Lower bounds on the homology of Vietoris-Rips complexes of hypercube graphs

We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $VR(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r<r'$ the inclusion $VR(Q_n;r)\hookrightarrow VR(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces $VR(Q_n;r)$. We provide lower bounds on the ranks of homology groups of $VR(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}(VR(Q_n;r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of \emph{homology propagation}: if $q\ge 1$ and if $p$ is the smallest integer for which $rank H_q(VR(Q_p;r))\neq 0$, then $rank H_q(VR(Q_n;r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot rank H_q(VR(Q_p;r))$ for all $n \ge p$. When $r\le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_q(VR(Q_n;r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of $VR(Q_n;r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators.


Introduction
Let Q n be the vertex set of the hypercube graph, equipped with the shortest path metric.In other words, Q n can be thought of the set of all 2 n binary strings of 0's and 1's equipped with the Hamming distance, or alternatively, as the set {0, 1} n ⊆ R n equipped with the ℓ 1 metric.
In this paper, we study the topology of the Vietoris-Rips simplicial complexes of Q n .Given a metric space X and a scale r ≥ 0, the Vietoris-Rips simplicial complex VR(X; r) has X as its vertex set, and a finite subset σ ⊆ X as a simplex if and only if the diameter of σ is at most r.Originally introduced for use in algebraic topology [24] and geometric group theory [8,17], Vietoris-Rips complexes are now a commonly-used tool in applied and computational topology in order to approximate the shape of a dataset [9,13].Important results include the fact that nearby metric spaces give nearby Vietoris-Rips persistent homology barcodes [12,11], that Vietoris-Rips complexes can be used to recover the homotopy types of manifolds [18,19,26,20], and that Vietoris-Rips persistent homology barcodes can be efficiently computed [7].Nevertheless, not much is known about Vietoris-Rips complexes of manifolds or of simple graphs at large scale parameters, unless the manifold is the circle [3], unless the graph is a cycle graph [2,5], or unless one restricts attention to 1-dimensional homology [25,16].
Let VR(Q n ; r) be the Vietoris-Rips complex of the vertex set of the n-dimensional hypercube at scale parameter r.The homotopy types of VR(Q n ; r) are known for r ≤ 3 (and otherwise mostly unknown); see Table 1.For r = 0, VR(Q n ; 0) is the disjoint union of 2 n vertices, and hence homotopy equivalent to a wedge sum (2 n −1)-fold wedge sum of zero-dimensional spheres.For r = 1, VR(Q n ; 1) is a connected graph (the hypercube graph), which by a simple Euler characteristic computation is homotopy equivalent to a ((n − 2)2 n−1 + 1)-fold wedge sum of circles.For r = 2, Adams and Adamaszek [4] prove that VR(Q n ; 2) is homotopy equivalent to a wedge sum of 3-dimensional spheres; see Theorem 2.4 for a precise statement which also counts the number of 3-spheres.For r = 3, Shukla proved in [22, Theorem A] that for n ≥ 5, the q-dimensional homology of VR(Q n ; 3) is nontrivial if and only if q = 7 or q = 4.The study of r = 3 was furthered by Feng [14] (based on work by Feng and Nukula [15]), who proved that VR(Q n ; 3) is always homotopy equivalent to a wedge sum of 7-spheres and 4-spheres; see Theorem 2.5 for a precise statement which also counts the number of spheres of each dimension.When r = n − 1, VR(Q n ; n − 1) is isomorphic to the boundary of the cross-polytope with 2 n vertices, and hence is homeomorphic to a sphere of dimension 2 n−1 − 1.For r ≥ n, the space space VR(Q n ; n) is a complete simplex, and hence contractible.However, there is an entire infinite "triangle" of parameters, namely r ≥ 4 and r ≤ n − 2, for which essentially nothing is known about the homotopy types of VR(Q n ; r).(See Table 2 for improved lower bounds for the r = 4 column.) In this paper, instead of focusing on a single value of r, we provide novel lower bounds on the ranks of homology groups of VR(Q n ; r) for all values of r.Some of these lower bounds are shown in blue in Table 1: using cross-polytopal generators, in Theorem 4.1 we prove rankH 2 r −1 (VR(Q n ; r)) ≥ 2 n−(r+1) n r+1 (although we show in Section 7 that for r ≥ 2 and n > 2r this does not constitute the entire reduced homology in all dimensions).This is the first result showing that the topology of VR(Q n ; r) is nontrivial for all values of r ≤ n − 1.Furthermore, we often show that VR(Q n ; r) is far from being contractible, with the rank of (2 r − 1)-dimensional homology tending to infinity exponentially fast as a function of n (with n increasing and with r fixed).
Our general strategy, which we refer to as homology propagation, is as follows.Let q ≥ 1. Suppose that one can show that the q-dimensional homology group H q (VR(Q p ; r)) is nonzero (for example, using a homology computation on a computer, or alternatively a theoretical result such as the mentioned cross-polytopal elements or the geometric generators of Section 7).Then we provide lower bounds on the ranks of the homology groups H q (VR(Q n ; r)) for all n ≥ p.In particular, in Theorem 6. 4 we prove that if p ≥ 1 is the smallest integer for which H q (VR(Q p ; r)) ̸ = 0, then Thus, a homology computation for a low-dimensional hypercube Q p has consequences for the homology of VR(Q n ; r) for all n ≥ p. See Table 2 for some consequences of this result and of related results.As we explain in Section 6.4, when r ≤ 3 our results are known to provide tight lower bounds on all Betti numbers of VR(Q n ; r).We take this as partial evidence that our novel results on the Betti numbers of VR(Q n ; r) for r ≥ 4 are likely to be good lower bounds, though we do not know how close they are to being tight as no upper bounds are known.Indeed, the main "upper bound" we know of on the Betti numbers of VR(Q n ; r) is a triviality result for 2-dimensional homology: Carlsson and Filippenko [10] prove that H 2 (VR(Q n ; r)) = 0 for all n and r.
For integers r < r ′ , we prove via a simple argument that the inclusion VR(Q n ; r) → VR(Q n ; r ′ ) is nullhomotopic.Therefore, there are no persistent homology bars of length longer than one, and all homological information about the filtration VR(Q n ; •) is determined by VR(Q n ; r) for individual integer values of r.
Though we have stated our results for Q n = {0, 1} n equipped with the ℓ 1 metric, we remark that these results hold for any ℓ p metric with 1 ≤ p < ∞.Indeed for x, y ∈ Q n , the i-th coordinates of x and y differ by either 0 or 1 for each 1 ≤ i ≤ n, and hence VR((Q n , ℓ p ); r) = VR((Q n , ℓ 1 ); r p ). So, our results can be translated into any ℓ p metric by a simple reparametrization of scale.
We expect that some of our work could be transferred over to provide results for Čech complexes of hypercube graphs, as studied in [6], though we do not pursue that direction here.
We begin with some preliminaries in Section 2. In Section 3 we review contractions, and we prove that VR(Q n ; •) has no persistent homology bars of length longer than one.In Section 4 we use cross-polytopal generators to prove rankH 2 r −1 (VR(Q n ; r)) ≥ 2 n−(r+1) n r+1 .We introduce concentrations in Section 5, which we use to prove our more general forms of homology propagation in Section 6.In Section 7 we prove the existence of novel lower-dimensional homology generators, and we conclude with some open questions in Section 8.
2.1.Homology.All homology groups will be considered with coefficients in Z or in a field.The rank of a finitely generated abelian group is the cardinality of a maximal linearly independent subset.We let β q denote the q-th Betti number of a space, i.e., the rank of the q-dimensional homology group.Definition 2.1.Given n ∈ {1, 2, . ..}, the hypercube graph Q n is the metric space {0, 1} n , equipped with the ℓ 1 metric.In particular, the elements of the space are n-tuples (a 1 , a 2 , . . ., a n ) = (a i ) i∈[n] with a i ∈ {0, 1}, and the ℓ 1 distance is defined as In other words, the distance between two n-tuples is the number of coordinates in which they differ.
For a ∈ Q n , its antipodal point ā is given as ā = (1, 1, . . ., 1) − a.In particular, ā is the furthest point in Q n from a and thus shares no coordinate with a. Observe that d(a, ā) = n.

