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Homology of transitive digraphs and discrete spaces Yu. V. Muranova, R. Jimenezb a University of Warmia and Mazury in Olsztyn, Olsztyn, Poland b Institute of Mathematics, National Autonomous University of Mexico, Oaxaca, Mexico
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 8, DOI: https://doi.org/10.4213/sm9842 
Bibliographic databases:
    § 1. IntroductionTransferring results of differential geometry and topology to discrete objects and, in particular, to the category of digraphs became a topical problem recently. This is related to problems in noncommutative geometry and the development of discrete methods in problems of mathematical physics; see [1]–[3]. One of the main research tools is the theory of differential forms on the algebra of functions defined on discrete sets. The corresponding cohomology theory on the category of digraphs was developed in [4]. The cohomology of digraphs is closely related to the cohomology of simplicial complexes and Hochschild cohomology; see [5]–[7]. In [5] a functor assigning an associative algebra with unity $A(S)$ to a locally finite simplex $S$ was constructed, and this functor was proved to induce a multiplication-preserving isomorphism between the simplicial cohomology of $S$ and the Hochschild cohomology of $A(S)$. The use of digraph cohomology gives, in particular, a new proof of an isomorphism between the cohomology of a simplicial complex $S$ and the Hochschild cohomology of the algebra $A(S)$: see [7]. Subsequently (see [8] and [9]), a theory of path homology on arbitrary discrete sets was developed which, on the category of digraphs, is dual to the cohomology theory in [4]. Path homology has many properties similar to classical homology theories, including functoriality and homotopy invariance (see [10] and [11]); however, this theory has no analogues in classical algebraic topology. On the category of digraphs, a homotopy invariant and functorial theory of singular cubic homology has been developed, which is similar to the singular cubic homology theory of topological spaces (see [12]). Singular cubic homology is constructed using digraphs-cubes, which are analogues of topological cubes. On the category of digraphs singular cubic homology is not isomorphic to path homology in the general case. An example of such a digraph and the calculation of its homology were presented in [12], Propositions 12 and 13. Despite the fact that path homology has no direct analogues in continuous algebraic topology, it is closely related to the Alexandroff homology of discrete topological spaces; see [13] and [14]. The discrete topological spaces defined by Alexandroff in [14] carry the natural structure of partially ordered sets on the set of their elements: see [15], §§ 4.2 and 6.2. Since partially ordered sets are naturally identified with transitive digraphs (see [16] and [17]), we obtain an equivalence between discrete topological spaces and transitive digraphs. In view of this equivalence Alexandroff’s homology theory of discrete topological spaces is carried over to transitive digraphs. Moreover, as shown in [13], on the category of transitive digraphs path homology coincides with Alexandroff homology. Note also that the Alexandroff homology of a discrete space $X$ is realized as the simplicial homology of a simplicial complex defined functorially for the space $X$ (see [18]). Subsequently, algebraic topology on discrete spaces was developed by many authors (see, for example, [19]–[21]). In this paper we prove that on the category of transitive digraphs singular cubic homology coincides with path homology (and therefore, with Alexandroff homology). We also develop as follows new theories of singular cubic homology on the category of digraphs and the category of discrete spaces. Every digraph cube can uniquely be completed to a transitive digraph having a natural cube structure. Thus, we obtain a new singular digraph homology theory based on such transitive digraph cubes. It follows from the definition of a completed cube that the resulting singular homologies on transitive digraphs coincide with the singular homologies defined previously. However, in the general case these singular homology groups are different, as follows from an example given in our paper. Using the equivalence between the transitive digraphs and the discrete topological spaces, we obtain the definition of a discrete topological space representing an $n$-dimensional cube in the category of discrete topological spaces. This implies a natural definition of singular cubic homology in the category of discrete topological spaces which is based on cubes that are discrete spaces. We use the notation of [12]. The needed information about simplicial complexes and their homologies can be found in [22]–[24]. § 2. Discrete spaces, partially ordered sets and digraphsIn this section we recall the main definitions used in this paper (see [14]–[17]). Definition 1. By a discrete topological space we mean a topological space with the separability axiom $T_0$, so that for any two distinct points $x, y\in X$ at least one of them has a neighborhood not containing another point, in which the intersection of any family of open sets is open. Definition 2. A set $X$ is said to be partially ordered if the set of its elements is equipped with an order relation $x< y$, which is transitive ($x<y$ and $y<z$ imply that $x< z$), asymmetric (relations $x<y$ and $y<x$ are incompatible), and anti-reflexive (the relation $x<x$ is inconsistent). A partially ordered set is locally finite if for every $x\in X$ each of the sets $\{y\in X\mid x<y\}$ and $\{y\in X\mid y<x\}$ is finite. Recall how the equivalence between the discrete spaces and the partially ordered sets is established (see [14], § 3.2). Given two different points $x$ and $y$ in a discrete space $X$, we set $x<y$ if and only if $x$ lies in the closure of the set consisting of the point $y$. Conversely, a partially ordered set $X$ defines a discrete space in the following way. Given a point $y\in X$, we define the closure $\overline{\{y\}}$ of the set $\{y\}$ by
$$
\begin{equation*}
\overline{\{y\}}=\{x\in X\mid x=y\ \text{or}\ x< y\}.
\end{equation*}
\notag
$$
The closure of an arbitrary nonempty subset $M$ of $X$ is defined as the union of the closures of its points, and the closure of an empty set is an empty set. Note that, according to Definition 1, the union of any family of closed sets in a discrete topological space is closed. In this paper we consider only locally finite partially ordered sets and the corresponding locally finite discrete spaces. Definition 3. A digraph $G=(V_{G},E_{G})$ is defined as the set $V_G$, whose elements are called vertices, and a subset $E_{G}{\subset}\{V_{G}\,{\times}\, V_{G}\}$ of ordered pairs $(x,y)=x\to y$ of different vertices, which are called oriented edges. A digraph $G=(V_G,E_G)$ is said to be transitive if the condition $((x\to y)\in E_G)\,\&\, ((y\to z)\in E_G)$ implies that $(x\to z)\in E_G$. A digraph is said to be locally finite if for every vertex $x\in V_G$ both $\{y\in V_G\mid (x\to y)\in E_G\}$ and $\{y\in V_G\mid (y\to x)\in E_G\}$ are finite sets. Every partially ordered set $(X, <)$ can be regarded as a transitive digraph $G=(V_G, E_G)$ in which the vertex set $V_G$ coincides with $X$ and $(x\to y)\in E_G$ if $x<y$. Conversely, every transitive digraph $G=(V_G, E_G)$ can be regarded as a partially ordered set $(V_G, <)$, where $x<y$ if $(x\to y)\in E_G$. This correspondence and the equivalence between discrete spaces and partially ordered sets defined above (see [14], § 3.2) define an equivalence between the locally finite discrete spaces and the locally finite transitive digraphs. In our paper we use the terminology of digraphs in all cases when the transition to discrete spaces or partially ordered sets follows directly from the results in this section. § 3. Singular homology and path homology of digraphsIn this section we recall the definitions of the singular cubic homology and path homology of digraphs (see [8]–[12]). Definition 4. For two digraphs $G$ and $H$ their direct product $\Pi =G\,\Box\, H$ is a digraph with vertex set $V_{\Pi}=V_{G}\times V_{H}$ and the set of oriented edges specified by the condition that $[(x,y)\to (x',y') ] \in E_{\Pi}$ if and only if $x=x'$ and $y\to y'$ or $x\to x'$ and $y=y'$. We denote the digraph consisting of the single vertex 0 by $I^0$. Consider the digraph $I=(0\to 1)$ and, for $n\geqslant 1$, define the $n$-dimensional graph cube by $I^n =\underbrace{I\Box \dots \Box I}_{n}$. Note that for $n\geqslant 1$ every vertex of $I^n$ is specified by a sequence of zeros and ones of length $n$. An oriented edge $a\to b$ in such a cube exists only when the the vertex $b$ is obtained from $a$ by replacing ‘0’ by ‘1’ in precisely one position. For an arbitrary digraph $G$ the singular cubic homology groups $H_*^{\mathrm{c}}(G,R)$ with coefficients in a unitary ring $R$ were defined in [12] using the cubes $I^n$. Recall the definition of a map in the category of digraphs. Definition 5. Let $G$ and $H$ be two digraphs. A map of digraphs $f\colon G\to H$ is a map of the vertices $f\colon V_{G}\to V_{H}$ for which the condition $(v\to w)\in E_G$ implies that $f(v)=f(w)$ or $(f(v)\to f(w))\in E_H$. For $n\geqslant 0$, by a singular $n$-dimensional cube in a digraph $G$ we mean an arbitrary map $\phi\colon I^n\to G$. Let us define some $R$-modules. Let $Q_{-1}=0$, and let $Q_n=Q_n(G)$ for $n\geqslant 0$ be the free $R$-module whose generators are provided by the singular $n$-dimensional cubes in $G$. Given a singular $n$-cube $\phi$, we denote the generator of the module $Q_n$ corresponding $\phi$ by $\phi^{\Box}$. We define embeddings $F_{1\varepsilon}^{0}(0)\colon I^0\!\to\! I^1$ on the vertices by the formula ${F_{1\varepsilon}^{0}(0)\!=\!(\varepsilon)}$. For $n\geqslant 2$, $1\leqslant j\leqslant n$, and $\varepsilon =0,1$ we define embeddings $F_{j\varepsilon}^{n-1}\colon I^{n-1}\to I^{n}$ by the formula
$$
\begin{equation}
F_{j\varepsilon}^{n-1}(c_{1},\dots ,c_{n-1})=(c_{1},\dots ,c_{j-1},\varepsilon ,c_{j},\dots ,c_{n-1}).
