HOMOTOPY LIE GROUPS

JESPER M. MØLLER

Abstract. Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson [13], represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector [32, 33] in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller’s proof of the Sullivan conjecture [25] and Lannes’ division functors [22]. Today, with Dwyer and Wilkerson’s implementation of Rector’s vision, the tantalizing classification theorem seems to be within grasp. Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory.

1. Introduction

The aim of this report is to advertise the discovery by W.G. Dwyer and C.W. Wilkerson of a remarkable class of spaces called homotopy Lie groups or p-compact groups. These purely homotopy theoretic objects capture the essence of the idea of a Lie group. I shall here focus on [13] where Dwyer and Wilkerson introduce homotopy Lie groups and prove (1.1) and (1.3). Subsequent developments [11, 12, 14], possibly leading to a classification theorem in the near future, will be briefly described in the final section.

A p-compact group is a pointed topological space, BX, with all of its homotopy theory concentrated at the prime p, whose loop space X = ΩBX satisfies a cohomological finiteness condition. The p-compact r-torus BT = K(ℤ_{p}, 2)^{r} is an example of a p-compact group.

A maximal torus for the p-compact group BX is a map Bi: BT → BX of a p-compact torus BT to BX satisfying an injectivity and a maximality condition.

Theorem 1.1. [13, 8.13, 9.4] Any p-compact group admits a maximal torus, unique up to conjugacy.

The Weyl group, W_{T} (X), associated to the maximal torus Bi: BT → BX is, when X is connected, faithfully represented in the vector space H_{2}(BT; ℚ_{p}) and the algebra homomorphism induced by the W_{T} (X)-invariant map Bi,

H^{∗}(Bi; ℚ_{p}): H^{∗}(BX; ℚ_{p}) → H^{∗}(BT; ℚ_{p})W_{T} (X) (1.2)

takes the p-adic rational cohomology of BX into the invariant ring of W_{T} (X).

1991 Mathematics Subject Classification. 55P35, 55R35.

Key words and phrases. Finite loop space, p-compact group, maximal torus, Weyl group.

Theorem 1.3. [13, 9.7] Let X be a connected p-compact group with maximal torus T ⟶ X. Then:

(1) T and X have the same rank.

(2) The Weyl group W_{T} (X) is faithfully represented as a reflection group in the ℚ_{p}-vector space H_{2}(BT; ℚ_{p}).

(3) The homomorphism ( 1.2) is an isomorphism.

The precise definition of a p-compact group, which is motivated by the concept of a finite loop space, follows shortly.

1.4. Finite loop spaces. Suppose G is a compact Lie group. Take a free, contractible G-space EG and define BG = EG/G to be the orbit space. Then the associated fibre sequence

ΩEG ⟶ ΩBG ⟶ G ⟶ EG ⟶ BG

contains a homotopy equivalence ΩBG ⟶ G.

This phenomenon is embedded in the general concept of a finite loop space.

Definition 1.5. A finite loop space is a connected, pointed space BX such that X = ΩBX is homotopy equivalent to a finite CW-complex.

Note that X — by definition — is the loop space of BX. It is customary, though ambiguous, to refer to the finite loop space BX by its underlying space X and then call BX the classifying space of X.

We have already seen that compact Lie groups are finite loop spaces. The classifying space of SU(2), for instance, is the infinite quaternionic projective space B SU(2) = ℍP^{∞}. However, the class of finite loop spaces is much larger. A striking example was provided by Rector [31] who found an uncountable family of homotopically distinct finite loop spaces BX with X homotopy equivalent to SU(2). In other words, the homotopy type SU(2) supports uncountable many distinct loop space structures.

Rector’s example destroyed all hopes of a classification theorem in the spirit of compact Lie groups — as long as one sticks to integral spaces, that is. The situation looks brighter in the category of 𝔽_{p}-local spaces.

1.6. Notation. In the following, p denotes a fixed prime number, 𝔽_{p} the field with p elements, ℤ_{p} the ring of p-adic integers, and ℚ_{p} = ℤ_{p} ⊗ ℚ the field of p-adic numbers.

H^{∗}(−) denotes singular cohomology with 𝔽_{p}-coefficients, H^{∗}(−; 𝔽_{p}), while H^{∗}(−; ℚ_{p}) denotes H^{∗}(−; ℤ_{p}) ⊗ ℚ (and not singular cohomology with ℚ_{p}- coefficients).

A space K is 𝔽_{p}-finite if H^{∗}(K) is finite dimensional over 𝔽_{p}. A map A ⟶ B is an 𝔽_{p}-equivalence if it induces an isomorphism H^{∗}(B) ⟶ H^{∗}(A) on H^{∗}(−).

1.7. 𝔽_{p}-local spaces. A space K is 𝔽_{p}-local if any 𝔽_{p}-equivalence A ⟶ B induces a homotopy equivalence map(B, K) ⟶ map(A, K) of mapping spaces.

𝔽_{p}-local spaces exist — the classifying space K(𝔽_{p}, i) for the functor H^{i}(−) is an obvious example. In fact, any space can be made 𝔽_{p}-local in a minimal and functorial way.

Theorem 1.8. (Bousfield [4, 3.2]) The exists a functor K ⇝ K_{p}, of the (homotopy) category of CW-complexes into itself, with a natural transformation η_{K}: K → K_{p} such that η_{K} is an 𝔽_{p}-equivalence and K_{p} is 𝔽_{p}-local.