2.3.
Vietoris-Rips complexes.A Vietoris-Rips complex is a way to "thicken" a metric space, as we describe via the definitions below.Definition 2.3.Given r ≥ 0 and a metric space X the Vietoris-Rips complex VR(X; r) is the simplicial complex with vertex set X, and with a finite subset σ ⊆ X being a simplex whenever diam(σ) ≤ r.This is the closed Vietoris-Rips complex, since we are using the convention ≤ instead of <.But, since the metric spaces Q n are finite, all of our results have analogues if one instead considers the open Vietoris-Rips complex that uses the < convention.
In [4] it was proven that VR(Q n ; 2) is homotopy equivalent to a wedge sum of 3-dimensional spheres: Theorem 2.4 (Theorem 1 of [4]).For n ≥ 3, we have the homotopy equivalence See [21] for some relationships between this result and generating functions.
In [14], it was proven that VR(Q n ; 3) is always homotopy equivalent to a wedge sum of 7-spheres and 4-spheres: Theorem 2.5 (Theorem 24 of [14]).For n ≥ 5, we have the homotopy equivalence 2.4.Embeddings of hypercubes.For k a positive integer, let [k] = {1, 2, . . ., k}.Given p ∈ [n − 1] there are many isometric copies of Q p in Q n .For any subset S ⊆ [n] of cardinality p we can isometrically embed Q p in Q n , using set S as its variable coordinates, and leaving the rest of the entries fixed.In more detail, we define an isometric embedding ι b S : is the smallest isometric copy of a cube (i.e., the image of Q p ′ via some map ι) containing B.
For our purposes we will only consider isometric embeddings Q p → Q n (also denoted by Q p ≤ Q n ) that retain the order of coordinates, although any permutation of coordinates of Q p results in a nonconstant isometry of Q p and thus a different isometric embedding into Q n .With this convention of retaining the coordinate order, there are n p 2 n−p isometric embeddings ι :

Contractions and the persistent homology of hypercubes
In this section we prove the following results.First, fix the scale r ≥ 0, and let p ≤ n.An isometric embedding Q p → Q n gives an inclusion VR(Q p ; r) → VR(Q n ; r), which is injective on homology in all dimensions.Alternatively, fix dimension n, and consider integer scale parameters r < r ′ .The inclusion VR(Q n ; r) → VR(Q n ; r ′ ) is nullhomotopic, and hence the filtration VR(Q n ; •) has no persistent homology bars of length longer than one.These results follow from the properties of contractions, which we introduce now.

Contractions
Our interest in contractions stems from the fact that if a contraction X → A exists, then the homology of a Vietoris-Rips complex of A maps injectively into the homology of the corresponding Vietoris-Rips complex of X: Proposition 3.1 ( [27]).If f : X → A is a contraction, then the embedding A → X induces injections on homology H q (VR(A; r)) → H q (VR(X; r)) for all integers q ≥ 0 and scales r ≥ 0.
We prove that the projections from a higher-dimensional cube to a lower-dimensional cube in Section 2.4 are contractions: of cardinality p, and offset b as in Section 2.4, the following hold: (1) Maps π S and π b S are contractions.x, y) is the number of components in which x and y differ.On the other hand, d(π b S (x), π b S (y)) is the number of components from S in which x and y differ (and the same holds for map π S instead of π b S as well).Thus d(x, y) ≤ d(π b S (x), π b S (y)) with: p as their coordinates outside S agree, yielding (1) and (3)(a), and p , the number of coordinates outside of S on which x and y disagree is at least 1 (due to x, y not being in the same Q * p ) and at most n − p (which is the cardinality of [n] \ S).Item (2) follows from the observation that: • d(x, π b S (y)) is the number of components from S in which x and y differ, and • d(π b S (y), y) is the number of components from [n] \ S in which x and y differ, as x ∈ Q b p .