\end{equation}
\tag{3.1}
$$
In what follows we write $F_{j\varepsilon}$ instead of $F_{j\varepsilon }^{n-1}$ if the dimension is clear from the context. We denote by $I_{j\varepsilon}=I_{j\varepsilon}^{n-1}$ the $(n-1)$-dimensional face that is the image of the morphism $F_{j\varepsilon}^{n-1} $. For any singular $n$-cube $\phi\colon I^n\to G$, $n\geqslant 1$, singular cubes $\phi_{j\varepsilon}$ of dimension $n-1$, where $1\leqslant j\leqslant n$ and $\varepsilon =0,1$, are defined by
$$
\begin{equation}
\phi_{j\varepsilon}=\phi_{j\varepsilon}^{n-1}= ( \phi \circ F_{j\varepsilon})\colon I^{n-1}\to G.
\end{equation}
\tag{3.2}
$$
Let $\partial^{\mathrm{c}}=0\colon Q_0\to Q_{-1}$. For $n\geqslant 1$ we define a homomorphism $\partial^{\mathrm{c}}\colon Q_{n}\to Q_{n-1}$ on the basis elements $\phi^{\Box}$ by
$$
\begin{equation}
\partial^{\mathrm{c}}(\phi^{\Box})=\sum_{j=1}^{n}(-1)^{j}( \phi_{j0}^{\Box }-\phi_{j1}^{\Box}).
\end{equation}
\tag{3.3}
$$
Let $T^j\colon I^n\to I^{n-1}$, $n\geqslant 1$, $1\leqslant j \leqslant n$, denote the standard projection onto the $j$th face. A singular $n$-cube $\phi\colon I^n\to G$, $n\geqslant 1$, is said to be degenerate if there exist an $(n-1)$-dimensional singular cube $\psi\colon I^{n-1}\to G$ and a projection $T^j\colon I^n\to I^{n-1}$ such that $\phi=\psi \circ T^j$. We obtain the subcomplex $B_*(G)\subset Q_*(G)$ generated by the degenerate singular cubes and the quotient complex $\Omega_{\ast}^{\mathrm{c}}(G)= Q_{\ast}(G)/B_{\ast}(G)$. The homology group $H_{k}(\Omega_{\ast}^{\mathrm{c}}(G))$ is called the singular cubic homology group of dimension $k$ of the digraph $G$ and is denoted by $H_{k}^{\mathrm{c}}(G)= H_{k}^{\mathrm{c}}(G,R) $. By adding oriented edges, the digraph $I^n$ can be extended to a transitive cubic digraph, which we denote by $\widehat I^n$. The embedding of digraphs $F_{j\varepsilon}^{n-1}\colon I^{n-1}\to I^{n}$, defined for $1\leqslant j\leqslant n$ and $\varepsilon =0,1$, continues to an embedding
$$
\begin{equation*}
\widehat F_{j\varepsilon}^{n-1}\colon \widehat I^{n-1}\to \widehat I^{n}.
\end{equation*}
\notag
$$
Given a digraph $G$ and $n\geqslant 0$, we denote by $\widehat Q_{n}=\widehat Q_{n}(G)$ the free $R$-module generated by all morphisms $\widehat I^n\to G$, which we call singular transitive $n$-cubes. Let $\widehat{\phi}^{\,\Box}$ be the generator of the module $\widehat Q_{n}$ given by the morphism $\widehat{\phi}$. We write $\widehat Q_{-1}=0$. The homomorphism $\widehat {\partial}^{\mathrm{c}}\colon \widehat Q_{n}\to \widehat Q_{n-1}$ is defined on the basis elements $\widehat{\phi}^{\,\Box}$ of the module $\widehat Q_n$ by using the embeddings of faces $\widehat F_{j\varepsilon}$ in the standard way; see [23]. Thus, a chain complex $\widehat Q_{\ast}(G)$ is well defined. For $n\geqslant 1$ the projection $T^j\colon I^n\to I^{n-1}$ onto the $j$th face extends to a map $\widehat T^j\colon \widehat I^n\to \widehat I^{n-1}$. Now the concept of a degenerate transitive $n$-cube is defined similarly to [23], and we obtain the subcomplex $\widehat B_*(G)\subset \widehat Q_*(G)$ generated by degenerate transitive cubes and the quotient complex $\widehat{\Omega}_{\ast}^{\mathrm{c}}(G)=\widehat Q_{\ast }(G)/\widehat B_{\ast}(G)$. The homology group $H_{k}(\widehat{\Omega}_{\ast}^{\mathrm{c}}(G))$ is called the singular transitive cubic homology group of dimension $k$ of the digraph $G$; it is denoted by $\widehat H_{k}^{\mathrm{c}}(G)=\widehat H_{k}^{\mathrm{c}}(G,R) $. Proposition 1. Let $G$ be a transitive digraph. Then the homology groups $\widehat H_{*}^{\mathrm{c}}(G)$ and $H_{*}^{\mathrm{c}}(G)$ are naturally isomorphic for any coefficient ring. Proof. The operations of projection onto a face and embedding a face are compatible for the transitive cube $\widehat I^n$ and its subgraph $I^n$. For an arbitrary transitive digraph $G$ there is a one-to-one correspondence between the singular cubes $\varphi\colon I^n\to G$ and the transitive singular cubes. This correspondence is natural with respect to morphisms of transitive digraphs. This completes the proof of the proposition. In the following example we use the concept of homotopy in the category of digraphs: see [10] and [11]. For $n\geqslant 0$ we define a linear digraph $I_{n}=(V_{I_n}, G_{I_n})$ as follows. Let $V_{I_n}=\{0,1,\dots,n\}$, and assume that for every $i=0,1,\dots, n-1$ there is exactly one oriented edge $i\to i+1$ or $i+1\to i$. Then, in particular, $I$ is the linear digraph $0\to 1$. Two digraph maps $f,g\colon G\to H$ are said to be homotopy equivalent if there exists a linear digraph $I_{n}$ and a map $F\colon G\,{\Box}\, I_{n}\to H$ such that $F|_{G\Box \{0\}}=f$ and $F|_{G\Box \{n\}}=g$, where $G\,{\Box}\, \{i\} $ is identified with $G$. In this case we write $f\simeq g$, and the map $F$ is called a homotopy between $f$ and $g$. A digraph is said to be contractible if it is homotopy equivalent to the digraph consisting of a single vertex. Now we present an example of a digraph $G$ for which the homology groups $\widehat{H}^{\mathrm{c}}_*(G)$ and ${H}^{\mathrm{c}}_*(G)$ are different. Examples 1. Let $R=\mathbb Z$. Consider the digraph $G\cong I^2$ of the form In this case, according to [10], Example 