HOMOTOPY THEORY OF CLASSIFYING SPACES

OF COMPACT LIE GROUPS

by Stefan Jackowski, James McClure, and Bob Oliver

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod’s problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.

The study of maps between classifying spaces goes back to Hurewicz [Hur, p. 219], who in 1935 showed that

[X, Y] ≅ Hom(π_{1}X, π_{1}Y)/ Inn(π_{1}Y)

for any pair of aspherical spaces X and Y. This result might suggest a hope that all maps between the classifying spaces of any pair of compact Lie groups should be induced by homomorphisms. Much later however, in 1970, Sullivan [Su] provided the first counterexamples, by constructing maps (called “unstable Adams operations”) which did not agree even in rational cohomology with any map induced by a homomorphism. Sullivan’s work then led to a careful investigation by Hubbuck, Mahmud, and Adams ([Hub1], [AM], [Ad], [AM2]) of the effect that maps between classifying spaces could have in rational cohomology.

It was the proof of the Sullivan conjecture by Miller ([Mil], [DMN]) and Carlsson [Ca], and subsequent work by Lannes ([La1], [La2]) which provided the basis for the large amount of current activity and progress in this subject. It led, for example, to the theorems of Dwyer & Zabrodsky [DZ] and Notbohm [No], which describe [BP, BG] for any p-toral group P and any compact Lie group G. And that, together

Typeset by AMS-TEX

Second author partially supported by NSF grant DMS–9002525.

with a decomposition, due to the authors, of BG as a homotopy direct limit of BP ’s for certain p-toral P ⊆ G, has now yielded more general results about mapping sets [BG, BG′]. In particular, we obtained a complete description of [BG, BG] when G is connected and simple by showing that homotopy between maps is detected by rational cohomology in this case.

Decompositions of BG have also led to proofs of uniqueness theorems for classifying spaces in many cases. More precisely, for certain compact Lie groups G and certain primes p, any p-complete space X for which H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) (as algebras over the Steenrod algebra) is homotopy equivalent to BG_{p̂}. Results of this form have been proven by Dwyer, Miller & Wilkerson ([DMW1], [DMW2]), and more recently by Notbohm (unpublished).

Classifying spaces of compact connected Lie groups are notable by (in many cases) having polynomial algebras as their mod p cohomology, and in having loop spaces with the homotopy type of finite complexes. Hence the study of classifying spaces is closely connected to the problem of constructing other spaces with finite loop space, or with given polynomial algebras (with given Steenrod operations) as their mod p cohomology. The first “exotic” examples of such spaces were pcomplete spaces whose 𝔽_{p}-cohomology is polynomial on one generator of dimension 2n (i.e., deloopings of S^{2n−1} at p), constructed whenever n|(p − 1) by Sullivan [Su]. More examples were then given by Quillen [Q2, §10], Clark & Ewing [CEw], and Zabrodsky [Za]. The general decomposition of classifying spaces constructed in [JM2] then helped to motivate the development, by Lannes, Aguade [Ag], and Dwyer & Wilkerson [DW2], of more systematic ways of constructing new spaces of this type as homotopy direct limits of familiar spaces.

CONTENTS OF THE PAPER:

1. Prerequisites on homotopy direct limits

2. Homotopy decompositions of classifying spaces

3. Maps between classifying spaces

4. Higher inverse limits

5. Self maps of BG

6. Homotopical uniqueness of classifying spaces

7. Realizations of polynomial algebras

1. Prerequisites on homotopy direct limits

In this section we recall the construction of homotopy colimits of diagrams of spaces, a construction which plays a central role throughout this survey. The first approach is based on a construction of Segal [Se]. Let C be a small category (which can be a topological category, as defined in [Se]), and let F : C → T op be a covariant functor into the category of (compactly generated) topological spaces. Define a new category C_{F}, whose objects are pairs (C, x) for C ∈ Ob(C) and x ∈ F (C), and where

Mor_{C_{F}} ((C, x), (C′, x′)) = {f ∈ Mor_{C}(C, C′) : f(x) = x′}.

We regard C_{F} as a topological category with the compact-open topology. Then the homotopy colimit of F, hocolim_{⟶} (F), is defined to be the nerve (geometric realization) BC_{F}. This construction is clearly functorial with respect to continuous natural transformations of functors on C, and with respect to pullbacks of functors via continuous functors C′ → C. The natural transformation from F to the point functor induces a map p_{F} : hocolim_{⟶} (F) → BC.

More concretely, hocolim_{⟶} (F) is a kind of bar construction: an identification space of the form

hocolim_{⟶}_{C}(F) = (∐_{n≥0} ∐_{x_{0}→··→x_{n}} F (x_{0}) × Δ^{n})/ ∼,

where each face or degeneracy map between the x_{0} → ⋯ → x_{n} gives rise to an obvious identification between the corresponding F (x_{0}) × Δ^{n}. Regarded in this fashion, the projection p_{F} : hocolim_{⟶} (F) → BC is just induced by projecting each F (x_{0}) × Δ^{n} to its second factor. Note that each fiber (point inverse) of p_{F} has the form F (x) for some x ∈ Ob(C).

A different but equivalent approach to defining hocolim_{⟶} (F) is given by Bousfield & Kan in [BK]. They consider functors F : C → S_{∗} defined on an arbitrary small category C with values in the category of simplicial sets, and regard homotopy colimits as a special case of the more general construction of a balanced product (tensor product) X ×_{C} F of F with a contravariant functor X : C → S_{∗}. An elegant treatment of this viewpoint on homotopy colimits and limits is given in Hopkins [Ho].

Important examples of homotopy colimits include:

(a) Double mapping cylinders Let C be the “push-out” category:

C = (y_{1} ← x → y_{2}).

Then the homotopy colimit of a functor F : C → T op is just the double mapping cylinder of the maps F (y_{1}) ← F (x) → F (y_{2}).

(b) Mapping telescopes Let C = ℕ be the category obtained from the directed set of natural numbers. Then for any functor F : C → T op, there is a homotopy equivalence hocolim_{⟶} (F) ≃ Tel(F (1) → F (2) → …) (cf. [BK, XII.3.5]). (Note however that this is not a homeomorphism.)

(c) The Borel construction Fix a group G, and let O_{1}(G) be the category of G: the category with single object ∗, and with End(∗) ≅ G. (This notation is motivated by that used later for the orbit categories O(G) and O_{p}(G).) Then a functor F : O_{1}(G) → T op is just a G-space F (∗); and (with an appropriate definition of EG) hocolim_{⟶} (F) is homeomorphic to the Borel construction EG ×_{G} F (∗). If we allow topological categories, then the last equivalence remains true for continuous actions of topological groups.

As noted above, the natural transformation into the constant (point) functor induces a map p_{F} : hocolim_{⟶} (F) → BC, whose fibers all have the form F (−). When C is a discrete category, the Leray spectral sequence for p_{F} provides a means of calculating the cohomology of hocolim_{⟶} (F). If f : X → Y is any map (satisfying certain simple conditions), then the Leray spectral sequence for f is a spectral sequence of the form

E_{2}^{pq} = H^{p}(Y; H^{q}(f)) ⇒ H^{p+q}(X).

Here, H^{q}(f) is a sheaf over Y, whose stalk over y ∈ Y is H^{q}(f^{−1}y) (cf. [Br, §IV.6] for details). When this is applied to the map p_{F}, then the cohomology groups for these sheaves over BC are isomorphic to the higher derived functors of inverse limits of the (contravariant) functors

H^{q}(F (−)) : C → Ab.

(This can be shown using Lemma 4.2 below.) We thus get the following theorem.

Theorem 1.1. For every covariant functor F : C → T op, there is a spectral sequence

E_{2}^{pq} = lim_{⟵}^{p}_{C}(H^{q}(F (−))) ⇒ H^{p+q}(hocolim_{⟶}(F)).

For a more explicit statement, and a proof of Theorem 1.1, see [BK, XII.4.5]. Higher derived functors of inverse limits play an important role throughout this subject, and will be discussed in more detail in Section 4. Note that in example (a) the spectral sequence reduces to the usual Meyer-Vietoris exact sequence; in (b) to the Milnor lemma [Mln]; and in (c) to the Serre spectral sequence of the fibration F (∗) → EG ×_{G} F (∗) → BG.

Our own interest in homotopy colimits stems from their use as a tool for constructing maps and homotopies. For example, we will be constructing and classifying maps BG → X, for appropriate compact Lie groups G, by decomposing BG as a homotopy colimit of simpler spaces, and then comparing maps defined on the homotopy colimit with maps defined on the individual spaces. This is done using the next two theorems.

Theorem 1.2. Fix a discrete category C, and a functor F : C → T op. Let X be any other space, and consider the map

R : [hocolim_{⟶}_{C}(F), X] → lim_{⟵}_{c∈C}[F (c), X]

defined by restriction. Fix an element f̂ = (f_{c})_{c∈C} ∈ lim_{⟵}[F (−), X]; and define

contravariant functors α_{n} : C → Ab (all n ≥ 1) by setting

α_{n}(c) = π_{n}(map(F (c), X)_{f_{c}})

for all c ∈ Ob(C). Then f̂ ∈ Im(R) if the groups lim_{⟵}^{n+1}(α_{n}) vanish for all n ≥ 1;

and R^{−1}(f̂) contains at most one element if the groups lim_{⟵}^{n}(α_{n}) vanish for all n ≥ 1.

Theorem 1.2 can be proved by a standard obstruction-theory argument in which one inducts over the simplicial filtration of the homotopy colimit. However, one must be quite careful about basepoints, and about what happens when α_{1}(c) is not abelian. These points are handled in detail by Wojtkowiak in [Wo].

Theorem 1.2 is a special case of the following spectral sequence, constructed by Bousfield & Kan ([BK, XII.4.1 & XI.7.1] and [Bo]). As we will show later, this can often be used for explicit calculations involving R^{−1}(f̂), even when the obstructions of Theorem 1.2 do not all vanish.

Theorem 1.3. Let F, R, and f̂ be as in Theorem 1.2. Let map(hocolim_{⟶} (F), X)f̂

be the union of the components of the mapping space lying in R^{−1}(f̂). Define α_{n} : C → Ab by setting

α_{n}(c) = π_{n}(map(F (c), X)_{f_{c}})

for all c. Then there is a spectral sequence of the form

E_{−p,q}^{2} = lim_{⟵}^{p}_{C}(α_{q}) ⇒ π_{q−p}(map(hocolim_{⟶}(F), X)f̂)

which converges strongly if there is an N such that lim_{⟵}^{p}(α_{q}) vanishes for all q and all p > N.

Bousfield & Kan construct the spectral sequence of Theorem 1.3 as a special case of a spectral sequence for a homotopy inverse limit (and with a less restrictive convergence condition: see [BK, XI.7.1 and IX.5.3]). A more direct proof can be given as follows. The skeletal filtration of the homotopy colimit induces a tower of fibrations of mapping spaces, and the spectral sequence of Theorem 1.3 is the spectral sequence of this tower of fibrations. There is an explicit chain complex which calculates lim_{⟵}^{i} (see Lemma 4.2 below), and by using arguments similar to

Wojtkowiak’s in [Wo] one can show that the E_{2}-term in the spectral sequence is the homology of this chain complex.

We have seen above that, for any discrete group G, BG is the homotopy direct limit of the functor F : O_{1}(G) → T op which sends the single object ∗ to a point. Theorem 1.3 therefore gives a spectral sequence

E_{−p,q}^{2} = lim_{⟵}^{p}_{O_{1}(G)}(π_{q}(X)) ≅ H^{p}(G; π_{q}(X)) ⇒ π_{q−p}(map(BG, X)),

and one might hope that this would be a useful tool for proving results about maps out of BG. The problem is that this spectral sequence will in general have lots of differentials, and these seem to be impossible to compute even in the simplest cases: e.g., when G = ℤ/2 and X = BSO(3). Our solution to this problem is to consider more complicated decompositions of BG for which the spectral sequence of Theorem 1.3 collapses.

2. Homotopy decompositions of classifying spaces

Most recent results involving the homotopy theory of the classifying spaces of all but the simplest compact Lie groups are based on decompositions of BG as direct limits of simpler spaces—usually the classifying spaces of other groups. The first such decompositions were given by mapping telescopes; while later results involved taking homotopy colimits of more complicated diagrams. In particular, there are now two general ways to decompose an arbitrary BG, at a given prime p, in such a way that the pieces are classifying spaces of subgroups of G. These two general decompositions have both proved quite useful for answering different types of questions about BG.

