DIAGRAMS OF CLASSIFYING SPACES AND k-FOLD BOOLEAN ALGEBRAS

ERIC BABSON AND DMITRY N. KOZLOV

Abstract. In this paper we study the problem of determining the homology groups of a quotient of a topological space by an action of a group. The method is to represent the original topological space as a homotopy limit of a diagram, and then act with the group on that diagram. Once it is possible to understand what the action of the group on every space in the diagram is, and what it does to the morphisms, we can compute the homology groups of the homotopy limit of this quotient diagram.

Our motivating example is the symmetric deleted join of a simplicial complex. It can be represented as a diagram of symmetric deleted products. In the case where the simplicial complex in question is a simplex, we perform the complete computation of the homology groups with ℤ_{p} coefficients. For the infinite simplex the spaces in the quotient diagram are classifying spaces of various direct products of symmetric groups and diagram morphisms are induced by group homomorphisms. Combining Nakaoka’s description of the ℤ_{p}-homology of the symmetric group with a spectral sequence, we reduce the computation to an essentially combinatorial problem, which we then solve using the braid stratification of a sphere. Finally, we give another description of the problem in terms of posets and complete the computation for the case of a finite simplex.

1. Introduction

Assume that we have a topological space B and a group G acting on it. We wish to compute the homology groups of B/G. Sometimes it is possible to find a diagram D such that B is homotopy equivalent to holim D and G acts on D in a sufficiently simple way that we can understand the spaces and morphisms of the quotient diagram D/G. We can then compute the homology groups of holim (D/G) (for example using the Mayer-Vietoris spectral sequence) and obtain the desired answer.

Our motivating example is the n-fold deleted join of a simplicial complex C and G = S_{n} acting on B by permuting the terms. We then represent B as the homotopy limit of a diagram D, where spaces are various deleted products of C, and subsequently act with S_{n} on D.

Date: April 29, 1997 Research at MSRI was supported in part by NSF grant DMS-9022140. The first author was supported by an NSF Postdoctoral Fellowship. The second author was supported by the Swedish Science Council Postdoctoral Fellowship M-PD 11292-303.

We perform the complete computation in the case where C is a simplex. The diagram for the infinite case consists of different Eilenberg-MacLane spaces (of various direct products of symmetric groups) combined to obtain the symmetric deleted join of the infinite simplex.

In order to actually compute the homology groups, we combine the description of the homology groups, with coefficients in ℤ_{p}, of the symmetric group given by M. Nakaoka, [N60] with the standard spectral sequence converging to the homotopy limit of a diagram (cf. [Da]). This spectral sequence collapses at the second step. We compute the tableaux E_{2}^{∗,∗} using a geometric argument involving stratification of a sphere by hyperplanes from the braid arrangement.

The topological spaces that we study can also be described as classifying spaces of certain posets with a natural combinatorial description. We use this language to exhibit the sequence of symmetric deleted joins of simplices and to prove that their homology strongly converges to that of the computed example. As a byproduct, we obtain a family of small finite dimensional complexes, whose homology essentially coincides with the initial segment of the homology of the symmetric group.

Here is a brief outline of the contents of the paper.

Section 2. The general framework of the studied problem is described in terms of categories, double categories and colimits.

Section 3. We describe how these abstract results specialize to our context.

Section 4. We explain how to decompose a (symmetric) deleted join of a simplicial complex C as a diagram consisting of (symmetric) deleted products of C.

Section 5. In this long section we compute the homology groups of the symmetric deleted join of a simplex.

Section 6. We interpret our results in terms of posets (k-fold Boolean algebras) and the finite dimensional approximations.

2. The general framework

In this section we describe the general framework of our studies in terms of category theory. We refer to [Ma] as the general reference and [Bé,Er,Pa] for the information concerning double categories. Some of the lemmata are stated without proof and should be straightforward to check.

When K is a category, we denote by O(K) the set of its objects and by M(K) the set of its morphisms. Sometimes O will be considered as the set of identities in M. If a, b ∈ O(K), we denote by M_{K}(a, b) the set of all morphisms from a to b. If m ∈ M_{K}(a, b), we write ∂^{•}m = a and ∂_{•}m = b. When K_{1} and K_{2} are two categories we denote by F(K_{1}, K_{2}) the set of all functors between K_{1} and K_{2}. Finally, we let T (F_{1}, F_{2}) denote the set of all natural transformations between two functors F_{1}, F_{2} ∈ F(K_{1}, K_{2}).

In the following, P denotes the category of all posets, Top - the category of all topological spaces, Cat - the category of all categories and 1 the terminal category with one object and only the identity morphism. Δ denotes the realization functor Δ : Cat → Top mapping a category to its realization space, see [Q73, Q78, Se].

Definition 2.1. Let S and R be categories such that objects of R have a name Name. A functor F : S → R is called a diagram of Name over S.

For example, when S is a poset and R = Top we talk about a diagram of topological spaces over a poset. If D = (F : S ⟶ R) is a diagram we write F = F^{D}, S = S^{D} and R = R^{D}. When it does not lead to confusion, we simply use D instead of F^{D}.

Definition 2.2. Define the functor 𝔻 : Cat → Cat. If K ∈ O(Cat), 𝔻K is the category of all diagrams of objects of K. More specifically, 𝔻K is defined by:

• O(𝔻K) = {D |R^{D} = K};

• M_{𝔻K}(A, B) = {(f, τ) | f ∈ F(S^{A}, S^{B}) and τ ∈ T (F^{A}, F^{B} ∘ f)}. Furthermore, for any functor F : K → L, 𝔻F is a functor from 𝔻K to 𝔻L defined by 𝔻F (A) = F ∘ A and 𝔻F ((f, τ)) = (f, F ∘ τ).

When T = (f, τ) ∈ M_{𝔻K}(A, B), write f = f^{T}, τ = τ^{T}.

Define the functor Σ ∈ F(𝔻K, Cat) by Σ(X) = S^{X}.

Definition 2.3. Let X ∈ O(𝔻Cat), X = (F : S → Cat). A sink of X is L ∈ O(Cat) together with a collection of morphisms {λ_{s}^{L} ∈ M_{Cat}(F (s), L)}_{s∈O(S)}, such that if α ∈ M_{S}(s_{1}, s_{2}) then λ_{s_{2}}^{L} ∘ α = λ_{s_{1}}^{L}. When L is universal with respect to this property we call it the colimit of X and write L = lim X.

Definition 2.4. Let K be a category and G a finite group. We say that G acts on K if there is a diagram X = (F : G → Cat), such that the object of G maps to K. For brevity we shall identify F (g) with g, for g ∈ G. The quotient category K/G is the colimit of X (it always exists since Cat is cocomplete).

Let X ∈ O(𝔻Cat). We define an equivalence relation ∼ on the set of morphisms ∐_{σ∈O(S^{X})} M(F^{X}(σ)) by saying that n ∼ n′ if there exists m ∈ M(S^{X}), such that F^{X}(m)(n) = n′. Square brackets [ ] or [ ]_{X} will denote the ∼-equivalence classes.

The following condition will be crucial for our applications.

Condition A. If X ∈ O(𝔻Cat), ∂^{•}b = ∂^{•}b′ and b′ ∈ [b] then b′ = b.

Lemma 2.5. Let X ∈ O(𝔻Cat) and assume that X satisfies condition A. Then lim X can be explicitly described in terms of equivalence classes: M(lim X) = {[f] | f ∈ ∐_{σ∈O(S^{X})} M(F^{X}(σ))}. Clearly [f] ∈ M([∂^{•}f], [∂_{•}f]).

When A is a category and a ∈ O(A) we denote by (a ↓ A) the category of all objects under a, see [Ma]. (a ↓ A) is defined by:

• O(a ↓ A) = {f | f ∈ M(A) and ∂^{•}f = a};

• M_{(a↓A)}(f_{1}, f_{2}) = {h | h ∈ M_{A}(∂_{•}f_{1}, ∂_{•}f_{2}) and h ∘ f_{1} = f_{2}}.

We define a functor s̃d ∈ F(Cat, Cat) by: s̃d(S) = S′, where O(S′) = M(S) and M_{S′}(f, g) = {(h, k) | h, k ∈ M(S), k ∘ g ∘ h = f}. We call s̃d(S) the morphism category of S.

Definition 2.6. Let us define the subdivision functor sd : 𝔻K → 𝔻(Cat × K).

(1) Take A ∈ O(𝔻K). Set S^{sdA} = s̃d(S^{A}). For objects set F^{sdA}(f) = (∂_{•}f ↓ S^{A}) × F^{A}(∂^{•}f), and for morphisms F^{sdA}((h, k)) = (h∘) × F (k), where h∘ is the contravariant functor acting by composition with h.

(2) If T ∈ M_{𝔻K}(A, B) then f^{sdT} ((h, k)) = (f^{T} (h), f^{T} (k)) and τ_{m}^{sdT} = f̃ × τ_{∂^{•}m}^{T} where f̃ is a functor from (b ↓ S) to (f(b) ↓ R) induced by f.

