CLASSIFYING SPACES AND HOMOLOGY DECOMPOSITIONS

W. G. DWYER

Abstract. We use homotopy methods to study the problem of calculating the cohomology of the classifying space BG of a finite group G. The basic procedure is to express BG, up to cohomology, as a homotopy colimit of classifying spaces of smaller groups. Doing this involves constructions and techniques from several different areas of topology: simplicial objects, the homotopy theory of categories, equivariant homotopy theory, . . ..

Contents

1. Introduction 1 2. Classifying spaces 3 3. Simplicial complexes and simplicial sets 6 4. Simplicial spaces and homotopy colimits 17 5. Nerves of categories and the Grothendieck construction 24 6. Homotopy orbit spaces 28 7. Homology decompositions 30 8. Sharp homology decompositions. Examples 35 9. Reinterpreting the homotopy colimit spectral sequence 39 10. Bredon homology and the transfer 42 11. Acyclicity for G-spaces 46 12. Non-identity p-subgroups 48 13. Elementary abelian p-subgroups 49 14. Appendix 52 References 54

1. Introduction

In these notes we will discuss a particular technique for trying to understand the classifying space BG of a finite group G. The technique is especially useful for studying the cohomology of BG, but it can also serve other purposes. The approach is pick a prime number p and construct BG, up to mod p cohomology, by gluing together classifying spaces of proper subgroups of G. A construction like this is called a homology decomposition of BG; in principle it gives an inductive way to obtain information about BG from information about classifying spaces of smaller groups. We look at these homology decompositions both to study them for their own sake and to illustrate how to use some basic topological techniques.

Date: March 2, 1998.

Section 2 introduces classifying spaces, shows how to construct them, and establishes some notation for their cohomology. Section 3 faces the issue that for our purposes it is much easier to work with combinatorial models for topological spaces than with topological spaces themselves. There is an extended attempt to motivate the particular combinatorial models we will use… they are called simplicial sets… and the section ends with a description of the simplicial sets that correspond to classifying spaces. Section 4 gives a systematic account of a whole class of gluing constructions called homotopy colimits. Homotopy colimits are most easily understood with the help of simplicial spaces. From the point of view of §3 simplicial spaces look a little peculiar, but there is no real way of avoiding them, and they lead to a very pretty spectral sequence for the homology of a homotopy colimit (4.16). Section 5 focuses on the nerve of a category; this is the simplest kind of homotopy colimit, namely the homotopy colimit of a constant functor with value a one-point space. It turns out that there is a way using the “Grothendieck Construction” to represent other more complex homotopy colimits as nerves. The advantage of doing this is that it is easy (by using natural transformations between functors) to construct combinatorial homotopies between maps of nerves. In contrast, constructing explicit combinatorial homotopies between maps of arbitrary simplicial sets is an error-prone business. Section 6 introduces another specialized homotopy colimit (the homotopy orbit space) associated to a group action.

In §7, at last, the main characters come on stage: there is a definition of homology decomposition in terms of homotopy colimits, and a simple explanation (using the results of §5) for the fact that any collection of subgroups of G which satisfies a certain “ampleness” property (7.7) gives rise to three distinct homology decompositions for BG. Section 8 points out that some homology decompositions, called sharp decompositions, are nicer than others, in that a sharp decomposition gives a formula for the homology of BG instead of just a spectral sequence. In this section there are also some examples of homology decompositions, although only a few trivial examples are fully worked out. The rest of the paper is devoted to working out some nontrivial cases. Section 9 shows that the spectral sequence associated to a homology decomposition can be interpreted as an “isotropy spectral sequence” associated to the action of G on a space X. The following sections (§10, §11) give a homological interpretation of the E^{2}-page of the isotropy spectral sequence, and develop methods for proving that this E^{2}-page collapses in such a way that H_{∗} BG appears along the vertical axis and all other groups are zero. If such a collapse takes place, the corresponding homology decomposition in sharp. Sections 12 and 13 apply this homological machinery to study two specific collections of subgroups of G, the collection of all nontrivial p-subgroups, and the collection of all nontrivial elementary abelian p-subgroups. It turns out that both collections are ample (as long as p divides the order of G), and that many of the six homology decompositions associated to these collections are sharp. The appendix contains a result on G-spaces which is referred to in §10.

2. Classifying spaces

Let G be a discrete group. Later on, we will assume that G is finite.

2.1. Definition. A classifying space for G is a pointed connected CW– complex B such that π_{1}B is isomorphic to G and π_{i}B is trivial for i > 1.

2.2. Remark. In practice, we will usually assume that a classifying space B for G comes with a chosen isomorphism ι_{B} : π_{1}B ≈ G.

2.3. Example. The circle S^{1} is a classifying space for the infinite cyclic group ℤ; real projective space ℝP^{∞} is a classifying space for ℤ/2. Let ℤ/p^{k} act on the unit sphere S^{2n−1} in ℂ^{n} in such a way that the generator acts by multiplying each coordinate by e^{2πi/k}. The union X = ∪_{n}S^{2n−1}/(Z/p^{k}) is a classifying space for ℤ/p^{k}. If X is a classifying space for G ahd Y is a classifying space for K, then X×Y is a classifying space for G × K.

2.4. Theorem. Any discrete group G has a classifying space.

Sketch of proof. Start with a presentation of G by generators {x_{α}} and relations {r_{β}}. Let B_{0} be a one-point space, and B_{1} a space obtained from B_{0} by attaching a 1-cell e_{α} for each x_{α}, so that B_{1} is a bouquet of circles and π_{1}B_{1} is isomorphic to the free group F (x_{α}) on the symbols {x_{α}}. Construct B_{2} from B_{1} by attaching a 2-cell e_{β} for each relation r_{β}; the attaching map for e_{β} should be the word in F (x_{α}) given by r_{β}. By the van Kampen theorem, π_{1}B_{2} is isomorphic in a natural way to G. Now by induction build B_{n} from B_{n−1} (n > 2) by attaching n-cells to B_{n−1} via maps {f_{γ} : S^{n−1} → B_{n−1}} which run through a collection of generators for π_{n−1}B_{n−1}. It is not hard to see that π_{1}B_{n} ≈ G and π_{i}B_{n} ≈ 0 for 2 ≤ i ≤ n − 1. Let B = ∪_{n}B_{n}. Since the sphere S^{k} is compact, any map f : S^{k} → B has image contained in B_{n} for some n, and so f is null homotopic for k ≥ 2. It follows from this that B is a classifying space for G.

With minor adjustments, the above method can be used to make more general constructions. Let X be a pointed space, W a pointed finite CW–complex, and Σ^{k}W = S^{k} ∧ W the k-fold suspension of W. Let X_{1} = X and by induction build X_{n} (n ≥ 2) from X_{n−1} by attaching a copy of a cone on Σ^{k}W for each map Σ^{k}W → X (k ≥ 0). Let X_{∞} = ∪X_{k}. Then any map f : Σ^{k}W → X_{∞} has image contained in X_{n} for some n, and so f is null homotopic; in fact X_{∞} is in a certain sense universal with respect to this mapping property [4]. The space X_{∞} is denoted P_{W} (X). One can remove the requirement that W be a finite complex by iterating the process transfinitely. One further generalization (replacing W by a map f : U → V) yields a functor L_{f}; essentially all known homotopy theoretic localization functors are of the form L_{f} for suitable f [5, 15].

For the rest of this section we will let BG denote some chosen classifying space for G.

2.5. Proposition. If X is any classifying space for G, then there is up to homotopy a unique basepoint-preserving map f : BG → X such that ι_{BG} = ι_{X} · f_{#} (cf. 2.2). The map f is a homotopy equivalence.

Sketch of proof. Let f send the basepoint of BG to the basepoint of X and each 1-cell e_{α} of BG to a loop in X representing (ι_{X})^{−1}ι_{BG}(⟨e_{α}⟩). Now extend over the 2-skeleton of BG by using the fact that any relation among the homotopy classes {⟨e_{α}⟩} in π_{1} BG also holds among their images in π_{1}X, and extend over higher skeleta of BG by using the fact that the higher homotopy groups of X are trivial. A similar argument gives a basepoint-preserving homotopy between f and any other map f′ of the same type. The map f is obviously a weak equivalence (i.e. induces isomorphisms π_{i} BG ≈ π_{i}X); since the domain and range of f are CW–complexes, f is a homotopy equivalence.

A similar argument shows that if Y is any pointed connected CW–complex, then the space of pointed maps Y → BG is weakly equivalent to the discrete set of homomorphisms π_{1}Y → G. What is the homotopy type of the space of all maps Y → BG?

Suppose that X is a pointed, connected CW–complex, and that X̃ is its universal cover. The group G = π_{1}(X) acts freely on X̃ (let’s say on the left) by covering transformations. Suppose that M is a right G-module and that N is a left G-module. For any space Y, let S_{∗}(Y) denote the singular chain complex of Y. Observe that S_{∗}(X̃) is a chain complex of free modules over ℤ[G]; this follows from the fact that if σ is a singular simplex of X, the singular simplices of X̃ which lift σ are permuted freely and transitively by G.

2.6. Definition. The homology of X with coefficients in M, denoted H_{∗}(X; M), is the homology of the chain complex M ⊗_{ℤ[G]} S_{∗}(X̃). The cohomology of X with coefficients in N, denoted H^{∗}(X; N) is the cohomology of the cochain complex Hom_{ℤ[G]}(S_{∗}(X̃), N).

These are sometimes called twisted or local coefficient (co-)homology groups of X. If ℤ is a trivial G-module, there is a natural isomorphism

ℤ ⊗_{ℤ[G]} S_{∗}(X̃) ≈ S_{∗}(X) .

If M is an arbitrary trivial G-module, the resulting isomorphisms

M ⊗_{ℤ[G]} S_{∗}(X̃) ≈ M ⊗_{ℤ} ℤ ⊗_{ℤ[G]} S_{∗}(X̃) ≈ M ⊗_{ℤ} S_{∗}(X)

show that H_{∗}(X; M) as defined above agrees with the usual singular cohomology of X with coefficients in M. A similar result holds for cohomology.

The antiautomorphism of G sending g to g^{−1} gives a way of passing from right Gmodules to left G-modules, and vice versa. For this reason the distinction between right G-modules and left G-modules is not too important, and from now on we will not always pay attention to it.

If X = BG, then X̃ is contractible, so S_{∗}(X̃) is a free resolution of the trivial G-module ℤ over ℤ[G] [21, V.11]. In this case elementary homological algebra gives formulas for the homology and cohomology groups of X.

2.7. Proposition. Suppose that M is a right G-module and that N is a left G-module. Then there are natural isomorphisms

H_{∗}(BG; M) ≈ Tor_{∗}^{ℤ[G]}(M, ℤ)
H^{∗}(BG; N) ≈ Ext_{ℤ[G]}^{∗}(ℤ, N)

For a general connected CW–complex X with fundamental group G there is a first-quadrant spectral sequence converging to H_{∗}(X; M) whose E_{2}-page depends in an algebraic way on the group G and the G-modules H_{∗}(X̃; ℤ) and M. Can you construct this spectral sequence and identify its E_{2}-page?

2.8. Remark. The twisted homology H_{∗}(BG; M) (resp. the twisted cohomology H^{∗}(BG; N)) is sometimes denoted H_{∗}(G; M) (resp. H^{∗}(G; N)) and called the group homology of G with coefficients in M (resp. the group cohomology of G with coefficients in N). In theory there’s a possibility of confusing H_{∗}(G; M) and H^{∗}(G; M) with the homology or cohomology of the discrete space G, but in practice this is not a problem. The groups H_{∗}(G; M) have an algebraic interpretation; they are (2.7) the left derived functors of the construction which assigns to any G-module M the abelian group M′ = M ⊗_{ℤ[G]} ℤ, i.e., the largest quotient M′ of M which is a trivial G-module. Similarly, H^{∗}(G; N) is the collection of right derived functors of the construction N ↦ N^{G} = Hom_{ℤ[G]}(ℤ, N).

Let G be a finitely generated abelian group. Compute H^{∗}(BG; 𝔽_{p}) and H^{∗}(BG; ℚ) as rings. What can you say if G is not finitely generated?

2.9. Remark. Fix a prime number p, and let 𝔽_{p} denote the field with p elements. Let G be a finite group. In these notes we will construct various maps f : A → BG which approximate BG up to (co- )homology, in the sense that f induces isomorphisms on H_{∗}(–; 𝔽_{p}) or H^{∗}(–; 𝔽_{p}). These approximations can be used to either to calculate the (co-)homology of BG or for other purposes (for instance, to study the set of homotopy classes of maps BG → X for appropriate spaces X).

One could also look for maps f : A → BG which induce isomorphisms on H_{∗}(–; M) or on H^{∗}(–; N) for suitable 𝔽_{p}[G]-modules M and N. Let G be a finite group, X a CW–complex with a map h : π_{1}(X) → G, and X′ the regular covering space of X corresponding to ker(h). Show that H_{∗}(X; 𝔽_{p}[G]) is isomorphic to H_{∗}(X′; 𝔽_{p})^{⊕n}, where n is the index of im(h) in G. Conclude that f : A → BG induces an isomorphism H_{∗}(A; M) → H_{∗}(BG; M) for all 𝔽_{p}[G]-modules M if and only if π_{1}(f) is surjective and the homotopy fibre of f has the 𝔽_{p}-homology of a point. Is it necessary to assume that G is finite? What happens if f induces an isomorphism H^{∗}(BG; N) → H^{∗}(A; N) for all 𝔽_{p}[G]-modules N? What happens if BG is replaced by another space Y?

2.10. Notation. From now on, all homology and cohomology will have have coefficients in 𝔽_{p}, unless some other coefficients are specified. If there is a group G involved, we will always assume that 𝔽_{p} has a trivial G-action.

3. Simplicial complexes and simplicial sets

In order to get much further with our project of approximating BG up to homology (2.9), we need to develop a method of describing spaces combinatorially. CW–complexes have nice topological properties, but it is not clear how to give explicit constructions of arbitrary large cell complexes.

Abstract simplicial complexes. An abstract simplicial complex K is a pair (V_{K}, S_{K}), where V_{K} is a set and S_{K} is a collection of finite subsets of V_{K}. The elements of V_{K} are the vertices of K and the elements of S_{K} are the simplices of K. The collection S_{K} is required to be closed under passage to subsets: if σ ∈ S_{K} and σ′ ⊂ σ, then σ′ ∈ S_{K}. To avoid carrying around extra baggage, we also insist that if v ∈ V_{K} the singleton subset {v} should belong to S_{K}. The subsets of a simplex σ are called its faces.

Associated to an abstract simplicial complex K is a space |K|, called the geometric realization of K, obtained as the space of formal linear combinations

∑_{v∈V_{K}} t_{v}v , (3.1)

where 0 ≤ t_{v} ≤ 1, supp{t_{v}} ∈ S_{K}, and ∑ t_{v} = 1. (Here supp{t_{v}} = {v | t_{v} ≠ 0}). The topology on |K| is given as follows [25, §1.1–3]: for each simplex σ ∈ S_{K} the subspace ∑_{v∈σ} t_{v}v is topologized as a subspace of ℝ^{card(σ)}, and then |K| is given the finest topology with respect to which the inclusions of these subspaces are continuous. A simplex σ ∈ S_{K} with card(σ) = n+1 is called an n-simplex of K, because the subspace of |K| corresponding to σ is a topological n-simplex.

Abstract simplicial complexes form the objects of a category ASC, in which a map K → L is a map of sets f : V_{K} → V_{L} such that for each simplex σ of K, f(σ) is a simplex of L. Such a map induces a continuous map f : |K| → |L| according to the formula

f(∑ t_{v}v) = ∑ t_{v}f(v) .

