Rigidity of Balanced Minimal Cycle Complexes

Barnett’s lower bound theorem [2] asserts that the boundary complex $\Delta$ of a simplicial $d$-polytope satisfies the inequality

where $f_{i}(\Delta)$ denotes the number of $i$-dimensional faces of $\Delta$. Subsequently this was generalized to pseudomanifolds [7, 12]. Several variants of the lower bound theorem have been investigated (e.g. [9, 17]). In this paper we discuss the balanced lower bound theorem due to Goff, Klee, Novik [9] and Klee and Novik [13]. A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced (or completely balanced) if its graph $G(\Delta)$ is $d$-colorable. The balanced lower bound theorem [13] asserts that, for $d\geq 3$, every balanced normal $(d-1)$-pseudomanifold $\Delta$ satisfies the inequality

where $h_{1}(\Delta)=f_{0}(\Delta)-d$, $h_{2}(\Delta)=f_{1}(\Delta)-(d-1)f_{0}(\Delta)+\binom{d}{2}$.

A connection between the lower bound theorem and rigidity theory was pointed out by Kalai [12] (see also [10]). Kalai [12] noted that the inequality (1) follows immediately from the (generic) rigidity of the graph of $\Delta$ in $\mathbb{R}^{d}$. Goff, Klee, Novik [9] pointed out that the inequality (2) holds if $G(\Delta)[\kappa^{-1}(T)]$ is rigid in $\mathbb{R}^{3}$ for every $3$-set $T\subseteq[d]:=\{1,\ldots,d\}$, where $G[W]$ denotes the subgraph of $G$ induced by $W$ and $\kappa$ is the proper $d$-coloring of $G(\Delta)$. Klee and Novik [13] showed this rigidity result, and thus the balanced lower bound theorem, for balanced normal $(d-1)$-pseudomanifolds for $d\geq 3$. This rigidity result for balanced normal pseudomanifolds [13, Lemma 3.5] is based on the rigidity theorem of pseudomanifolds by Fogelsanger [7]. Fogelsanger introduced a superclass of pseudomanifolds, called minimal cycle complexes, and show that minimal cycle complexes admit a decomposition into minimal cycle complexes in such a way that the decomposition behaves nicely with respect to edge contractions. He used this decomposition together with vertex splitting [20] and gluing to show that the graph of any minimal $(d-1)$-cycle complex is rigid in $\mathbb{R}^{d}$ for $d\geq 3$. The remarkable aspect of this proof of rigidity is that the inductive proof works within the same dimension, so it is applicable to a class of simplicial complexes which is not closed under taking links. Fogelsanger’s idea is recently used to show the global rigidity of pseudomanifolds [4] and $\mathbb{Z}_{2}$-symmetric rigidity of $\mathbb{Z}_{2}$-symmetric pseudomanifolds [5].

In this paper, we give a new application of Fogelsanger’s idea to prove rigidity results of balanced simplicial complexes. We generalize the rigidity result [13, Lemma 3.5] of balanced normal pseudomanifolds to balanced minimal cycle complexes (Theorem 4.1). Using an argument by Goff, Klee, Novik [9], this immediately implies the balanced lower bound theorem for minimal cycle complexes (Corollary 4.4).

Balanced simplicial complexes, and more broadly $\bm{a}$-balanced simplicial complexes, have also been studied in the theory of Stanley-Reisner ring [16]. For $\bm{a}=(a_{1},\ldots,a_{m})\in\mathbb{Z}_{>0}^{m}$ with $\sum_{i=1}^{m}a_{i}=d$, a $(d-1)$-dimensional simplicial complex $\Delta$ on the vertex set $V(\Delta)$ is $\bm{a}$-balanced if there is a map $\kappa:V(\Delta)\rightarrow[m]$ satisfying $|F\cap\kappa^{-1}(i)|\leq a_{i}$ for any $F\in\Delta$ and $i\in[m]$. We call such a map $\kappa$ an $\bm{a}$-coloring of $\Delta$. Given a simplicial complex $\Delta$, the Stanley-Reisner ring of $\Delta$ is $\mathbb{R}[\Delta]:=\mathbb{R}[x_{v}:v\in V(\Delta)]/I_{\Delta}$, where $I_{\Delta}$ is the ideal generated by monomials $\prod_{v\in G}x_{v}$ over all $G\not\in\Delta$. For $\bm{a}\in\mathbb{Z}_{>0}^{m}$ and an $\bm{a}$-balanced simplicial complex $\Delta$ with an $\bm{a}$-coloring $\kappa$, one can define an $\mathbb{N}^{m}$-graded algebra structure of $\mathbb{R}[\Delta]$ by $\deg x_{v}=\bm{e}_{\kappa(v)}$, where $\bm{e}_{i}\in\mathbb{N}^{m}$ denotes the $i$th unit coordinate vector. Stanley [16] showed that, for an $\bm{a}$-balanced $(d-1)$-dimensional simplicial complex $\Delta$, $\mathbb{R}[\Delta]$ has a system of parameters $\theta_{1},\ldots,\theta_{d}$ such that exactly $a_{i}$ of $\theta_{j}$’s are of degree $\bm{e}_{i}$ for $i\in[m]$. Such a system of parameters is called an $\bm{a}$-colored s.o.p. for $\mathbb{R}[\Delta]$. Cook et al. [11] showed that if $\Delta$ is a balanced simplicial $2$-sphere, there is a $(1,1,1)$-colored s.o.p. $\Theta=(\theta_{1},\theta_{2},\theta_{3})$ for $\mathbb{R}[\Delta]$ and a linear form $\omega\in\mathbb{R}[\Delta]_{1}$ such that the multiplication map $(\times\omega):(\mathbb{R}[\Delta]/\Theta\mathbb{R}[\Delta])_{1}\rightarrow(\mathbb{R}[\Delta]/\Theta\mathbb{R}[\Delta])_{2}$ is bijective. We conjecture that the statement of Cook et al. [11] holds for a larger class of simplicial complexes and any $\bm{a}\in\mathbb{Z}_{>0}^{m}$ with a few exceptions as follows.