The idea of decomposing the classifying space of a Lie group into a telescope of the classifying spaces of its subgroups goes back to Adams [Ad], who approximated the classifying spaces of p-toral groups via their finite p-subgroups. This idea was developed further by Feshbach [Fe], who in his work on the Segal conjecture for compact Lie groups was interested in extensions of tori by finite groups, and used their finite subgroups for a similar approximation. More precisely, Feshbach showed that for any compact Lie group G whose identity component G_{0} is a torus, there exists a chain of finite subgroups F_{1} ⊆ F_{2} ⊆ ⋯ ⊆ G such that the natural map Tel(BF_{i}) → BG induces an isomorphism in cohomology with arbitrary finite coefficients.

The approach of Adams and Feshbach was extended to arbitrary compact Lie groups by Friedlander & Mislin ([FM1], [FM2], [FM3]). For any compact Lie group G and any prime p not dividing the order of π_{0}(G), they found a chain of finite groups F_{1} ⊆ F_{2} ⊆ … (not necessary subgroups of G), and a compatible family of maps f_{i} : BF_{i} → BG (not necessarily induced by homomorphisms), such that the limit map f : Tel(BF_{i}) → BG induces an isomorphism in cohomology with coefficients in any finite π_{0}(G)-module of order not divisible by p. For example, when G = U(n) (and p is any prime), then this takes the form of a map

Tel(BGL_{n}(𝔽_{p^{m}})) ≃ BGL_{n}(𝔽̄_{p}) → BU(n)

(where 𝔽̄_{p} denotes the algebraic closure of 𝔽_{p}).

The use of homotopy colimit decompositions other than telescopes was first introduced by Dwyer, Miller, & Wilkerson for the particular case of G = SO(3). Consider the following two diagrams, where D(8) denotes the dihedral subgroup of order 8, Σ_{4} the octahedral group, and O(2) the normalizer of the maximal torus SO(2):

SO(3)/D(8) ⟶ SO(3)/O(2) ; ↓ ; SO(3)/Σ_{4} (1)  BD(8) ⟶ BO(2) ; ↓ ; BΣ_{4} (2)

It is not hard to check that the homotopy pushout of diagram (1) is 𝔽_{2}-acyclic. Diagram (2) is obtained by applying the Borel construction EG ×_{G} − to (1); and hence its homotopy pushout is 𝔽_{2}-homology equivalent to EG ×_{G} pt = BG = BSO(3).

There is another, closely related, decomposition of BSO(3) at the prime p = 2, which is taken over a more complicated category, but more useful because the pieces are simpler. Consider the category C which consists of two objects, 1 and 2, and where End(1) ≅ Σ_{3} (the symmetric group of order 6), End(2) ≅ 1, | Mor(1, 2)| = 3 (the maps being permuted by End(1) in the obvious way), and Mor(2, 1) = ∅. Consider the functor F : C → T op given by

F (1) = ESO(3)/((ℤ/2)^{2}) ≃ B(ℤ/2)^{2} and F (2) = ESO(3)/O(2) ≃ BO(2)

(with the obvious induced maps). In other words, F defines a diagram of the following form:

F = (Σ_{3} ⟲ ESO(3)/((ℤ/2)^{2})⟶ ; ⟶ ; ⟶ESO(3)/O(2))  (3)

A cohomology computation, using Theorem 1.1, shows that there is an 𝔽_{2}-homology equivalence hocolim_{⟶} (F) → BSO(3). Alternatively, it can be shown directly that hocolim_{⟶} (F) must be homotopy equivalent to the homotopy pushout of Dwyer, Miller, & Wilkerson in (2) above. For more information on converting homotopy colimits to homotopy pushouts in certain cases, cf. Słomińska [Sl].

This approximation of BSO(3) at the prime 2 can, unlike the homotopy-pushout decomposition, be generalized to all compact Lie groups and all primes. In fact, this can be done in two essentially different ways, which will be described in the next two subsections.

2a. Approximation via p-toral subgroups.

Recall that a compact Lie group P is p-toral if its identity component P_{0} is a torus and P/P_{0} is a p-group. The p-toral groups play a similar role for compact Lie groups to that played by p-groups for finite groups. For example, the “Sylow theorem” says that any compact Lie group G contains a unique conjugacy class of maximal p-toral subgroups, that any p-toral subgroup is contained in a maximal one, and that a p-toral subgroup P ⊆ G is maximal if and only if p ∤ χ(G/P). For any given maximal torus T ⊆ G and any Sylow p-subgroup W_{p} of the Weyl group W = N(T)/T, the extension

1 → T → N_{p}(T) → W_{p} → 1

is a maximal p-toral subgroup.

For any G, let O(G) be the orbit category: the (topological) category whose objects are the orbits G/H for closed subgroups H ⊆ G, and where Mor_{O(G)}(G/H, G/K) is the space of all G-maps. Let O_{p}(G) be the full subcategory of orbits G/P for p-toral P ⊆ G. We want to approximate BG as a homotopy colimit taken over O_{p}(G) or some appropriate subcategory.

When G is finite, then this is motivated in part by the theorem of Cartan & Eilenberg [CE, Theorem XII.10.1], which says that for any ℤ_{(p)}[G]-module A,

H^{∗}(BG; A) ≅ lim_{⟵}_{G/P ∈O_{p}(G)} H^{∗}(BP; A).  (4)

More precisely, H^{∗}(BG; A) is the inverse limit of the functor H^{∗}(EG ×_{G} −; A) (defined on O_{p}(G)). Formula (4) suggests that BG is 𝔽_{p}-homology equivalent to the homotopy direct limit of the BP ’s—and this can be confirmed by showing that the higher limits of the system in (4) all vanish (see [Mis, §2]).

The obvious way to extend this to the case of compact Lie groups is to take the limit of the BP ’s taken over all p-toral P ⊆ G (i.e., taken over O_{p}(G)). The main problem with doing this is that we want to work over a discrete category, and preferably a finite one. For example, we saw that when G = SO(3) and p = 2, it suffices to consider the 2-toral subgroups (ℤ/2)^{2} and O(2).

Definition 2.1. A closed subgroup P ⊆ G is called p-stubborn if it is p-toral, if N(P)/P is finite, and if N(P)/P does not contain any nontrivial normal p-subgroup. We let R_{p}(G) denote the full subcategory of O_{p}(G) whose objects are those G/P for p-stubborn P ⊆ G.

For any G and p, the category R_{p}(G) is finite in the sense that all morphism sets are finite, and there are finitely many isomorphism classes of objects (i.e., finitely many conjugacy classes of p-stubborn subgroups). When G = SO(3), it is easy to see that the only 2-stubborn subgroups are the groups (ℤ/2)^{2} and O(2) already mentioned above. When p ∤ |W | (W = N(T)/T), then the only p-stubborn subgroups of G are the maximal tori: i.e., R_{p}(G) is equivalent to the category with one object with automorphism group W.

For finite G, the p-stubborn subgroups are exactly the same as the p-radical subgroups which play a role in the classification of finite simple groups. They have also been used by Puig [Pu] for computing cohomology groups. In fact, Puig showed that H^{∗}(G; A) (for any ℤ_{(p)}[G]-module A) is the inverse limit of the H^{∗}(P; A) taken over all essential p-subgroups P ⊆ G: where the essential subgroups form a proper subclass of the radical (stubborn) subgroups.

The next theorem is our main approximation result for classifying spaces using p-stubborn subgroups. Here, G-T op denotes the category of spaces with G-action.

Theorem 2.2. Fix any G and p, and let I : R_{p}(G) ↪ G-T op be the inclusion. Then the map

hocolim_{⟶}_{R_{p}(G)}(EG ×_{G} I) ≅ EG ×_{G} (hocolim_{⟶}_{R_{p}(G)}(I)) ⟶ BG

induces an isomorphism of cohomology with Z_{(p)}-coefficients.

Note, in Theorem 2.2, that the individual terms in the homotopy colimit have the form EG ×_{G} (G/P) ≅ EG/P ≃ BP for p-stubborn subgroups P ⊆ G.

When p ∤ |W |, then by the above remarks (G/T is the only object in R_{p}(G)), the homotopy colimit has the form EW ×_{W} BT ≃BN(T). So in this case, Theorem 2.2 recovers a well known result of Borel.

As discussed earlier (diagrams (1) and (2) above), the original decomposition of BSO(3) as a homotopy pushout of certain orbits was found by first constructing an 𝔽_{2}-acyclic SO(3)-complex as a homotopy pushout, and then applying the Borel construction. Similarly, when proving Theorem 2.2, it suffices to show that the homotopy colimit hocolim_{⟶}_{R_{p}(G)} (I) is Z_{(p)}-acyclic. The difficult part of doing this is the following result about transformation groups:

Theorem 2.3. For any compact Lie group G and any prime p, there exists a finite dimensional 𝔽_{p}-acyclic G-CW-complex X such that all isotropy groups G_{x} for x ∈ X are p-stubborn.

Theorem 2.2 follows from Theorem 2.3 by first constructing a map X → hocolim_{⟶} (I), and then showing that it is an 𝔽_{p}-homology equivalence by showing that it restricts to an 𝔽_{p}-homology equivalence on all fixed point sets of isotropy subgroups. The key observation when doing this is that for any full subcategory C ⊆ O(G), (hocolim_{⟶}_{C} (I))^{H} is contractible for all G/H in C. For the details of these arguments, see [JMO, Section 1].

The G-CW-complex X of Theorem 2.3 is constructed in two steps. One first constructs a G-CW-complex all of whose isotropy subgroups are p-toral, and then one eliminates those which are not p-stubborn (this is the origin of the term pstubborn). The proof parallels Oliver’s construction of fixed-point free actions on acyclic complexes [Ol1], and also uses some arguments from his proof of the Conner conjecture [Ol2].

2b. Approximation via centralizers of elementary abelian p-subgroups.

Diagram (3) above admits a second generalization to arbitrary compact Lie groups. For any G, we let A_{p}(G) be the category introduced by Quillen [Q1], whose objects are nontrivial elementary abelian p-subgroups, and whose morphisms are restrictions of inner automorphisms of G. Note that SO(3) has precisely two conjugacy classes of nontrivial elementary abelian subgroups, and that C(ℤ/2) ≅ O(2) and C((ℤ/2)^{2}) ≅ (ℤ/2)^{2}. Here, C(−) denotes the centralizer of a subgroup in SO(3). Thus, diagram (3) above can be considered as being indexed over A_{2}(SO(3)), and defined by the functor which sends every elementary abelian 2-subgroup to the classifying space of its centralizer. This viewpoint has been generalized in [JM2] to arbitrary compact Lie groups, as follows.

On the opposite category A_{p}^{o}(G), we define a functor

EG ×_{G} (G/C(−)) : A_{p}^{o}(G) → T op

which assigns to every elementary abelian subgroup A ∈ A_{p}^{o}(G) the space EG ×_{G} (G/C(A)) ≃ BC(A). Projection maps induce a natural transformation a from the functor EG ×_{G} (G/C(−)) to the constant functor BG.

Theorem 2.4. [JM2, Theorem 1.3] The projection maps EG ×_{G} (G/C(−)) ↠ BG induce a map

α_{G} : hocolim_{⟶}_{A_{p}^{o}(G)}(EG ×_{G} (G/C(−))) ⟶ BG

which is an 𝔽_{p}-cohomology isomorphism.

The proof in [JM2] of Theorem 2.4 is based on the Bousfield-Kan spectral sequence (Theorem 1.1 above) for computing the cohomology of the homotopy colimit. One first proves, using the Becker-Gottlieb transfer and Feshbach’s double coset formula, that

Res : H^{∗}(BG; 𝔽_{p}) → lim_{⟵}_{A_{p}^{o}(G)} H^{∗}(EG ×_{G} G/C(−); 𝔽_{p})

is an isomorphism. It then remains to show that the higher inverse limits

lim_{⟵}^{i}_{A_{p}^{o}(G)}(H^{∗}(EG ×_{G} G/C(−); 𝔽_{p}))

all vanish for i ≥ 1. This is again a consequence of the existence of transfer maps for the functor H^{∗}(EG ×_{G} G/C(−); 𝔽_{p}). It turns out to be a Mackey functor in a sense close to the one introduced by Dress (cf. [Dr, §1]). A general, algebraic theorem is proven in [JM2] (Theorem 4.6 below), which says that all higher inverse limits vanish for such functors.

