^{e}(BX_{i}(q)) ⟶ F_{p}^{e}(BX_{i}) .

Thus, we obtain that every elementary abelian p-subgroup (E, μ) of BX_{i}(q) factors as a subgroup of t_{X_{i}}: E ≤ t_{X_{i}} and μ ≃ ν|_{BE}. There is a distinguished subgroup ℤ/p ≅ Z ≤ t_{X_{i}} such that, Z ≤ t_{X_{i}} ≤ T_{X_{i}} ≤ SU_{p} ≅ C_{X_{i}}(Z, ν|_{BZ}). If E ≤ t_{X_{i}} is not conjugate to Z in X_{i}, then the centralizer C_{X_{i}}(E, ν|_{BE}) is a p-compact group whose Weyl group, the point-wise stabilizer of E ≤ T_{X_{i}}, W_{X_{i}}(E), has order not divisible by p. In X_{i}(q), we obtain:

Proposition 10.1. There is one conjugacy class of elements of order p in X_{i}(q), (Z, g|_{BZ}), such that the centralizer is

C_{X_{i}(q)}(Z, g|_{BZ}) ≃ SU_{p}(q)

and contains (S, f):

\xymatrix{BS \ar[d]_{Bincl} \ar[dr]^{f} \\ BSU_{p}(q) \ar[r] & BX_{i}(q)}

as Sylow p-subgroup of SU_{p}(q).

If E ≤ t_{X_{i}} represents another conjugacy class of elementary abelian p-subgroups, then

C_{X_{i}(q)}(E, g|_{BE}) ≃ T_{ℓ}^{p−1} ⋊ W_{X_{i}}(E)

where the order of W_{X_{i}}(E) is not divisible by p. Furthermore, the diagram

\xymatrix{BT_{ℓ}^{(p−1)} \ar[r]^{Bincl} \ar[d]_{Bincl} & BS \ar[d]^{f} \\ BC_{X_{i}(q)}(E, g|_{BE}) \ar[r]^{j} & BX_{i}(q)}

is commutative up to homotopy, where j: BC_{X_{i}(q)}(E, g|_{BE}) → BX_{i}(q) is the natural map induced by evaluation.

Proof. For Z ≤ t_{X_{i}}, we have C_{X_{i}(q)}(Z, g|_{BZ}) ≅ SU_{p}(q) by Corollary 7.4.

If E ≤ t_{X_{i}} be another subgroup, not conjugated to Z, then the centralizer in X_{i} is the connected non-modular p-compact group BC_{X_{i}}(E, ν|_{BE}) ≃ B(T_{X_{i}}⋊W_{X_{i}}(E))_{p}^{∧}, and then, first, Corollary 7.4 implies that BC_{X_{i}(q)}(E, g|_{BE}) ≃ BC_{X_{i}}(E, ν|_{BE})(q), and secondly, Theorem 9.7 gives BC_{X_{i}}(E, ν|_{BE})(q) ≃ B(T_{ℓ}^{p−1} ⋊ W_{X_{i}}(E))_{p}^{∧}.

Finally, we use the inclusions BE → Bt_{X_{i}} → BS ⟶^{f} BX_{i}(q) in order to compare the centralizers of E and t_{X_{i}} in S and X_{i}(q):

\xymatrix{BT_{ℓ}^{p−1} ≃ BC_{S}(t_{X_{i}}) \ar[dr]|{≃}^{f_{♯}} \ar[rr]^{≃} &  & BC_{S}(E) \ar[d]^{f_{♯}} \ar[r] & BS \ar[dr]^{f} \\  & BC_{X_{i}(q)}(t_{X_{i}}, g) \ar[r] & BC_{X_{i}(q)}(E, g|_{BE}) \ar[rr] &  & BX_{i}(q) .  □}

Proposition 10.2. For i = 12, 29, 31, 34, the natural map

hocolim_{F_{p}^{e}(BX_{i}(q))^{op}} BC_{X_{i}(q)} → BX_{i}(q)  (28)

is a mod p homology equivalence.

Proof. According to Theorem E the cohomology rings of BX_{i}(q) are:

H^{∗}(BX_{12}(q); 𝔽_{3}) ≅ P [x_{12}, x_{16}] ⊗ E[y_{11}, y_{15}] ,
H^{∗}(BX_{29}(q); 𝔽_{5}) ≅ P [x_{8}, x_{16}, x_{24}, x_{40}] ⊗ E[y_{7}, y_{15}, y_{23}, y_{39}] ,
H^{∗}(BX_{31}(q); 𝔽_{5}) ≅ P [x_{16}, x_{24}, x_{40}, x_{48}] ⊗ E[y_{15}, y_{23}, y_{39}, y_{47}] ,
H^{∗}(BX_{34}(q); 𝔽_{7}) ≅ P [x_{12}, x_{24}, x_{36}, x_{48}, x_{60}, x_{84}] ⊗ E[y_{11}, y_{23}, y_{35}, y_{47}, y_{59}, y_{83}] ,

and they embed in the invariant rings H^{∗}(BX_{i}(q); 𝔽_{p}) ⊆ H^{∗}(BT_{ℓ}^{p−1}; 𝔽_{p})^{G_{i}}. These invariant rings are described in the Example 9.6. It turns out that the above inclusion is an isomorphism if i = 29, 31, 34, but it is not surjective i = 12.

The centralizers of elementary abelian p-subgroups of BX_{i}(q) are described in Proposition 10.1. The centralizer, C_{X_{i}(q)}(E, g|_{BE}), of an elementary abelian p-subgroup E ≤ t_{X_{i}} in X_{i}(q) is either SU_{p}(q) or a non-modular p-compact group.

In cases i = 29, 31, 34, H^{∗}(C_{X_{i}(q)}(E, g|_{BE}); 𝔽_{p}) ≅ H^{∗}(BT_{X_{i}}; 𝔽_{p})^{W (E)} is satisfied by Theorem E and examples 9.2 and 9.3, hence we meet the conditions of Proposition 7.9 and the map (28) is a mod p homology equivalence.

In the case i = 12, the Proposition 7.9 does not apply, so we will need a separate analysis. The p-compact group X_{12}, p = 3, is also denoted DI_{2} = X_{12}, because G_{12} ≅ GL(2, 3) and H^{∗}(BDI_{2}; 𝔽_{3}) ≅ H^{∗}(BT^{2}; 𝔽_{3})^{GL(2,3)} ≅ 𝔽_{3}[x_{12}, x_{16}] is the rank two Dickson algebra at p = 3. It admits two conjugacy classes of elementary abelian p-subgroups, one of rank one and another of rank two, hence so does BDI_{2}(q), as well. We have equivalences of categories

F_{p}^{e}(BDI_{2}) ≅ F_{p}^{e}(BDI_{2}(q)) ≅ 𝕀(1)

with Aut_{𝕀(1)}(0) = GL(2, 3), Aut_{𝕀(1)}(1) = N_{GL(2,3)}(Σ_{3})/Σ_{3} ≅ ℤ/2, where N_{GL(2,3)}(Σ_{3}) = Σ_{3} × ℤ/2, and Mor_{𝕀(1)}(1, 0) = Σ_{3}\GL(2, 3), Mor_{𝕀(1)}(0, 1) = ∅. The centralizers diagram BC_{DI_{2}(q)} is described in the picture

[figure: BSU_{3}(q); ℤ/2; BT_{ℓ}^{2}; GL(2,3)^{op}; Σ_{3}^{op}\GL(2,3)^{op}] .  (29)

The Bousfield-Kan spectral sequence

E_{2}^{i,j} ≅ \underleftarrow{lim}^{i}_{𝕀(1)} H^{j}(BC_{DI_{2}(q)}; 𝔽_{3}) ⟹ H^{i+j}(hocolim_{𝕀(1)^{op}}BC_{DI_{2}(q)}; 𝔽_{3})

computes the cohomology of the homotopy colimit hocolim_{𝕀(1)^{op}} BC_{DI_{2}(q)}.