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Since each of the projections π : Q n → Q p is a contraction by Lemma 3.2, Proposition 3.1 then implies that each of the embeddings Q p → Q n induces an injective map on homology H q (VR(Q p ; r)) → H q (VR(Q n ; r)) for all dimensions q.
3.2.Persistent homology of hypercubes.The emphasis in modern topology is often on persistent homology arising from the Vietoris-Rips filtration.However, in the setting of Vietoris-Rips complexes of hypercubes, persistent homology does not provide any more information beyond the homology groups at fixed scale parameters.Indeed, the following proposition implies that for any integers r < r ′ , the inclusion VR(Q n ; r) → VR(Q n ; r + 1) induces a map that is trivial on homology.Proposition 3.3.For any positive integers n and r, the natural inclusion VR(Q n ; r) → VR(Q n ; r + 1) is homotopically trivial.
Proof.We first claim that the inclusion VR( . In order to prove the claim we will show that the two maps are contiguous in VR(Q n ; r + 1) (i.e., for each simplex σ ∈ VR(Q n ; r) the union σ ∪ π [n−1] (σ) is contained in a simplex of VR(Q n ; r + 1)), which implies that the two maps are homotopic.
Let σ ∈ VR(Q n ; r).By definition diam(σ) ≤ r.As π [n−1] (σ) is obtained by dropping the final coordinate we also have diam(π , and the claim is proved.We proceed inductively, proving that each projection π , by the same argument as above.As a result, the embedding VR(

Homology bounds via cross-polytopes and maximal simplices
Fix a scale r ≥ 2, and consider an isometric embedding ι : The aim of this section is to prove not only that the induced map VR(Q r+1 ; r) → VR(Q n ; r;) is injective on (2 r − 1)dimensional homology, but also that different (ordered) embeddings ι produce independent homology generators.Let us explain this in detail.
We first observe that VR(Q r+1 ; r) is homeorphic to a (2 r − 1)-dimensional sphere, i.e.VR(Q r+1 ; r) ∼ = S 2 r −1 .The reason is that each vertex x ∈ Q r+1 is connected by an edge in VR(Q r+1 ; r) to every vertex of Q r+1 except for x, the antipodal vertex.Therefore, after taking the clique complex of this set of edges, we see that VR(Q r+1 ; r) is isomorphic (as simplicial complexes) to the boundary of the cross-polytope with 2 r+1 vertices.This cross-polytope is a 2 r -dimensional ball in 2 r -dimensional Euclidean space, and therefore its boundary is a sphere of dimension 2 r − 1.In particular, rankH 2 r −1 (VR(Q r+1 ; r)) = 1.
Since VR(Q r+1 ; r) is the boundary of a cross-polytope, there is a convenient (2 r − 1)-dimensional cycle γ generating H 2 r −1 (VR(Q r+1 ; r)).Define the set of maximal antipode-free simplices as The cycle γ is defined as the sum of appropriately oriented elements of A r .The space Q r+1 consists of 2 r+1 points, which can be partitioned into 2 r pairs of mutually antipodal points.If a subset of Q r+1 contains exactly one point from each such pair, it is of cardinality 2 r .Thus A r consists of sets of cardinality 2 r .Given x ∈ Q r+1 , the only element of Q r+1 which disagrees with x on all r + 1 coordinates is x.As a result each element of A r is of diameter at most r and thus a simplex of VR(Q r+1 ; r).Observe also that any element of A r is a maximal simplex of VR(Q r+1 ; r): adding any point to such a simplex would mean the presence of an antipodal pair, and so the diameter would thus grow to r + 1.
As explained above, the embeddings ι : Q r+1 → Q n induce injections on homology by Lemma 3.2 and Proposition 3.1.The fact that these embeddings give independent homology generators is formalized in the following statement, which is also the main result of this section. .
The proof will be provided at the conclusion of the section.Recall that 2 n−(r+1) n r+1 is the number of different (ordered) embeddings ι : Q r+1 → Q n .We will use maximal simplices and the pairing between homology and cohomology in order to prove that these 2 n−(r+1) n r+1 different embeddings provide independent cross-polytopal generators for homology.Proposition 4.2.Suppose K is a simplicial complex and σ is a maximal simplex of dimension p in K.If there is a p-cycle α in K in which σ appears with a non-trivial coefficient λ, then any representative p-cycle of [α] also contains σ with the same coefficient λ.
Proof.As σ is maximal, the p-cochain mapping σ to 1 and all other p-simplices to 0 is a p-cocycle denoted by ω σ .Utilizing the cap product we see that for each representative α 2 could also be proved directly.If α and α ′ are homologous p-cycles, then their difference is a boundary of a (p + 1)-chain.The later cannot contain σ since σ is maximal; hence the coefficients of σ in α and in α ′ coincide.We emphasize the cohomological proof because we will use the cochain ω σ again.
We next focus on the construction of maximal simplices of VR(Q r+1 ; r) which are furthermore also maximal simplices in VR(Q n ; r).The following is a simple criterion identifying such a simplex as a maximal simplex in VR(Q n ; r); see Figure 1.(We recall that the local diameter of σ ⊆ Q n at a point w ∈ σ is defined as localDiam(σ, w) = max z∈σ d(w, z).) Proof.Assume a point x ∈ Q n \ σ is added to σ.We will show that this increases the diameter of σ beyond r, by repeatedly using Lemma 3.2.
, and also the local diameter assumption implies there exists We now construct maximal simplices σ in VR(Q r+1 ; r) that, by Proposition 4.4, will remain maximal in VR(Q n ; r).We recall that the cubic hull cHull(σ) is the smallest isometric copy of a cube containing σ.That the convex hull of σ is all of Q r+1 will later be used to give the independence of homology generators in the proof of Theorem 4.1.
Proof.We first prove the case when r ≥ 2 is odd (before afterwards handling the case when r ≥ 3 is odd).Define σ as the collection of vertices in Q r+1 whose coordinates contain an even number of values 1.As r + 1 is odd, this means x ∈ σ iff x / ∈ σ, so σ ∈ A r .We proceed by determining the local diameter.Let y ∈ σ and define y ′ by taking ȳ and flipping one of its coordinates.Then y ′ ∈ σ as it has an even number of ones, and it disagrees with y on all coordinates except the flipped one, hence d(y, y ′ ) = r.So localDiam(σ, y) = r for all y ∈ σ.
It remains to show that cHull(σ) = Q r+1 .If cHull(σ) ⊊ Q r+1 , there would be a single coordinate shared by all the points of σ.However, as r ≥ 2 we can prescribe any single coordinate as we please, and then fill in the rest of the coordinates to obtain a vertex of σ: • if the chosen coordinate was 1, fill another coordinate as 1 and the rest as 0; • if the chosen coordinate was 0, fill all other coordinates as 0. Next, we handle the case when r ≥ 3 is odd.Let τ be the the maximal simplex in Q r obtained in the proof of the even case.Define Formally speaking, σ = ι r , with the associated index set being ∈ σ.We proceed by determining the local diameter.Take x = y × {i} ∈ σ.As localDiam(τ, y) = r − 1, there exists It remains to show that cHull(σ) = Q r+1 .Similarly as in the proof of the even case, this follows from the fact that as r ≥ 3 we can prescribe any single coordinate as we please, and then fill in the rest of the coordinates to obtain a vertex of σ: • the last coordinate can be chosen freely by the construction of σ; • any of the first r coordinates can be choosen freely by the construction and by the even case.