Theorem 2.4 was the starting point for a generalization obtained by Dwyer & Wilkerson [DW1]. The key observation for their generalization was a theorem of Dwyer & Zabrodsky [DZ], stated as Theorem 3.1 below. In particular, their theorem says that when A is an elementary abelian p-group and G is a compact Lie group, then conjugacy classes of homomorphisms A → G can be identified with homotopy classes of maps BA → BG. Inclusions correspond to the maps f : BA → BG such that the induced homomorphism H^{∗}(f^{∗}; 𝔽_{p}) makes H^{∗}(BA; 𝔽_{p}) into a finitely generated H^{∗}(BG; 𝔽_{p})-module. Moreover, for any inclusion i : A ↪ G, there is a natural map e_{i} : BC_{G}(A) → map(BA, BG)_{Bi} into the connected component containing the map Bi; and e_{i} becomes a homotopy equivalence after p-completion.

These remarks suggest a definition of the category of “elementary abelian subgroups of an arbitrary space”. For any X, let A_{p}^{o}(X), be the category whose objects are homotopy classes of those maps f : BA → X, for which A is a nontrivial elementary abelian p-group and H^{∗}(f; 𝔽_{p}) makes H^{∗}(BA; 𝔽_{p}) into a finitely generated H^{∗}(X; 𝔽_{p})-module. Morphisms in A_{p}^{o}(X) are homotopy classes Bφ : BA′ → BA of maps over X. A functor A_{p}^{o}(X) → T op is defined, which assigns to every map f : BA → X its connected component map(BA, X)_{f} in the mapping space. When X = BG for a compact Lie group G, then this functor is naturally 𝔽_{p}-homology equivalent (under the equivalence of categories A_{p}^{o}(BG) = A_{p}^{o}(G)) to the functor EG ×_{G} G/C(−) defined above. This motivates the following theorem:

Theorem 2.5. (Dwyer & Wilkerson [DW1]) Let X be a space whose cohomology H^{∗}(X; 𝔽_{p}), as an unstable algebra over the Steenrod algebra, is isomorphic to the subalgebra of elements fixed by an action of a finite group on H^{∗}(BT; 𝔽_{p}) for some torus T. Then the evaluation map induces an 𝔽_{p}-homology equivalence:

α_{X} : hocolim_{⟶}_{A_{p}^{o}(G)}(map(B−, X)_{−}) → BG

The proof of Theorem 2.5 follows from a purely algebraic result which involves unstable modules over the Steenrod algebra. Similarly to the proof in [JM2] of Theorem 2.4 above, the proof of Dwyer & Wilkerson is based on the existence of certain transfer maps.

Dwyer & Wilkerson found various other conditions on X which imply that α_{X} is an 𝔽_{p}-homology equivalence. They also noted (at the end of [DW1]) that this is not the case when p = 2 and X = Bℤ/2∨Bℤ/2. It would be quite interesting to have a characterization of the spaces X for which α_{X} is an 𝔽_{p}-homology equivalence, since such spaces could be expected to have many of the good properties of classifying spaces.

3. Maps between classifying spaces

The title of this section was first used by Adams & Mahmud [AM], and since then by many authors. At the beginning of their paper, Adams and Mahmud explained their interest in the subject as follows: “Let G and G′ be compact connected Lie groups … The object of this paper is to study maps f : BG → BG′, since this seems to be a case of the homotopy classification problem which is both particularly interesting and particularly favourable.” This section provides an update of this problem, and a confirmation of the prediction of Adams and Mahmud.

The results of Adams & Mahmud, both in [AM] and in their later papers [Ad] and [AM2], classified maps BG → BG′ only up to what was detected by rational cohomology or K-theory, and constructed maps only after inverting certain primes in BG′. The main tools which now make it possible to study such maps more precisely are a series of consequences of the proof of the Sullivan conjecture ([Mil], [Ca], [La2]). The results of principal importance are those of Dwyer & Zabrodsky [DZ], and their extension by Notbohm [No].

As mentioned in the introduction, the construction by Sullivan of unstable Adams operations destroyed the hope that all maps BG→BG′ might be homotopic to maps induced by homomorphisms. In contrast, the next theorem says that in the special case when G is p-toral (i.e., an extension of a torus by a finite p-group), then not only are all maps induced by homomorphisms, but all homotopies between maps are induced by conjugations, and the higher homotopies also have a group theoretic description.

For any G and G′, define Rep(G, G′) = Hom(G, G′)/ Inn(G′); i.e., the set of homomorphisms ρ : G → G′ modulo conjugation in G′. Since Bα ≃ Id_{BG′} for any α ∈ Inn(G′), there is a well defined map

B : Rep(G, G′) → [BG, BG′]

defined by sending ρ to Bρ.

Theorem 3.1. (Dwyer & Zabrodsky [DZ], Notbohm [No]) Fix a p-toral group P, and any compact Lie group G. Then

B : Rep(P, G) →^{≅} [BP, BG]

is a bijection, and the completion map

[BP, BG] ↣ [BP, BG_{p̂}]

is injective. Furthermore, for any ρ : P → G,

π_{∗} (map(BP, BG_{p̂})_{Bρ}) ≅ π_{∗} (map(BP, BG)_{Bρ}) _{ p̂} ≅ π_{∗} (BC_{G}(Im(ρ))) ⊗ ℤ̂_{p}.

Theorem 3.1 was shown by Dwyer & Zabrodsky when P is a finite p-group, and extended by Notbohm to the case when P is p-toral.

The homotopy isomorphisms in Theorem 3.1 are induced by the map BC_{G}(Im(ρ)) → map(BP, BG), which is in turn the adjoint to the map B (Im(ρ) × P → G).

The following corollary to Theorem 3.1 will also be useful in the discussion below.

Corollary 3.2. (Notbohm [No]) Let T be a torus, and let G be any compact connected Lie group. Then two maps f, f′ : BT → BG are homotopic if and only if H^{∗}(f; ℚ) = H^{∗}(f′; ℚ).

Again let G and G′ be compact Lie groups, where G′ is connected. For each prime p and each p-toral subgroup P ⊆ G (in particular, for each p-stubborn subgroup), Theorem 3.1 describes the set of homotopy classes of maps BP → BG′, as well as the homotopy type of each component of map(BP, BG′_{p̂}). This is exactly what is needed when applying Theorem 1.2 or 1.3 to describe the set of homotopy classes of maps from the homotopy colimit of the BP ’s to BG′_{p̂}; and hence (using Theorem

2.2) from BG to BG′_{p̂}.

We are now ready to outline our general strategy for applying these techniques to construct and classify maps BG → BG′, when G and G′ are compact Lie groups and G′ is connected. Of particular interest are the cases where G is connected or finite. As will be seen shortly, the condition that G′ be connected is needed since we have to be able to apply Sullivan’s arithmetic pullback square for localizations and completions of BG′.

Now fix G and G′ as above. Choose maximal tori T ⊆ G and T′ ⊆ G′, and let W = N(T)/T and W′ = N(T′)/T′ denote the Weyl groups. The construction and classification of maps BG → BG′ is carried out in three steps:

Step 1 Admissible maps.

For the purposes of this survey, we define an admissible map to be a homomorphism ϕ : T → T′, for which there exists a homomorphism ϕ̄ : W → W′ such that ϕ is ϕ̄-equivariant. As noted below, this is more restrictive than the definition of Adams & Mahmud in [AM]. But the ones we consider are the only ones which can induce maps BG → BG′ which are defined globally (rather than after a finite localization).

Theorem 3.3. (Adams & Mahmud [AM, Corollary 1.11]) For any f : BG → BG′, there exists an admissible map ϕ : T → T′ such that the following square commutes up to homotopy:

BT ⟶^{Bϕ} BT′
↓^{incl}  ↓^{incl}
BG ⟶^{f} BG′

(1)

If ϕ′ : T → T′ is any other homomorphism for which this square commutes up to homotopy, then ϕ′ = w ∘ ϕ for some element w ∈ W′. And conversely, if ϕ : T → T′ is any admissible map, then for some n there is a map f : BG → BG′[1/n] which makes (1) commute up to homotopy.

In fact, Adams and Mahmud showed only that the square in Theorem 3.3 commutes in rational cohomology. But by Notbohm’s lemma (Corollary 3.2 above), this is equivalent to its being homotopy commutative.

Adams & Mahmud in [AM] actually define an admissible map to be a linear map ϕ̃ : T̃ → T̃′ between the universal covers, such that for some n ≥ 1, n · ϕ̃ covers a homomorphism ϕ : T → T′ which is admissible in the sense defined above. Using this definition, their theorem says that square (1) above induces a one-to-one correspondence between

(i) homomorphisms H^{∗}(BG′; ℚ) → H^{∗}(BG; ℚ) which are induced by maps f : BG → BG′[1/n] (for some n ≥ 1); and

(ii) W′-conjugacy classes of admissible maps ϕ. ϕ̃: T̃ → T̃′.

We are now left with the problem: given an admissible map ϕ : T → T′, is it induced by some f : BG → BG′? And if so, by how many homotopy classes of maps? Since we want to answer these by replacing BG by a p-local approximation, we must first consider extensions of ϕ to maps BG → BG′_{p̂} for the individual primes p. The following proposition describes how this is done.

Proposition 3.4. Fix an admissible map ϕ : T → T′, and let [BG, BG′]_{ϕ}, and [BG, BG′_{p̂}]_{ϕ}, be the sets of homotopy classes of maps whose restrictions to BT are homotopic to Bϕ. Then

[BG, BG′]_{ϕ} ≅ ∏_{p||W |}[BG, BG′_{p̂}]_{ϕ}.

The proof of Proposition 3.4 is based on Sullivan’s arithmetic pullback square (applied to the simply connected space BG′), together with the facts that that H^{∗}(BG; ℚ) vanishes in odd dimensions, and that BG′_{ℚ} is a product of Eilenberg-Maclane spaces. The details are given in [JMO, Theorem 3.1]. The reason for taking the product only over primes dividing the order of W is that [BG, BG′_{p̂}]_{ϕ} always has order 1 when p ∤ |W | (see Theorem 3.6 below).

Step 2 R_{p}-invariant representations.

We now fix a prime p, and consider the problem of determining the set of homotopy classes of maps BG → BG′_{p̂} which extend a given admissible map ϕ : T → T′. Fix a maximal p-toral subgroup N_{p}(T) ⊆ G: i.e., a subgroup for which N_{p}(T)/T is a p-Sylow subgroup of N(T)/T.

By an R_{p}-invariant representation on G we mean an element

ρ ∈ Rep(N_{p}(T), G′) = Hom(N_{p}(T), G′)/ Inn(G′)

with the property that the restrictions of ρ combine to form an element

ρ̂ = (ρ | P)_{G/P ∈R_{p}(G)} ∈ lim_{⟵}_{G/P ∈R_{p}(G)} Rep(P, G′).

Equivalently, ρ : N_{p}(T) → G′ is R_{p}-invariant if for any p-toral P ⊆ G and any two homomorphisms i_{1}, i_{2} : P ↣ N_{p}(T) induced by inclusions and conjugation in G, ρ ∘ i_{1} is conjugate (in G′) to ρ ∘ i_{2}.

By Theorem 3.1, [BP, BG′_{p̂}] ⊇ [BP, BG′] ≅ Rep(P, G′) for all p-toral P. So for any f : BG → BG′_{p̂} which comes from a global map BG → BG′, f | BN_{p}(T) ≃ Bρ for some unique R_{p}-invariant representation ρ : N_{p}(T) → G′. The question now is: given an admissible map ϕ : T → T′, can it be extended to an R_{p}-invariant representation, and if so to how many?

When G′ is one of the matrix groups U(n), SU(n), Sp(n), or O(n), then R_{p}- invariant representations are relatively easy to construct: character theory can be used to verify that the appropriate pairs of homomorphisms are conjugate. For example, assume G = G′ = U(n), fix p ≤ n, and let T ⊆ G be the group of diagonal matrices. For some k prime to n!, consider the admissible map ϕ_{k} : T → T given by ϕ_{k}(t) = t^{k}. Then N(T) is the group of monomial matrices (one nonzero entry in every row and column). Let ρ_{k} : N_{p}(T) → G, be the homomorphism defined by raising each entry in a matrix to the k-th power. Then for every x ∈ N_{p}(T), Tr(ρ_{k}(x)) = Tr(x^{k}); and using this relation one easily checks that ρ_{k} is R_{p}-invariant. Later in this section, we will show how the ρ_{k} can be used to give a new construction of Sullivan’s unstable Adams operations on BU(n).