A correspondence between Stanley-Reisner ring theory and (skeletal) rigidity theory has been noted in [14, 18]. In this correspondence, under the normalization $\omega=\sum_{v\in V(\Delta)}x_{v}$, a linear system of parameters $(\theta_{1},\ldots,\theta_{d})$ for $\mathbb{R}[\Delta]$ is identified with a point configuration $p:V(\Delta)\rightarrow\mathbb{R}^{d}$ through $\theta_{i}=\sum_{v\in V(\Delta)}p(v)_{i}x_{v}$ for $i\in[d]$ (see Corollary 5.3). Thus Conjecture 1.1 is equivalently formulated as the problem of finding an infinitesimally rigid realization of $\bm{a}$-balanced minimal cycle complexes with non-generic point configurations. Let $G$ be a graph, $\kappa:V(G)\rightarrow[m]$ a map, and $\bm{a}\in\mathbb{Z}_{>0}^{m}$ an integer vector with $\sum_{i=1}^{m}a_{i}=d$. We consider $\mathbb{R}^{d}$ as the direct product of $\mathbb{R}^{a_{i}}$ over all $i=1,\ldots,m$. Then each $x\in\mathbb{R}^{d}$ is denoted by $x=(x_{1},\ldots,x_{m})$ with $x_{i}\in\mathbb{R}^{a_{i}}$. For $i=1,\ldots,m$, let $H_{i}:=\{x=(x_{1},\ldots,x_{m}):x_{j}=0~{}(j\neq i)\}\subseteq\mathbb{R}^{d}$. We say that a point configuration $p:V(G)\rightarrow\mathbb{R}^{d}$ is $(\kappa,\bm{a})$-sparse if $p(v)\in H_{\kappa(v)}$ for all $v\in V(G)$, and $G$ is $(\kappa,\bm{a})$-sparse rigid if $(G,p)$ is infinitesimally rigid in $\mathbb{R}^{d}$ for some $(\kappa,\bm{a})$-sparse point configuration $p$. For example, if $\bm{a}=(1,\ldots,1)$ and $\kappa$ is a $d$-coloring, then a $(\kappa,\bm{a})$-sparse point configuration is the one such that each vertex of color $i$ is realized on the $i$-th coordinate axis. Conjecture 1.1 can be restated as follows.

We will explain the equivalence of Conjecture 1.1 and Conjecture 1.2 in Section 5. The assumption on $\bm{a}$ in Conjecture 1.2 is necessary. Cook et al. [11] pointed out that there is a $(2,1)$-balanced simplicial $2$-sphere $\Delta$ and its $(2,1)$-coloring $\kappa$ such that $G(\Delta)$ is not $(\kappa,\bm{a})$-sparse rigid. As we will see in Example 7.6, the construction of Cook et al. [11] can be extended to higher dimension. In this paper we confirm Conjecture 1.2 for the following cases. In Theorem 6.1, we prove that Conjecture 1.2 holds if $a_{i}\geq 2$ for all $i\in[m]$ using Fogelsanger’s idea. In Theorem 7.2, we then apply the standard coning argument and show that Conjecture 1.2 holds if $d\geq 4$ and $\Delta$ is a homology $(d-1)$-manifold. The base cases in the coning argument are Gluck’s theorem [8], the result of Cook et al. [11, Theorem 1.1] and Theorem 6.1 for $\bm{a}=(2,2)$.