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We are now in position to prove the main result of this section, Theorem 4.1, which states that rankH Proof of Theorem 4.1.For notational convenience, let k = 2 n−(r+1) n r+1 .There are k isometric copies of Q r+1 in Q n obtained via embeddings ι, which we enumerate as C 1 , C 2 , . . ., C k .For each i: (1) Let σ i be the maximal simplex in C i obtained from Proposition 4.4 and Lemma 4.5.
(2) Let [α i ] be the mentioned cross-polytopal generator of H 2 r −1 (VR(C i ; r)), and recall the coefficient of σ i in α i is 1. (3) Let ω i be the (2 r − 1)-cochain on Q n mapping σ i to 1 and the rest of the (2 r − 1)-dimensional simplices to 0. As σ i is a maximal simplex in Q n , the cochains ω i are cocycles.(4) Note that σ i is not contained as a term in α j for any i ̸ = j.Indeed, if that was the case, σ i would be contained in the lower-dimensional cube via the cap product we obtain λ j = 0 by (4) above.Hence the rank of

More contractions: concentrations
Up to now the only contractions that we have utilized are the projections π S .In order to establish additional homology bounds we need to employ a new kind of contractions called concentrations.The idea of a such maps in low-dimensional settings is shown in Figures 2 and 3. We proceed with an explanation of the general case.
Let n > k be positive integers.Choose a = (a k+1 , a k+2 , . . ., a n ) ∈ {0, 1} n−k and let C denote the copy of Working towards a contraction we define a concentration map f : Q n → C of codimension n − k by the following rule: (1) f | C = Id C , and ( In particular, we concentrate the k-th coordinate of Q n \ C to 1 (although we might as well have used 0).Permuting the coordinates of Q n generates other concentration maps.In order to discuss the properties of concentration functions it suffices to consider the concentrations defined as f above.

Proof. (i) We verify the claim by a case analysis:
For x, y ∈ C we have as f is the identity on C. For x, y / ∈ C the quantity d(x, y) is the number of coordinates in which x and y differ, while d(f (x), f (y)) is the number of coordinates amongst the first k − 1 in which x and y differ.Thus d(f (x), f (y)) ≤ d(x, y).
Let x ∈ C, y / ∈ C. Then d(x, y) is the sum of the following two numbers: • The number of coordinates amongst the first k coordinates in which x and y differ.
• The number of coordinates amongst the last n − k coordinates in which x and y differ.Note that this quantity is at least 1 as y / ∈ C.
On the other side, d(f (x), f (y)) is less than or equal to the sum of the following two numbers: • The number of coordinates amongst the first k − 1 coordinates in which x and y differ.
• The number 1 if the k-th coordinate of x does not equal 1.The concentration onto the bottom left Q 1 , mapping all hollow-square vertices to the solid square vertex, isometrically identifies the three bold copies of Q 1 , and sends the other copies of Q 1 to a point.In this case the concentration is of codimension 2, i.e., maps As the codimension t increases, the number of isometrically identified subcubes is exponential 2 t for projections and linear t + 1 for concentrations.The exponential increase leads to weaker lower bound in Theorem 6.2 than the linear increase in Theorems 6.3 and 6.4.
Together we obtain d(f (x), f (y)) ≤ d(x, y).This covers all possible cases.We conclude that f is a contraction.
(ii) The n − k + 1 copies of Q k in question are the ones of the form for p ∈ {k, k + 1, . . ., n}.The case p = k shows that C is one of these copies.Note that the part is contained in C and thus f is identity on it.On the other hand, the part by retaining the first k − 1 coordinates.Together these two parts form C.
has one of the first k − 1 coordinates (say, the p-th coordinate) constant.Thus the same holds for its image via f and consequently, its image is contained in the corresponding Q k−1 ⊆ Q n , i.e., the one having the p-th coordinate constant, and having the last n − k coordinates prescribed as in C. □

