arXiv subjects

Bruno Benedetti

Publications and source records attributed to Bruno Benedetti.

At least 19 recordsLinked to original sources

arXiv2026

Skeleton Chordalities

We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) $\Delta$ skeleton-E-chordal $\Rightarrow$ $\Delta^\vee$ vertex-decomposable $\Rightarrow$ $\Delta$ skeleton-clique-chordal. Moreover, for subflag complexes, $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ $\Delta^\vee$ vertex-decomposable. (For $d=1$ this boils down to ``$G$ chordal $\Longleftrightarrow$ $G^\vee$ vertex-decomposable'', a result closely related to Fr\"oberg's theorem.) (2) For subflag complexes, $\Delta$ is skeleton-E-chordal $\Longleftrightarrow$ it splits as $\Delta = \Delta_1 \cup \Delta_2$, with each $\Delta_i$ a skeleton-E-chordal induced subcomplex of $\Delta$, and with $\Delta_1 \cap \Delta_2$ a complex whose $1$-skeleton is a clique. (This generalizes ``$G$ chordal $\Longleftrightarrow$ $G$ splits as a union of chordal graphs that intersect in a common clique''). (3) $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ every nonempty induced subcomplex of $\Delta$ has a skeleton-E-simplicial vertex. (Generalizes ``$G$ chordal $\Leftrightarrow$ every nonempty induced subgraph has a simplicial vertex''.) (4) $\Delta$ underclosed $\Rightarrow$ $\Delta$ skeleton-weakly-chordal and weakly-closed. (Generalizes ``$G$ interval $\Rightarrow$ $G$ chordal and co-comparability''.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)

math.CO↗

arXiv2022

On the dual graph of Cohen-Macaulay algebras

Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)

math.AC↗

arXiv2022

Higher-dimensional counterexamples to Hamiltonicity

For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Gr\"unbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10 n^{\alpha}$, for some constant $\alpha<1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.

math.CO↗

arXiv2021

Random Simple-Homotopy Theory

We implement an algorithm RSHT (Random Simple-Homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions. For triangulated d-manifolds with d < 7, we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, (14k+1)-vertex triangulations of Bing's houses with k rooms, which all can be deformed to a point using only six pure elementary expansions.

cs.CG↗

arXiv2021

2-LC triangulated manifolds are exponentially many

We introduce "$t$-LC triangulated manifolds" as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion interpolates between the class of LC manifolds introduced by Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most $2^{d^2 \, N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove that there are at most $2^{\frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds with $N$ facets. This extends to all dimensions an intuition by Mogami for $d=3$. We also introduce "$t$-constructible complexes", interpolating between constructible complexes (the case $t=1$) and all complexes (case $t=d$). We show that all $t$-constructible pseudomanifolds are $t$-LC, and that all $t$-constructible complexes have (homotopical) depth larger than $d-t$. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen--Macaulay.

math.CO↗

arXiv2021

Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are: (1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result "unit-interval connected graphs are traceable".) (2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that "unit-interval 2-connected graphs are Hamiltonian".) (3) Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Groebner basis for a diagonal term order which is compatible with the traceability of the complex. (This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa-Vandebogert.) (4) Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution. (This extends the result by Kiani and Saeedi-Madani that "only the complete graph has a binomial edge ideal with linear resolution".) (5) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.

math.CO↗

arXiv2020

Non-ridge-chordal complexes whose clique complex has shellable Alexander dual

A recent conjecture that appeared in three papers by Bigdeli--Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the $k$-skeleton of the simplex is extendably shellable, for any $k$. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.

math.CO↗

arXiv2020

Sparse handlebody decompositions and non-finiteness of $g_3=0$

We prove that a PL manifold admits a handle decomposition into handles of index $\le k$ if and only if $M$ is $k$-stacked, i.e., it admits a PL triangulation in which all $(d-k-1)$-faces are on $\partial M$. We use this to solve a problem posed in 2008 by Kalai: In any dimension higher than four, there are infinitely many homology-spheres with $g_3 =0$.

math.GT↗

arXiv2020

Linear embeddings of contractible and collapsible complexes

(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a contractible 2-dimensional cell complex) is "hard to embed in $\mathbb{R}^3$", in the sense that lots of linear subdivisions are necessary. (2) For any d, we show that all collapsible d-complexes with n facets linearly embed in $\mathbb{R}^{2d}$ after less than n barycentric subdivisions. This is best possible, as cones over non-planar graphs do not topologically embed in $\mathbb{R}^{3}$.

math.MG↗

arXiv2019

Collapsibility of CAT(0) spaces

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All CAT(0) cube complexes are collapsible. (2) Any triangulated manifold admits a CAT(0) metric if and only if it admits collapsible triangulations. (3) All contractible d-manifolds ($d \ne 4$) admit collapsible CAT(0) triangulations. This discretizes a classical result by Ancel--Guilbault.

