Lie Groups and p-Compact Groups

W. G. Dwyer

Abstract. A p-compact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of p-compact groups, with the intention of illustrating the fact that many classical structural properties of compact Lie groups depend only on homotopy theoretic considerations.

1 From compact Lie groups to p-compact groups

The concept of p-compact group is the culmination of a series of attempts, stretching over a period of decades, to isolate the key homotopical characteristics of compact Lie groups. It has been something of a problem, as it turns out, to determine exactly what these characteristics are. Probably the first ideas along these lines were due to Hopf [10] and Serre [31].

1.1. Definition. A finite H-space is a pair (X, m), where X is a finite CW– complex with basepoint ∗ and m : X × X → X is a multiplication map with respect to which ∗ functions, up to homotopy, as a two-sided unit.

The notion of compactness in captured here in the requirement that X be a finite CW–complex. To obtain a structure a little closer to group theory, one might also ask that the multiplication on X be associative up to homotopy. Finite H-spaces have been studied extensively; see [18] and its bibliography. Most of the results deal with homological issues. There are a few general classification theorems, notably Hubbuck’s theorem [11] that any path-connected homotopy commutative finite H-space is equivalent at the prime 2 to a torus; this is a more or less satisfying analog of the classical result that any connected abelian compact Lie group is a torus. Experience shows, though, that there is little hope of understanding the totality of all finite H-spaces, or even all homotopy associative ones, on anything like the level of detail that is achieved in the theory of compact Lie groups. The problem is that there are too many finite H-spaces; the structure is too lax.

Stasheff pointed out one aspect of this laxity [32] that is particularly striking when it comes to looking at finite H-spaces as models for group theory. He discovered a whole hierarchy of generalized associativity conditions, all of a homotopy theoretic nature, which are satisfied by a space with an associative multiplication but not necessarily by a finite H-space. These are called A_{n}-conditions (n ≥ 1); a space is an H-space if it satisfies condition A_{2} and homotopy associative if it satisfies condition A_{3}. Say that a space X is an A_{∞}-space if it satisfies condition A_{n} for all n. The following proposition comes from combining [32] with work of Milnor [22] [21] and Kan [17]; it suggests that that A_{∞}-spaces are very good models for topological groups. From now on we will use the term equivalence for spaces to mean weak homotopy equivalence.

1.2. Proposition. If X is a path-connnected CW–complex, the following four conditions imply one another:

1. X is an A_{∞}-space,

2. X is equivalent to a topological monoid,

3. X is equivalent to a topological group, and

4. X is equivalent to the space ΩY of based loops on some 1-connected pointed space Y.

In fact, there are bijections between homotopy classes of the four structures. There is a similar result for disconnected X, in which conditions 1 and 2 are expanded by requiring that an appropriate multiplication on π_{0}X make this set into a group.

If X is a topological group as in 1.2(3), then the space Y of 1.2(4) is the ordinary classifying space BX. Proposition 1.2 leads to the following convenient formulation of the notion “finite A_{∞}-space” or “homotopy finite topological group”. This definition appears in a slightly different form in work of Rector [30].

1.3. Definition. A finite loop space is a triple (X, BX, e), where X is a finite CW–complex, BX is a pointed space, and e : X → Ω BX is an equivalence.

Finite loop spaces appear as if they should be very good homotopy theoretic analogs of Lie groups, but one of the very first theorems about them was pretty discouraging. Rector proved in [29] that there are an uncountable number of distinct finite loop space structures on the three-sphere S^{3}. In other words, he showed that there are an uncountable number of homotopically distinct spaces Y with ΩY ≃ S^{3}. This is in sharp contrast to the geometric fact that up to isomorphism there is only one Lie group structure on S^{3}. It suggests that the theory of finite loop spaces is unreasonably complicated.