Homology bounds via contractions
In Section 4 we showed how the appearance of cross-polytopal homology classes in certain dimensions of the Vietoris-Rips complexes of cubes generate independent homology elements in Vietoris-Rips complexes of higher-dimensional cubes.Applying Proposition 3.1 to the canonical projections implies that the homology of each smaller subcube embeds.However, the independence of homology classes arising from various subcubes was proved using maximal simplices; this argument depended heavily on the fact that convenient (cross-polytopal) homology representatives were available to us.In this section we aim to provide an analogous result for homology in any dimension, without prior knowledge of homology generators.
Example 6.1.As a motivating example, consider the graph VR(Q 3 ; 1).The cube Q 3 contains six subcubes Q 2 and each VR(Q 2 ; 1) ≃ S 1 has the first Betti number equal to 1.However, the first Betti number of Q 3 is not 6 but rather 5, demonstrating that homology classes of various subcubes might in general interfere, i.e., not be independent.In Theorem 4.1 we proved there is no such interference in a specific case (involving cross-polytopes).In the more general case of this section, we prove lower bounds even when independence may not hold.
The general setting of this section: Fix r, q ∈ {1, 2, . ..}.Let p ≥ 1 be the smallest integer for which H q (VR(Q p ; r)) ̸ = 0. We implicitly assume that such a p exists, i.e., that H q (VR(Q n ; r)) is non-trivial for some n.
The cube Q n contains 2 n−p n p canonical Q p subcubes.We aim to estimate the rank of the homomorphism induced by the map consisting of the natural inclusions of all of the Q p subcubes.
6.1.Homology bounds via projections.We will start with the simplest argument to demonstrate how contractions, in this case projections, may be used to lower bound the homology.
Theorem 6.2.Let q ≥ 1.If p is the smallest integer for which H q (VR(Q p ; r)) ̸ = 0, then for n ≥ p, Proof.Let the Q * p denote the n p 2 n−p different p-dimensional subcubes of Q n (say as * varies from 1 to n p 2 n−p ).For a subset S ⊆ [n] of cardinality p, the projection π S : Q n → Q p is an isometry on 2 n−p of these subcubes Q * p (the ones having exactly the coordinates S as the free coordinates).For each such S choose one of these cubes and designate it as Q p,S , thus marking n p copies of Q p ⊆ Q n .For each S let a 1,S , a 2,S , . . ., a rankHq(VR(Qp;r)),S denote a largest linearly independent collection in H q (VR(Q p,S ; r)).We claim that the collection {a i,S } of cardinality n p • rankH q (VR(Q p ; r)) is linearly independent in for some coefficients λ i,S .Fix a subset S ′ ⊆ [n] of cardinality p, and to the equality above apply the map on H q induced by the projection π S ′ : Q n → Q p .As p is the minimal dimension of a cube in which H q is non-trivial on Vietoris-Rips complexes, and as π S ′ maps all of the Q * p to a smaller-dimensional cube except for the ones with exactly the coordinates in S ′ as the free coordinates, we obtain (π S ′ ) * (a i,S ) = 0 for all S ̸ = S ′ , where * denotes the induced map on homology.As π S ′ | Q p,S ′ → Q p is a bijection on the corresponding Vietoris-Rips complexes, and induces an isomorphism homology, we have reduced our equation to Consequently, λ i,S ′ = 0 for all i, as {a i,S ′ } forms a linearly independent collection in H q (VR(Q p ; r)) by definition.As S ′ was arbitrary we conclude λ i,S = 0 for all i and S, and thus the claim holds.□ 6.2.Codimension 1 homology bounds via concentrations.The following result states that in codimension 1 (i.e., when we increase the dimension of the cube by 1 from p to p + 1), all but at most one of the subcubes induce independent inclusions on homology.Example 6.1 shows that all subcubes need not induce independent inclusions of homology (and we see that Theorem 6.3 is tight in the case of Example 6.1).
Theorem 6.3.Let q ≥ 1.If p is the smallest integer for which H q (VR(Q p ; r)) ̸ = 0, then Proof.The cube Q p+1 contains 2p + 2 subcubes Q p , which we enumerate as Q p,1 , Q p,2 , . . ., Q p,2p+2 .For each 1 ≤ j ≤ 2p + 1, let {a i,j | 1 ≤ i ≤ rankH q (VR(Q p ; r))} denote a largest linearly independent collection in H q (VR(Q p,j ; r)).We claim that the collection i,j λ i,j • a i,j = 0 for some coefficients λ i,j with 1 ≤ i ≤ rankH q (VR(Q p ; r)) and 1 ≤ j ≤ 2p + 1.Note that there are no representatives from Q p,2p+2 .Choose a subcube Q p−1 ⊆ Q p,2p+2 .It is the intersection of Q p,2p+2 and another p-dimensional cube of the form Q p, * , say Q p,1 .We apply the concentration map f corresponding to these choices using Proposition 5.1: (i) f is bijective on exactly two p-cubes: Q p,2p+2 and Q p,1 .
(ii) f maps all other p-cubes to cubes of dimension less than p.By the choice of p (as the first dimension of a cube in which nontrivial p-dimensional homology appears in its Vietoris-Rips complex), we get f * (a i,j ) = 0 for all i > 1. (iii) As a result, after applying the induced map f * on homology, Equation (1) simplifies to j λ j • a 1,j = 0. Consequently, λ 1,j = 0 for all i, as {a 1,j } forms a linearly independent collection in H q (VR(Q p,1 ; r)) by definition.We keep repeating the procedure of the previous paragraph: • Choose a subcube Q p,j , whose corresponding coefficients λ i,j have been determined to be zero, and choose a neighboring subcube Q p,j ′ (i.e., a subcube with a common (p − 1)-dimensional cube), whose corresponding coefficients λ i,j ′ have not yet been determined to be zero.• Apply the concentration map corresponding to these two p-dimensional cubes to deduce that the coefficients λ i,j ′ also equal zero.Any cube Q p,j ′ can be reached from Q p,2p+2 by an appropriate sequence of cubes (i.e., Q p,2p+2 , a (p − 1)dimensional subcube thereof, an enclosing p-dimensional cube, a (p − 1)-dimensional subcube thereof, . . ., Q p,j ′ ).Therefore, we can eventually deduce that λ i,j = 0 for all i and j.Hence the rank bound holds due to the setup of Equation ( 1).□ 6.3.General homology bounds via concentrations.In this subsection we will generalize the argument of Theorem 6.3 to deduce a lower bound on homology on all subsequent larger (not just codimension one) cubes.The core idea is the following.In the previous subsection we were able to "connect" p-dimensional cubes by concentrations.Each chosen concentration was bijective on exactly two adjacent cubes of the form Q p sharing a common (p − 1)-dimensional cube; see item (i) in the proof of Theorem 6.3.If the coefficients of Equation (1) corresponding to one of the two copies of Q p were known to be trivial, then the homological version of the said concentration transformed Equation (1) so that only the coefficients corresponding to the other of the two cubes Q p were retained; see item (ii).These coefficients were then deduced to be trivial by their definition as the resulting equation contained only terms arising from a single Q p , see item (iii).
In this subsection we generalize this argument to all higher-dimensional cubes.Instead of concentrations isolating 2 adjacent copies of Q p (as happens in codimension 1), the concentrations will in general isolate n − p + 1 (i.e., codimension plus one; see Proposition 5.1(ii)) copies of Q p .The main technical question is thus to determine how many of the subcubes are independent in the above sense.Theorem 6.4.Let q ≥ 1.If p is the smallest integer for which H q (VR(Q p ; r)) ̸ = 0, then for n ≥ p, In particular cases, Theorem 6.4 reduces to the following.For n = p we get the tautology that rankH q (VR(Q p ; r)) is at least as large as itself.For n = p + 1 we recover Theorem 6.3: Proof of Theorem 6.4.The cube Q n consists of two disjoint copies of Q n−1 ; see Figure 5: • the rear one with the last coordinate 0, denoted by Q 0 n−1 , and • the front one with the last coordinate 1, denoted by Q 1 n−1 .We partition the Q p subcubes of Q n into three classes: • The ones contained in the rear Q 0 n−1 where vertices have last coordinate 0, denoted by R n .
A sketch of the proof of Theorem 6.4.In each step the thick dashed lines represent copies of Q p in M n yielding new independent homology classes within the Vietoris-Rips complex of Q n , in addition to the established independent homology classes (thick solid lines) arising from certain copies of Q p , denoted by F n , within the front face The multiplicative factor in the theorem is the total number of the thick edges, both dashed and solid ones.
• The ones contained in the front Q 1 n−1 where vertices have last coordinate 1, denoted by F n .• The ones contained in the middle passage between them, denoted by We will prove that the following Q p subcubes of Q n induce independent embeddings on homology H q of Vietoris-Rips complexes: the elements of M n (dashed cubes in Figure 5) and the elements of F n that have inductively been shown to include independent embeddings on homology H q of Vietoris-Rips complexes on Q 1 n−1 (bold cubes in Figure 5).The initial cases of the inductive process have been discussed in the paragraph before the proof.For n = p + 1 this is Theorem 6.3.
The cardinality of Each such subcube has the last coordinate constantly 0. Taking a union with a copy of the same Q p−1 subcube with the last coordinates changed to 1, we obtain a Q p subcube in M n .It is apparent that all elements of M n arise this way.Let us enumerate the elements of M n as Q M p,j with 1 ≤ j ≤ 2 n−p n−1 p−1 .For each such j let {a i,j | 1 ≤ i ≤ rankH q (VR(Q p ; r))} denote a largest linearly independent collection in H q (VR(Q M p,j ; r)).The cardinality of the copies of Q p in F n that have inductively been shown to include independent embeddings on homology H q of Vietoris-Rips complexes on For each such j let {b i,j | 1 ≤ i ≤ rankH q (VR(Q p ; r))} denote a largest linearly independent collection in H q (VR(Q F p,j ; r)).Assuming the equality for some coefficients λ i,j , µ i,j , we claim that all coefficients equal zero.This will prove the theorem as the number of involved terms equals ).We will first prove that the coefficients λ i,j are all zero.Fix some 1 ≤ j ≤ 2 n−p n−1 p−1 , and let D be the copy of 4 one can visualize D as the solid round vertex, and C as the edge between the two solid vertices.)By Proposition 5.1: • f maps any Q p subcube of Q n that contains D bijectively onto C = Q M p,j .All such subcubes except for C are contained in R n .
• f maps all of the other Q p subcubes of Q n to lower-dimensional subcubes.These two observations imply that applying the induced map f * on homology to Equation (2), we obtain i λ i,j • a i,j = 0.By the choice of {a i,j } i as an independent collection of homology classes for H q (VR(Q M p,j ; r)), we obtain λ i,j = 0 for all i.Since this can be done for any 1 ≤ j ≤ 2 n−p n−1 p−1 , we have λ i,j = 0 for all i and j.
We have thus reduced Equation ( 2) to i,j µ i,j • b i,j = 0. Let π S : Q n → Q n−1 be the projection that forgets the last coordinate of each vector (explicitly, S = [n − 1] ⊆ [n]).Note that the restrictions of π S to Q 0 n and to Q 1 n are bijections.Hence, after applying the induced map (π S ) * on homology, the inductive definition of the b i,j implies that µ i,j = 0 for all i and j. □ 6.4.Comparison with known results.In this subsection we demonstrate that our lower bounds agree with actual ranks of homology in many known cases.In particular, for r = 1, 2, 3, if we assume that we know the homotopy types or Betti numbers for the first few cases (n ≤ r + 1 or n ≤ r + 2), then we show that our lower bounds on the Betti numbers of VR(Q n ; r) are tight (i.e.optimal) for all n and for all dimensions of homology.For r = 4 we explain the best lower bounds we know on the Betti numbers of VR(Q n ; 4), which are based on homology computations by Ziqin Feng.Since the homotopy types of VR(Q n ; 4) are unknown for n ≥ 6, we do not know if these bounds are tight.For a summary see Table 2.
6.4.1.The case r = 1.Assuming the obvious homeomorphism VR(Q 2 ; 1) ∼ = S 1 , the lower bound of Theorem 6.4 with p = 2 gives where the last step is explained in Appendix A.1.This inequality is actually an equality, as one can see via an Euler characteristic computation.Indeed, VR(Q n ; 1) has 2 n vertices and n2 n−1 edges, and so the Euler characteristic is 2 n − n2 n−1 .As VR(Q n ; 1) is connected, the rank of H 1 (VR(Q n ; 1)) equals n2 n−1 − 2 n + 1. See [10, Proposition 4.12] for a related computation.
6.4.2.The case r = 2.We know that the embedding of each individual subcube induces an injection on homology.Our results provide the lower bounds on the rank of the map on homology induced by the inclusion of all subcubes Q p (where p is the dimension of the first appearance of q-dimensional homology H q ).The upper bound for homology obtained in this way is 2 n−p n p rankH q (VR(Q p ; r)), where the the multiplicative constant is the number of all Q p subcubes on Q n .These possible generators are all independent in the case of cross-polytopal generators (Theorem 4.1).In case this bound is exceeded we can thus deduce that certain new homology classes appear that are not generated by the embeddings of Q p subcubes.
In the case r = 2, H 3 (VR(Q 3 ; 2)) has rank one and is generated by a cross-polytopal element.Even though Q 4 has only 2 4  1 = 8 subcubes Q 3 , we have rankH 3 (VR(Q 4 ; 2)) = 9 > 8.This indicates the appearance of a homology class α not generated by embedded homologies of Q 3 subcubes; see Section 7 for a description of this "geometric" generator.This new homology class contributes to the homology of higher-dimensional cubes in the same way as the homology described by Theorem 6.4.We formalize this in the next subsection; see Theorem 6.5.Together, this cross-polytopal generator and this geometric generator α explain all of the homology when r = 2: (a) The cross-polytopal elements provide 2 n−3 n 3 independent generators for H 3 (VR(Q n ; 2)) (Theorem 4.1).(b) The non-cross-polytopal element α provides Table 2.For each of r = 1, 2, 3, 4, the pair of columns represent the comparison between (left) known homotopy types and homology groups of VR(Q n ; r) with (right) the bounds arising from our results.The bold red spheres are the initial cross-polytopal spheres that induce the red lower bounds on Betti numbers due to Theorem 4.1.For r = 1, 2, 3, the bold blue and violet spheres induce the blue and violet lower bounds on Betti numbers due to Theorems 6.4 and 6.5.Observe that the total lower bounds match the known Betti numbers for r = 1, 2, 3.The homology computations for VR(Q 6 ; 4) by Ziqin Feng induce the lower bounds on Betti numbers of VR(Q n ; 4) for n ≥ 6 by Theorems 6.4 and 6.5.Theorem 7.1(iii) states that in each column r ≥ 2, we have at least one homology class (such as the features in blue) that is not induced from a red cross-polytopal sphere.
Thus, the combined lower bound says that the rank of H 3 (VR(Q n ; 2)) is at least Since VR(Q n ; 2) ≃ cn S 3 (Theorem 2.4), this combined lower bound explains all of the homology when r = 2. 4 independent generators for H 4 (VR(Q n ; 3)) (Theorem 6.4).The total lower bound equals the actual rank of all homology of VR(Q n ; 3), due to Theorem 2.5 which says 6.4.4.The case r = 4. Ziqin Feng at Auburn University has computed the homology of VR(Q 6 ; 4).To do so, he used the Easley Cluster at Auburn University (a system for high-performance and parallel computing), about 180 GB of memory, and the Ripser software package [7].His computations show that This computation is shown in the first r = 4 column in Table 2, and the consequences implied by this computation and by our results are shown in the second r = 4 column in that table.
denote the natural inclusion of all the Q m subcubes of Q n .Given a positive integer q, let the homomorphism (Ψ m,n ) * : be the induced map on homology.
Our previous results have provided lower bounds on the rank of maps (Ψ p,n ) * , where p is the smallest parameter for which H q (VR(Q p ; r)) is non-trivial.Our next result explains how an analogous result also holds for other maps (Ψ m,n ) * .Theorem 6.5.Let q ≥ 1.Let Note that (Ψ p−1,p ) * = 0 by the definition of p, and so we recover Theorem 6.4 by setting m = p in Theorem 6.5.
Proof.The proof proceeds by induction on m.We will actually prove The base case of the induction at m = p is Theorem 6.4, since in this case im(Ψ p−1,p ) * = 0 as H q (VR(Q p−1 ; r)) = 0.It remains to show the inductive step.Our proof is essentially the same as the proof of Theorem 6.4 applied to ρ m,n instead of to H q (VR(Q n ; r)).
The cube Q n consists of two disjoint copies of Q n−1 ; see Figure 5: • the rear one with the last coordinate 0, denoted by Q 0 n−1 , and • the front one with the last coordinate 1, denoted by Q 1 n−1 .We partition the Q m subcubes of Q n into three classes: • The ones contained in the rear Q 0 n−1 where vertices have last coordinate 0, denoted by R n .• The ones contained in the front Q 1 n−1 where vertices have last coordinate 1, denoted by F n .• The ones contained in the middle passage between them, denoted by We will prove that the following Q m subcubes of Q n induce independent homology in ρ m,n : the elements of M n (dashed cubes in Figure 5) and the elements of F n that have inductively been shown to include independent embeddings in H q (VR(Q 1 n−1 ; r))/im(Ψ m−1,n−1 ) * (bold cubes in Figure 5).The base case of the inductive process is Theorem 6.4.
The cardinality of Each such subcube has the last coordinate constantly 0. Taking a union with a copy of the same Q m−1 subcube with the last coordinates changed to 1, we obtain a Q m subcube in M n .It is apparent that all elements of M n arise this way.Let us enumerate the elements of in ρ m,n for some coefficients λ i,j , µ i,j , we claim that all coefficients equal zero.This will prove the theorem as the number of involved terms equals n i=m 2 i−m i−1 m−1 • R m .We will first prove that the coefficients λ i,j are all zero.Fix some 1 ≤ j ≤ 2 n−m n−1 m−1 , and let D be the copy of Let f be any concentration Q n → C := D × {0, 1}.(For example, in Figure 4 one can visualize D as the solid round vertex, and C as the edge between the two solid vertices.)By Proposition 5.1: • f maps all of the other Q m subcubes of Q n to lower-dimensional subcubes, and hence the map These two observations imply that applying the induced map f * on homology to Equation (3), we obtain i λ i,j • a i,j = 0.By the choice of {a i,j } i as an independent collection of homology classes, we obtain λ i,j = 0 for all i.Since this can be done for any 1 ≤ j ≤ 2 n−p n−1 p−1 , we have λ i,j = 0 for all i and j.We have thus reduced Equation (3) to i,j µ i,j • b i,j = 0. Let π S : Q n → Q n−1 be the projection that forgets the last coordinate of each vector (explicitly, S = [n − 1] ⊆ [n]).Note that the restrictions of π S to Q 0 n and to Q 1 n are bijections.Hence, after applying the induced map (π S ) * on homology, the inductive definition of the b i,j as being independent in H q (VR(Q 1 n−1 ; r))/im(Ψ m−1,n−1 ) * implies that µ i,j = 0 for all i and j. □