Note these categorical consequences of the definition and the theorem: • Any 𝔽_{p}-equivalence between 𝔽_{p}-local spaces is a homotopy equivalence.

• A map is an 𝔽_{p}-equivalence if and only if its 𝔽_{p}-localization is a homotopy equivalence.

• K is 𝔽_{p}-local if and only if η_{K}: K → K_{p} is a homotopy equivalence. If K is nilpotent or is connected and either H_{1}(K; 𝔽_{p}) = 0 or π_{1}(K) is finite, then

[5, VI.5.3, VII.3.2, VII.5.1] [13, §11] the Bousfield localization, K_{p}, coincides with the (perhaps more familiar) Bousfield–Kan localization, (𝔽_{p})_{∞}K. In particular [5, VI.5.2], π_{i}(K_{p}) ≅ π_{i}(K) ⊗ ℤ_{p} when K is connected, pointed and nilpotent with finitely generated abelian homotopy groups. (This is far from true without the nilpotency hypothesis (2.2).)

1.9. Homotopy Lie groups. By design, actually inescapably [7], the uncountably many finite loop spaces BX of Rector’s example all 𝔽_{p}-localize to the standard (B SU(2))_{p}. This observation indicates that “𝔽_{p}-local finite loop spaces” are better behaved than integral loop spaces. The problem, however, is that this term is absurd as finite complexes are unlikely to be 𝔽_{p}-local. The solution proposed in [13] is to replace the topological finiteness criterion in (1.5) by a cohomological one.

Definition 1.10. [13, 2.2] A p-compact group is an 𝔽_{p}-local space BX such that X = ΩBX is 𝔽_{p}-finite.

Again, it is customary to use X, by definition the loop space of BX, when referring to the p-compact group BX, the classifying space of X.

2. Examples of p-compact groups

Let G be any compact Lie group whose component group π_{0}(G) is a p-group. Define BĜ = (BG)_{p}. Then Ĝ is a p-compact group with H^{∗}(BĜ) = H^{∗}(BG), π_{0}(Ĝ) = π_{0}(G), and π_{i}(Ĝ) = π_{i}(G) ⊗ ℤ_{p} for all i ≥ 1 [13, §11].

This example includes all finite p-groups such as the trivial group {1} and the cyclic p-groups ℤ/p^{n}, n ≥ 1.

2.1. Toral groups. When applied to an r-torus S = SO(2)^{r}, the above construction produces a p-compact r-torus T = Ŝ. The classifying space BT = K(ℤ_{p}, 2)^{r} is an Eilenberg-MacLane space with homotopy in dimension two and H^{∗}(BT) = 𝔽_{p}[t_{1}, …, t_{r}] is polynomial on r generators of degree two.

Alternatively, BT = (BŤ)_{p}, where Ť = (ℤ/p^{∞})^{r} is a p-discrete r-torus.

More generally, a p-compact toral group P is a p-compact group with BP = (BP̌)_{p}, where P̌, a p-discrete toral group, is an extension of a p-discrete torus Ť by a finite p-group π. Note that the 𝔽_{p}-localized sequence BT ⟶ BP ⟶ Bπ is [5, II.5.1] a fibration sequence as π necessarily acts nilpotently on H_{i}(BŤ; 𝔽_{p}).

Since map(BP, BX) ≃ map(BP̌, BX) for any p-compact group X and P̌ is the union of an ascending chain of finite p-groups, results for finite p-groups can often, by discrete approximation [13, §6], be extended to p-compact toral groups.

2.2. Exotic examples. Call a connected p-compact group exotic if it is not of the form Ĝ for any connected compact Lie group G. Sullivan spheres and, more generally, (many) Clark–Ewing p-compact groups are exotic.

Proposition 2.3. (Sullivan) Assume that n > 2 is an integer dividing p−1. Then the 𝔽_{p}-local sphere (S^{2n−1})_{p} is a p-compact group.

The construction goes as follows: The cyclic group ℤ/n acts on Ť = ℤ/p^{∞} as ℤ/n < Aut(Ť) ≅ ℤ_{p}^{∗} when n|(p − 1). Define BX = (BŇ)_{p} where Ň = Ť⋊ℤ/n is the semidirect product. The computation

H^{∗}(BX) = H^{∗}(BŇ) = H^{∗}(BŤ)^{ℤ/n} = 𝔽_{p}[t]^{ℤ/n} = 𝔽_{p}[t^{n}],

which uses (1.7) and the fact that n is prime to p, shows that the mod p cohomology of BX is polynomial on one generator of degree 2n. Thus the 𝔽_{p}-local space BX is (2n − 1)-connected [5, I.6.1] and its loop space X is (2n − 2)-connected with H^{∗}(X) abstractly isomorphic to H^{∗}(S^{2n−1}). The Hurewicz theorem tells us that this abstract isomorphism is realizable by an 𝔽_{p}-equivalence S^{2n−1} ⟶ X, i.e. (1.7) by a homotopy equivalence (S^{2n−1})_{p} ⟶ X_{p} = X.