Lemma 2.7. Assume that X ∈ O(𝔻Cat) and X satisfies condition A, then 𝔻s̃d(X) satisfies condition A.

Definition 2.8. Let K be a category with products and colimits and N ∈ F(Cat, K) a functor. N lim is the functor from 𝔻K to K defined by the following composition:

𝔻K ⟶^{sd} 𝔻(Cat × K) ⟶^{𝔻(N ×id)} 𝔻K ⟶^{lim} K.

Note that if N is the trivial functor to the terminal object in K then N lim = lim.

Definition 2.9. If N = id, resp. N = Δ (then K = Top), we call N lim the poset limit, resp. homotopy limit, and use the notation N lim = plim , resp. holim .

When D is a diagram of posets over a poset, the first part of the Definition 2.9 specializes to the one given in [WZŽ]. Namely, the set of the elements of plim D is {(p, e) | p ∈ S^{D}, e ∈ F^{D}(p)} and the partial ordering is defined by: (p, e) ≥ (p′, e′) iff p ≥ p′ and f_{pp′}(e) ≥ e′ where f_{pp′} = F^{D}(p → p′).

Proposition 2.10. Assume that we are given two categories K and K′ and functors N ∈ F(Cat, K), N′ ∈ F(Cat, K′) and W ∈ F(K, K′), such that W ∘ N = N′ and W preserves colimits and products. Then the following diagram commutes

𝔻K ⟶^{sd} 𝔻(Cat × K) ⟶^{𝔻(N ×id)} 𝔻K  ⟶^{lim}  K
𝔻W ↓  (I)  ↓ 𝔻(id × W) (II)  ↓ 𝔻W (III)  ↓ W
𝔻K′  ⟶^{sd} 𝔻(Cat × K′) ⟶^{𝔻(N′×id)} 𝔻K′  ⟶^{lim}  K′

Proof. The commutativity of the first square follows from the definition of sd. Square (II) commutes, because W preserves products. Square (III) commutes, since W preserves colimits.

Corollary 2.11. For every diagram D = (F : S → Cat) we have

holim (𝔻Δ(D)) ≃ Δ(plim D).

Proof. Since Δ has a weak homotopy inverse, see [FL], it preserves colimits up to homotopy.

Note. If D satisfies condition A, the last square commutes exactly, so we get equality: holim (𝔻Δ(D)) = Δ(plim D).

Note. In the special case when S is a poset and K = P see Simplicial Model Lemma, [WZŽ, Proposition 3.19].

For an arbitrary category K, the barycentric subdivision of K is the face poset of Δ(K). We denote it Bd K. Bd K consists of chains of nonidentity morphisms in K ordered by composition. This notion can be generalized to diagrams.

Definition 2.12. Given a diagram D ∈ O𝔻K, the barycentric subdivision of D is a diagram, which we will denote D^{Bd}. Set S^{D^{Bd}} = Bd S^{D}. For α = (x_{1} ⟵^{m_{1}} …

⟵^{m_{t−1}} x_{t}) ∈ O(Bd S^{D}) with x_{i} ∈ O(S^{D}) and m_{i} ∈ M_{S^{D}}(x_{i+1}, x_{i}), we define F^{D^{Bd}(α)} =

F^{D}(x_{t}). For all pairs α, β ∈ O(Bd S), where α is as above and β < α, i.e. β = (x_{i_{1}} ← … ← x_{i_{v}}) for some 1 ≤ i_{1} < ⋯ < i_{v} ≤ t, the set M_{S^{D^{Bd}}}(α, β) has exactly one element t and F^{D^{Bd}}(t) is id if x_{t} = x_{i_{v}} and m_{i_{v}} ∘ m_{i_{v+1}} ∘ ⋯ ∘ m_{t−1} otherwise.

Definition 2.13. Let K be some given category. 𝔻^{2}K is a category defined by: • O(𝔻^{2}K) = {(C, f) | C is a double category, f ∈ F(C, K^{2})};

• M_{𝔻^{2}K}((C_{1}, f_{1}), (C_{2}, f_{2})) = {(f, τ) | f ∈ F(C_{1}, C_{2}), τ ∈ T (f_{1}, f_{2} ∘ f)}. Here K^{2} is the double category whose morphisms are the commuting squares in K.

Although this provides a general definition of 𝔻^{2}K, for our purposes we need to assume that conditions 1 - 3 below are satisfied. The morphisms of C can be composed in two different ways, which we refer to as ⋄ and ∗ operations.

Before we formulate the conditions we will need some additional notations. C^{∗}, resp. C^{⋄}, denotes the category which is obtained from the double category C by taking the same set of morphisms, but considering only ⋄, resp. ∗, operations. [C]^{∗}, resp. [C]^{⋄}, is the set of all ∗-, resp. ⋄-connected components. If a ∈ M(C), [a]^{∗}, resp. [a]^{⋄}, is the ∗-, resp. ⋄-connected component containing a. For a ∈ M(C), ∂^{∗}(a), ∂_{∗}(a), ∂^{⋄}(a), ∂_{⋄}(a) will denote upper and lower ∗-identities, resp. left and right ⋄- identities. C_{1}^{∗} denotes the subcategory of C^{∗} consisting of ⋄-identities. If f is a double category functor f : C → K^{2}, f^{∗}, f_{∗}, f^{⋄} and f_{⋄} denote the restrictions of f to the left, right, upper and lower morphisms. Clearly, f^{∗}, f_{∗} ∈ F(C^{∗}, K), f^{⋄}, f_{⋄} ∈ F(C^{⋄}, K).

Condition 1. If a, b ∈ M(C), a is a ⋄-identity and a ∗ b exist, then a ∗ b is a ⋄-identity again. The same is true if we swap ∗ and ⋄.

Condition 2. If a, b, c ∈ MC and a ∗ c, a ⋄ b exist, then there exists d ∈ MC, such that c ⋄ d and b ∗ d exist. The same is true if we swap ∗ and ⋄ or the order of the first operand. The uniqueness of d follows from

Condition 3. If b′ ∈ [b]^{⋄} ⊆ MC and ∂^{•}b = ∂^{•}b′ then b = b′ and the same is true if we swap ∗ and ⋄.

We shall now define a functor φ_{⋄} : 𝔻^{2}K → 𝔻𝔻K.

(1) Let (C, f) ∈ O(𝔻^{2}K). Define a new category S by taking M(S) = [C]^{⋄}. For A, B ∈ M(S), we call C the composition of A and B if there exist a ∈ A, b ∈ B and c ∈ C, such that a ∗ b = c. In this case, we simply write C = A ∗ B.

Let us check that S is well defined as a category. Assume that a′ ∈ A, b′ ∈ B and c′ = a′ ∗ b′. We need to prove that c′ ∈ C. Take a_{1} ∈ A, such that a ⋄ a_{1} and a_{1} ⋄ a′ exist. According to the condition 2 there exist b_{1}, b′_{1} ∈ B, such that b ⋄ b_{1}, b_{1} ⋄ b′_{1}, a ∗ b_{1}, and a′ ∗ b′_{1} exist. Since b′_{1}, b′ ∈ [b]^{⋄} we have b′_{1} = b′ by condition 3. Take c_{1} = a_{1} ∗ b_{1}, then c ⋄ c_{1} and c_{1} ⋄ c′ exist and hence c′ ∈ C.

In particular, objects of S are just ⋄-connected components formed by ∗-identities (according to condition 1, a ⋄-connected component which has a single ∗-identity, must entirely consist of ∗-identities).

Observe that, according to conditions 2 and 3, whenever [a]^{⋄} ∗ [b]^{⋄} exists, there exists a unique c ∈ [b]^{⋄}, such that a ∗ c exists. We denote this c by γ_{[b]^{⋄}}(d). Clearly γ_{[c^{⋄}]} : [∂^{∗}c]^{⋄} → [c]^{⋄}.

Let us now define F ∈ F(S, 𝔻K). Each A ∈ O(S) is a ⋄-subcategory of C^{⋄}. Define F (A) = f^{⋄}|_{A}.

Furthermore, take B ∈ M_{S}(A, A′), (which means that A∗B and B ∗A′ exist). Let us define F (B) = (B̃, τ), where B̃ ∈ F(A, A′) (considering A and A′ as subcategories of C^{⋄}) and τ ∈ T (F (A), F (A′) ∘ B̃). Let a ∈ A, we define B̃(a) = γ_{A′}(γ_{B}(a)). By condition 2, B̃ is a functor from A to A′. For a ∈ A and b ∈ B, such that a ∗ b exists, we define τ_{a} = f^{∗}(b). Clearly τ ∈ T (F (A), F (A′) ∘ B̃).