Geometric generators
For r ≥ 0, let f n : VR(Q n ; r) → [0, 1] n be the map defined by sending each vertex of Q n to the corresponding point in {0, 1} n ⊆ [0, 1] n , and then by extending linearly to simplices.Let n(r) ∈ {0, 1, . ..} be the smallest integer n such that f n : VR(Q n ; r) → [0, 1] n is not surjective.The fact that this is welldefined follows from Lemma 7.5, which proves that n(r) ≤ 2r + 1.
The main theorem in this section is the following.
Theorem 7.1.For all r ≥ 2, i: There exists some m ≤ n(r) such that π m−1 (VR(Q m ; r)) ̸ = 0; ii: There exists some k ≤ m such that H k−1 (VR(Q m ; r)) ̸ = 0; iii: Not all of the above non-trivial homotopy group (resp.homology group) is generated by the initial cross-polytopal spheres, i.e., the image of the induced map (Ψ r+1,m ) * of Vietoris-Rips complexes Statement ii above is true for homology taken with any choice of coefficients.
An important consequence of this theorem is the following.Together, statements i-iii imply that for each r ≥ 2, there is a new topological feature in VR(Q m ; r) that is not induced from an inclusion VR(Q r+1 ; r) → VR(Q m ; r).In Table 2, these appear as the new blue S 3 feature in VR(Q 4 ; 2) ≃ ∨ 9 S 3 , and as the new blue S 4 feature in VR(Q 5 ; 3) ≃ ∨ 10 S 7 ∨ S 4 .
In the following example, when r = 2, we see that we can take k = m = n(r).However, we do not know if this is the case in general.
We now build up towards the proof of Theorem 7.1.We will use the following notation: 1 By convention, the interior of a vertex is that vertex.and note it is a map between topological (N − 1)-spheres.Observe that ψ is subcube-preserving, i.e., ∀Q j < Q N : ψ(ConvQ j ) ⊆ ConvQ j .Map ψ : ∂C N → ∂C N is homotopic to the identity, as is demonstrated by the linear homotopy H : ∂C N × I → ∂C N , H(x, t) = (1 − t)φ(x) + tx, which is well-defined by the subcube-preserving property.As ψ is homotopically non-trivial, so is φ.Since VR(Q N ; r) is path connected, this implies π N (VR(Q N ; r)) is non-trivial regardless of which basepoint is used, giving i. ii: By the Hurewicz theorem and Lemma 7.5(3), the first non-trivial homotopy group of VR(Q m ; r) is isomorphic to the corresponding homology group with integer coefficients in the same dimension.By i, the mentioned dimension is at most m − 1.
iii: Let m ∈ {1, 2, n(r)} be the parameter from the proof of i that satisfies π m−1 VR(Q mr ; r) ̸ = 0.For r > 2 we claim that m − 1 < 2 r − 1.Indeed, m − 1 ≤ n(r) − 1 ≤ 2r < 2 r − 1 by Lemma 7.5 (2).So for r > 2, the dimension (m − 1 or lower) of the homotopy and homology groups in i and ii is lower than the dimension 2 r − 1 of the Ψ mr r+1 induced invariants, giving iii.Finally, in the case r = 2, we have m = 2 and VR(Q 4 ; 2) ≃ ∨ 9 S 3 by [4].Since Q 4 contains only 8 copies of Q 3 , the image of Ψ 4  3 induced map on π 3 is of rank at most 8; thus the claim iii follows also for r = 2. □