Rector’s method was interesting. For any space X, Bousfield and Kan (also Sullivan) had constructed a rationalization X_{ℚ} of X, and 𝔽_{p}-completions X_{p̂} (p a prime); if X is a simply connected space with finitely generated homotopy groups, then π_{i}X_{ℚ} ≅ ℚ ⊗ π_{i}X and π_{i}X_{p̂} ≅ ℤ_{p} ⊗ π_{i}X. (Here ℤ_{p} is the ring of p-adic integers.) For such spaces there is a homotopy fibre square on the left

\begin{matrix}X & ⟶ & ∏_{p} X_{p̂} \\ ↓ & & ↓ \\ X_{ℚ} & ⟶ & (∏_{p} X_{p̂})_{ℚ} \end{matrix}  \begin{matrix}π_{i}X & ⟶ & ∏_{p} ℤ_{p} ⊗ π_{i}X \\ ↓ & & ↓ \\ ℚ ⊗ π_{i}X & ⟶ & ℚ ⊗ (∏_{p} ℤ_{p} ⊗ π_{i}X) \end{matrix}

which is a geometric reflection of the algebraic pullback diagrams on the right. This fibre square, called the arithmetic square [33] [3], amounts to a recipe for reconstituting X from its 𝔽_{p}-completions by mixing in rational glue. Rector constructed an uncountable number of loop space structures on S^{3} by taking the standard Lie group structure on S^{3}, 𝔽_{p}-completing to get “standard” loop space structures on each of the spaces (S^{3})_{p̂}, and then regluing these standard structures over the rationals in an uncountable number of exotic different ways. In particular, all of his loop space structures become standard after 𝔽_{p}-completion at any prime p. Later on [9] it became clear that this last behavior is unavoidable, since up to homotopy there is only one loop space structure on the space (S^{3})_{p̂}.

Apparently, then, the theory of finite loop spaces simplifies after 𝔽_{p}- completion, and it is exactly this observation that leads to the definition of pcompact group. The definition uses some terminology. We will say that a space Y is 𝔽_{p}-complete if the 𝔽_{p}-completion map Y → Y _{ p̂} is an equivalence; if Y is simply connected and H_{∗}(Y; 𝔽_{p}) is of finite type, then Y is 𝔽_{p}-complete if and only if the homotopy groups of Y are finitely generated modules over ℤ_{p}. We will say that Y is 𝔽_{p}-finite if H_{i}(Y; 𝔽_{p}) is finite-dimensional for each i and vanishes for all but a finite number of i (in other words, if H_{∗}(Y; 𝔽_{p}) looks like the 𝔽_{p}-homology of a finite CW–complex).

1.4. Definition. Suppose that p is a fixed prime number. A p-compact group is a triple (X, BX, e), where X is a space which is 𝔽_{p}-finite, BX is a pointed space which is 𝔽_{p}-complete, and e : X → Ω BX is an equivalence.

Here the idea of “compactness” is expressed in the requirement that X be 𝔽_{p}-finite. Assuming in addition that BX is 𝔽_{p}-complete is equivalent to assuming that X is 𝔽_{p}-complete and that π_{0}X is a finite p-group.

1.5. Example. If G is a compact Lie group such that π_{0}G is a p-group, then the 𝔽_{p}-completion of G is a p-compact group.

The definition of p-compact group is a homotopy theoretic compromise between between the inclination to stay as close as possible to the notion of Lie group, and the desire for an interesting and manageable theory. The reader should note that it is the remarkable machinery of Lannes [19] which makes p-compact groups accessible on a technical level. For instance, the machinery of Lannes lies behind the uniqueness result of [9] referred to above.

Organization of the paper. In section 2 we describe a general scheme for translating from group theory to homotopy theory. Sections 3 and 4 describe the main properties of p-compact groups; almost all of these are parallel to classical properties of compact Lie groups [4]. The final section discusses examples and conjectures.

It is impossible to give complete references or precise credit in a short paper like this one. The basic results about p-compact groups are in [6], [7], [8], [23], and [25]. There is a treatment of compact Lie groups based on homotopy theoretic arguments in [4]. The interested reader should look at the survey articles [20], [24], and [26], as well as their bibliographies, for additional information.