The computation of the E_{2}-term follows from Proposition 9.11. Since N_{GL(2,3)}(Σ_{3}) ≅ Σ_{3} × ℤ/2 and H^{∗}(GL(2, 3); A) ≅ H^{∗}(N_{GL(2,3)}(Σ_{3}); A) ≅ H^{∗}(Σ_{3}; A), for any GL(2, 3)-module A, there is an exact sequence

0 → \underleftarrow{lim}^{0}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) → H^{∗}(BSU(3)(q); 𝔽_{3})^{ℤ/2} ⊕ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{GL(2,3)}

→ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}×ℤ/2} → \underleftarrow{lim}^{1}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) → 0 , (30)

and \underleftarrow{lim}^{i}_{𝕀(1)} BC_{DI_{2}(q)} = 0 if i ≥ 2.

The invariant rings H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{GL(2,3)} and H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}} as well as the restriction R: H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{GL(2,3)} ↪ H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}} have been described in examples 9.4 and 9.6. The cohomology of BSU(3)(q) is identified with the subalgebra P [x_{4}, x_{6}] ⊗ E[y_{3}, y_{5}] of H^{∗}(BT_{ℓ}^{2}; 𝔽_{3})^{Σ_{3}}. The cokernel of the inclusion is isomorphic to P [x_{6}]y_{4}, and then the exact sequence (30) is simplified to

0 → \underleftarrow{lim}^{0}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) → \frac{P [x_{12}, x_{16}] ⊗ E[y_{10}, y_{11}, y_{15}]}{(y_{11}y_{15} − x_{16}y_{10}, y_{10}y_{11}, y_{10}y_{15})}

⟶^{R̄} (P [x_{6}]y_{4})^{ℤ/2} → \underleftarrow{lim}^{1}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) → 0 ,

and (P [x_{6}]y_{4})^{ℤ/2} = P [x_{6}^{2}](x_{6}y_{4}) which is in the image of R̄. It follows that

\underleftarrow{lim}^{0}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) ≅ P [x_{12}, x_{16}] ⊗ E[y_{11}, y_{15}]

and \underleftarrow{lim}^{i}_{𝕀(1)} BC_{DI_{2}(q)} = 0 if i ≥ 1, so, therefore the Bousfield-Kan spectral sequence collapses to an isomorphism

H^{∗}(hocolim_{𝕀(1)^{op}}BC_{DI_{2}(q)}; 𝔽_{3}) ≅ \underleftarrow{lim}^{0}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) ≅ P [x_{12}, x_{16}] ⊗ E[y_{11}, y_{15}] ;

that is, hocolim_{𝕀(1)^{op}} BC_{DI_{2}(q)} → BG_{2}(q) is a sharp homology decomposition at the prime 3 and

H^{∗}(DI_{2}(q); 𝔽_{3}) ≅ \underleftarrow{lim}^{0}_{𝕀(1)} H^{∗}(BC_{DI_{2}(q)}; 𝔽_{3}) ≅ P [x_{12}, x_{16}] ⊗ E[y_{11}, y_{15}] .  □

Theorem 10.3. (S, f) is a Sylow p-subgroup for BX_{i}(q), the fusion system F_{(S,f)}(BX_{i}(q)) of the space BX_{i}(q) over the p-subgroup (S, f) is saturated, and

(S, F_{(S,f)}(BX_{i}(q)), L_{(S,f)}(BX_{i}(q)))

is a p-local finite group with classifying space

|L_{(S,f)}(BX_{i}(q))|_{p}^{∧} ≃ BX_{i}(q) .

Proof. It is a consequence of Theorem 4.6, using the above propositions 10.1 and 10.2. □

Now, we will go deeper into the structure of the fusion system F = F_{(S,f)}(BX_{i}(q)). We have seen that the fusion category of elementary abelian p-subgroups is equivalent to that of the p-compact group X_{i}; in particular, every elementary abelian p-subgroup is toral; that is, F-conjugate to a subgroup of T_{ℓ}^{(p−1)}. If we denote Z = Z(S) the center of S, then (10.1) BC_{X_{i}(q)}(Z) = BSU_{p}(q)_{p}^{∧} ≃ BSL_{p}(q)_{p}^{∧}, so, the centralizer fusion system C_{F}(Z) over C_{S}(Z) = S coincides with the fusion system of SL_{p}(q) over S. Hence, we can identify S with the Sylow p-subgroup of SL_{p}(q) and then use the notation of Example 3.6. Recall from 3.6 that any centric radical subgroup of S in C_{F}(Z) is conjugate to either S, T_{ℓ}^{(p−1)}, or an extraspecial group Γ_{1}(ξ^{r}), r = 0, …, p − 1.

Proposition 10.4. Any centric radical subgroup of S in F = F_{(S,f)}(BX_{i}(q)) is conjugate to one of the groups in the table:

Q Out_{F}(Q) Conditions
T_{ℓ}^{(p−1)}  G_{i}
S  ℤ/(p − 1) × ℤ/(p − 1)
Γ_{1}  GL_{2}(p)
Γ_{1}(ξ)  SL_{2}(p)  if ℓ > 1 or p > 3.

(31)

Proof. The proof is divided in four steps, where we first determine a set of representatives for centric radical subgroups of S in F, and then refine it to a minimal set of representatives and compute their automorphisms groups in F.

Step 1: Toral and non toral centric radical subgroups. T_{ℓ}^{p−1} is centric in F and Out_{F}(T_{ℓ}^{p−1}) ≅ G_{i} is p-reduced, hence T_{ℓ}^{p−1} is also radical in F. No other subgroup of T_{ℓ}^{p−1} is centric, so for any other centric and radical subgroup Q ≤ S in F, there is a morphism of extensions

[figure: Q_{0}; Q; ℤ/p; T_{ℓ}^{p−1}; S; ℤ/p]  (32)

where Q_{0} = T_{ℓ}^{p−1} ∩ Q.

We are assuming that Q is centric, hence the center Z ≅ ℤ/p of S should be contained in Q_{0}. But if Q_{0} = Z, then Q ≅ ℤ/p × ℤ/p is elementary abelian and then toral in F, hence it would not be centric. Thus Z ≠ Q_{0} and the center of Q is Z(Q) = Q_{0}^{ℤ/p} = Z. In particular, every automorphism of Q restricts to an automorphism of Z, so we obtain a homomorphism Aut_{F}(Q) → Aut_{F}(Z). The kernel is composed of automorphisms of Q that restrict to the identity in Z; that is, automorphisms of Q in the centralizer fusion system C_{F}(Z), hence we have an exact sequence

1 → Aut_{C_{F} (Z)}(Q) → Aut_{F}(Q) → Aut_{F}(Z)  (33)

where Aut_{F}(Z) ≤ ℤ/p − 1 lifts to Aut_{F}(T_{ℓ}^{(p−1)}) and Aut_{F}(S) as unstable Adams operations (the center of G_{i}). Thus, if Q is radical in C_{F}(Z), then it is radical in F.