Clark and Ewing [6] observed that applicability of Sullivan’s construction is not restricted to the rank one case. Let Ť be a p-discrete r-torus and W < Aut(Ť) ≅ GL_{r}(ℤ_{p}) a finite group of order prime to p acting on Ť. Put BX = (BŇ)_{p} where Ň = Ť⋊W. The invariant ring

H^{∗}(BX) = H^{∗}(BŇ) = H^{∗}(BŤ)^{W} = 𝔽_{p}[t_{1}, …, t_{r}]^{W}

is, essentially by the Shephard–Todd theorem [2, 7.2.1] [21, §23], a finitely generated polynomial algebra if and only if W is a reflection group in GL_{r}(ℚ_{p}). If this is the case, H^{∗}(X) is an exterior algebra on finitely many odd degree generators. In particular, X is 𝔽_{p}-finite and BX a p-compact group. These Clark–Ewing pcompact groups are fairly well understood [8].

The Clark–Ewing p-compact group associated to any non-Coxeter group from the list [6] of irreducible p-adic reflection groups is (7.6) exotic. This scheme produces many exotics if p is odd but none for p = 2 as the only non-Coxeter 2-adic reflection group, W (DI(4)), number 24 on the list and of rank 3, has even order.

To come up with an exotic 2-compact group, a much more sophisticated approach is required. In their landmark paper [18], Jackowski and McClure showed how to decompose BG, for any compact Lie group G, as a generalized pushout of classifying spaces of subgroups (proper subgroups if the center of G is trivial). Dwyer and Wilkerson realized that a similar decompositon applies in the case of p-compact groups [11, §8] and that this could be used in the construction of an exotic 2compact group.

Theorem 2.4. [12] There exists a connected 2-compact group DI(4) such that H^{∗}(B DI(4); 𝔽_{2}) is isomorphic (as an algebra over the Steenrod algebra) to the rank 4 mod 2 Dickson algebra H^{∗}(B(𝔽_{2})^{4}; 𝔽_{2})^{GL_{4}(𝔽_{2})} and H^{∗}(B DI(4); ℚ_{2}) is isomorphic to the invariant ring of W (DI(4)).

Assuming that such a space exists, it is possible [10] to read off from its cohomology a finite diagram that looks like the cohomological image of a diagram of spaces. Some effort is required to verify that the picture seen in cohomology actually is realizable on the level of spaces. The generalized pushout of this diagram is the exotic 2-compact group.

Since G_{1} := SO(3) = DI(2) and G_{2} = DI(3), it is tempting to put G_{3} = DI(4). No 2-compact group deserves the name DI(n) with n ≥ 5 [19, 35].

2.5. Cohomological invariants. If the space K is 𝔽_{p}-finite, also H^{∗}(K; ℚ_{p}) is finite dimensional over ℚ_{p}, and the Euler characteristic

χ(K) = ∑(−1)^{i} dim_{𝔽_{p}} H^{i}(K) = ∑(−1)^{i} dim_{ℚ_{p}} H^{i}(K; ℚ_{p})

and the cohomological dimension

cd(K) = max{i | H^{i}(K) ≠ 0}

are defined [13, 4.3, 6.13].

For a connected p-compact group X, in particular, H^{∗}(X; ℚ_{p}) is a connected finite dimensional Hopf algebra so, by Borel [3] or Milnor–Moore [26], H^{∗}(X; ℚ_{p}) = E(x_{1}, …, x_{r}) is an exterior algebra on finitely many generators of odd degree, |x_{i}| = 2d_{i} − 1, and H^{∗}(BX; ℚ_{p}) = ℚ_{p}[y_{1}, …, y_{r}] is a polynomial algebra on generators of even degree, |y_{i}| = 1 + |x_{i}|, 1 ≤ i ≤ r. The number, r = rk(X), of generators is [13, 5.9] the rank of X. The cohomological dimension of X is [14, 3.8] given by cd(X) = max{i | H^{i}(X; ℚ_{p}) ≠ 0} = ∑_{i=1}^{r}(2d_{i} − 1). For instance, cd(Ĝ) = dim G, rk(Ĝ) is the rank of G, a p-compact r-torus has rank r, cd(P) = rk(P) if (and only if) P is a p-compact toral group, rk(S^{2n−1}) = 1 and cd(S^{2n−1}) = 2n − 1, while rk(DI(4)) = 3 and [2, Appendix A] cd(DI(4)) = 27.

Exercise 2.6. The trivial p-compact group has Euler characteristic χ({1}) = 1. The empty space has Euler characteristic χ(∅) = 0. For a connected p-compact group X, X is trivial ⇔ χ(X) ≠ 0 ⇔ rk(X) = 0 [13, 5.10].

3. Morphisms

A p-compact group morphism f: X → Y is a based map Bf: BX → BY between the classifying spaces. The trivial morphism 0: X → Y is the constant map B0: BX → BY, and the identity morphism 1: X → X is the identity map B1: BX → BX.

Note the fibration sequence

X ⟶^{f} Y ⟶ Y/f ⟶ BX ⟶^{Bf} BY (3.1)

where Y/f, or Y/X when f is understood, denotes the homotopy fibre of Bf.

Two morphisms f, g: X → Y are conjugate if the maps Bf, Bg: BX → BY are freely homotopic and Rep(X, Y) = π_{0} map(BX, BY) = [BX, BY] denotes the set of conjugacy classes of morphisms of X to Y.

3.2. Monomorphisms, epimorphisms, and isomorphisms. The morphism f: X → Y is a monomorphism if Y/X is 𝔽_{p}-finite, an epimorphism if Y/X is the classifying space of some p-compact group, and an isomorphism if Y/X is contractible.

