# Cuts and flows of cell complexes

## 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 complex### Mathematics Subject Classification

05C05 05C21 05C50 05E45 11H06## 1 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 \((d-1)\)-cochains gives a model of discrete flow on \(\Sigma \), generalizing the chip-firing 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].

*cut and flow spaces*and

*cut and flow lattices*of a cell complex \(\Sigma \) by

*cocritical group*\(K^*(\Sigma )\), and the

*cutflow group*\(\mathbb {Z}^n/(\mathcal {C}\oplus \mathcal {F})\). We show that the critical and cocritical groups are, respectively, isomorphic to the discriminant groups of the cut lattice and flow lattice, and that the cutflow group mediates between them with an “error term” given by homology. Specifically, if \(\dim \Sigma = d\), then we have the short exact sequences:

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 torsion-weighted 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 chip-firing game. Our central goal is to extend the theory from graphs to higher-dimensional spaces. Thus, we work in the setting of a finite CW complex, regarded as the higher-dimensional 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., \(\Sigma _{(i)}=\Sigma _i\cup \Sigma _{i-1}\cup \cdots \cup \Sigma _0\). Again, in keeping with simplicial-complex 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 codimension-1 skeleton, i.e., \(\Gamma _{(d-1)}=\Sigma _{(d-1)}\). 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, \(\partial _i(\Sigma ;R):C_i(\Sigma ;R) \rightarrow C_{i-1}(\Sigma ;R)\) and \(\partial ^{*}_i(\Sigma ;R):C_{i-1}(\Sigma ;R) \rightarrow C_i(\Sigma ;R)\); 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 vertex-edge 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 top-dimensional boundary map (which can be any integer matrix).

*acyclic in codimension one*if \(\tilde{H}_{d-1}(\Sigma ;R)=0\). For a graph (\(d=1\)), both \(\mathbb {Q}\)- and \(\mathbb {Z}\)-acyclicity in codimension one are equivalent to connectedness. The \(i\)th reduced Betti number is \(\tilde{\beta }_i(\Sigma )=\dim \tilde{H}_i(\Sigma ;\mathbb {Q})\), and the \(i\)th torsion coefficient \(\mathbf{t}_i(\Sigma )\) is the cardinality of the torsion subgroup \(\mathbf{T}(\tilde{H}_i(\Sigma ;\mathbb {Z}))\). We will frequently use the fact that

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

*cellular spanning forest*(CSF) of \(\Sigma \) to be a subcomplex \(\Upsilon \subseteq \Sigma \) such that \(\Upsilon _{(d-1)}=\Sigma _{(d-1)}\) and

^{1}of a graph \(G\): respectively, it is acyclic, has \(c\) components, and has \(n-c\) edges, where \(n\) and \(c\) are the numbers of vertices and components of \(G\). Just as in the graphic case, any two of the conditions (2a), (2b), (2c) together imply the third; the proof is just a slight modification of the proof of [15, Proposition 3.5]. An equivalent and perhaps simpler definition is that a subcomplex \(\Upsilon \subseteq \Sigma \) is a cellular spanning forest if and only if its \(d\)-cells correspond to a column basis for the cellular boundary matrix \(\partial =\partial _d(\Sigma )\); however, the definition focusing on integral homology is frequently the most useful (see, e.g., Remark 4.15).

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 higher-dimensional 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.

*up-down*,

*down-up*and

*total*Laplacian operators

^{2}on \(\Sigma \) by

*complexity*of \(\Sigma \) as

*cellular matrix-tree theorem*[16, Theorem 2.8] states that if \(\Sigma \) is \(\mathbb {Q}\)-acyclic in codimension one and \(L_{\bar{\Upsilon }}\) is the submatrix of \(L^{\mathrm {ud}}_{d-1}(\Sigma )\) obtained by deleting the rows and columns corresponding to the facets of a \((d-1)\) spanning tree \(\Upsilon \), then

### 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 finite-dimensional 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.

