On homology theories of cubical digraphs

We prove the equivalence of the singular cubical homology and the path homology on the category of cubical digraphs. As a corollary we obtain new relations between the singular cubical homology of digraphs and simplicial homology.


Introduction
The path homology theory and the singular cubical homology theory for the category of digraphs were introduced recently in the papers [1], [2], [3], and [4].In this category, there is a natural transformation of the cubical homology theory to the path homology theory, that induces an isomorphism of homology groups in dimensions 0 and 1.Additionally, in [1] is given an example of a digraph for which the path homology are trivial in dimension 2 but singular cubical homology are non-trivial in this dimension.
In this paper we prove the equivalence of the singular cubical homology and the path homology theories on the category of cubical digraphs.As an intermediate result we prove that the image of every map of a digraph cube to a cubical digraph is contractible.As a corollary we obtain a relation of the singular cubical homology of digraphs to simplicial homology.
The paper is organized as follows.In Section 2, we recall the basic definitions from graph theory and describe some properties of singular cubical homology H c * and the path homology H p * on the category of digraphs [1], [2], [3], and [4].
In Section 3, we recall the definition of cubical digraph from [4] and prove contractibility of the image of a digraph cube in a cubical digraph for any digraph map.Then we state and prove the main result of the paper: Theorem 1.1.On the category of cubical digraphs the singular cubical homology theory is equivalent to the path homology theory.
Then we obtain several corollaries that describe relation of the singular cubical homology theory of digraphs to simplicial homology.

Singular cubical and path homology theories
In this Section we give necessary preliminary material about digraphs and homology theories on the category of digraphs.We shall consider only finite digraphs in the paper.For two vertices v, w ∈ V G , we write v − → = w if either v = w or v → w.
A subgraph H of a digraph G is a digraph whose set of vertices is a subset of that of G and set edges of H is the subset of edges of G.In this case we write G ⊂ H.
An induced subgraph H of a digraph G is a digraph whose set of vertices is a subset of that of G and the edges of H are all those edges of G whose adjacent vertices belong to H.

