1 Introduction↩︎

In [1], Barcelo, Caparo, and White introduced the discrete cubical homology groups¹ \[H_n(G)\] of a graph \(G\). These homology groups capture the higher-dimensional structure beyond that seen in classical homologies (where one considers the homology of the \(1\)-dimensional CW complex associated to a graph, or the homology of a simplicial complex, such as the clique complex, associated to a graph). In contrast to these classical homologies, discrete cubical homology probes a graph using discrete cubes of all dimensions, yielding an interesting theory both from a theoretical perspective and for its applications.

Discrete cubical homology, together with its associated homotopy theory [3], raises interesting questions in higher category theory. For instance, computational considerations of discrete cubical homology led to the result that the homology of a cubical set is unchanged when a subcomplex consisting of connection cubes is quotiented out from the chain complex [4]. Similar studies motivated the comparison between homotopy \(n\)-types of graphs and those of cubical and simplicial sets [5], [6], [7]. On the other hand, from an applied viewpoint, discrete cubical homology has also been shown to be more robust to noise than the homology of the flag complex of a graph, making it useful for persistent homology and topological data analysis [8].

Despite its conceptual appeal and broad applications, discrete cubical homology is shockingly difficult to compute in degrees greater than \(1\), even for some very simple graphs. The primary reason for this is that the hypotheses for excision in discrete cubical homology are much more constrained than in the topological or simplicial contexts [1]. While discrete cubical homology of a finite graph is in principle finitely computable, in practice, even with the implementation of efficient algorithms, actual machine computation of many simple graphs is limited to degrees less than or equal to 3 [9]. This difficulty highlights the limited understanding of the nature of the higher-dimensional graph cells contributing to nontrivial homology classes.

At present, it is only known that higher-dimensional cells cannot arise in the absence of triangles and squares in the graph. Indeed, a key result of [10] shows that graphs containing no triangles or squares have trivial discrete cubical homology in dimensions \(2\) and higher. This result is consistent with earlier work in discrete homotopy theory, where triangles and squares play the role of \(2\)-cells: the discrete fundamental group agrees with the fundamental group of the CW complex obtained by attaching \(2\)-cells along all triangles and squares [3]. Therefore, triangle-free and square-free graphs represent homotopy \(1\)-types. Homotopy \(n\)-types for \(n\geq 2\) continue to be rather mysterious objects of study [6], though Carranza and Kapulkin have recently shown the striking result that every homotopy \(n\)-type can be abstractly represented by a graph [7].

These results naturally lead to the question of whether there exists an intrinsic notion of dimension for graphs that governs the vanishing of discrete cubical homology in higher degrees. A natural candidate is the cubical dimension of a graph, defined as the maximal dimension of an injective singular cube it contains. It is unclear whether this notion alone is sufficient: in principle, non-cubical subgraphs could contribute to homology in degrees exceeding the cubical dimension, though we know of no actual examples of this phenomenon occurring.

In this paper, we will define a degree filtration on the discrete cubical singular chains of a graph which arises from consideration of the maximal injective dimension of the facets of a singular cube. Associated to this filtration is a degree spectral sequence \[E^1_{p,q} (G) = H^{[p]}_{p+q}(G) \Rightarrow H_{p+q}(G)\] where the homology groups \(H^{[p]}_*(G)\) are the homology of the \(p\)th associated graded of our filtration and satisfy \[H^{[>n]}_n(G) = 0.\] If \(G\) has cubical dimension \(d\), then \[H^{[>d]}_n(G) = 0.\]

We will define the injective homology \(H^{inj}_*(G)\) to be the edge of the \(E_2\)-page of this spectral sequence given by the homology of the chain complex \[H^{[0]}_0(G) \xleftarrow{d_1} H^{[1]}_1(G) \xleftarrow{d_1} \cdots.\] There are maps \[\label{eq:edge} H_n(G) \twoheadrightarrow E^{\infty}_{n,0}(G) \hookrightarrow H^{inj}_n(G).\tag{1}\]