(2) Let (f, τ) ∈ M_{𝔻^{2}K}((C_{1}, f_{1}), (C_{2}, f_{2})). Define φ_{⋄}(f, τ) to be the morphism between φ_{⋄}(C_{1}, f_{1}) and φ_{⋄}(C_{2}, f_{2}) as follows. Let φ_{⋄}(C_{1}, f_{1}) = (F_{1} : S_{1} → 𝔻K), φ_{⋄}(C_{2}, f_{2}) = (F_{2} : S_{2} → 𝔻K). f induces a map from S_{1} to S_{2} defined by f̃: [a]^{⋄} → [fa]^{⋄}. This map is well defined and is a functor, since f ∈ F(C_{1}, C_{2}). Furthermore, define τ̃ ∈ T (F_{1}, F_{2} ∘ f̃) by letting τ̃_{[a]^{⋄}} be a restriction of (f̄, τ̄) to the connected component [a]^{⋄}, where f̄ and τ̄ are f and τ considered as functor, resp. natural transformation of corresponding ∗-categories.

The functor ϕ_{∗} is defined analogously.

Lemma 2.14. If X ∈ O(𝔻^{2}K), then 𝔻Σ(ϕ_{⋄}X) satisfies condition A.

Lemma 2.15. If L ∈ F(K, K′) and X ∈ O(𝔻K), then lim 𝔻L(X) = L(lim X) iff there is some sink k for X satisfying L(k) = lim 𝔻L(X).

Definition 2.16. Take X ∈ O(𝔻𝔻K), Y a sink for X and y ∈ O(S^{Y}). X_{y} ∈ O(𝔻K) is defined by:

• O(S^{X_{y}}) = ∐_{σ∈O(S^{X})}{u ∈ O(F^{X}(σ)) | λ_{σ}^{Y} (u) = y};

• M_{S^{Xy}} (u, u′) = {m ∈ M_{S^{X}} | F^{X}(m)(u) = u′}.

• F^{X_{y}}(u) = F^{F^{X}(σ)}(u).

Lemma 2.17. If X ∈ O(𝔻𝔻K), 𝔻Σ(X) satisfies condition A, and Y ∈ O(𝔻K) is a sink for X, then Y = lim X iff (α) Σ(Y) = lim(𝔻Σ(X)); (β) lim X_{y} = F^{Y} (y), for all y ∈ O(S^{Y}).

Lemma 2.18. If X ∈ O(𝔻𝔻K), 𝔻Σ(X) satisfies condition A, σ ∈ O(S^{X}), and m ∈ M(F^{X}(σ)), then (𝔻sdX)_{[m]} = (∂_{•}m ↓ S^{F^{X}(σ)}) × X_{[∂^{•}m]}.

Theorem 2.19. Let K be a cocomplete category and N ∈ F(Cat, K). Assume that if X ∈ O(𝔻K) and a ∈ O(K), then a × lim X = lim(a × X), where a × X ∈ O(𝔻K) is defined by F^{a×X}(m) = id_{a} × F^{X}(m).

Then squares (I), (III) and (I+II) of the following diagram commute.

𝔻^{2}K ⟶^{φ_{⋄}}𝔻𝔻K ⟶^{𝔻 sd}𝔻𝔻(Cat × K) ⟶^{𝔻 𝔻(N ×id)}𝔻𝔻(K × K)⟶^{𝔻μ}𝔻𝔻K ⟶^{𝔻 lim} 𝔻K
φ_{∗}↓ (I) ↓ lim  (II)  lim ↓ (III) ↓ lim
𝔻𝔻K ⟶^{𝔻 lim} 𝔻K ⟶^{sd} 𝔻(Cat × K) ⟶^{𝔻 (N ×id)} 𝔻(K × K) ⟶^{μ} 𝔻K ⟶^{lim} K

Proof. Let us first check that the square (I) commutes. Let (C, f) ∈ O(𝔻^{2}K). This follows from Lemma 2.17 since 𝔻 lim ϕ_{∗}(C, f) is a sink for ϕ_{⋄}(C, f). Condition α follows from Lemma 2.5 giving S^{ϕ_{∗}(C,f)} = S^{lim ϕ_{⋄}(C,f)} = [C]^{∗}. Condition β follows from the observation that (ϕ_{⋄}(C, f))_{[a]^{∗}} = F^{ϕ_{∗}(C,f)}([a]^{∗}).

Let us now show that square (I+II) commutes. Let H = μ ∘ (F × id) ∘ sd. Pick X ∈Im ϕ_{⋄}, X ∈ O(𝔻𝔻K). According to Lemma 2.14, 𝔻Σ(X) satisfies condition A. According to Lemma 2.7, 𝔻Σ(𝔻H(X)) satisfies condition A. According to Lemma 2.5, lim X and lim 𝔻H(X) have explicit descriptions in terms of the equivalence classes.

We want to show that lim 𝔻H(X) = H(lim X). For that we need to check conditions (α) and (β) of Lemma 2.17.

Check of (α). Follows from the explicit description of the colimit. Namely, the objects of Σ(H(lim X)) are equivalence classes [m] of morphisms m ∈ M(Σ(F^{X}(σ))), for σ ∈ O(S^{X}).

Check of (β). (H lim X)([m]) = N (∂_{•}m ↓ S^{F^{X}(σ)}) × F^{lim X}([∂^{•}m]) = N (∂_{•}m ↓ S^{F^{X}(σ)})×lim X_{[∂^{•}m]}, where the first equality follows from the definition of H and the second from Lemma 2.17.

(lim 𝔻H(X))([m]) = lim(𝔻H(X)_{[m]}) = lim(N (∂_{•}m ↓ S^{F^{X}(σ)} × X_{[∂^{•}m]}) = N (∂_{•}m ↓ S^{F^{X}(σ)}) × lim X_{[∂^{•}m]}, where the first equality follows from Lemma 2.17, second from Lemma 2.18 and third from Lemma 2.7.

Square (III) commutes in general, see [Ma, IX 8], but can also be checked using Lemma 2.17.

3. Diagrams of posets and the group action

A partially ordered set P can be viewed as a category. The objects of that category are just the elements of the poset and the morphism f : {x} ⟶ {y} between the objects x, y ∈ P exists and is unique iff x > y. Categories arising in this way are characterized by the property |M_{P} (x, y)| + |M_{P} (y, x)| ≤ 1 for all x, y ∈ O(P).

A poset map is a functor between two posets viewed as categories, or equivalently an order preserving map.

The diagrams which will turn out to be important in this paper, are diagrams of posets. Rephrasing Definition 2.1 a diagram of posets D over a posset P is a collection of posets {D(x) | x ∈ P } and poset maps {f_{xy} : D(x) ⟶ D(y) | x, y ∈ P, x ≥ y} such that f_{yz} ∘ f_{xy} = f_{xz}. In the previous notation S^{D} = P, R^{D} = P, F^{D}(x) = D(x) and F^{D}(x → y) = f_{xy} (i.e. in this case the category K is a category of all posets with poset maps serving as morphisms).

Note. In this situation the elements of plim D are pairs (x, e) with x ∈ P and e ∈ D(x).

Proposition 3.1. Δ(plim D^{Bd}) ≃ Δ(plim D).

Proof. Let A = plim D^{Bd}, B = plim D and P = S^{D}. We shall show that posets A and B are homotopy equivalent. Let us describe a map ϕ : A ⟶ B. Pick α = x_{1} < ⋯ < x_{t} ∈ P^{Bd} and e ∈ D^{Bd}(α). By Definition 2.12 D^{Bd}(α) is a copy of D(x_{t}) so we can define ϕ((α, e)) = (x_{t}, e).

To verify that ϕ is actually a poset map take (α, e), (α′, e′) ∈ A, such that (α, e) ≥ (α′, e′). Assume α = x_{1} < ⋯ < x_{t} and e ∈ D(x_{t}) = D^{Bd}(α). Since α′ ≤ α we can write α′ = x_{i_{1}} < ⋯ < x_{i_{m}}, where i_{1} < ⋯ < i_{m} ≤ t, clearly f_{x_{t}x_{im}}(e) ≥ e′. We have ϕ(α, e) = (x_{t}, e) and ϕ(α′, e′) = (x_{i_{m}}, e′), hence from what is said above it follows that ϕ(α, e) ≥ ϕ(α′, e′).

Let us now show that ϕ satisfies Quillen’s conditions, [Q78], and hence induces a homotopy equivalence. Let x ∈ P, e ∈ D(x) and consider the poset ϕ^{−1}(B_{≤(x,e)}). Let α ∈ P^{Bd} be a chain in P consisting of a single element x. Clearly e ∈ D^{Bd}(α) and ϕ((α, e)) = (x, e). We claim that there always exists a join of (α, e) with any other element of ϕ^{−1}(B_{≤(x,e)}). In this case ϕ^{−1}(B_{≤(x,e)}) is join-contractible, which implies that it is contractible.