Several examples are given in [JMO2], where we construct R_{p}-invariant representations (both to matrix groups and to the exceptional Lie group F_{4}), extending some of the admissible maps studied by Adams & Mahmud in [AM]. But we do still lack general techniques for doing this. For example, this is the missing step if we want to construct the unstable Adams operations for the exceptional Lie groups using these methods.

Step 3 Computation of higher limits.

An R_{p}-invariant representation ρ determines a family of maps, compatible up to homotopy, from the BP ≃ EG/P to BG′_{p̂}. The obstructions to extending these to a map from hocolim_{⟶} (EG/P) to BG′_{p̂}—and hence from BG to BG′_{p̂}—have already been described in Theorems 1.2 and 1.3 above.

It will be convenient to adopt a different notation for higher inverse limits. We will see in the next section that these groups may be thought of as cohomology groups of the underlying category with twisted coefficients. Accordingly, from now on, we write H^{∗}(C; F) rather than lim_{⟵}_{C}^{∗}(F) to denote the higher inverse limits of F : C → Ab.

For a given ρ, define functors

Π_{1}^{ρ} : R_{p}(G) → p-groups  and  Π_{n}^{ρ} : R_{p}(G) → ℤ_{(p)}-mod (n ≥ 2)

by setting

Π_{n}^{ρ}(G/P) = π_{n} (map(BP, BG′_{p̂})_{Bρ|P} ) ≅ \{ \begin{matrix}π_{1}(BC_{G′}(Im(ρ))) & if n = 1 \\ [π_{n}(BC_{G′}(Im(ρ)))]_{p̂} & if n ≥ 2. \end{matrix}

(Note that for any p-toral P ⊆ G′, π_{0}(C_{G′}(P)) is a p-group by [JMO, Proposition A.4].) Theorems 1.2 and 1.3 now take the following form:

Theorem 3.5. For any R_{p}-invariant representation ρ, Bρ extends to a map BG → BG′_{p̂} if the higher limits H^{n+1}(R_{p}(G); Π_{n}^{ρ}) vanish for all n ≥ 1; and the extension is unique if H^{n}(R_{p}(G); Π_{n}^{ρ}) = 0 for all n ≥ 1. Furthermore, there is a spectral sequence

E_{2}^{pq} = H^{p}(R_{p}(G); Π_{q}^{ρ}) ⇒ π_{q−p}(map(BG, BG′)_{ρ}),

where map(−, −)_{ρ} is the space of maps which extend Bρ.

In Proposition 4.11 below, we will see that there is a number k = k(G, p), such that H^{i}(R_{p}(G); F) = 0 for all i > k and any F : R_{p}(G) → ℤ_{(p)}-mod. So the spectral sequence of Theorem 3.5 always converges strongly.

The last step when constructing maps BG → BG′, or checking whether they are unique, is thus to compute the higher limits H^{∗}(R_{p}(G); Π_{∗}^{ρ}). A priori, one might expect the computation of these higher limits to be quite hard in general. However, as will be seen in the next section (Theorems 4.8 and 4.9), we have succeeded in developing some very powerful tools which are successful in making these computations in many cases.

As a first simple illustration of the application of the methods in this section, we show how they apply in the case where p ∤ |W |.

Theorem 3.6. [AM] If G is connected and p ∤ |W |, then any admissible map ϕ : T → T′ lifts to a unique map f : BG → BG′_{p̂}.

Proof. This is shown by Adams & Mahmud in [AM, Theorem 1.10]; but we note here how it follows from the theory just presented. When p ∤ |W |, then N_{p}(T) = T, and the only p-stubborn subgroups of G are the maximal tori (any p-stubborn subgroup P ⊆ G is contained in a maximal torus, and N(P)/P must be finite). In other words, R_{p}(G) is equivalent to the category with one object G/T, with End(G/T) ≅ W.

In particular, ϕ is automatically R_{p}-invariant. Also, for each i, j ≥ 1,

H^{i}(R_{p}(G); Π_{j}^{ϕ}) ≅ H^{i}(W; Π_{j}^{ϕ}(G/T)_{p̂}) = 0

(again since p ∤ |W |). So ϕ extends to a unique map BG → BG′_{p̂} by Theorems 3.5 and 2.2 above. □

As a second example, set G = G′ = U(n), fix p and k such that p|n! and (k, n!) = 1; and consider the R_{p}-invariant representation ρ_{k} : N_{p}(T) → G defined above. Then for each i ≥ 1, Π_{i}^{ρ_{k}} ≅ Π_{i}, where Π_{i} is defined by setting Π_{i}(G/P) = π_{i}(BC_{G}(P))_{p̂} for G/P in R_{p}(G). Also, H^{j}(R_{p}(G); Π_{i}) = 0 for all i, j ≥ 1 by Lemma 5.3 below. Theorem 3.5 thus applies to show that ρ_{k} extends to a unique map ψ^{k} : BG → BG_{p̂}. Since ψ^{k}|BT ≃ Bϕ_{k}, where ϕ_{k} : T → T is the k-th power map, ψ^{k} is an unstable Adams operation of degree k on BU(n) (see Definition 5.1 below).

As a last example, we use these procedures to classify maps BΓ → BSU(2) and BΓ → BSO(3) for any finite group Γ. Note that in these cases, Proposition 3.4 takes the form

[BΓ, BSU(2)] ≅ ∏_{p||Γ|}[BΓ, BSU(2)_{p̂}] and [BΓ, BSO(3)] ≅ ∏_{p||Γ|}[BΓ, BSO(3)_{p̂}].

Example 3.7. Fix a prime p and a finite group Γ, and let Γ_{p} ⊆ Γ be a Sylow p-subgroup. We consider maps BΓ → BSU(2)_{p̂}.

(i) A homomorphism ρ : Γ_{p} ⟶ SU(2) is R_{p}-invariant if and only if for any pair of elements g, h ∈ Γ_{p} conjugate in Γ, ρ(g) and ρ(h) are conjugate in SU(2).

(ii) Every R_{p}-invariant representation ρ : Γ_{p} → SU(2) lifts to a map f_{ρ} : BΓ → BSU(2)_{p̂} which is unique up to homotopy.

The above description of [BΓ, BSU(2)] was shown by Mislin and Thomas [MT, Theorem 3.2], in the case when Γ has periodic cohomology. In the same paper [MT, Theorem 2.6], they also describe [BΓ, BG] for any arbitrary compact connected Lie group G and any periodic group Γ satisfying the “2-normalizer condition”.

In all of the examples described in [MT] or in Example 3.7 above, homotopy classes of maps BΓ → BG are detected by their restrictions to Sylow subgroups of Γ; i.e., by the sets Rep(Γ_{p}, G). When G = SO(3), this is not always the case.

Example 3.8. Fix a prime p and a finite group Γ, and let Γ_{p} ⊆ Γ be a Sylow p-subgroup. We consider maps BΓ → BSO(3)_{p̂}.

(i) A homomorphism ρ : Γ_{p} ⟶ SO(3) is R_{p}-invariant if and only if for any pair of elements g, h ∈ Γ_{p} conjugate in Γ, ρ(g) and ρ(h) are conjugate in SO(3).

(ii) Every R_{p}-invariant representation ρ : Γ_{p} → SO(3) lifts to a map f_{ρ} : BΓ → BSO(3)_{p̂}; and the lifting is unique up to homotopy if p is odd or if Im(ρ) is abelian. In all other cases, there are at most two homotopy classes of maps BΓ → BSO(3)_{p̂} which lift ρ.

(iii) Assume that p = 2 and Im(ρ) ≅ D(2^{k}) for k ≥ 3. Let H_{1}, H_{2} ⊆ Im(ρ) be representatives for the two conjugacy classes of subgroups (ℤ/2)^{2} ⊆ D(2^{k}), and set P_{i} = ρ^{−1}(H_{i}). Then ρ has two distinct liftings BΓ \overrightarrow{⟶} BSO(3)_{ˆ2} if and only if both maps

N(P_{i})/P_{i} ⟶^{conj} Out(P_{i}/ Ker(ρ)) ⟶_{≅}^{ρ} Aut(H_{i}) ≅ Σ_{3}

are onto, and P_{1} and P_{2} are not conjugate in Γ.

Proofs. Point (i) in each case follows from the definition of R_{p}-invariance; and the fact that when G = SU(n) or SO(2n + 1), two homomorphisms H → G are conjugate in G if and only if they have the same character.

Points (ii) and (iii) follow upon showing that for each n ≥ 1, H^{m}(R_{p}(Γ); Π_{n}^{ρ}) = 0 for m ≥ 2, and that H^{1}(R_{p}(Γ); Π_{1}^{ρ}) has order one or two as indicated. This is an easy consequence of Theorems 4.8 and 4.9 and Proposition 4.10 in the next section. □

Note in particular that when p = 2 or 3, then any monomorphism from Γ_{p} to SO(3) or SU(2) is R_{p}-invariant. Also, when Γ = GL_{3}(𝔽_{2}) and ρ : Γ_{2}≅D(8)↣SO(3) is an injection, then point (iii) in Example 3.7 applies to show that there are two distinct homotopy classes of maps BΓ → BSO(3)_{ˆ2} which extend ρ. Another easy consequence of these examples is that there are exactly 12 homotopy classes of maps from BSL(2, 𝔽_{5}) to BSU(2) (compare with [Ad, Proposition 1.18]).

The reason the descriptions of [BΓ, BSO(3)] and [BΓ, BSU(2)] are so simple is in part because they are matrix rings (so conjugacy is easily determined), but mostly because their subgroups are well known and easily described. Presumably, similar results can be found (but with more complicated formulations) for maps to (for example) BSU(3) or BSp(2). However, it seems unlikely that any general result (or even conjecture) about [BG, BG′] can be formulated, neither for G and G′ arbitrary (distinct) compact connected Lie groups, nor for G finite and G′ connected.

4. Higher inverse limits

Let C be an arbitrary small category. We write C-mod for the abelian category of contravariant functors M : C → Ab. The reason for this terminology is that when C is the category O_{1}(Γ) (recall that this category has a single object Γ/1 and End(Γ/1) ≅ Γ)) then C-mod is just the usual category of ℤ[Γ]-modules. This example will be used throughout this section to illustrate some of the abstract categorical notions via examples related to finite groups.

There is a functor lim_{⟵} : C-mod → Ab which assigns to every M its inverse limit,

i.e., the group of compatible families of elements (x_{c})_{c∈Ob(C)}. We want to study the derived functors lim_{⟵}^{i} of the inverse limit (cf. [GZ, Appx. 2 §3], where the dual

construction is described in detail). For example, for any ℤ[Γ]-module M, regarded as a functor on O_{1}(Γ), lim_{⟵}_{O_{1}(Γ)}(M) can be identified with the fixed point subgroup

M^{Γ}, and thus the higher derived functors are isomorphic to H^{∗}(Γ; M). In order to emphasize this analogy with group cohomology, we shall denote the functors lim_{⟵}^{i} from now on by

H^{i}(C; −) : C-mod → Ab.

The following proposition is just a special case of the usual long exact sequence induced by derived functors.

Proposition 4.1. For any small category C, and any short exact sequence of functors 0 → M′ → M → M′′ → 0 in C-mod, there exists a functorial long exact sequence

⋯ → H^{i}(C; M′) → H^{i}(C; M) → H^{i}(C; M′′) → H^{i+1}(C; M) → ….

Let ℤ denote the constant functor. One sees easily that for any M in C-mod,

lim_{⟵}_{C}(M) ≅ Hom_{C-mod}(ℤ, M).

Hence, instead of taking an injective resolution for M when computing its higher inverse limits, one can choose a single projective resolution P_{∗} of ℤ, and then H^{∗}(C; M) ≅ H^{∗}(Hom_{C-mod}(P_{∗}, M)) for all M. One way to do this is the following.