The injective homology promises to be much more computable than the discrete cubical homology, because it only involves injective singular cubes. Indeed, we will show that if \(CW^\Box(G)\) is the CW complex whose \(n\)-cells correspond to the subgraphs of \(G\) isomorphic to \(n\)-cubes, then there is a surjection of chain complexes \[\label{eq:cell} C^{\emph{cell}}_n(CW^\Box(G)) \twoheadrightarrow H^{[n]}_n(G)\tag{2}\] resulting in a map \[\label{eq:comparison} H_n(CW^\Box(G)) \to H^{inj}_n(G).\tag{3}\] We then are motivated to ask:

We will see that sometimes affirmative answers to Question 1 derive from combinatorial conditions on the graph \(G\). For example, the results of [10] imply that \[H_n(G) \cong H_n(CW^\Box(G))\] for triangle-free square-free graphs.² Moreover, it follows from [11] (see also [3]) that for triangle-free graphs \(G\) we have³ \[H_1(G) \cong H_1(CW^\Box(G)).\]

The groups \(H^{[0]}_{\ge 1}(G)\) are all zero for trivial reasons. We will adapt the “covering space” technique used in [10] to prove that for \(G\) triangle-free we have (Theorem 2) \[H^{[1]}_{\ge 2}(G) = 0.\] It follows that if \(G\) is a triangle-free graph of cubical dimension \(2\), there are isomorphisms \[H_n(G) \cong \begin{cases} H_n^{inj}(G), & n \le 2, \\ H_n^{[2]}, & n > 2. \end{cases}\]

The notion of monophobic cube neighborhoods, introduced in [12], provides a different generalization of the triangle-free square-free condition considered by [10], where one considers graphs in which every \(n\)-cube has a monophobic neighborhood.

Indeed, a triangle-free and square-free graph is precisely one in which every edge (viewed as a \(1\)-cube) has a monophobic neighborhood (such graphs are referred to as \(1\)-monophobic in [12]). In this paper, we introduce the class of \(n\)-quasimonophobic graphs (see Definition 1). It will turn out that a graph is \(1\)-quasimonophobic if and only if it is triangle-free.

We will show (Theorem 5) that if \(G\) is \(n\)-quasimonophobic, then the map (2 ) gives an isomorphism \[\label{eq:celliso} H^{[n]}_n(G) \cong C^{\emph{cell}}_n(CW^\Box(G)).\tag{4}\] We will then deduce (Theorem 6) that for \(G\) \(n\)-quasimonophobic, the map (2 ) induces an injection \[H_n(CW^\Box(G)) \hookrightarrow H^\mathit{inj}_n(G)\] and this map is an isomorphism if \(G\) is \((n-1)\)-quasimonophobic.

We then deduce our main result (Theorem 7) from the degree spectral sequence. Suppose that \(G\) is \((\le 2)\)-quasimonophobic. Then the map (1 ) and map (2 ) induce an isomorphism \[H_2(G) \cong H_2(CW^\Box(G)).\] This appears to be the first systematic result computing discrete cubical \(H_2\) with the exception of that of [10], which establishes that \(H_2(G) = 0\) for \(G\) triangle-free and square-free.

As an application, we compute the second discrete cubical homology of the Greene spheres \(G^{sph}_n\). The graphs \(G^{sph}_n\) (\(n \ge 4\)) are a family of graphs of cubical dimension \(2\) which were constructed by Greene to model the \(2\)-sphere in the setting of discrete cubical homology. Specifically, the graph \(G^{sph}_4\) was introduced in [2], where the authors prove that \(H_2(G^{sph}_4) \ne 0\), and machine computation is used in [9] to show that⁴ \[H_2(G^{sph}_4) \cong \mathbb{Z}.\] The Greene spheres turn out to be \(n\)-quasimonophobic for every \(n\), and hence are amenable to Theorem 7. We will deduce \[\label{eq:Greene} H_2(G^{sph}_n) = \mathbb{Z}\qquad \text{for n \ge 4}.\tag{5}\] Presumably, machine computations could be used to reproduce (5 ) for small \(n\).

