Cuts and flows of cell complexes
 254 Downloads
 4 Citations
Abstract
We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe, and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite’s constant.
Keywords
Cut lattice Flow lattice Critical group Spanning forest Cell complexMathematics Subject Classification
05C05 05C21 05C50 05E45 11H061 Introduction
This paper is about vector spaces and integer lattices of cuts and flows associated with a finite cell complex. Our primary motivation is the study of critical groups of cell complexes and related group invariants. The critical group of a graph is a finite abelian group the order of which is the number of spanning forests. The definition was introduced independently in several settings, including arithmetic geometry [28], physics [12], and algebraic geometry [2] (where it is also known as the Picard group or Jacobian group). It has received considerable recent attention for its connections to discrete dynamical systems, tropical geometry, and linear systems of curves; see, e.g., [3, 4, 7, 22].
In a previous work [17], the authors extended the definition of the critical group to a cell complex \(\Sigma \) of arbitrary dimension. To summarize, the critical group \(K(\Sigma )\) can be calculated using a reduced combinatorial Laplacian, and its order is a weighted enumeration of the cellular spanning trees of \(\Sigma \). Moreover, the action of the critical group on cellular \((d1)\)cochains gives a model of discrete flow on \(\Sigma \), generalizing the chipfiring and sandpile models; see, e.g., [4, 12].
Bacher et al. first defined the lattices \(\mathcal {C}\) and \(\mathcal {F}\) of integral cuts and flows for a graph [2]. By regarding a graph as an analogue of a Riemann surface, they interpreted the discriminant groups \(\mathcal {C}^\sharp /\mathcal {C}\) and \(\mathcal {F}^\sharp /\mathcal {F}\), respectively, as the Picard group of divisors and as the Jacobian group of holomorphic forms. In particular, they showed that the critical group \(K(G)\) is isomorphic to both \(\mathcal {C}^\sharp /\mathcal {C}\) and \(\mathcal {F}^\sharp /\mathcal {F}\). Similar definitions and results appear in the work of Biggs [4].
Before proving these results, we study the cut space (Sect. 4), the flow space (Sect. 5), and the cut and flow lattices (Sect. 6) in some detail. In order to do this, we begin in Sect. 3 by describing and enumerating cellular spanning forests of an arbitrary cell complex, generalizing our earlier work [15, 16]. Similar results were independently achieved, using different techniques, by Catanzaro, Chernyak and Klein [8]. Our methods and results are very close to those of Lyons [29], but our technical emphasis is slightly different.
Every cellular spanning forest \(\Upsilon \) naturally gives rise to bases of the cut space (Theorem 4.8) and the flow space (Theorem 5.5). In the graphic case, these basis vectors are simply signed characteristic vectors of fundamental cocircuits and circuits in the graphic matroid, and they always form integral bases for the cut and flow lattices. For a general cellular complex, the supports of basis vectors are given by cocircuits and circuits in the cellular matroid of \(\Sigma \) (i.e., the matroid represented by the columns of \(\partial \)), but their entries are not determined by the matroid. We prove that the basis vectors can be scaled so that their entries are torsion coefficients of homology groups of certain subcomplexes (Theorems 4.11 and 5.3). Under certain conditions on \(\Upsilon \), these bases are in fact integral bases for the cut and flow lattices (Theorems 6.1 and 6.2). Although the matroid data alone are not enough to extend the theory of [2] to arbitrary cell complexes, the perspective of matroid theory will frequently be useful.
The idea of studying cuts and flows of matroids goes back to Tutte [33]. More recently, Su and Wagner [32] define cuts and flows of a regular matroid (i.e., one represented by a totally unimodular matrix \(M\)); when \(M\) is the boundary matrix of a cell complex, this is the case where the torsion coefficients are all trivial. Su and Wagner’s definitions coincide with ours; their focus, however, is on recovering the structure of a matroid from the metric data of its flow lattice.
In the final section of the paper, we generalize a theorem of Kotani and Sunada [26], who observed that a classical inequality for integer lattices, involving Hermite’s constant (see, e.g., [27]), could be applied to the flow lattice of a graph to give a bound for girth and complexity. We prove the corresponding result for cell complexes (Theorem 9.2), where “girth” means the size of a smallest circuit in the cellular matroid (or, topologically, the minimum number of facets supporting a nonzero homology class), and “complexity” is the torsionweighted count of cellular spanning trees.
2 Preliminaries
In this section, we review the tools needed throughout the paper: cell complexes, cellular spanning trees and forests, integer lattices, and matroids.
2.1 Cell complexes
Our work is motivated by algebraic graph theory, including critical groups, cut and flow spaces and lattices, and the chipfiring game. Our central goal is to extend the theory from graphs to higherdimensional spaces. Thus, we work in the setting of a finite CW complex, regarded as the higherdimensional analogue of a graph. Accordingly, we begin by reviewing some of the topology of cell complexes; for a general reference, see [23, p. 5]. The reader more familiar with simplicial complexes may safely consider that special case throughout.
Throughout the paper, \(\Sigma \) will denote a finite CW complex (which we refer to simply as a cell complex) of dimension \(d\). We adopt the convention that \(\Sigma \) has a unique cell of dimension \(1\) (as though it were an abstract simplicial complex); this will allow our results to specialize correctly to the case \(d=1\) (i.e., that \(\Sigma \) is a graph). We write \(\Sigma _i\) for the set of \(i\)dimensional cells in \(\Sigma \), and \(\Sigma _{(i)}\) for the \(i\)dimensional skeleton of \(\Sigma \), i.e., \(\Sigma _{(i)}=\Sigma _i\cup \Sigma _{i1}\cup \cdots \cup \Sigma _0\). Again, in keeping with simplicialcomplex terminology, a cell of dimension \(d\) is called a facet.
Unless otherwise stated, every \(d\)dimensional subcomplex \(\Gamma \subseteq \Sigma \) will be assumed to have a full codimension1 skeleton, i.e., \(\Gamma _{(d1)}=\Sigma _{(d1)}\). Accordingly, for simplicity of notation, we will often make no distinction between the subcomplex \(\Gamma \) itself and its set \(\Gamma _d\) of facets.
The symbol \(C_i(\Sigma )=C_i(\Sigma ;R)\) denotes the group of \(i\)dimensional cellular chains with coefficients in a ring \(R\). The \(i\)dimensional cellular boundary and coboundary maps are, respectively, \(\partial _i(\Sigma ;R):C_i(\Sigma ;R) \rightarrow C_{i1}(\Sigma ;R)\) and \(\partial ^{*}_i(\Sigma ;R):C_{i1}(\Sigma ;R) \rightarrow C_i(\Sigma ;R)\); we will write simply \(\partial _i\) and \(\partial ^{*}_i\) whenever possible.