math.MG↗

arXiv2018

A Cheeger-type exponential bound for the number of triangulated manifolds

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger's finiteness theorem. Further consequences of our work are: (1) There are exponentially many geometric triangulations of $S^d$. (2) There are exponentially many convex triangulations of the d-ball.

math.CO↗

arXiv2017

Regularity of Line Configurations

We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of the coordinate ring of the arrangement. We also prove that every (d-1)-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth $\ge 3$. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang.

math.AG↗

arXiv2017

Barycentric subdivisions of convex complexes are collapsible

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d<4. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in $\mathbb{R}^d$ is simplicially collapsible after d-2 derived subdivisions at most. This presents progress on an old question by Goodrick.

math.CO↗

arXiv2017

Regulating Hartshorne's connectedness theorem

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity $\le 3$, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.

math.AG↗

arXiv2016

Extremal examples of collapsible complexes and random discrete Morse theory

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, there are collapsible (and shellable) simplicial $d$-complexes with only one free face. Also, there are non-evasive $d$-complexes with only two free faces. (Both results are optimal in all dimensions.) (2) Optimal discrete Morse vectors need not be unique. We explicitly construct a contractible, but non-collapsible $3$-dimensional simplicial complex with face vector $f=(106,596,1064,573)$ that admits two distinct optimal discrete Morse vectors, $(1,1,1,0)$ and $(1,0,1,1)$. Indeed, we show that in every dimension $d\geq 3$ there are contractible, non-collapsible simplicial $d$-complexes that have $(1,0,\dots,0,1,1,0)$ and $(1,0,\dots,0,0,1,1)$ as distinct optimal discrete Morse vectors. (3) We give a first explicit example of a (non-PL) $5$-manifold, with face vector $f=(5013,72300,290944,$ $495912,383136,110880)$, that is collapsible but not homeomorphic to a ball. Furthermore, we discuss possible improvements and drawbacks of random approaches to collapsibility and discrete Morse theory. We will introduce randomized versions \texttt{random-lex-first} and \texttt{random-lex-last} of the \texttt{lex-first} and \texttt{lex-last} discrete Morse strategies of \cite{BenedettiLutz2014}, respectively --- and we will see that in many instances the \texttt{random-lex-last} strategy works significantly better than Benedetti--Lutz's (uniform) \texttt{random} strategy. On the theoretical side, we prove that after repeated barycentric subdivisions, the discrete Morse vectors found by randomized algorithms have, on average, an exponential (in the number of barycentric subdivisions) number of critical cells asymptotically almost surely.

math.CO↗

arXiv2016

Mogami manifolds, nuclei, and 3D simplicial gravity

Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer would imply a much-desired exponential upper bound for the total number of 3-balls (and 3-spheres) with N tetrahedra. Here we provide a negative answer: many 3-balls are not Mogami. On the way to this result, we characterize the Mogami property in terms of nuclei, in the sense of Collet-Eckmann-Younan: "The only three-dimensional Mogami nucleus is the tetrahedron".

math-ph↗

arXiv2014

Smoothing discrete Morse theory

After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, and conversely, every PL handle vector is also a discrete Morse vector on some PL triangulation. It follows that in dimension up to 7, every discrete Morse vector on some PL triangulation is also a smooth Morse vector; the vice versa is true in all dimensions. This revises and improves a result by Gallais. Some further consequences of our work are: (1) For $d \ne 4$, every simply connected smooth d-manifold admits locally constructible triangulations. In contrast, the Mazur 4-manifold has no locally constructible triangulation. (This solves a question by Zivaljevic and completes work by the author and Ziegler.) (2) The Heegaard genus of 3-manifolds can be characterized as the smallest integer g for which some triangulation of the manifold has discrete Morse vector (1,g,g,1). (This allows for heuristics to bound the Heegaard genus of any 3-manifold.) (3) Some non-PL 5-spheres admit discrete Morse functions with only 2 critical faces. (This result, joint with Adiprasito, completes the Sphere Theorem by Forman.)

math.GT↗

arXiv2014

Discrete Morse Theory Is At Least As Perfect As Morse Theory

In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show that some subdivision of the triangulation admits a boundary-critical discrete Morse function with c_i interior critical faces of dimension d-i. This dualizes and extends a recent result by Gallais. Further consequences of our work are: (1) Every simply connected smooth d-manifolds (except possibly when d=4) admits a locally constructible triangulation. (This solves a problem by Zivaljevic.) (2) Up to refining the subdivision, the classical notion of geometric connectivity can be translated combinatorially via the notion of collapse depth.

math.DG↗