Step 2: Non-abelian centric radical subgroups, all of which abelian characteristic subgroups are cyclic. Assume that all abelian characteristic subgroups of Q are cyclic, then a theorem of Hall implies that Q is the central product of an extraspecial group Γ of exponent p and a cyclic group C, where the elements or order p in C, Ω_{1}(C), coincide with the center Z(Γ) of Γ (cf. [29, Chap. 5, 4.9, 5.3]).

The faithful irreducible representations of the central product of an extraspecial group Γ or order p^{1+2r} and a cyclic group of order p^{ℓ} over the algebraic closure of a field of q elements, (q, p) = 1, have degree p^{r}, and there are exactly p^{ℓ−1}(p − 1) inequivalent representations in this degree.

Hence, only the case r = 1 can appear in GL_{p}(q). We denote Γ_{1} the extraspecial group of order p^{3} and exponent p, and Γ_{k} the central product ℤ/p^{k} ∘ Γ_{1}. The different irreducible faithful representations of Γ_{k} in GL_{p}(q) are obtained by composing with the extension to Γ_{k} of the automorphisms of ℤ/p^{k}, (ℤ/p^{k})^{∗}. Thus, there is at most one subgroup isomorphic to Γ_{k} in GL_{p}(q), up to conjugation. A subgroup of GL_{p}(q) isomorphic to Γ_{1} is described in Example 3.5. Since C_{GL_{p}(q)}(Γ_{1}) = Z(GL_{p}(q)) ≅ GL_{1}(q), Γ_{k} is a subgroup of GL_{p}(q) if and only if ℤ/p^{k} < GL_{1}(q). Hence Γ_{ℓ}, ℓ = ν_{p}(1 − q), is the biggest one that can occur in GL_{p}(q) (see Example 3.5).

Finally, the intersection of Γ_{ℓ} with SL_{p}(q), and hence, of any conjugate of Γ_{ℓ}, is isomorphic to Γ_{1}, and there are exactly p conjugacy classes of such subgroups Γ_{1}(ξ^{r}) (see Example 3.6). These are radical in C_{F}(Z), and so, therefore, they are also radical in F.

Step 3: Non-abelian centric radical subgroups having non-cyclic abelian characteristic subgroups. Assume now that Q contains a non-cyclic abelian characteristic group. If Q is radical in C_{F}(Z), then it is radical in F. Now, we assume also that Q is not radical in C_{F}(Z).

We can view Q ≤ S as subgroups of SL_{p}(q) and GL_{p}(q), for an appropriate prime power q such that S is the Sylow p-subgroup of SL_{p}(q): ℓ = ν_{p}(1 − q). Write N = N_{GL_{p}(q)}(Q). The arguments of [4, (4A)] show that (up to conjugacy in GL_{p}(q))

Q ≤ N ∩ (ℤ/p^{k} ≀ ℤ/p) ◁ N

for some k ≤ ℓ, or, taking the intersection with SL_{p}(q)

Q ≤ N̄ ∩ S_{k} ◁ N̄

where S_{k} = (ℤ/p^{k} ≀ ℤ/p) ∩ SL_{p}(q) ≤ S and N̄ = N ∩ SL_{p}(q) = N_{SL_{p}(q)}(Q), an then

Inn Q ≤ (N̄ ∩ S_{k})/Z(Q) ◁ Aut_{C_{F} (Z)}(Q)

where N̄/C_{SL_{p}(q)}(Q) = Aut_{C_{F} (Z)}(Q). We will see that (N̄ ∩ S_{k})/Z(Q) is still normal in Aut_{F}(Q).

Assume that φ ∈ Aut_{F}(Q) restricts to Z as the unstable Adams operation ψ^{ζ}, ζ a (p−1)st root of unity. If ψ^{1/ζ}(Q) = Q′ ≤ S, then ψ^{1/ζ} ∘φ: Q → Q′ is a morphism of F, that restricted to Z is trivial, hence a morphism of C_{F}(Z). Since, we have assumed that Q is not radical in C_{F}(Z), ψ^{1/ζ} ∘φ should be obtained as composition of restrictions of automorphisms of centric radical subgroups of C_{F}(Z), by Alperin fusion theorem [11, A.10]. This is the fusion system of SL_{p}(q), and the Sylow p-subgroup S itself is the only centric radical that contains Q, hence, there is χ ∈ Aut_{C_{F} (Z)}(S) with χ|_{Q} = ψ^{1/ζ} ∘ φ, hence φ = ψ^{ζ} ∘ χ|_{Q} extends to an automorphism ψ^{ζ} ∘ χ of Aut_{F}(S). Notice that ψ^{ζ}(S_{k}) = S_{k} and also χ(S_{k}) = S_{k}, hence, if g ∈ S_{k} normalizes Q, we have φ ∘ c_{g} ∘ φ^{−1} = c_{φ(g)}, with φ(g) ∈ N̄ ∩ S_{k}. This proves that we have

Inn Q ≤ (N̄ ∩ S_{k})/Z(Q) ◁ Aut_{F}(Q)

and since Q is radical in F, Q = S_{k}.

We claim that only the case S_{k} = S is radical. First we compute the normalizer of ℤ/p^{k} ≀ ℤ/p in GL_{p}(q). The subgroup (ℤ/p^{k})^{p} is a characteristic subgroup of ℤ/p^{k} ≀ ℤ/p, for it is the only abelian subgroup of index p, hence, N_{GL_{p}(q)}(ℤ/p^{k} ≀ ℤ/p) ≤ N_{GL_{p}(q)}((ℤ/p^{k})^{p}). It is not difficult to compute N_{GL_{p}(q)}((ℤ/p^{k})^{p}) = GL_{1}(q)≀Σ_{p}, the group of invertible matrices with only one non-trivial entry in each line and column. By direct computation one can obtain that N_{GL_{p}(q)}(ℤ/p^{k} ≀ ℤ/p) = GL_{1}(q) · (ℤ/p^{k} ≀ N_{Σ_{p}}(ℤ/p)), where GL_{1}(q) is identified with the subgroup of all diagonal matrices of GL_{p}(q); that is, the center of GL_{p}(q).

Call N_{k} = N_{GL_{p}(q)}(ℤ/p^{k} ≀ ℤ/p) ∩ SL_{p}(q). We have N_{k} ≅ B_{k} ⋊ N_{Σ_{p}}(ℤ/p), with

B_{k} = { (z · x_{1}, …, z · x_{p}) ∈ GL_{1}(q)^{p} | x_{i} ∈ Z/p^{k}, z^{p}x_{1} … x_{p} = 1 }

and N_{SL_{p}(q)}(S_{k}) = N_{k}. Notice that, when k < ℓ, S_{k} has index p in the Sylow p-subgroup B_{k} ⋊ ℤ/p, and this is normal in N_{k}, hence only S = S_{ℓ} is radical in SL_{p}(q).