When \(\Sigma \) is a graph (i.e., a cell complex of dimension 1), its top boundary map is a familiar object, namely its signed vertexedge incidence matrix (with respect to some edge orientation). In this article, our goal will be to extract combinatorial information about an arbitrary cell complex from its topdimensional boundary map (which can be any integer matrix).
While many definitions and results can be stated purely algebraically (e.g., in terms of chain complexes over \(\mathbb {Z}\)), we regard the underlying object of interest as the cell complex (see Remark 4.2).
2.2 Spanning forests and Laplacians
In the case that \(\Sigma \) is \(\mathbb {Q}\)acyclic in codimension one, this definition specializes to our earlier definition of a cellular spanning tree [16, Definition 2.2].
There are two main reasons that enumeration of spanning forests of cell complexes is more complicated than for graphs. First, many properties of graphs can be studied component by component, so that one can usually make the simplifying assumption of connectedness; on the other hand, a higherdimensional cell complex cannot in general be decomposed into disjoint pieces that are all acyclic in codimension one. Second, for complexes of dimension greater than or equal to two, the possibility of torsion homology affects enumeration.
2.3 Lattices
Starting in Sect. 6, we will turn our attention to lattices of integer cuts and flows. We review some of the general theory of integer lattices; see, e.g., [1, Chap. 12], [21, Chap. 14], [24, Chap. IV].
A lattice \(\mathcal {L}\) is a discrete subgroup of a finitedimensional vector space \(V\); that is, it is the set of integer linear combinations of some basis of \(V\). Every lattice \(\mathcal {L}\subseteq \mathbb {R}^n\) is isomorphic to \(\mathbb {Z}^r\) for some integer \(r\le n\), called the rank of \(\mathcal {L}\). The elements of \(\mathcal {L}\) span a vector space denoted by \(\mathcal {L}\otimes \mathbb {R}\). For \(\mathcal {L}\subseteq \mathbb {Z}^n\), the saturation of \(\mathcal {L}\) is defined as \(\hat{\mathcal {L}}=(\mathcal {L}\otimes \mathbb {R})\cap \mathbb {Z}^n\). An integral basis of \(\mathcal {L}\) is a set of linearly independent vectors \(v_1,\dots ,v_r\in \mathcal {L}\) such that \(\mathcal {L}=\{c_1v_1+\cdots +c_rv_r :c_i\in \mathbb {Z}\}\). We will need the following fact about integral bases of lattices; the equivalences are easy consequences of the theory of free modules (see, e.g., [1, Chap. 12], [24, Chap. IV]):
Proposition 2.1

(a) Every integral basis of \(\mathcal {L}\) can be extended to an integral basis of \(\mathbb {Z}^n\).

(b) Some integral basis of \(\mathcal {L}\) can be extended to an integral basis of \(\mathbb {Z}^n\).

(c) \(\mathcal {L}\) is a summand of \(\mathbb {Z}^n\), i.e., \(\mathbb {Z}^n\) can be written as an internal direct sum \(\mathcal {L}\oplus \mathcal {L}'\).

(d) \(\mathcal {L}\) is the kernel of some group homomorphism \(\mathbb {Z}^n\rightarrow \mathbb {Z}^m\).

(e) \(\mathcal {L}\) is saturated, i.e., \(\mathcal {L}=\hat{\mathcal {L}}\).

(f) \(\mathbb {Z}^n/\mathcal {L}\) is a free \(\mathbb {Z}\)module, i.e., its torsion submodule is zero.
Proposition 2.2

(a) If the columns of \(M\) form an integral basis for the lattice \(\mathcal {L}\), then the columns of \(M(M^TM)^{1}\) form the corresponding dual basis for \(\mathcal {L}^\sharp \).

(b) The matrix \(P=M(M^TM)^{1}M^T\) represents orthogonal projection from \(\mathbb {R}^n\) onto the column space of \(M\).

(c) If the greatest common divisor of the \(r \times r\) minors of \(M\) is 1, then \(\mathcal {L}^\sharp \) is generated by the columns of \(P\).
2.4 The cellular matroid
Many ideas of the paper may be expressed efficiently using the language of matroids. For a general reference on matroids, see, e.g., [31]. We will primarily consider cellular matroids. The cellular matroid of \(\Sigma \) is the matroid \(\mathcal {M}(\Sigma )\) represented over \(\mathbb {R}\) by the columns of the boundary matrix \(\partial \). Thus, the ground set of \(\mathcal {M}(\Sigma )\) naturally corresponds to the \(d\)dimensional cells \(\Sigma _d\), and the matroid records which sets of columns of \(\partial \) are linearly independent. If \(\Sigma \) is a graph, then \(\mathcal {M}(\Sigma )\) is its usual graphic matroid, while if \(\Sigma \) is a simplicial complex, then \(\mathcal {M}(\Sigma )\) is its simplicial matroid [10].
While the language of matroids will frequently be useful, it is important to point out that most of the objects of interest to us, such as the cut and flow lattices and the critical group of a cell complex \(\Sigma \), are not purely combinatorial invariants of its cellular matroid \(\mathcal {M}(\Sigma )\). (See [32] for more on this subject, and [11, 19] for generalizations of matroids that contain finer arithmetic information). As an example, the summands in Eq. (3) are indexed by the bases of \(\mathcal {M}(\Sigma )\), but the summands themselves are not part of the matroid data. (On the other hand, when \(\Sigma \) is a graph, all summands are 1.)
Linear algebra  Graph  Matroid  Cell complex 

Column vectors  Edges  Ground set  Facets 
Independent set  Acyclic subgraph  Independent set  Acyclic subcomplex 
Min linear dependence  Cycle  Circuit  Circuit 
Basis  Spanning forest  Basis  CSF 
Set meeting all bases  Disconnecting set  Codependent set  Cut 
Min set meeting all bases  Bond  Cocircuit  Bond 
Rank  # edges in spanning forest  Rank  # facets in CSF 
Here “codependent” means dependent in the dual matroid.
3 Enumerating cellular spanning forests
In this section, we study the enumerative properties of cellular spanning forests of an arbitrary cell complex \(\Sigma \). Our setup is essentially the same as that of Lyons [29], Sect. 6], but the combinatorial formulas we will need later, namely Propositions 3.2 and 3.4, are somewhat different. As a corollary, we obtain an enumerative result, Proposition 3.5, which generalizes the simplicial and cellular matrixtree theorems of [15] and [16] (in which we required that \(\Sigma \) be \(\mathbb {Q}\)acyclic in codimension one). The result is closely related, but not quite equivalent, to Lyons’ generalization of the cellular matrixtree theorem [29, Corollary 6.2], and to [8, Corollary D].
The arguments require some tools from homological algebra, in particular, the long exact sequence for relative homology and some facts about the torsionsubgroup functor. The details of the proofs are not necessary to understand the constructions of cut and flow spaces in the later sections.