For any c ∈ Ob(C), let A_{c} : C → Ab be the functor where for each x ∈ Ob(C), A_{c}(x) is the free abelian group with basis Mor_{C}(x, c). For any f : x → y in C, A_{c}(f) : A_{c}(y) → A_{c}(x) is induced by the obvious map between bases. Then for any M in C-mod,

Hom_{C-mod}(A_{c}, M) ≅ M(c).

In particular, A_{c} is projective. One now checks that there is a projective resolution of ℤ

⋯ ⟶ ⨁_{x_{0}→x_{1}→x_{2}} A_{x_{0}} ⟶ ⨁_{x_{0}→x_{1}} A_{x_{0}} ⟶ ⨁_{x} A_{x} ⟶ ℤ ⟶ 0

where the boundary maps are alternating sums of maps induced by face maps in the nerve of C. For example, when C = O_{1}(Γ), then this is the usual bar resolution for Γ. This resolution leads to the following lemma, which is sometimes useful for proving results about higher limits.

Lemma 4.2. [BK, XI,6.2] For any M in C-mod, H^{∗}(C; M) is the cohomology of the cochain complex

0 ⟶ ∏_{x} M(x) ⟶ ∏_{x_{0}→x_{1}} M(x_{0}) ⟶ ∏_{x_{0}→x_{1}→x_{2}} M(x_{0}) ⟶ …,

whose boundary maps are alternating sums of homomorphisms induced by the face maps in the nerve of C.

In order to describe one of the other properties of higher limits analogous to group cohomology, we first need to define the Kan extension of a functor. For any functor F : C → D, the restriction functor F^{∗} : D-mod → C-mod has a right adjoint

F_{∗} : C-mod → D-mod.

For any M in C-mod, the functor F_{∗}M is called the right Kan extension of M along F (cf. [HS, IX.5] or [GZ, Appx. 2, §3]). If F is an embedding onto a full subcategory, then F^{∗} ∘ F_{∗} = Id [HS, Proposition IX.5.2]; i.e., the Kan extension provides an ordinary extension from functors on C to functors on D. If F has a right adjoint functor G : D → C, then F_{∗} = G^{∗}. But in general, F_{∗} does not have to be induced by any functor D → C.

As a special case, consider a pair of groups H ⊆ Γ, and the induced functor F : O_{1}(H) → O_{1}(Γ). In this case, for an arbitrary H-module A, F_{∗}(A) ≅ Hom_{ℤH}(ℤΓ, A) is just the usual induced representation. The following result from [JM2] thus generalizes Shapiro’s lemma for group cohomology.

Lemma 4.3. [JM2, Lemma 3.1] Let F : C → D be a functor such that F_{∗} : C-mod → D-mod preserves epimorphisms. Then there is a natural isomorphism

H^{∗}(C; M) ≅ H^{∗}(D; F_{∗}M).

This lemma can be shown by adapting the usual proof of Shapiro’s lemma; or alternatively using the Leray spectral sequence of the functor F : C → D. Note that if F_{∗} = G^{∗} for some functor G : D → C, then F must preserve epimorphisms.

Of particular interest are the functors whose higher limits all vanish.

Definition 4.4. A functor M ∈ C-mod is called acyclic if H^{∗}(C; M) = 0 for all i > 0.

Below, we will review various criteria obtained in [JM2] and [JMO] for acyclicity of functors. The starting point for these results is the following simple observation:

Lemma 4.5. A functor M is acyclic if either of the following condition holds:

(a) M splits off as a direct summand of a acyclic functor

(b) M admits a filtration 0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M such that M_{i}/M_{i−1} is acyclic for any i = 1, …, n.

We begin with a result based on Lemma 4.5(a). Note first that every functor on a category with final object is acyclic. This, together with Lemma 4.3, provides us with a large class of acyclic functors, provided that the category C satisfies some mild assumptions. The splitting needed to apply Lemma 4.5(a) is provided by an additonal structure on a functor: the Mackey structure (i.e., transfer maps satisfying the double coset formula). The idea of abstract Mackey functors, generalizing the relations among the restriction and induction maps for representation rings, was introduced by Dress (cf. [Dr, Section 1]) in the context of abstract induction theory. In [JM2], the first two authors define “proto-Mackey functors” (a slightly weaker notion than Dress’, in particular not assuming the existence of an initial object in the category); and prove a general criterion for the vanishing of their higher limits [JM2, Corollary 5.16]. Here we quote only one sample application which is often useful.

Theorem 4.6. Let p be a prime, and let C be either the category O_{p}(Γ) for a finite group Γ, or A_{p}^{o}(G) for a compact Lie group G. Then every proto-Mackey functor C → ℤ_{(p)}-mod is acyclic.

An important example is the functor Γ/P ↦ M^{P} on O_{p}(Γ), which is defined for any Γ-module M. This functor possesses an evident transfer map and is therefore acyclic by Theorem 4.6. (Note, however, that in this particular case the composite of transfer and restriction is multiplication by the index, so one can give an elementary proof of acyclicity—see for example Lemma 2.1 of [Mis]).

To apply Lemma 4.5(b) to the computation of higher limits over orbit categories, we consider the atomic functors: i.e., functors F : C → Ab which vanish on all but possibly one single isomorphism class of objects. It is easy to see that every functor on an orbit category—or more generally on a category in which all endomorphisms are isomorphisms—admits filtrations whose quotient functors are atomic. Note that an atomic functor is determined by a single End(C)-module for some object C ∈ Ob(C). The surprising thing is that in some interesting cases, the higher limits of an atomic functor can be described solely in terms of this module without refering to the whole category C.

In order to make this explicit, we introduce a new invariant Λ^{∗}(Γ; M), for a finite group Γ and a Z_{(p)}[Γ]-module M.

Definition 4.7. For any prime p, any finite group Γ, and any ℤ_{(p)}[Γ]-module M, let F_{M} : O_{p}(Γ) → Ab be the atomic functor concentrated on the free orbit Γ/1 with F_{M}(Γ/1) = M; and set

Λ^{∗}(Γ; M) = H^{∗}(O_{p}(Γ); F_{M}).

Note that Λ^{∗}(Γ; M) depends implicitly on the prime p, as well as on Γ and M. Also, by [JMO, Corollary 1.8], H^{∗}(O_{p}(Γ); F) ≅ H^{∗}(R_{p}(Γ); F) for any finite group Γ and any F in O_{p}(Γ)-mod (so Λ^{∗}(Γ; M) could just as easily be defined as the higher limits of a functor over R_{p}(Γ)).

We now concentrate on functors in R_{p}(Γ)-mod, where Γ is an arbitrary compact Lie group. The change of categories lemma (Lemma 4.3) implies easily:

Theorem 4.8. [JMO, Lemma 5.4] For any atomic functor F : R_{p}(G) → ℤ_{(p)}-mod concentrated on the orbit G/P,

H^{∗}(R_{p}(G); F) ≅ Λ^{∗}(N(P)/P; F (G/P)).

This method of filtering the higher limits H^{∗}(R_{p}(G); −) looks quite convenient, but it is not yet clear why it should be very useful for concrete calculations. The reason is that for many such computations, most of the groups Λ^{∗}(Γ; M) which occur are zero. This is due mainly to the following vanishing result, shown in [JMO, Propositions 5.5 and 6.1].

Theorem 4.9. Fix a prime p, a finite group Γ, and a ℤ_{(p)}[Γ]-module M. Then Λ^{∗}(Γ; M) = 0 (for all ∗ ≥ 0) if p||Ker[Γ → Aut(M)]|, or if Γ contains a nontrivial normal p-subgroup.

The following theorem describes some of the other basic properties of the functors Λ^{∗}, and gives some feel for how these groups behave:

Proposition 4.10. Fix a prime p, a finite group Γ, and a ℤ_{(p)}[Γ]-module M.

(a) If p ∤ |Γ|, then Λ^{0}(Γ; M) ≅ M^{Γ}, and Λ^{i}(Γ; M) = 0 for i ≥ 1. If p||Γ|, then Λ^{0}(Γ; M) = 0.

(b) Λ^{∗}(Γ; M) ≅ Λ^{∗}(Γ/H; M) if H ◁ Γ is the kernel of the Γ-action on M and p ∤ |Γ|.

(c) Let Γ_{p} ⊆ Γ be a Sylow p-subgroup, and assume |Γ_{p}| = p. Then Λ^{1}(Γ; M) ≅ M^{N(Γ_{p})}/M^{Γ}, and Λ^{i}(Γ; M) = 0 for i ≠ 1.

Proposition 4.10 is shown in [JMO]: the first two parts in Proposition 6.1, and the third in Proposition 6.2.

The functors Λ^{∗} enable us to show the following finiteness result which holds for an arbitrary functor F : R_{p}(G) → ℤ_{(p)}-mod:

Proposition 4.11. For any G and p, there is some k = k(G, p) such that for any functor F : R_{p}(G) → ℤ_{(p)}-mod, H^{∗}(R_{p}(G); F) = 0 for i > k. If G is finite and its Sylow p-subgroup has order p^{m}, then we can take k = m.

By [JMO, Proposition 1.6], any R_{p}(G) contains only finitely many isomorphism classes of objects (i.e., G contains finitely many conjugacy classes of p-stubborn subgroups). So upon filtering F by atomic functors, Proposition 4.11 is reduced to proving the corresponding result for the Λ^{∗}(Γ; M), for finite groups Γ. And this is shown via induction on the order of Γ, and using the acyclicity (Theorem 4.6) of the functor F′_{M} in O_{p}(Γ)-mod defined by F′_{M}(Γ/P) = M^{P}.

One important consequence of Proposition 4.11 is that the spectral sequences for spaces of maps BG → X_{p̂} (Theorems 1.3 and 3.5) always converge strongly.

5. Self maps of BG

In Section 3, we discussed our general strategy for constructing and classifying maps between classifying spaces. Those procedures are very effective in many concrete situations. But it seems unlikely that there is any completely general description of the sets [BG, BG′] for arbitrary pairs G, G′ of compact connected Lie groups.

One case where there are general results of this type is that of self maps: either arbitrary self maps of BG when G is connected and simple, or ℚ-equivalences BG→BG when G is any compact connected Lie group. It is these results which are the subject of this section.

We start by defining the unstable Adams operations on BG. As mentioned earlier, it was Sullivan’s construction of unstable Adams operations on BU(n) which gave the first examples of maps between classifying spaces not induced by homomorphisms.

Let U = ⋃_{n=1}^{∞} U(n) be the infinite dimensional unitary group, and let T_{∞}⊆U be the subgroup of all diagonal matrices. The Adams operation ψ^{k} : K(X)→K(X) determines a (homotopy class of) maps BU→BU, also denoted ψ^{k}; and ψ^{k}|BT_{∞}≃Bϕ_{k} where ϕ_{k} : T_{∞}→T_{∞}⊆U is the k-th power homomorphism. In particular, the induced homomorphism H^{2i}(ψ^{k}; ℚ) is multiplication by k^{i} for each i. This motivates the following definition.

Definition-Proposition 5.1. Let G be any compact connected Lie group. A selfmap f : BG→BG is called an unstable Adams operation of degree k if any of the following equivalent conditions hold:

(i) For each i ≥ 0, H^{2i}(f; ℚ) is multiplication by k^{i}.

(ii) For any maximal torus T ⊆G, the following square commutes up to homotopy:

\begin{matrix}BT & ⟶^{incl} & BG \\ ^{Bϕ_{k}}↓ & & ^{f}↓ \\ BT & ⟶^{incl} & BG \end{matrix}  (ϕ_{k}(t) = t^{k})  (1)

(iii) For some embedding (equivalently, all embeddings) G↪U, the following square commutes up to homotopy:

\begin{matrix}BG & ⟶ & BU \\ ^{f}↓ & _{ψ^{k}}↓ \\ BG & ⟶ & BU \end{matrix}  (2)

Note that by Notbohm’s lemma (Corollary 3.2 above), square (1) commutes up to homotopy if and only if it commutes in rational cohomology. Similarly, square (2) commutes up to homotopy if and only if it commutes in rational cohomology: this follows from arguments involving the Chern character.

Unstable Adams operations were first constructed by Sullivan [Su], for G = U(n) or SU(n), and any k prime to n!. His idea was to use étale homotopy to relate the profinite completion BU(n)ˆ to the structure of BU(n) as a variety over ℂ without topological structure—and then consider the action of the Galois group Aut(ℂ) on it. This gave unstable Adams operations on the p-completions BU(n)_{p̂}; and they were then combined to get self maps of BU(n), using Sullivan’s arithmetic pullback square [Su, §3.4]. This construction was later extended by Wilkerson [Wi] to the case where G is an arbitrary compact connected Lie group and k is prime to the order of its Weyl group.