We also explain more generally how to “geometrically” produce nontrivial classes in \(H_n(G)\) when \(G\) is \(n\)-quasimonophobic (Theorem 8).

Organization of the paper

In Section ¿sec:sec_[prelim]? we set our basic definitions, and recall the definition of discrete cubical homology. In Section ¿sec:sec_[dss]? we define the degree filtration and the degree spectral sequence. We show in Section 4 that triangle-free graphs \(G\) have \(H^{[1]}_{\ge 2}(G) = 0\). In Section 5 we introduce the notion of \(n\)-quasimonophobicity, and show that for such graphs \(G\) we have \(H^{[n]}_n(G) \cong C^{\emph{cell}}_n(CW^\Box(G))\), and deduce our main results. In Section ¿sec:sec_[examples]? we apply our results to compute \(H_2(G^{sph}_n)\). We also give an example to show the necessity of quasimonophobicity for Theorem 7 and contemplate possible generalization of Theorem 7 for quasimonophobic graphs. We end this section with a geometric construction of classes of \(H_n(G)\), and which we prove are non-trivial when \(G\) is \(n\)-quasimonophobic. Acknowledgments

The first author thanks Curtis Greene and Georg Wille for helpful discussions; in particular, Wille brought Example 1 to her attention. The authors were also grateful for the comments Chris Kapulkin provided on an earlier draft. This material is based in part upon work supported by the National Science Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778, while the first author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the year 2024. This material is also based upon work supported by the National Science Foundation under Award No. DMS-2506564.

2 Preliminaries {#sec:sec_[prelim]}↩︎

Throughout this document, the term “graph” denotes a simple undirected graph – one without loops or multiple edges. We denote a graph as a pair \((V,E)\) of vertex set \(V\) and edge set \(E\), where each edge is a two-point subset of \(V\). We sometimes denote the vertex and edge sets as \(V(G)\) and \(E(G)\), respectively.

A graph map \[f\colon G\to G'\] from a graph \(G=(V,E)\) to another graph \(G'=(V',E')\) is a map \(f\colon V\to V'\) between their vertex sets that maps every edge to an edge or a vertex. We shall say \(f\) is an isomorphism if it has an inverse. We will let \[\mathop{\mathrm{Map}}(G,G')\] denote the set of all graph maps from \(G\) to \(G'\).

The image \(f(G)\) or \(\operatorname{Im}(f)\) of a graph map \(f\colon G\to G'\) is the subgraph of \(G'\) with vertex set \(f(V)\) and only those edges of \(G'\) that are images of the edges in \(G\).

We shall write \[G \subseteq G'\] to indicate that \(G\) is a subgraph of \(G'\), in the sense that \(V(G) \subseteq V(G')\) and \(E(G) \subseteq E(G')\). A subgraph \(G \subseteq G'\) is induced if for each pair of vertices \(v,w \in V(G)\), an edge connects them in \(G\) if and only if an edge connects them in \(G'\).

The neighborhood graph \(N(v)\) of a vertex \(v\) in a graph \(G\) is the induced subgraph of \(G\) containing \(v\) and the vertices adjacent to \(v\). We say that a graph map \(f: G \to G'\) is a local isomorphism for all vertices \(v\in V(G)\), \(f\) restricts to an isomorphism \[f\big\vert_{N(v)}: N(v) \to N(f(v)).\]

For \(n\geq 0\), we will let \(I^n\) denote the discrete \(n\)-cube graph, with the vertex set \(\{0,1\}^n\) and two vertices are adjacent if they differ in exactly one coordinate. In particular, \(I^0\) is the graph with a single vertex and no edges.