Example 3.3. {1} ⟶ X is a monomorphism with X/{1} = X, X ⟶ {1} is an epimorphism with {1}/X = BX, 1: X → X is an isomorphism with X/X = {1}, and the diagonal Δ: X → X^{n} is a monomorphism since X^{n}/X is homotopy equivalent to X^{n−1}. There exists [12, 1.8] a monomorphism of { \atop Spin(7)} to DI(4).

These definitions are motivated by

Example 3.4. Let f: G → H be a monomorphism (an epimorphism) of compact Lie groups. The homotopy fibre of the induced map Bf: BG → BH is H/f(G) (B(ker f)) so the corresponding p-compact group morphism f̂: Ĝ → Ĥ is a monomorphism (an epimorphism). (Not all morphisms between Ĝ and Ĥ are induced from homomorphisms between G and H [16].)

A diagram X ⟶ Y ⟶ Z of p-compact group morphisms is a short exact sequence if BX ⟶ BY ⟶ BZ is a fibration sequence. Any p-compact group sits in a short exact sequence of the form X_{0} ⟶ X ⟶ π_{0}(X) where X_{0} is the identity component of X; the identity component of a p-compact toral group, for instance, is a p-compact torus (2.1).

Exercise 3.5. Let f: X → Y and g: Y → Z be morphisms.

(1) If f and g are monomorphisms, then g ∘ f is a monomorphism.

(2) If X is a p-compact toral group and g ∘ f a monomorphism, then f is a monomorphism.

(3) Assume that X ⟶^{f} Y ⟶^{g} Z is a short exact sequence. Show that f is a monomorphism and g an epimorphism. Show also that if X is a p-compact r-torus and Z a p-compact s-torus, then Y is a p-compact (r + s)-torus.

To be fair, part (2) of this exercise, requiring the theory of kernels [13, 7.1–7.3], is highly nontrivial. (The condition on X can be removed [13, 9.11].)

An inspection of the Serre spectral sequence for the left segment of (3.1) yields

Proposition 3.6. [13, 6.14] If f: X → Y is a monomorphism, then cd(Y) = cd(X) + cd(Y/X).

3.7. Nontrivial elements. The existence of nontrivial elements in nontrivial pcompact groups represents the first and decisive step in constructing the maximal torus.

Theorem 3.8. [13, 5.4, 5.5, 7.2, 7.3] Let X be a nontrivial p-compact group.

(1) There exists a monomorphism ℤ/p ⟶ X.

(2) If X is connected, there exists a monomorphism S ⟶ X from a p-compact 1-torus S to X.

Note that (2) implies (1): In case the identity component X_{0} is nontrivial, use (2) to get (3.5) a monomorphism ℤ/p ⟶ S ⟶ X_{0} ⟶ X. Otherwise, (1) reduces to obstruction theory. (A sketch of the proof of (3.8) will be presented in §6.)

Analogously, any nontrivial, connected compact Lie group contains a copy of SO(2) (in its maximal torus).

Exercise 3.9. Use (3.8) and Lannes theory [22] to show that BX is 𝔽_{p}-finite only if X is trivial. Next show that a p-compact group morphism which is both a monomorphism and an epimorphism, is an isomorphism.

4. Homotopy Fixed Point Spaces

Let π be a finite p-group and K a space. A π-space with underlying space K is a fibration K_{hπ} ⟶ Bπ over Bπ with fibre K. A π-map is a map u_{hπ}: K_{hπ} → L_{hπ} over Bπ.

The homotopy orbit space is the total space, K_{hπ}, and the homotopy fixed point space, K^{hπ}, is the space of sections (which may very well be empty). These spaces are related by the evaluation map Bπ × K^{hπ} ⟶ K_{hπ}.

For brevity, a π-space will often be denoted by its underlying space and a π-map by its restriction to the underlying spaces.

Example 4.1. The trivial π-space with underlying space K is the trivial fibration K × Bπ ⟶ Bπ with homotopy orbit space K_{hπ} = Bπ × K and homotopy fixed point space K^{hπ} = map(Bπ, K).

The homotopy fixed point construction K^{hπ} is functorial in both variables:

• For any π-map u: K → L, composition with u_{hπ}: K_{hπ} → L_{hπ} determines a map u^{hπ}: K^{hπ} → L^{hπ}.

• For any subgroup κ < π, any π-space is also a κ-space. The inclusion ι: κ → π induces a map K_{hι}: K_{hκ} → K_{hπ} over Bι: Bπ → Bκ and a map K^{hι}: K^{hπ} → K^{hκ} of homotopy fixed point spaces.

The homotopy orbit space and the homotopy fixed point space are homotopy invariant constructions in that any π-map u: K → L, which is an ordinary (nonequivariant) homotopy equivalence, induces homotopy equivalences u_{hπ}: K_{hπ} → L_{hπ} and u^{hπ}: K^{hπ} → L^{hπ}.

4.2. Exactness. Let U denote the (ordinary, nonequivariant) homotopy fibre of a π-map u: K → L (where L is assumed to be connected), or, equivalently, the homotopy fibre of u_{hπ}: K_{hπ} → L_{hπ}. The pullback diagram

[figure: U_{hπ}; K_{hπ}; u_{hπ}; Bπ; l; L_{hπ}]

shows that any homotopy fixed point l ∈ L^{hπ} makes U into a π-space such that

Proposition 4.3. [13, 10.6] U ⟶ K ⟶ L is a fibration sequence of π-maps between π-spaces and U^{hπ} ⟶ K^{hπ} ⟶ L^{hπ} is a fibration sequence of homotopy fixed point spaces (where l ∈ L^{hπ} serves as base point).