In this case we write G H.
A directed path p = (a 1 , α 1 , a 2 , α 2 , . . ., α n , a n+1 ) in a digraph G is a sequence of vertices a i and arrows α i such that α i = (a i → a i+1 ).The number of arrows fitting in path is called length of the path and is denoted by |p|.The vertex a 1 is the origin of the path and the vertex a n+1 is the end of the path.
The set of all digraphs with digraph maps form the category of digraphs that will be denoted by D. Definition 2.3.For digraphs G, H define their Box product Π = G H as a digraph with a set of vertices V Π = V G × V H and a set of arrows E Π given by the rule (x, y) → (x , y ) if x = x and y → y , or x → x and y = y , where x, x ∈ V G and y, y ∈ V H .
Fix n ≥ 0. Denote by I n a digraph with the set of vertices V = {0, 1, . . ., n} and, for i = 0, 1, . . .n − 1, there is exactly one arrow i → i + 1 or i + 1 → i and there are no others arrows.Such digraph we call a line digraph and a direct line digraph if additionally all arrow have the form i → i + 1.There are only two line digraphs with two vertices.We denote the digraph 0 → 1 by I.
For n ≥ 0, a standard n-cube digraph I n is defined as follows.For n = 0 we put I 0 = {0} -one-vertex digraph.For n ≥ 1, I n is given by a set V of 2 n vertices such that any vertex a ∈ V can be identified with a sequence a = (a 1 , . . ., a n ) of binary digits so that a → b if and only if the sequence b = (b 1 , . . ., b n ) is obtained from a = (a 1 , . . ., a n ) by replacing a digit 0 by 1 at exactly one position.The digraph 0 → 1 is an 1-cube and we call a square any digraph that is isomorphic the standard 2-cube digraph.
We shall call an n-cube digraph any digraph that is isomorphic to the standard n-cube.Note an n-cube digraph is isomorphic to the digraph The notion of homotopy in the category of digraphs was introduced in [2].Now we recall several definitions which we shall use in the paper.Definition 2.4.Two digraph maps f, g : G → H are called homotopic if there exists a line digraph I n with n ≥ 1 and a digraph map where we identify G {0} and G {n} with G by the natural way.In this case we shall write f g.The map F is called a homotopy between f and g.
In the case n = 1 we refer to the map F as an one-step homotopy.
Definition 2.5.Digraphs G and H are called homotopy equivalent if there exist maps In Then r is a deformation retraction, the digraphs G and H are homotopy equivalent, and i, r are their homotopy inverses.
Now we recall the definitions of path homology groups from [4] and singular cubical homology groups from [1] on digraphs with the group of coefficients Z.Let V be a finite set, whose elements will be called vertices.An elementary p-path on a finite set V is any (ordered) sequence i 0 , ..., i p of p + 1 vertices of V that will be denoted by e i 0 ...ip .Denote by Λ p = Λ p (V ) the free abelian group generated by all elementary p-paths e i 0 ...ip .The elements of Λ p are called p-paths.Thus each p-path v ∈ Λ p has the form where v i 0 i 1 ...ip ∈ Z are the coefficients of v.
An elementary p-path e i 0 ...i p (p ≥ 1) is called regular if i k = i k+1 for all k.For p ≥ 1, let I p be the subgroup of Λ p that is spanned by all irregular e i 0 ...i p and we set and with the chain map that is induced by ∂.
Now we define paths on a digraph G = (V, E).Let e i 0 ...i p be a regular elementary p-path on V .It is called allowed if i k−1 → i k for any k = 1, ..., p, and non-allowed otherwise.For p ≥ 1, denote by A p = A p (G) the subgroup of R p spanned by the allowed elementary p-paths, that is, A p = span e i 0 ...ip : i 0 ...i p is allowed .
and set A −1 = 0.The elements of A p are called allowed p-paths.
Consider the following subgroup of A p (p ≥ 0) The elements of Ω p are called ∂-invariant p-paths, and we obtain a chain complex The homology groups of the digraph G are defined as In what follows, we will refer to H p (G) as the path homology groups of a digraph G.
We can define a natural augmentation which is an epimorphism and ε • ∂ = 0. Now we recall the construction of the cubical singular homology theory of digraphs from [1]. Let Fix n ≥ 1.For any 1 ≤ j ≤ n and = 0, 1, consider the following inclusion of digraphs: for n ≥ 2, and the free abelian group generated by all singular n-cubes in G, and denote φ the singular n-cube φ as the element of the group Q n .For n ≥ 1 and 1 ≤ p ≤ n, and (2.6) For n ≥ 1, define a homomorphism ∂ c : Q n → Q n−1 on the basis elements φ by the rule and ∂ c = 0 for n = 0. Then (∂ c ) 2 = 0 and the groups Q n (G) form a chain complex which we denote For n ≥ 1 and 1 ≤ p ≤ n, consider the natural projection T p : I n → I n−1 on the p-face I n−1 defined as follows.For n = 1, T 1 is the unique digraph map I 1 → I 0 .For n ≥ 2, we have on the set of vertices T p (i n , . . ., i 1 ) = (i n , . . ., i p+1 , i p−1 , . . ., i 1 ).The singular n-cube φ : which is an epimorphism and ε Recall the basic properties of the path and the singular cubical homology groups (see [ 4] and [1]).
• The groups H c * (X) and H * (X) are functors from the category D to the category of abelian groups.
• Let f g : X → Y be two homotopic digraph maps.Then the induced homomorphisms f * , g * of homology groups are equal for k ≥ 0 for both theories.