If (α′, e′) ∈ ϕ^{−1}(B_{≤(x,e)}) with α′ = x_{1} < ⋯ < x_{t} then x_{t} ≤ x. Further (α, e) ∨ (α′, e′) = (β, e), where β = (x_{1} < ⋯ < x_{t} < x) if x_{t} ≠ x and β = α′ otherwise. Indeed, (β, e) ≥ (α, e) and (β, e) ≥ (α′, e′), since β ≥ α, β ≥ α′ and f_{xx_{t}}(e) ≥ e′. Moreover assume that for some (β̃, ẽ) ∈ ϕ^{−1}(B_{≤(x,e)}) we have that (β̃, ẽ) ≥ (α, e) and (β̃, ẽ) ≥ (α′, e′). Then β̃ must include elements x_{1}, …, x_{t} as well as x, hence β̃ ≥ β. Let y be the maximal element from the chain β, then y ≥ x and f_{yx}(ẽ) ≥ e (since (β̃, ẽ) ≥ (α, e)), so we can conclude that (β̃, ẽ) ≥ (β, e).

In what follows we assume that P/G is a poset and that the conditions of Lemma 2.5 hold.

There is a natural poset map p : P ⟶ P/G which maps x ∈ P to an orbit a ∈ P/G such that x ∈ a. Obviously every chain in P is mapped to some chain in P/G. As the following lemma shows, the converse is true as well.

Lemma 3.2. For every chain a = (a_{1} < ⋯ < a_{t}) in P/G there exists at least one chain x = (x_{1} < ⋯ < x_{t}) in P such that p(x) = a.

Proof. We induct on t. If t = 1 then there is nothing to prove. If t = 2 then the statement is a direct consequence of Lemma 2.5.

Assume t ≥ 3 and choose a chain a = (a_{1}, …, a_{t}) with a_{i} ∈ P/G. By induction there is a chain x_{1} < ⋯ < x_{t−1} with x_{i} ∈ P such that p(x_{i}) = a_{i}. According to Lemma 2.5, we can choose y < z ∈ P such that p(y) = a_{t−1}, p(z) = a_{t}. Since p(y) = p(x_{t−1}) there must exist g ∈ G such that g(y) = x_{t−1}. Put x_{t} = g(z). Since the action of G is order-preserving, we have y < z ⇒ g(y) < g(z) ⇒ x_{t−1} < x_{t}. Hence, we have found a chain x_{1} < ⋯ < x_{t} with the required properties.

The action of G on the poset P naturally induces an action of G on the order simplicial complex Δ(P) as a topological space, namely the action on the vertices is given and we extend it over every simplex linearly. We denote the quotient space by Δ(P)/G. The map p described above induces a map p^{∗} : Δ(P) ⟶ Δ(P)/G.

In general Δ(P)/G will not be a simplicial complex but some simplicial poset, see [St], with every face being a simplex spanned by its vertices, but some sets of vertices spanning more than one simplex. The following lemma shows that under some conditions we obtain a simplicial complex.

Proposition 3.3. If G acts on P in the sense of the Definition 2.4 then the action of G on P induces an action on the barycentric subdivision Bd P and this new action satisfies the separability condition A.

Proof. Let Q = Bd P. Choose two chains x = (x_{1}, …, x_{t}) and y = (y_{1}, …, y_{t}) in Q such that p(x) = p(y). Thus x_{t} = (χ_{1}, …, χ_{m}) and y_{t} = (γ_{1}, …, γ_{m}) are chains in P with p(x_{t}) = p(y_{t}). Choose g ∈ G such that g(x_{t}) = y_{t}. Since x_{i} is a subchain of (χ_{1}, …, χ_{m}), y_{i} is a subchain of (γ_{1}, …, γ_{m}), and the action of g preserves order, we have g(x) = y.

4. How to decompose a symmetric deleted join as a diagram

Let P be a poset with ˆ0. We call two elements x, y ∈ P disjoint if P_{≤x}∩P_{≤y} = {ˆ0}, in other words from z ≤ x, z ≤ y it follows that z = ˆ0. Denote P̂ = P \ {ˆ0}.

Definition 4.1. Let A be a strictly increasing t-tuple of natural numbers, i.e. A = (a_{1}, …, a_{t}), where a_{1} < ⋯ < a_{t}, a_{i} ∈ ℕ. The deleted join of P is a poset P^{[A]} consisting of all t-tuples (x_{1}, …, x_{t}) with x_{i} ∈ P, such that x_{i} and x_{j} are disjoint for 1 ≤ i < j ≤ t. The order relations are given by (y_{1}, …, y_{t}) < (x_{1}, …, x_{t}) iff y_{i} < x_{i} for all i ∈ [t]. The terms of the deleted join are in natural bijection with elements of A. When A = {1, …, n} we simply write P^{[n]} instead of P^{[A]}.

When the poset P is the face lattice of some simplicial complex, our definition coincides with the one for the deleted join of a simplicial complex given by K. Sarkaria in [Sa]. In the example we compute in the subsequent sections, P will be a (possibly infinite) Boolean algebra, i.e. a face lattice of a simplex.

Definition 4.2. The deleted product of P is a poset P̂^{[A]}, which is obtained from P^{[A]} by imposing the additional condition that in every t-tuple (x_{1}, …, x_{t}) we have x_{i} ≠ ˆ0, for i ∈ [t]. Again P̂^{[{1,...,n}]} = P̂^{[n]}.

Let B̂_{n} = B_{n} \ {ˆ0}. For a poset P with ˆ0 the deleted join P^{[n]} can be decomposed as a diagram D_{n} of posets over B̂_{n} with posets assigned to different elements of B_{n} in the following way: to A ⊆ [n] we assign D_{n}(A) = P̂^{[A]}, to every element of A there is exactly one corresponding term in the product. If B ⊂ A, the map p_{AB} : D_{n}(A) ⟶ D_{n}(B) is a projection of P̂^{[A]} onto P̂^{[B]} which is given by forgetting the terms which corresponds to elements in A \ B.

Let us show that the diagram D_{n} is really a decomposition of P^{[n]}.

Proposition 4.3. For any poset P and positive integer n, plim D_{n} = P^{[n]}.

Proof. Recall that the elements of plim D_{n} have the form (A, e), where A = {a_{1}, …, a_{t}} ⊆ [n] and e = (x_{a_{1}}, …, x_{a_{t}}) ∈ D_{n}(A) = P̂^{[A]}. We define a map on the elements of the posets ϕ : plim D_{n} → P^{[n]} by ϕ(A, (x_{a_{1}}, …, x_{a_{t}})) = (y_{1}, …, y_{n}), where y_{a_{i}} = x_{a_{i}} for i ∈ [t] and y_{j} = ˆ0 otherwise. It is obvious that ϕ is a bijection on the sets of elements of plim D_{n} and P^{[n]}. Furthermore, ϕ is clearly order-preserving and hence a poset isomorphism.

Let D_{n}^{Bd}, be the barycentric subdivision of D_{n}. By Definition 2.12 the underlying poset of D_{n}^{Bd} is Bd (B̂_{n}) and for every x = (S_{1} ⊂ S_{2} ⊂ ⋯ ⊂ S_{t}) ∈ Bd (B̂_{n}) we have D_{n}^{Bd}(x) = P̂^{[S_{t}]}. If y = (S_{i_{1}} ⊂ S_{i_{2}} ⊂ ⋯ ⊂ S_{i_{m}}) < x for some 1 ≤ i_{1} < i_{2} < ⋯ < i_{m} ≤ t, according to the same definition, the map f_{xy} : D_{n}^{Bd}(x) → D_{n}^{Bd}(y) is defined by

f_{xy} = \{ \begin{matrix}id, & if S_{i_{m}} = S_{t}, \\ projection map P̂^{[S_{t}]} ↪ P̂^{[S_{im}]}, & if S_{i_{1}} ⊃ S_{1}. \end{matrix}

Now let us define an action of the symmetric group S_{n} on the diagram D_{n}^{Bd}. Fix π ∈ S_{n}. As definitions 2.2 and 2.4 suggest, we shall describe a corresponding pair (m, τ) ∈ M_{𝔻K}(D_{n}^{Bd}, D_{n}^{Bd}). m ∈ F(Bd (B̂_{n}), Bd (B̂_{n})) is defined by

m : (S_{1} ⊂ S_{2} ⊂ ⋯ ⊂ S_{t}) ⟶ (π(S_{1}) ⊂ π(S_{2}) ⊂ ⋯ ⊂ π(S_{t})).

Further, we have to define τ_{x} : D_{n}^{Bd}(x) → D_{n}^{Bd}(m(x)). Unwinding definitions gives us D_{n}^{Bd}(x) = P̂^{[S_{t}]} and D_{n}^{Bd}(m(x)) = P̂^{[π(S_{t})]}. The map π : S_{t} → π(S_{t}) induces poset isomorphism τ_{x} : P̂^{[S_{t}]} → P̂^{[π(S_{t})]}, which is exactly the map that we are looking for.