*dual lattice*of \(\mathcal {L}\) by

*integral*if it is contained in its dual; for instance, any subgroup of \(\mathbb {Z}^n\) is an integral lattice. The

*discriminant group*(or

*determinantal group*) of an integral lattice \(\mathcal {L}\) is \(\mathcal {L}^\sharp / \mathcal {L}\); its cardinality can be calculated as \(\det M^TM\), for any matrix \(M\) whose columns form an integral basis of \(\mathcal {L}\). We will need the following facts about bases and duals of lattices:

**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].

*cut*if deleting \(B\) from \(\Sigma \) increases its codimension-one homology, i.e., \(\tilde{\beta }_{d-1}(\Sigma {\setminus } B)>\tilde{\beta }_{d-1}(\Sigma )\). A cut \(B\) is a

*bond*if \(r(\Sigma {\setminus } B) = r(\Sigma )-1\), but \(r((\Sigma {\setminus } B) \cup \sigma )=r(\Sigma )\) for every \(\sigma \in B\). That is, a bond is a minimal cut. In matroid terminology, a bond of \(\Sigma \) is precisely a cocircuit of \(\mathcal {M}(\Sigma )\), i.e., a minimal set that meets every basis of \(\mathcal {M}(\Sigma )\). Equivalently, a bond is the complement of a flat of rank \(r(\Sigma )-1\). If \(\Upsilon \) is a cellular spanning forest (i.e., a basis of \(\mathcal {M}(\Sigma )\)) and \(\sigma \in \Upsilon _d\) is a facet, then the

*fundamental bond*of the pair \((\Upsilon ,\sigma )\) is

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 matrix-tree 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 matrix-tree 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 torsion-subgroup 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 \(d-1\) such that \(\Gamma _{(d-2)}=\Sigma _{(d-2)}\). 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 \(k<d-2\).

**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 \(|\Gamma _{d-1}|=|\Sigma _{d-1}|-r\). This quantity may be zero (in which case the only relatively acyclic subcomplex is \(\Sigma _{(d-2)}\)). A relatively acyclic subcomplex is precisely the complement of a \((d-1)\)-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 \(\tilde{H}_{d-1}(\Sigma ;\mathbb {Q})=0\), then \(\Gamma \) is relatively acyclic if and only if it is a cellular spanning forest of \(\Sigma _{(d-1)}\).

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) \(\tilde{H}_{d-1}(\Upsilon ,\Gamma ;\mathbb {Q})=0\).

(d) \(\Upsilon \) is a cellular spanning forest of \(\Sigma \) and \(\Gamma \) is relatively acyclic.

*Proof*

*torsion subgroup*of a finitely generated abelian group \(A\) is defined as the subgroup

**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 maximal-rank 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 matrix-forest 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

*cut space*and \(i\)-

*flow space*of \(\Sigma \) are defined, respectively, as the spaces of cellular coboundaries and cellular cycles:

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 \((d-1)\)-cells is the complement of a relatively acyclic \((d-1)\)-subcomplex.

This is immediate from Proposition 3.2. Recall that if \(\tilde{H}_{d-1}(\Sigma ;\mathbb {Q})=0\), then “relatively acyclic \((d-1)\)-subcomplex” is synonymous with “spanning tree of the \((d-1)\)-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 \(\partial _{d-1}\) [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 codimension-one homology, or equivalently a cocircuit of the cellular matroid.) The hard part is to identify the entries of these cut-vectors. 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 top-dimensional 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 cut-vectors

*support*of a vector \(v=(v_1,\dots ,v_n)\in \mathbb {R}^n\) is the set

**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 inclusion-minimal 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 cut-vector 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 cut-vector are given by certain minors of the down-up 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 \(\dim U=r-1\); 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 cut-vector 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 cut-vector is really the pair \((\Upsilon ,\sigma )\).

**Definition 4.7**

*(uncalibrated) characteristic vector*of the bond \({{\mathrm{\mathsf {bo}}}}(\Upsilon , \sigma )\) is

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*

*equatorial bipyramid*is the two-dimensional simplicial complex \(\Theta \) with facet set \(\{123,124,125,134,135,234,235\}\) (Fig. 1a). Let \(\Upsilon \) be the simplicial spanning tree with facets \(\{123,124,234,135,235\}\) (unfolded in Fig. 1b). Then

### 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 \(\mathbf{t}_{d-1}(\Upsilon ')\) 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, \(\mathbf{t}_{d-1}(\Upsilon ')\) divides \(\det \partial _{S,\Upsilon '}\) in every summand of Eq. (12). Therefore, it divides \(\det L^{\mathrm {du}}_{\Upsilon ,\Upsilon '}\), and \(\mu _\Upsilon =\pm \det L^{\mathrm {du}}_{\Upsilon ,\Upsilon '}/\mathbf{t}_{d-1}(\Upsilon ')\) is an integer.