3.12, the digraph $G$ is contractible, ${{H}^{\mathrm{c}}_0(G)=\mathbb Z}$, and the other homology groups ${H}^{\mathrm{c}}_i(G)$ are trivial. Let us calculate the group $\widehat{H}^{\mathrm{c}}_1(G)$. Let $\psi_i$, $i=0,1,2,3$, denote the singular nondegenerate zero-dimensional cube $\psi_i(0)=i\in V_G$, $0\in V_{\widehat{I}^0}$. Then There are only four nondegenerate maps $\phi_1$, $\phi_2$, $\phi_3$ and $\phi_4$ of the digraph $\widehat{I}^1=I^1=(0\to 1)$ into $G$. On the vertex set they are specified as follows: $\phi_1(0)=0$, $\phi_1(1)=1$; $\phi_2(0)=0$, $\phi_2(1)=2$; $\phi_3(0)=1$, $\phi_3(1)=3$; $\phi_4(0)=2$ and $\phi_4(1)=3$. Therefore, $\widehat{\Omega}^{\mathrm{c}}_1(G)=\langle \phi_1^{\Box}, \phi_2^{\Box},\phi_3^{\Box},\phi_4^{\Box}\rangle$. Let $a=\phi_1^{\Box}-\phi_2^{\Box}+ \phi_3^{\Box}-\phi_4^{\Box}$. Then we obtain It can readily be seen that the kernel $\operatorname{Ker}[ \partial^1 \colon \widehat{\Omega}^{\mathrm{c}}_1(G)\to \widehat{\Omega}^{\mathrm{c}}_0(G)]$ is generated by the element $a$, that is, $\operatorname{Ker} \partial^1=\mathbb Z$. Consider the transitive digraph square $\widehat{I}^2$ presented below: If the restriction of $\gamma\colon \widehat{I}^2\to G$ to the oriented edge $(00\to 11)$ is a degenerate map, then $\gamma$ is degenerate. Otherwise, just one of the following conditions holds:1) $\gamma(00\to 01)= \gamma(00\to 10)=\gamma(00\to 11)$, $\gamma(01\to 11)= \gamma(10\to 11)=\gamma(11)$; 2) $\gamma(10\to 11)= \gamma(01\to 11)=\gamma(00\to 11)$, $\gamma(00\to 01)= \gamma(00\to 10)=\gamma(11)$. It follows now from the definition of the differential that $\partial^2=0$. Hence $\widehat{H}_1^{\mathrm{c}}(G, \mathbb Z)=\mathbb Z$. We note that the result thus obtained implies that the groups $\widehat{H}_*^{\mathrm{c}}$ are not homotopy invariant. Recall the definition of the path homology of a digraph with coefficients in a commutative unital ring $R$ (see [8] and [11]). By a regular path on the set $V$ we mean an arbitrary finite sequence $i_0,i_1, \dots , i_n$ of elements of $V$, where $i_k\ne i_{k+1}$. Given a digraph $G=(V,E)$ and $n\geqslant 0$, consider the module $\mathcal R_n=\mathcal R_n(V)$ generated by the elements $e_p=e_{i_0\dots i_n}$, where $p=(i_0,\dots , i_n)$ is a regular path on $V$. Let $\mathcal R_{-1}=0$. Then the boundary homomorphism $\partial\colon \mathcal R_{p}\to \mathcal R_{p-1}$ is defined by the equation
$$
\begin{equation*}
\partial e_{i_{0}\dots i_{p}}=\sum_{q=0}^{p}( -1) ^{q}e_{i_{0}\dots \widehat{i_{q}}\dots i_{p}}
\end{equation*}
\notag
$$
for $p\geqslant 1$, and $\partial\colon \mathcal R_0\to \mathcal R_{-1}$ is the zero map. Then $\partial^{2}=0$, and so $\mathcal R_{\ast} $ is a chain complex. We consider the submodules $\mathcal{A}_{p}=\mathcal{A}_{p}(G)\subset \mathcal{R}_{p}$ generated by the elements $e_{i_0\dots i_p}$ such that the path $i_0,\dots, i_p$ is admissible, that is, $i_j\to i_{j+1}$ for $0\leqslant j\leqslant p-1$. We set $\mathcal{A}_{-1}=0$. Then the modules
$$
\begin{equation*}
\Omega_{p}=\Omega_{p}( G) =\{v\in \mathcal{A} _{p}\colon \partial v\in \mathcal{A}_{p-1}\}
\end{equation*}
\notag
$$
with the induced differential define a chain complex $\Omega_*$, whose homology is called the path homology of the digraph $G$ and is denoted by $H_*(G)=H_*(G,R)=H_*(\Omega_*)$. Let $X$ be a finite discrete $T_0$-space, and therefore a strictly partially ordered set. Consider the simplicial complex $K(X)$ whose vertices are the points in $X$ and simplexes are totally ordered subsets of points in $X$; see [14]. Let $G(X)$ be the digraph whose vertices are the points of $X$ and for two vertices $x$ and $y$ a directed edge $x\to y$ exists if and only if $x< y$ (see [15] and [16]). We note that in this way we obtain a transitive digraph without cycles. The following statement is a reformulation of a result due to Alexandroff [14]. Theorem 1. There is an isomorphism between the simplicial homology of the complex $K(X)$ and the path homology of the digraph $G(X)$. In a similar way, given a finite transitive digraph $G$ without cycles, we define a simplicial complex $K(G)$ whose vertices are the vertices of the digraph $G$ and $p$-simplexes are regular admissible oriented paths of length $p$ of the digraph. Corollary 1. Let $G$ be a finite transitive digraph. Then there is an isomorphism between the simplicial homology of the complex $K(G)$ and the path homology of the digraph $G$. § 4. Isomorphism of homology groups of transitive digraphsIn this section we consider finite transitive acyclic digraphs. Note that, given a digraph $G$, for all $n\geqslant 0$ the groups $\Omega_n^{\mathrm{c}}(G)$ and $\Omega_n(G)$ are finitely generated. A singular cube $\phi\colon I^n\to G$ defines a generator of the group $\Omega_{n}^{\mathrm{c}}(G)$, which we denote by $\phi^{\Box}$. The complexes $\Omega_{*}^{\mathrm{c}}(G)$ and $\Omega_{*}(G)$ are equipped with the natural augmentations $\epsilon^{\mathrm{c}}\colon\Omega_{0}^{\mathrm{c}}(G)\to \mathbb Z$ and $\epsilon\colon\Omega_{0}(G)\to \mathbb Z$, where
$$
\begin{equation*}
\epsilon^{\mathrm{c}}\Bigl(\sum k_i \phi_i\Bigr)=\sum k_i\quad\text{and}\quad \epsilon\Bigl(\sum k_i p_i\Bigr)=\sum k_i, \qquad k_i\in \mathbb Z.