We don’t require f to be a monomorphism, so it might be necessary to rearrange terms on the right hand side of the above equality and add up the coefficients of any given vertex of L in order to obtain a sum in standard form (3.1).

3.2. Examples. The abstract n-simplex D[n] is the abstract simplicial complex whose vertex set is n = {0, 1, …, n} and whose collection of simplices is the set of all subsets of n. The geometrical realization |D[n]| is a standard topological n-simplex. The automorphism group of D[n] is the symmetric group Σ_{n+1}, which acts by permuting vertices. There is a unique map D[n] → D[0], which on taking geometrical realizations gives the map from a topological n-simplex to a one-point space.

Describing spaces in terms of abstract simplicial complexes has a direct geometric appeal [31, §3] [25, §1], but for a couple of reasons it is difficult to use the category of abstract simplicial complexes as a foundation for homotopy theory. One problem is that the categorical product in ASC is not well-behaved, in the sense that it does not commute with geometric realization.

8 W. G. DWYER

If X and Y are objects of some category C, an object Z of C is called a (categorical) product of X and Y if Z is provided with morphisms (called projection maps) pr_{1} : Z → X and pr_{2} : Z → Y such that for any object W of C, composition with these maps induces a bijection

Hom_{C}(W, Z) → Hom_{C}(W, X) × Hom_{C}(W, Y) . It is easy to check that if Z and Z′ are two products for X and Y, then there is a unique isomorphism Z → Z′ which is compatible with the projection maps, and so it is common to speak of “the” product of X and Y. See [22, III.4] for an extended discussion of products and, more generally, limits.

Suppose that K and L are abstract simplicial complexes. Let V_{K⊗L} denote the product V_{K} × V_{L}, let pr_{1} : V_{K⊗L} → V_{K} and pr_{2} : V_{K⊗L} → V_{L} be the (set theoretic) projection maps, and let S_{K⊗L} denote the collection of all subsets σ of V_{K⊗L} such that pr_{1}(σ) ∈ V_{K} and pr_{2}(σ) ∈ V_{L}.

3.3. Lemma. In the above situation, the abstract simplicial complex K ⊗ L = (V_{K⊗L}, S_{K⊗L}) is the product of K and L in the category ASC. In general, the natural map |K ⊗ L| → |K| × |L| is not a homeomorphism.

Proof. The first statement in the lemma is obvious. For the second, it’s only necessary to notice that D[1] ⊗ D[1] contains the 3-simplex {(0, 0), (0, 1), (1, 0), (1, 1)}.

A study of the example of D[1] ⊗ D[1] suggests that the categorical product K ⊗ L of two abstract simplicial complexes is too big because it is forced to be a target of too many maps. One way to deal with this problem is to reduce the number of maps in the category by introducing some extra structure on the objects. It turns out that a good way to do this is to introduce a total ordering on the vertex set of each simplex.

Ordered simplicial complexes. An ordered simplicial complex K is an abstract simplicial complex (V_{K}, S_{K}) with the property that V_{K} is furnished with a partial ordering [31, 1.1] which restricts to a total ordering on each simplex of K. The geometric realization |K| of an ordered simplicial complex is defined to be the geometric realization of the underlying simplicial complex. Ordered simplicial complexes form the objects of a category OSC, in which the morphisms are the maps K → L of simplicial complexes which respect the partial orderings on the vertex sets.

3.4. Remark. A set with a partial ordering is called a partially ordered set, or poset for short.

3.5. Examples. The ordered n-simplex Δ[n] is the ordered simplicial complex whose vertex set is the ordered set n (3.2) and whose collection of simplices is the set of all subsets of n. The geometrical realization |Δ[n]| is a standard topological n-simplex. The automorphism group of Δ[n] is trivial, since there are no nonidentity order-preserving bijections n → n. As in 3.2, there is a unique map Δ[n] → Δ[0] which corresponds under geometric realization to the map from a topological n-simplex to a one-point space.

3.6. Example. Suppose that S is a poset. Associated S is a “largest possible” ordered simplicial complex Cx(S) with vertex poset S; this is the complex whose simplices consist of all of the totally ordered subsets of S. Given any simplicial complex K, ordered or not, let S = S_{K} be the set of simplices of K, ordered under inclusion. In this case the ordered simplicial complex Cx(S) is called the barycentric subdivision [31, p. 123–124] of K, and denoted sd K. The geometric realization of sd K is homeomorphic to the geometric realization of K.

Suppose that K and L are ordered simplicial complexes. Let V_{K×L} denote the product V_{K} ×V_{L}, where the product is taken in the category of posets (i.e., (v, w) ≤ (v′, w′) if v ≤ v′ and w ≤ w′). Let pr_{1} : V_{K×L} → V_{K} and pr_{2} : V_{K×L} → V_{L} be the projection maps, and let S_{K×L} denote the collection of all totally ordered subsets σ of V_{K×L} such that pr_{1}(σ) ∈ S_{K} and pr_{2}(σ) ∈ S_{L}.

The next proposition shows that replacing the category ASC by OSC solves the problem of products.

3.7. Proposition. In the above situation, the ordered simplicial complex K × L = (V_{K×L}, S_{K×L}) is the product of K and L in the category of ordered simplicial complexes. The natural map |K × L| → |K| × |L| is a homeomorphism.

Proof. The first statement in the lemma is obvious. The indicated natural map takes a point ∑ c_{k}(v_{k}, w_{k}) of |K × L| to the point

(∑ c_{k}v_{k}, ∑ c_{k}w_{k})

of |K| × |L|. The inverse homeomorphism |K| × |L| → |K × L| takes (∑ s_{i}v_{i}, ∑ t_{j}w_{j}) to the point ∑ c_{k}p_{k} determined as follows. Relabel so that that the vertices of K and L which appear with a nonzero coefficient are

v_{0} < v_{1} < ⋯ < v_{m} and w_{0} < w_{1} < ⋯ < w_{n} .

Let S_{i} = ∑_{i′≤i} s_{i′} and T_{j} = ∑_{j′≤j} t_{j′}. Note that S_{m} = T_{n} = 1. Let

C_{0} ≤ C_{1} ≤ ⋯ ≤ C_{m+n}

be the sequence that results from arranging the (n + m + 2) numbers T_{i} and S_{j} in order and deleting one of the repeated 1’s at the high end. Then p_{k} = (v_{a(k)}, w_{b(k)}), where a(k) is the cardinality of the set {i | S_{i} ≤ C_{k}} and b(k) is the cardinality of {j | T_{j} ≤ C_{k}}. The coefficient c_{k} is C_{k} − C_{k−1}. See [14, p. 68].

3.8. Remark. The best way to understand the above proposition is to figure out how it gives triangulations of products of low-dimensional simplices. Start with |Δ[1]| × |Δ[1]| and go on to |Δ[1]| × |Δ[2]|. The process works surprisingly well, and is less complicated than it seems to be at first sight. The top-dimensional simplices of Δ[p] × Δ[q] are indexed by (p, q)-shuffles [23, p. 17].

3.9. Remark. The careful reader will worry about whether the “inverse homemorphism” described in the proof of 3.7 is continuous. In fact it is not continuous unless K and L satisfy some finiteness conditions or (which is usually the case in homotopy theory) the product |K| × |L| is tacitly given the compactly generated topology [32].

Now that we have handled products, we have to deal with a second deficiency of the category ASC, a deficiency which is inherited by OSC; this is the poor behavior of pushouts (or more generally, of colimits).

Given a diagram Z ⟵^{g} X ⟶^{f} Y in some category C, a pushout for the diagram is an object P of C together with a commutative square (on the left)

\begin{matrix}X & ⟶^{f} & Y \\ ^{g}↓ & & ^{u}↓ \\ Z & ⟶^{v} & P \end{matrix}  \begin{matrix}X & ⟶^{f} & Y \\ ^{g}↓ & _{u′}↓ \\ Z & ⟶^{v′} & P′ \end{matrix}

such that for any commutative square on the right there is a unique morphism t : P → P′ with tu = u′ and tv = v′. See [22, p. 66]; any two pushouts for the same diagram are isomorphic in a unique way respecting the pushout structure maps. The intuitive content of this notion is that the pushout P is obtained by gluing Y and Z together along X. See [22, III.1] for a discussion of the more general notion of colimit. Colimits are often used to construct complicated objects from simpler ones; the approximations we construct for BG (see 2.9) will be built as certain colimits in which the constituents have the homotopy type of BH for various subgroups H of G.

If L ← K → L′ is a diagram of ordered simplicial complexes, one would hope to be able to form its pushout W in such a way that |W | would be homeomorphic to the pushout of |L| ← |K| → |L′|. This is impossible. Let ∂Δ[n] be the subcomplex of Δ[n] containing all simplices except {0, 1, … n}. The pushout of the diagram

Δ[0] ← ∂Δ[n] → Δ[n] .

in OSC is isomorphic to Δ[0]: just calculate with vertices, and observe that any ordered simplicial complex with one vertex is isomorphic to Δ[0]. The pushout of the corresponding geometric diagram

\begin{aligned}|Δ[0]| & ⟵ & |∂Δ[n]| & ⟶ & |Δ[n]| \\ ^{≈}↓ & & ^{≈}↓ & & ^{≈}↓ \\ ∗ & ⟵ & S^{n−1} & ⟶ & D^{n} \end{aligned} (3.10)

is the n-sphere. The problem here is that a simplex in an ordered simplicial complex is by definition determined by its vertices, so that identifying vertices with one another automatically collapses any simplices which they form. In the topological category, however, it is possible to identify all of the vertices of a simplex to a point, or even to collapse the whole boundary of a simplex to a point, without collapsing the simplex itself.

The solution to this problem is to stop thinking of a simplex as determined by its vertices, and instead to think of it as an indivisible object which has vertices (and faces of other dimensions) associated to it in some explicit way. Of course these vertices (and other faces) are again indivisible objects of the same general type. This leads to the notion of a simplicial set.

Simplicial sets. Let K be an ordered simplicial complex. We think of a map σ : Δ[n] → K as a kind of generalized simplex of K. Note that the maps Δ[n] → K which give monomorphisms n → V_{K} correspond bijectively to ordinary simplices of K, but it is a bad idea to restrict attention to these special kinds of maps, because the don’t behave functorially.

What does “they don’t behave functorially” actually mean?

A generalized simplex σ of K corresponds to a face of σ′ if σ = σ′·i for some monomorphism i : Δ[n] → Δ[m]. Similarly, σ is obtained from σ′ by repeating vertices if σ = σ·j for some epimorphism j : Δ[n] → Δ[m]. This suggests considering the following structure.

3.11. Definition. Let Δ denote the full subcategory of OSC with objects Δ[n] (n ≥ 0). A simplicial set X is a functor

X : Δ^{op} → Sets .

The category Δ^{op} is the opposite category of Δ [22, p.33]. A functor with domain Δ^{op} is the same thing as a contravariant functor with domain Δ.

Simplicial sets form a category Sp in which a morphism is a natural transformation between functors. If X is a simplicial set, we write X_{n} for the set X(Δ[n]) and call this the set of n-simplices of X. The above considerations provide a functor Sing : OSC → Sp given by

Sing(K)_{n} = Hom_{OSC}(Δ[n], K) .

It is easy to see that the category Δ can be identified with the category whose objects are the ordered sets n (n ≥ 0) and whose maps are the set maps f : n → m which are weakly order preserving (in the sense that f(i) ≤ f(j) if i ≤ j); this amounts to noticing that a map Δ[n] → Δ[m] is completely determined by its effect on vertices. One can then check that the morphisms in Δ are generated by injective maps d^{i} : n − 1 → n and surjective maps s^{i} : n + 1 → n (0 ≤ i ≤ n) given by the following formulas:

(3.12) d^{i}(j) = \begin{cases}j & if j < i \\ j + 1 & if j ≥ i \end{cases} s^{i}(j) = \begin{cases}j & if j ≤ i \\ j − 1 & if j > i \end{cases}

These morphisms satisfy a few obvious relations

(3.13)  d^{j}d^{i} = d^{i}d^{j−1} if i < j s^{j}s^{i} = s^{i}s^{j+1} if i ≤ j s^{j}d^{i} = d^{i}s^{j−1} if i < j s^{j}d^{j} = identity = s^{j}d^{j+1} s^{j}d^{i} = d^{i−1}s^{j} if i > j + 1 .

and it is possible to verify that all relations between composites of the d^{i}’s and s^{j}’s are consequences of these specific ones [23, p. 4]. This leads to another way to look at a simplicial set X. Recall that X is a contravariant functor on Δ; let d_{j} = X(d^{j}) and s_{j} = X(s^{j}). Then X is effectively a collection of sets X_{n} (n ≥ 0) together with maps d_{i} : X_{n} → X_{n−1} and s_{i} : X_{n} → X_{n+1} (0 ≤ i ≤ n) satisfying a list of identities [23, p. 1] which are the opposites of the ones in (3.13). For completeness, we will write these identities out:

d_{i}d_{j} = d_{j−1}d_{i} if i < i
s_{i}s_{j} = s_{j+1}s_{i} if i ≤ j
d_{i}s_{j} = s_{j−1}d_{i} if i < j
d_{j}s_{j} = identity = d_{j+1}s_{j}
d_{i}s_{j} = s_{j}d_{i−1} if i > j + 1 .

(3.14)

The d_{i}’s are called face operators and the s_{i}’s degeneracy operators. If x is an n-simplex of X, then d_{i}x is an (n − 1)-simplex of X which is the i’th face of x.

3.15. Remark. More generally, if C is some category, a simplicial object in C is by definition just a functor Δ^{op} → C (see 3.11); these simplicial objects form a category, in which an arrow is a natural transformation of functors. A simplicial set is a simplicial object in Sets. A simplicial object X in C amounts to a collection {X_{n}}_{n≥0} of objects of C together with morphisms d_{i} : X_{n} → X_{n−1} and s_{i} : X_{n} → X_{n+1} (0 ≤ i ≤ n) satisfying the identities 3.14.

If C is the category of abelian groups (or more generally the category of modules over a ring R), then the category of simplicial objects over C is equivalent (via a normalization functor [23, 22.4]) to the category of nonnegatively graded chain complexes over C. The idea of “chain complex” does not extend very well to more general categories C, but the idea of “simplicial object” does. Studying these simplicial objects gives a way to generalize homological algebra to nonabelian settings; following Quillen [29], this generalization is often called homotopical algebra. Since simplicial sets are closely connected to topological spaces, ordinary homotopy theory can be thought of as the homotopical algebra of the category of sets! Another interesting example is [26], where Quillen studies the homotopical algebra of the category of commutative rings.

The geometric realization construction for ordered simplicial complexes extends to a similar construction for simplicial sets, although the description of this extended construction is a little more complicated. Let Δ_{n} = |Δ[n]| be the topological n-simplex, and for each morphism d^{i} : Δ[n] → Δ[n + 1] or s^{i} : Δ[n] → Δ[n − 1] (3.12) denote by the same symbols the induced continuous maps Δ_{n} → Δ_{n+1} and Δ_{n} → Δ_{n−1}. (The map d^{i} : Δ_{n} → Δ_{n+1} is the i’th face inclusion, and s^{i} : Δ_{n} → Δ_{n−1} is a linear collapse obtained by pinching together the i’th and (i + 1)’st vertices.) If X is a simplicial set, the geometric realization |X| is the space obtained from the disjoint union

⋃_{n} X_{n} × Δ_{n}

(here X_{n} is treated as a space with the discrete topology) by making the identifications

(d_{i}x, p) ∼ (x, d^{i}p) for (x, p) ∈ X_{n} × Δ_{n−1}
(s_{i}x, p) ∼ (x, s^{i}p) for (x, p) ∈ X_{n−1} × Δ_{n} (3.16)

See [23, III]. A simplex x ∈ X_{n} is said to be degenerate if x = s_{i}x′ for some i and some x′ ∈ X_{n−1}. The space |X| is a CW–complex in which the n-cells correspond to the nondegenerate n-simplices of X [23, 14.1]. If X = Sing(K) for an ordered simplicial complex K, then the nondegenerate simplices of X are essentially the ordinary simplices of K, and |X| is homeomorphic to |K|.