4.4. Exponential Laws. The exponential law in its simple form reads

map(K_{hπ}, A) = map(K, A)^{hπ}, (4.5)

and in slightly more general form,

K^{hπ} = (K^{hκ})^{h(π/κ)} (4.6)

as K_{hπ} = (K_{hκ})_{h(π/κ)} for any normal subgroup κ ◁ π.

The standing assumption made here that π be finite is not essential; π can (and will) be replaced by a p-discrete toral group or even a p-compact toral group.

5. Centralizers

Let P be a p-compact toral group, Y any p-compact group, and g: P → Y a morphism of P to Y.

The centralizer of g, C_{Y} (g), or C_{Y} (P) when g is understood, is the loop space of BC_{Y} (g) = map(BP, BY)_{Bg}, the mapping space component containing Bg. Note the evaluation map BC_{Y} (g) × BP ⟶ BY. Base point evaluation, BC_{Y} (g) ⟶ BY, in particular, provides the first nontrivial example of a monomorphism.

Theorem 5.1. [13, 5.1, 5.2, 6.1] C_{Y} (g) is a p-compact group and C_{Y} (g) ⟶ Y is a monomorphism.

The difficulty here is to show that C_{Y} (g) and Y/C_{Y} (g) are 𝔽_{p}-finite spaces. (It is unknown if this remains true with P replaced by a general p-compact group).

5.2. Central maps. The morphism g: P → Y is said to be central if (1) C_{Y} (g) ⟶ Y is an isomorphism, or,

(2) g extends to a morphism Y × P ⟶ Y which is the identity on Y. These two conditions are equivalent as the adjoint of a morphism as in (2) is an inverse to the evaluation monomorphism in (1).

Example 5.3. [15] [7, 2.5] [34, 9.6] [11, 12.5] Let C_{G}(f) be the centralizer of a homomorphism f: π → G of a finite p-group π into a compact Lie group G whose component group is a finite p-group. Then π_{0}(C_{G}(f)) is a p-group [17, A.4] and there is an isomorphism

{ \atop C_{G}(f)} ⟶ CĜf̂

which is adjoint to the 𝔽_{p}-localization of the map BC_{G}(f) × Bπ ⟶ BG induced by the homomorphism C_{G}(f) × π ⟶ G. Thus f̂: π → Ĝ is a central morphism of p-compact groups if f: π → G is central as a homomorphism of Lie groups.

Here are two more examples of central morphisms.

Theorem 5.4. [13, 5.3, 6.1] The constant morphism 0: P → Y is central.

This first example is an immediate consequence of the Sullivan conjecture as proved by H. Miller [25].

In contrast to the very deep Theorem 5.4, nothing more than elementary obstruction theory is needed for the second example of a central morphism.

Lemma 5.5. The identity map 1: S → S of a p-compact torus S is central.

In other words, S is abelian. (See §8 for more on central maps and abelian p-compact groups). In fact, any morphism into S is central.

Consider a morphism g: S → Y defined on a p-compact torus. Identifying S with the centralizer of the identity morphism, composition of maps provides a morphism C_{Y} (g) × S ⟶ C_{Y} (g). The restriction to the second factor is a central factorization g′: S → C_{Y} (g) of g through its own centralizer.

Lemma 5.6. [13, 8.2, 8.3] Suppose that g: S → Y is a monomorphism of a pcompact torus S to Y. Then there exists a short exact sequence

S ⟶^{g′} C_{Y} (g) ⟶ C_{Y} (g)/g′

of p-compact groups such that S ⟶^{g′} C_{Y} (g) ⟶ Y is conjugate to g.

Note that (5.6) asserts the existence of a classifying space B(C_{X}(g)/g′) for the homogeneous space C_{X}(g)/g′.

Exercise 5.7. Any monomorphism S ⟶ S is an isomorphism.

6. Algebraic Smith theory

Suppose that π is a finite p-group and that the p-compact group classifying spaces BX and BY are π-spaces. Let f: X → Y be a monomorphism such that Bf: BX → BY is a π-map. Choose a base point y ∈ (BY)^{hπ} and equip Y/X with the corresponding π-space structure (4.3) such that Y/X ⟶ BX ⟶ BY is a fibration sequence of π-maps and (Y/X)^{hπ} ⟶ (BX)^{hπ} ⟶ (BY)^{hπ} a fibration sequence of homotopy fixed point spaces.

Algebraic Smith theory, based on work by J. Lannes and his collaborators and concerned with the cohomological properties, in particular the Euler characteristic (2.5), of the fibre (Y/X)^{hπ}, can be summarized as follows.

Theorem 6.1. [13, 4.5, 4.6, 5.7] [9] [24] Under the above assumptions the following hold:

(1) (Y/X)^{hπ} is 𝔽_{p}-finite.

(2) χ((Y/X)^{hπ}) = χ(Y/X) mod p.

(3) χ((Y/X)^{hπ}) = Λ(Y/X; π) if π is cyclic.

The Lefschetz number of (6.1(3)) is the alternating sum

Λ(Y/X; π) = ∑_{i=0}^{∞}(−1)^{i} trace H^{i}(ξ; ℚ_{p})

where H^{i}(ξ; ℚ_{p}) is the automorphism of H^{i}(Y/X; ℚ_{p}) induced by any generator ξ of the cyclic group π.