\end{equation*}
\notag
$$
We define a chain map
$$
\begin{equation}
\tau_*\colon \Omega_*^{\mathrm{c}}(G)\to \Omega_*(G)
\end{equation}
\tag{4.1}
$$
as follows. For a singular zero-dimensional cube $\phi\colon I^0=0\to i\in V_G$ we set $\tau_0(\phi^{\Box})=e_i\in \Omega_0(G)$. Thus we obtain an augmentation-preserving map (4.1) in dimension zero. For $n\geqslant 1$ the vertices of the digraph $I^n$ are numbered by sequences of zeros and ones of length $n$. An oriented edge $i\to j$ between two vertices exists if and only if the vertex $j$ is obtained from $i$ by replacing zero by one in exactly one position. We call the vertex $(0,\dots, 0)$ of the cube the initial vertex, and we call $(1, \dots, 1)$ the terminal vertex. We write an oriented path $p$ from the initial to terminal vertex in the following form:
$$
\begin{equation}
p=(i_0\to i_{1}\to \dots \to i_n), \qquad i_j\in V_{I_n},
\end{equation}
\tag{4.2}
$$
where for $1\leqslant j\leqslant n$ the vertex $i_{j}$ is obtained from $i_{j-1}$ by changing the coordinate $\pi(j)$, $1\leqslant \pi(j)\leqslant n$, from ‘0’ to ‘1’. We denote the sign of the permutation
$$
\begin{equation*}
\pi(p)=\begin{pmatrix} 1& 2& \dots & n\\ \pi(1)&\pi(2)&\dots &\pi(n) \end{pmatrix}
\end{equation*}
\notag
$$
by $\sigma(p)$. Consider the element where the sum is taken over all paths in the cube going from the initial to terminal vertex. This element is the generator of the module $\Omega_n(I^n)$: see [12]. In the cube $I^n$, $n\geqslant 1$, we consider the path
$$
\begin{equation}
p_{\#}=p_{\#}^n= ((0\dots 0)\to (10\dots 0)\to (110\dots 0)\to \cdots \to(1\dots1))
\end{equation}
\tag{4.4}
$$
of length $n$ from the initial vertex of the cube to the terminal vertex. Note that $\sigma(p_{\#})=+1$. For every $n$-dimensional singular cube $\phi\colon I^n\to G$ we define the morphism of chain complexes by
$$
\begin{equation}
\tau_{\ast}\colon \Omega_{\ast}^{\mathrm{c}}(G)\to \Omega_{\ast}(G), \qquad \tau_{n}(\phi^{\Box}):=\phi_{\ast}(\omega_n),
\end{equation}
\tag{4.5}
$$
where $\phi_*\colon \Omega_{\ast}(I^n)\to \Omega_{\ast}(G)$ is the morphism of chain complexes induced by $\phi$ (see [12]). For $n\geqslant 0$ we define an $n$-dimensional digraph simplex $\Delta_n$ with vertex set $V_{\Delta_n}=\{0,\dots , n\}$ and set of directed edges $E_{\Delta_n}=\{i\to j\mid i, j\in V_{\Delta^n}\colon i<j\}$. A total order is naturally defined on the vertex set of this digraph, and there is only one admissible regular path of length $n$, namely, the path $p_{\Delta}= p_{\Delta}^n=(0\to 1\to \dots \to n)$. There are natural embeddings of faces:
$$
\begin{equation*}
\sigma^i_{n-1}=\lambda_{n-1}^{01\dots\widehat{i}\dots n}\colon \Delta_{n-1}\to \Delta_{n}, \qquad \lambda_{n-1}^{01\dots\widehat{i}\dots n}(k)= \begin{cases} k & \text{for} \ k< i, \\ k+1 & \text{for} \ k\geqslant i. \end{cases}
\end{equation*}
\notag
$$
In particular, the morphism $\sigma^n_{n-1}=\lambda_{n-1}^{01\dots (n-1)}$ takes vertices of the digraph $\Delta_{n-1}$ to the vertices of $\Delta_{n}$ that have the same numbers. For $n\geqslant 0$ we define a digraph morphism $\pi^n\colon I^n\to \Delta_n$ recursively as follows. Let $\pi^0\colon (0)\to (0)$ and $\pi^1\colon (0\to 1)\to (0\to 1)$ be the natural isomorphisms. Suppose that the morphism $\pi^k\colon I^k\to \Delta_k$ is already defined for all $k$, $1\leqslant k<n$. Consider the embedding $F_{n0}\colon I^{n-1}\to I^n$ in (3.1) which specifies an isomorphism between the digraph $I^{n-1}$ and the face $I_{n0}^{n-1}\subset I^n$ of the cube defined by the condition that there is a zero in the last, $n$th position. Now we define a morphism $\pi^n\colon I^n\to \Delta_n$ on the vertex set by
$$
\begin{equation}
\pi^n(w)= \begin{cases} \sigma^n_{n-1} \pi^{n-1}F_{n0}^{-1}(w),&w\in V_{I_{n0}^{n-1}}, \\ n,& w\notin V_{I_{n0}^{n-1}}, \end{cases}=\begin{cases} \sigma^n_{n-1}\pi^{n-1}(v), & w=(v0)\in V_{I^n}, \\ n, & w=(v1)\in V_{I^n}, \end{cases}
\end{equation}
\tag{4.6}
$$
where $v\in V_{I^{n-1}}$ and $n\in V_{\Delta_n}$ is the maximal vertex of $\Delta_n$. For $n\geqslant 1$ vertices of the $n$-dimensional cube $I^n$ are specified by sequences of zeros and ones of length $n$, and the cube $I^0$ consists of the single vertex $0$. In these coordinates the map $\pi^n$ in (4.6) is constructed from the map $\pi^{n-1}$ as follows. At the vertex $(v0)$ of $I^{n}$ obtained from a vertex $(v)$ of the cube $I^{n-1}$ by adding a zero in the last position we have $\pi^n(v0)=\sigma^n_{n-1}\pi^{n-1}(v)$, and at the vertex $(v1)$ of $I^{n}$ obtained from the vertex $(v)$ of $I^{n-1}$ by adding a one in the last position we have $\pi^n(v1)=n$. Thus, the map $\pi^n$ is extended from the face $I^{n-1}_{n0}$, identified with $I^{n-1}$, to the whole of the cube $I^{n}$. Lemma 1. i) The singular cube $\pi^n\colon I^n\to \Delta_{n}$ is nondegenerate for $n\geqslant 0$. ii) Let $v\in V_{I^n}$ be a vertex of the digraph $I^n$, $n\geqslant 1$. Then $\pi^n(v)=0\in V_{\Delta_n}$ for $v=(0\dots 0)\in V_{I^n}$. Given $k$, $1\leqslant k\leqslant n$, for a vertex $v$ of the form the equality $\pi^n(v)=k\in V_{\Delta_n}$ holds. Proof. The first assertion can readily be proved by contradiction using induction on $n$. The second assertion follows directly from the definition of $\pi^n$. This completes the proof of the lemma. Proposition 2. The morphism $\pi^n\colon I^n\,{\to}\, \Delta_n$ maps the path $p_{\#}$ onto the path $p_{\Delta}$ in a one-to-one way. Any other admissible regular path of length $n$ in $I^n$ is taken to an irregular path. That is, on any regular path of length $n$ distinct from $p_{\#}$ there are two consecutive vertices $i_{j}$ and $ i_{j+1}$ for which $\pi^n(i_j)=\pi^n(i_{j+1})$. Proof. It follows from the definitions of the path $p_{\#}^n$ and morphism $\pi^n$ and from Lemma 1 that The first assertion is proved.