1.6. Terminology. There are a few basic topological issues which it is worth pointing out. We assume that all spaces have been replaced if necessary by equivalent CW–complexes. If f : X → Y is a map of spaces, then Map(X, Y)_{f} is the component containing f of the space of maps X → Y. The space Aut_{f}(X) is the space of self-equivalences of X over Y; to obtain homotopy invariance, this is constructed by replacing f by an equivalent Serre fibration f′ : X′ → Y and forming the space of self homotopy equivalences X′ ⟶^{∼} X′ which commute with f′.

The notation H_{ℚ_{p}}^{∗}(Y) stands for ℚ ⊗ H^{∗}(Y; ℤ_{p}); this is a variant of rational cohomology which is better-behaved than ordinary rational cohomology for spaces Y which are 𝔽_{p}-complete.

2 A dictionary between group theory and homotopy theory

We now set up a dictionary which will allow us to talk about p-compact groups in ordinary algebraic terms. We begin with concepts that apply to loop spaces in general (a loop space is a triple (X, BX, e) with e : X ⟶^{∼} Ω BX) and then specialize to p-compact groups. From now on we will refer to a loop space or p-compact group (X, BX, e) as a space X with some (implicit) extra structure.

2.1. Definition. Suppose that X and Y are loop spaces.

• A homomorphism f : X → Y is a pointed map Bf : BX → BY. Two homomorphisms f, f′ : X → Y are conjugate if Bf and Bf′ are homotopic.

• The homogeneous space Y/f(X) (denoted Y/X if f is understood) is the homotopy fibre of Bf.

• The centralizer of f(X) in Y, denoted C_{Y} (f(X)) or C_{Y} (X), is the loop space Ω Map(BX, BY)_{Bf}.

• The Weyl Space W_{Y} (X) is the space Aut_{Bf}(BX); this is in fact a a loop space, essentially because it is an associative monoid under composition (1.2). The normalizer N_{Y} (X) of X in Y is the loop space of the homotopy orbit space of the action of W_{Y} (X) on BX by composition.

• A short exact sequence X → Y → Z of loop spaces is a fibration sequence BX → BY → BZ; Y is said to be an extension of Z by X.

2.2. Remark. If X and Y are discrete groups, treated as loop spaces via 1.2, and f : X → Y is an ordinary homomorphism, then the above definitions specialize to the usual notions of coset space, centralizer, normalizer, and short exact sequence, at least if X → Y is injective. It is not hard to see that in general there are natural loop space homomorphisms C_{Y} (X) → N_{Y} (X) → Y; the homomorphism C_{Y} (X) → Y, for instance, amounts to the map Map(BX, BY)_{Bf} → BY given by evaluation at the basepoint of BX. There is always a short exact sequence X → N_{Y} (X) → W_{Y} (X).

The key additional definitions for p-compact groups are the following ones.

2.3. Definition. A p-compact group X is a p-compact torus if X is the 𝔽_{p}- completion of an ordinary torus, and a p-compact toral group if X is an extension of a finite p-group by a p-compact torus. If f : X → Y is a homomorphism of p-compact groups, then f is a monomorphism if Y/f(X) is 𝔽_{p}-finite.

3 Maximal tori and cohomology rings

If X is a p-compact group, a subgroup Y of X is a p-compact group Y and a monomorphism i : Y → X (i is called a subgroup inclusion). In general, if f : Y → X is a homomorphism of p-compact groups, the associated loop space homomorphism g : C_{X}(Y) → X is not obviously a subgroup inclusion; it is not even clear that C_{X}(Y) is a p-compact group. For special choices of Y, though, the situation is nicer.

3.1. Proposition. Suppose that f : Y → X is a homomorphism of p-compact groups, and that Y is a p-compact toral group. Then C_{X}(Y) → X is a subgroup inclusion.

A p-compact group is said to be abelian if the natural map C_{X}(X) → X is an equivalence.

3.2. Proposition. A p-compact group is abelian if and only if it is the product of a p-compact torus and a finite abelian p-group. If A is an abelian p-compact group and f : A → X is a homomorphism, then f naturally lifts over the subgroup inclusion C_{X}(A) → X to a homomorphism f′ : A → C_{X}(A).