The centralizer of S_{k} in SL_{p}(q) is C_{SL_{p}(q)}(S_{k}) = Z ≅ ℤ/p and then Aut_{C_{F} (Z)}(S_{k}) ≅ Aut_{SL_{p}(q)}(S_{k}) ≅ N_{k}/Z. (B_{k}/Z)⋊ℤ/p is normal in N_{k}/Z, and, since the Adams operations ψ^{ζ}, ζ a (p − 1)st root of unity, act internally in B_{k}, (B_{k}/Z) ⋊ ℤ/p is also a normal of Aut_{F}(S_{k}):

Inn S_{k} = S_{k}/ℤ/p ◁ (B_{k}/ℤ/p) ⋊ ℤ/p ◁ Aut_{F}(S_{k})

thus, S_{k} is radical in the fusion system F if and only if k = ℓ; that is, only the case S_{k} = S is radical. In this case we have obtained Aut_{F}(S) ≅ N_{ℓ}/Z ⋊ ℤ/(p − 1), where ℤ/(p − 1) on the right is generated by the Adams operations of exponent a primitive (p − 1)st root of unity, and Out_{F}(S) ≅ ℤ/(p − 1) × ℤ/(p − 1), given by the Adams operations and N_{Σ_{p}}(ℤ/p)/ℤ/p.

Step 4: Minimal set of representatives and automorphism groups. It remains to check which of those are F-conjugate to one of the others in the list and also to compute their Fautomorphisms.

For Q = S the restriction Aut_{F}(Q) → Aut_{F}(Z) is split because unstable Adams operations extend to S. Moreover, since they are realized by the center of G_{i}, the F-automorphisms of S are given by conjugation in the normalizer N_{ℓ,i} of the maximal finite torus T_{ℓ}^{(p−1)}. We have seen already that the same is true for Q = T_{ℓ}^{(p−1)}.

Finally, we analyse the case Q = Γ_{1}(ξ^{r}), r = 0, …, p − 1. Assume that φ ∈ Aut_{F}(Q) and that the restriction to the center Z is the unstable Adams operation ψ^{z}. This extends to an F-automorphism of S. Write Q′ = ψ^{z}(Q). Then χ = ψ^{z} ∘ φ^{(−1)} : Q → Q′ is a homomorphism of F that restricts to the identity in Z, hence it belongs to the centralizer fusion system C_{F}(Z). In other words, every automorphism φ ∈ Aut_{F}(Q) is the composite of an isomorphism χ: Q → Q′ of C_{F}(Z) and a unstable Adams operation ψ^{z}.

It is then enough to compute the effect of unstable Adams operations on the family of subgroups Γ_{1}(ξ^{r}). It turns out that unstable Adams operations restrict to automorphisms of Γ_{1} = Γ_{1}(ξ^{0}) so that Out_{F}(Γ_{1}) = GL_{p}(q), while, for p > 3 or ℓ > 1, they conjugate Γ_{1}(ξ^{r}) for r = 1, …, p − 1 to each other and Out_{F}(Γ_{1}(ξ) = SL_{p}(q). □

Corollary 10.5. The fusion system of BX_{i}(q) is

F_{(S,f)}(BX_{i}(q)) = ⟨ F_{N_{ℓ,i}}(S) ; F_{Γ_{1}}(GL_{2}(p)) , F_{Γ_{1}(ξ)}(SL_{2}(p)) ⟩ ,

for p > 3 or ℓ > 1, and F_{(S,f)}(BX_{12}(q)) = ⟨ F_{N_{1,i}}(S) ; F_{Γ_{1}}(GL_{2}(p)) ⟩, for p = 3 and ℓ = 1, where N_{ℓ,i} = N_{X_{i}(q)}(T_{ℓ}^{(p−1)}) ≅ T_{ℓ}^{(p−1)} ⋊ G_{i}.

Proof. It is a consequence of Proposition 10.4 and Alperin’s fusion theorem for saturated fusion systems (see section 3). □

We end this section with a case by case study in order to determine which spaces BX_{i}(q) are p-completed classifying spaces of finite groups and which cases correspond to exotic examples of p-local finite groups.

We first observe that S contains no proper strongly closed subgroups in F = F_{(S,f)}(BX_{i}(q)) and so, according to [11, 9.2], if F BX_{i}(q) is the p-completed classifying space of a finite group, this group is almost simple.

In fact, a strongly closed subgroup of S in F is a normal subgroup P of S such that no element of P is F-conjugate to any element in S \ P. Now, if P is non trivial it contains at least an element of order p, and this is F-conjugate to an element of order p in T_{ℓ}^{(p−1)}. Now, the maximal elementary abelian p-subgroup t of T_{ℓ}^{(p−1)} turns out to be an irreducible G_{i}-module, hence t ⊂ P and since the cycle of order p generating S/T_{ℓ}^{(p−1)} is conjugate to an element of t, the extension of t by this cycle is in P. Thus we have a diagram of extensions

[figure: P_{T}; P; ℤ/p; T_{ℓ}^{(p−1)}; S; ℤ/p]

where t ≤ P_{T} = P ∩ T. Now S/P ≅ T_{ℓ}^{(p−1)}/P_{T} is abelian. The abelianization of S is seen to be Z/p × ℤ/p, and then we obtain that T_{ℓ}^{(p−1)}/P_{T} is either trivial or has order p. It follows that all elements of order up to p^{ℓ−1} of T_{ℓ}^{(p−1)} belong to P_{T}. Taking the quotient by

this subgroup we obtain an inclusion of G_{i}-modules \overline{P_{T}} ≤ \overline{T_{ℓ}^{(p−1)}}, but again, this last is an

irreducible G_{i}-module, hence \overline{P_{T}} = \overline{T_{ℓ}^{(p−1)}}, and then P = S.

Example 10.6. BX_{29}(q) at p = 5 and BX_{34}(q) at p = 7 are classifying spaces of exotic p-local finite groups.

We have seen that the Sylow subgroup does not contain any proper strongly closed subgroup in F_{(S,f)}(BX_{i}(q)), hence if this is the p-completed classifying space of a finite group G, then G is almost simple [11, 9.2]. A complete list of almost simple groups with a Sylow subgroup of the characteristics of S is provided by [11, Proposition 9.5]. No one in the list contains G_{29} or G_{34} as automorphisms of T_{ℓ}^{(p−1)} induced by conjugation in the group. Hence X_{29}(q) at p = 5 and X_{34}(q) at p = 7 are exotic.

Example 10.7. BX_{12}(q) at p = 3 is the 3-completed classifying space of a twisted Chevalley group of type F_{4}. More precisely, if ℓ = ν_{3}(q^{2} − 1), there is a positive integer n such that also ℓ = ν_{3}(1 + 2^{2n+1}) and then BX_{12}(q) ≃ B(^{2}F_{4}(2^{2n+1}))_{3}^{∧}.