For each \(n \ge 0\), define the singular cubical \(n\)-chains on \(G\) to be the free abelian group \[Q_n(G) := \mathbb{Z}\mathop{\mathrm{Map}}(I^n,G).\] generated by the singular \(n\)-cubes \[\sigma \colon I^n \to G.\] For \(1 \le i \le n\), the front and back \(i\)-faces of such a singular \(n\)-cube \(\sigma\) are denoted as \(f_i^-\sigma\) and \(f^+_i\sigma\), where \(f_i^-\sigma\) is obtained by fixing the \(i\)-th coordinate equal to \(0\), while the back \(i\)-face \(f^+_i\sigma\) is obtained by fixing the \(i\)-th coordinate equal to \(1\). These face operators satisfy the face relation: for \(\sigma\) a singular \(n\)-cube and \(1 \le i < j \le n\), we have \[\begin{align} \label{eq95face32relation} f_i^af_j^b\sigma=f^b_{j-1}f^a_i\sigma. \end{align}\tag{6}\]

The boundary operator \(\partial_n \colon Q_n(G) \to Q_{n-1}(G)\) is defined on a singular \(n\)-cube \(\sigma\) by \[\partial_n \sigma = \sum_{i=1}^n (-1)^i (f_i^- \sigma - f^+_i \sigma).\] We have \(\partial_n \circ \partial_{n+1} = 0\), so that \((Q_*(G), \partial_*)\) forms a chain complex. However, as in the case with singular cubical chains for topological spaces, the homology of this chain complex is not the desired discrete cubical homology of \(G\) (e.g. if \(G\) has 1 vertex, \(H_n(Q_*(G))\) is nonzero in every degree). We instead are required to normalize by modding out by degeneracies.

A singular \(n\)-cube \(\sigma\) is said to be degenerate, if there exists an index \(i\) such that \(f_i^- \sigma = f^+_i \sigma\). The boundary of a degenerate cube is a linear combination of degenerate cubes, so the degenerate singular cubes form a subcomplex \[D_*(G) \subseteq C_*(G).\] We define the normalized singular cubical chains to be the quotient complex \[C_*(G) := Q_*(G)/D_*(G).\] The discrete cubical homology is defined to be the homology of this quotient complex: \[H_n(G) := H_n(C_*(G)).\]

It should be noted that even for graphs with few vertices, the rank of these normalized singular cubical chain groups typically grows super-exponentially with respect to dimension, making the computation of discrete cubical homology highly challenging.

3 The degree spectral sequence {#sec:sec_[dss]}↩︎

We define the degree of a singular \(n\)-cube \[\sigma\colon I^n\to G\] to be the dimension of the maximal injective facet of \(\sigma\).

Let \[C_n^k (G) \le C_n(G)\] denote the subgroup generated by singular \(n\)-cubes of degree \(k\) or below. This results in a degree filtration \[C^0_*(G) \le C^1_*(G) \le \cdots\] on the chain complex \(C_*(G)\). We will denote the associated graded of this filtration by \[C_n^{[k]} (G) := C_n^k (G)/C_n^{k-1} (G).\] We let \[H^{[k]}_*(G) := H_*(C^{[k]}_*(G))\] denote the homology of this quotient complex.

The cubical dimension of a graph \(G\) is the dimension of the maximal injective singular cube in \(G\). If \(G\) has cubical dimension \(d\), we clearly have \[C^{[k]}_*(G) = 0 \quad \text{for k > d}\] and therefore \[H^{[k]}_*(G) = 0 \quad \text{for k > d}.\]

Associated to the degree filtration is a convergent spectral sequence (see for example, [13]) \[E^1_{p,q} = H^{[p]}_{p+q}(G) \Rightarrow H_{p+q}(G)\] which we will refer to as the degree spectral sequence.

Since for all \(k>n\), we have \(C_n^k (G)=C_n (G)\), it follows that \[H_n^{[k]} (G)=0 \quad \text{for }k > n.\] For \(n>0\), the degree-\(0\) singular \(n\)-cubes are degenerate, hence \[H_n^{[0]} (G)=0.\] For a graph \(G\) with cubical dimension \(d\), \(H_n^{[p]} (G)= 0\), for \(p>d\). The degree spectral sequence is shown in Figure 1, where we write \(H_{p+q}^{[p]}\) for \(H_{p+q}^{[p]} (G)\).