Conclusion and open questions
We conclude with a description of some open questions.We remind the reader of questions from [4], which ask if VR(Q n ; r) is always a wedge of spheres, what the homology groups and homotopy types of VR(Q n ; r) are for 3 ≤ r ≤ n − 2, and if VR(Q n ; r) collapses to its (2 r − 1)-skeleton (which would imply that the homology groups H q (VR(Q n ; r)) are zero for q ≥ 2 r ).Below we pose some further questions.
The first four questions are related to the geometric generators in Section 7. Understanding the answers to any of them would provide further information about parameters of Theorem 7.1.) ∈ R n .For example, the tetrahedron τ 1 in Example 7.2 is a set of four vertices that forms a balanced subset.If ( 1 2 , 1 2 , . . ., 1  2 ) is in the image of f n : VR(Q n ; r) → [0, 1] n , then does there necessarily exist a balanced subset of {0, 1} n of diameter at most r?One of the reasons we ask this question is that the answers to the StackExchange post [1] place constraints on the smallest diameter for a balanced subset of the n-dimensional cube.
The remaining questions are more general.Question 8.5.In Section 4 we described cross-polytopal homology generators.In Section 7 we described geometric homology generators.In Non-Example 7.4 we described homology generators, due to computations by Ziqin Feng, that are neither cross-polytopal in the sense of Section 4 (arising from an isometric embedding Q r+1 → Q n ) nor geometric in the sense of Section 7. What other types of homology generators are there for H q (VR(Q n ; r))? Question 8.6.Our main results show how the homology (and persistent homology) of VR(Q p ; r) for 1 ≤ p ≤ m place lower bounds on the Betti numbers of VR(Q n ; r) for all n ≥ m.For every r ≥ 1, is there some integer m(r) such that our induced lower bounds are tight for all n ≥ m(r) and for all homology dimensions?Question 8.7.The group of symmetries of the n-dimensional cube is the hyperoctahedral group.How does this group act on the homology H q (VR(Q n ; r))? Question 8.8.What homology propogation results can be proven for Čech complexes of hypercube graphs, as studied in [6]?