**Theorem 4.11**

*characteristic vector of*\(B\)

*with respect to*\(A\) as

*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*

*characteristic vector*of the pair \((\Upsilon ,\sigma )\) by

*Example 4.13*

Let \(\Theta \) be the bipyramid of Example 4.9. Every cellular spanning forest \(\Upsilon \subseteq \Theta \) is torsion-free. Moreover, the relatively acyclic subcomplexes \(\Gamma \) that appear in Eq. (11) are the spanning trees of the 1-skeleton \(\Theta _{(1)}\) (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 cut-vectors, 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 cut-vectors 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 cut-vector 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 (\(d-1\))-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 2-sphere, 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

*cut lattice*and

*flow lattice*of \(\Sigma \) are defined as

**Theorem 6.1**

*Proof*

**Theorem 6.2**

Suppose that \(\Sigma \) has a cellular spanning forest \(\Upsilon \) such that \(\tilde{H}_{d-1}(\Upsilon ;\mathbb {Z})=\tilde{H}_{d-1}(\Sigma ;\mathbb {Z})\). 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 torsion-free (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**

*critical group*of \(\Sigma \) is

Note that the second and third terms in the definition are equivalent because \(\ker \partial _{d-1}\) is a summand of \(C_{d-1}(\Sigma ;\mathbb {Z})\) 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 \(\ker \partial _{d-1}/{{\mathrm{im}}}\partial _{d}\partial ^{*}_{d}\) 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**

*cocritical group*\(K^*(\Sigma )\) is

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 \(K_i(\Psi )=K_{d-i}^*(\Sigma )\) 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*

*Step 1: Construct the bottom row of*(14). The inclusions \({{\mathrm{im}}}\partial _d\partial ^{*}_d\subseteq {{\mathrm{im}}}\partial _d\subseteq \ker \partial _{d-1}\) give rise to the short exact sequence

*Step 2: Construct the top row of*(14). Let \(r={{\mathrm{rank}}}\partial _d\), let \(\{v_1,\dots ,v_r\}\) be an integral basis of \(\mathcal {C}\), and let \(V\) be the matrix with columns \(v_1,\dots ,v_r\). By Proposition 2.2, the dual basis \(\{v_1^*,\dots ,v_r^*\}\) for \(\mathcal {C}^\sharp \) consists of the columns of the matrix \(W=V(V^TV)^{-1}\). Let \(\psi \) be the orthogonal projection \(\mathbb {R}^n\rightarrow {{\mathrm{Cut}}}(\Sigma )\), which is given by the matrix \(P=WV^T=V(V^TV)^{-1}V^T\) (see Proposition 2.2). Then

*Step 3: Describe the vertical maps in*(14). The maps \(\alpha \) and \(\beta \) are each induced by \(\partial _d\) in the following ways. First, the image of the cutflow group under \(\partial _d\) is

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 \(\partial _d\,\mathcal {C}^\sharp \subseteq {{\mathrm{im}}}_\mathbb {R}\partial _d\subseteq \ker _\mathbb {R}\partial _{d-1}\). Second, observe that for every \(w\in \mathcal {C}^\sharp \) and \(\rho \in C_{d-1}(\Sigma ;\mathbb {Z})\), 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, \(\partial _d\,\mathcal {C}^\sharp \subseteq C_{d-1}(\Sigma ;\mathbb {Z})\). It follows that \(\partial _d\) maps \(\mathcal {C}^\sharp \) to \((\ker _\mathbb {R}\partial _{d-1})\cap C_{d-1}(\Sigma ;\mathbb {Z})=\ker _\mathbb {Z}\partial _{d-1}\), hence defines a map \(\beta :\mathcal {C}^\sharp /\mathcal {C}\rightarrow \ker _\mathbb {Z}\partial _{d-1}/{{\mathrm{im}}}_\mathbb {Z}\partial _d\partial ^{*}_d\). 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 left-hand square commutes. The map \(\gamma \) is then uniquely defined by diagram-chasing.

The snake lemma now implies that \(\ker \gamma =0\). Since the groups \(\mathbf{T}(\tilde{H}_{d-1}(\Sigma ;\mathbb {Z}))\) 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 \(\tilde{H}_{d-1}(\Sigma ;\mathbb {Z})\) is torsion-free, 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 Cohen-Macaulay (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 \(\tilde{H}_{d-1}(\Sigma ;\mathbb {Z})\) is torsion-free. 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 \(\tilde{H}_{d-1}(\Upsilon ;\mathbb {Z})=\tilde{H}_{d-1}(\Sigma ;\mathbb {Z})=0\) (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 torsion-weighted 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

*Hermite constant*\(\gamma _n\) is defined as the maximum value of

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. H98230-12-1-0274. 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.

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