Let us prove by induction that any other admissible regular path of length $n$ is mapped to an irregular path. For $n=0,1$ there are no admissible regular paths other than $p_{\#}^n$ from the initial vertex of $I^n$ to the terminal one. For $n=2$ there is only one such path, $p_2=(00\to 01\to 11)$, which is taken to the irregular path $\pi^2(p_2)=(0\to 2\to 2)$. Let $n\geqslant 3$. We assume that we have proved the assertion for all $k < n$. Consider an admissible regular path $p$ of length $n$ from the initial to terminal vertex in $I^n$ such that $p\ne p_{\#}^n$. We write it in the form (4.2): $p=(i_0\to i_1\to \dots\to i_{n-1}\to i_n)$. Two cases are possible. In the first case the path $(i_0\to i_1\to \dots\to i_{n-1})$ of length $n-1$ lies in the face $I_{n0}^{n-1}$ defined by the condition that the last coordinate of all vertices is zero. In this case just one, the last vertex $i_n=(1\dots 1)$ of $p$ does not lie in $I_{n0}^{n-1}$, and the last edge has the form
$$
\begin{equation*}
(i_{n-1}\to i_n)=((\underbrace{1\dots 1}_{n-1}0)\to (\underbrace{1\dots 1}_{n-1}1)).
\end{equation*}
\notag
$$
Therefore, the path $(i_0\to i_1\to \dots \to i_{n-1})$ does not coincide with $p_{\#}^{n-1}$, and by the induction assumption the path $\pi^{n-1}(i_0\to i_1\to \dots\to i_{n-1})$ is irregular. Therefore, the path $\pi^{n}(i_0\to i_1\to \dots\to i_{n-1}\to i_n)$ is also irregular in this case.
In the other case there exists $m\geqslant 1$ such that the vertices $i_{n-m}, i_{n-m+1}$, …, $i_n$ do not lie in the face $I_{10}^{n-1}$, while the previous vertices $i_0,\dots ,i_{n-m-1}$ lie in $I_{10}^{n-1}$. By definition Thus, $\pi^{n}(i_0\to \dots\to i_n)$ is irregular in the second case. This completes the proof of the proposition.Let $G$ be a transitive digraph without cycles. Any admissible regular path ${p=(i_0\to \dots \to i_n)}$ in $G$ defines an induced subgraph $\Delta_n^p=\Delta_n^{i_0\dots, i_n}\subset G$ with vertices $i_0, \dots, i_n\in V_{\Delta^p_n}\subset V_G$, which is isomorphic to $\Delta_n$ by means of the isomorphism $\lambda_n^p= \lambda^{i_0\dots i_n}_n\colon \Delta_n\to \Delta_n^p$ defined on the vertex set by the formula $\lambda_n(j)=i_j$ for $0\leqslant j \leqslant n$, $j\in V_{\Delta_n}$. For $n\geqslant 0$ we define a homomorphism $\theta_n\colon \Omega_n(G)\to \Omega_n^{\mathrm{c}}(G)$ as follows. For $n=0$ we write
$$
\begin{equation*}
\theta_0(e_i)=[\lambda^i\pi^0]^{\Box}, \quad \text{where } \lambda^i\pi^0\colon I^0=0\to i\in V_G.
\end{equation*}
\notag
$$
For $n\geqslant 1$ and the basis element $e_p=e_{i_0\dots i_n}$ corresponding to an admissible regular path $p=(i_0\to\dots\to i_n)$ of length $n$ we write
$$
\begin{equation}
\theta_n(e_{i_0\dots i_n})\colon = [\lambda_n^p\circ \pi^n]^{\Box} =[\lambda_n^{i_0\dots i_n}\circ \pi^n]^{\Box}.
\end{equation}
\tag{4.7}
$$
Recall that if $\phi^{\Box}\in Q_n(G)$ is the element given by a singular cube $\phi\colon I^n\to G$, $n\geqslant 1$, $1\leqslant j\leqslant n$, then the faces of $\phi^{\Box}$ are well defined by
$$
\begin{equation*}
\phi_{j\varepsilon}^{\Box}=( \phi \circ F_{j\varepsilon})^{\Box }\in Q_{n-1}(G),
\end{equation*}
\notag
$$
where the embeddings $F_{j\varepsilon}$ are defined in (3.1). Let $Q_{-1}=0$. For $n\geqslant 1$ the homomorphism $\partial^{\mathrm{c}}\colon Q_{n}\to Q_{n-1}$ is defined at the basis elements of $\phi^{\Box}$ by
$$
\begin{equation*}
\partial^{\mathrm{c}}(\phi^{\Box})=\sum_{j=1}^{n}(-1)^{j}( \phi_{j0}^{\Box }-\phi_{j1}^{\Box})
\end{equation*}
\notag
$$
and therefore it defines an induced boundary morphism $\partial^{\mathrm{c}}\colon \Omega^{\mathrm{c}}_n(G)\to \Omega^{\mathrm{c}}_{n-1}(G)$. Here the index $j$, $1\leqslant j\leqslant n$, denotes the number of the face, and $0$ or $1$ corresponds to the fixed coordinate on this face. Let us prove a technical lemma. Lemma 2. Let $\pi^n\colon I^n\to \Delta_n$ be the digraph morphism defined above. Then for $n\geqslant 1$, $1\leqslant j\leqslant n$ and $\varepsilon =0,1$ the morphism is well defined and specifies an $(n-1)$-dimensional singular cube in $\Delta_n$. The following formulae hold: Among the singular cubes $\pi_{j\varepsilon}^n$, the singular cubes described in (4.8) and only them are nondegenerate. Proof. For $n=1$ and $j=1$ the singular cubes
$$
\begin{equation*}
\pi_{10}^1=\pi^1\circ F_{10}=\lambda^0_0\pi^0\colon 0\to 0
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\pi_{11}^1=\pi^1\circ F_{11}=\lambda_0^1\pi^0\colon 0\to 1
\end{equation*}
\notag
$$
are nondegenerate, and there are no other singular cubes for $n=1$. We prove that for $n\geqslant 2$ the following diagram is commutative: We obtain the first equality in (4.8). Let $v=(c_1\dots c_{n-1})\in V_{I^{n-1}}$. Consider two cases. In the first case let $c_{n-1}=0$ and $ v=(c_1\dots c_{n-2} 0)= (w0)\in V_{I^{n-1}}$. Then
$$
\begin{equation}
\pi^n\circ F_{11}(v)= \pi^n(1c_1\dots c_{n-2}0) \stackrel{(4.6)} {=} \pi^{n-1}(1c_1\dots c_{n-2}).
\end{equation}
\tag{4.10}
$$
By Lemma 1 the last expression yields
On the other hand we find that
$$
\begin{equation*}
\sigma^{0}_{n-1}\pi^{n-1}(c_1\dots c_{n-2}c_{n-1})=\sigma^{0}_{n-1}\pi^{n-1}(c_1\dots c_{n-2} 0).