The paper is organized as follows. In Section 2, preliminaries on simplicial complexes and rigidity are given. In Section 3, we summarize the basic properties of Fogelsanger decomposition for minimal cycle complexes. In Section 4, we prove the rigidity of subgraph of $G(\Delta)$ induced by colors and derive the balanced lower bound theorem for a balanced minimal cycle complex $\Delta$. In Section 5, we discuss the equivalence of Conjecture 1.1 and Conjecture 1.2. In Section 6, we prove Conjecture 1.2 in the case when $a_{i}\geq 2$ for all $i$. In Section 7, we prove Conjecture 1.2 for homology manifolds. In Section 8, we give further observations related to Conjecture 1.2.

Let $\Delta$ be a simplicial complex and $uv\in E(\Delta)$. We define a simplicial complex $\Delta/uv$ by

The operation $\Delta\rightarrow\Delta/uv$ is called an edge contraction of $uv$ (onto $u$). We remark that we do not allow multiple copies of the same faces, so for $G\in\operatorname{lk}_{\Delta}(u)\cap\operatorname{lk}_{\Delta}(v)$, faces $G+u$ and $G+v$ of $\Delta$ are identified under the edge contraction of $uv$.

Let $\Gamma$ be an abelian group and let $\Delta$ be a simplicial complex on the vertex set $[n]$. An $i$-chain is a $\Gamma$-coefficient formal sum of $i$-faces. For an $i$-chain $c=\sum_{F}a_{F}F$, define an $(i-1)$-chain $\partial c$ by $\partial c:=\sum_{F}a_{F}\partial F$, where for $F=\{x_{1},\ldots,x_{i+1}\}$ ($x_{1}<\cdots<x_{i+1}$), $\partial F:=\sum_{j=1}^{i+1}(-1)^{j}F\setminus\{x_{j}\}$. An $i$-cycle is an $i$-chain $c$ satisfying $\partial c=0$. An $i$-chain $c^{\prime}=\sum_{F}b_{F}F$ is a subchain of an $i$-chain $c=\sum_{F}a_{F}F$ if for each $i$-face $F$, $b_{F}=a_{F}$ or $b_{F}=0$ holds. An $i$-cycle $c$ is a minimal $i$-cycle if its only subchains which are $i$-cycles are $c$ and $0$. The support of an $i$-chain $c=\sum_{F}a_{F}F$ is $\operatorname{supp}c:=\{F:a_{F}\neq 0\}\subseteq\binom{[n]}{i+1}$, where $\binom{[n]}{i+1}$ is the set of all $(i+1)$-subsets of $[n]$.

A $(d-1)$-dimensional simplicial complex $\Delta$ is called a minimal $(d-1)$-cycle complex (over $\Gamma$) if (i) $\Delta=\langle\{F\}\rangle$ for a $d$-set $F$ or (ii) $\Delta=\langle\operatorname{supp}c\rangle$ for a nonzero minimal $(d-1)$-cycle $c$. If the dimension is clear from the context, it is simply called a minimal cycle complex. We remark that although minimal cycle complexes are originally defined as simplicial complexes satisfying (ii), we include the case (i) to make the presentation simple. It is possible to unify (i) and (ii) by using multicomplex formulation as in [4]. A minimal cycle complex satisfying (i) (resp. (ii)) is said to be trivial (resp. nontrivial). A minimal cycle complex is always pure. Minimal cycle complexes may have singular point, so the class of minimal cycle complexes are not closed under taking links.

Pseudomanifolds are minimal cycle complexes over $\mathbb{Z}_{2}$ [4]. $2$-CM complexes over a field $\bm{k}$ are a minimal cycle complexes over a free $\bm{k}$-module of finite rank [15]. Other examples of minimal cycle complexes arise in the context of simplicial matroid. A $(d-1)$-simplicial matroid on a subset $\mathcal{E}$ of $\binom{[n]}{d}$ over a field $\bm{k}$ is a matroid such that $\{F_{1},\ldots,F_{k}\}\subseteq\mathcal{E}$ is independent if and only if $\{\partial F_{1},\ldots,\partial F_{k}\}$ is linearly independent. A simplicial complex spanned by the circuit of $(d-1)$-simplicial matroid over $\bm{k}$ is a minimal $(d-1)$-cycle complex over $\bm{k}$. A matroid on a ground set $E$ is connected if for any $e,f\in E$, there is a circuit containing both $e$ and $f$. A pure $(d-1)$-dimensional simplicial complex whose facets define a connected $(d-1)$-simplicial matroid over $\bm{k}$ is a minimal $(d-1)$-cycle complex over a free $\bm{k}$-module of finite rank. Thus a simplicial complex having a convex ear decomposition [3] is a minimal cycle complex.

We list basic properties of minimal cycle complexes.