Let \(\Sigma \) be a \(d\)dimensional cell complex with rank \(r\). Let \(\Gamma \subseteq \Sigma \) be a subcomplex of dimension less than or equal to \(d1\) such that \(\Gamma _{(d2)}=\Sigma _{(d2)}\). Thus, the inclusion map \(i:\Gamma \rightarrow \Sigma \) induces isomorphisms \(i_*:\tilde{H}_k(\Gamma ;\mathbb {Q})\rightarrow \tilde{H}_k(\Sigma ;\mathbb {Q})\) for all \(k<d2\).
Definition 3.1
The subcomplex \(\Gamma \subseteq \Sigma \) is called relatively acyclic if in fact the inclusion map \(i:\Gamma \rightarrow \Sigma \) induces isomorphisms \(i_*:\tilde{H}_k(\Gamma ;\mathbb {Q})\rightarrow \tilde{H}_k(\Sigma ;\mathbb {Q})\) for all \(k<d.\)
By the long exact sequence for relative homology, \(\Gamma \) is relatively acyclic if and only if \(\tilde{H}_d(\Sigma ;\mathbb {Q})\rightarrow \tilde{H}_d(\Sigma ,\Gamma ;\mathbb {Q})\) is an isomorphism and \(\tilde{H}_k(\Sigma ,\Gamma ;\mathbb {Q})=0\) for all \(k<d\). These conditions can occur only if \(\Gamma _{d1}=\Sigma _{d1}r\). This quantity may be zero (in which case the only relatively acyclic subcomplex is \(\Sigma _{(d2)}\)). A relatively acyclic subcomplex is precisely the complement of a \((d1)\)cobase (a basis of the matroid represented over \(\mathbb {R}\) by the rows of the boundary matrix \(\partial \)) in the terminology of Lyons [29].
Two special cases are worth noting. First, if \(d=1\), then a relatively acyclic complex consists of one vertex in each connected component. Second, if \(\tilde{H}_{d1}(\Sigma ;\mathbb {Q})=0\), then \(\Gamma \) is relatively acyclic if and only if it is a cellular spanning forest of \(\Sigma _{(d1)}\).
For a matrix \(M\), we write \(M_{A,B}\) for the restriction of \(M\) to rows indexed by \(A\) and columns indexed by \(B\).
Proposition 3.2

(a) The \(r\times r\) square matrix \(\hat{\partial }=\partial _{R,\Upsilon }\) is nonsingular.

(b) \(\tilde{H}_d(\Upsilon ,\Gamma ;\mathbb {Q})=0\).

(c) \(\tilde{H}_{d1}(\Upsilon ,\Gamma ;\mathbb {Q})=0\).

(d) \(\Upsilon \) is a cellular spanning forest of \(\Sigma \) and \(\Gamma \) is relatively acyclic.
Proof
Lemma 3.3
Proof
Since \(C\) is finite, we have \(\ker j={{\mathrm{im}}}h\subseteq \mathbf{T}D\), and so replacing \(D,E\) with their torsion summands preserves exactness. The same argument implies that we can replace \(D',E'\) with \(\mathbf{T}D',\mathbf{T}E'\).
Second, note that \(A,A',B,B'\) all have the same rank (since the rows are exact, \(C,C'\) are finite, and \(\alpha \) is an isomorphism). Hence, \(f(A)\) is a maximalrank free submodule of \(B\); we can write \(B=\mathbf{T}B\oplus F\), where \(F\) is a free summand of \(B\) containing \(f(A)\). Likewise, write \(B'=\mathbf{T}B'\oplus F'\), where \(F'\) is a free summand of \(B'\) containing \(f'(A')\). Meanwhile, \(\beta \) is surjective, and hence must restrict to an isomorphism \(F\rightarrow F'\), which induces an isomorphism \(F/f(A)\rightarrow F'/f'(A')\). Abbreviating this last group by \(G\), we obtain the desired diagram (7) . Since \(\ker g={{\mathrm{im}}}f\subseteq F\), the map \(g:\mathbf{T}B\oplus G\rightarrow \mathbf{T}C\) is injective, proving exactness of the first row; the second row is exact by the same argument. Exactness of each row implies that the alternating product of the cardinalities of the groups is 1, from which the Eq. (8) follows. \(\square \)
Proposition 3.4
Proof
As a consequence, we obtain a version of the cellular matrixforest theorem that applies to all cell complexes (not only those that are \(\mathbb {Q}\)acyclic in codimension one).
Proposition 3.5
Proof
Remark 3.6
4 The cut space
There are two natural ways to construct bases of the cut space of a graph, in which the basis elements correspond to either (a) vertex stars or (b) the fundamental circuits of a spanning forest (see, e.g. [21, Chap. 14]). The former is easy to generalize to cell complexes, but the latter involves more work.
First, if \(G\) is a graph on vertex set \(V\) and \(R\) is a set of (“root”) vertices, one in each connected component, then the rows of \(\partial \) corresponding to the vertices \(V{\setminus } R\) form a basis for \({{\mathrm{Cut}}}_1(G)\). This observation generalizes easily to cell complexes:
Proposition 4.1
A set of \(r\) rows of \(\partial \) forms a row basis if and only if the corresponding set of \((d1)\)cells is the complement of a relatively acyclic \((d1)\)subcomplex.
This is immediate from Proposition 3.2. Recall that if \(\tilde{H}_{d1}(\Sigma ;\mathbb {Q})=0\), then “relatively acyclic \((d1)\)subcomplex” is synonymous with “spanning tree of the \((d1)\)skeleton.” In this case, Proposition 4.1 is also a consequence of the fact that the matroid represented by the rows of \(\partial _d\) is dual to the matroid represented by the columns of \(\partial _{d1}\) [16, Proposition 6.1].
The second way to construct a basis of the cut space of a graph is to fix a spanning tree and take the signed characteristic vectors of its fundamental bonds. In the cellular setting, it is not hard to show that each bond supports a unique (up to scaling) vector in the cut space (Lemma 4.4) and that the fundamental bonds of a fixed cellular spanning forest give rise to a vector space basis (Theorem 4.8). (Recall from Sect. 2 that a bond in a cell complex is a minimal collection of facets whose removal increases the codimensionone homology, or equivalently a cocircuit of the cellular matroid.) The hard part is to identify the entries of these cutvectors. For a graph, these entries are all 0 or \(\pm 1\). In higher dimension, this need not be the case, but the entries can be interpreted as the torsion coefficients of certain subcomplexes (Theorem 4.11). In Sect. 5, we will prove analogous results for the flow space.
Remark 4.2
Although many of our results may be stated in terms of algebraic chain complexes over \(\mathbb {Z}\) (integer boundary matrices), we use the language of cell complexes. (This is a difference only in terminology, not the generality of the results, since every integer matrix is the topdimensional boundary matrix of some cell complex.) Thus, definitions and results about column bases, row bases, rank, etc., can be interpreted topologically in terms of cellular spanning trees and creating and puncturing holes in cell complexes (see Example 4.9 and Fig. 1). This is analogous to the situation in algebraic graph theory, where results that can be stated in terms of matrices are often more significant in terms of trees, cuts, flows, etc.