\end{equation*}
\notag
$$
Similarly to the previous calculation we see that $\pi^{n-1}(c_1\dots c_{n-2}0)$ is equal to
Now it follows from the definition of $\sigma^0_{n-1}$ that in the first case diagram (4.9) is commutative. In the second case let $c_{n-1}=1, v=(c_1\dots c_{n-2} 1)= (w1)\in V_{I^{n-1}}$. By the definition of $\pi^n$ we obtain On the other hand let us calculate
$$
\begin{equation*}
\sigma^{0}_{n-1}\pi^{n-1}(c_1\dots c_{n-2}c_{n-1}) =\sigma^{0}_{n-1}\pi^{n-1}(c_1\dots c_{n-2}1)=\sigma^0_{n-1}(n-1)=n.
\end{equation*}
\notag
$$
Thus, the first equation in (4.8) is proved.
To prove the other equation in (4.8) for $n\geqslant 2$, we consider the following diagram and prove its commutativity for $1\leqslant j\leqslant n$: Let $v=(c_1\dots c_{n-1})\in V_{I^{n-1}}$. Consider two cases. Similarly to the considerations in the first case, let $c_{n-1}=0$ and $ v=(c_1\dots c_{n-2} 0)= (w0)\in V_{I^{n-1}}$. Then
$$
\begin{equation}
\pi^n\circ F_{j0}(v)= \pi^n(c_1\dots c_{j-1}0c_j c_{n-2}0) =\pi^{n-1}(c_1\dots c_{j-1}0c_j c_{n-2}).
\end{equation}
\tag{4.13}
$$
By Lemma 1 the last expression yields
$$
\begin{equation}
\begin{aligned} \, \notag \pi^n\circ F_{j0}(v) &= \pi^n(c_1\dots c_{j-1}0c_j c_{n-2}0) =\pi^{n-1}(c_1\dots c_{j-1}0c_j c_{n-2}) \\ &=\begin{cases} 0 & \text{if }c_k=0 \ \text{for }0\leqslant k\leqslant n-2, \\ k,\ 1\leqslant k \leqslant j-1, &\text{if }c_k=1\text{ and } c_m=0\text{ for } m>k, \\ k+1,\ j\leqslant k \leqslant n-2, &\text{if } c_k=1\text{ and } c_m=0\text{ for } m>k. \\ \end{cases} \end{aligned}
\end{equation}
\tag{4.14}
$$
On the other hand, let us calculate $\lambda_{n-1}^{01\dots \widehat{j}\dots n}\pi^{n-1}(c_1\dots c_{j-1}c_j c_{n-2}0)$. Similarly to the previous considerations we see that $\pi^{n-1}(c_1\dots c_{j-1}c_j c_{n-2}0)$ is equal to
Thus,
$$
\begin{equation}
\begin{aligned} \, \pi^{n-1}(v)&=\pi^{n-1}(c_1\dots c_{n-2}0)\notag \\ &=\begin{cases} 0 &\text{if } c_k=0\text{ for } 0\leqslant k\leqslant n-2, \\ k &\text{if } c_k=1\text{ and } c_m=0\text{ for } 1\leqslant k \leqslant n-2\text{ and } m>k. \end{cases} \end{aligned}
\end{equation}
\tag{4.15}
$$
Now, applying $\lambda_{n-1}^{01\dots \widehat{j}\dots n}$ to (4.15), where $1\leqslant j\leqslant n$, we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\lambda_{n-1}^{01\dots \widehat{j}\dots n}\pi^{n-1}(c_1\dots c_{n-2}0) \\ &\ =\begin{cases} 0 & \text{if } c_k=0 \text{ for } 0\leqslant k\leqslant n-2, \\ k &\text{if } c_k=1 \text{ and } c_m=0\text{ for } 1\leqslant k \leqslant n-2, 1\leqslant k\leqslant j-1, \text{ and } m>k, \\ k+1 &\text{if } c_k=1 \text{ and } c_m=0\text{ for } 1\leqslant k \leqslant n-2, k\geqslant j, \text{ and } m>k. \end{cases} \end{aligned}
\end{equation}
\tag{4.16}
$$
Equalities (4.14) and (4.16) imply that in the first case diagram (4.12) is commutative.
In the second case let $c_{n-1}=1$ and $v=(c_1\dots c_{n-2} 1)= (w1)\in V_{I^{n-1}}$. Consider two subcases. 1) Let $j=n$. Then from the definition of $\pi^n$ we obtain
$$
\begin{equation}
\pi^n\circ F_{n0}(v)=\pi^n(c_1\dots c_{j-1}0 c_jc_{n-2}10)=n-1.
\end{equation}
\tag{4.17}
$$
On the other hand
$$
\begin{equation*}
\lambda_{n-1}^{01\dots (n-1)}\pi^{n-1}(c_1\dots c_{n-2}1) =\lambda_{n-1}^{01\dots n}(n-1)=n-1.
\end{equation*}
\notag
$$
That is, in this subcase the commutativity of (4.12) is proved.
2) Let $j\ne n$. From the definition of $\pi^n$ we obtain
$$
\begin{equation}
\pi^n\circ F_{j0}(v)=\pi^n(c_1\dots c_{j-1}0 c_jc_{n-2}1)=n.
\end{equation}
\tag{4.18}
$$
On the other hand, for $1\leqslant j < n$ we calculate as follows:
$$
\begin{equation*}
\lambda_{n-1}^{01\dots \widehat{j}\dots n}\pi^{n-1}(c_1\dots c_{n-2}c_{n-1}) =\lambda_{n-1}^{01\dots \widehat{j}\dots n}\pi^{n-1}(c_1\dots c_{n-2}1) =\lambda_{n-1}^{01\dots \widehat{j}\dots n}(n-1)=n.
\end{equation*}
\notag
$$
Thus, the diagram (4.12) is commutative, and the second equation in (4.8) is proved. Since the singular cube $\pi^{n-1}\colon I^{n-1}\colon \Delta_{n-1}$ is nondegenerate by Lemma 1 and the maps $\lambda$ in (4.8) are isomorphisms, the singular cubes in (4.8) are nondegenerate.
We claim that for $1 < j \leqslant n$ the singular cubes $\pi^{n}_{j1}$ are degenerate. For $n=2$ we have one cube of this kind, namely, $\pi^2_{21}\colon I^1\to \Delta_2$, which is degenerate.Let the singular cubes $\pi^{n-1}_{j1}$ be degenerate for $1<j\leqslant n-1$. This means that for each morphism
$$
\begin{equation}
\pi^{n-1}_{j1}\colon I^{n-2}\to \Delta_{n-1}, \qquad 1<j\leqslant n-1,
\end{equation}
\tag{4.19}
$$
the cube $I^{n-2}$ has two opposite faces glued naturally together by the map $\pi^{n-1}_{j1}$.
Consider $\pi^n_{j1}\colon I^{n-1}\to \Delta_n$, $1< j\leqslant n$. The following cases are possible. (1) Let $j=n$. Then by (4.6),
$$
\begin{equation}
\pi_{n1}^n(v)= \pi^n\circ F_{n1}(v)=\pi^n(v1)=n\in \Delta_n, \qquad v\in V_{I^{n-1}}.
\end{equation}
\tag{4.20}
$$
That is, $\pi_{n1}^n$ is degenerate.