Another approach to constructing unstable Adams operations was given by Friedlander in [Fr2], who showed that BG can be approximated by classifying spaces of Lie groups over fields of finite characteristic. For example, when k = p is a prime, then BU(n)_{q̂}≃BGL_{n}(𝔽̄_{p})_{q̂} for all primes q≠p [Fr1]. The Frobenius automorphism of 𝔽̄_{p} then induces an unstable Adams operation (ψ^{p})_{q̂} of degree p on BU(n)_{q̂}.

Neither of these constructions gave any indication as to whether the unstable Adams operations are unique up to homotopy. The following theorem answers this question, and is one of the main results in [JMO].

Theorem 5.2. [JMO, Theorem 4.3] Let G be any compact connected Lie group. Then for each k, there is up to homotopy at most one unstable Adams operation ψ^{k} : BG→BG of degree k.

For G = SU(2), this result was proved by Mislin [Mis].

The proof of Theorem 5.2 provides another illustration of how the program outlined in Section 3 can be applied. To keep things short, we assume k ≠ 0 (the case k = 0 is in fact simpler, but must be considered separately).

Fix a maximal torus T ⊆G, and let W = N(T)/T be the Weyl group. We must show that the admissible map ϕ_{k} : T →T, where ϕ_{k}(t) = t^{k}, has at most one lifting to a map BG→BG. By Proposition 3.4, it suffices to show that for each p||W |, there is at most one lifting of ϕ_{k} to a map BG→BG_{p̂}.

Fix p, and let W_{p} = N_{p}(T)/T be a Sylow p-subgroup of W = N(T)/T. Then N_{p}(T) is a maximal p-toral subgroup of G. The first step is to show that ϕ_{k} extends to at most one R_{p}-invariant representation (up to conjugacy in G). If we fix one R_{p}-invariant representation ρ_{k} : N_{p}(T)→G and compare the other representations to it, we obtain a one-to-one correspondence between the set of all extensions of ϕ_{k} to representations N_{p}(T)→G and the group H^{1}(W_{p}; T). Under this correspondence, the R_{p}-invariant representations all lie in the image of H^{1}(W; T)_{(p)}↪H^{1}(W_{p}; T). Also, H^{1}(W; T) is a 2-group, and is detected by restriction to the reflections in W. Using this, it is straightforward to check that any other R_{p}-invariant representation must represent the trivial element in H^{1}(W; T), and hence is conjugate to ρ_{k}. For the details, see the proof in [JMO, Proposition 3.5].

It remains to show that Bρ_{k} : BN_{p}(T)→BG_{p̂} extends to at most one map BG→BG_{p̂}. By Theorem 3.5 above, this means showing that the higher limits H^{i}(R_{p}(G); Π_{i}^{ρ_{k}}) vanish for all i ≥ 1. One first checks that Π_{i}^{ρ_{k}} ≅ Π_{i} as functors R_{p}(G) → ℤ_{(p)}-mod, where Π_{i}(G/P) ≅ π_{i}(BC_{G}(P))_{p̂}. The latter homotopy group can actually be determined explicitly, since when P is p-stubborn, the centralizer C_{G}(P) is equal to the center Z(P) (see [JMO, Lemma 1.5]), and is in particular the product of a torus and a finite abelian group. Note, for example, that Π_{i} = 0 for i ≥ 3, and that Π_{1}(G/P) is abelian for all P.

The techniques described in Section 4 are now used to show that H^{m}(R_{p}(G); Π_{n}) = 0 for all m, n ≥ 1 (see [JMO, Sections 5 & 6], where this is shown for simply connected G). To illustrate these computations, we sketch the proof when G = U(n) (and the same proof works when G = SU(n) or Sp(n)).

Lemma 5.3. Fix any prime p, and set G = U(n). Then for any i ≥ 1,

H^{j}(R_{p}(G); Π_{i}) ≅ \{ \begin{matrix}0 & if j > 0 \\ π_{i}(BZ(G))_{p̂} & if j = 0. \end{matrix}

Proof. The proof is based on results and notation from Section 4. Recall in particular Theorem 4.8: if F is a p-local functor on R_{p}(G), and vanishes except on one orbit type G/P, then H^{∗}(R_{p}(G); F) ≅ Λ^{∗}(NP/P; F (G/P)).

Assume first that P ⊆G is a p-stubborn subgroup such that

Λ^{∗}(N(P)/P; Π_{i}(G/P)) ≠ 0.

We want to show that P contains a maximal torus. By Theorem 4.9, the kernel of the action of N(P)/P on Π_{i}(G/P) ≅ π_{i}(BZ(P))_{p̂} must have order prime to p. In particular, p ∤ |[N(P)∩C_{G}(Z(P))]/P |; and so P is a maximal p-toral subgroup of C_{G}(Z(P)) (see [JMO, Lemma A.2]). But Z(P) is abelian, and hence (since G = U(n)) is contained in a maximal torus. So C_{G}(Z(P)), and hence P, must contain maximal tori.

Now let Π̂_{i} ⊆ Π_{i} : R_{p}(G) → ℤ_{(p)}-mod be the subfunctor defined by setting Π̂_{i}(G/P) = Π_{i}(G/P) if P contains a maximal torus, and Π̂_{i}(G/P) = 0 otherwise. By what was shown in the last paragraph,

Λ^{∗}(NP/P; (Π_{i}/Π̂_{i})(G/P)) = 0

for each G/P in R_{p}(G). So H^{∗}(R_{p}(G); Π_{i}/Π̂_{i}) = 0 by Lemma 4.5 and Theorem 4.8; and H^{∗}(R_{p}(G); Π_{i})≅H^{∗}(R_{p}(G); Π̂_{i}) by Proposition 4.1.

Now fix a maximal torus T, and let W = N(T)/T be the Weyl group. Let α_{i} : R_{p}(W) → ℤ_{(p)}-mod be the functor

α_{i}(W/Q) = Π_{i}(G/P) ≅ π_{i}(BZ(P)) ≅ π_{i−1}(T^{Q}). (Q = P/T ⊆ W)

It is not hard to see that

H^{∗}(R_{p}(G); Π̂_{i}) ≅ H^{∗}(R_{p}(W); α_{i})

(regard R_{p}(W) as a subcategory of R_{p}(G)). But α_{i} is a Mackey functor in the sense of [JM2], and so its higher limits vanish by Theorem 4.6 above. More precisely, by [JMO, Proposition 5.2],

H^{j}(R_{p}(W); α_{i}) ≅ \{ \begin{matrix}0 & if j > 0 \\ π_{i−1}(T_{W}) & if j = 0. \end{matrix}

Since T^{W} = Z(G) (G = U(n)), this completes the proof. □

We are now ready to describe the set of homotopy classes of self maps of BG when G is connected and simple. Note that for such G, Rep(G, G) ≅ {0}⨿ Out(G), where 0 denotes the trivial homomorphism and Out(G) the outer automorphism group. By the smash product of two monoids M_{1}, M_{2} with zero element is meant the quotient monoid

(M_{1} × M_{2})/⟨(x_{1}, 0) = (0, 0) = (0, x_{2}) : x_{i} ∈ M_{i}⟩.

Theorem 5.4. Let G be any compact connected simple Lie group with maximal torus T and Weyl group W. Then there is a bijection

β : ({0} ⨿ Out(G)) ∧ {k ≥ 0 : k = 0 or (k, |W |) = 1} ⟶^{≅}[BG, BG]

of monoids with zero element, which sends (α, k) to ψ^{k}∘Bα (for any unstable Adams operation ψ^{k} of degree k). In particular, for any f, f′ : BG→BG, the following are equivalent:

(1) f and f′ are homotopic

(2) f|BT ≃f′|BT : BT →BG

(3) H^{∗}(f; ℚ) = H^{∗}(f′; ℚ).

Theorem 5.4 was proven in several stages. Hubbuck [Hub1], [Hub2] and Mahmud (see [AM]) showed, using methods involving the Steenrod algebra, that each f : BG→BG is the composite of an unstable Adams operation and a map induced by an automorphism of G. Later, Ishiguro [Is] showed that unstable Adams operations of type ψ^{k} exist only for k = 0 or k prime to the order of the Weyl group of G. A shorter proof of these results, using Notbohm’s theorem about p-toral groups (Theorem 3.1 above) is given in [JMO, Theorem 3.4].

As mentioned before, the existence of unstable Adams operations ψ^{k} : BG→BG, whenever (k, |W |) = 1, was shown by Sullivan [Su], Wilkerson [Wi] and Friedlander [Fr2]. And the last step in the proof of Theorem 5.4 was the result in [JMO] about uniqueness of unstable Adams operations, stated in Theorem 5.2 above.

The case of self maps of BG for an arbitrary compact connected Lie group G is much more complicated. In [JMO, Example 7.1], we construct two maps f, f′ : BSU(3)→BSO(8) which induce the same map on rational cohomology but are not homotopic. In particular, this shows that homotopy classes of self maps of BG are not detected by rational cohomology when G = SU(3)×SO(8) (i.e., conditions (1) and (3) in Theorem 5.4 are not equivalent in this case). However, if one restricts attention to ℚ-equivalences BG→BG for arbitrary connected G, then one does get a strong result, analogous to Theorem 5.4. One way of formulating it is the following.

Theorem 5.5. [JMO3] Let G be any compact connected Lie group, and let T ⊆G be a maximal torus. Then two ℚ-equivalences f, f′ : BG→BG are homotopic if and only if they have the same effect in rational cohomology, if and only if f|BT ≃f′|BT, if and only if f and f′ are extensions of the same surjective admissible map ϕ : T ↠T.

The set of ℚ-equivalences is described explicitly as follows. For each prime p||W |, let G⟨p⟩ ⊆ G be the subgroup generated by the simple summands of G whose Weyl group has order divisible by p. For each n ≥ 1, let H_{n} ◁ G be the product of all simple components (normal subgroups) of G isomorphic to SO(2n+1). Then a surjective admissible map ϕ : T ↠T can be extended to a ℚ-equivalence BG→BG if and only if the following two conditions hold:

(a) p ∤ |Ker(ϕ) ∩ G⟨p⟩| for all p||W |; and

(b) ϕ(H_{n} ∩ T) = H_{n} ∩ T for all n ≥ 1.

In other words, there is a one-to-one correspondence between ℚ-equivalences f : BG→BG and admissible surjections ϕ : T ↠T which satisfy conditions (a) and (b) above. There is also a version of Theorem 5.5 dealing with self equivalences of BG_{p̂} (also shown in [JMO3]).

The following example shows why condition (b) is needed in Theorem 5.5. Fix n ≥ 1, and set G = SO(2n+1)×Sp(n). The standard maximal tori T_{1}⊆SO(2n+1) and T_{2}⊆Sp(n) can be identified in a way such that the map ϕ which switches them is an admissible automorphism of the maximal torus T_{1}×T_{2} = T ⊆G. But ϕ cannot extend to a self map of BG, since BSO(2n+1) and BSp(n) have distinct homotopy types at the prime 2.

6. Homotopical uniqueness of classifying spaces

In the last section, we saw that very complete results can be obtained about self maps of classifying spaces of compact connected simple Lie groups. This is not their only unusual property. It turns out that in many cases, the p-completed homotopy types of classifying spaces of compact connected Lie groups are fully determined by their 𝔽_{p}-cohomology as algebras over the Steenrod algebra A_{p}. This is of course a very special phenomenon, even among other classifying spaces: for example, Bℤ/p^{2} and Bℤ/p^{3} have isomorphic cohomology in this sense, and are p-complete, but they are not homotopy equivalent.

Proving the uniqueness of BG in this sense means realizing an isomorphism H^{∗}(X; 𝔽_{p})≅H^{∗}(BG; 𝔽_{p}) of A_{p}-algebras by a map BG→X_{p̂}. The starting point for doing this is a theorem of Lannes (Theorem 6.1 below), which says that when V is an elementary abelian p-group and X satisfies certain mild conditions, then any A_{p}-algebra map H^{∗}(X; 𝔽_{p})→H^{∗}(BV; 𝔽_{p}) can be realized by a map BV →X_{p̂} (unique up to homotopy).