4 Degree-1 homology cycles in a triangle-free graph↩︎

In this section, our main goal is to prove that \(H_n^{[1]}\) of a triangle-free graph is trivial for \(n\geq 2\) (Theorem 3). This result extends the main result of [10], which asserts that the discrete cubical homology of triangle- and square-free graphs is trivial in dimension \(2\) and higher.

To provide context and to clarify how the arguments of [10] carry over to our setting, we first give a streamlined approach to the main result of [10]. The technique these authors used to prove Theorem 1 is based on a combinatorial analog of the notion of covering space in the category of graphs.⁵

Given graph \(G\), we will let \(G_\mathbb{R}\) denote the usual \(1\)-dimensional CW complex (topological graph) whose \(0\)-cells are given by the vertices of \(G\), and \(1\)-cells are given by the edges of \(G\). Let \[p: \widetilde{G}_\mathbb{R}\to G_\mathbb{R}\] denote the universal cover. The CW structure on \(G_\mathbb{R}\) lifts to a CW structure on \(\widetilde{G}_\mathbb{R}\), giving \(\widetilde{G}_\mathbb{R}\) the structure of a topological forest (a disjoint union of topological trees). Let \(\widetilde{G}\) be the combinatorial forest associated to this topological forest. Then the local homeomorphism \(p\) arises from a local isomorphism of graphs \[\widetilde{G} \to G.\] If \(G\) is path connected, \(\widetilde{G}\) is a tree, and the fundamental group \[\pi := \pi_1(G_\mathbb{R},v)\] acts freely on \(\widetilde{G}\) through deck transformations. The following theorem is proven in [10].

In Theorem 1, the graph \(G\) is required to be triangle- and square-free because this is the condition required to guarantee the existence of lifts of all singular \(n\)-cubes. Topologically, the obstruction to lifting a singular \(n\)-cube \[\sigma: I^n \to G\] is the non-triviality of the image of the induced map \[\sigma_*: \pi_1(I^n_\mathbb{R}) \to \pi_1(G_\mathbb{R}).\]

We deduce the following (compare with [10]).

We deduce the following generalization of Theorem 1⁶

The main result of this section is the following:

The degree spectral sequence immediately gives the following corollary.

5 \(H_n^{[n]}\) of \(n\)-quasimonophobic graphs↩︎

In this section, we show that for a certain class of graphs, which we call \(n\)-quasimonophobic graphs, the homology group \(H_n^{[n]}(G)\) admits a particularly simple description. We begin by introducing the notion of \(n\)-quasimonophobic graphs.

The concept of monophobic cube neighborhoods was introduced by Babson, Greene, and Jamil in [12], a sufficient condition for nontriviality of certain elements in the discrete homotopy groups and in the discrete cubical homology groups of a graph \(G\) is established. In the present section, we give the definition of \(n\)-quasimonophobic graphs and develop several results building on those in [12], to facilitate the computation of \(H_n^{[n]}\) for \(n\)-quasimonophobic graphs.

A rigid \(n\)-cube in a graph is an induced subgraph isomorphic to \(I^n\), the standard discrete \(n\)-cube. A singular \(n\)-cube \(\sigma\) is said to be supported by a rigid \(n\)-cube if the image of \(\sigma\) is equal to that cube.

A rigid \(n\)-cube \(Q\) is monophobic in a graph \(G\) if, whenever a singular \((n+1)\)-cube contains a face supported by \(Q\), it also contains at least one additional face supported by \(Q\). This relates to the framework of [12] in the sense that a rigid \(n\)-cube \(Q\subseteq G\) is monophobic if and only if it posesses a monophobic neighborhood.

We shall say that a rigid \(n\)-cube \(Q\) is quasimonophobic in a graph \(G\) if, whenever a noninjective singular \((n+1)\)-cube contains a face supported by \(Q\), it also contains at least one additional face supported by \(Q\).

Note the following:

• If a graph is \(n\)-monophobic, then it is \(n\)-quasimonophobic.

• Every graph is \(0\)-quasimonophobic, but the only graphs which are \(0\)-monophobic are those without edges.

• A graph is \(1\)-monophobic if and only if it is triangle-free and square-free, and it is \(1\)-quasimonophobic if and only if it is triangle-free.