4.1 A basis of cutvectors
Proposition 4.3
[31, Proposition 9.2.4]. Let \(M\) be an \(r\times n\) matrix with rowspace \(V\subseteq \mathbb {R}^n\), and let \(\mathcal {M}\) be the matroid represented by the columns of \(M\). Then the cocircuits of \(\mathcal {M}\) are the inclusionminimal elements of the family \({{\mathrm{Supp}}}(V) := \{{{\mathrm{supp}}}(v) :v\in V{\setminus }\{0\}\}\).
Lemma 4.4
Proof
Suppose that \(v,w\) are vectors in the cut space, both supported on \(B\), that are not scalar multiples of each other. Then there is a linear combination of \(v,w\) with strictly smaller support; this contradicts Proposition 4.3. On the other hand, Proposition 4.3 also implies that \({{\mathrm{Cut}}}_B(\Sigma )\) is not the zero space; therefore, it has dimension 1. \(\square \)
We now know that for every bond \(B\), there is a cutvector supported on \(B\) that is uniquely determined up to a scalar multiple. As we will see, there is a choice of scale so that the coefficients of this cutvector are given by certain minors of the downup Laplacian \(L=L^{\mathrm {du}}_d(\Sigma )=\partial ^{*}\partial \) (Lemma 4.6); these minors (up to sign) can be interpreted as the cardinalities of torsion homology groups (Theorem 4.11).
Lemma 4.5
Let \(B\) be a bond of \(\Sigma \) and let \(U\) be the space spanned by \(\{\partial \sigma :\sigma \in \Sigma _d{\setminus } B\}\). In particular, \(U\) is a subspace of \({{\mathrm{im}}}\partial \) of codimension one. Let \(V\) be the orthogonal complement of \(U\) in \({{\mathrm{im}}}\partial \), and let \(v\) be a nonzero element of \(V\). Then \({{\mathrm{supp}}}(\partial ^{*}v)=B\).
Proof
First, we show that \(\partial ^{*}v\ne 0\). To see this, observe that the column space of \(\partial \) is \(U + \mathbb {R}v\), and so the column space of \(\partial ^{*}\partial \) is \(\partial ^{*}U + \mathbb {R}\partial ^{*}v\). However, \({{\mathrm{rank}}}(\partial ^{*}\partial )={{\mathrm{rank}}}\partial =r\), and \(\dim U=r1\); therefore, \(\partial ^{*}v\) cannot be the zero vector. Second, if \(\sigma \in \Sigma _d {\setminus } B\), then \(\partial \sigma \in U\), and so \(\left\langle \partial ^{*}v,\sigma \right\rangle = \left\langle v,\partial \sigma \right\rangle = 0.\) It follows that \({{\mathrm{supp}}}(\partial ^{*}v)\subseteq B\), and in fact \({{\mathrm{supp}}}(\partial ^{*}v)=B\) by Proposition 4.3. \(\square \)
Lemma 4.6
Proof
Equation (10) does not provide a canonical cutvector associated to a given bond \(B\), because \(\partial ^{*}v\) depends on the choice of \(A\) and \(\sigma \). On the other hand, the bond \(B\) can always be expressed as a fundamental bond \({{\mathrm{\mathsf {bo}}}}(\Upsilon ,\sigma )\) (equivalently, fundamental cocircuit; see Eq. 4 in Sect. 2.4) by taking \(\sigma \) to be an arbitrary facet of \(B\) and taking \(\Upsilon =A\cup \sigma \), where \(A\) is a maximal acyclic subset of \(\Sigma {\setminus } B\). This observation suggests that the underlying combinatorial data that gives rise to a cutvector is really the pair \((\Upsilon ,\sigma )\).
Definition 4.7
The next result is the cellular analogue of [21, Lemma 14.1.3].
Theorem 4.8
The family \(\{\bar{\chi }(\Upsilon ,\sigma ) :\sigma \in \Upsilon \}\) is an \(\mathbb {R}\)vector space basis for the cut space of \(\Sigma \).
Proof
Let \(\sigma \in \Upsilon \). Then \({{\mathrm{supp}}}\bar{\chi }(\Upsilon ,\sigma )={{\mathrm{\mathsf {bo}}}}(\Upsilon ,\sigma )\) contains \(\sigma \), but no other facet of \(\Upsilon \). Therefore, the set of characteristic vectors is linearly independent, and its cardinality is \(\Upsilon _d=r=\dim {{\mathrm{Cut}}}_d(\Sigma )\). \(\square \)
Example 4.9
4.2 Calibrating the characteristic vector of a bond
Let \(\varepsilon ^A_{\sigma ,\sigma '}\) be the relative sign of \(\partial \sigma ,\partial \sigma '\) with respect to \(\partial A\); that is, it is \(+1\) or \(1\) according to whether \(\partial \sigma \) and \(\partial \sigma '\) lie on the same or on the opposite sides of the hyperplane in \({{\mathrm{im}}}\partial \) spanned by \(\partial A\). In the language of oriented matroids, this sign is simply a product of the entries corresponding to \(\partial \sigma \) and \(\partial \sigma '\) in one of the cocircuits corresponding to the hyperplane and determines the relative signs of a basis orientation on \(A \cup \sigma \) and \(A \cup \sigma '\) [6, Sect. 3.5].
Proposition 4.10
Proof
A corollary of the proof is that \(\mu _\Upsilon \) is an integer, for the following reason. The number \(\mathbf{t}_{d1}(\Upsilon ')\) is the gcd of the \(r \times r\) minors of \(\partial _{\Upsilon '}\), or equivalently the \(r \times r\) minors of \(\partial \) using columns \(\Upsilon '\). In other words, \(\mathbf{t}_{d1}(\Upsilon ')\) divides \(\det \partial _{S,\Upsilon '}\) in every summand of Eq. (12). Therefore, it divides \(\det L^{\mathrm {du}}_{\Upsilon ,\Upsilon '}\), and \(\mu _\Upsilon =\pm \det L^{\mathrm {du}}_{\Upsilon ,\Upsilon '}/\mathbf{t}_{d1}(\Upsilon ')\) is an integer.
Theorem 4.11
Proof
Apply the formula of Proposition 4.10 to the formula of Definition 4.7 for the characteristic vector, and factor the integer \(\mu _\Upsilon \) out of every coefficient. Meanwhile, replacing \(\sigma \) with a different facet \(\sigma '\in B\) merely multiplies all coefficients by \(\varepsilon ^A_{\sigma ',\rho }/\varepsilon ^A_{\sigma ,\rho }=\varepsilon ^A_{\sigma ,\sigma '}\in \{\pm 1\}\). \(\square \)
Remark 4.12
Example 4.13
Let \(\Theta \) be the bipyramid of Example 4.9. Every cellular spanning forest \(\Upsilon \subseteq \Theta \) is torsionfree. Moreover, the relatively acyclic subcomplexes \(\Gamma \) that appear in Eq. (11) are the spanning trees of the 1skeleton \(\Theta _{(1)}\) (see Sect. 3), which is the graph \(K_5\) with one edge removed; Accordingly, we have \(\mu _\Upsilon =\tau (\Theta _{(1)})=75\), and so the calibrated characteristic vectors are as given in Example 4.9, with all factors of 75 removed.