The 3-completed classifying space of the twisted Chevalley group ^{2}F_{4}(2^{2n+1}) can be described at p = 3 as B(^{2}F_{4}(2^{2n+1})) ≃ BF α(F_{4}), for α = φ ∘ ψ^{2^{n}}, where φ is the Friedlander’s exceptional isogeny of F_{4}. φ has the effect of reflecting the Dynkin diagram of F_{4} by sending the short roots to the long roots and the long roots to 2 times short roots. Furthermore, φ^{2} ≃ ψ^{2}, and then we can choose ζ a square root of −2 in ℤ_{3} such that β = φ ∘ ψ^{1/ζ} is a self equivalence of BF_{4} at p = 3 of order two and 2^{n}ζ ≡ 1 mod 3. We can write α = β ∘ ψ^{2^{n}ζ}, and then, by Proposition 6.2, BF α(F_{4}) ≃ (BF_{4})^{hβ}(2^{n}ζ). In [13] it is shown that (BF_{4})^{hβ} ≃ BX_{12}, hence BX_{12}(2^{n}ζ) ≃ B(^{2}F_{4}(2^{2n+1}))_{3}^{∧}. Since ψ^{−1} belongs to the Weyl group of X_{12}, BX_{12}(q) ≃ BX_{12}(−q), and then, according to Theorem C, the homotopy type of BX_{12}(±q) does only depend on ℓ = ν_{3}(q^{2} − 1), thus, if we choose n with ℓ = ν_{3}(q^{2} − 1) = ν_{3}(1 − 2^{n}ζ) = ν_{3}(1 + 2^{2n+1}), then we have

BX_{12}(q) ≃ BX_{12}(2^{n}ζ) ≃ B(^{2}F_{4}(2^{2n+1}))_{3}^{∧} .

The local structure of ^{2}F_{4}(2^{2n+1}), also called Ree groups of characteristic two, was studied by Malle [37].

Example 10.8. For any 5-adic unit, q ∈ ℤ_{5}^{∗}, BX_{31}(q) at p = 5 is the 5-completed classifying space of a Chevalley group of type E_{8}, namely, BX_{31}(q) ≃ BE_{8}(2^{2m+1})_{5}^{∧} if ν_{5}(q^{4} − 1) = ν_{5}(1 + 2^{4m+2}).

Let i = \sqrt{−1} be a primitive 4th root of unity. Since ψ^{i} belongs to the Weyl group of X_{31}, we can assume that q ≡ 1 mod 5 for otherwise we can multiply q by an appropriate power of i and still have BX_{31}(q) ≃ BX_{31}(i^{r}q). Moreover, according to Theorem C, the homotopy type of BX_{31}(q) will only depend on ℓ = ν_{5}(q^{4} − 1).

We fix a prime power q_{0} with q_{0} ≡ ±2 mod 5 and ℓ = ν_{5}(±iq−1) = ν_{5}(q_{0}^{4}−1) = ν_{5}(q_{0}^{2}+1), where we choose +i or −i in order that the equality makes sense.

We can write q_{0} = i · (−i · q_{0}), where now −i · q_{0} ≡ ±1 mod p. Since ψ^{−1} belongs to the Weyl group of E_{8}, we can apply Proposition 6.2 and get BE_{8}(q_{0}) ≃ (BE_{8})^{hψ^{i}}(−iq_{0}). Now we have seen in Example 5.15(2), that (BE_{8})^{hψ^{i}} ≃ BX_{31}, so, therefore

BE_{8}(q_{0}) ≃ BX_{31}(−iq_{0}) ≃ BX_{31}(q_{0}) ,

and this last is homotopy equivalent to BX_{31}(q) by our choice of q_{0} with ν_{5}(q_{0}^{4}−1) = ν_{5}(q^{4}−1).

Similar considerations can be made, more generally, at any prime p such that p ≡ 1 mod 4; that is, any prime at which X_{31} can be defined, and then obtain that BE_{8}(q_{0}) ≃ BX_{31}(q_{0}) for a prime power q_{0} with q_{0}^{2} + 1 ≡ 0 mod p.

The local structure of E_{8}(q) was described in [35].

Remark 10.9. One can easily obtain natural maps BX_{i}(q^{p^{r}}) → BX_{i}(q^{p^{r+1}}), that at the level of maximal finite torii induces an inclusion T_{ℓ+r}^{(p−1)} ≤ T_{ℓ+r+1}^{(p−1)}, and then obtain that the p-compact group X_{i} can be reconstructed by means of a telescope construction

BX_{i} ≃ hocolim_{q} BX_{i}(q) .

In particular, BX_{12} = BDI_{2} and BX_{31} are telescopes of classifying spaces of finite Chevalley groups.

11. Finite Chevalley versions of generalized p-adic Grassmannians

Let p be an odd prime, m ≥ 1, r ≥ 1, and n ≥ 1 with r|m|(p − 1). We denote by diag(a_{1}, …, a_{n}) an n×n matrix with entries a_{1} through a_{n} in the diagonal and zero otherwise. Define the group

A(m, r, n) = \{ diag(a_{1}, …, a_{n}) | a_{i}^{m} = 1, (a_{1} … a_{n})^{\frac{m}{r}} = 1 \} ≤ GL_{n}(ℤ_{p})

and

G(m, r, n) = A(m, r, n)Σ_{n} ≤ GL_{n}(ℤ_{p})

where Σ_{n} is identified with the subgroup of permutation matrices in GL_{n}(ℤ_{p}). Every group G(m, r, n) is a pseudoreflection group.

Each group G(m, r, n) is realized as the Weyl group of a 1-connected p-compact group X(m, r, n), whose cohomology is

H^{∗}(BX(m, r, n); 𝔽_{p}) = H^{∗}(BT (X(m, r, n)); 𝔽_{p})^{G(m,r,n)} ≅ P [x_{1}, …, x_{n−1}, e]

with deg(x_{i}) = 2mi and deg(e) = \frac{2mn}{r}. They are usually referred to as the generalized p-adic Grassmannians. This family of p-compact groups, as we have defined it, includes some classical Grassmannians, namely BX(1, 1, n) ≃ BU(n), BX(2, 2, n) ≃ BSO(2n), and BX(2, 1, n) ≃ BSO(2n + 1). Furthermore, X(m, r, 1) ≃ Ŝ_{p}^{2( \frac{m}{r})−1} are the Sullivan spheres. The existence of X(m, r, n) for other values of m, r, and n is shown in [54, 52].

For m ≥ 2 and n ≥ 2, the groups G(m, r, n) form the family 2a in the list of Clark-Ewing. If n = 1, the groups G(m, r, n) ≅ ℤ/(\frac{m}{r}) are cyclic and appear as family 3 in the Clark-Ewing list, the Weyl groups of the Sullivan spheres. For m = 1, G(1, 1, n) ≅ Σ_{n} is the symmetric group, which is a pseudoreflection group as Weyl group of GL(n, ℂ), or U(n), but as a such it is not irreducible, hence it is not in the Clark-Ewing list. Family 1 in the list corresponds to Σ_{n} as Weyl group of SU(n).

We are interested in the finite Chevalley versions of the generalized p-adic Grassmannians: the spaces BX(m, r, n)(q), defined by the the pullback diagram

\xymatrix{BX(m, r, n)(q) \ar[r]^{ι} \ar[d]_{ι} & BX(m, r, n) \ar[d]^{Δ} \\ BX(m, r, n) \ar[r]^{(1,ψ^{q})} & BX(m, r, n) × BX(m, r, n) .}  (34)

Remark 11.1. Many cases already appear in the literature (cf. [25, 28, 54]). We can extract the following equivalences, up to p-completion, for a prime power q, coprime to p:

(1) BSU(n + 1)(q) ≃ BSL_{n+1}(q).