14 W. G. DWYER

From the categorical point of view, the geometric realization of a simplicial set X is a curious construction; it is built by gluing, but it certainly is not given as a colimit over Δ. The best way to understand it is as a coend [22, p. 223], or equivalently as a kind of tensor product over Δ of the covariant functor n ↦ Δ_{n} with the contravariant functor n ↦ X_{n}.

All small limits and colimits exist in the category Sp and can be constructed dimensionwise; see [22, p. 111] for limits, and the dual (i.e. opposite-category) version for colimits. For example, the categorical product X × Y of two simplicial sets is given by

(X × Y)_{n} = X_{n} × Y_{n}

with simplicial operators d_{i} = d_{i}^{X} × d_{i}^{Y} and s_{i} = s_{i}^{X} × s_{i}^{Y}. If Z ⟵^{g} X ⟶^{g} Y is a diagram of simplicial sets then the pushout P of the diagram exists, and P_{n} can be calculated as the pushout of the diagram Z_{n} ⟵^{g_{n}} X_{n} ⟶^{f_{n}} Y_{n} in the category of sets.

The category of simplicial sets solves the two problems we encountered above. First of all, for any simplicial sets X and Y the natural map

|X × Y | → |X| × |Y |

is a homeomorphism [23, 14.3], at least if |X| × |Y | is given a suitable topology (3.9). This is an extension of the result above for ordered simplicial complexes, since if K and L are objects of OSC, there is an isomorphism Sing(K) × Sing(L) ≈ Sing(K × L). Secondly, the realization functor Sp → Top commutes with pushouts, and, even better, with all colimits [23, 16.1] [22, p. 115]. Note how (3.10) works in Sp. The pushout of the diagram

Sing(Δ[0]) ← Sing(∂Δ[n]) → Sing (Δ[n])

in Sp is a simplicial set X which has only two nondegenerate simplices, one of dimension 0 and one of dimension n; X is an object of Sp which is genuinely new: it does not come from OSC. The geometrical realization |X| is obtained from the topological n-simplex Δ_{n} by collapsing its boundary to a point, and is homeomorphic to the n-sphere.

3.17. Remark. From now on we will identify the ordered simplicial complex Δ[n] with the corresponding simplicial set Sing(Δ[n]) and call it the standard n-simplex. The set Δ[n]_{k} of its k-simplices is the set of nondecreasing sequences ⟨i_{0}, …, i_{k}⟩ of elements of n.

3.18. Homotopy and homology of simplicial sets. For the rest of this paper we will refer to simplicial sets as spaces, and sets with topologies as topological spaces. This is partially for convenience, and partially to emphasize the fact that in almost everything we do here, point-set topology plays no significant role; what is important are the combinatorial patterns in which simplices are matched with one another. In general, the category Sp of simplicial sets is a very convenient one to use for building explicit combinatorial models of topological spaces (or for making some kinds of homotopy theoretic manipulations [6]).

3.19. Remark. We will think of the category of sets as embedded in Sp by the functor which assigns to a set S the constant simplicial set (i.e. constant functor Δ^{op} → Sets) with value S. These simplicial sets are called discrete spaces. Equivalently, this functor assigns to S a coproduct of copies of Δ[0], one copy for each element of S. The geometric realization of this simplicial set is homeomorphic to the set S itself (endowed with the discrete topology).

The homotopical relationship between Sp and Top is very close; besides a realization functor Sp → Top, there is a singular functor Sing : Top → Sp given by

Sing(X)_{n} = Hom_{Top}(Δ_{n}, X) .

Almost by definition, the cellular homology of the CW–complex | Sing(X)| is the singular homology of X, so it is not surprising that the more or less obvious natural map of spaces

| Sing(X)| → X

is an isomorphism on homology [23, 16.2] or even a weak homotopy equivalence. The realization functor is left adjoint [22, p .81] to the singular functor, and this gives a simple explanation for the fact that the realization functor commutes with colimits [22, p. 114–115]. For a thorough discussion of simplicial theory from a modern point of view see [17].

3.20. Definition. A map f of spaces is said to be an equivalence or weak equivalence if the geometric realization of f is a weak equivalence of topological spaces.

More generally, the homotopy groups π_{∗}X of a space X can be defined either in terms of the ordinary homotopy groups of |X|, or by a direct combinatorial formula [23, §3] [23, 26.7]. Needless to say the combinatorial formula is pretty complicated, since so far it has not led to explicit formulas for π_{∗}(Δ[n]/∂Δ[n])! A map of spaces is a weak equivalence in the above sense if and only if it induces an isomorphism on homotopy groups for every choice of basepoint. Can you give a combinatorial definition of the set of components of a space X? How about a definition of the fundamental group of X (based at a vertex)?

Before considering the homology of a space, we have to make an algebraic definition.

3.21. Definition. Suppose that A is a simplicial abelian group (3.15). The normalization of A, denoted N(A), is the chain complex obtained by defining

N(A)_{n} = A_{n}/(s_{0}A_{n−1} + ⋯ + s_{n−1}A_{n−1}

and letting ∂ : N(A)_{n} → N(A)_{n−1} be the quotient map [23, 22.2] induced by the alternating sum of face operators

∑_{i=0}^{n} d_{i} : A_{n} → A_{n−1} . (3.22)

3.23. Remark. The homology groups of N(A) are denoted H_{∗} N(A) or sometimes, for brevity, just H_{∗}(A). It is possible to form another chain complex C(A) by setting C(A)_{n} = A_{n} and letting ∂ : C(A)_{n} → C(A)_{n−1} be given by formula (3.22) (without passing to any quotient). This change does not make much of a difference: the obvious surjection C(A) → N(A) is a chain map which induces an isomorphism on homology groups [23, 22.2].

3.24. Definition. If X is a space, let ℤ[X] denote the simplicial abelian group obtained by applying to X (dimensionwise) the free abelian group functor from the category of sets to the category of abelian groups. If M is an abelian group, let M[X] denote the simplicial abelian group M ⊗_{ℤ} (ℤ[X]). The homology of X with coefficients in M, denoted H_{∗}(X; M) is defined to be the homology of the chain complex N(M[X]).

How would you define H^{∗}(X; M)?

3.25. Remark. If Y is a subspace of X, the relative homology homology H_{∗}(X, Y; M) is defined to be the homology of the chain complex

N(M[X]/M[Y]) ≈ N(M[X])/N(M[Y]) .

It is not hard to see that H_{∗}(X, Y; M) is exactly the cellular homology, with coefficients in M, of the pair (|X|, |Y |) of CW–complexes.

If A is a simplicial abelian group, there is a natural isomorphism [23, 22.1]

π_{∗}(A) ≈ H_{∗}(N(A)) . In particular, if X is a space, |ℤ[X]| is another space whose homotopy groups are the ordinary (singular or cellular) integral homology groups of |X|. Compare this to the Dold-Thom theorem [9], which ties the homology of a topological space Y to the homotopy of the infinite symmetric product SP^{∞}(Y). There is quite a similarity here: ℤ[X] is the simplicial free abelian group on X, while SP^{∞}(Y) is the topological free abelian monoid on Y.

3.26. Classifying spaces. An action of a (discrete) group G on a space X amounts to an actions of G on each of the sets X_{n} (n ≥ 0) which commute with all face and degeneracy operators. An action of G on X is said to be free if the induced action of G on each X_{n} is free. A space X is said to be weakly contractible if the unique map X → ∗ = Δ[0] is a weak equivalence, or, in other words, if |X| is a contractible topological space. Suppose that E is a weakly contractible space with a free action of G. It is not hard to see that the natural map

|E| → |E/G|

is a principal covering map with group G, and so |E/G| is a topological classifying space (2.1) for G. In the combinatorial context we have chosen to work in, we will call E/G itself a classifying space for G. Later on (5.10) we will describe a simple functorial construction for classifying spaces.

4. Simplicial spaces and homotopy colimits

The construction of the homotopy colimit is motivated by the fact that ordinary colimits are not well-behaved with respect to weak equivalences. For instance, consider the following commutative diagram of topological spaces (where D^{n} is the n-disk and S^{n−1} its boundary sphere).

\begin{aligned}D^{n} & ⟵ & S^{n−1} & ⟶ & D^{n} \\ ↓ & & ^{=}↓ & & ↓ \\ ∗ & ⟵ & S^{n−1} & ⟶ & ∗ \end{aligned} (4.1)

All three vertical arrows are weak equivalences (even homotopy equivalences) but the colimit of the top row is homeomorphic to S^{n}, the colimit of the bottom row is a one-point space ∗, and the map S^{n} → ∗ induced by the diagram is not a weak equivalence. The same sort of thing can happen with spaces (i.e., simplicial sets (3.18)); just consider the diagram

\begin{aligned}Δ[n] & ⟵ & ∂Δ[n] & ⟶ & Δ[n] \\ ↓ & & ↓ & & ↓ \\ ∗ & ⟵ & ∂Δ[n] & ⟶ & ∗ \end{aligned} (4.2)

where in this case ∗ = Δ[0]. In elementary homotopy theory you usually learn that it is the colimit of the top row in (4.1) is the “correct” homotopy pushout: if you are interested in homotopy types, then before taking the pushout of a diagram of topological spaces you should first replace the maps by weakly equivalence cofibrations. There is a parallel principle in Sp; before taking the pushout of a diagram of simplicial sets, you should first replace the maps by weakly equivalent injections. What is gained by such replacements is homotopy invariance (cf. 4.14). The general homotopy colimit construction we will describe below makes these procedures systematic, and gives a way to generalize them to colimits more complicated than pushouts. The result is that we will be able to use a single language to discuss whole families of homotopy invariant ways of gluing spaces together. Ultimately we are interested in ways of constructing BG, at least up to mod p homology, by gluing together classifying spaces of subgroups of G.

See [12, §10] for a conceptual approach to homotopy pushouts. From the point of view presented there (which originally appeared in [6]), the homotopy pushout is a derived functor of the pushout [12, 10.7], in a sense closely related to the way Tor is a derived functor of ⊗ [12, 9.6]. There is a similar interpretation of general homotopy colimits.

The homotopy colimit is defined in terms of a construction which is also useful for other purposes. This is the construction of the realization of a simplicial space.

Simplicial spaces and their realizations. Suppose that X is a simplicial space (3.15), i.e., a simplicial object in the category of spaces.

4.3. Definition. The realization of X is defined to be the space constructed by taking the disjoint union

⋃_{n} X_{n} × Δ[n] (4.4)

and making the analogues of identifications (3.16).

4.5. Remark. There is another very different way to understand such a realization. The object X itself is a sequence {X_{n}} of spaces, with “horizontal” maps d_{i}^{h} : X_{n} → X_{n−1}, s_{i}^{h} : X_{n} → X_{n−1}. Each X_{n} is itself a sequence of sets X_{m,n} with “vertical” maps d_{i}^{v} : X_{m,n} → X_{m−1,n}, s_{i}^{v} : X_{m,n} → X_{m−1,n}. Combining the two directions allows X to be considered as a rectangular array {X_{m,n}} of sets with both horizontal and vertical simplicial operators. The following proposition may be hard to believe, but it is easy to check.

4.6. Proposition. If X is a simplicial space, then the realization of X is naturally isomorphic to the space diag(X) with diag(X)_{n} = X_{n,n}. The face operators d_{i} and degeneracy operators s_{i} of diag(X) are given in terms of those of X by the formulas d_{i} = d_{i}^{h} · d_{i}^{v}, s_{i} = s_{i}^{h} · s_{i}^{v}.

This proposition states that if the simplicial space X is interpreted as a functor

X : Δ^{op} × Δ^{op} → Sets ,

then the realization of X is isomorphic to the functor Δ^{op} → Sets obtained by composing X with the diagonal functor Δ^{op} → Δ^{op} × Δ^{op}.

This is the reason for the notation “diag(X)”. In fact, because of 4.6, the realization of a simplicial space X is often called the diagonal of X.

4.7. A homology spectral sequence. For a simplicial space X there is an increasing filtration

F_{0} diag(X) ⊂ ⋯ ⊂ F_{n} diag(X) ⊂ ⋯

of diag(X) given by letting F_{n} diag(X) be the image (4.4) of the space ∪_{i≤n}X_{i}×Δ[n]. This gives increasing filtrations of the simplicial abelian group ℤ[X] (3.18) and of the chain complex N(ℤ[X]). Associated to this filtration is a spectral sequence for H_{∗}(diag(X); ℤ). It turns out to be a first quadrant spectral sequence of homological type (e.g. the differential d^{r} has bidegree (−r, r−1)). We will denote this spectral sequence E_{i,j}^{r}(X; ℤ); if there is a coefficient module M the corresponding spectral sequence is written E_{i,j}^{r}(X; M). The E^{2}-page is described by the following proposition. Let H_{n}(X; M) denote the simplicial abelian group obtained by applying the functor H_{n}(–; M) (cf. 3.24) dimensionwise to X (so that H_{n}(X; M)_{j} = H_{n}(X_{j}; M)) and let H_{m} H_{n}(X; M) denote the m’th homology group of H_{n}(X; M) (cf. 3.23).

4.8. Proposition. [6, XII 5.7] If X is a simplicial space, there are natural isomorphisms

E_{i,j}^{2}(X; M) ≈ H_{i} H_{j}(X; M) .

The diagonal construction for a simplicial space has the following basic homotopy invariance property. It’s not hard to give a direct proof by examining how diag(X) is glued together from the spaces spaces X_{n} × Δ[n].

4.9. Proposition. [6, XII 4.2, 4.3] If f : X → Y is a map of simplicial spaces which induces weak equivalences (3.20) X_{i} → Y_{i} (i ≥ 0), then f induces a weak equivalence diag(X) → diag(Y).

4.10. Homotopy colimits. Suppose that D is a small category. Consider the poset n as a category with one morphism i → j if i ≤ j, and no other morphisms. The singular complex or nerve N(D) of D is the space given by

N(D)_{n} = Hom_{Cat}(n, D) .

Here Cat is the category in which the objects are small categories and the morphisms are functors. More concretely, an n-simplex σ of N(D) is just a length n sequence

σ(0) ⟶^{α_{1}} σ(1) ⟶^{α_{2}} ⋯ ⟶^{α_{n}} σ(n) (4.11)

of composable arrows in D. The face and degeneracy operators of N(D) are given by composition with the functors determined by the formulas (3.12). In other words, d_{i}σ is obtained from σ by leaving out σ(0) if i = 0, by composing α_{i+1} with α_{i} if 0 < i < n, and by leaving out σ(n) if i = n; s_{i}σ is obtained by inserting the identity map σ(i) → σ(i) between α_{i} and α_{i+1}.

4.12. Example. Any poset S can be treated as a category in which the objects are the elements of S and in which there is exactly one morphism x → y if x ≤ y (there are no other morphisms). The nerve N(S) is then the same (3.19) as the ordered simplicial complex associated to S in 3.6.