• If \(G\) has cubical dimension \(d\), it is tautologically \((>d)\)-monophobic (and therefore \((>d)\)-quasimonophobic).

In this section, we focus on the degree-\(n\) quotient complex \(C^{[n]}_*(G)\) associated with an \(n\)-quasimonophobic graph \(G\) and are primarily concerned with its homology in dimension \(n\): \[H_n^{[n]}(G) = H_n(C_*^{[n]}(G)).\] All boundary computations pertaining to the boundary operator \[\partial^{[n]}_{n+1} : C_{n+1}^{[n]}(G) \to C_n^{[n]}(G)\] are understood to be taken modulo singular cubes of degree less than \(n\), and only injective singular \(n\)-cubes can contribute nontrivially to \(H_n^{[n]}(G)\).

Now, we proceed to show that \(H_n^{[n]}\) of \(n\)-quasimonophobic graphs is given by \(C_n^{[n]}(G)\) modulo the equivalence relation defined in the following theorem.

We will prove Theorem 4 through a sequence of lemmas. These lemmas were originally proven in [15] in the context of graphs which are digital images; we include the proofs here for completeness.

Note that \(\sigma\) and \(\gamma\) are related by a finite sequence of compositions of cube automorphisms \(T_i, \;T_{i,j} \colon I^n \to I^n, \quad i,j \in \{1,\ldots,n\}, \;i<j,\) where \(T_i\) reflects the \(i\)-th coordinate via \(t_i \mapsto 1 - t_i\), and \(T_{i,j}\) permutes the \(i\)-th and \(j\)-th coordinates. To prove the desired result, we show that \(\sigma\), \(\sigma \circ T_i\), and \(\sigma \circ T_{i,j}\) are homologous to each other, up to a sign, where the homology consists solely of singular \((n+1)\)-cubes of degree \(n\).

The following two lemmas show that the chain \(\sigma + \sigma \circ T_{i,j}\) is a boundary in \(C_n^{[n]}(G)\). To this end, we first define the cube automorphism \(R_{i,j} = T_{i,j} \circ T_j,\) which acts on the cube \(I^n\) as a quarter-turn rotation in the anticlockwise direction. Then Lemma 3 shows that \(\sigma - \sigma \circ R_{i,j}\) is a boundary in \(C_n^{[n]}(G)\); subsequently, Lemma 4 establishes that \(\sigma + \sigma \circ T_{i,j}\) is likewise a boundary in \(C_n^{[n]}(G)\).

These lemmas conclude the proof of Theorem 4.

We are now ready to prove one of the main result of this section, which provides a convenient description of \(H_n^{[n]}(G)\) for \(n\)-quasimonophobic graphs.

The proof of Theorem 5 is somewhat involved. We pause to explain the strategy of the argument. For any graph \(G\), \(C_{n-1}^{[n]}(G)\) is trivial, therefore, the the \(n\)-th homology group \(H_n^{[n]}(G)\) of the quotient complex \(C^{[n]}(G)\), is given by \[H_n^{[n]}(G) \;=\; \frac{C_n^{[n]}(G)}{\partial_{n+1}^{[n]}\!\left(C_{n+1}^{[n]}(G)\right)} \;=\; \frac{C_n^{[n]} (G) \big/\sim}{\partial_{n+1}^{[n]}\!\left(C_{n+1}^{[n]}(G)\right)\big/\sim}.\] To obtain the desired result, we will use monophobicity arguments, following [12], to show that for an \(n\)-quasimonophobic graph \(G\), the group \[\partial_{n+1}^{[n]}\!\left(C_{n+1}^{[n]}(G)\right)\big/\sim\] is trivial.