On the other hand, \(\mu _\Upsilon \) is not necessarily the greatest common factor of the entries of each uncalibrated characteristic vector, as the following example illustrates.
Example 4.14
Remark 4.15
As an illustration of where torsion plays a role, and of the principle that the cellular matroid \(\mathcal {M}(\Sigma )\) does not provide complete information about cutvectors, let \(\Sigma \) be the the complete \(2\)dimensional simplicial complex on \(6\) vertices, which has complexity \(6^6=46656\) [25, Theorem 1]. Most of the cellular spanning trees of \(\Sigma \) are contractible topological spaces, hence \(\mathbb {Z}\)acyclic, and the calibrated cutvectors obtained from them have all entries equal to 0 or \(\pm 1\). On the other hand, \(\Sigma \) has twelve spanning trees \(\Upsilon \) homeomorphic to the real projective plane (so that \(\tilde{H}_1(\Upsilon ;\mathbb {Z})\cong \mathbb {Z}_2\)). For any facet \(\sigma \in \Upsilon \), we have \({{\mathrm{\mathsf {bo}}}}(\Upsilon ,\sigma )=\Sigma _2{\setminus }\Upsilon _2\cup \{\sigma \}\), and the calibrated cutvector contains a \(\pm 2\) in position \(\sigma \) and \(\pm 1\)’s in positions \(\Sigma {\setminus }\Upsilon \).
Remark 4.16
When \(\Sigma \) is a graph and \(\Upsilon \) is a spanning forest, \(\mu _\Upsilon \) is just the number of vertices of \(\Sigma \). Then, for any edge \(\sigma \) in \(\Upsilon \), the vector \(\chi _{\Upsilon }(\sigma )\) is the usual characteristic vector of the fundamental bond \({{\mathrm{\mathsf {bo}}}}(\Upsilon ,\sigma )\).
Remark 4.17
5 The flow space
In this section, we describe the flow space of a cell complex. We begin by observing that the cut and flow spaces are orthogonal to each other.
Proposition 5.1
The cut and flow spaces are orthogonal complements under the standard inner product on \(C_d(\Sigma ;\mathbb {R})\).
Proof
First, we show that the cut and flow spaces are orthogonal. Let \(\alpha \in {{\mathrm{Cut}}}_d={{\mathrm{im}}}\partial ^{*}_d\) and \(\beta \in {{\mathrm{Flow}}}_d=\ker \partial _d\). Then \(\alpha = \partial ^{*}\gamma \) for some (\(d1\))chain \(\gamma \), and \(\langle \alpha ,\beta \rangle = \langle \partial ^{*}\gamma ,\beta \rangle = \langle \gamma ,\partial \beta \rangle = 0\) by Eq. (9).
It remains to show that \({{\mathrm{Cut}}}_d\) and \({{\mathrm{Flow}}}_d\) have complementary dimensions. Indeed, let \(n=\dim C_d(\Sigma ;\mathbb {R})\); then \(\dim {{\mathrm{Flow}}}_d = \dim \ker \partial _d = n\dim {{\mathrm{im}}}\partial _d = n\dim {{\mathrm{im}}}\partial ^{*}_d = n\dim {{\mathrm{Cut}}}_d\). \(\square \)
Next we construct a basis of the flow space whose elements correspond to fundamental circuits of a given cellular spanning forest. Although cuts and flows are in some sense dual constructions, it is easier in this case to work with kernels than images, essentially because of Proposition 2.1. As a consequence, we can much more directly obtain a characteristic flow vector whose coefficients carry topological meaning.
We need one preliminary result from linear algebra.
Proposition 5.2
Proof
Recall that a set of facets \(C\subseteq \Sigma _d\) is a circuit of the cellular matroid \(\mathcal {M}(\Sigma )\) if and only if it corresponds to a minimal linearly dependent set of columns of \(\partial _d\). Applying Proposition 5.2 with \(N=\partial _C\) (i.e., the restriction of \(\partial \) to the columns indexed by \(C\)), we obtain a flow vector whose support is exactly \(C\). We call this the characteristic vector \(\varphi (C)\).
Theorem 5.3
Proof
Example 5.4
For a cellular spanning forest \(\Upsilon \) and facet \(\sigma \not \in \Upsilon \), let \({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma )\) denote the fundamental circuit of \(\sigma \) with respect to \(\Upsilon \), that is, the unique circuit in \(\Upsilon \cup \sigma \).
Theorem 5.5
Proof
The flow space is the kernel of a matrix with \(\Sigma _d\) columns and rank \(\Upsilon _d\), so its dimension is \(\Sigma _d\Upsilon _d\). Therefore, it is enough to show that the \(\varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma ))\) are linearly independent. Indeed, consider the matrix \(W\) whose rows are the vectors \(\varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma ))\); its maximal square submatrix \(W'\) whose columns correspond to \(\Sigma {\setminus }\Upsilon \) has nonzero entries on the diagonal but zeroes elsewhere. \(\square \)
Example 5.6
Recall the bipyramid of Example 4.9, and its spanning tree \(\Upsilon \). Then \({{\mathrm{\mathsf {ci}}}}(\Upsilon , 125) = \{125, 123, 135, 235\}\), and \({{\mathrm{\mathsf {ci}}}}(\Upsilon , 134) = \{134, 123, 124, 234\}\). If we instead consider the spanning tree \(\Upsilon ' = \{124, 125, 134, 135, 235\}\), then \({{\mathrm{\mathsf {ci}}}}(\Upsilon ', 123) = \{123, 125, 135, 235\}\), and \({{\mathrm{\mathsf {ci}}}}(\Upsilon ', 234) = \{234, 124, 125, 134, 135, 235\}\). Each of these circuits is homeomorphic to a 2sphere, and the corresponding flow vectors are the homology classes they determine. Furthermore, each of \(\{\varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon , 125)), \varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon , 134))\}\) and \(\{\varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon ', 123)), \varphi ({{\mathrm{\mathsf {ci}}}}(\Upsilon ', 234))\}\) is a basis of the flow space.
6 Integral bases for the cut and flow lattices
Theorem 6.1
Proof
Theorem 6.2
Suppose that \(\Sigma \) has a cellular spanning forest \(\Upsilon \) such that \(\tilde{H}_{d1}(\Upsilon ;\mathbb {Z})=\tilde{H}_{d1}(\Sigma ;\mathbb {Z})\). Then \(\{\hat{\varphi }({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma )):\sigma \not \in \Upsilon \}\) is an integral basis for the flow lattice \(\mathcal {F}(\Sigma )\).