Fogelsanger [7] pointed out that a minimal $(d-1)$-cycle can be decomposed into minimal $(d-1)$-cycles in such a way that it behaves nicely with respect to edge contractions. We summarize the properties of this decomposition below, which is necessary in the later discussion, in terms of minimal cycle complexes. See [4, 7] for more details.

We remark that, as in Lemma 3.2 (b), for the minimal cycle complexes $\Delta_{1}^{+},\ldots,\Delta_{m}^{+}$ given in Lemma 3.2, $\Delta_{i}^{+}$ may have a face not in $\Delta$. We also remark that the decomposition given in Lemma 3.2 is not unique.

We use the notation $[d]:=\{1,\ldots,d\}$. A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced (or completely balanced) if $G(\Delta)$ is $d$-colorable, or equivalently, there is a map $\kappa:V(\Delta)\rightarrow[d]$ such that for any face $F\in\Delta$, $|F\cap\kappa^{-1}(i)|\leq 1$ for $i\in[d]$. Such a coloring is called a proper coloring of $\Delta$. A proper coloring of a strongly connected simplicial complex is unique up to the permutation of colors and is called the proper coloring of $\Delta$. The boundary complex of a $d$-dimensional cross-polytope $\mathcal{C}_{d}^{*}:=\operatorname{conv}\{\pm\bm{e}_{i}:i\in[d]\}$, where $\bm{e}_{i}$ is the $i$th unit coordinate vector in $\mathbb{R}^{d}$, is an example of a balanced $(d-1)$-simplicial complex.

Let $\Delta$ be a balanced, strongly connected, $(d-1)$-dimensional simplicial complex and $\kappa$ be the proper coloring of $\Delta$. For $T\subseteq[d]$, the $T$-rank selected subcomplex of $\Delta$ is $\Delta_{T}:=\Delta[\kappa^{-1}(T)]$. Klee and Novik [13] showed that $G(\Delta_{T})$ is rigid in $\mathbb{R}^{|T|}$ for any $T\subseteq[d]$ with $|T|\geq 3$ if $\Delta$ is a balanced normal $(d-1)$-pseudomanifold with $d\geq 3$. We extend this result to the class of minimal $(d-1)$-cycle complexes as follows.

For a pure simplicial complex $\Delta$, we say that a vertex subset $U\subseteq V(\Delta)$ is $(\geq k)$-transversal if $|U\cap F|\geq k$ for every facet $F\in\Delta$. Since, for a proper coloring $\kappa$ of a pure simplicial complex $\Delta$ and $T\subseteq[d]$, $\kappa^{-1}(T)\subseteq V(\Delta)$ is $(\geq|T|)$-transversal, Theorem 4.1 is a corollary of the following lemma.

The balanced analogue of the lower bound theorem [2] has been investigated in [9, 13]. For positive integers $n,d$ with $n$ divisible by $d$, a stacked cross-polytopal $(d-1)$-sphere on $n$ vertices is the connected sum of $\frac{n}{d}-1$ copies of the boundary complex of the cross-polytope $\mathcal{C}_{d}^{*}$. A stacked cross-polytopal $(d-1)$-sphere $\Delta$ satisfies $2h_{2}(\Delta)=(d-1)h_{1}(\Delta)$, and the balanced lower bound theorem [13] asserts that $2h_{2}(\Delta)\geq(d-1)h_{1}(\Delta)$ holds for any balanced normal $(d-1)$-pseudomanifolds with $d\geq 3$. Goff, Klee, Novik [9] showed that for a balanced pure simplicial complex $\Delta$, this inequality follows from the rigidity of $G(\Delta_{T})$ in $\mathbb{R}^{3}$ for every $3$-set $T\subseteq[d]$. Hence Theorem 4.1 implies the following generalization of balanced lower bound theorem to balanced minimal cycle complexes.

For an $\mathbb{N}$-graded algebra $A$, the $i$th homogeneous component is denoted as $A_{i}$. An element of $A_{1}$ is called a linear form. For a sequence of linear forms $\Theta=(\theta_{1},\ldots,\theta_{d})$ of $\mathbb{R}[\Delta]$, the ideal of $\mathbb{R}[\Delta]$ generated by $\theta_{1},\ldots,\theta_{d}$ is denoted as $\Theta\mathbb{R}[\Delta]$. For a $(d-1)$-dimensional simplicial complex $\Delta$, a sequence of $d$ linear forms $\theta_{1},\ldots,\theta_{d}\in\mathbb{R}[\Delta]_{1}$ is called a linear system of parameters (l.s.o.p. for short) for $\mathbb{R}[\Delta]$ if $\dim_{\mathbb{R}}\mathbb{R}[\Delta]/\Theta\mathbb{R}[\Delta]<\infty$. For Stanley-Reisner ring, the following criterion of an l.s.o.p. is known (see [16]).