A subgroup Y of X is said to be an abelian subgroup if Y is abelian, or a torus in X if Y is a p-compact torus. If Y′ is another subgroup of X, Y′ is said to be contained in Y up to conjugacy if the homomorphism Y′ → X lifts up to conjugacy to a homomorphism Y′ → Y.

3.3. Definition. A torus T in X is said to be a maximal torus if any other torus T′ in X is contained in T up to conjugacy.

We will say that an abelian subgroup A of X is self-centralizing if the map A → C_{X}(A) is an equivalence. If Z is a space which is 𝔽_{p}-finite, the Euler characteristic χ(Z) is the usual alternating sum of the ranks of the 𝔽_{p} homology groups of Z.

3.4. Proposition. Suppose that X is a p-compact group and that T is a torus in X. Then T is maximal if and only if χ(X/T) ≠ 0. If X is connected, then T is maximal if and only if T is self-centralizing.

3.5. Proposition. Any p-compact group X has a maximal torus T, unique up to conjugacy.

A space is said to be homotopically discrete if each of its components is contractible.

3.6. Proposition. Suppose that X is a p-compact group with maximal torus T. Then the Weyl space W_{X}(T) is homotopically discrete, and π_{0}W_{X}(T), with the natural composition operation, is a finite group.

If T is a maximal torus for X, the finitely generated free ℤ_{p}-module π_{1}T is called the dual weight lattice L_{X} of X; its rank as a free module is the rank rk(X) of X. The finite group appearing in 3.6 is called the Weyl group of X and denoted W_{X}; by definition, W_{X} acts on L_{X}.

3.7. Definition. If M is a finitely generated free module over a domain R (such as ℤ_{p}), an automorphism α of M is said to be a reflection (or sometimes a pseudoreflection or generalized reflection) if the endomorphism (α − Id) of M has rank one. A subgroup of Aut(M) is said to be generated by reflections if it is generated as a group by the reflections it contains.

3.8. Proposition. Suppose that X is a connected p-compact group of rank r. Then the action of W_{X} on L_{X} is faithful and represents W_{X} as a finite subgroup of GL_{r}(ℤ_{p}) generated by reflections.

3.9. Proposition. Suppose that X is a connected p-compact group with maximal torus T, Weyl group W, and rank r. Then the cohomology rings H_{ℚ_{p}}^{∗}(BT) and H_{ℚ_{p}}^{∗}(BX) are polynomial algebras over ℚ_{p} of rank r, and the natural restriction map H_{ℚ_{p}}^{∗}(BX) → H_{ℚ_{p}}^{∗}(BT)^{W} is an isomorphism.

3.10. Proposition. If X is a p-compact group, then the cohomology ring H^{∗}(BX; 𝔽_{p}) is finitely generated as an algebra over 𝔽_{p}.

4 Centers and product decompositions

A product decomposition of a p-compact group X is a way of writing X up to homotopy as a product of two p-compact groups, or, equivalently, a way of writing BX up to homotopy as a product of spaces. The most general product theorem is the following one.

4.1. Proposition. If X is a connected p-compact group, then there is a natural bijection between product decompositions of X and product decompositions of L_{X} as a module over W_{X}.

In general, connected p-compact groups are constructed from indecomposable factors in much the same way that Lie groups are, by twisting the factors together over a finite central subgroup.

4.2. Definition. A subgroup Y of a p-compact group X is said to be normal if the usual map N_{X}(Y) → X is an equivalence. The subgroup Y is central if the usual map C_{X}(A) → X is an equivalence.

If the subgroup Y of X is normal, then there is a loop space structure on X/Y (because X/Y is equivalent to the Weyl space W_{X}(Y)) and a short exact sequence Y → X → X/Y of p-compact groups.

4.3. Proposition. Any central subgroup of a p-compact group X is both abelian and normal; moreover, there exists up to homotopy a unique maximal central subgroup Z_{X} of X (called the center of X). The center of X can be identified as C_{X}(X).