Proof
By the hypothesis on \(\Upsilon \), the columns of \(\partial \) indexed by the facets in \(\Upsilon \) form a \(\mathbb {Z}\)basis for the column space. That is, for every \(\sigma \not \in \Upsilon \), the column \(\partial _\sigma \) is a \(\mathbb {Z}\)linear combination of the columns of \(\Upsilon \); equivalently, there is an element \(w_\sigma \) of the flow lattice, with support \({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma )\), whose coefficient in the \(\sigma \) position is \(\pm 1\). However, then \(w_\sigma \) and \(\hat{\varphi }({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma ))\) are integer vectors with the same linear span, both of which have the gcd of their entries equal to 1; therefore, they must be equal up to sign. Therefore, retaining the notation of Theorem 5.5, the matrix \(W'\) is in fact the identity matrix, and it follows that the lattice spanned by the \(\hat{\varphi }({{\mathrm{\mathsf {ci}}}}(\Upsilon ,\sigma ))\) is saturated, so it must equal the flow lattice of \(\Sigma \). \(\square \)
If \(\Sigma \) is a graph, then all its subcomplexes and relative complexes are torsionfree (equivalently, its incidence matrix is totally unimodular). Therefore, Theorems 6.1 and 6.2 give integral bases for the cut and flow lattices, respectively. These are, up to sign, the integral bases constructed combinatorially in, e.g., [21, Chap. 14].
7 Groups and lattices
In this section, we define the critical, cocritical, and cutflow groups of a cell complex. We identify the relationships between these groups and to the discriminant groups of the cut and flow lattices. The case of a graph was studied in detail by Bacher, de la Harpe and Nagnibeda [2] and Biggs [4], and is presented concisely in [21, Chap. 14].
Throughout this section, let \(\Sigma \) be a cell complex of dimension \(d\) with \(n\) facets, and identify both \(C_d(\Sigma ;\mathbb {Z})\) and \(C^d(\Sigma ;\mathbb {Z})\) with \(\mathbb {Z}^n\).
Definition 7.1
Note that the second and third terms in the definition are equivalent because \(\ker \partial _{d1}\) is a summand of \(C_{d1}(\Sigma ;\mathbb {Z})\) as a free \(\mathbb {Z}\)module. This definition coincides with the usual definition of the critical group of a graph in the case \(d=1\), and with the authors’ previous definition in [17] in the case that \(\Sigma \) is \(\mathbb {Q}\)acyclic in codimension one (when \(\ker \partial _{d1}/{{\mathrm{im}}}\partial _{d}\partial ^{*}_{d}\) is its own torsion summand).
Definition 7.2
The cutflow group of \(\Sigma \) is \(\mathbb {Z}^n/(\mathcal {C}(\Sigma )\oplus \mathcal {F}(\Sigma ))\).
In order to define the cocritical group of a cell complex, we first need to introduce the notion of acyclization.
Definition 7.3
An acyclization of \(\Sigma \) is a (\(d+1)\)dimensional complex \(\Omega \) such that \(\Omega _{(d)}=\Sigma \) and \(\tilde{H}_{d+1}(\Omega ;\mathbb {Z})=\tilde{H}_d(\Omega ;\mathbb {Z})=0\).
Algebraically, this construction corresponds to finding an integral basis for \(\ker \partial _d(\Sigma )\) and declaring its elements to be the columns of \(\partial _{d+1}(\Omega )\) (so in particular \(\Omega _{(d+1)}=\tilde{\beta }_d(\Sigma )\)). Topologically, it corresponds to filling in just enough \(d\)dimensional cycles with \((d+1)\)dimensional faces to remove all \(d\)dimensional homology. The definition of acyclization and equation (1) together imply that \(\tilde{H}^{d+1}(\Omega ;\mathbb {Z}) = 0\); that is, \(\partial ^{*}_{d+1}(\Omega )\) is surjective.
Definition 7.4
It is not immediate that the group \(K^*(\Sigma )\) is independent of the choice of \(\Omega \); we will prove this independence as part of Theorem 7.7. For the moment, it is at least clear that \(K^*(\Sigma )\) is finite, since \({{\mathrm{rank}}}\partial ^{*}_{d+1}={{\mathrm{rank}}}L^{\mathrm {du}}_{d+1}={{\mathrm{rank}}}C_{d+1}(\Omega ;\mathbb {Z})\). In the special case of a graph, \(L^{\mathrm {du}}_{d+1}\) is the “intersection matrix” defined by Kotani and Sunada [26]. (See also [5, Sects. 2, 3].)
Remark 7.5
As in [17], one can define critical and cocritical groups in every dimension by \(K_i(\Sigma )=\mathbf{T}(C_i(\Sigma ;\mathbb {Z})/{{\mathrm{im}}}\partial _{i+1}\partial ^{*}_{i+1})\) and \(K^*_i(\Sigma )= \mathbf{T}(C_i(\Sigma ;\mathbb {Z})/{{\mathrm{im}}}\partial ^{*}_i\partial _i)\). If the cellular chain complexes of \(\Sigma \) and \(\Psi \) are algebraically dual (for example, if \(\Sigma \) and \(\Psi \) are Poincaré dual cell structures on a compact orientable \(d\)manifold), then \(K_i(\Psi )=K_{di}^*(\Sigma )\) for all \(i\).
We now come to the main results of the second half of the paper: the critical and cocritical groups are isomorphic to the discriminant groups of the cut and flow lattices, respectively, and the cutflow group mediates between the critical and cocritical groups, with an “error term” given by homology.
Theorem 7.6
Proof
The cellular boundary map \(\partial _d\) also gives rise to the map \(\beta :\mathcal {C}^\sharp /\mathcal {C}\rightarrow K(\Sigma )\), as we now explain. First, note that \(\partial _d\,\mathcal {C}^\sharp \subseteq {{\mathrm{im}}}_\mathbb {R}\partial _d\subseteq \ker _\mathbb {R}\partial _{d1}\). Second, observe that for every \(w\in \mathcal {C}^\sharp \) and \(\rho \in C_{d1}(\Sigma ;\mathbb {Z})\), we have \(\left\langle \partial w,\rho \right\rangle =\left\langle w,\partial ^{*}\rho \right\rangle \in \mathbb {Z}\), by equation (9) and the definition of dual lattice. Therefore, \(\partial _d\,\mathcal {C}^\sharp \subseteq C_{d1}(\Sigma ;\mathbb {Z})\). It follows that \(\partial _d\) maps \(\mathcal {C}^\sharp \) to \((\ker _\mathbb {R}\partial _{d1})\cap C_{d1}(\Sigma ;\mathbb {Z})=\ker _\mathbb {Z}\partial _{d1}\), hence defines a map \(\beta :\mathcal {C}^\sharp /\mathcal {C}\rightarrow \ker _\mathbb {Z}\partial _{d1}/{{\mathrm{im}}}_\mathbb {Z}\partial _d\partial ^{*}_d\). Since \(\mathcal {C}^\sharp /\mathcal {C}\) is finite, the image of \(\beta \) is purely torsion, hence contained in \(K(\Sigma )\). Moreover, \(\beta \) is injective because \((\ker \partial _d)\cap \mathcal {C}^\sharp =\mathcal {F}\cap \mathcal {C}^\sharp =0\) by Proposition 5.1.