Now, as we have an action of S_{n} on D_{n}^{Bd} we can consider a new diagram D̃_{n} = D_{n}^{Bd}/S_{n}. The underlying poset of D̃_{n} is (by coincidence) again B̂_{n}, namely it is Bd(B̂_{n})/S_{n}. For x = {a_{1}, …, a_{t}} ∈ B̂_{n} with a_{1} < ⋯ < a_{t} square (I) of 2.19 asserts in our case that

D̃_{n}(x) = P̂^{[a_{t}]}/S_{a_{1}} × S_{a_{2}−a_{1}} × ⋯ × S_{a_{t}−a_{t−1}},

where S_{a_{1}} acts on the first a_{1} terms of P̂^{[a_{t}]}, S_{a_{2}−a_{1}} acts on the next a_{2} − a_{1} terms and so on.

5. An application: symmetric deleted join of a simplex

Let us now specify the choice of the partially ordered set P. Take P = B_{∞}. Following the terminology of Section 6, P^{[n]} = B_{∞}^{[n]} = B_{∞,n}^{o}. According to Theorem 6.2(a), Δ(B_{∞,n}^{o}) is an infinite contractible simplicial complex, hence the spaces of the diagram Δ(D̃_{n}) are given by

Δ(D̃_{n}(X)) = Δ(P^{[a_{m}]}/S_{a_{1}} × S_{a_{2}−a_{1}} × ⋯ × S_{a_{m}−a_{m−1}}) =

= Δ(P^{[a_{m}]})/S_{a_{1}} × S_{a_{2}−a_{1}} × ⋯ × S_{a_{m}−a_{m−1}} ≃ K(S_{a_{1}} × ⋯ × S_{a_{m}−a_{m−1}}, 1) ≃

≃ K(S_{a_{1}}, 1) × ⋯ × K(S_{a_{m}−a_{m−1}}, 1), (5.1)

where X = {a_{1}, …, a_{m}}.

Let us understand the structure of the morphisms of the diagram Δ(D̃_{n}). For x ≥ y ∈ B̂_{n}, denote X = D^{Bd}_{n}(x), Y = D^{Bd}_{n}(y). Both X and Y are contractible and the diagram morphism f : X → Y is G-equivariant and therefore induces a map f/G : X/G → Y/G. We have two fibre bundles H → X → X/G and K → Y → Y/G, where H and K are subgroups of G, such that either H ⊆ K or K ⊆ H. Since the map f is G-equivariant, it induces a map between the two long exact sequences of homotopy groups associated to the fibre bundles. It follows from the naturality of these long exact sequences that the map f/G is induced by the homomorphism ι : H → K, where ι is inclusion if H ⊆ K and restriction if K ⊆ H.

Thus the diagram D̃_{n} consists of classifying spaces for various direct products of symmetric groups glued together by the maps that are induced by the homomorphisms between these direct products.

In the remainder of this section we compute H_{∗}(Δ(plim D̃_{n}), ℤ_{p}). All homology groups will have coefficients in ℤ_{p}, unless the contrary is explicitly stated. According to the Proposition 2.11, H_{∗}(Δ(plim D̃_{n})) = H_{∗}(holim (Δ(D̃_{n}))). To calculate the homology groups of holim (Δ(D̃_{n})), we set up a spectral sequence (E_{r}^{∗,∗})_{r=1}^{∞}, using the following filtration:

F_{0} = ∅, F_{m} = holim D̃_{n}|_{B̂_{n}(m)},

where B̂_{n}(m) is the m-skeleton of B̂_{n} a poset which consists of all nonempty subsets of [n] of cardinality at most m. Clearly ∅ = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = holim D̃_{n}.

To understand the differentials in this spectral sequence, let us describe a cell structure on holim D̃_{n}. Let X ⊂ [n]. Since every D̃_{n}(X) is a classifying space of a direct product of some symmetric groups, it can be represented by a simplicial complex (the particular structure of that complex is not important in our context). The building blocks of holim D̃_{n} are spaces of the form Δ((B̂_{n})_{≤X}) × D̃_{n}(X). The space T_{X} = Δ((B̂_{n})_{≤X}) can be viewed as a simplex, i.e. a simplicial complex having exactly one largest cell T. The cells of T_{X} × D̃_{n}(X) are direct products of the cells from the two simplicial complexes. Let c be a cell of D̃_{n}(X) and t be a cell of T_{X}, then

∂ (t × c) = ∂ t × c + t × ∂ c.  (5.2)

Let us describe the entries in E_{1}^{∗,∗}. First of all we have

H_{∗}(F_{m}, F_{m−1}) = ⨁_{X⊆[n],|X|=m} H_{∗}(T_{X} × D̃_{n}(X), δT_{X} × D̃_{n}(X)),  (5.3)

where δT_{X} is the boundary of the simplex T_{X}, i.e. δT_{X} = T_{X} \T. From (5.2) and (5.3) we see that relative to F_{m−1} we have ∂ (t × c) = 0 unless t is the unique largest cell T of T_{X}, in which case ∂ (T × c) = T × ∂ c. This means that the boundary operator in C_{∗}(T_{X} ×D̃_{n}(X), δT_{X} ×D̃_{n}(X)) is identical to the one in C_{∗}(D̃_{n}(X)) up to the shift by the dimension of the simplex T_{X}, that is Z_{∗}(T_{X} ×D̃_{n}(X), δT_{X}) = {T ×c | c ∈ Z_{∗}(D̃_{n})} and the same for B_{∗}. Hence

E_{1}^{m,d−m} = H_{d}(F_{m}, F_{m−1}) = ⨁_{X⊆[n],|X|=m} H_{d+m−1}(D̃_{n}(X)).  (5.4)

Furthermore, in holim D̃_{n}, whenever t is not the largest cell of T_{X}, say t corresponds to some Y ⊂ X, the cell t × c gets identified with t × f_{X,Y} (c). In order to describe d_{1} - the first differential of our spectral sequence we introduce an auxiliary diagram R_{X,c} for any X ⊆ [n] and c ∈ Z_{∗}(D̃_{n}(X)). The underlying poset of R_{X,c} is (B̂_{n})_{<X}. For Y ⊂ X, we have R_{X,c}(Y) = f_{X,Y} (c). For Y_{2} ⊂ Y_{1} ⊂ X, the diagram morphism between R_{X,c}(Y_{1}) and R_{X,c}(Y_{2}) is induced by f_{Y_{1},Y_{2}}. We define f(c) = holim R_{X,c} and f̃(c) = holim (R_{X,c}|_{P_{x}}), where P_{x} = (B̂_{n})_{<X} \ {coatoms}.

Let X = {a_{1}, … a_{m}} and c ∈ Z_{∗}(D̃_{n}(X)), d_{1} is given by

d_{1}(T × c) = (δT_{X} × f(c), δ̃T_{X} × f̃(c)) = ∑_{i=m}^{t}(−1)^{i} T_{i} × f_{i}(c),  (5.5)

where f_{i} = f_{X,X\{a_{i}}}, T_{i} is a face corresponding to X \ {a_{i}}, and δ̃T_{X} = δT_{X} \ {T_{1}, …, T_{m}}.

We can immediately show that the spectral sequence collapses after the second step. Namely, take such c ∈ Z_{∗}(D̃_{n}(X)) that d_{1}(c) = 0, this means that f_{i}(c) = 0 for i ∈ [m]. Let Y ⊂ X and pick i ∈ X \ Y, then f_{X,Y} = f_{X\{i},Y} ∘ f_{i} = 0. This means that R_{X,c}(Y) = 0 for any Y ⊂ X and hence f(c) = holim R_{X,c} = 0, therefore ∂ (c) = T × ∂ c + ∂ T × f(c) = 0, which means that the sequence collapses.

In order to understand the maps f_{i} on the homology level better we should first understand the homology of the symmetric group. For that we give an excerpt from the work of M. Nakaoka, [N60, N61], describing the homology groups H_{∗}(K(S_{n}, 1), ℤ_{p}) = H_{∗}(S_{n}).

Theorem 5.1. ([N60, Corollary 5.9])

For any group G, n and q positive integers, we have

H_{q}(S_{n}, G) = H_{q}(S_{n−1}, G) ⊕ H_{q}(S_{n}, S_{n−1}, G).

The following is a comprised version of sections 6.1 and 6.2 of [N60].

Given a prime p, we denote by Q(p) the set of all sequences J = (j_{1}, …, j_{l}), l > 0, of positive integers satisfying the following conditions:

1. j_{k} = 0 or −1 mod 2(p − 1) for 1 ≤ k ≤ l,

2. j_{k} ≤ pj_{k+1} for 1 ≤ k < l,

3. j_{1} > (p − 1)(j_{2} + ⋯ + j_{l}).

For each element J ∈ Q(p), define the dimension and the rank by

D(J) = j_{1} + ⋯ + j_{l}, R(J) = p^{l}.

Let Q(P)_{r}^{d} denote the set of all elements a ∈ Q(P), such that D(a) = d and R(a) = r, obviously Q(P) = ∪_{d,r}Q(P)_{r}^{d}. Let U(Q(P)) be the ℤ_{p}-algebra generated by all elements a ∈ Q(P) subject to the relations

ab = (−1)^{D(a)D(b)}ba.  (5.6)

In particular, if D(a) is odd and p ≠ 2, then a^{2} = 0.