(2) BU(n)(q) ≃ BX(1, 1, n)(q) ≃ BGL_{n}(q).

(3) BX(m, 1, n)(q) ≃ BGL_{mn}(q).

(4) BX(2, 2, n)(q) ≃ BSO(2n)(q) ≃ BSO_{2n}^{+}(q).

According to Remark 6.6, we have that, also for any p-adic unit q, BSU(n + 1)(q), BX(m, 1, n)(q) and BX(2, 2, n)(q) are homotopy equivalent to classifying spaces of finite groups, up to p-completion.

These also include the cases BX(m, 2, n)(q), that can be reduced to BX(2, 2, n)(q′) using propositions 5.13 and 6.2, so they are also equivalent, up to p-compeltion, to classifying spaces of orthogonal groups over finite fields.

The above observations will be used as the starting point of the induction arguments that we will develop in the rest of this section in order to study the structure of the finite Chevalley versions BX(m, r, n)(q), for q ≡ 1 mod p, q ≠ 1, and general values of m, r, and n.

Fix q ≡ 1 mod p, q ≠ 1. The p-compact groups X(m, r, n) are polynomial, hence propositions 7.5 and 7.6 apply. The maximal elementary abelian p-subgroup of X(m, r, n), (t_{X}, ν), factors as a p-subgroup, (t_{X}, g), of X(m, r, n)(q), and the maximal finite torus of X(m, r, n)(q) is

BT_{ℓ}^{n} ≃ BC_{X(m,r,n)(q)}(t_{X}, g)

where ℓ = ν_{p}(q − 1). The Weyl group is W_{X(m,r,n)(q)}(T_{ℓ}^{n}) ≅ G(m, r, n), and the extension N_{X(m,r,n)(q)}(T_{ℓ}^{n}) ≅ T_{ℓ}^{n} ⋊G(m, r, n) sits in the maximal torus normalizer of X(m, r, n), making the following diagram homotopy commutative:

\xymatrix{BN_{X(m,r,n)(q)}(T_{ℓ}^{n}) \ar[r] \ar[d] & BN_{X(m,r,n)}(T^{n}) \ar[d] \\ BX(m, r, n)(q) \ar[r]^{ι} & BX(m, r, n) .}

Corollary 7.7 implies that the functor

ι_{♯}: F_{p}^{e}(X(m, r, n)(q)) → F_{p}^{e}(X(m, r, n))  (35)

is an equivalence of categories. The next result is a description of the centralizers of elementary abelian p-subgroups.

Proposition 11.2. Let p be an odd prime, m ≥ 1, r ≥ 1, n ≥ 1 with r|m|(p − 1), and q ≡ 1 mod p, q ≠ 1. Then,

(1) any elementary abelian p-subgroup h: BE → BX(m, r, n)(q), factors through the maximal finite torus, and

(2) for any subgroup E ≤ t_{x} ≤ T_{ℓ}^{n}, the centralizer of (E, g|_{BE}) in X(m, r, n)(q),

BC_{X(m,r,n)(q)}(E, g|_{BE}) ≃ BX(m, r, n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q) ,

n = n_{0} + n_{1} + ⋯ + n_{s}, is determined by the point-wise stabilizer of E ≤ T_{ℓ}^{n} in the Weyl group G(m, r, n), G(m, r, n)(E) ≅ G(m, r, n_{0}) × Σ_{n_{1}} × ⋯ × Σ_{n_{s}}.

Proof. All elementary abelian p-subgroups of X(m, r, n) are toral, hence the same is true for X(m, r, n)(q) by the equivalence (35). If E ≤ t_{X}, by Corollary 7.4, the restriction of ψ^{q} to the centralizer of (E, g|_{BE}), is ψ^{q} again, ψ^{q}|_{C_{X(m,r,n)(q)}(E,g|_{BE})} = ψ^{q}, and

BC_{X(m,r,n)(q)}(E, g|_{BE}) ≃ BC_{X(m,r,n)}(E, ν|_{BE})(q) .

The centralizers C_{X(m,r,n)}(E, ν|_{BE}) are known to be connected p-compact groups of maximal rank, with Weyl group G(m, r, n_{0}) × Σ_{n_{1}} × ⋯ × Σ_{n_{s}}, the point-wise stabilizer of E in T^{n} by the action of the Weyl group G(m, r, n):

BC_{X(m,r,n)}(E, ν|_{BE}) ≃ BX(m, r, n_{0}) × BU(n_{1}) × ⋯ × BU(n_{s}) ,

thus,

BC_{X(m,r,n)}(E, ν|_{BE})(q) ≃ BX(m, r, n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q)

contains the same maximal finite torus T_{ℓ}^{n} as X(m, r, n)(q), ℓ = ν_{p}(q−1), n = n_{0}+n_{1}+⋯+n_{s} and the Weyl group is G(m, r, n_{0}) × Σ_{n_{1}} × ⋯ × Σ_{n_{s}} (see propositions 7.5 and 7.6). □

Proposition 11.3. Let p be an odd prime, m ≥ 1, r ≥ 1, n ≥ 1 with r|m|(p − 1), and q ≡ 1 mod p, q ≠ 1. The natural map

hocolim_{F_{p}^{e}(X(m,r,n)(q))^{op}} BC_{X(m,r,n)(q)} ⟶ BX(m, r, n)(q)

is a mod p homology equivalence.

Proof. According to Theorem E and Example 9.5

H^{∗}(BX(m, r, n)(q); 𝔽_{p}) ≅ H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{G(m,r,n)} ≅ P [x_{1}, …, x_{n−1}, e] ⊗ E[y_{1}, …, y_{n−1}, u]

with deg(x_{i}) = 2mi, deg(e) = \frac{2mn}{r}, deg(y_{i}) = 2mi − 1, and deg(u) = \frac{2mn}{r} − 1.

Since this is true for all values of m, r, n, we obtain from Proposition 11.2 that also, for every elementary abelian p-subgroup E ≤ t_{X},

H^{∗}(BC_{X(m,r,n)(q)}(E, g|_{BE}); 𝔽_{p}) ≅ H^{∗}(BT_{ℓ}^{n}; 𝔽_{p})^{G(m,r,n)(E)}

where G(m, r, n)(E) is the point-wise stabilizer of E in T_{ℓ}^{n}, by the action of the Weyl group G(m, r, n). So, then, the result follows from Proposition 7.9. □

Fix a Sylow p-subgroup of N_{X(m,r,n)(q)}(T_{ℓ}^{n}), S_{n,ℓ} ≅ ℤ/p^{ℓ} ≀ S_{n}, where S_{n} is the Sylow psubgroup of the symmetric group Σ_{n}. Call f the composition BS_{n,ℓ} → BN_{X(m,r,n)(q)}(T_{ℓ}^{n}) → BX(m, r, n)(q), Thus (S_{n,ℓ}, f) is a p-subgroup of BX(m, r, n)(q).

We will denote by

F(m, r, n, q) = F_{(S_{n,ℓ},f)}(BX(m, r, n)(q))

the fusion system of BX(m, r, n)(q) over (S_{n,ℓ}, f) and by

L(m, r, n, q) = L_{(S_{n,ℓ},f)}(BX(m, r, n)(q)) ,

the associated centric linking system. Recall that the underlying category of F(m, r, n, q) is equivalent to F_{p}(BX(m, n, r)(q)).