(2) Let $2\leqslant j\leqslant n-1$. Then, as above, for $v=(c_1\dots c_{n-1})\in V_{I^{n-1}}$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \pi_{j1}^n(v) &= \pi^n\circ F_{j1}(v)=\pi^n(c_1\dots c_{j-1}1c_j\dots c_{n-1}) \\ \notag &=\begin{cases} \pi^{n-1}(c_1c_2\dots c_{j-1}1c_j\dots c_{n-2}) &\text{for}\ c_{n-1}=0, \\ n & \text{for}\ c_{n-1}=1, \end{cases} \\ \notag &=\begin{cases} \pi^{n-1}\circ F_{j1}(c_1c_2\dots c_{j-1}c_j\dots c_{n-2}) &\text{for}\ c_{n-1}=0, \\ n & \text{for}\ c_{n-1}=1, \end{cases} \\ &=\begin{cases} \pi^{n-1}_{j1}(c_1c_2\dots c_{j-1}c_j\dots c_{n-2}) &\text{for}\ c_{n-1}=0, \\ n & \text{for}\ c_{n-1}=1. \end{cases} \end{aligned}
\end{equation}
\tag{4.21}
$$
For the map $\pi^{n-1}_{j1}$ we have $2\leqslant j \leqslant n-1$. Therefore, this map is degenerate according to the induction assumption. Thus, it follows from (4.21) that the restriction of $\pi^n_{j1}$ to the $(n-1)$-face of the cube $I^n$ defined by the condition $c_{n-1}=0$ is degenerate, and the restriction of $\pi^n_{j1}$ to the $(n-1)$-face of $I^n$ defined by $c_{n-1}=1$ is the trivial map to the vertex $n\in \Delta_n$. Consequently, $\pi^n_{j1}$ is degenerate for $2\leqslant j\leqslant n$, and the restriction of the map $\pi^n_{j0}$ to the $(n-1)$-face of $I^n$ defined by $c_1=1$ is the trivial map to the vertex $n\in \Delta_n$. Therefore, $\pi^n_{j0}$ is nondegenerate for $1\leqslant j\leqslant n$. This completes the proof of Lemma 2. Proposition 3. Let $G$ be a transitive digraph without cycles. Then for $n\geqslant 1$ the following commutative diagram of modules and homomorphisms holds: Hence $\theta_*$ is a morphism of chain complexes $ \Omega_*(G)\to \Omega_*^{\mathrm{c}}(G)$. Proof. It follows from the definition of homomorphism $\theta_n$ that it is sufficient to prove the commutativity of the diagram in which the module $\Omega_n(\Delta_n)$ is generated by the single basis element $e_{01\dots n}$. For ${n\geqslant 1}$, by the definition of $\theta_{n-1}$ in (4.7) and the definition of $\partial$ we have
On the other hand, by the definition of $\partial^{\mathrm{c}}$ in (3.3) and the definition of $\theta_n$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \partial^{\mathrm{c}}\circ \theta_{n} (e_{0\dots n}) &=\partial^{\mathrm{c}}[(\lambda_{n}^{0\dots n}\pi^n)^{\Box}]= \partial^{\mathrm{c}}[\pi^n]^{\Box} =\sum_{j=1}^{n}(-1)^{j}( [\pi^n_{j0}]^{\Box }-[\pi^n_{j1}]^{\Box}) \\ \notag &\!\!\!\!\!\!\stackrel{\text{Lemma }2} {=}\biggl(\sum_{j=1}^{n}(-1)^{j} [\pi^n_{j0}]^{\Box}\biggr) + [\pi^n_{11}]^{\Box} \\ \notag &=\biggl(\sum_{j=1}^{n}(-1)^{j} [\lambda^{01\dots \widehat{j}\dots n}_{n-1}\pi^{n-1}]^{\Box} \biggr) + [\lambda^{1\dots n}_{n-1}\pi^{n-1}]^{\Box} \\ &=\sum_{j=0}^{n} (-1)^j [\lambda_{n-1}^{(0\dots \widehat j\dots n)}\circ \pi^{n-1}]^{\Box}. \end{aligned}
\end{equation}
\tag{4.25}
$$
The assertion of the proposition follows from (4.24) and (4.25). Theorem 2. The homomorphism $\theta_0$ preserves augmentation. The homomorphisms $\theta_n$ define a morphism of chain complexes $\theta_*\colon \Omega_*(G)\to \Omega_*^{\mathrm{c}}(G)$, which is the right inverse of $\tau_*$, that is, Proof. The first assertion is obvious. For $n\geqslant 1$ consider the following diagram of modules and homomorphisms: in which the left-hand square is commutative by Proposition 3 and the right-hand square is commutative according to [12].
For $n\geqslant 1$ we obtain
$$
\begin{equation}
\tau_n\theta_n(e_{i_0\dots i_n}) =\tau_n[(\lambda_n^{i_0\dots i_n}\pi^n)^{\Box}]=e_{i_0\dots i_n},
\end{equation}
\tag{4.27}
$$
since by Lemma 2 only the image $\lambda_n^{i_0\dots i_n}\pi^n(p_{\#})$ of the path $p_{\#}$ is a nondegenerate path in the image $\lambda_n^{i_0\dots i_n}\pi^n(I^n)$. Moreover, this path coincides with $i_0\to i_1\to \dots \to i_n$ by definition. This completes the proof of the theorem. Recall the theorem on acyclic supports, which we need below; see [22], § 3.4, and [24], § 1.2.1. Let $C_*$ be a chain complex of finitely generated free Abelian groups, and let $C_p=0$ for $p< $0. In this case $C_*$ is called a geometric chain complex. Fix a basis in every group $C_p$. Given two basis elements $b\in C_{p-1}$ and $b' \in C_p$, we write $b\prec b'$ if $b$ is involved with nonzero coefficient in the expansion of $\partial (b')$ with respect to the basis. We denote by $\widetilde{C}_*$ the complex $C_*$ with the augmentation homomorphism $\varepsilon \colon C_0\to \mathbb Z$ given by the formula
$$
\begin{equation*}
\varepsilon \biggl(\sum_i k_ib_i\biggr)= \sum_i k_i, \quad\text{where } k_i\in \mathbb Z\text{ and the } b_i \text{ are basis elements of} \ C_0.