Maps of cube to cubical digraph
In this section we reformulate slightly the definition of a cubical digraph from [4] and prove that the image of a cube in a cubical digraph is contractible.Then we prove Theorem 1.1.
Recall, that any vertex of a a cube I n is given by a sequence of binary numbers (a 1 , . . ., a n ).For any arrow a → b in a digraph cube I n we have also the arrow in I n where right sequence of binary numbers presents a vertex in I n which has only one non-trivial element 1 on a place i.We say that two arrows α = (a → b) and β = (c → d) of I n are parallel and write α||β if In the opposite case we shall call two arrows orthogonal.
An arrow α ∈ E I n defines two (n − 1) -faces of I n : the face I 0 = I α 0 that contains origin vertices of the arrows that are parallel to α and the face I 1 = I α 1 that contains end vertices of the arrows that are parallel to α.Note that any arrow that is orthogonal to α lies in I 0 or in I 1 .
For the digraph cube I m there is a natural partial order on the set of its vertices V I m that is defined as follows: we write a ≤ b if there exists a directed path with the origin vertex Note that the set of all paths from a to b in I a,b coincides with the set of all paths from a to b in G.It is easy to see that cubical digraphs with digraph maps form a category.Now we prove that the image of a cube I n in any cubical digraph is contractible.Note, that this statement is not true in general case.Proof.The image f (I n ) is connected as the image of the connected graph.Let s = (0, . . ., 0) ∈ V I n be the origin vertex and z = (1, . . ., 1) ∈ V I n be the end vertex of is isomorphic to a m-dimensional cube which we denote J = J m ∼ = I m where m = Δ(f (s), f (z)).Hence, without loss of generality, we can suppose that G = I f (s),f (z) = J that is f (s) = (0, . . ., 0) ∈ V J , f (z) = d = (1, . . ., 1) ∈ V J .We prove the statement of the Theorem using induction on dimension m.
The base of induction by m.For m = 0, 1, 2 the statement is trivial since any connected subgraph of the digraphs J 0 , J 1 , and J 2 is contractible.
The step of induction by m.Suppose that the statement of the Theorem is proved for every map I n → J m−1 .Consider the case J = J m where m ≥ 3 and d = (1, . . ., 1) ∈ V J is the end vertex of the cube J. Since d = f (z) ∈ Image(f ), there exists a nonempty set of arrows Γ ⊂ E J defined as follows The set Γ consists of arrows in E J with the end vertex d that are lying in the image of the map f .Let γ = (c → d) ∈ Γ be an arrow such that Note that α is defined may be by a non unique way.For for ease of references we formulae the following result.Proof.Follows immediately from definition of k in (3.2).The arrow γ defines two (m − 1) -dimensional faces J 0 and J 1 of the cube J with c =∈ V J 0 , d ∈ V J 1 and we have the natural projection π : J → J 0 along the arrow γ.Let H be a subgraph of I n .We define subgraphs K 0 , K 1 , K ⊂ J which depend on the map f : I n → J and H ⊂ I n as follows: It is easy to see that for an arrow (v → w) ∈ E J we have: For technical reasons we introduce the following definition.
2), and the digraphs K, K 0 , K 1 ⊂ J are defined in (3.3).We say that a subgraph H ⊂ I n satisfies to the Π -condition if the following properties are satisfied defines (n − 1)-face I 0 = I s,x and opposite (n-1)-face I 1 of the cube I n .Let a = (0, . . ., 0) be the origin vertex of J (and hence origin vertex of J 0 ) and b be the origin vertex of J 1 .Then a → b is parallel γ = (c → d).We have and, hence, by (3.3) for H = I n , we have f (I 0 ) ⊂ K 0 .Let t be a vertex of There exists an unique vertex r The same line of arguments as above gives the vertices r, r ∈ V I 0 such that (r → t) and r → t are parallel to α and, hence, π(τ This proves condition (2) of (3.5).Thus Π-condition is satisfied for the cube I n and k = 0.
The induction step.By inductive assumption we have that any map f : Thus, without loss of generality, we can suppose that (3.7) From now we put y 0 = y ∈ V I n and let the vertex y i is obtained from y by replacing the last coordinate "1" in y by "0", and i-th coordinate "0" of y by "1" for 1 ≤ i ≤ k.For example, We define also Then, as before, f (I 0 ) ⊂ K 0 ⊂ J 0 .Consider a vertex t ∈ V I n and t / ∈ V I 0 that has the form where at least one coordinate b j is "1".If at least one coordinate b j is zero we obtain that t ∈ I s,z j I n where ).
The (n−1)-dimensional subcube I s,z j ⊂ I n contains the vertices x and t.Moreover Δ(x, z j ) = k and there is an arrow Hence, by the inductive assumption, the map f | Is,z j : I s,z j → J satisfies the Π-condition.Now consider a vertex t for which all (k + 1)-coordinates b j are equal "1" such that t / ∈ I x,z .This means that at least one of the first (n − k − 1)-coordinates a i is "0".Recall that (k + 1) ≥ 2. Thus consider the vertices where a i ∈ {0, 1}.Consider a directed path p in the digraph I 0 from the vertex r ∈ V I 0 to the vertex x ∈ V I 0 of the length l = |p| ≥ 1 (since t / ∈ I x,z ).Write this path in the following form p = (r → x 1 → x 2 → . . .x l−1 → x l = x) ⊂ I r,x ⊂ I 0 .
Consider a directed path q from the vertex r ∈ V I 0 to the vertex t of the length k+1 = |q| ≥ 2. Note that q lies in the digraph I r,t of dimension k + 1. Write this path in the following form Any such two paths p and q defines an unique subgraph of the graph I n that has the following form Now we prove, using induction in the length l = |q| ≥ 1 the following statement.(L): For every path q and every path p, as above, there is a path (that may be is equal to p) such that q and p defines the subgraph (similarly above) and at least one of the following conditions is satisfied (3.11) The base of induction for (L), the case l = 1.Consider the unique path p = (r → x) The subcube of I n that is defined by the digraph on (3.13). path .9).This finishes the proof of the inductive step and, statement (L) is proved.
It follows from the statement (L) that image w = f (t) and images of all arrows with end or origin t lay in the image of the subcube I r,z j with Δ(x, z j ) = Δ(r, z j ) = k which satisfies to Π-condition by the inductive assumption in k.Hence the cube I n satisfies to Π-condition and the Proposition is proved.Now we finish the proof of Theorem 3.3.Since digraph I n satisfies the Π-condition then Proposition 2.7 and (3.5) implies that restriction π| K of the projection π : J m → J m−1 0 to the image K of the map f is well defined deformation retraction to K 0 .But K 0 is contractible by the inductive assumption in m.Thus Theorem 3.3 is proved.