We linearly extend functions D and R to all the monomials in U(Q(P)):

D(ab) = D(a) + D(b), R(ab) = R(a) + R(b),  (5.7)

for monomials a, b ∈ U(Q(P)). Again, for a monomial ϕ ∈ U(Q(P)) we call D(ϕ) and R(ϕ) the dimension and the rank of ϕ. We denote by U_{r}^{d}(Q(P)) the submodule generated by all monomials of dimension d and rank r.

Theorem 5.2. ([N60, Theorem 6.3]). H_{d}(S_{m}, S_{m−1}; ℤ_{p}) = U_{m}^{d}(Q(p)), hence, because of the Theorem 5.1, we have H_{d}(S_{m}, ℤ_{p}) = ⨁_{r≤m} U_{r}^{d}(Q(p)).

Theorem 5.2 can be rephrased as follows: the Betti number β_{d}(S_{m}, S_{m−1}) is equal to the number of choices (J_{1}, …, J_{t}), J_{i} ∈ Q(P) such that

(1) ∑_{i=1}^{t} R(J_{i}) = m;

(2) if J_{i} = J_{j} for some i ≠ j then either D(J_{i}) is even or p = 2 (we call this the nonvanishing condition).

Let k and m be positive integers. Naturally defined group homomorphisms ι : S_{k} ⟶ S_{k+m} and μ : S_{k} × S_{m} ⟶ S_{k+m} (for π_{1} ∈ S_{k}, π_{2} ∈ S_{m}, μ(π_{1} × π_{2}) acts as π_{1} on the first k elements of [k + m] and as π_{2} on the last m elements of [k + m]; ι(π_{1}) = μ(π_{1} × id)) induce homology maps ι_{∗} : H_{∗}(S_{k}) ⟶ H_{∗}(S_{k+m}) and μ_{∗} : H_{∗}(S_{k} × S_{m}) ⟶ H_{∗}(S_{k+m}). Nakaoka has combinatorially described these maps: ι_{∗} is a monomorphism (see [N60, Theorem 5.8] and description following it) and μ_{∗} coincides with the multiplication operation in the algebra U(Q(P)) (see [N61, Lemma 1.2,(4)]).

Now, using these results, we are ready to resume the computation.

Since the spectral sequence collapses, H_{∗}(plim D̃_{n}) = H_{∗}(E_{1}^{∗,∗}, d_{1}), where (E_{1}^{∗,∗}, d_{1}) is the chain complex described by (5.4) and (5.5). We denote this complex by Δ_{n} and shall now describe its combinatorial structure.

The vector space C_{∗}(Δ_{n}) has a basis B which can be indexed by triples (J, π, f), where J = {J_{1}, …, J_{t}}, J_{i} ∈ Q(P); π = (π_{1}, …, π_{m}); f : [t] ⟶ [m] satisfying the following conditions:

Conditions A.

(1) ∑_{i=1}^{t} R(J_{i}) ≤ ∑_{r=1}^{m} π_{i} ≤ n;

(2) ∑_{f(i)=j} R(J_{i}) ≤ π_{j}, for all j ∈ [m];

(3) the set {J_{i}}_{f(i)=j} satisfies the nonvanishing condition for all j ∈ [m]. Combinatorially, conditions (2) and (3) mean that we distribute J_{i}’s into blocks

with sizes given by π such that the total rank of J_{i}’s which are put into the same block does not exceed its size and no two equal J_{i}’s of odd rank are put into the same block (putting in the same block simply means taking product).

Let d = ∑_{i=1}^{t} D(J_{i}) then (J, π, f) ∈ C_{d+m−1}(Δ_{n}). It is easy to read off the boundary map of Δ_{n} from the description above:

∂ (J, π, f) = ∑_{i=1}^{m−1}(−1)^{i}(J, π^{i}, f^{i}) + (−1)^{m}ϵ,  (5.8)

where π^{i} = (π_{1}, …, π_{i} + π_{i+1}, …, π_{m}) and

f^{i}(j) = \begin{cases}f(j), & if f(j) ≤ i \\ f(j) − 1, & if f(j) ≥ i + 1. \end{cases}

The ϵ above is given by the rule:

• if im f ⊆ [m − 1] then ϵ = (π′, f′), where π′ = (π_{1}, …, π_{m−1}) and f′ = f|_{[m−1]};

• if m ∈ im f, then ϵ = 0.

It is easy to see that sets π = {π_{1}, …, π_{m}}, satisfying condition A(1) above, are in bijection with nonempty subsets of [n] (simply define {a_{1}, …, a_{m}} ⊆ [n] by a_{i} = ∑_{j=1}^{i} π_{j}). Thus the basis B can be split into groups indexed by nonempty subsets of [n]. This is exactly the picture we see on the first step of our spectral sequence above, compare (5.1) and (5.4).

To compute the second, terminal step of the spectral sequence we shall regroup our basis elements. The key observation is that the boundary operator ∂ does not change J, see (5.8). Hence, if we fix the set J = (J_{1}, …, J_{t}), the chain complex Δ_{n}(J) = span((J′, π, f) | (J′, π, f) ∈ B, J′ = J) is a subcomplex of Δ_{n}. Its basis elements can be simply indexed by pairs (π, f) (of course satisfying conditions A). In what follows let l_{i} = |J_{i}| and r_{i} = p^{l_{i}}.

Clearly, the homology of Δ_{n} splits as a direct sum

H_{k}(Δ_{n}) = ⨁ H_{k−d}(Δ_{n}(J_{1}, …, J_{t}))  (5.9)

where r = ∑_{i=1}^{t} r_{i} and d = ∑_{i=1}^{q} D(J_{i}) and the sum is taken over all sets {J_{1}, …, J_{t}} satisfying r ≤ n.

It turns out that the parameters l_{i} (or equivalently r_{i}) described above are excessive for our computation. Let us describe yet another chain complex which will be isomorphic to Δ_{n}(J_{1}, …, J_{t}). The chain complex that we shall now describe will be denoted Δ_{n}(O, E, t), where O = {O_{1}, …, O_{u}}, E = {E_{1}, …, E_{v}}, such that O_{i}, E_{j} ⊆ [t], sets {O_{i}} and {E_{j}} are disjoint and ∑_{i=1}^{u} |O_{i}| + ∑_{j=1}^{v} |E_{j}| ≤ t.

Description of Δ_{n}(O, E, t). The elements of the basis of C_{∗}(Δ_{n}(O, E, t)) are indexed by pairs (π, f) with π = (π_{1}, …, π_{m}) and f : [t] ⟶ [m], satisfying following conditions:

Conditions B.

(1) t ≤ ∑_{i=1}^{m} π_{i} ≤ n;

(2) |f^{−1}(j)| ≤ π_{j} for all j ∈ [m];

(3) Assume that x < y. If x, y ∈ O_{i} for some i ∈ [u] then f(x) < f(y). If x, y ∈ E_{i} for some i ∈ [v] then f(x) ≤ f(y).

Clearly, the difference from the conditions A is that the rank function R has been replaced by constant 1 and the set O, resp. E, keeps track of the sets of mutually equal elements of J which have odd, resp. even, dimension. As before (π, f) ∈ C_{d+m−1}(Δ(O, E, t)), where d = ∑_{i=1}^{t} D(J_{i}).

The differential in this new chain complex is defined in the same way as above, see (5.8).

Lemma 5.3. The chain complexes Δ_{n}(J_{1}, …, J_{t}) and Δ_{n′}(O, E, t) are isomorphic. Here n′ = n + t − (r_{1} + ⋯ + r_{t}), O = {O_{1}, …, O_{u}} and E = {E_{1}, …, E_{v}}. If p ≠ 2, O_{i}’s, resp. E_{i}’s, are all sets of elements of J which are mutually equal and have odd, resp. even, dimension. If p = 2, then O_{i}’s are all sets of elements of J which are mutually equal and E = ∅.

Proof. An isomorphism ϕ is defined by ϕ(π, f) = (π′, f), where π′ = (π_{1} − α_{1}, …, π_{t} − α_{t}), here α_{j} = ∑_{f(i)=j}(r_{i} − 1).

Proposition 5.4. Δ_{n}(O, E, t) is acyclic unless t = n and v = 0, i.e. E = ∅, in which case Δ_{n}(O, E, t) has homology of an (n − 1)-sphere, that is

H̃_{i}(Δ_{n}(O, ∅, n)) = \begin{cases}ℤ, & if i = n − 1, \\ 0, & otherwise. \end{cases}  (5.10)

Proof. Let A_{n+1} be the braid arrangement in ℝ^{n+1} and let Λ_{n+1} be the collection of cells of the stratification of the sphere S^{n+1} by A_{n+1}. We shall now describe how to associate a cell from Λ_{n+1} to an element of the basis of Δ_{n}(O, E, t).