Let U denote the category of unstable modules over the Steenrod algebra A_{p}, and let K be the category of unstable A_{p}-algebras. For example, for any space X, H^{∗}(X; 𝔽_{p}) is in both K and U. For each elementary abelian p-group V, Lannes and Zarati constructed a functor T_{V} : K → K which is left adjoint to the tensor product functor −⊗H^{∗}(BV; 𝔽_{p}). In fact, T_{V} can be regarded as a functor either from K to itself or from U to itself; and as a functor U → U it preserves exact sequences and tensor products. For more details, see either [La1] or [La2].

For any space X, the evaluation map map(BV, X)×BV →X induces a map of cohomology

H^{∗}(X; 𝔽_{p}) → H^{∗}(map(BV, X); 𝔽_{p}) ⊗ H^{∗}(BV; 𝔽_{p});

and this is adjoint to a homomorphism

T_{V} (H^{∗}(X; 𝔽_{p})) → H^{∗}(map(BV, X); 𝔽_{p}).  (1)

(Indeed, the existence of this homomorphism was the reason for introducing the functor T_{V} in the first place.) Also, for any algebra K in K,

(T_{V} K)^{0} ≅ map[Hom_{K}(K, H^{∗}(BV)), 𝔽_{p}].

The following theorem thus describes cases in which (1) is an isomorphism.

Theorem 6.1. (Lannes [La2, Théorèmes 0.4–0.6]) Let V be an elementary abelian p-group, and let X be a p-complete space such that H^{∗}(X; 𝔽_{p}) is finite in each dimension. Then the following hold:

(i) [BV, X] ≅ Hom_{K} (H^{∗}(X; 𝔽_{p}), H^{∗}(BV; 𝔽_{p})) .

(ii) Assume that either T_{V} (H^{∗}(X; 𝔽_{p})) vanishes in dimension 1; or that there is some p-complete space Z and a map Z→ map(BV, X) such that the induced map T_{V} (H^{∗}(X; 𝔽_{p}))→H^{∗}(Z; 𝔽_{p}) is an isomorphism. Then the map (1) above is an isomorphism; i.e.,

H^{∗}(map(BV, X); 𝔽_{p}) ≅ T_{V} (H^{∗}(X; 𝔽_{p})).

Lannes’ theorem (part (i), at least) is needed to prove the theorem of Dwyer & Zabrodsky and Notbohm (Theorem 3.1 above), which plays such an important role when describing maps between classifying spaces. Lannes’ theorem has thus appeared indirectly in previous sections (via the use of Theorem 3.1), but in this one and the next it will be used explicitly.

Theorem 6.1 says in particular that if X and Y are p-complete spaces such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(Y; 𝔽_{p}) as A_{p}-algebras, then map(BV, X) and map(BV, Y) also have the same cohomology (componentwise) as A_{p}-algebras. When X = BG for some compact connected Lie group G, then this (with an appropriate choice of V) can often be used to construct a “maximal torus” BT →Y analogous to the maximal torus in G. Examples of this procedure will be given in the sketches of the proofs of Theorems 6.3 and 6.5 below.

We now turn to uniqueness results for classifying spaces. The ones discussed here are all of the form: for certain G and p, a p-complete space X is homotopy equivalent to BG_{p̂} if its 𝔽_{p}-cohomology is isomorphic to that of BG (as algebras over the Steenrod algebra A_{p}). The first result of this type was for the group G = SU(2):

Theorem 6.2. (Dwyer, Miller, & Wilkerson [DMW1]) Let p be any prime, and let X be any p-complete space such that H^{∗}(X; 𝔽_{p})≅H^{∗}(BSU(2); 𝔽_{p}) as A_{p}-algebras.

Then X≃BSU(2)_{p̂}.

For p = 2, Theorem 6.2 is proved using the pushout decomposition of BSO(3) described in diagram (2) of Section 2. First, a 2-complete space Y is constructed, together with a fibration K(ℤ/2, 1)→X→Y, such that H^{∗}(Y; 𝔽_{2}) ≅ H^{∗}(BSO(3); 𝔽_{2}). Then, Y is included in a homotopy commutative square

\begin{matrix}BD(8) & ⟶ & BO(2) \\ ↓ & & ↓ \\ BΣ_{4} & ⟶ & Y \end{matrix}  (2)

(and this is of course the hard part). Square (2) induces a map BSO(3)_{ˆ2} → Y, which is shown to be an 𝔽_{2}-homology equivalence and hence a homotopy equivalence. And this map is then lifted to a homotopy equivalence BSU(2)_{ˆ2}≃X.

If X is a space such that H^{∗}(X; 𝔽_{p})≅H^{∗}(BSU(2); 𝔽_{p}) (as A_{p}-algebras) for all primes p, then Theorem 6.2 says only that X has the same genus as BSU(2); i.e., X_{p̂}≃BSU(2)_{p̂} for all p. Results of Rector [Re] show that the genus of BSU(2) contains uncountably many distinct homotopy types.

The next theorem is a special case of a general existence and uniqueness theorem proven in [DMW2]. That full theorem will be stated in Theorem 7.1 below.

Theorem 6.3. (Dwyer, Miller, & Wilkerson [DMW2]) Let G be a compact connected Lie group, and let p be any prime which does not divide the order of the Weyl group of G. Then for any p-complete space X such that H^{∗}(X; 𝔽_{p})≅H^{∗}(BG; 𝔽_{p}) as A_{p}-algebras, X≃BG_{p̂}.

Theorem 6.3 is proven by showing that any p-complete X for which

H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) ≅ H^{∗}(BT; 𝔽_{p})^{W}

is homotopy equivalent to the Borel construction of W acting on BT_{ p̂} (and BG_{p̂}≃EW ×_{W} BT_{ p̂} when p ∤ |W |: see the discussion following Theorem 2.2 above).

Let V ⊆T be the subgroup of elements of order p. Lannes’ theorem (Theorem 6.1) is used to construct a map i : BV →X which induces the appropriate map in cohomology, and then to show that map(BV, X)_{i}≃BT_{ p̂}. This gives a “maximal torus” for X; i.e., a map BT_{ p̂}→X which induces the usual map H^{∗}(BG; 𝔽_{p})↣H^{∗}(BT; 𝔽_{p}) in cohomology. The rest of the proof involves showing that i can be extended to a map EW ×_{W} BT_{ p̂}→X defined on the Borel construction.

The most general uniqueness results so far are given in recent work of Notbohm. It is not yet clear how generally his methods apply, but it seems quite possible that they could lead to a proof of the following conjecture.

Conjecture 6.4. (Notbohm) Let G be a connected compact Lie group, and let p be any prime such that H^{∗}(BG; ℤ) is p-torsion free. Then for any p-complete space X such that H^{∗}(X; 𝔽_{p})≅H^{∗}(BG; 𝔽_{p}) as A_{p}-algebras, X≃BG_{p̂}.

In fact, no counterexamples to Conjecture 6.4 are known, even when H^{∗}(BG; ℤ) is not p-torsion free.

The following theorem describes some of the cases of the conjecture which have been proven by now. The proof has not yet appeared as a preprint, but Notbohm has been helpful enough to send us the details.

Theorem 6.5. (Notbohm) Conjecture 6.4 holds when p is an odd prime and G is one of the groups U(n), SU(n), Sp(n), or SO(n).

Notbohm has also proven the conjecture in a number of other cases, including many of those involving exceptional simple Lie groups.

In order to give a feeling for the techniques used to prove Theorem 6.5, we sketch the proof in the case when p is odd and G≅U(n). The proofs in the other cases differ only in occasional details. The same proof can also be used to prove Theorem 6.3 (but it is much simpler in that case).

Fix a p-complete space X, and let φ : H^{∗}(X; 𝔽_{p}) ⟶^{≅} H^{∗}(BG; 𝔽_{p}) be an isomorphism of A_{p}-algebras. Let T ⊆G be the standard maximal torus of diagonal matrices, and let W ≅ Σ_{n} be the Weyl group. The goal is to construct an 𝔽_{p}- cohomology equivalence BG→X realizing φ. This is done using the approximation of BG via classifying spaces of p-stubborn subgroups (Theorem 2.2).

The first step is to construct a map f_{T} : BT →X such that f_{T} ∘Bw≃f_{T} for each w∈W ⊆ Aut(G), and such that the following “big triangle” commutes:

H^{∗}(BT; 𝔽_{p})

f_{T}^{∗} ↗ ↖ (incl)^{∗}

H^{∗}(X; 𝔽_{p}) ⟶_{≅}^{φ} H^{∗}(BG; 𝔽_{p})

To do this, let V ≅ (ℤ/p)^{n} be the p-torsion subgroup in T. Let i ∈ [BV, X] be the component corresponding to the inclusion BV ↪BG, under the identification

\begin{aligned}[BV, X] & ≅ Hom_{K} (H^{∗}(X; 𝔽_{p}), H^{∗}(BV; 𝔽_{p})) \\ & ≅ Hom_{K} (H^{∗}(BG; 𝔽_{p}), H^{∗}(BV; 𝔽_{p})) ≅ [BV, BG] \end{aligned}

of Theorem 6.1(i). With proper choice of basepoint x∈BV, the evaluation map

e_{x} : map(BV, X)_{i} → X

is W -equivariant (with the trivial action on X). Also,

H^{∗}(map(BV, X)_{i}; 𝔽_{p}) ≅ H^{∗}(map(BV, BG)_{incl}; 𝔽_{p})  (Theorem 6.2(ii))
≅ H^{∗}(BC_{G}(V); 𝔽_{p}) ≅ H^{∗}(BT; 𝔽_{p}).  (Theorem 3.1)

This isomorphism between the 𝔽_{p}-cohomology rings of map(BV, X)_{i} and BT is equivariant with respect to the W -actions on the two spaces. Notbohm then shows that H^{2}(map(BV, X)_{i}; ℤ̂_{p})≅H^{2}(BT; ℤ̂_{p}) as ℤ̂_{p}[W]-modules; and the isomorphism induces a homotopy equivariant homology equivalence [map(BV, X)_{i}]_{p̂}→BT_{ p̂}. Since X is p-complete, e_{x} factors through a map f_{T} : BT →X with the required properties.

The next step is to extend f_{T} to a map f_{NT} : BN(T)→X, and show that the “little triangle”

H^{∗}(BN(T); 𝔽_{p})

f_{NT}^{∗} ↗ ↖ (incl)^{∗}

H^{∗}(X; 𝔽_{p}) ⟶_{≅}^{φ} H^{∗}(BG; 𝔽_{p})

commutes. Here, f_{NT} : BN(T) → X is constructed by obstruction theory, by regarding BN(T) as the homotopy colimit of EN(T)/T ≃BT over the category O_{1}(W) of the free orbit W/1. By Theorem 1.2 above, the obstructions to doing this lie in the higher limits

H^{i+1}(W; π_{i}(map(BT, X)_{f_{T}})_{p̂}) ≅ H^{i+1}(W; π_{i}(BT)_{p̂}) ≅ H^{i+1}(Σ_{n}; (ℤ̂_{p})^{n}) = 0.

The commutativity of the little triangle then follows upon showing that H^{∗}(BU(n); 𝔽_{p})→H^{∗}(BT; 𝔽_{p}) has a unique lifting to H^{∗}(BN(T); 𝔽_{p}) (as maps of A_{p}-algebras).

Now, for each p-stubborn subgroup P ⊆N(T), set f_{P} = f_{NT} |BP. The third step is to show that the f_{P} are compatible up to homotopy; i.e., that they define an element in the inverse limit lim_{⟵}_{R_{p}(G)}[BP, X]. This is the trickiest part of the proof.

It requires a choice of conjugacy class representatives for the p-stubborn subgroups which have “sufficiently large” intersection with T, and a lemma which compares the mapping spaces map(BA, X) and map(BA, BT) for any abelian p-toral group A.

The last step is to show that the obstructions to extending the f_{P} to a map

BG ⟵^{∼} hocolim_{⟶}_{G/P ∈R_{p}(G)}(EG/P) ⟶^{f} X

all vanish. By Theorem 1.2, they lie in the higher limits

H^{i+1}(R_{p}(G); α_{i}),  where  α_{i}(G/P) = π_{i}(map(BP, X)_{f_{P}}).