Suppose now that F : D → Sp is a functor. (Such an F is sometimes called a diagram of spaces with the shape of D.) The simplicial replacement of F is the simplicial space ∐_{∗} F which in dimension n consists of the coproduct (i.e. disjoint union)

(∐_{∗} F)_{n} = ∐_{σ∈N(D)_{n}} F (σ(0)) .

The horizontal degeneracy operator s_{i} : (∐_{∗} F)_{n} → (∐_{∗} F)_{n+1} maps the space F (σ(0)) indexed by σ to F ((s_{i}σ)(0)) by the identity map; the horizontal face operator d_{i} maps F (σ(0)) to F ((d_{i}σ)(0)) by the identity map if i > 0 and by the map α_{0} if i = 0.

The simplicial space defined here as ∐_{∗} F differs from the one described in [6, XII.5.1] by an automorphism of the category Δ^{op}. This is a technical point which does not affect any of its properties.

4.13. Definition. Let D be a small category and F : D → Sp a functor. The homotopy colimit of F is the space hocolim(F) given by diag(∐_{∗} F).

4.14. Remark. The homotopy colimit construction is functorial, in the sense that a natural transformation τ : F → F′ induces a map

hocolim τ : hocolim F → hocolim F′ .

It follows from 4.9 that homotopy colimits have a strong homotopy invariance property: if τ induces a weak equivalence τ_{d} : F (d) → F′(d) for each object d of D, then hocolim(τ) is a weak equivalence.

The homotopy colimit construction is also functorial in D, in the sense that if j : D′ → D is a functor and j^{∗}(F) denotes the composite F · j, then there is a natural map

hocolim j^{∗}(F) → hocolim F .

4.15. Remark. The space hocolim(F) is constructed [6, XII, §2] by taking

• for each object d_{0} of D a copy of F (d_{0}),

• for each arrow α_{0} : d_{0} → d_{1} of D a copy of F (d_{0}) × Δ[1], and in general,

• for each chain d_{0} → ⋯ → d_{n} of composable arrows in D a copy of F (d_{0}) × Δ[n],

and making appropriate identifications. The identifications amount to

• collapsing F (d_{0}) × Δ[n] to something smaller if it arises from a chain d_{0} → ⋯ → d_{n} containing an identity map, and

• identifying the subspace F (d_{0}) × ∂Δ[n] of F (d_{0}) × Δ[n] with an appropriate subspace of hocolim(F) arising from chains of smaller length.

There is a more or less obvious natural map

hocolim F → colim F .

This map is not usually a weak equivalence [6, XII.2.5].

4.16. Remark. It follows from 4.7 and the construction of hocolim(F) that for any abelian group M there is a natural spectral sequence E_{i,j}^{r}(F; M) converging to H_{∗}(hocolim F; M). This is called the Bousfield-Kan homology spectral sequence of the homotopy colimit. Let Ab be the category of abelian groups and Ab^{D} the category of functors D → Ab (with morphisms being natural transformations). According to [6, XII.5.7] the E^{2}-page E_{i,j}^{2}(F; M) can be described by the formula

E_{i,j}^{2}(F; M) = colim_{i} H_{j}(F; M) .

where

• colim is the colimit functor Ab^{D} → Ab,

• colim_{i} is the i’th left derived functor of colim, and

• H_{j}(F; M) is the functor D → Ab obtained by composing F with the homology functor H_{j}(–; M).

What are the projective objects of Ab^{D}? Show that there are enough of them to construct left derived functors. See [16, p. 154]. There is a parallel cohomology spectral sequence

E_{2}^{i,j}(F; M) = lim^{i} H^{j}(F; M) ⇒ H^{i+j}(hocolim F; M) ,

where

• lim is the limit functor Ab^{D^{op}} → Ab,

• lim^{i} is the i’th right derived functor of lim, and

• H^{j}(F; M) is the functor D^{op} → Ab obtained by composing F with the cohomology functor H^{j}(–; M). This cohomological version appears more frequently in the literature than the homological one. If M = 𝔽_{p}, for instance, this cohomology spectral sequence is the 𝔽_{p}-dual of E_{i,j}^{r}(F; M).

The left derived functors of colimit are sometimes easier to calculate than you might think. Suppose that D is the pushout category of 4.18 below. Show that if

F = (A ⟵^{u} B ⟶^{v} C)

is a functor from D to abelian groups, then

colim_{i} F = \begin{cases}coker B ⟶^{(u,−v)} A ⊕ C & i = 0 \\ ker B ⟶^{(u,−v)} A ⊕ C & i = 1 \\ 0 & otherwise \end{cases}

Conclude that the homology spectral sequence for homotopy pushouts amounts to the usual Mayer-Vietoris sequence. If G is the category of a group G (5.10) and M is a G-module, treated as a functor from G to abelian groups, observe that colim M = colim_{0} M = H_{0}(G; M) (group homology, as in 2.8) and conclude that colim_{i} M is isomorphic to H_{i}(G; M).

Examples of homotopy colimits. How complicated a particular homotopy colimit construction turns out to be depends mostly on the shape of the indexing category D. In the following examples, we assume that F is a functor from D to Sp.

4.17. Homotopy coproducts. If D is a trivial category with a collection {d_{α}} of objects and no nonidentity morphisms, then hocolim(F) is ∐_{α} F (d_{α}) and the map hocolim(F) → colim(F) is an isomorphism. For example, let D be an arbitrary category, let d be an object of D, and let i_{d} : {d} → D be the inclusion functor whose domain is the trivial category with the single object d. By naturality (4.14) there is an induced map

j_{d} : F (d) = hocolim i_{d}^{∗}(F) → hocolim F .

This shows that each space which appears as a value of the functor F has a natural map to hocolim F.

4.18. Homotopy pushouts. Let 0 and 1 denote the two copies of Δ[0] inside the simplicial interval Δ[1]. If D is the category

a ⟵^{f} b ⟶^{g}c

then hocolim(F) is isomorphic to the space obtained from the coproduct

F (a) ∐ (F (b)_{f} × Δ[1]) ∐ F (b) ∐ (F (b)_{g} × Δ[1]) ∐ F (c)

by gluing • F (b)_{f} × 1 to F (a) by F (f),

• F (b)_{f} × 0 to F (b) by the identity map of F (b),

• F (b)_{g} × 0 to F (b) by the identity map of F (b), and

• F (b)_{g} × 1 to F (c) by F (g). This is a slightly modified form of the usual double mapping cone construction of the homotopy pushout. The map from hocolim(F) to colim(F) is not usually a weak equivalence.

Sequential colimits. If D is the category given by the poset of nonnegative integers

1 → 2 → ⋯ → n → ⋯

then, although hocolim(F) on the face of it looks large and complicated, the map hocolim(F) → colim(F) is always a weak equivalence. The same result holds if D is replaced by a more general “right filtering” category [6, XII.3.5].

4.19. Nerves. If F is the constant functor ∗ with value the one-point space, then hocolim(F) is isomorphic to the nerve N(D). More generally, if F is the constant functor with value X, then hocolim F is isomorphic to X × N(D). For any F, the unique natural transformation F → ∗ induces a map hocolim F → N(D).

In general, the map q : hocolim F → N(D) is a kind of “singular fibration” in which, after geometric realization, the fibre over each point of the interior of a nondegenerate simplex σ is homeomorphic to |F (σ(0))|. One form of Quillen’s Theorem B [28, p. 97] is the statement that if F carries each morphism of D to a weak equivalence, then |q| is up to homotopy a fibration in which the fibre over the point corresponding to an object d of D is weakly equivalent to |F (d)|.

Simplicial objects. [6, XII.3.4] If D = Δ^{op}, so that F is a simplicial space, then hocolim(F) is weakly equivalent to diag(F). In particular, any space X is weakly equivalent to hocolim(F), where F : Δ^{op} → Sets has F (n) = X_{n} (see 3.19 for how to treat a set as a space).

We will see other examples of homotopy colimits later on.

5. Nerves of categories and the Grothendieck construction

The nerve of a category (4.10) is a very special kind of homotopy colimit (4.19). In this section we will describe some special properties of nerves, and then give a construction (due to Thomason [34]) which which allows many homotopy colimits to be interpreted as the nerves of auxiliary categories.

Properties of the nerve. Suppose that X and Y are spaces (3.18). Two maps f, g : X → Y are said to be simplicially homotopic if there is a map

H : X × Δ[1] → Y

such that the restriction of H to X × 0 (4.18) is f and the restriction of H to X × 1 is g (see [23, 5.1] for a mercilessly combinatorial formulation). The notion of simplicial homotopy is unsatisfactory in some ways; in particular, it does not give an equivalence relation on the set of maps from X to Y. Nevertheless, since the geometric realization functor preserves products and the realization of Δ[1] is a unit interval, the following proposition is clear.

5.1. Proposition. Suppose that f and g are maps of spaces. If f is simplicially homotopic to g, then |f| is homotopic (in the topological sense) to |g|. In particular, f is a weak equivalence (3.20) if and only if g is.

In practice, explicit simplicial homotopies arise most frequently by applying the nerve construction to natural transformations between functors.

5.2. Proposition. Suppose that F and G are functors between small categories. If F and G are related by a natural transformation, then N(F) is simplicially homotopic N(G). In particular, N(F) is a weak equivalence if and only if N(G) is.

Proof. The fact that F and G are related by a natural transformation is equivalent to the statement that F and G extend to a functor

τ : D × 1 → D′ .

(Here 1 is the category of the poset 1 (4.10)). By construction the functor N(–) : Cat → Sp preserves products, and N(1) is exactly Δ[1], so N(τ) gives the required simplicial homotopy.

5.3. Remark. Functors and natural transformations are much easier to work with than simplicial maps and simplicial homotopies. This basically explains why we are interested in nerves of categories.

Proposition 5.2 has some immediate consequences.

5.4. Proposition. If F : D → D′ is an equivalence of categories, then N(F) : N(D) → N(D)′ is a weak equivalence of spaces.

Proof. If F′ : D′ → D is an inverse equivalence, then the composites F F′ and F′F are naturally equivalent to the respective identity functors.

The conclusion of 5.4 continues to hold under the weaker assumption that there is a functor F′ : D′ → D which is either left adjoint [22, p .81] or right adjoint to F. In either case, the composites F F′ and F′F are connected to the respective identity functors by natural transformations. These natural transformations are not necessarily natural equivalences, but they still provide simplicial homotopies. (5.2).

An object d of a category D is said to be an initial object (resp. a terminal object) if for any object d′ of D there is exactly one morphism d → d′ (resp. d′ → d). For instance, the empty topological space is an initial object of Top, and any one-point space is a terminal object.

5.5. Proposition. If D has either an initial object or a terminal object, then N(D) is weakly contractible.

Proof. Let ∗ denote the trivial category with one object and only the identity morphism. There is a unique functor F : D → ∗, as well as a unique functor F′ : ∗ → D sending the object of ∗ to d. The composite F F′ is the identity functor of ∗. If d is either an initial or a terminal object of D, there is an obvious natural transformation connecting the identity functor of D to the composite F′F. By 5.2, N(D) is weakly equivalent to N(∗) = Δ[0].

5.6. Example. If S is a poset with a maximal or minimal element, then the nerve of S (4.12) is weakly contractible.

An argument similar to the previous one gives the following.

5.7. Proposition. If the identity functor of D is connected to some constant functor D → D by zigzag of natural transformations, then N(D) is weakly contractible.

A constant functor is one which takes the same value d_{0} on each object of D and sends all morphisms of D to the identity map of d_{0}. The hypothesis of the proposition means that there exists some chain of natural transformations

F_{0} ← F_{1} → F_{2} ← ⋯ → F_{n}

in which F_{0} is the identity functor and F_{n} is a constant functor.

The following result is a bit trickier to prove with category theoretic arguments.

5.8. Proposition. [28, p. 86] The nerve of D is weakly equivalent in a natural way to the nerve of D^{op}

5.9. Remark. It is pretty clear that there is in general no functor D → D^{op} inducing a weak equivalence of nerves. Quillen’s proof of 5.8 proceeds by introducing a third category S(D) and two functors s : S(D) → D^{op}, t : S(D) → D, both of which do induce weak equivalences when the nerve construction is applied. All of this data depends functorially on D; this is the meaning of the phrase “in a natural way” in 5.8. Another way to prove 5.8 is to notice that the geometric realization of N(D) is naturally homeomorphic to the geometric realization of N(D)^{op}.

We can now give a few more examples of nerves.

5.10. Classifying spaces as nerves. Let G be a discrete group. The category of G is the category G with one object ∗, and with the monoid of maps ∗ → ∗ isomorphic to G. For example, a functor F : G → Sp is essentially a space X = F (∗) with an action (3.26) of G. We claim that the nerve N(G) is a classifying space BG. To see this, let EG denote the category whose objects are the elements x of G, and in which there is exactly one morphism between any two objects, and let EG = N(EG). The space EG is weakly contractible (5.5), because every object of EG is both initial and terminal. The group G acts on EG (via functors) by the rule g · x = gx. The induced action of G on EG is free, and it is easy to see that the quotient (EG)/G is isomorphic to N(G). Therefore (3.26), N(G) is a classifying space for G. From now on we will let BG denote this specific classifying space. Note that BG is functorial in G, in the sense that a homomorphism G → H induces a functor G → H and hence a map BG → BH.

5.11. Groupoids. A groupoid is a small category in which every morphism is invertible. Suppose that D is a groupoid. For any object d of D the vertex group G_{x} is the group Hom_{D}(x, x). There is an obvious functor G_{x} → D taking the unique object of G_{x} to x. Choose representative {x_{α}} of isomorphism classes of objects from D, and consider the functor

∐_{α} G_{x_{α}} → D .

It is not hard to see that this functor is an equivalence of categories [22, p .91] and so induces a weak equivalence on nerves. This gives a weak equivalence

∐_{α} BG_{x_{α}} → N(D) .

In other words, the nerve of a groupoid is a disjoint union of classifying spaces of groups; there is one component for each isomorphism class of objects in the groupoid, and this component has the homotopy type of the classifying space of the group of automorphisms of any object in the isomorphism class.

In calculating with nerves we will sometimes use the following elementary observation.

5.12. Proposition. Suppose that G is a group acting on a category D, and that D^{G} is the fixed subcategory (that is, the subcategory of D containing those objects and morphisms fixed by the action). Then the natural map

N(D^{G}) → (N(D))^{G}

is an isomorphism.

The Grothendieck construction. Suppose that F : D → Sets is a functor. The transport category Tr(F) of F is the category whose objects consist of pairs (d, x) where x ∈ F (d). A map (d, x) → (d′, x′) is a morphism f : d → d′ in D such that F (f)(x) = x′. These morphisms compose according to the composition of morphisms in D. The homotopical meaning of Tr(F) is very simple.

5.13. Proposition. For any functor F : D → Sets there is a natural isomorphism

N(Tr(F)) ≈ hocolim N(F) .

Proof. Check that an n-simplex of hocolim N(F) amounts to a pair (σ, x), where σ is an n-simplex (4.11) of the nerve of D and x ∈ F (σ(0)). These are exactly the n-simplices of N(Tr(F)).

5.14. Example. Let X be a space, considered as a functor X : Δ^{op} → Sets. The transport category Tr(X) can be thought of as a category whose objects are the simplices x of X; a morphism x → x′ is a suitable morphism Φ of Δ^{op} (i.e. a suitable composite of face and degeneracy operators (3.14)) such that Φ(x) = x′. The nerve of Tr(X) is isomorphic to hocolim X, and so is weakly equivalent to X (4). This shows that every space can be constructed up to weak equivalence as the nerve of a category.