Equivalence of homology theories on cubical digraphs
In this section we prove our main result -Theorem 1.1, that is stated below as Theorem 4.5.For that we use the Acyclic Carrier Theorem from homology theory (see, for example, [5, §3.4] and [6, §1.2.1]).Recall that a chain complex C * is called non-negative if C p = 0 for p < 0 and is called free if C p are finitely generated free abelian groups for all p.We say that C * is a geometric chain complex if it is non-negative, free, and if a basis B p is chosen in the group C p for any p ≥ 0. For example, any finite simplicial complex gives rise to a geometric chain complex, where B p consists of all p-simplexes.
Let C * be a geometric chain complex with fixed bases B p .For b ∈ B p−1 and b ∈ B p , we write b ≺ b if b enters with a non-zero coefficient into the expansion of ∂b in the basis B p−1 .The augmentation homomorphism ε : C 0 → Z is defined as We have only two different possibilities for the φ * (w n ).In the first case, φ is an isomorphism on its image G φ = I s,t ∼ = I n with s = φ(0, . . ., 0), t = φ(1, . . ., 1) where (0, . . ., 0) ∈ V I n is the origin vertex, and (1, . . ., 1) ∈ V I n is the end vertex of the cube I n .Note that for any isomorphism ψ : I n → I n we have ψ * (w n ) = ±w n .Hence in this case subgraph G φ ⊂ G coincides with the subgraph cube G χ s,t ⊂ G and by (4.4) we have where χ s,t : I n → D s,t = G φ .That is In the second case, the image of φ does not contain any cube of dimension n and, hence φ * (w n ) = 0. Consequently, we have θ n • φ * (w n ) = 0 ∈ E * (φ ).Then the claim follows from the Acyclic Carriers Theorem 4.2.
Theorem 4.5.For any finite cubical digraph G, the chain maps τ * and θ * are homotopy inverses and, hence, induce isomorphisms of homology groups Proof.It follows from Propositions 4.3 and 4.4 that the chain maps τ * and θ * are homotopy inverses.Now the statement of the Theorem follows.
Corollary 4.6.Let Δ be a finite simplicial complex.Consider a digraph G Δ (see [4]) with the set of vertices given by the set of all simplexes from Δ, and s → t (t, s ∈ Δ) iff s ⊃ t and dim s = dim t + 1.
Then the graph G Δ is a cubical digraph and where H * (Δ) are the simplicial homology groups of Δ.
Proof.Indeed, it is proved in [4] that path homology groups H * (G Δ ) are isomorphic to simplicial homology groups H * (Δ).
of ordered pairs (v, w) of vertices which are called arrows and are denoted v → w.The vertex v = orig (v → w) is called the origin of the arrow and the vertex w = end(v → w) is called the end of the arrow.
a and the end vertex b.Now we introduce a distance Δ(a, b) for a pair of vertex a, b ∈ I n that is defined only for comparable pair of vertices.Let a ≤ b be two vertices then as follows from definition of the cube digraph the length of the path p from a to b does not depend on the choice of the path, and we put Δ(a, b) = Δ(b, a) : = |p|.We shall call the vertex a = (0, . . ., 0) of a cube origin vertex and the vertex d = (1, . . ., 1) end vertex.It follows immediately from the definition of a cube digraph that the for any vertex x the distances Δ(a, x) and Δ(x, d) are well defined.For an arrow α = (x → y) we define Δ(α, d) : = Δ(y, d) where d is end vertex of the cube.Let a ≤ b be a pair of comparable vertices of I n for which there is a direct path p from a to b. Denote by I a,b induced subgraph of I n with the set of vertices {c ∈ V I n |a ≤ c ≤ b}.Clearly, I a,b is isomorphic to a digraph cube I k , where k = |p| = Δ(a, b).

