The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere has the componentwise minimal $h$-vector. It is also verified that the $l$-skeleton of a flag $(d-1)$-dimensional 2-CM complex is $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-sphere is $2(d-l)$-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.
We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.
A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Bjorner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of a-balanced CM complexes, while the second is the theorem of Novik (necessity) and Browder (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.
A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifol…
Let G_0,...,G_k be finite abelian groups and let G_0*...*G_k be the join of the 0-dimensional complexes G_i. We give a characterization of the integral k-coboundaries of subcomplexes of G_0*...*G_k in terms of the Fourier transform on the group G_0 \times ... \times G_k. This leads to an extension of a recent result of Musiker and Reiner on a topological interpretation of the cyclotomic polynomial.
The flag f-vectors of three-colored complexes are characterized. This also characterizes the flag h-vectors of balanced Cohen-Macaulay complexes of dimension two, as well as the flag h-vectors of balanced shellable complexes of dimension two.
The class of $(d-1)$-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the $h$-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the $h$-numbers of flag $m$-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.
We show that if a $d$-dimensional Cohen-Macaulay complex is, in a certain sense, sufficiently "close" to being balanced, then there is a $d$-dimensional balanced Cohen-Macaulay complex having the same $f$-vector. This in turn provides some partial evidence for a conjecture of Kalai on the $f$-vectors of Cohen-Macaulay flag complexes.
We revisit several known versions of the Dehn--Sommerville relations in the context of: homology manifolds, semi-Eulerian complexes, general simplicial complexes, balanced semi-Eulerian complexes and general completely balanced complexes. In addition, we present Dehn--Sommerville relations for reciprocal complexes and general balanced simplicial complexes; which slightly generalize some of the previous results. Our proofs are uniform, and are based on two simple evaluations of the $\tilde h$-polynomial: one that recovers the $\tilde f$-polynomial, and one that counts faces according to certain…
Given a finite CAT(0) cubical complex, we define a flag simplicial complex associated to it, called the crossing complex. We show that the crossing complex holds much of the combinatorial information of the original cubical complex: for example, hyperplanes in the cubical complex correspond to vertex links in the crossing complex, and the crossing complex is balanced if and only if the cubical complex is cubically balanced. The most significant result is that the sets of $f$-vectors of CAT(0) cubical complexes and flag simplicial complexes are equal, up to an invertible linear transformation.
We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that rank selected subcomplexes of balanced simplicial complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial compex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings $S/I$, where $S$ is a polynomial ring and $I\subseteq S$ is an homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they on…
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial $d$-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every $m \geq d+2$ and any closed combinatorial $d$-manifold $M$, two $m$-colored triangulations of $M$ can be connected by …
Doran, Jensen and Giansiracusa showed a bijection between homogeneous elements in the Cox ring of $\overline{M}_{0,n}$ not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of $\overline{M}_{0,n}$, we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of $\overline{M}_{0,n}$ for large $n…
We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector satisfy several inequalities. We apply these results to the study of $P$-partitions of double posets, and weak colorings of mixed graphs.
Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $μ_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $μ_{k}(X)$ for $k\geq d$ and $μ_{d-1}(X)$. In particular, we establish the following vanishing result: If $μ_{d-1}(X)>(1-\binom{k+1}{d}^{-1})n$, then $\tilde{H}^{j}(X;\mathbb{R})=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval and Montejano f…
Given any order ideal $U$ consisting of color-squarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional simplicial complex $Δ_{\mathrm{bal}}(U)$ that we call a balanced squeezed complex. In fact, these complexes have properties similar to squeezed balls as introduced by Kalai and the more general squeezed complexes, introduced by the authors. We show that any balanced squeezed complex is vertex-decomposable and that its flag $h$-vector can be read off from the underlying order ideal. Moreover, we describe explicitly its Stanley-Reisner ideal $I…
Given any finite simplicial complex Δ, we show how to construct a new simplicial complex Δ_χ that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex Δ_χ is precisely the f-vector, denoted f(Δ), of the original complex Δ. We deduce this result by relating f(Δ) with the graded Betti numbers of the Alexander dual of Δ_χ. Our construction generalizes the "whiskering" construction of Villarreal, and Cook and Nagel. As a corollary of our work, we add a new equivalent statement to a theorem of Björner, Frankl, and Stanley that classifies the f-vectors o…
We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.
Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $τ\subset V$ such that $τ\notin X$ but $σ\in X$ for any $σ\subsetneq τ$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension la…
Let $G$ be a finite group of order $n$ and for $1 \leq i \leq k+1$ let $V_i=\{i\} \times G$. Viewing each $V_i$ as a $0$-dimensional complex, let $Y_{G,k}$ denote the simplicial join $V_1*\cdots*V_{k+1}$. For $A \subset G$ let $Y_{A,k}$ be the subcomplex of $Y_{G,k}$ that contains the $(k-1)$-skeleton of $Y_{G,k}$ and whose $k$-simplices are all $\{(1,x_1),\ldots,(k+1,x_{k+1})\} \in Y_{G,k}$ such that $x_1\cdots x_{k+1} \in A$. Let $L_{k-1}$ denote the reduced $(k-1)$-th Laplacian of $Y_{A,k}$, acting on the space $C^{k-1}(Y_{A,k})$ of real valued $(k-1)$-cochains of $Y_{A,k}$. The $(k-1)$-th …
A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are $(i+1)$-banner but not $i$-banner are constructed. It is shown that several theorems for flag complexes have appropriate $i$-banner analogues. Among them are (1) the codimension-$(i+j-1)$ skeleton of an $i$-banner homology sphere $Δ$ is $2(i+j)$-Cohen--Macaulay for all $0\leq j\leq \dimΔ+1-i$, and (2) for every $i$-banner simplicial compl…
Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $σ\notin X$ such that $τ\in X$ for any $τ\subsetneq σ$. For a $k$-dimensional simplex $σ$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $δ_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $μ_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $μ_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: \[…
We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $Δ$, we show that there exists a balanced complex with the same $f$-vector as $Δ$, and whose top-dimensional Betti number is at least that of $Δ$, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of $Δ$ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the $f$-vector of $Δ$, in terms of the top-dimensional Betti n…
We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $β_1\neq 0$ and $β_2=0$, where $β_i$'s are the Betti numbers, computed with coefficients in $\mathbb Q$.
Let $M$ be a closed triangulable manifold, and let $Δ$ be a triangulation of $M$. What is the smallest number of vertices that $Δ$ can have? How big or small can the number of edges of $Δ$ be as a function of the number of vertices? More generally, what are the possible face numbers ($f$-numbers, for short) that $Δ$ can have? In other words, what restrictions does the topology of $M$ place on the possible $f$-numbers of triangulations of $M$? To make things even more interesting, we can add some combinatorial conditions on the triangulations we are considering (e.g., flagness, balancedness, et…
We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Δ$, we show that its $γ$-vector $γ^Δ=(1,γ_1,γ_2,\ldots)$ satisfies: \begin{align*} γ_j=0,\text{ for all } j>γ_1, \quad γ_2\leq\binom{γ_1}{2}, \quad γ_{γ_1}\in\{0,1\}, \quad \text{ and }γ_{γ_1-1}\in\{0,1,2,γ_1\}, \end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Δ$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $γ$-vectors of flag ho…