The analogous Euler characteristic formulas were known to be true in classical Smith theory dealing with fixed point spaces for (reasonable) group actions on finite complexes.

I refrain from commenting on the proof of (6.1) but refer to [23] for more detailed information.

A particularly advantageous situation arises when the finite p-group π can be replaced by a p-discrete torus (2.1) Ť.

Corollary 6.2. [13, 4.7, 5.7] [9] Suppose that Bf: BX → BY is a Ť-map. Then:

(1) χ((Y/X)^{hA}) = χ(Y/X) for any finite subgroup A < Ť.

(2) (Y/X)^{hŤ} ≠ ∅ if χ(Y/X) ≠ 0.

It is unknown if (1) also holds for infinite subgroups, such as Ť itself.

As a demonstration of the power of Smith theory, I now sketch the proof of (3.8). Let X be a connected, nontrivial p-compact group so that (2.6) rk(X) > 0.

Consider the map

(6.3) map(Bι, BX): map(Bℤ/p^{n+1}, BX) → map(Bℤ/p^{n}, BX)

induced by the inclusion ι: ℤ/p^{n} → ℤ/p^{n+1}, n ≥ 0. Viewing BX as a trivial ℤ/p^{n+1}-space and Bℤ/p^{n} = (ℤ/p)_{hℤ/p^{n+1}} as the total space of a p-fold covering map of Bℤ/p^{n+1}, we get (4.1, 4.5)

map(Bℤ/p^{n+1}, BX) = (BX)^{hℤ/p^{n+1}}
map(Bℤ/p^{n}, BX) = map(ℤ/p, BX)^{hℤ/p^{n+1}} = (BX^{p})^{hℤ/p^{n+1}}

and map(Bι, BX) = (BΔ)^{hℤ/p^{n+1}} reveals itself as induced by the p-fold diagonal (3.3) BΔ: BX → map(ℤ/p, BX) = BX^{p}, a ℤ/p^{n+1}-map with fibre X^{p}/X.

By (6.1), the homotopy fibre (X^{p}/X)^{hℤ/p^{n+1}} of (6.3) is 𝔽_{p}-finite and the remarkable, but straightforward, Lefschetz number calculation [13, 5.11],

χ((X^{p}/X)^{hℤ/p^{n+1}}) = Λ(X^{p}/X; ℤ/p^{n+1}) = p^{rk(X)},

shows (2.6) that it is nonempty and noncontractible.

If all maps of Bℤ/p to BX were inessential, then (5.4) map(Bι, BX) would be a homotopy equivalence of the total space map(Bℤ/p, BX) = BC_{X}(0) to the base space BX of (6.3) with n = 0 and the homotopy fibre would be contractible — but it is not, so there must exist an essential map Bf_{1} : Bℤ/p → BX, i.e. [13, §7] a monomorphism f_{1} : ℤ/p → X. Since the homotopy fibre over Bf_{1} of (6.3) with n = 1 is nonempty, f_{1} extends (up to conjugacy) to a monomorphism [13, §7] f_{2} : ℤ/p^{2} → X. Proceeding inductively, we obtain a morphism f_{∞}: ℤ/p^{∞} → X that restricts to a monomorphism on ℤ/p^{n} for all n ≥ 1. The 𝔽_{p}-localization of Bf_{∞} is (2.1) a monomorphism f: S → X of a p-compact 1-torus S to X.

This proves (3.8). Also (5.1) is a quick consequence of (6.1).

7. Maximal tori and Weyl groups

Let X be any p-compact group. The maximal torus of X is constructed by an inductive procedure.

If X(= C_{X}({1})/{1}) is not homotopically discrete, it is (3.8) the target of a monomorphism S_{1} ⟶ X defined on a p-compact 1-torus S_{1}. This monomorphism factors through its own centralizer (5.6) to give a short exact sequence

S_{1} ⟶ C_{X}(S_{1}) ⟶ C_{X}(S_{1})/S_{1}

of p-compact groups.

If C_{X}(S_{1})/S_{1} is not homotopically discrete, it is (3.8) the target of a monomorphism S_{2}/S_{1} ⟶ C_{X}(S_{1})/S_{1} defined on a p-compact 1-torus S_{2}/S_{1}. Pull back along this monomorphism induces a commutative diagram of p-compact group morphisms

[figure: S_{1}; S_{2}; S_{2}/S_{1}; S_{1}; C_{X}(S_{1}); C_{X}(S_{1})/S_{1}]

where S_{2} is (3.5) a p-compact 2-torus and the middle arrow a monomorphism (C_{X}(S_{1})/S_{2} ≃ \frac{C_{X}(S_{1})/S_{1}}{S_{2}/S_{1}} is 𝔽_{p}-finite). Thus X is the target of a monomorphism (3.5) S_{2} ⟶ C_{X}(S_{1}) ⟶ X defined on a p-compact 2-torus.

By dimension considerations (3.6), this inductive procedure eventually stops at a maximal torus for X where

Definition 7.1. [13, 8.8, 8.9] A maximal torus is a monomorphism i: T → X of a p-compact torus T to X such that C_{X}(T)/T is a homotopically discrete p-compact group.

We have thus established the existence part of (1.1).