For related results see [24] (every connected space is weakly equivalent to the nerve of a category with a single object, i.e. to the classifying space of a monoid) and [20] (every connected space is homology equivalent to the classifying space of a group).

5.15. Example. Let G be a group and S a G-set, considered as a functor S : G → Sets (5.10). The transport category of S is the groupoid (5.11) whose objects are the elements of S and in which a morphism s → s′ is an element of g ∈ G such gs = s′. The isomorphism classes of objects in this category correspond to the orbits of the action of G on X, and the vertex group of an object x ∈ S is the isotropy subgroup G_{x}. By 5.11, Tr(S) (or hocolim S) is equivalent to a disjoint union ∐_{α} BG_{x_{α}}, where {x_{α}} runs through a set of orbit representatives for the action of G on S.

Proposition 5.13 has a wonderful generalization, due to Thomason. Suppose that D is a small category, and that F : D → Cat is a functor. The Grothendieck Construction on F, denoted Gr(F), is the category whose objects are the pairs (d, x) where d is an object of D and x is an object of F (d). An arrow (d, x) → (d′, x′) in Gr(F) is a pair (f, g), where f : d → d′ is a morphism in D and g : (F (f))(x) → x′ is a morphism in F (d′). Arrows compose according to the rule (f, g) · (f′, g′) = (f′′, g′′), where f′′ is the composite f · f′ and g′′ is the composite of g with the image of g′ under the functor F (f). 5.16. Theorem. [34, 1.2] Suppose that D is a small category and F : D → Cat is a functor. Let Gr(F) be the Grothendieck Construction on F. Then there is a natural weak equivalence

hocolim N(F) ⟶^{∼} N(Gr(F)) .

Variations. Suppose that F : D → Cat is a functor. Let F^{op} denote the composite of F with the “opposite” construction Cat → Cat; note that F^{op} is again a functor D → Cat. It follows from 5.16, 5.8, and the homotopy invariance of homotopy colimits (4.14) that the four categories Gr(F), Gr(F)^{op}, Gr(F^{op}) and Gr(F^{op})^{op} all have nerves which are weakly equivalent in a natural way to hocolim N(F).

6. Homotopy orbit spaces

In this section we give a description of the homotopy orbit space, which is a particular kind of homotopy colimit that plays a key role in our description of homology decompositions.

Suppose that X is a G-space (i.e. a space with an action of the group G), and denote by the same letter X the corresponding (5.10) functor G → Sp. 6.1. Definition. The homotopy orbit space of the action of G on X, denoted X_{hG}, is the space

X_{hG} = hocolim_{G} X .

6.2. Remark. It is easy to check that the space X_{hG} is isomorphic to the quotient space

(X × EG)/G ,

where EG is the free contractible G-space of 5.10 and G acts diagonally on the product X × EG. In particular there is a fibration sequence

X → X_{hG} → (EG)/G = BG . (6.3)

(See [23, II.7] and [30] for information about fibrations of simplicial sets). The map X_{hG} → (∗)_{hG} = BG is induced by the unique G-map X → ∗. The space X_{hG} is sometimes called the fibration over BG associated to the action of G on X or the Borel construction of the action of G on X. Note that colim_{G} X is the actual orbit space of the action of G on X, so that there is a natural map (4.15)

X_{hG} → X/G

from the homotopy orbit space to the usual orbit space. The property which more or less characterizes the homotopy orbit space construction, and distinguishes it from the usual orbit space is that (like the homotopy pushout) the homotopy orbit space is homotopy invariant (4.14): if f : X → Y is a map of G-spaces which is an ordinary weak equivalence of spaces, then (f)_{hG} is a weak equivalence.

The homology spectral sequence (4.16) of X_{hG} can be identified with the Serre spectral sequence of 6.3; it has the form

E_{i,j}^{2} = H_{i}(BG; H_{j}(X; M)) ⇒ H_{i+j}(X_{hG}; M) .

6.4. Example. If S is a transitive G-set, then by 5.15 the space S_{hG} is weakly equivalent to BG_{s}, where G_{s} is the isotropy subgroup of some element s ∈ S.

Let GSp denote the category of G-spaces. If D is a small category and F : D → GSp is a functor, then, either by inspection or by general naturality principles, hocolim(F) is in a natural way a G-space. The following proposition is basic for our approach to homology decompositions.

6.5. Proposition. Suppose that G is a group, D is a small category, and F : D → GSp is a functor. Then there is a natural isomorphism of spaces

(hocolim F)_{hG} ≈ hocolim(F_{hG}) .

Proof. Check using 4.6 that the n-simplices of both spaces correspond to triples (x, σ, τ) where τ is an n-simplex of N(G) (5.10), σ is an n-simplex of N(D), and x is an n-simplex of σ(0).

7. Homology decompositions

Suppose that G is a discrete group. Recall (2.9) that p is a fixed prime number.

7.1. Definition. A homology decomposition for BG consists of a mod p homology isomorphism

hocolim F ⟶_{p}^{∼} BG

where D is a small category, F : D → Sp is a functor, and, for each object d of D, F (d) is weakly equivalent to BH_{d} for some subgroup H_{d} of G.

7.2. Remark. In any reasonable homology decomposition, the composite map

BH_{d} ≃ F (d) → hocolim F → BG

(see 4.17) agrees in some homotopical sense with the map BH_{d} → BG induced (5.10) by the inclusion of H_{d} in G as a subgroup.

7.3. Remark. Finding a good homology decomposition is a process of striking a balance. On one extreme, if D is the trivial category with one object and F is the constant functor with value BG, there is a weak equivalence (even an isomorphism)

hocolim F ≈ BG .

This is a homology decomposition in which all of the complexity is concentrated in the functor F. On the other extreme, if D is category of the group G (or the transport category (5.14) of BG) and F is the constant functor with value the one-point space (= B{e}), there is also a weak equivalence hocolim F ≃ BG . In this case F is trivial, and the complexity of the decomposition is concentrated in the shape of the category D. Neither of these decompositions is very interesting. In the most useful homology decompositions the category D is simple, the subgroups H_{d} are small, and the focus of the structure is p-primary, in the sense that the map hocolim F → BG is a mod p homology isomorphism but not a weak equivalence.

We will get all of our homology decompositions by the method described in the following proposition.

7.4. Proposition. Suppose that D is a small category and that F is a functor from D to the category of transitive G-sets. Suppose that the natural map (6.2)

(7.5) (hocolim F)_{hG} → BG is a mod p homology isomorphism. Then there is a homology decomposition

hocolim(F_{hG}) ⟶_{p}^{∼} BG .

Proof. It is easy to see that the natural maps F (d)_{hG} → BG induce a map hocolim(F_{hG}) → BG which under the equivalence of 6.5 corresponds to the mod p homology isomorphism (7.5). In order to finish the proof, then, all we have to do is check that each space F (d)_{hG} is weakly equivalent to BH_{d} for some subgroup H_{d} of G. This follows from 6.4.

Any homology decomposition can be constructed as in 7.4. Can you see how to prove it?

Proposition 7.4 suggests that to construct a homology decomposition for BG, the first thing to look for is a G-space X with the property that the natural map X_{hG} → BG is a mod p homology equivalence. We will find such spaces by looking at complexes associated to collections of subgroups of G.

7.6. Definition. A collection C of subgroups of G is a set of subgroups of G which is closed under conjugation, in the sense that if H ∈ C and g ∈ G, then gHg^{−1} ∈ C.

A collection C of subgroups is a poset with respect to inclusion of one subgroup in another, i.e., with respect to the convention that H ≤ H′ if H ⊂ H′. Let K_{C} be the associated ordered simplicial complex (3.6), or equivalently the nerve of the category (4.12) associated to C. We will let K_{C} denote this category, so that K_{C} = N(K_{C}). The n-simplices of K_{C} are just chains

H_{0} ⊂ H_{1} ⊂ ⋯ ⊂ H_{n}

of elements of C; in particular, if G is finite then K_{C} is a finite simplicial complex. The group G acts on C by conjugation (if g ∈ G and H ∈ C then g · H = gHg^{−1}). This action preserves the inclusion relation and passes to an action of G on K_{C}.

7.7. Definition. A collection C of subgroups of G is said to be ample if the natural map

(K_{C})_{hG} → BG (7.8)

is a mod p homology isomorphism.

7.9. Example. If K_{C} is contractible the collection C is ample, since then the map 7.8 is a weak equivalence (4.14). The homology decomposition maps (7.1) associated below to such a collection are weak equivalences, not just mod p homology isomorphisms.

At this point it is probably not clear that there exist any ample collections of subgroups, Before looking for some, we want to motivate the search. Given a collection C, we will associate to it three functors

α̃_{C} : (A_{C})^{op} → GSets
β̃_{C} : O_{C} → GSets
δ̃_{C} : s̄dK_{C} → GSets

(7.10)

Of course, one issue will be to define the domain categories of these functors. We will then prove the following proposition. (The proof involves combining 5.13 with 5.4. The most interesting feature of it is the way in which working with functors and natural transformations makes it possible to construct what amount to explicit simplicial homotopies (§5) with only a small amount of work.)

7.11. Proposition. If C is an ample collection of subgroups of G, then all three functors of (7.10) satisfy the hypotheses of 7.4, and the corresponding functors

\begin{aligned}α_{C} & = (α̃_{C})_{hG} \\ β_{C} & = (β̃_{C})_{hG} \\ δ_{C} & = (δ̃_{C})_{hG} \end{aligned}

give homology decomposition for BG. Conversely, if any one of these functors gives a homology decomposition for BG, then C is ample.

The three decompositions of 7.11 are called the centralizer decomposition, the subgroup decomposition, and the normalizer decomposition associated to C. The names come from the fact that

• the values of α_{C} have the homotopy type of BC, where C is the centralizer in G of some element of C,

• the values of β_{C} have the homotopy type of BH, where H is an element of C, and

• the values of δ_{C} have the homotopy type of BN, where N is an intersection in G of normalizers of elements of C.

The centralizer decomposition. The C-conjugacy category A_{C} is the category in which the objects are pairs (H, Σ), where H is a group and Σ is a conjugacy class of monomorphisms i : H → G with i(H) ∈ C. A morphism (H, Σ) → (H′, Σ′) is a group homomorphism j : H → H′ which under composition carries Σ′ into Σ.

CLASSIFYING SPACES 33

We are going to want to take homotopy colimits over A_{C} (actually over its opposite category) and, given the way we have defined A_{C}, this is not possible: the category A_{C} is not small. This difficulty is not serious, since A_{C} is equivalent to a small category. What details have to be worked out to make this precise? Can you find an explicit small subcategory of A_{C} which is equivalent to A_{C}?

There is a functor

α̃_{C} : (A_{C})^{op} → GSets

which assigns to each object (H, Σ) the set Σ itself, on which G acts transitively by conjugation. Here is the first third of 7.11.

7.12. Proposition. The collection C is ample if and only if the natural map

(hocolim α̃_{C})_{hG} → BG

is a mod p homology isomorphism.

Proof. According to 5.13, the space X_{C}^{α} = hocolim α̃_{C} is the nerve of a category X_{C}^{α} whose objects consist of pairs (H, i), where H is a group and i : H → G is a monomorphism with i(H) ∈ C. A map (H, i) → (H′, i′) is a homomorphism j : H → H′ such that i′j = i. The action of G on hocolim α̃_{C} corresponds to the action of G on X_{C}^{α} obtained by letting g · (H, i) = (H, gig^{−1}). There is a functor

u : X_{C}^{α} → K_{C}

which sends (H, i) to i(H), a functor which by inspection is G-equivariant. To finish the proof it is enough to show that u induces a weak equivalence from the nerve of X_{C}^{α} to K_{C}. In fact, if this is true, then by 6.2 there is a commutative diagram

(hocolim α̃_{C})_{hG} = (X_{C}^{α})_{hG} ⟶^{≃} (K_{C})_{hG}
↓  ↓
BG  ⟶^{=} BG

in which by 4.14 the top arrow is a weak equivalence. This implies that the left hand arrow is an isomorphism on mod p homology if and only if the right hand one is.

To see that u has the desired property, use 5.4 and note that u is an equivalence of categories. In fact, there is a functor u′ : K_{C} → X_{C}^{α} which sends an object H ∈ C to the pair (H, i) where i is the inclusion map H → G. The composite uu′ is the identity map of K_{C}, while the composite u′u is connected to the identity map of X_{C}^{α} by an obvious natural equivalence.

7.13. Remark. If H ⊂ G is a subgroup, let C_{G}(H) denote the centralizer of H in G. By 6.4, the centralizer decomposition functor α_{C} = (α̃_{C})_{hG} assigns to any object (H, Σ) of A_{C} a space which, for any i ∈ Σ, is weakly equivalent to BC_{G}(i(H)).

The subgroup decomposition. The C orbit category O_{C} is the category whose objects are the G-sets G/H, H ∈ C, and whose morphisms are G-maps. There is a functor β̃_{C} : O_{C} → GSets which assigns to G/H the G-set G/H itself. Here is the second third of 7.11.

7.14. Proposition. The collection C is ample if and only if the natural map

(hocolim β̃_{C})_{hG} → BG

is a mod p homology isomorphism.

Proof. By 5.13, the space X_{C}^{β} = hocolim β̃_{C} is the nerve of a category X_{C}^{β} whose objects consist of pairs (G/H, x), where G/H is a coset space of G with H ∈ C, and x is an element of G/H. A morphism (G/H, x) → (G/H′, x′) is a G-map f : G/H → G/H′ such that f(x) = x′. The action of G on X_{C}^{β} corresponds to the action of G on X_{C}^{β} obtained by letting g · (G/H, x) = (G/H, gx). There is a functor

v : X_{C}^{β} → K_{C}

which sends (G/H, x) to the isotropy subgroup G_{x}, and by inspection this functor is G-equivariant. As in the proof of 7.12, it is enough to show that v is an equivalence of categories. This is easy; there is a functor v′ : K_{C} → X_{C}^{β} which sends H ∈ C to the pair (G/H, eH), where e ∈ G is the identity element. The composite vv′ is the identity functor of K_{C}, and the composite v′v is connected to the identity functor of X_{C}^{β} by an obvious natural equivalence.

7.15. Remark. The subgroup decomposition functor β_{C} = (β̃_{C})_{hG} assigns to the object G/H of O_{C} the space (G/H)_{hG}, which by 6.4 is weakly equivalent to BH.

The normalizer decomposition. This one is a little trickier. Let s̄dK_{C} be the category of “orbit simplices” for the action of G on the simplicial complex K_{C}. The objects of s̄dK_{C} are the orbits σ̄ of the action of G on the simplices of K_{C}, and there is one morphism σ̄ → σ̄′ if for some simplices σ ∈ σ̄ and σ′ ∈ σ̄′, σ′ is a face of σ. (This might look backwards, but it’s necessary to define morphisms this way in order to get the functor δ̃_{C} below.) Note that here we are definitely thinking of K_{C} as a simplicial complex, and not as a simplicial set; degenerate simplices play no role. An object of s̄dK_{C} is an equivalence class, under the action of G by conjugation, of chains

H_{0} ⊊ H_{1} ⊊ ⋯ ⊊ H_{n} (7.16)

such that each H_{i} belongs to C.

There is a functor

δ̃_{C} : s̄dK_{C} → GSets

which assigns to an orbit σ̄ the transitive G-set provided by σ̄ itself. Here is the last part of 7.11.

7.17. Proposition. The collection C is ample if and only if the natural map

(hocolim δ̃_{C})_{hG} → BG

is a mod p homology isomorphism.