2. 2 .
Hypercubes.Hypercubes are among the simplest examples of product spaces.

Definition 2 . 2 .
Given a metric space X and a finite subset A ⊆ X, the diameter of A is diam(A) = max a,b∈A d(a, b).The local diameter of A at a point a ∈ A equals localDiam(A, a) = max b∈A d(a, b).
and • a ′ i = b i otherwise.Given a fixed set S, there are 2 n−p such embeddings ι S , each associated to a different offset b.Let π S : Q n → Q p be the map projecting onto the coordinates in S. Then π s • ι S = id Qp for any map ι S (i.e., for any choice of an offset b).Given an offset (b i ) i∈[n]\S , let Q b p denote the image of ι b S corresponding to the offset b, and let π b

Figure 3 .Proposition 5 . 1 .
Figure 3. Two concentrations of codimension one.In both cases there are two codimension one subcubes that are being mapped isometrically (the thick Q 1 on the left and the shaded Q 2 on the right).On the left we have n = 2, k = 1, and a 2 = 0, and on the right we have n = 3, k = 2, and a 3 = 0.

Figure 4 .
Figure 4. (Left)The projection onto the front bottom Q 1 isometrically identifies the four bold copies of Q 1 , and sends other copies of Q 1 to a point.(Right) The concentration onto the bottom left Q 1 , mapping all hollow-square vertices to the solid square vertex, isometrically identifies the three bold copies of Q 1 , and sends the other copies of Q 1 to a point.In this case the concentration is of codimension 2, i.e., mapsQ 3 → Q 1 .As the codimension t increases, the number of isometrically identified subcubes is exponential 2 t for projections and linear t + 1 for concentrations.The exponential increase leads to weaker lower bound in Theorem 6.2 than the linear increase in Theorems 6.3 and 6.4.

6. 4 . 5 . 6 . 5 .
The case of general r.In Section 7 we prove that in each column of Tables1or 2 with r ≥ 2, a new homology generator appears in VR(Q n ; r) with n ≥ r + 1, i.e., below the diagonal entry n = r + 1 where the cross-polytopal generator appears.Examples of these new homology generators are in the items (b) above for r = 2 and 3. Propagation of non-initial homology.For positive integers m < n let Ψ m,n :