Example 3 . 2 .Figure 1 :Theorem 3 . 3 .
Figure 1: The map f : I 3 → G with non-contractible image.Theorem 3.3.Let f : I n → G be a digraph map to a cubical digraph.Then the image f (I n ) ⊂ G is contractible.

Lemma 3 . 4 .
For every vertex v ∈ V I n with Δ(v, z) ≤ k we have f (v) = d.Hence the cube I y,z I n is mapped by f into the vertex d.

. 5 ) 3 . 6 .
Proposition Consider the map f : I n → J = J m with m ≥ 3. Let k and γ are defined in (3.2) and let us consider the same designations as above.Then the cube I n satisfies to Π-condition.Proof.Induction in k ≥ 0. The base of induction, k = 0. Hence y = z = (1, . . ., 1) ∈ V I n is the end vertex of I n and n ≥ m ≥ 3. The arrow α complex C * is called acyclic if all homology groups of the augmented complex C * are trivial.Let C * and D * be two geometric complexes with augmentation homomorphism ε and ε , respectively.A chain map φ * :C * → D * is called augmentation preserving if ε φ 0 (c) = ε(c) for any c ∈ C 0 .withcoefficients (±1) where the maps V p : I n−1 → I n are the inclusions.Hence the digraph G φ•Vp is a subgraph of G φ and, hence, the chain complex E * (φ• V p ) = Ω c * G φ•Vp is a subcomplex of E * (φ ).Thus for the basic singular cube b ∈ Ω c n−1 (G) and b ≺ φ we obtain that b = (φ • V p ) E * (b) = E * (φ • V p ) ≺ E * (φ ).Hence we have the algebraic acyclic carrier function E from Ω c * (G) to itself.Now we prove, that the chain maps θ * • τ * and Id from Ω c * (G) to itself are carried by the function E. Consider a basic element φ ∈ Ω c n (G).Then Id φ ∈ φ ∈ Ω c * (G φ ) = E * (φ ) (4.11) since image of φ is the digraph G φ .Hence the chain map Id : Ω c n (G) → Ω c n (G) is carried by the algebraic carrier function E. By (4.3) and (4.4), we have θ n • τ n φ = θ n (φ * (w n )) , φ : I n → G. (4.12) this case we shall write H G and the maps f and g are called homotopy inverses of each other.A digraph G is called contractible if G { * } where { * } is a one-vertex digraph.