Theorem 11.4. If q is a p-adic unit such that q ≡ 1 mod p, q ≠ 1, and ℓ = ν_{p}(1 − q), then, (S_{n,ℓ}, f) is a Sylow p-subgroup for BX(m, r, n)(q) and

(S_{n,ℓ}, F(m, r, n, q), L(m, r, n, q))

is a p-local finite group with classifying space

|L(m, r, n, q)|_{p}^{∧} ≃ BX(m, r, n)(q) .

Proof. We proceed by induction on n, the p-rank of X(m, r, n)(q). For n < p, X(m, r, n) is a non-modular p-compact group, and then, X(m, r, n)(q) is the p-completed classifying space of a finite group (see 9.7). Also, for BX(1, 1, n) ≃ BU(n)_{p}^{∧}, Remark 11.1 characterizes BX(1, 1, n)(q) as p-completed classifying spaces of finite groups. In all that cases, the conclusion of the theorem is clearly satisfied (see section 3).

Assume, that n is large and that the theorem holds for every n_{0} < n. That is, for every n_{0} < n, BX(m, r, n_{0})(q) is the classifying space of the p-local finite group (S_{n_{0},ℓ}, F(m, r, n_{0}, q), L(m, r, n_{0}, q)). The result about BX(m, r, n)(q) will follow from Theorem 4.6. We will show that the space BX(m, r, n)(q) and its p-subgroup (S_{n,ℓ}, f) meet the conditions of 4.6. Condition (1) of 4.6 is satisfied by Proposition 7.1.

Condition (2a) of 4.6 amounts to show that if E ≤ t_{X}, then the centralizer BC_{X(m,r,n)(q)}(E, g|_{BE}) is the classifying space of a p-local finite group. This follows by the induction hypothesis. In fact, by 11.2, there is a homotopy equivalence BC_{X(m,r,n)(q)}(E, g|_{BE}) ≃ BX(m, r, n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q), for n = n_{0} + n_{1} + … n_{s}, a non-trivial decomposition of n into positive summands, and by the induction hypothesis and [11, 1.4] this is the classifying space of the p-local finite group defined as the product

(S_{n_{0},ℓ}, F(m, r, n_{0}, q), L(m, r, n_{0}, q)) × (S_{n_{1},ℓ}, F(1, 1, n_{1}, q), L(1, 1, n_{1}, q)) × ⋯ × (S_{n_{s},ℓ}, F(1, 1, n_{s}, q), L(1, 1, n_{s}, q)) .

Condition (2b) of 4.6 establishes that Sylow p-subgroups of centralizers of elementary abelian subgroups of BX(m, r, n)(q) factor through (S_{n,ℓ}, f). This is proved by reducing the question to unitary groups, obtained as centralizers of the center of S_{n,ℓ}.

Let Z ≅ ℤ/p denote the diagonal elements of order p in T_{ℓ}^{n} ≅ (ℤ/p^{ℓ})^{n} ≤ S_{n,ℓ}. Then, the point-wise stabilizer of Z in T_{ℓ}^{n} by the action of G(m, r, n) is Σ_{n} and therefore, according to Proposition 11.2, BC_{BX(m,r,n)(q)}(Z, g|_{BZ}) ≃ BU(n)(q).

By naturality of the construction of the normalizer of the maximal finite torus, we obtain a diagram

\xymatrix{BN_{U(n)(q)}(T_{ℓ}^{n}) \ar[r] \ar[d] & BN_{X(m,r,n)(q)}(T_{ℓ}^{n}) \ar[d] \\ BU(n)(q) \ar[r]^{Bj_{n}} & BX(m, r, n)(q)}

hence a factorization of (S_{n,ℓ}, f):

\xymatrix{ & BS_{n,ℓ} \ar[dl]_{f′} \ar[dr]^{f} \\ BU(n)(q) \ar[rr]^{Bj_{n}} &  & BX(m, r, n)(q) .}  (36)

Choose any other subgroup E ≤ t_{X} ≤ S_{n,ℓ}. Assume that the point-wise stabilizer of E in T_{ℓ}^{n} by the action of G(m, r, n) is G(m, r, n)(E) ≅ G(m, r, n_{0})×Σ_{n_{1}} ×⋯×Σ_{n_{s}}. Define E′ = Z ·E ≤ t_{X}, then, the point-wise stabilizer of E′ will be G(m, r, n)(E′) ≅ Σ_{n_{0}} ×Σ_{n_{1}} ×⋯×Σ_{n_{s}}. The inclusions E ≤ E′ ≥ Z induce a commutative diagram of centralizers

\xymatrix{BC_{X(m,r,n)(q)}(E′, g|_{BE′}) \ar[r]^{Bj_{n}^{♯}} \ar[d] & BC_{X(m,r,n)(q)}(E, g|_{BE}) \ar[d] \\ BC_{X(m,r,n)(q)}(Z, g|_{BZ}) \ar[r]^{Bj_{n}} & BX(m, r, n)(q) .}  (37)

Now, BC_{X(m,r,n)(q)}(E, g|_{BE}) ≃ BX(m, r, n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q) with Sylow p-subgroup S_{n_{0},ℓ} × ⋯ × S_{n_{s},ℓ} while BC_{X(m,r,n)(q)}(E′, g|_{BE′}) ≃ BU(n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q) and from the above discussion we have a factorization

\xymatrix{B(S_{n_{0},ℓ} × ⋯ × S_{n_{s},ℓ}) \ar[r] \ar@{-->}[dr] & BU(n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q) \ar[d]^{Bj_{n}^{♯}≃Bj_{n_{0}}×1×···×1} \\  & BX(m, r, n_{0})(q) × BU(n_{1})(q) × ⋯ × BU(n_{s})(q) .}  (38)

Diagrams (36), (37), and (38) provide a homotopy commutative diagram

\xymatrix{B(S_{n_{0},ℓ} × ⋯ × S_{n_{s},ℓ}) \ar[rr] \ar[r] \ar@{.>}[d]_{Bρ} & BC_{X(m,r,n)(q)}(E′, g|_{BE′}) \ar[r]^{Bj_{n}^{♯}} \ar[d] & BC_{X(m,r,n)(q)}(E, g|_{BE}) \ar[d] \\ BS_{n,ℓ} \ar[rr] \ar[r] & BC_{X(m,r,n)(q)}(Z, g|_{BZ}) \ar[r]^{Bj_{n}} & BX(m, r, n)(q)}

where the existence of the homomorphism ρ: S_{n_{0},ℓ} × ⋯ × S_{n_{s},ℓ} → S_{n,ℓ} making homotopy commutative the left square is obtained because S_{n,ℓ} is a Sylow p-subgroup of U(n)(q).

We have proved that BX(m, r, n)(q) and (S_{n,ℓ}, f) satisfy the conditions (1) and (2) of Theorem 4.6, and therefore, that (S_{n,ℓ}, f) is a Sylow p-subgroup of BX(m, r, n)(q) and (S_{n,ℓ}, F(m, r, n, q), L(m, r, n, q)) is a p-local finite group.

Finally, BX(m, r, n)(q) is the classifying space |L(m, r, n, q)|_{p}^{∧} according to Proposition 11.3 and Theorem 4.6. □

Proposition 11.5. X(m, r, n)(q) is a exotic p-local finite group if r > 2, n ≥ p.