One first shows that map(BP, X) and map(BP, BG) have the same homotopy (at least in the relevant components); so that α_{i}(G/P) ≅ π_{i}(BC_{G}(P))_{p̂} for each p-stubborn P ⊆G. Then α_{i} ≅ Π_{i} in the notation of [JMO, §§4-5], and H^{j}(R_{p}(G); Π_{i}) = 0 for all i, j ≥ 1 by Lemma 5.3 above.

7. Realizations of polynomial algebras

For any compact connected Lie group G, H^{∗}(BG; 𝔽_{p}) is a polynomial algebra over 𝔽_{p} for almost all (and in many cases all) primes p. It is thus natural to ask which other finitely generated polynomial algebras over 𝔽_{p} can be realized as the cohomology algebras of spaces.

Recall that if G is a compact connected Lie group with maximal torus T and Weyl group W, and p is prime to the order of W, then H^{∗}(BG; 𝔽_{p})≅H^{∗}(BT; 𝔽_{p})^{W}, and BG_{p̂} has the homotopy type of the Borel construction EW ×_{W} BT_{ p̂}. In other words, BG_{p̂} can be constructed in this case as a homotopy colimit involving only the space BT_{ p̂}. This procedure was extended by Clark & Ewing [CEw], to realize many other polynomial algebras as cohomology algebras of spaces. Later, Adams & Wilkerson [AW] proved that the methods of Clark & Ewing apply in the following generality:

Theorem 7.1. [AW], [DMW2] Fix a prime p, and let K^{∗} be any A_{p}-algebra which is polynomial on generators of degrees prime to p. Then there exists a simply connected p-complete space X, unique up to homotopy, such that H^{∗}(X; 𝔽_{p})≅K^{∗}.

The existence part of Theorem 7.1 is due to Adams & Wilkerson [AW]. They prove that (for p odd) any such A_{p}-algebra K^{∗} has the form K^{∗}≅H^{∗}(BT; 𝔽_{p})^{W} for some finite group W, some torus T, and some action of W on H^{∗}(BT; 𝔽_{p}). The results of Clark & Ewing [CEw] are then used to realize the W -action as an action on BT_{ p̂}; and K^{∗} is the cohomology of the Borel construction EW ×_{W} BT_{ p̂}. In fact, Adams & Wilkerson also gave necessary and sufficient conditions for any polynomial algebra with A_{p}-algebra structure to be the fixed point set of some finite group acting on H^{∗}(BT; 𝔽_{p}).

The uniqueness part of Theorem 7.1 is shown in [DMW2, Theorem 1.1]. The proof is essentially the same as that sketched in Section 6 (Theorem 6.3) for the special case when K^{∗}≅H^{∗}(BG; 𝔽_{p}) for some G.

Just as in the case of maps between classifying spaces and the uniqueness problem, the Borel construction must be replaced by more general homotopy colimits if one wants further results. The first examples of “exotic” spaces with polynomial 𝔽_{p}-cohomology, where p does divide the degrees of some generators, were given by Zabrodsky [Za] (who used an ad hoc construction) and Aguade [Ag] (who used homotopy colimits to recover Zabrodsky’s examples and create new ones). Here, “exotic” means that they are not the classifying spaces of compact Lie groups. The following theorem describes the general setup used when realizing these spaces.

Theorem 7.2. (Aguade [Ag]) Fix a prime p, a pair H⊆G of finite groups, and a finite 𝔽_{p}[G]-module V. Assume that H^{∗}(G; M)≅H^{∗}(H; M) (induced by the restriction map) for any 𝔽_{p}[G]-module M. Set P^{∗} = S^{∗}(V), the graded symmetric algebra, where V = P^{2}. Assume we are given a G-space X_{0} for which H^{∗}(X_{0}; 𝔽_{p})≅P^{∗} (as 𝔽_{p}[G]-algebras), a space X_{1} with H^{∗}(X_{1}; 𝔽_{p}) ≅ (P^{∗})^{H}, and a map f : X_{0}/H→H_{1} such that H^{∗}(f; 𝔽_{p}) is the inclusion (P^{∗})^{H}↪P^{∗}. Then there is a space X such that H^{∗}(X; 𝔽_{p}) ≅ (P^{∗})^{G} as algebras over the Steenrod algebra.

Aguade applied Theorem 7.2 to realize seven different algebras as cohomology algebras of spaces. For example, the theorem can be applied to the following triples (G, p, V), where V has the obvious structure as an 𝔽_{p}[G]-module:

(GL_{2}(𝔽_{3}), 3, (𝔽_{3})^{2}), (W (E_{7}), 5, (𝔽_{5})^{7}), (W (E_{7}), 7, (𝔽_{7})^{7}), (W (E_{8}), 7, (𝔽_{7})^{8}).

Here, W (E_{i}) denotes the Weyl group of E_{i}.

To prove Theorem 7.2, Aguade considered the category I = I(G, H), defined for any pair H⊆G, with two objects 0 and 1 and such that End(0)≅G, End(1) = 1, Mor(0, 1)≅G/H, and Mor(1, 0) = ∅. The spaces X_{0} and X_{1} in the hypotheses of the theorem thus define a functor F : I → T op; and one easily sees that H^{0}(I; H^{∗}(F (−); 𝔽_{p}))≅P^{G}. The assumption H^{∗}(G; M)≅H^{∗}(H; M) is used when showing that all higher inverse limits of the functor H^{∗}(F (−); 𝔽_{p}) vanish, and hence (using Theorem 1.1 above) that H^{∗}(hocolim_{⟶} (F); 𝔽_{p})≅P^{G}.

Aguade’s examples were all constructed at odd primes. The first example of such a space at the prime 2 was constructed recently by Dwyer & Wilkerson.

Theorem 7.3. (Dwyer & Wilkerson [DW2]) There exists a 2-complete space BD(4) whose 𝔽_{2}-cohomology is the ring of rank 4 Dickson invariants over 𝔽_{2}. In other words, as algebras over the Steenrod algebra,

H^{∗}(BD(4); 𝔽_{2}) ≅ H^{∗}(B(ℤ/2)^{4}; 𝔽_{2})^{GL_{4}(𝔽_{2})}.

The space BD(4) is also constructed as the homotopy colimit of simpler spaces, but this time over a much more complicated category: one which was motivated by the general decompositions in Theorems 2.4 and 2.5 above. Theorem 2.5 suggests that if BD(4) exists, then it should be the homotopy colimit of a diagram over the category of “injective” maps BV →BD(4); or (using Lannes’ theorem) the category whose objects are A_{2}-algebra homomorphisms

H^{∗}(B(ℤ/2)^{4}; 𝔽_{2})^{GL_{4}(𝔽_{2})} → H^{∗}(BV; 𝔽_{2})

where V is an elementary abelian 2-group and H^{∗}(BV) is a finitely generated module over the Dickson algebra. This category is equivalent to the category A, whose objects are the four vector spaces A_{i} = (𝔽_{2})^{i} for 1≤i≤4, and whose morphisms are the monomorphisms. A functor F : A^{o} → T op_{h} is then constructed, where T op_{h} is the homotopy category, and where F (A_{i}) = BC_{Spin(7)}(A_{i}) for an appropriate choice of embeddings A_{1}⊆A_{2}⊆A_{3}⊆A_{4}⊆ Spin(7). For example, F (A_{1}) = Spin(7), and F (A_{4}) = A_{4} ≅ (ℤ/2)^{4}. This step — defining F and checking that it is a well defined functor to T op_{h} — is the hardest part of the construction of BD(4).

Once F has been constructed, one wants to define BD(4) = hocolim_{⟶}_{A} (F), and then use the spectral sequence of Theorem 1.1 to show that H^{∗}(BD(4); 𝔽_{2}) is the rank 4 Dickson algebra. The calculations in this last step are similar to those used to prove Theorem 2.5. The problem is that homotopy colimits are defined only for functors defined to T op; not to the homotopy category. So it is also necessary to study the obstructions to lifting F to a functor F̃ : A → T op. It is this problem which will be discussed throughout the rest of the section.

Let F : C → T op_{h} be an arbitrary functor defined on a discrete category C. By a homotopy lifting of F is meant a functor F̃ : C → T op, together with a fixed homotopy equivalence α(x) : F̃(x) ⟶^{≃} F (x) for all x∈ Ob(C), such that for each x→y in C the obvious square commutes up to homotopy.

The obstructions to constructing a lifting have been studied by Dwyer & Kan: first for the general lifting problem in [DK1], and then for a certain specialized case in [DK2]. The problem of constructing a homotopy lifting of F is in fact equivalent to the problem of constructing the “homotopy colimit” of F in some appropriate sense. If F does lift to a functor F̃ : C → T op, then hocolim_{⟶} (F̃) is a homotopy colimit for F. Conversely, once p_{F} : hocolim_{⟶} (F) → C has been constructed with

the right properties, then F̃ can be defined by letting F̃(x) (for x∈ Ob(C)) be the pullback in the following square:

F̃(x) ⟶ hocolim_{⟶}_{C} (F)
↓  ↓^{p_{F}}
B(C↓x) ⟶^{BΦ_{x}}  BC

Here, C↓x is the category over x (Ob(C↓x) = {(y→x) ∈ Mor(C)}), and Φ_{x} : C↓x → C is the obvious functor.

A functor F : C → T op_{h} is called centric if for each f : x→y in C, the map

Aut(F (x))_{1} := map(F (x), F (x))_{Id} ⟶^{F (f)∘−} map(F (x), F (y))_{F (f)}

is a homotopy equivalence. This is a particularly convenient case to work with, because the obstructions to lifting F are actually higher inverse limits of functors on C. And many of the diagrams constructed when working with classifying spaces, including the decomposition of BG via p-stubborn subgroups (Theorem 2.2), and the functor F : A → T op_{h} constructed by Dwyer & Wilkerson and described above, are centric.

The next theorem describes the obstructions to diagram lifting in the special case where the homotopy functor is centric.

Theorem 7.4. (Dwyer & Kan [DK2]) Let F : C → T op_{h} be a centric functor from a (small) discrete category C to the homotopy category. For each i ≥ 1, let α_{i} : C → Ab denote the contravariant functor α_{i}(x) = π_{i}(Aut(F (x))_{1}). Then the obstructions to the existence of a homotopy lifting of F to F̃ : C → T op lie in the higher inverse limits lim_{⟵}^{i+2}(α_{i}), and the obstructions to the uniqueness of such a

lifting lie in the higher limits lim_{⟵}^{i+1}(α_{i}).

As suggested above, the most direct way to “see” how the obstructions in Theorem 7.4 arise is to try directly to construct an appropriate space hocolim_{⟶} (F). More precisely, we want to construct an identification space

hocolim_{⟶}(F) = ( ∐_{n≥0} ∐_{x_{0}→··→x_{n}} F (x_{0}) × Δ^{n})/ ∼,

where each face map between the x_{0}→ ⋯ →x_{n} gives rise to an identification between the corresponding F (x_{0})×Δ^{n}. The “1-skeleton” hocolim_{⟶}^{(1)}(F) is easily constructed, by taking the disjoint union of the F (x) (for all x∈ Ob(C)), and then attaching one copy of F (x) × Δ^{1} for each morphism x→y. The 2-skeleton is then obtained by attaching a copy of F (x) × Δ^{2} for each x ⟶^{f} y ⟶^{g} z in C, using the fact that F (gf) ≃ F (g) ∘ F (f) (F being a functor to the homotopy category).

The first obstructions to this procedure occur when constructing the 3-skeleton. For each sequence x_{0}→x_{1}→x_{2}→x_{3}, F (x_{0}) × Δ^{3} can be attached to the 2-skeleton, except for F (x_{0})×D^{2}, where D^{2} ⊆ ∂Δ^{3} is a small disk near the last vertex.

The obstruction to completing the attachment lies in

π_{1} (map(F (x_{0}), F (x_{3}))_{F (x_{0}→x_{3})}) ≅ π_{1} (Aut(F (x_{0}))_{1}) = α_{1}(x_{0}).

Thus, for each 3-simplex x_{0}→x_{1}→x_{2}→x_{3} in BC, we get an obstruction in α_{1}(x_{0}). Using Lemma 4.2 above, these obstructions can be combined to give an element in lim_{⟵}_{C}^{3}(α_{1}). This is the first obstruction to constructing hocolim_{⟶} (F); and the higher obstructions are obtained in a similar way.

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