Let (π, f) be such an element with π = (π_{1}, …, π_{m}) and f : [t] → [m]. First we add an artificial block π_{m+1} to π such that |π_{m+1}| = n + 1 − t. The function f shows how J_{i}’s are distributed in blocks π_{1}, …, π_{m}. Let us augment the set {J_{1}, …, J_{t}} by n − t elements called ∗_{t+1}, …, ∗_{n+1}, which we distribute into the blocks π_{1}, …, π_{m+1} filling the places left over from J_{i}’s. More formally, the function f̃: [n + 1] → [m + 1] is uniquely defined by

(1) f̃|_{[t]} = f;

(2) f̃|_{{t+1,...,n+1}} is increasing and |f̃^{−1}(j)| = π_{j} for all j ∈ [m + 1].

The cell c_{π,f} ∈ Λ_{n+1} associated to (π, f) is given by the equations on x_{1}, …, x_{n+1} which are induced by the function f̃ in the obvious manner: if f̃(i) = f̃(j) then x_{i} = x_{j}; if f̃(i) < f̃(j) then x_{i} < x_{j}.

Let C = ∪_{(π,f)}c_{π,f}, where the union is taken over whole basis of Δ_{n}(O, E, t). It is not difficult to see that the boundary operator of the chain complex Δ_{n}(O, E, t) coincides with the natural boundary operator in Λ_{n+1}, hence Δ_{n}(O, E, t) ≃ (A_{n+1}, A_{n+1} \ C) (considering A_{n+1} relative to A_{n+1} \ C means exactly that we limit our attention to the cells in C). The whole set C can be described by the following inequalities:

Inequalities C.

(1) x_{i} < x_{n+1} if i ∈ [t] (since we have to make sure that n + 1 is in the block π_{m+1} and none of the J_{i}’s is);

(2) x_{i} ≤ x_{j} if t + 1 ≤ i < j ≤ n + 1;

(3) x_{i} < x_{j} if i, j ∈ O_{k} for some k ∈ [u];

(4) x_{i} ≤ x_{j} if i, j ∈ E_{k} for some k ∈ [v].

Clearly, if all of the inequalities were non-strict, then C would be a closed convex set, homeomorphic to an (n − 1)-ball T. However, even in our case it is clear that (A_{n+1}, A_{n+1} \ C) ≃ (T, Z), where Z is a piece of the boundary of that ball. If n = t and E = ∅ then we only have strict inequalities, hence Z is the full boundary of T and (T, Z) ≃ (S^{n−1}, ∗) where ∗ is a single point, which proves (5.10).

Let us now show that otherwise Z is contractible (that would mean that (T, Z) is acyclic and hence so is Δ_{n}(O, E, t)). We call an inequality from C essential if it does not follow from other inequalities from C. Z can be seen as a union of convex sets (Z_{i})_{i∈I}, each Z_{i} is defined by changing one of the essential inequalities in (1) or (3) to an equality and making other inequalities non-strict. Let us show that all of these convex sets have a common point.

If t ≥ 1, let α be the cell defined by x_{t+1} ≤ ⋯ ≤ x_{n+1} = x_{1} = ⋯ = x_{t}, clearly α ∈ Z_{i}, for all i ∈ I, since all of the strict inequalities (which are turned to equalities in Z_{i}) contain only elements from the set {1, …, t, n + 1}.

Assume now that t = n but E ≠ ∅, say c, d ∈ E_{1}, c < d. Let α be defined by x_{c} ≤ x_{1} = ⋯ = x_{c−1} = x_{c+1} = ⋯ = x_{n+1}. The only strict inequality in C containing x_{c} is x_{c} < x_{n+1}, which is a consequence of x_{c} ≤ x_{d} and x_{d} < x_{n+1}, hence it is not an essential inequality. Therefore we may conclude that also in this case α ∈ Z_{i} for all i ∈ I.

Since in both cases ∩_{i∈I}Z_{i} and Z_{i}’s are convex, Z = ∪_{i∈I}Z_{i} is contractible. This finishes the proof of the proposition.

Let us now proceed to the Main Theorem. Recall Nakaoka’s description of the relative homology H_{∗}(S_{n}, S_{n−1}; ℤ_{p}). To obtain H_{∗}(plim D̃_{n}, ℤ_{p}) we only need to make one modification in his description: the dimension function should be different: D̃(a) = D(a) + 1 for all a ∈ Q(P).

Let the algebra with this new dimension function be denoted by Ũ(Q(P)), then

Main Theorem 5.5.

H_{q}(Δ(plim D̃_{n}), ℤ_{p}) = Ũ_{n}^{q+1}(Q(p)).  (5.11)

Proof. According to (5.9) we have

H_{q}(Δ_{n}) = ⨁ H_{q−d}(Δ_{n}(J_{1}, …, J_{t})),  (5.12)

where d = ∑_{i=1}^{t} D(J_{i}), r = ∑_{i=1}^{t} R(J_{i}) and the sum is taken over all t-tuples (J_{1}, …, J_{t}) satisfying r ≤ n. Furthermore, Lemma 5.3 asserts that

H_{q−d}(Δ_{n}(J_{1}, …, J_{t})) = H_{q−d}(Δ_{n′}(O, E, t)),  (5.13)

where n′ = n + t − r.

Combining (5.12), (5.13) and Proposition 5.4 we can conclude that β_{q}(Δ_{n}) is equal to the number of all t-tuples (J_{1}, …, J_{t}) such that E = ∅, r = n and d = q + 1 − t, i.e. d̃= q, where d̃= ∑_{i=1}^{t} D̃(J_{i}), which proves this theorem.

6. Homology of the k-fold Boolean algebras

Throughout this section n is either a positive integer or ∞.

Definition 6.1. Let k be a positive integer and n either a positive integer or ∞. We define posets B_{n,k}, which we call k-fold Boolean algebras, as follows. Elements of B_{n,k} are sets, each consisting of k disjoint, possibly empty, finite subsets of [n] (here [∞] = ℕ), in other words, {S_{1}, …, S_{k}} such that S_{i} ⊆ [n], S_{i} ∩ S_{j} = ∅ for i ≠ j and |S_{i}| < ∞. The partial ordering is given by the rule: {S′_{1}, …, S′_{k}} ≤ {S_{1}, …, S_{k}} if and only if there exists a permutation π of the set [k] such that S′_{i} ⊆ S_{π_{i}} for all i ∈ [k].

Using the terminology of Section 4, let us define B_{n,k}^{o} = B_{n}^{[k]} and B̂_{n,k}^{o} = B̂_{n}^{[k]}. Then the Definition 6.1 can be reformulated as B_{n,k} = B_{n,k}^{o}/S_{k}. Analogously, we define B̂_{n,k} = B̂_{n,k}^{o}/S_{k}. The latter can be directly defined just like B_{n,k} with an extra condition that the sets S_{i} should be non-empty.

The following theorem is the main result of this section.

Theorem 6.2.

(a) Δ(B_{n,k}^{o}) is homotopy equivalent to a wedge of (n − 1)-dimensional spheres if n < ∞ and Δ(B_{∞,k}^{o}) is contractible;

(b) homology groups of Δ(B_{n,k}) with integral coefficients converge to those of Δ(B_{∞,k}), more precisely

H̃_{i}(B_{n,k}, ℤ) = \begin{cases}H̃_{i}(B_{∞,k}, ℤ) & for 0 ≤ i ≤ n − 2, \\ ℤ^{μ(B_{n,k})} & for i = n − 1, \\ 0 & otherwise; \end{cases}  (6.1)

(c) H_{q}(B_{∞,k}, ℤ_{p}) = Ũ_{k}^{q+1}(Q(p));

(d) same as (a) with B_{n,k}^{o} and B_{∞,k}^{o} replaced by B̂_{n,k}^{o} and B̂_{∞,k}^{o}, and (n − 1)- dimensional replaced by (n − k)-dimensional;

(e) same as (b) with B_{n,k} and B_{∞,k} replaced by B̂_{n,k} and B̂_{∞,k} , and (n − 1)- dimensional replaced by (n − k)-dimensional.

We need some preparation before we can proceed with the proof of Theorem 6.2. The following generalizes the definitions of B_{n,k} and B_{n,k}^{o}.

Definition 6.3. Let t and k be integers, 0 ≤ t ≤ k and n ∈ [∞]. An element of the poset B_{n,k,t} is a pair consisting of a t-tuple (S_{1}, …, S_{t}) (the ordered part) and a set {S_{t+1}, …, S_{k}} (the unordered part) such that S_{i} ⊆ [n], S_{i} ∩ S_{j} = ∅ for i ≠ j and S_{i} is finite. The partial ordering is given by the rule:

(S′_{1}, …, S′_{t}){S′_{t+1}, …, S′_{k}} ≤ (S_{1}, …, S_{t}){S_{t+1}, …, S_{k}}

if and only if S′_{i} ⊆ S_{i} for i = 1, …, t and there exists a permutation π of the set {t + 1, …, k} such that S′_{i} ⊆ S_{π_{i}} for i = t + 1, …, k.