Proof. By 5.13, the space X_{C}^{δ} = hocolim δ̃_{C} is the nerve of a category X_{C}^{δ} whose objects are the simplices σ of K_{C}. There is one morphism σ → σ′ if σ′ is a face of σ and no other morphisms. The category X_{C}^{δ} is the opposite of the category whose nerve is the barycentric subdivision (3.6) of K_{C}. There are G-equivariant homeomorphisms (5.9, 3.6)

|X_{C}^{δ}| ≅ | N((X_{C}^{δ})^{op})| ≅ |K_{C}| .

However, these homeomorphisms are not realized by functors or simplicial maps, so there is some more work to be done. We leave this to the reader: the problem is to devise a way of relating X_{C}^{δ} to K_{C} inside the category Sp, by maps which are G-equivariant and are weak equivalences.

7.18. Remark. If H is a subgroup of G, let N_{G}(H) denote the normalizer of H in G. By 6.4, the normalizer decomposition functor δ_{C} = (δ̃_{C})_{hG} assigns to the orbit of the simplex (7.16) of K_{C} a space which has the homotopy type of B(∩_{i}N_{G}(H_{i}))

8. Sharp homology decompositions. Examples

Associated to any homology decomposition for BG (7.1) is a first quadrant mod p homology spectral sequence (4.16)

E_{i,j}^{2}(F) = colim_{i} H_{j}(F) ⇒ H_{i+j}(BG) .

8.1. Definition. A homology decomposition for BG is sharp if its homology spectral sequence collapses onto the vertical axis, in the sense that E_{i,j}^{2} = 0 for i > 0.

The usefulness of a sharp homology decomposition functor F is that it gives an isomorphism

(8.2) colim H_{∗}(F) = colim_{0} H_{∗}(F) ⟶^{≈} H_{∗} BG .

This is essentially a formula for H_{∗} BG in terms of the homology groups of subgroups of G.

It turns out that sharp homology decompositions are sometimes easier to recognize than arbitrary ones. One way in which we will show that a collection C of subgroups is ample (7.7) is to show that one of the three functors α_{C}, β_{C}, or δ_{C} associated to C gives a sharp homology decomposition for BG. The other two functors then give homology decompositions too (although these may or may not be sharp).

Examples of homology decompositions. From now on we will assume that G is a finite group.

8.3. Definition. An ample collection C of subgroups of G is said to be centralizer-sharp (resp. subgroup-sharp, normalizer-sharp) if the centralizer decomposition (resp. subgroup decomposition, normalizer decomposition) associated to C is sharp.

Here are some examples of homology decompositions.

8.4. Example: C = {{e}}. Suppose that C contains only the trivial subgroup of G. Then K_{C} is a one-point space, so C is ample. Moreover, C is both centralizer-sharp and normalizer-sharp, since both the centralizer and normalizer decomposition diagrams reduce to the trivial diagram with BG as its only constituent. The subgroup decomposition category O_{C} has only one object, and the group G is the space of self-maps of this object. The functor β_{C} assigns to this object the trivial one-point space. By 6.2, the homology spectral sequence of β_{C} has

E_{i,j}^{2} = \begin{cases}H_{i}(G) & j = 0 \\ 0 & j > 0 \end{cases} .

This spectral sequence collapses, but onto the wrong axis! The collection C is subgroup-sharp if and only if the reduced mod p homology of BG is trivial.

If G is a finite group, the reduced mod p homology of BG is trivial if and only if the order of G is prime to p. Can you prove this? (Hint: the hard part is to show that if the mod p (co-)homology is trivial, then the order of G is prime to p. Suppose that the order of G is not prime to p. Take a faithful complex representation ρ of G, and try computing the Chern classes of the restriction of ρ to a cyclic subgroup of G of order p.)

8.5. Example: C = {G}. Suppose that C contains only G itself. Again K_{C} is has only one point, so C is ample. The collection C is both subgroup-sharp and normalizer-sharp, since both the subgroup and normalizer decomposition diagrams reduce to the trivial diagram with BG as its only constituent. Let Z be the center of G. The category A_{C} has one object whose group of self-maps is G/Z and the functor α_{C} assigns to this object the space BZ. By 6.2, the homology spectral sequence of α_{C} is the Lyndon-Hochschild-Serre spectral sequence of the group extension Z → G → G/Z. From this it is possible to prove that C is centralizer-sharp if and only if Z contains a Sylow p-subgroup of G; this is the case if and only if G is the product of an abelian p-group and a group of order prime to p.

Can you prove these last statements?

8.6. Other trivial examples. If C contains either {e} or G, then K_{C} is contractible, because the category associated to C (4.12) has either {e} as an initial object or G as a terminal object (see 5.5). In particular, C is ample. The corresponding homology decompositions are in some sense circular, since (as suggested in the above two examples) G itself is somehow encoded in each of the three homotopy colimits associated to C.

Suppose that C is the set of conjugates of H, where H is some subgroup of G different from {e} and from G itself. Can you think of examples of subgroups H for which C is ample? What are the corresponding three decomposition functors? What sharpness properties does C have? Think first about the special case in which H is normal, so that C = {H}.

8.7. Non-identity p-subgroups. Suppose that p divides the order of G, and let C be the collection of all non-identity p-subgroups of G. In one form or another it has been known for a long time that C is ample. We will prove this in §12, and show in addition that C is centralizer-sharp, subgroup-sharp, and normalizer-sharp; Most of the technical results that go into proving sharpness are due originally to Webb [35] [36]. The colimit formula (8.2) for H_{∗} BG that arises from the associated subgroup decomposition (§7) is essentially the 𝔽_{p}-dual of the formula of Cartan and Eilenberg [8, p. 259] expressing H^{∗} BG as the set of stable elements in the cohomology of a Sylow p-subgroup of G.

8.8. Elementary abelian p-subgroups. A finite abelian group is said to be an elementary abelian p-group if it is a module over 𝔽_{p}. Suppose that p divides the order of G, and let C be the collection of all nonidentity elementary abelian p-subgroups of G, and C′ the collection of all non-identity p-subgroups of G. It is a theorem of Quillen that the natural map K_{C} → K_{C′} is a weak equivalence [2, 6.6.1].

The proof is a simple but clever manipulation with categories, functors, and natural transformations. Can you figure out what it is?

Since C′ is ample (8.7), so is C. In fact, C is both centralizer-sharp and normalizer-sharp; by the duality between homology and cohomology, it cannot possibly be subgroup-sharp unless the mod p cohomology of G is detected on elementary abelian p-subgroups.

The mod p cohomology of G is detected on elementary abelian subgroups if for every nonzero x ∈ H^{∗} BG there is an elementary abelian p-subgroup V of G such that the restriction of x to H^{∗} BV is nonzero. Can you think of a finite group G such that H^{∗} BG is not detected on elementary abelian p-subgroups? Quillen showed [27] that if G is the group GL_{n}(𝔽_{q}) (where q is relatively prime to p) or if G is a symmetric group, then the mod p cohomology of BG is detected on elementary abelian subgroups. It seems reasonable to guess that for such groups the collection of nontrivial elementary abelian p-subgroups is subgroup-sharp. It would be interesting to have a proof.

We will discuss this and related collections in 13. The question of centralizer sharpness was first studied by Jackowski and McClure [18], later by Dwyer and Wilkerson [13] and by Benson [3]. The main ingredient necessary to prove normalizer sharpness was provided by Brown [7, p. 268].

Let p = 2, and let G be the group SL_{3}(𝔽_{q}), where q is odd. Show that up to conjugacy there are only two nontrivial elementary abelian p-subgroups of G. What are the corresponding homology decompositions?

8.9. p-centric subgroups. A p-subgroup P of G is said to be p-centric if the center of P is a Sylow p-subgroup of C_{G}(P). This is equivalent to the condition that C_{G}(P) is the direct product of the center of P and a group of order prime to p. Let C be the collection of all p-centric subgroups of G. Then C is ample [11] as well as subgroup-sharp [10].

Suppose that G is a group with the property that the Sylow p-subgroups of G are abelian. Argue that up to congugacy there is only one p-centric subgroup of G. What are the associated homology decomposistions? Is it obvious that the subgroup decomposition is sharp? Recover a certain classical theorem of Swan.

8.10. Subgroups both p-centric and p-stubborn. A p-subgroup P of G is said to be p-stubborn if N_{G}(P)/P has no non-identity normal psubgroups. Let C be the collection of all subgroups of G which are both p-centric (8.9) and p-stubborn. Then C is ample [11] and subgroup-sharp [10]. The role of p-stubborn subgroups was pointed out by Jackowski, McClure and Oliver [19], and independently (in a different context and with different terminology) by Bouc [2, 6.6.6].

9. Reinterpreting the homotopy colimit spectral sequence

In this section we point out that the Bousfield-Kan homology spectral sequence (4.16) associated to a homotopy colimit can be interpreted as a Leray spectral sequence. For the particular homotopy colimits that give rise to the homology decompositions of §7, these Leray spectral sequences are the isotropy spectral sequences associated to G-spaces. This makes it possible to study the spectral sequences (e.g., in order to show that a decomposition is sharp) by making geometric manipulations with G-spaces.

We first introduce the Leray spectral sequence, then we specialize to the isotropy spectral sequence, and finally we explain the relationship of the isotropy spectral sequence to homology decompositions.

The Leray spectral sequence. The n-skeleton sk_{n} B (n ≥ 0) of a space B is the subobject of B generated by all simplices of dimension ≤ n.

9.1. Definition. Let f : X → B be a map of spaces, and let X_{n} denote f^{−1}(sk_{n} B). The Leray spectral sequence of f is the (mod p) homology spectral sequence associated to the filtration

X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯

of X.

This spectral sequence is usually indexed in such a way that E_{i,j}^{1} = H_{i+j}(X_{i}, X_{i−1}). Since X_{n} contains sk_{n} X, E_{i,j}^{1} = 0 for j < 0 and this is a first quadrant, strongly convergent homology spectral sequence. In particular, the differential d^{r} has bidegree (−r, r − 1). Here are a few examples.

9.2. Collapse map. If Y is a subspace of X, the E^{2}-term of the Leray spectral sequence of f : X → X/Y vanishes except for the groups E_{0,0}^{2} = H_{0} X, E_{0,j}^{2} = H_{j}(Y) (j > 0) and E_{i,0}^{2} = H_{i}(X/Y) (i > 0). The various differentials running from the horizontal axis to the vertical axis give the connecting homomorphisms in the long exact homology sequence of the pair (X, Y).

9.3. Fibration. If f : X → Y is a fibration of spaces [23], the Leray spectral sequence of f can be identified with the Serre spectral sequence [30] of |f|.

9.4. Homotopy colimit. Suppose that D is a small category and F : D → Sp is a functor. The unique natural transformation from F to the constant functor ∗ with value the one-point space induces a map

f : hocolim F → hocolim ∗ ≈ N(D)

The Leray spectral sequence of f can be identified from E^{2} onwards with the Bousfield-Kan homology spectral spectral sequence (4.16) of the homotopy colimit. This can be seen by inspecting the definitions (4.7, 9.1) and seeing that the two spectral sequences arise from the same filtration of hocolim F.

The isotropy spectral sequence. We will be interested in a specific Leray spectral sequence associated to a G-space X.

9.5. Definition. The isotropy spectral sequence of a G-space X is the Leray spectral sequence of the map (6.2)

f : X_{hG} = (EG × X)/G → X/G .

9.6. Remark. Consider the diagram

{BG_{x}} ; ↓ ; X ⟶ X_{hG} ⟶^{q} (∗)_{hG} = BG ; ^{f}↓ ; X/G (9.7)

The horizontal sequence is a fibration sequence, and the Leray spectral sequence of q is the usual Serre spectral sequence converging to H_{∗}(X_{hG}; M). Equivalently, this is the Bousfield-Kan homology spectral sequence associated to the description of X_{hG} as a homotopy colimit (6.1). The vertical sequence is meant to suggest that the geometric fibres of f in general differ from point to point, and that the fibre over a simplex x̄ ∈ X/G can be identified up to homotopy with BG_{x}, where G_{x} is the isotropy subgroup of a simplex x ∈ X above x̄. The Leray spectral sequence of f is the isotropy spectral sequence of X, and it too converges to H_{∗}(X_{hG}).

9.8. Examples. If G acts freely on X then the map f of (9.7) is an equivalence with contractible fibres, and the isotropy spectral sequence of X collapses onto the horizontal axis. More generally, suppose that K is a normal subgroup of G which acts trivially on X, and that the quotient group W = G/K acts freely on X. Then the map f of 9.7 is a fibration with fibre BK, and the isotropy spectral sequence of X can be identified with the Serre spectral sequence of the fibration BK → X_{hG} → X_{hW}.

9.9. The E^{2}-page of the isotropy spectral sequence. The isotropy spectral sequence of the G-space X can be constructed as the homology spectral sequence (4.7) of the simplicial space Y given by

Y_{n} = (X_{n})_{hG} .

Here we are treating X_{n} as a discrete space (3.19) with a G-action; the horizontal (4.5) face and degeneracy operators are induced by the face and degeneracy operators of X.

According to 4.8 and 3.23, the E^{2}-page of the isotropy spectral sequence can be constructed in the following way. For each j ≥ 0 one can form a chain complex ⟨H_{j} (X_{∗})_{hG}, d⟩ which in dimension i ≥ 0 contains the group H_{j} (X_{i})_{hG} and has boundary map d induced by taking the alternating sum of the maps

H_{j} (X_{i})_{hG} → H_{j} (X_{i−1})_{hG}

induced by the (i+1) face operators X_{i} → X_{i−1}. The group in position E_{i,j}^{2}-term of the isotropy spectral sequence is then the i’th homology group of ⟨H_{j} (X_{∗})_{hG}, d⟩.

Homology decomposition spectral sequences. We now show that the Bousfield-Kan homology spectral sequence of any homotopy colimit constructed by the technique of 7.4 can be interpreted as the isotropy spectral sequence of an associated G-space. This will allow us to work with the Bousfield-Kan spectral sequence (which is pretty abstract) by manipulating G-spaces (which are much easier to handle). The basis for this is the following proposition.

9.10. Proposition. Suppose that D is a small category and that F is a functor from D to the category of transitive G-sets. Let X(F) denote the G-space hocolim F. Then the Bousfield-Kan homology spectral sequence for the homology of hocolim(F)_{hG} can be identified in a natural way with the isotropy spectral sequence of X(F).

Proof. There is a commutative diagram

hocolim(F)_{hG} ⟶ hocolim(F/G) ≈ N(D)
≈↓  ^{≈}↓
(hocolim F)_{hG} ⟶  (hocolim F)/G

in which both vertical arrows are isomorphisms (see 6.5 for the left arrow). The Bousfield-Kan homology spectral sequence of F is the Leray spectral sequence of the upper map (9.4) and the isotropy spectral sequence of X(F) is the Leray spectral sequence of the lower one.

We now establish notation for the G-spaces associated to the decompositions of §7.

9.11. The centralizer decomposition. Let X_{C}^{α} denote the G-space given by hocolim α̃_{C}. As in the proof of 7.12, X_{C}^{α} is is the nerve of the category X_{C}^{α} whose objects are pairs (H, i), where H is a group and i : H → G is a monomorphism with i(H) ∈ C. The Bousfield-Kan homology spectral sequence associated to α_{C} is the isotropy spectral sequence of the action of G on X_{C}^{α}.