\end{equation*}
\notag
$$
The complex $C_*$ is said to be acyclic if all the homology groups of $\widetilde{C}_*$ are equal to zero. A chain map $\phi_*\colon C_*\to C'_*$ preserves augmentation if $\varepsilon' \phi_0 (c)=\varepsilon (c)$ for each element $c\in C_0$. Definition 6. i) An algebraic support function $E$ from a geometric chain complex $C_*$ to a chain complex $D_*$ defines for every basis element $b\in C_n$, $n\geqslant 0$, a chain subcomplex $E(b)=E_*(b)\subset D_*$ such that the relation $b\prec b'$ implies that ${E_*(b) \subset E_*(b')}$. ii) The function $E$ is called acyclic if every subcomplex $E(b)$ is acyclic. iii) A chain map $f_*\colon C_*\to D_*$ is said to be transferred by the algebraic support function $E$ if $f_n(b)\in E_*(b)$ for each basis element $b \in C_n$. Theorem 3 (on acyclic supports). Let $f_*,g_*\colon C_*\to D_*$ be chain maps of geometric chain complexes that preserve augmentation, and let these maps be transferred by an acyclic support function $E$. Then $f_*$ and $g_*$ are chain homotopy equivalent. Let $G$ be a transitive digraph without cycles. Consider a singular cube $\phi\colon {I^n\!\to\! G}$. Each path $p$ of length $n$ from the starting vertex $(0,\dots,0)$ of the cube $I^n$ to the terminal vertex $(1,\dots,1)$ is mapped by $\phi$ to some path $\phi(p)$ of length at most $n$. Let $\phi(p)=(i_0\dots i_{m})$, where $m\leqslant n$. Since $G$ is a transitive digraph, the subgraph simplex $\Delta_m^{\phi(p)}=\Delta_m^{i_0\dots i_m}\subset G$ containing the path in question is well defined. Consider the subgraph $\Upsilon(\phi)$ of the digraph $G$ equal to the union of all such simplexes in $G$,
$$
\begin{equation}
\Upsilon(\phi)= \bigcup_{p\in \mathbf P} \Delta_m^{\phi(p)},
\end{equation}
\tag{4.28}
$$
where $\mathbf P$ denotes the set of paths in the cube $I^n$ from the initial vertex $(0,\dots,0)$ of $I^n$ to $(1,\dots,1)$. We note that the image of $\phi(I^n)$ lies in the digraph $\Upsilon(\phi)$. Lemma 3. The digraph $\Upsilon(\phi)$ is contractible (homotopy equivalent to a single-point digraph) for any singular cube $\phi\colon I^n\to G$ in the transitive digraph $G$ without cycles. Proof. It follows from the definition that the digraph $\Upsilon(\phi)$ is the union of the digraphs of simplexes that have the initial vertex $\phi(0,\dots,0)$ and the terminal vertex $a=\phi(1,\dots, 1)$. It follows from the definition of a simplex digraph that for any vertex $x\in V_{\Upsilon(\phi)}$ there is an edge $(x\to a)\in E_{\Upsilon(\phi)}$. Therefore, $\Upsilon(\phi)$ in (4.28) is contractible according to Corollary 3.7 in [10]. This completes the proof of the lemma. Proposition 4. Let $G$ be a transitive digraph without cycles. Then the chain maps $\theta_*\circ \tau_* \colon \Omega^{\mathrm{c}}_*(G)\to \Omega^{\mathrm{c}}_*(G)$ and the identity map $ \operatorname{Id}\colon \Omega^{\mathrm{c}}_*(G)\to \Omega^{\mathrm{c}}_*(G)$ are chain homotopy equivalent. Proof. The chain complex $\Omega_*^{\mathrm{c}}(G)$ is geometric, and the chain maps $\theta_*\circ \tau_*$ and $ \operatorname{Id}$ preserve augmentation. Given a singular cube $\phi\colon I^n\to G$, consider the subgraph $\Upsilon(\phi)\subset G$ in (4.28). For every basis singular cube $\phi^{\Box}\in \Omega_*^{\mathrm{c}}(G)$ we define a subcomplex $E_*(\phi^{\Box})$ of the chain complex $\Omega_*^{\mathrm{c}}(G)$ by
$$
\begin{equation}
E_*(\phi^{\Box})\overset{\mathrm{def}}={\Omega}_*^{\mathrm{c}}(\Upsilon(\phi))\subset \Omega_*^{\mathrm{c}}(G).
\end{equation}
\tag{4.29}
$$
To prove that the resulting complex $E_*(\phi^{\Box})$ is acyclic, it is sufficient to prove that the digraph $\Upsilon(\phi)$ is contractible, and this holds by Lemma 3.
Now we check that $E$ is an algebraic support function, that is, condition (i) in Definition 6 is satisfied. Consider a basis element $\phi^{\Box}\in \Omega_*^{\mathrm{c}}(G)$ given by $\phi\colon I^n\to G$, $n\geqslant 0$. Then $\partial(\phi^{\Box})$ is a sum of the basis elements $(\phi\circ F_{j\epsilon})^{\Box}$ taken with coefficients $\pm 1$, where the maps $F_{j\epsilon}\colon I^{n-1}\to I^n$ are embeddings. According to (4.28) and (4.29), the digraph $\Upsilon(\phi\circ F_{j\epsilon})$ is a subgraph of $\Upsilon(\phi)$. Thus, the chain complex $E_*((\phi\circ F_{j\epsilon})^{\Box})=\Omega^{\mathrm{c}}_*(\Upsilon(\phi\circ F_{j\epsilon}))$ is a subcomplex of $E_*(\phi^{\Box})$. For $b\in \Omega^{\mathrm{c}}_{n-1}(G)$ and $b\prec \phi^{\Box}$ we have $b = (\phi\circ F_{j\epsilon})^{\Box}$ and
$$
\begin{equation*}
E_*(b)=E_*((\phi\circ F_{j\epsilon})^{\Box})\prec E_*(\phi^{\Box}).
\end{equation*}
\notag
$$
Therefore, $E$ is an acyclic support function from $\Omega_*^{\mathrm{c}}(G)$ to $\Omega_*^{\mathrm{c}}(G)$.
We check that the chain maps $\theta_*\circ \tau_*$ and $\operatorname{Id}$ are transferred by the algebraic support function $E$. Let $\phi^{\Box}\in \Omega^{\mathrm{c}}_n(G)$ be a basis element. Then
$$
\begin{equation}
\operatorname{Id}(\phi^{\Box})=\phi^{\Box}\in {\Omega}_*^{\mathrm{c}}(\Upsilon(\phi)) =E_*(\phi^{\Box}),
\end{equation}
\tag{4.30}
$$
since the image of $\phi(I^n)$ lies in $\Upsilon(\phi)$. That is, the chain map $\operatorname{Id}\colon \Omega^{\mathrm{c}}_n(G)\to \Omega^{\mathrm{c}}_n(G)$ is transferred by $E$.
By the definitions (4.5) and (4.7) we have
$$
\begin{equation}
\theta_n\circ \tau_n(\phi^{\Box})=\theta_n(\phi_*(w_n)), \qquad \phi\colon I^n\to G.
\end{equation}
\tag{4.31}
$$
The element $\phi_*(w_n)$ is either a trivial element of $\Omega_n(G)$, or $\phi_*(w_n)$ is a sum of basis elements $\pm e_{i_0\dots i_n}$ for which the path $(i_0\dots i_n)=\phi(p)$, $p\in \mathbf P$, has length $n$, $i_0 =\phi(0,\dots,0)$ and $i_n=\phi(1,\dots, 1)$. Each of these paths $\phi(p)$ lies in $\Upsilon(\phi)$ according to (4.28). Therefore, applying the map $\theta_n$ to the above sum of paths, we obtain a sum of singular cubes, and the image of each of these coincides with one of the simplexes of dimension $n$ in (4.28). Thus,
$$
\begin{equation}
\theta_n\circ \tau_n(\phi^{\Box})\in {\Omega}_*^{\mathrm{c}}(\Upsilon(\phi)) =E_*(\phi^{\Box}).
\end{equation}
\tag{4.32}
$$
That is, the chain map $\theta_n\circ \tau_n\colon \Omega^{\mathrm{c}}_n(G)\to \Omega^{\mathrm{c}}_n(G)$ is transferred by $E$.
Now the assertion of Proposition 4 follows from the theorem on acyclic supports (Theorem 3). Theorem 4. For every transitive digraph $G$ without cycles the chain maps $\tau_*$ and $\theta_*$ are homotopy inverse to each other and therefore induce an isomorphism of homology groups This follows from Proposition 4 and Theorem 2. Corollary 2. For every transitive digraph $G$ without cycles the following isomorphism of homology groups holds: This follows from Proposition 1 and Theorem 4. The digraph $\widehat{I}^n$ is a transitive digraph without cycles, and it corresponds to some discrete-space cube $D^n$. The morphisms $\phi\colon I^n\to G$ of transitive digraphs define morphisms of the corresponding discrete spaces. The operations of restricting a morphism to a face of $\widehat{I}^n$, projection onto a face, and embedding faces define similar operations on the discrete cube. Thus, the singular cubic homology theory $H^d(X)$ is defined in the category of discrete $T_0$ spaces. The role of singular cubes here is played by continuous maps $\phi\colon D^n\to X$. Corollary 3. For any discrete $T_0$ space $X$ there is an isomorphism of homology groups where $H_*(X)$ denotes the Alexandroff homology groups. 
Citation in format AMSBIB
\Bibitem{MurJim23} |
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