Clearly B_{n,k,0} = B_{n,k} and B_{n,k,n} = B_{n,k,n−1} = B_{n,k}^{o}.

Observe that the lower intervals of B_{n,k,t} are boolean algebras, hence every B_{n,k,t} is a face poset of a regular CW complex.

Lemma 6.4. If t ≥ 1 then the poset B_{n,k,t} is EL-shellable, see [BW83], in particular Δ(B_{n,k,t}) is homotopy equivalent to a wedge of spheres of dimension n − 1.

Proof. Assume that t ≥ 1 and k ≥ 2. Our labeling will use integer labels. Observe that every edge (x, y) in the Hasse diagram of B_{n,k,t} is either of the form (x, ˆ1) or a merging of one of the sets of x with some element a ∈ [n]. Since t > 0 we have a certain set in x which is first in the ordering of the corresponding t-tuple. In the rest of the proof we often refer to the sets of x or y as blocks.

We divide all edges in the Hasse diagram into three groups and impose the labels in every group separately:

(1) edges (x, y), where y ≠ ˆ1, and the element a described above is merged to the first block of x, we label such edge with −a;

(2) edges from some coatom x to ˆ1, we label these edges with 0;

(3) edges (x, y), where y ≠ ˆ1, and the element a described above is merged to a block of x which is not the first one, we label such edge with a.

Let us now show that these labels give us an EL-labeling. Consider an interval (x, y). We divide the further proof into two cases.

Case 1. y ≠ ˆ1. The increasing chain is given by merging first all elements which have to be merged to the first block in decreasing order and then all other elements in increasing order. This chain is obviously lexicographically least. It is also uniquely defined, since one has to merge the elements to the first block first in the unique way, and then there is a unique way to merge the rest of the elements to the other blocks (creating a new nonempty block is considered as a merging to an empty block).

Case 2. y = ˆ1. The last label in any maximal chain in (x, y) has the label 0, hence every increasing chain should have only nonpositive labels. This means that we have to merge all the elements which are not already in one of the blocks of x to the first block, and we have to do it in decreasing order. Hence again the increasing chain is unique and lexicographically least.

We have shown that for t > 0 the posets B_{n,k,t} are EL-shellable, hence B_{n,k,t} are shellable and the simplicial complexes Δ(B_{n,k,t}) are homotopy equivalent to a wedge of spheres.

The following technical lemma is needed for the transition from the sequence of finite posets to their infinite limit. Lemma 6.5. Assume that we have a family of finite simplicial complexes {K_{i}}_{i=1}^{∞} and an infinite simplicial complex K_{∞}, such that K_{i} is a subcomplex of K_{j}, whenever i < j or j = ∞, and K_{∞} = ∪_{i=1}^{∞}K_{i}.

(1) If K_{i} is t_{i}-connected and lim_{i→∞} t_{i} = ∞ then K_{∞} is contractible.

(2) Assume that dim K_{i} = i − 1 and that for t ≥ 2, K_{t} is homotopy equivalent to a regular CW complex obtained from K_{t−1} by pasting on a number of (t − 1)-cells.

Then H̃_{i}(K_{∞}) = H̃_{i}(K_{t}), for all i = 0, 1, 2, …, t − 2.

Proof. (1) The image A of a mapping of any compact space (in particular an n-sphere) into K_{∞} lies in a union of finite number of cells. Hence for sufficiently large N, A ⊆ K_{N} and t_{N} ≥ n, which means that A can be shrunk to a point in K_{N}, hence also in K_{∞}. Thus all homology groups of K_{∞} are trivial and therefore K_{∞} is contractible.

(2) Obvious.

Proof of Theorem 6.2. (a) Setting t = n in Lemma 6.4 we get that Δ(B_{n,k}^{o}) is homotopy equivalent to a wedge of (n−1)-spheres, in particular it is (n−2)-connected. Furthermore, Lemma 6.5 with K_{i} = Δ(B_{i,k}^{o}) asserts that Δ(B_{∞,k}^{o}) is contractible.

(b) Recall that B_{n,k} is a face lattice of some regular CW complex. Denote this complex by Δ̃_{n,k}. Obviously Δ̃_{n,k} ≃ Δ(B_{n,k}). Let us study how the complex Δ̃_{n+1,k} differs from Δ̃_{n,k}.

It is clear that Δ̃_{n+1,k} has a vertex x, represented by a set ({n + 1}, ∅, …, ∅), such that a cell of Δ̃_{n+1,k} is not a cell of Δ̃_{n,k} if and only if it contains x. In other words Δ̃_{n+1,k} = Δ̃_{n,k} ∪ stΔ̃_{n,k}(x). It is easy to see that stΔ̃_{n,k}(x) ∩ Δ̃_{n,k} = lk Δ̃_{n,k}(x) ≃ Δ(B_{n−1,k,1}). The latter is, by Lemma 6.4, homotopy equivalent to a wedge of (n−2)- spheres. This means that Δ̃_{n+1,k} is homotopy equivalent to a space obtained from Δ̃_{n,k} by pasting on a number of (n − 1)-dimensional cells (one cell over each (n − 2)- sphere in Δ(B_{n−1,k,1})). Thus the conditions of (2) of Lemma 6.5 are satisfied and hence H̃_{i}(B_{n,k}, ℤ) = H̃_{i}(B_{∞,k}, ℤ) for 0 ≤ i ≤ n − 2.

That H̃_{n−1}(B_{n,k}, ℤ) = ℤ^{μ(B_{n,k})} follows from the fact that Δ(B_{∞,k}) is ℚ-acyclic.

(c) Clearly Δ(B_{∞,k}) ≃ Δ(plim D̃_{k}), so Main Theorem 5.5 applies.

(d) We shall assume that n ≥ k + 2, the other cases are obvious. B_{n,k}^{o} can be viewed as a face poset of a regular CW complex. Denote this complex by C. Let P_{i} be the subposet of B_{n,k}^{o} consisting of elements (S_{1}, …, S_{k}), such that S_{i} = ∅, where i ∈ [k]. Clearly, P_{i} is a face poset of C_{i}, where C_{i} is a regular CW subcomplex of C. Let T = ∪_{i=1}^{k}C_{i}.

There is a projection map p from C to a k-simplex S, defined by mapping a k-tuple (S_{1}, …, S_{k}) to its support subset of [k], under which T is mapped to the boundary of S. Δ(B̂_{n,k}) is the preimage of the biggest cell of that simplex and therefore H_{∗}((Δ(B̂_{n,k})) × (S, δS)) = H_{∗}(C, T). To show that Δ(B̂_{n,k}) is homotopy equivalent to a wedge of spheres, it is enough to show that π_{1}(Δ(B̂_{n,k})) = 0 and that H_{∗}(C, T) is zero except for the highest dimension.

We know that C_{i_{1}} ∩ ⋯ ∩ C_{i_{t}} is (n − t − 2)-connected so by the Nerve Lemma, T = ∪_{i=1}^{k}C_{i} is (n − 3)-connected. Thus T has the homology of a wedge of spheres of dimension one less than that of C. Using Mayer-Vietoris long exact sequence one sees that H_{∗}(C, T) (and hence H_{∗}(Δ(B̂_{n,k}))) can be non-zero only in the highest dimension.

Let us now show that π_{1}(Δ(B̂_{n,k})) = 0. Let R be the rank selection of B̂_{n,k} where we select the elements of rank at most 3. Assume that Δ(R) is simply connected. We can add elements of B̂_{n,k} \ R in some linear extension order, to finally form the full B̂_{n,k}. When we add an element x ∈ B̂_{n,k} \ R, we are gluing a cone over Δ((B̂_{n,k})_{<x}), which is connected, since it is a direct product of simplices. Therefore, by repetitive use of Seifert - Van Kampen theorem, Δ(B̂_{n,k}) is simply connected.

Let us now show that π_{1}(R) = 0. R is a face poset of a CW complex W. W can be described combinatorially as follows: vertices are ordered k-tuples of different elements of [n], edges are pairs of such ordered k-tuples differing in exactly one coordinate, and the two-dimensional faces are all triangles and squares which are formed by these edges. Assume that π_{1}(R) ≠ 0 and take the shortest path which cannot be contracted. It must contain at least two changes of the same coordinate. Moving the chosen path along squares we can permute the coordinates of any two consecutive changes. Thus we can retract our path to a path of the same length where two consecutive changes occur in the same coordinate. Now, we can use the proper triangle and replace these two steps by just one step, combining these two changes into one change. We thereby obtain a shorter path and hence a contradiction.

(e) S_{k} acts freely on both Δ(B̂_{∞,k}^{o}) and Δ(B̂_{n,k}^{o}). Furthermore, Δ(B̂_{∞,k}^{o}) is contractible and Δ(B̂_{n,k}^{o}) is (n − k − 1)-connected, hence Δ(B̂_{∞,k}) ≃ K(S_{k}, 1) and H̃_{i}(B̂_{n,k}^{o}) = H̃_{i}(K(S_{k}, 1)), for i = 0, …, n − k − 1.

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