Every element of \(\mathbb {R}^n\) can be written uniquely as \(c+f\) with \(c\in {{\mathrm{Cut}}}(\Sigma )\) and \(f\in {{\mathrm{Flow}}}(\Sigma )\). The map \(\psi \) is orthogonal projection onto \({{\mathrm{Cut}}}(\Sigma )\), so \(\partial _d(c+f)=\partial _d c=\partial _d(\psi (c+f))\). Hence, the lefthand square commutes. The map \(\gamma \) is then uniquely defined by diagramchasing.
The snake lemma now implies that \(\ker \gamma =0\). Since the groups \(\mathbf{T}(\tilde{H}_{d1}(\Sigma ;\mathbb {Z}))\) and \(\mathbf{T}(\tilde{H}^d(\Sigma ;\mathbb {Z}))\) are abstractly isomorphic by equation (1), in fact \(\gamma \) must be an isomorphism and \({{\mathrm{coker}}}\gamma =0\) as well. Applying the snake lemma again, we see that all the vertical maps in (14) are isomorphisms. \(\square \)
Theorem 7.7
Proof
We will now show that \(\mathcal {F}^\sharp /\mathcal {F}\cong K^*(\Sigma )\). To see this, observe that \(\partial ^{*}_{d+1}(\mathcal {F}^\sharp )=\partial ^{*}_{d+1}({{\mathrm{colspace}}}(Q))={{\mathrm{colspace}}}(A^TQ)={{\mathrm{colspace}}}(A^T)={{\mathrm{im}}}\partial ^{*}_{d+1}=C_{d+1}(\Omega )\) (by the construction of an acyclization). In addition, \(\ker \partial ^{*}_{d+1}\) is orthogonal to \(\mathcal {F}^\sharp \), hence their intersection is zero. Therefore, \(\partial ^{*}_{d+1}\) defines an isomorphism \(\mathcal {F}^\sharp \rightarrow C_{d+1}(\Omega )\). Moreover, the same map \(\partial ^{*}_{d+1}\) maps \(\mathcal {F}=\ker \partial _d={{\mathrm{im}}}\partial _{d+1}\) surjectively onto \({{\mathrm{im}}}\partial ^{*}_{d+1}\partial _{d+1}\). \(\square \)
Corollary 7.8
If \(\tilde{H}_{d1}(\Sigma ;\mathbb {Z})\) is torsionfree, then the groups \(K(\Sigma )\), \(K^*(\Sigma )\), \(\mathcal {C}^\sharp /\mathcal {C}\), \(\mathcal {F}^\sharp /\mathcal {F}\), and \(\mathbb {Z}^n/(\mathcal {C}\oplus \mathcal {F})\) are all isomorphic to each other.
Corollary 7.8 includes the case that \(\Sigma \) is a graph, as studied by Bacher, de la Harpe and Nagnibeda [2] and Biggs [4]. It also includes the combinatorially important family of CohenMacaulay (over \(\mathbb {Z}\)) simplicial complexes, as well as cellulations of compact orientable manifolds.
Example 7.9
Suppose that \(\tilde{H}_d(\Sigma ;\mathbb {Z})=\mathbb {Z}\) and that \(\tilde{H}_{d1}(\Sigma ;\mathbb {Z})\) is torsionfree. Then the flow lattice is generated by a single element, and it follows from Corollary 7.8 that \(K(\Sigma )\cong K^*(\Sigma )\cong \mathcal {F}^\sharp /\mathcal {F}\) is a cyclic group. For instance, if \(\Sigma \) is homeomorphic to a cellular sphere or torus, then the critical group is cyclic of order equal to the number of facets. (The authors had previously proved this fact for simplicial spheres [17, Theorem 3.7], but this approach using the cocritical group makes the statement more general and the proof transparent.)
Example 7.10
Example 7.11
8 Enumeration
For a connected graph, the cardinality of the critical group equals the number of spanning trees. In this section, we calculate the cardinalities of the various group invariants of \(\Sigma \).
Examples 7.10 and 7.11 both indicate that \(K(\Sigma )\cong \mathcal {C}^\sharp /\mathcal {C}\) should have cardinality equal to the complexity \(\tau (\Sigma )\). Indeed, in Theorem 4.2 of [17], the authors proved that \(K(\Sigma )=\tau (\Sigma )\) whenever \(\Sigma \) has a cellular spanning tree \(\Upsilon \) such that \(\tilde{H}_{d1}(\Upsilon ;\mathbb {Z})=\tilde{H}_{d1}(\Sigma ;\mathbb {Z})=0\) (in particular, \(\Sigma \) must be not merely \(\mathbb {Q}\)acyclic, but actually \(\mathbb {Z}\)acyclic, in codimension one). Here, we prove that this condition is actually not necessary: for any cell complex, the order of the critical group \(K(\Sigma )\) equals the torsionweighted complexity \(\tau (\Sigma )\). Our approach is to determine the size of the discriminant group \(\mathcal {C}^\sharp /\mathcal {C}\) directly, then use the short exact sequences of Theorems 7.6 and 7.7 to calculate the sizes of the other groups.
Theorem 8.1
Proof
By Theorems 7.6 and 7.7, it is enough to prove that \(\mathcal {C}^\sharp /\mathcal {C}=\tau _d(\Sigma )\).
Dually, we can interpret the cardinality of the cocritical group as enumerating cellular spanning forests by relative torsion (co)homology, as follows:
Theorem 8.2
Note that the groups \(\tilde{H}^{d+1}(\Omega ,\Upsilon ;\mathbb {Z})\) and \(\tilde{H}_d(\Omega ,\Upsilon ;\mathbb {Z})\) are all finite, by definition of acyclization.
Proof
Remark 8.3
9 Bounds on combinatorial invariants from lattice geometry
As observed by Kotani and Sunada [26], if \(\mathcal {L}=\mathcal {F}\) is the flow lattice of a connected graph, then the shortest vector in \(\mathcal {F}\) is the characteristic vector of a cycle of minimum length; therefore, the numerator in equation (17) is the girth of \(G\). Meanwhile, \(\mathcal {F}^\sharp /\mathcal {F}\) is the number of spanning trees. We now generalize this theorem to cell complexes.
Definition 9.1
Let \(\Sigma \) be a cell complex. The girth and the connectivity are defined as the cardinalities of, respectively, the smallest circuit and the smallest cocircuit of the cellular matroid of \(\Sigma \).