Let i: T → X be a maximal torus such that Bi: BT → BX is a fibration. The Weyl space W_{T} (X) is the topological monoid of selfmaps of BT over BX. As a space, W_{T} (X) is the fibre over Bι of map(BT, Bι): map(BT, BT) → map(BT, BX), i.e. (5.5)

W_{T} (X) = (X/T)^{hT} = ∐_{w} C_{X}(i)/C_{T} (w) = ∐_{w} C_{X}(T)/T (7.2)

where the disjoint union indexed by all w ∈ Rep(T, T), necessarily (3.5, 5.7) central automorphisms, with i ∘ w conjugate to i. The right hand side shows that the Weyl space is homotopically discrete.

Definition 7.3. [13, 9.6] The Weyl group W_{T} (X) is the component group (!) π_{0}W_{T} (X) of the Weyl space.

The homotopy fixed point space (X/T)^{hT} from (7.2), by discrete approximation (2.1) homotopy equivalent [13, 6.1, 6.7] to (X/T)^{hA} for some finite subgroup A < Ť, is (6.1) 𝔽_{p}-finite and the computation (6.2)

χ(X/T) = χ((X/T)^{hA}) = χ((X/T)^{hT}) = |W_{T} (X)| (7.4)

shows that the Euler characteristic of the homogeneous space X/T equals the order of the Weyl group; in particular, χ(X/T) > 0.

Another application of (6.2) now yields the uniqueness part of (1.1): Suppose that i_{1} : T_{1} → X and i_{2} : T_{2} → X are maximal tori. The fact that (X/T_{2})^{hŤ_{1}} = (X/T_{2})^{hT_{1}} ≠ ∅ ≠ (X/T_{1})^{hŤ_{2}} = (X/T_{1})^{hT_{2}} means that there exist morphisms u: T_{1} → T_{2} and v: T_{2} → T_{1}, necessarily isomorphisms (3.5, 5.7), such that i_{1} is conjugate to i_{2} ∘ u and i_{2} to i_{1} ∘ v.

Approaching the key results of [13], we now specialize to connected p-compact groups.

Proposition 7.5. [13, 9.1] Let X be a connected p-compact group and T ⟶ X a maximal torus. Then the morphism T ⟶ C_{X}(T) is an isomorphism.

Again, the proof uses Smith theory.

With C_{X}(T)/T = {1}, (7.2) shows that the monoid morphism W_{T} (X) ⟶ Rep(T, T) = [BT, BT] is injective, i.e. that W_{T} (X) is faithfully represented in H_{2}(BT; ℚ_{p}) := H_{2}(BT; ℤ_{p}) ⊗ ℚ. The W_{T} (X)-invariant map Bi: BT → BX induces an algebra map

H^{∗}(Bi; ℚ_{p}): H^{∗}(BX; ℚ_{p}) → H^{∗}(BT; ℚ_{p})W_{T} (X)

into the invariant ring of this faithful representation. By means of a transfer map, generalizing the Becker–Gottlieb transfer [1] to fibrations with 𝔽_{p}-finite fibres and ingeniously constructed [13, 9.13] via the Kan–Thurston theorem [20], Dwyer and Wilkerson show that H^{∗}(Bi; ℚ_{p}) is injective. The extension H^{∗}(BX; ℚ_{p}) ⊆ H^{∗}(BT; ℚ_{p}) is finite since the fibre X/T is 𝔽_{p}-finite so these two rings have identical Krull dimensions, i.e. rk(T) = rk(X) (1.3.(1)). The culmination of [13] is (1.3.(2)) asserting that H^{∗}(BX; ℚ_{p}) is isomorphic to the invariant ring of the Weyl group. The fact that the invariant ring is polynomial (2.5) implies, by the classical Shephard–Todd theorem [2, 7.2.1], that (1.3.(3)) W_{T} (X) is represented as a reflection group in the vector space H_{2}(BT; ℚ_{p}). Thus W_{T} (X) must be isomorphic to a product of irreducible p-adic reflection groups from the Clark–Ewing list [2, 7.1] and H^{∗}(BX; ℚ_{p}) must be isomorphic to a tensor product of the corresponding graded invariant rings.

Example 7.6. Suppose that G is a compact Lie group with π_{0}(G) a p-group. Then any Lie theoretic maximal torus T ⟶ G induces a maximal torus T̂ ⟶ Ĝ of the pcompact group Ĝ. The associated Weyl groups are isomorphic. The Weyl group of a p-compact toral group P is π_{0}(P). The Weyl group of the Sullivan sphere (S^{2n−1})_{p} is ℤ/n and the Weyl group of the Clark–Ewing p-compact group (B(Ť⋊W))_{p} is W. The Weyl group of DI(4) is W (DI(4)), abstractly isomorphic to the product of a cyclic group of order two and the simple group of order 168.

Exercise 7.7. Modify the above construction of the maximal torus for a p-compact group to obtain an unconventional construction of the maximal torus for a compact Lie group [23] [13, 1.2].

While the p-adic rational cohomology H^{∗}(BX; ℚ_{p}) is under control, it is quite another matter with the cohomology algebra H^{∗}(BX) with coefficients in 𝔽_{p}. The difference between the two situations is that H^{∗}(BX; ℚ_{p}) embeds into the polynomial ring H^{∗}(BT; ℚ_{p}) while H^{∗}(BX) embeds into H^{∗}(BN_{p}(T)) where BN_{p}(T), the p-normalizer of the maximal torus [13, 9.8], is the Borel construction for the action of a Sylow p-subgroup of the Weyl group on BT. As a result, H^{∗}(BX) need not be polynomial but can actually be surprisingly complicated [36]. However, Dwyer and Wilkerson are able to settle the perhaps most basic structural question.