The center of X is maximal in the sense that up to conjugacy it contains any central subgroup A of X.

4.4. Proposition. If X is a connected p-compact group, the quotient X/Z_{X} has trivial center.

If X is connected, the quotient X/Z_{X} is called the adjoint form of X. For connected X there is a simple way to compute Z_{X} from what amounts to ordinary algebraic data associated to the normalizer N_{X}(T) of a maximal torus T in X.

4.5. Definition. A connected p-compact group X is said to be almost simple if the action of W_{X} on ℚ ⊗ L_{X} affords an irreducible representation of W_{X} over ℚ_{p}; X is simple if X is almost simple and Z_{X} = {e}.

4.6. Proposition. Any 1-connected p-compact group is equivalent to a product of almost simple p-compact groups. The product decomposition is unique up to permutation of factors.

4.7. Proposition. Any connected p-compact group with trivial center is equivalent to a product of simple p-compact groups. The product decomposition is unique up to permutation of factors.

4.8. Proposition. Any connected p-compact group is equivalent to a p-compact group of the form

(T × X_{1} × ⋯ × X_{n})/A

where T is a p-compact torus, each X_{i} is a 1-connected almost simple p-compact group, and A is a finite abelian p-subgroup of the center of the indicated product.

5 Examples and conjectures

Call a connected p-compact group exotic if it is not equivalent to the 𝔽_{p}-completion of a connected compact Lie group. The reader may well ask whether there are any exotic p-compact groups, or whether on the other hand the study of p-compact groups is just a way of doing ordinary Lie theory under artificially difficult circumstances. In fact, there are many exotic examples: Sullivan constructed loop space structures on the 𝔽_{p}-completions of various odd spheres S^{n} (n > 3) [33], and it is possible to do more elaborate things along the same lines, see, e.g., [1] and [5].

Conjecturally, the theory splits into two parts.

5.1. Conjecture. Any connected p-compact group can be written as a product X_{1} × X_{2}, where X_{1} is the 𝔽_{p}-completion of a compact Lie group and X_{2} is a product of exotic simple p-compact groups.

In addition, all of the exotic examples are conjecturally known.

5.2. Conjecture. The exotic simple p-compact groups correspond bijectively, up to equivalence, to the exotic p-adic reflection groups of Clark and Ewing [1].

Here a p-adic reflection group is said to be exotic if it is not derived from the Weyl group of a connected compact Lie group. For example, it would follow from 5.2 there is up to equivalence only one exotic simple 2-compact group, the one constructed in [5]. Closely related to the above conjectures is the following one.

5.3. Conjecture. Let X be a connected p-compact group with maximal torus T. Then X is determined up to equivalence by the loop space N_{X}(T).

This would the analog for p-compact groups of a Lie-theoretic result of Curtis, Wiederhold and Williams [2]. It is easy to see that the loop space N_{X}(T) is determined by the Weyl group W_{X}, the p-adic lattice L_{X}, and an extension class in H^{3}(W_{X}; L_{X}). Explicit calculation with examples shows that if p is odd the extension class vanishes. It would be very interesting to find a simple, direct way to construct a connected p-compact group X from N_{X}(T). For a connected compact Lie group G, this would (according to [2]) give a direct way to constuct the homotopy type of BG, or at least the 𝔽_{p}-completion of this homotopy type, from combinatorial data associated to the root system of G. All constructions of this type which are known to the author involve building a Lie algebra and then exponentiating it; this kind of procedure does not generalize to p-compact groups.

The strongest results along the lines of 5.3 are due to Notbohm [27] [28]. The theory of homomorphisms between general p-compact groups is relatively

undeveloped, though there is a lot of information available if the domain is a pcompact toral group or if the homomorphism is a rational equivalence [12] [14]. The general situation seems complicated [13], but it might be possible to find some analog for p-compact groups of the results of Jackowski and Oliver [15] on “homotopy representations” of compact Lie groups (see for instance [16]).

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William G. Dwyer

Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556 USA dwyer.1@nd.edu