9.12. The subgroup decomposition. Let X_{C}^{β} denote the G-space given by hocolim β̃_{C}. As in the proof of 7.14, this is the nerve of a category X_{C}^{β} whose objects consist of pairs (G/H, x) where H ∈ C and x ∈ G/H. The Bousfield-Kan homology spectral sequence associated to β_{C} is the isotropy spectral sequence of the action of G on X_{C}^{β}.

9.13. The normalizer decomposition. Let X_{C}^{δ} denote the space K_{C}. This is the nerve of the category X_{C}^{δ} associated to the poset C. It is possible to prove that X_{C}^{δ} is weakly G-equivalent (10.5) to the space hocolim δ̃_{C} (see the remarks in the proof of 7.17), and so the Bousfield-Kan homology spectral sequence associated to δ_{C} is the isotropy spectral sequence of the action of G on X_{C}^{δ}.

10. Bredon homology and the transfer

The previous section identified the Bousfield-Kan spectral sequence derived from a homology decomposition of BG as the isotropy spectral sequence of an associated G-space X. The E^{2}-term of this isotropy spectral sequence is a type of homological functor of X; in this section we point out exactly what sort of construction this homological functor is, and describe some of its properties.

10.1. Definition. Let H be a functor from the category of 𝔽_{p}[G]- modules to the category of 𝔽_{p} vector spaces. The functor H is said to be a coefficient functor for G if H preserves arbitrary direct sums. If K is a subgroup of G, the restriction of H to K, denoted H|_{K}, is the coefficient functor for K given by H|_{K}(A) = H(𝔽_{p}[G] ⊗_{𝔽_{p}[K]} A).

The functor H is said to preserve arbitrary direct sums if, for any set {M_{α}} of 𝔽_{p}[G]-modules, the natural map

⊕_{α}H(M_{α}) → H(⊕_{α}M_{α})

is an isomorphism.

10.2. Example. For each j ≥ 0 there is a coefficient functor H_{j}^{G} given by H_{j}^{G}(A) = H_{j}(G; A) (see 2.8). These are the coefficient functors we will interested in. If K is a subgroup of G and H = H_{j}^{G}, then H|_{K} ≈ H_{j}^{K}.

Prove the final statement; it’s a form of Shapiro’s lemma [37, 6.3.2].

10.3. Definition. Suppose that H is a coefficient functor for G and that X is a G-space. Let (C_{∗}^{G}(X; H), d) be the chain complex with CG_{n}(X; H) = H(𝔽_{p}[X_{n}]) (see 3.24) and with boundary d induced by the alternating sum of the face maps in X. The Bredon homology groups of X with coefficients in H, denoted H_{∗}^{G}(X; H), are defined to be the homology groups of C_{∗}^{G}(X; H).

10.4. Example. Suppose that X is a G-space, and let {H_{j}^{G}} be the coefficient functors from 10.2. The E^{2} page of the isotropy spectral sequence for X is then given by

E_{i,j}^{2} = H_{i}^{G}(X; H_{j}^{G}) .

This follows immediately from 9.10, 9.9, and the observation that for any G-set S, there is a natural isomorphism

H_{j}^{G}(𝔽_{p}[S]) ≈ H_{j}(S_{hG}) .

Prove this last statement, by combining additivity of H_{j}^{G} with the final sentence of 10.2. It is possible to use coefficients for Bredon homology more general than the functors allowed above; in fact, any functor from the orbit category of G to abelian groups can be used to construct a Bredon homology theory. For practical purposes, the “coefficient functors” above are the same as the “cohomological Mackey functors” of [38] or [33, p. 1928].

A key property of H_{∗}^{G}(X; H). The groups H_{∗}^{G}(X; H) have a certain basic invariance property.

10.5. Definition. A map f : X → Y of G-spaces is said to be a weak G-equivalence if f^{H} : X^{H} → Y^{H} is a weak equivalence for every subgroup H of G.

10.6. Remark. If X is a G-space, let Iso(X) denote the set of subgroups of G which appear as isotropy groups of simplices of X. By [11, 4.1], a map f : X → Y of G-spaces is a weak G-equivalence if and only if it induces a weak equivalence f^{H} : X^{H} → Y^{H} for all H ∈ Iso(X)∪Iso(Y).

10.7. Proposition. Suppose that f : X → Y is a a weak G-equivalence and that H is a coefficient functor (10.1). Then f induces isomorphisms H_{∗}^{G}(X; H) ≅ H_{∗}^{G}(Y; H).

A proof of this is sketched in §14.

10.8. The transfer. All of our techniques for dealing with the Gspaces from 9.11, 9.12, and 9.13 are based in one way or another on the transfer. We assume from now on that H is a coefficient functor (10.1), in practice one of the coefficient functors H_{j}^{G} (10.2).

Suppose that f : S → T is a map of G-sets. There is an induced map 𝔽_{p}[S] → 𝔽_{p}[T], also denoted f, as well as a map

f_{∗} = H(f) : H(𝔽_{p}[S]) → H(𝔽_{p}[T]) .

Say that f is finite-to-one if for each x ∈ T the set f^{−1}(x) is finite. For such an f there is a G-map τ(f) : 𝔽_{p}[T] → 𝔽_{p}[S], called the pretransfer, which sends x ∈ T to ∑_{y∈f^{−1}(x)} y. The induced map

τ_{∗}(f) : H(𝔽_{p}[T]) → H(𝔽_{p}[S])

is the transfer associated to f.

10.9. Example. Suppose that H and K are subgroups of G with H ⊂ K, and let H = H_{j}^{G} (10.2). Let f : G/H → G/K be the projection map. By Shapiro’s lemma [37, 6.3.2] there are isomorphisms

H(𝔽_{p}[G/H]) ≅ H_{j}(H) H(𝔽_{p}[G/K]) ≅ H_{j}(K) .

Under these identifications, f_{∗} : H_{j}(H) → H_{j}(K) is the map induced by the inclusion H ⊂ K and τ_{∗}(f) : H_{j}(K; M) → H_{j}(H; M) is the associated group homology transfer map [37, 6.3.9] [2, p. 67].

The transfer has the following basic properties, which are easy to verify by calculations with pretransfers. Recall that H is assumed to commute with direct sums.

10.10. Lemma. Suppose that f_{1} : S_{1} → T_{1} and f_{2} : S_{2} → T_{2} are maps of G-sets. If f_{1} and f_{2} are finite-to-one, then so is f_{1} ⨿ f_{2} : S_{1} ⨿ S_{2} → T_{1} ⨿ T_{2}, and τ_{∗}(f_{1} ⨿ f_{2}) = τ_{∗}(f_{1}) ⊕ τ_{∗}(f_{2}).

10.11. Lemma. Suppose that f_{1} : S_{1} → T and f_{2} : S_{2} → T are maps of G-sets. If f_{1} and f_{2} are finite-to-one, the so is f_{1} +f_{2} : S_{1} ⨿S_{2} → T, and τ_{∗}(f_{1} + f_{2}) = (τ_{∗}(f_{1}), τ_{∗}(f_{2}))

10.12. Remark. It follows from 10.9, 10.10, and 10.11 that if f : S → T is a map of G-sets which is finite-to-one, then τ_{∗}(f) can be computed in terms of a sum of transfers associated to the projections G/G_{x} ⊂ G/G_{f(x)}, x ∈ S. For convenience, we will sometimes call such a transfer the transfer associated to the inclusion G_{x} → G_{f(x)}.

10.13. Lemma. Suppose that f : S → T and g : T → R are maps of G-sets. If f and g are finite-to-one then so is g · f, and τ_{∗}(g · f) = τ_{∗}(f) · τ_{∗}(g).

10.14. Lemma. Suppose that

\begin{matrix}S′ & ⟶^{s} & S \\ ^{f′}↓ & & ^{f}↓ \\ T′ & ⟶^{t} & T \end{matrix}

is a pullback square of G-sets (i.e. a commutative diagram which induces an isomorphism from S′ to the pullback of S and T′ over T). Then if f is finite-to-one, so is f′, and τ_{∗}(f) · t_{∗} = s_{∗} · τ_{∗}(f′).

10.15. Definition. A map f : S → T of G-sets is said to be an even covering mod p if it is finite-to-one and the cardinality mod p of f^{−1}(x) does not depend on the choice of x ∈ T. The common value mod p of these inverse image cardinalities is called the degree of f and denoted deg(f).

10.16. Lemma. Suppose that f : S → T is a map of G-sets which is an even covering mod p. Then the composite f_{∗} · τ_{∗}(f) is the endomorphism of H(𝔽_{p}[T]) given by multiplication by deg(f).

10.17. Example. Suppose that H is a subgroup of G and that S is a G-set. The action map a : G ×_{H} S → S is finite-to-one and has degree given by the index of H in G. Moreover, if S′ → S is a map of G-sets, the diagram

G ×_{H} S′ ⟶ G ×_{H} S
a↓  ^{a}↓
S′ ⟶ S

is a pullback square. It follows that the maps τ_{∗}(a) give a natural map

H(𝔽_{p}[S]) ⟶^{τ_{∗}(a)} H(𝔽_{p}[G ×_{H} S])

on the category of G-sets. Moreover, the composite a_{∗} · τ_{∗}(a) is the endomorphism of H(𝔽_{p}[S]) given by multiplication by the index of H in G.

The notation “G ×_{H} S” above stands for the quotient of G × S by the equivalence relation “∼” given by (g, hs) ∼ (gh, s) for g ∈ G, s ∈ S, and h ∈ H. The action of G on this quotient is induced by the action of G on G × S obtained by setting g′ · (g, s) = (g′g, s). There is an isomorphism 𝔽_{p}[G ×_{H} S] ≈ 𝔽_{p}[G] ⊗_{𝔽_{p}[H]} 𝔽_{p}[S] of 𝔽_{p}[H]-modules. Show that there is also an isomorphism of G-sets G ×_{H} S ≈ (G/H) × S, where G acts diagonally on the product (G/H) × S.

11. Acyclicity for G-spaces

In this section we translate the question of whether a homology decomposition of BG is sharp into a question about the Bredon homology of the associated G-space (§9). We then study this second question. The symbol H denotes a coefficient functor (10.1) for G. A lot of the material in this section first appeared (in a slightly different form) in a paper of Webb [36].

11.1. Definition. A G-space X is said to be acyclic for H if the map X → ∗ induces an isomorphism H_{∗}^{G}(X; H) → H_{∗}^{G}(∗; H).

11.2. Remark. Note that H_{i}^{G}(∗; H) vanishes for i > 0 and H_{0}^{G}(∗; H) = H(𝔽_{p}). Let C be a collection of subgroups of G. By §9 and 10.4, the centralizer, subgroup and normalizer decompositions associated to C are sharp (8.1) if and only if the G-spaces X_{C}^{α}, X_{C}^{β}, and X_{C}^{δ} (respectively) are acyclic for the functors H_{j}^{G}, j ≥ 0. In fact, more is true: if any one of these three G-spaces is acyclic for all of the functors H_{j}^{G} (j ≥ 0), then C is ample, all three spaces correspond to homology decompositions for BG, and the particular decomposition corresponding to the acyclic G-space is sharp.

Note that there are functors

X_{C}^{α} → X_{C}^{δ} ← X_{C}^{β}

which induce G-maps

X_{C}^{α} → X_{C}^{δ} ← X_{C}^{β} . (11.3)

According to the arguments in the proofs of 7.12 and 7.14, these maps are weak equivalences. This reflects the fact that when it comes to the centralizer, subgroup, and normalizer diagrams, either all three give homology decomposition of BG or none of them do. However, the two maps of (11.3) are not usually weak G-equivalences. This reflects (10.7) the fact that the three decompositions can have different sharpness properties.

We have two ways to show that a G-space X is acyclic for H. In fact, the second method is a refinement of the first one.

The direct transfer method. This uses the fact that if K is a subgroup of G of index prime to p then the transfer exhibits H_{∗}^{G}(X; H) as a retract of H_{∗}^{K}(X; H|_{K}).

11.4. Theorem. Suppose that X is a G-space, H is a coefficient functor for G, and K is a subgroup of G of index prime to p. If X is acyclic as a K-space for H|_{K}, then X is acyclic as a G-space for H.

Proof. The transfers (10.8) associated to the maps q : G ×_{K} X_{n} → X_{n} provide a map t : C_{∗}^{G}(X; H) → C_{∗}^{K}(X; H|_{K}) (9.9). By 10.17 this map commutes with differentials, and the index assumption implies that the composite C_{∗}^{G}(X; H) ⟶^{t}C_{∗}^{K}(X; H|_{K}) ⟶^{q}C_{∗}^{G}(X; H) is an isomorphism. By naturality, then, the map H_{∗}^{G}(X; H) → H_{∗}^{G}(∗; H) is a retract of H_{∗}^{K}(X; H|_{K}) → H_{∗}^{K}(∗; H|_{K}), and the theorem follows from the fact that a retract of an isomorphism is an isomorphism.

11.5. The method of discarded orbits. This a more sophisticated version of the direct transfer method which exploits the fact that Korbits can be discarded if they do not contribute to the transfer.

11.6. Theorem. Let X be a G-space, K a subgroup of G of index prime to p, and Y a subspace of X which is closed under the action of K. Assume that Y is acyclic for H|_{K}, and that for each simplex x ∈ X \ Y the transfer map H(𝔽_{p}[G/G_{x}]) → H(𝔽_{p}[G/K_{x}]) is zero (cf. 10.12). Then X is acyclic for H.

Proof. The transfers associated to the maps q : G×_{K} X_{n} → X_{n} provide a map t : C_{∗}^{G}(X; H) → C_{∗}^{K}(X; H|_{K}) (§10). By 10.17 this map commutes with differentials, and the index assumption implies that the composite C_{∗}^{G}(X; H) ⟶^{t}C_{∗}^{K}(X; H|_{K}) ⟶^{q}C_{∗}^{G}(X; H) is an isomorphism. The transfer hypothesis shows that the image of t is actually in the subcomplex C_{∗}^{K}(Y; H|_{K}) of C_{∗}^{K}(X; H|_{K}). By naturality, the homology map H_{∗}^{G}(X; H) → H_{∗}^{G}(∗; H) is a retract of H_{∗}^{K}(Y; H|_{K}) → H_{∗}^{K}(∗; H|_{K}), and the theorem follows from the fact that a retract of an isomorphism is an isomorphism.

11.7. Example. Let X be a G-space, K a subgroup of G of index prime to p, and Y a subspace of X which is closed under the action of K. Suppose that Y is acyclic for the functors H_{j}^{G}, j ≥ 0. Assume finally that for each x ∈ X \ Y the transfer map H_{∗}(G_{x}) → H_{∗}(K_{x}) is zero. In light of 10.2, Theorem 11.6 implies that X is acyclic for the functors H_{j}^{G}, j ≥ 0.

A special case of the above is due to Webb [36] [1, V.3]; the proof here is essentially the same as his.

11.8. Corollary. Let X be a G-space and P a Sylow p-subgroup of G. Suppose that for every nonidentity subgroup Q of P the fixed point space X^{Q} is contractible, and that for any x ∈ X there is a (nonidentity) element of order p in G_{x}. Then X is acyclic for the functors H_{j}^{G}, j ≥ 0.

Proof. Let Y be the P -subspace of X consisting of simplices which are fixed by a nonidentity element of P. By 10.6, the map Y → ∗ is a weak P -equivalence, and so by 10.7 the space Y is acyclic for H_{j}(P; –). Moreover, for any j ≥ 0 and x ∈ X \ Y the transfer map

H_{j}(G_{x}) → H_{j}(P_{x}) = H_{j}({e})

is trivial. This is true for j > 0 because the target group is zero, and true for j = 0 because by assumption the norm map ∑_{g∈G_{x}} g : 𝔽_{p} → 𝔽_{p} is trivial. The result follows from 11.6 (cf. 11.7).