Theorem 7.8. [13, 2.3] H^{∗}(BX) is a finitely generated 𝔽_{p}-algebra for any pcompact group X.

Given (7.8) it is but a small step to verify [13, 1.1] the old conjecture that H^{∗}(BX) is a finitely generated 𝔽_{p}-algebra for any finite loop space X.

8. Classification

The classification scheme for p-compact groups, not yet completed, exhibits a pronounced analogy to Lie theory.

8.1. Centers. A p-compact group is abelian if its identity map is central. By (5.5), p-compact tori are abelian.

Theorem 8.2. [11, 1.1] [29, 3.1] A p-compact group is abelian if and only if it is isomorphic to a product of a finite abelian p-group and a p-compact torus.

If Z ⟶ X is a central monomorphism, then [11, 5.1] [29, 3.5] Z is abelian.

Theorem 8.3. [11, 1.2] [29, 4.4] For any p-compact group X there exists a central monomorphism Z(X) ⟶ X such that any central monomorphism into X factors, in an essentially unique way, through Z(X).

The terminal central monomorphism, esentially unique, of (8.3) is the center of X.

The (discrete approximation to the) center can be defined as the group of elements in (the discrete approximation to) C_{X}(T) that are central in X. Another candidate to the center title is the centralizer of the identity morphism. Fortunately, there is no discrepancy.

Theorem 8.4. [11, 1.3] The map BZ(X) ⟶ map(BX, BX)_{B1}, corresponding to the isomorphism C_{X}(Z(X)) ⟶ X, is a homotopy equivalence.

The highly nontrivial proof of (8.4) involves decomposing BX as a generalized pushout — a technique also applied in the proof of (2.4).

Example 8.5. Let G be a connected compact Lie group with Lie theoretic center Z(G). Then the maps

(BZ(G))_{p} ⟶ map(BĜ, BĜ)_{B1} ⟵ BZ(Ĝ)

adjoint to the 𝔽_{p}-localization of BZ(G) × BG ⟶ BG and BZ(Ĝ) × BĜ ⟶ BĜ, respectively, are homotopy equivalences [11, 1.4, 12.1] (8.4). The center of the p-compact group Ê_{6} is cyclic of order 3 if p = 3 and trivial otherwise.

For a connected p-compact group X [29, 5.2],

(8.6) rk(Z(X)) = dim_{ℚ_{p}}(π_{1}(X) ⊗ ℚ_{p}) = dim_{ℚ_{p}} H_{2}(BT; ℚ_{p})W^{T} (X)

so the center is finite if and only if the fundamental group is. The quotient pcompact group [13, 8.3] X/Z(X) has trivial center [11, 6.3] [29, 4.6].

The classification of connected p-compact groups essentially reduces to the simply connected case by the following result.

Theorem 8.7. [29, 5.4] For any connected p-compact group X, there exists a short exact sequence

A ⟶ Y × S ⟶ X

where A is a finite abelian p-group, A ⟶ Y × S ⟶^{pr_{1}} Y is a central monomorphism into the simply connected p-compact group Y, and S is a p-compact torus.

In (8.7), Y is the universal covering p-compact group of X, A is the torsion subgroup of π_{1}(X), and S = Z(X)_{0} is the identity component of the center.

8.8. Semisimplicity. Call a connected p-compact group with maximal torus T ⟶ X simple if the faithful representation of the Weyl group W_{T} (X) in H_{2}(BT; ℚ_{p}) is irreducible. Sullivan spheres and Clark–Ewing p-compact groups (2.2) are simple by design. Ĝ is simple for any connected compact simple Lie group G.

For any connected p-compact group, the W_{T} (X)-representation H_{2}(BT; ℚ_{p}) splits as a direct sum

H_{2}(BT; ℚ_{p}) = M_{1} ⊕ ⋯ ⊕ M_{n}

of irreducible representations. Provided the center Z(X) = 0 is trivial, all M_{i} are nontrivial (8.6) and [14, 1.5] this splitting of ℚ_{p}[W_{T} (X)]-modules descends to a splitting of ℤ_{p}[W_{T} (X)]-modules,

H_{2}(BT; ℤ_{p}) = L_{1} ⊕ ⋯ ⊕ L_{n}

where L_{i} = H_{2}(BT; ℤ_{p}) ∩ M_{i}. The splitting criterion [14, 1.4], guaranteeing the realizability as a splitting of X of any splitting of the ℤ_{p}[W_{T} (X)]-module H_{2}(BT; ℤ_{p}), now leads to the main result on semisimplicity.

Theorem 8.9. [14, 1.3] [30] Any connected p-compact group with trivial center is isomorphic to a product of simple p-compact groups.

For any simply connected p-compact group Y, Z(Y) is finite, and the decomposition of the center free quotient Y/Z(Y) into simple factors lifts to a decomposition of Y into simple factors [14, 1.6].

The decompositions of (8.7) and (8.9) are also useful for the classification of endomorphisms of p-compact groups [28] [27].

The final stage, still unresolved, of the classification scheme has to address existence and uniqueness of simple p-compact groups. At the core of this problem, two obstruction groups loom.

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Matematisk Institut, Universitetsparken 5, DK–2100 København Ø E-mail address: moller@math.ku.dk