Let \(\sigma\) be a singular \((n+1)\)-cube in \(G\) with degree equal to \(n\). To obtain the desired result, we analyze the boundary \(\partial_{n+1}^{[n]}(\sigma)\) in the quotient complex \(C^{[n]}_n(G)\). First note that \(\sigma\) is a noninjective singular cube with at least one face injective, that is, \(\sigma\) contains at least one face supported by a rigid \(n\)-cube in \(G\). Since \(G\) is \(n\)-quasimonophobic, for every face of \(\sigma\) supported by an \(n\)-cube, there is at least one more face of \(\sigma\), supported by the same \(n\)-cube. We show that, except possibly for four faces of \(\sigma\), all the remaining faces are noninjective and therefore vanish in \(C_n^{[n]}(G)\) (Lemma 8). We also prove that the injective faces of \(\sigma\) occur in pairs such that each face in a pair is related to the other by a sequence of compositions of operators of the form \(T_i\) or \(T_{i,j}\) (Lemma 7). Finally, we show that the relations among the injective faces of \(\sigma\) imply that the boundary \(\partial_{n+1}^{[n]}(\sigma)\) is trivial modulo the equivalence relation \(\sim\). This completes the proof.

Before proving Lemmas 7 and 8, we establish two preliminary results. The following result appears in [16], and a similar result is also proved in [12].

The preceding lemma shows that in case-2, faces supported by the same quasimonophobic \(n\)-cube are related by a composition of the form \(T_i T_{i,j}\). A different relation holds in case-1, as described in the next lemma; in that case, the faces are related by a composition of \(j-i-1\) factors.

The following lemma shows that all the other faces are noninjective.

This completes the proof of Theorem 5. We deduce the following.

6 Applications and questions {#sec:sec_[examples]}↩︎

Note that the computer computations of [9] actually establish that \[H_{3}(G^\mathrm{sph}_4) \cong H_{3}(CW^\Box(G^\mathrm{sph}_4)) = 0.\] We also saw that \(1\)-quasimonophobicity eliminated the contributions of \(H^{[1]}_{\ge 2}\) from the degree spectral sequence. Motivated by these data points, it is natural to ask the following:

The following example shows that the assumption of \(2\)-quasimonophobicity is necessary in Theorem 7.

Finally we explain how quasimonophobicity allows one to geometrically produce non-trivial classes in discrete cubical homology.

Suppose that \[X_\bullet: \Box^{op}_{inj} \to \mathrm{Sets}\] is a regular semi-cubical set in the sense that its geometric realization \(\lvert X \rvert\) is a regular CW complex: each characteristic map \[\Box^n \to \lvert X_\bullet \rvert\] is injective. Let \[G(X_\bullet)\] denote the graph given by its 1-skeleton. Suppose that \[x = \sum_i a_i \sigma_i \in C_n(X_\bullet) = \mathbb{Z}X_n\] is an \(n\)-cycle in the cubical \(n\)-chains of \(X_\bullet\).

The class \(x\) can be regarded as giving a cycle in the cellular chain complex \[x \in C^\mathrm{cell}_n(\lvert X_\bullet \rvert),\] giving a class \[[x] \in H_n(\lvert X_\bullet \rvert).\] By regularity, there is an inclusion of CW-complexes \[\label{eq:CWinc} \lvert X_\bullet \rvert \hookrightarrow CW^\Box (G(X_\bullet))\tag{12}\] and we let \[[x]_{CW} \in H_n(CW^\Box(G(X_\bullet)))\] denote the image of \([x]\) under this map. Alternatively, we may regard each \(\sigma_i \in X_n\) as a map \[\sigma_i: \Box^n_\bullet \to X_\bullet\] giving a map of graphs \[\sigma_i: I^n = G(\Box^n_\bullet) \to G(X_\bullet).\] The resulting class \[x = \sum_i a_i \sigma_i \in C_n(G(X_\bullet))\] is a cycle in the discrete cubical chains in \(G(X_\bullet)\), yielding a cycle \[[x]_\mathrm{graph} \in H_n(G(X_\bullet)).\] Under the maps (1 ) and (2 ) \[H_n(G(X_\bullet)) \to H_n^\mathit{inj}(G(X_\bullet)) \leftarrow H_n(CW^\Box(G(X_\bullet)))\] the classes \([x]_\mathrm{graph}\) and \([x]_{CW}\) map to the same element.