12. Non-identity p-subgroups

Let C be the collection of all non-identity p-subgroups of G. Assume that C is nonempty, i.e., that the order of G is divisible by p. In this section we will show that C is ample and that the three homology decompositions derived from C are sharp. In all three cases we use the method of Webb (11.8) to show that the spaces X_{C}^{δ}, X_{C}^{β}, and X_{C}^{α} are acyclic for the coefficient functors H_{j}^{G} (j ≥ 0) (11.2).

We first recall an elementary property of finite p-groups.

12.1. Lemma. Any nontrivial finite p-group has a nontrivial center.

Proof. Let P be a nontrivial finite p-group, and consider the action of P on itself by conjugation. The fixed point set of the action is the center C of P. Since each nontrivial orbit has order a power of p, card(C) is congruent mod p to card(P), so in particular card(C) is divisible by p. Since C certainly contains the identity element, it follows that C also contains nonidentity elements of P.

Let P denote a Sylow p-subgroup of G.

12.2. The normalizer decomposition. See Webb [36, 2.2.2]. We will show that X_{C}^{δ} is acyclic for the coefficient functors H_{j}^{G}. The first step is to show that for each nonidentity subgroup Q of P the space (X_{C}^{δ})^{Q} is contractible. By 9.13 and 5.12 (X_{C}^{δ})^{Q} is the nerve of the full subcategory D of X_{C}^{δ} generated by the objects H of X_{C}^{δ} (equivalently, elements H ∈ C) such that Q ⊂ N_{G}(H). The inclusions

H ⊂ H · Q ⊃ Q

provide a zigzag of natural transformations between the identity functor of D and the constant functor with value Q. The existence of this zigzag implies that N(D) is contractible (5.7).

A typical n-simplex Q_{0} ⊂ ⋯ ⊂ Q_{n} (Q_{i} ∈ C) of X_{C}^{δ} has isotropy subgroup ∩_{i}N_{G}(Q_{i}). It is clear that there is an element of order p in this isotropy subgroup; any element of Q_{0} will do. Now use 11.8.

12.3. Remark. The above result implies that the collection C of nontrivial p-subgroups of G is ample (see 11.2).

12.4. The centralizer decomposition. We again show that X_{C}^{α} is acyclic for the coefficient functors H_{j}^{G}. We first prove that for any nonidentity subgroup Q of P the space (X_{C}^{α})^{Q} is contractible. By 9.11 and 5.12, (X_{C}^{α})^{Q} is the nerve of the full subcategory D of X_{C}^{α} generated by the objects (H, i) with the property that Q ⊂ C_{G}(i(H)). Let Z be the center of Q and j : Z → G the inclusion. For an object (H, i) of D, let H′ denote the image of the product map H × Z → G and i′ : H′ → G the inclusion. The maps

(H, i) → (H′, i′) ← (Z, j)

give a zigzag of natural transformations between the identity functor of D and the constant functor with value (Z, j). As above, then, N(D) is contractible.

The isotropy subgroup of a typical n-simplex

H_{0} → H_{1} → ⋯ H_{n} → G (12.5)

of X_{C}^{α} has the form C_{G}(Q) for some Q ∈ C. It follows from 12.1 that such an isotropy subgroup contains a nonidentity element of order p. Now use 11.8.

12.6. The subgroup decomposition. Again, we show that X_{C}^{β} is acyclic for the coefficient functors H_{j}^{G}. As above the first problem is to show that for any non-identity subgroup Q of P the space (X_{C}^{β})^{Q} is contractible. By 9.12 and inspection (X_{C}^{β})^{Q} is the nerve of the full subcategory D of X_{C}^{β} generated by the pairs (x, G/H) with Q ⊂ G_{x}. The category D has (eQ, G/Q) as an initial element; in other words, for any object (x, G/H) of D there is a unique map (eQ, G/Q) → (x, G/H). This implies (5.5) that N(D) is contractible.

A typical simplex of X_{C}^{β} has as its isotropy subgroup a group of the form Q for some Q ∈ C. Any such isotropy subgroup contains an element of order p. Now, again, use 11.8.

13. Elementary abelian p-subgroups

In this section we prove several sharpness statements about collections of elementary abelian p-subgroups of G.

Non-identity elementary abelian p-subgroups. Let C be the collection of all non-identity elementary abelian p-subgroups of G. We will show that C is both both centralizer-sharp and normalizer-sharp. The arguments mimic the ones in §12. Let P be a Sylow p-subgroup of G. In each case we use the method of Webb (11.8) to show that the spaces X_{C}^{δ} and X_{C}^{α} are acyclic for the coefficient functors H_{j}^{G}.

Again, we begin with an elementary property of finite p-groups. It is proved by a counting argument like the one in the proof of 12.1

13.1. Lemma. Let P and Q be finite p-groups with Q ≠ {e}, and suppose that P acts on Q via group automorphisms. Then there exists a nonidentity element x in the center of Q such that x is fixed by the action of P.

13.2. The normalizer decomposition. Again, see [36, 2.2.2]. We follow 12.2. The first step is to show that for any non-identity subgroup Q of P the space (X_{C}^{δ})^{Q} is contractible. By 9.13 and 5.12, (X_{C}^{δ})^{Q} is the nerve of the full subcategory D of X_{C}^{δ} determined by the objects H of X_{C}^{δ} (equivalently, elements H ∈ C) such that Q ⊂ N_{G}(H). Let Z be the group of elements of exponent p in the center of Q, and given an object H of D, let H′ be the group of elements of exponent p in the center of QH. The inclusions

H ⊃ H ∩ H′ ⊂ H′Z ⊃ Z

give a zigzag of natural transformations between the identity functor of D and the constant functor with value Z. Lemma 13.1 implies that H ∩H′ is non the identity subgroup of G. By 5.7, N(D) is contractible.

As in 12.2, the isotropy subgroup of any simplex of X_{C}^{δ} contains an element of order p.

13.3. The centralizer decomposition. See [18], where the authors prove a similar theorem for compact Lie groups by a somewhat different argument. We follow the argument of 12.4. The first step is to show that if Q is a non-identity subgroup of P, then (X_{C}^{α})^{Q} is contractible. By 9.11 and 5.12, (X_{C}^{α})^{Q} is the nerve of the full subcategory D of X_{C}^{α} generated by objects (H, i) such that Q ⊂ C_{G}(i(H)). Let Z denote group of elements of exponent p in the center of Q and j : Z → G the inclusion. For an object (H, i) of D, let H′ denote the image of the product map H × Z → G and i′ : H′ → G the inclusion. The maps

(H, i) → (H′, i′) ← (Z, j) (13.4)

provide a zigzag of natural transformations between the identity functor of D and the constant functor with value (Z, j). As above, this implies that N(D) is contractible.

As in 12.4, the isotropy subgroup of any simplex of X_{C}^{α} contains an element of order p.

Smaller collections. If C is a collection of subgroups of G and K is a subgroup of G, let C ∩ 2^{K} denote the set of all elements of C which are subgroups of K. Clearly C ∩ 2^{K} is a collection of subgroups of K. We are aiming at the following theorem, as an illustration of how it is sometimes possible to work with smaller collections than the collection of all nontrivial elementary abelian p-subgroups.

13.5. Theorem. Let K be a subgroup of G of index prime to p, and C a collection of elementary abelian p-subgroups of G. If C ∩ 2^{K} is centralizer-sharp (as a K-collection) then C is centralizer-sharp (as a G-collection).

13.6. Example. Theorem 13.5 can be used as a substitute for the argument of 13.3 in showing that the collection C of all non-identity elementary abelian p-subgroups of G is centralizer-sharp (for the trivial module 𝔽_{p}). To see this, let P be a Sylow p-subgroup of G. It is enough to prove that the collection C′ = C ∩ 2^{P} of all non-identity elementary abelian p-subgroups of P is centralizer-sharp as a P -collection. We can derive this from 10.7 by showing that X_{C′}^{α} is P -equivariantly equivalent to a point. Let j : Z → P be the inclusion of the group of elements of exponent p in the center of P. If (H, i) is an object of X_{C′}^{α}, let H′ denote the image of the product map H × Z → P and i′ : H′ → P the inclusion. The maps

(H, i) → (H′, i′) ← (Z, j)

provide a zigzag of natural transformations between the identity functor of X_{C′}^{α} and the constant functor with value (Z, j). This zigzag respects the action of P on X_{C′}^{α}, and so, according to the arguments of §5, gives an equivariant contraction of X_{C′}^{α}.

13.7. Example. [3] It is possible to do better than the above. Let P be a Sylow p-subgroup of G, and let Z be any non-identity central elementary abelian p-subgroup of P. Let C be the smallest collection of subgroups of G which contains Z and has the property that if V ∈ C and V commutes with Z then ⟨Z, V ⟩ ∈ C. An argument virtually identical to the one in 13.6 shows that if C′ = C ∩ 2^{P}, then X_{C′}^{α} is P -equivariantly contractible. It follows from 13.5 and 10.7 that C is centralizer-sharp.

The proof of 13.5 depends on the following observation.

13.8. Lemma. Suppose that K is a subgroup of G and that V is an elementary abelian subgroup of G not entirely contained in K. Then the transfer map

H_{∗}(C_{G}(V); 𝔽_{p}) → H_{∗}(C_{G}(V) ∩ K; 𝔽_{p})

associated to C_{G}(V) ∩ K → C_{G}(V) (see 10.12) is zero.

Proof. Let C_{1} = C_{G}(V) ∩ K and C_{2} = C_{G}(V). Choose v ∈ V with v ∉ K, and let C′_{1} ≅ C_{1} × ⟨v⟩ be the subgroup of C_{2} generated by C_{1} and v. The inclusion C_{1} → C_{2} factors as the composite of f′ : C′_{1} → C_{2} with f : C_{1} → C′_{1}, so the transfer in question factors (10.13) as a parallel composite τ_{∗}(f)τ_{∗}(f′). However, the map τ_{∗}(f) is zero; this follows from the fact that the map

f_{∗} : H_{∗}(C_{1}; 𝔽_{p}) → H_{∗}(C′_{1}; 𝔽_{p})

is a monomorphism (f has a left inverse) and the fact that the composite τ_{∗}(f) · f_{∗} is multiplication by p (10.16).

Proof of 13.5. Let X be the G-space X_{C}^{α} = N(X_{C}^{α}); we have to show that X is acyclic for the functors H_{i}(G; –). The strategy is to use the method of discarded orbits (11.5). Let Y be the full subcategory of X_{C}^{α} determined by the objects (H, i) such that i(H) is a subgroup of K, and let Y = N(Y), so that Y is a subspace of X_{C}^{α}. The action of G on X_{C}^{α} restricts to an action of K on Y, and it is clear that Y is equivalent as a K-space to X_{C′}^{α}, where C′ = C ∩ 2^{K}. In particular, Y is by hypothesis acyclic for the functors H_{i}(K; –), i ≥ 0. Let x be a simplex of X \ Y as in 12.5, and let V be the image of H_{n} in G. Since V is not contained in K, Lemma 13.8 guarantees that the homology transfer map associated to the inclusion

K_{x} = C_{G}(V) ∩ K → G_{x} = C_{G}(V)

is zero. The desired result follows from 11.7.

14. Appendix

In this section, we sketch the proof of 10.7.

If (X, A) is a pair of G-spaces (i.e., A is a subspace of X), let C_{∗}^{G}(X, A; H) denote the quotient complex C_{∗}^{G}(X; H)/C_{∗}^{G}(A; H) and H_{∗}^{G}(X, A; H) its homology. It is clear that there is a long exact sequence relating H_{∗}^{G}(A; H), H_{∗}^{G}(X; H), and H_{∗}^{G}(X, A; H).

Let K be a normal subgroup of G. A pair (X, A) is said to be relatively free mod K if K acts trivially on the simplices of X not in A and G/K acts freely on these simplices. Let R = 𝔽_{p}[G/K]. If (X, A) is relatively free mod K, then the relative simplicial chain complex C_{∗}(X, A; 𝔽_{p}) is a chain complex of free R-modules, and there is an evident isomorphism C_{∗}^{G}(X, A; H) ≅ H(R) ⊗_{R} C_{∗}(X, A). The next lemma follows from basic homological algebra.

14.1. Lemma. Suppose that f : (X, A) → (Y, B) is a map between pairs of G-spaces which are relatively free mod K, and that f induces an isomorphism H_{∗}(X, A; 𝔽_{p}) ≅ H_{∗}(Y, B; 𝔽_{p}). Then f induces an isomorphism H_{∗}^{G}(X, A; H) ≅ H_{∗}^{G}(Y, B; H).

CLASSIFYING SPACES 53

Prove this. It follows from the fact that if R is a ring and f : C → C′ is a map of nonnegatively graded chain complexes over R such that

• both C and C′ are chain complexes of projective modules, and

• f induces an isomorphism on homology groups, then f is a chain homotopy equivalence.

We can now prove 10.7. Pick representatives {K_{i}}_{i=0}^{m} for the conjugacy classes of subgroups of G and label the representatives in such a way that if K_{i} is conjugate to a subgroup of K_{j} then i ≥ j. If Z is a G-space, write Z^{(n)} for the subspace of Z consisting of all z ∈ Z such that G_{z} is conjugate to one of the groups K_{i} for i ≤ n. Let the height of Z be the least integer n such that Z = Z^{(n)}. The proof will be by induction on the heights of the spaces X and Y involved. The result is easy to check if G acts trivially on X and Y, i.e., if the heights of these spaces are ≤ 0.

Assume by induction that the statement of 10.7 is true if the Gspaces involved have height ≤ n − 1. Suppose that X and Y are Gspaces of height ≤ n and that f : X → Y is a map which induces weak equivalences X^{H} → Y^{H} for all subgroups H of G. Let A = X^{(n−1)}, B = Y^{(n−1)}, K = K_{n} and N = N_{G}(K). We must prove that the map H_{∗}^{G}(X; H) → H_{∗}^{G}(Y; H) is an isomorphism.

It is easy to check that there is a map of pushout squares

\begin{matrix}G ×_{N} A^{K} & ⟶ & A \\ ↓ & & ↓ \\ G ×_{N} X^{K} & ⟶ & X \end{matrix}  ⟶  \begin{matrix}G ×_{N} B^{K} & ⟶ & B \\ ↓ & & ↓ \\ G ×_{N} Y^{K} & ⟶ & Y \end{matrix}

and that in these squares the vertical arrows are monic. By 10.6 the map A → B is a weak G-equivalence, so by induction and a long exact sequence argument it is enough to show that the map H_{∗}^{G}(X, A; H) → H_{∗}^{G}(Y, B; H) is an isomorphism. Given the above diagram of squares, this is equivalent to showing that the map

H_{∗}^{N}(X^{K}, A^{K}; H|_{N}) → H_{∗}^{N}(Y^{K}, B^{K}; H|_{N})

is an isomorphism. Since the maps A^{K} → B^{K} and X^{K} → Y^{K} are weak equivalences of spaces, the map H_{∗}(X^{K}, A^{K}; 𝔽_{p}) → H_{∗}(Y^{K}, B^{K}; 𝔽_{p}) is an isomorphism. The desired result now follows from 14.1, since the Nspace pairs (X^{K}, A^{K}) and (Y^{K}, B^{K}) are relatively free mod K. This last statement follows from the fact that all of the simplices which are added in going from A to X or from B to Y have isotropy group conjugate to K.

Can you use similar ideas to give a proof of 10.6?

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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA

E-mail address: dwyer.1@nd.edu