Notice that in the above hypothesis r|(p − 1), thus r > 2 can only occur with p ≥ 5, so that we are implicitely assuming also that p ≥ 5.

Proof. We wil first reduce the question to the rank p-case. Then we classify the centric radical subgroups in the fusion system of BX(m, r, p)(q) and show that they coincide with the p-local finite groups of [11, Example 9.4].

There is an elementary abelian p-subgroup E ≤ t_{X}, in X(m, r, n)(q), of rank n − p such that

C_{X(m,r,n)(q)}(E, g|_{BE}) ≅ X(m, r, p)(q) × U(1)_{p}^{∧}(q)^{n−p}

(see Proposition 11.2). Thus, if we assume that there is a finite group G such that BX(m, r, n)(q) ≃ BG_{p}^{∧}, then the map Bg|_{BE}: BE → BX(m, r, n)(q) ≃ BG_{p}^{∧} is induced by a homomorphism φ: E → G, and

BC_{G}(φ(E))_{p}^{∧} ≃ BX(m, r, p)(q) × BU(1)_{p}^{∧}(q)^{n−p} .

Since BU(1)_{p}^{∧}(q) ≃ Bℤ/p^{ℓ}, the projection BC_{G}(φ(E))_{p}^{∧} → BU(1)_{p}^{∧}(q)^{n−p} is the p-completion of the map induced by a homomorphism ρ: C_{G}(φ(E)) → (ℤ/p^{ℓ})^{n−p}. It has a section, also induced by a homomorphims σ: (ℤ/p^{ℓ})^{n−p} → C_{G}(φ(E)), hence ρ is an epimorphism. Therefore, we have a short exact sequence Ker ρ → C_{G}(φ(E)) → (ℤ/p^{ℓ})^{n−p} and an induced fibration B(Ker ρ)_{p}^{∧} → BC_{G}(φ(E))_{p}^{∧} → B(ℤ/p^{ℓ})^{n−p}, from which we obtain an equivalence B(Ker ρ)_{p}^{∧} ≃ BX(m, r, p)(q). This reduces the question to showing that X(m, r, p)(q) is an exotic p-local finite group.

We will show now that X(m, r, p)(q) coincide with the p-local finite groups constructed in [11, Example 9.4] in purelly algebraic terms. For this aim we will need to describe the centric and radical p-subgroups of X(m, r, p)(q).

Recall that T_{ℓ}^{p} ≅ (ℤ/p^{ℓ})^{p} is the maximal finite torus of X(m, r, p)(q) with Weyl group G(m, r, p) and they form a split extension

T_{ℓ}^{p} → N_{X(m,r,p)(q)}(T_{ℓ}^{p}) → G(m, r, p)

that contains S_{p,ℓ} = T_{ℓ}^{p} ⋊ ℤ/p ≤ N_{X(m,r,p)(q)}(T_{ℓ}^{p}), a Sylow p-subgroup of X(m, r, p)(q). For simplicity we will denote F = F(m, r, p, q), the fusion system of BX(m, r, p)(q) over (S_{p,ℓ}, f).

The center of the Sylow p-subgroup is Z(S_{p,ℓ}) ≅ ℤ/p^{ℓ} embeded diagonally in T_{ℓ}^{p}, and, if we write Z(t_{X}) for the elements of order p in Z(S_{p,ℓ}), then we obtain BC_{X(m,r,p)(q)}(Z(S_{p,ℓ}) ≃ BC_{X(m,r,p)(q)}(Z(t_{X})) ≃ BU(p)_{p}^{∧}(q) (see Proposition 11.2). We also know (see Remark 11.1) that BU(p)_{p}^{∧}(q) ≃ BGL_{p}(q_{0})_{p}^{∧} por a prime power q_{0} with ℓ = ν_{p}(1 − q) = ν_{p}(1 − q_{0}), hence we conclude that the centralizer fusion system C_{F}(Z(S_{p,ℓ})) coincides with the fusion system of GL_{p}(q_{0}), that has been described in Example 3.5.

The Sylow p-subgroup S_{p,ℓ} is clearly centric and radical. T_{ℓ}^{p} is centric and Out_{F}(T_{ℓ}^{p}) = G(m, r, p) hence it is also radical (p ≥ 5). Proper subgroups of T_{ℓ}^{p} are not centric, so we will look at subgroups Q ≤ S_{p,ℓ} not contained in T_{ℓ}^{p}. such a subgroup fits in an extension

[figure: Q_{0}; Q; ℤ/p; T_{ℓ}^{p}; S_{p,ℓ}; ℤ/p]

where Q_{0} = Q ∩ T_{ℓ}^{n}, and since Q is centric, Z(S_{p,ℓ}) ≤ Q_{0}. It turns out that this is actually a characteristic subgroup of Q, Hence there is an exact sequence of groups:

1 → Aut_{C_{F} (Z(S_{p,ℓ}))}(Q) → Aut_{F}(Q) → Aut_{F}(Z(S_{p,ℓ}))

where Aut_{F}(Z(S_{p,ℓ})) ≅ ℤ/r is given by the action of the Adams operations of exponents a rth root of unity.

Assume that Q is abelian. Then Q_{0} = Z(S_{p,ℓ}) and Q is either ℤ/p×Z(S_{p,ℓ}) or cyclic Z/p^{ℓ+1}. In the first case, Q is conjugated in F to a subgroup of T_{ℓ}^{n}, hence it is not centric while in the second case, it is conjugated to the group U_{ℓ+1} described in Example reffusionsystemGLpq. Adams operations do not act internally in U_{ℓ+1}, hence Out_{F}(U_{ℓ+1}) ≅ Out_{C_{F} (Z(S_{p,ℓ}))}(U_{ℓ+1}) ≅ ℤ/p and then U_{ℓ+1} is not radical in F.

Assume that Q is non-abelian. The same arguments as in 10.5 show that Q is either S_{p,ℓ} or Γ_{ℓ}, and both are radical in C_{F}(Z(S_{p,ℓ})). Thus we obtain that they complete the list of conjugacy classes of centric radical subgroups of S_{p,ℓ} in F.

In order to complete the picture it remains to compute the F-automorphisms of Γ_{ℓ}. We have Out_{C_{F} (Z(S_{p,ℓ}))}(Γ_{ℓ}) ≅ SL_{2}(p). Now, the Adams operations act internally in Γ_{ℓ} and we get Out_{F}(Γ_{ℓ}) ≅ SL_{2}(p).r.

By Alperin’s fusion theorem, a fusion system over S is generated by the automorphisms of its fully normalized centric radical subgroups in S. Since in our case al the automorphisms of T_{ℓ}^{p} are induced by conjugation in N_{X(m,r,p)(q)}(T_{ℓ}^{p}), we can write

F(m, r, p, q) = ⟨F_{N_{X(m,r,p)(q)}(T_{ℓ}^{p})}(S_{p,ℓ}); F_{Γ_{ℓ}}(SL_{2}(p).r)⟩

(see section 3) but this is precisely the definition of the fusion systems in [11, Example 9.4]. □

The cases BX(m, r, n)(q) with r = 1, 2 or n < p, are homotopy equivalent to p-completed classifying spaces of finite groups according to Theorem 9.7 and Remark 11.1.

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Matematisk Institut, Universitetsparken 5, DK–2100 København

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