Theorem 9.2
Proof
Every nonzero vector of the cut lattice \(\mathcal {C}\) contains a cocircuit in its support, so \(\min _{x\in \mathcal {C}{\setminus }\{0\}}\langle x,x\rangle \ge k\). Likewise, every nonzero vector of the flow lattice \(\mathcal {F}\) of \(\Sigma \) contains a circuit in its support, so \(\min _{x\in \mathcal {F}{\setminus }\{0\}}\langle x,x\rangle \ge g\). Meanwhile, \(\mathcal {C}^\sharp /\mathcal {C}=\tau \) and \(\mathcal {F}^\sharp /\mathcal {F}=\tau ^*\) by Theorem 8.1. The desired inequalities now follow from applying the definition of Hermite’s constant to the cut and flow lattices, respectively. \(\square \)
Footnotes
Notes
Acknowledgments
Third author was supported in part by a Simons Foundation Collaboration Grant and by National Security Agency Grant no. H982301210274. It is our pleasure to thank Andrew Berget, John Klein, Russell Lyons, Ezra Miller, Igor Pak, Dave Perkinson, and Victor Reiner for valuable discussions, some of which took place at the 23rd International Conference on Formal Power Series and Algebraic Combinatorics (Reykjavik, 2011). We are also grateful for the suggestions from an anonymous referee.
References
 1.Artin, M.: Algebra. PrenticeHall, Englewood Cliffs (1991)Google Scholar
 2.Bacher, R., de la Harpe, P., Nagnibeda, T.: The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France 125(2), 167–198 (1997)MATHMathSciNetGoogle Scholar
 3.Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)CrossRefMATHMathSciNetGoogle Scholar
 4.Biggs, N.L.: Chipfiring and the critical group of a graph. J. Algebr. Combin. 9(1), 25–45 (1999)CrossRefMATHMathSciNetGoogle Scholar
 5.Biggs, N.: The critical group from a cryptographic perspective. Bull. London Math. Soc. 39(5), 829–836 (2007)CrossRefMATHMathSciNetGoogle Scholar
 6.Björner, A., Vergnas, M.L., Sturmfels, B., White, N., Ziegler, G.: Oriented Matroids. Encyclopedia of Mathematics and its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)Google Scholar
 7.Bond, B., Levine, L.: Abelian Networks: Foundations and Examples. arXiv:1309.3445v1 [cs.FL] (2013).
 8.Catanzaro, M.J., Chernyak, V.Y., Klein, J.R.: Kirchhoff’s theorems in higher dimensions and Reidemeister torsion, Homol. Homotopy Appl. (in press) arXiv:1206.6783v2 [math.AT] (2012)
 9.Cohn, H., Kumar, A.: Optimality and uniqueness of the Leech lattice among lattices. Ann. Math. (2) 170(3), 1003–1050 (2009)CrossRefMATHMathSciNetGoogle Scholar
 10.Cordovil, R., Lindström, B.: Combinatorial Geometries, Encyclopedia of Mathematics and its Application. Simplicial matroids, vol. 29, pp. 98–113. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
 11.D’Adderio, M., Moci, L.: Arithmetic matroids, the Tutte polynomial and toric arrangements. Adv. Math. 232, 335–367 (2013)CrossRefMATHMathSciNetGoogle Scholar
 12.Dhar, D.: Selforganized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)CrossRefMATHMathSciNetGoogle Scholar
 13.Denham, Graham: The combinatorial Laplacian of the Tutte complex. J. Algebra 242, 160–175 (2001)CrossRefMATHMathSciNetGoogle Scholar
 14.Dodziuk, J., Patodi, V.K.: Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1–4, 1–52 (1977)Google Scholar
 15.Duval, A.M., Klivans, C.J., Martin, J.L.: Simplicial matrixtree theorems. Trans. Am. Math. Soc. 361, 6073–6114 (2009)CrossRefMATHMathSciNetGoogle Scholar
 16.Duval, A.M., Klivans, C.J., Martin, J.L.: Cellular spanning trees and Laplacians of cubical complexes. Adv. Appl. Math. 46, 247–274 (2011)CrossRefMATHMathSciNetGoogle Scholar
 17.Duval, A.M., Klivans, C.J., Martin, J.L.: Critical groups of simplicial complexes. Ann. Comb. 17, 53–70 (2013)CrossRefMATHMathSciNetGoogle Scholar
 18.Eckmann, B.: Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comment. Math. Helv. 17, 240–255 (1945)CrossRefMATHMathSciNetGoogle Scholar
 19.Fink, A., Moci, L.: Matroids over a ring, J. Eur. Math. Soc. (in press) arXiv:1209.6571v2 [math.CO] (2012)
 20.Friedman, J.: Computing Betti numbers via combinatorial Laplacians. Algorithmica 21(4), 331–346 (1998)CrossRefMATHMathSciNetGoogle Scholar
 21.Godsil, C., Royle, G.: Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207. Springer, New York (2001)CrossRefGoogle Scholar
 22.Haase, C., Musiker, G., Yu, J.: Linear systems on tropical curves. Math. Z 270(3–4), 1111–1140 (2012)CrossRefMATHMathSciNetGoogle Scholar
 23.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)Google Scholar
 24.Hungerford, T.W.: Algebra, Graduate Texts in Mathematics, vol. 73. Springer, New York (1974)Google Scholar
 25.Kalai, G.: Enumeration of \({\bf Q}\)acyclic simplicial complexes. Israel J. Math. 45(4), 337–351 (1983)CrossRefMATHMathSciNetGoogle Scholar
 26.Kotani, M., Sunada, T.: Jacobian tori associated with a finite graph and its abelian covering graphs. Adv. Appl. Math. 24(2), 89–110 (2000)CrossRefMATHMathSciNetGoogle Scholar
 27.Lagarias, J.: Point lattices, Handbook of Combinatorics, vol. 1. Elsevier, Amsterdam (1995)Google Scholar
 28.Lorenzini, D.J.: A finite group attached to the Laplacian of a graph. Discret. Math. 91(3), 277–282 (1991)CrossRefMATHMathSciNetGoogle Scholar
 29.Lyons, R.: Random complexes and \(\ell ^2\)Betti numbers. J. Topol. Anal. 1(2), 153–175 (2009)CrossRefMATHMathSciNetGoogle Scholar
 30.Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197–198, 143–176 (1994)CrossRefMathSciNetGoogle Scholar
 31.Oxley, J.: Matroid Theory. Oxford University Press, New York (1992)MATHGoogle Scholar
 32.Yi, S., Wagner, D.G.: The lattice of integer flows of a regular matroid. J. Comb. Theory Ser. B 100(6), 691–703 (2010)CrossRefMATHGoogle Scholar
 33.Tutte, W.T.: Lectures on matroids. J. Res. Nat. Bur. Stand. Sect. B 69B, 1–47 (1965)CrossRefMathSciNetGoogle Scholar