Given a graph \(G\) and a graph map \[\alpha: G(X_\bullet) \to G\] the image of \([x]_\mathrm{graph}\) under the map \(\alpha_*: H_n(G(X_\bullet)) \to H_n(G)\) gives “geometric” discrete cubical homology class \[\alpha_*[x]_\mathrm{graph} \in H_n(G).\] If the map \(\alpha\) is an injection, it induces an inclusion of CW complexes \[CW^\Box(G(X_\bullet)) \hookrightarrow CW^\Box(G),\] and the image of \([x]_{CW}\) under this map gives a homology class \[\alpha_*[x]_{CW} \in H_n(CW^\Box(G)).\]

For example, for \(n \ge 4\) there are clearly regular semi-cubical sets \(S^2(n)_\bullet\) whose realization is \(S^2\), we have \[G^\mathrm{sph}_{n} = G(S^2(n)_\bullet),\] and for \(n \ge 4\) the cannonical map gives a homeomorphism \[S^2 = \lvert S^2(n)_\bullet \rvert \xrightarrow{\approx} CW^\Box(G^\mathrm{sph}_n).\] The graph \(G^\mathrm{sph}_n\) is \(2\)-quasimonophonic, the fundamental class \[[S^2] \in H_2(S^2(n)_\bullet)\] has non-trivial image \[0 \ne [S^2]_{CW} \in H_2(CW^\Box(G^\mathrm{sph}_2)) \cong \mathbb{Z}\] and hence gives the (nontrivial) generator of \[[S^2]_\mathrm{graph} \in H_2(G^{sph}_n) \cong \mathbb{Z}.\]

However, in the case of \(n = 3\), while \(G^\mathrm{sph}_3 = I^3\) is still \(2\)-quasimonophobic, we have \[CW^\Box(G^\mathrm{sph}_3) = \Box^3\] and hence the image of \([S^2]\), \[[S^2]_{CW} \in H_2(CW^\Box(G^\mathrm{sph}_3)) = 0,\] is trivial. The element \[[S^2]_\mathrm{graph} \in H_2(G^\mathrm{sph}_3) = 0\] is also trivial, since \(G^\mathrm{sph}_3 = I^3\) is contractible.

Finally, we saw that the graph \(K_{2,3}\) is not \(2\)-quasimonophobic. There is a semi-cosimplicial set \(\widetilde{S}^2_\bullet\), obtained by gluing three \(\Box^2_\bullet\)’s together, with the property that:

• \(\lvert \widetilde{S}^2_\bullet \rvert \approx S^2\),

• \(G(\widetilde{S}^2_\bullet) = K_{2,3}\).

Since \(CW^\Box(K_{2,3}) \approx S^2\), the fundamental class \[[S^2] \in H_2(\widetilde{S}^2_\bullet)\] gives a non-trivial class \[0 \ne [S^2]_{CW} \in H_2(CW^\Box(K_{2,3})) = \mathbb{Z}.\] However, since \(K_{2,3}\) is contractible, the class \[[S^2]_\mathrm{graph} \in H_2(K_{2,3}) = 0\] is trivial. This demonstrates the necessity of the quasimonophobicity condition in Theorem 8.

References↩︎

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1. These are sometimes denoted \(\mathcal{H}_n^{\mathrm{Cube}}(G)\) — see [2].↩︎

2. For such graphs, \(CW^\Box(G)\) is nothing other than the \(1\)-dimensional CW complex associated to the graph \(G\).↩︎

3. Ender and Kapulkin instead consider the \(2\)-dimensional CW complex \(X_G\) obtained by filling in triangles and squares with \(2\)-cells, but for \(G\) triangle-free, \(X_G\) is just the \(2\)-skeleton of \(CW^\Box(G)\).↩︎

4. In fact, [9] also compute \(H_3(G^{sph}_4) = 0\).↩︎

5. This combinatorial notion of covering differs from the homotopical covering graph of [14], in that \(\widetilde{G}\) is always a tree.↩︎

6. Note that if \(G\) is square-free, then \(C^1_*(G) = C_*(G)\).↩︎