arXiv subjects
Publications and source records attributed to Bruno Benedetti.
Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes, and more generally, for all (connected) flag homology manifolds.
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. (2) The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naïveté, this approach turns out to be very successful even in the case of huge inputs. (3) In our view the existing libraries of examples in computational topology are `too easy' for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.
We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudin's classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing. As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulation of a sphere or of a ball is PL if and only if it becomes shellable after sufficiently many derived subdivisions. This improves on results by Whitehead, Zeeman and Glaser, and answers a question by Billera and Swartz. We also show that any contractible complex can be made collapsible by repeatedly taking products with an interval. This strengthens results by Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an example that this behavior extends to non-evasiveness, thereby answering a question of Welker.
We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 18 vertices) that is not locally constructible. All these examples are based on knotted subcomplexes with only three edges; the knots are the trefoil, the double trefoil, and the triple trefoil, respectively. The more complicated the knot is, the more distant the triangulation is from being polytopal, collapsible, etc. Further consequences of our work are: (1) Unshellable 3-spheres may have vertex-decomposable barycentric subdivisions. (This shows the strictness of an implication proven by Billera and Provan.) (2) For d-balls, vertex-decomposable implies non-evasive implies collapsible, and for d=3 all implications are strict. (This answers a question by Barmak.) (3) Locally constructible 3-balls may contain a double trefoil knot as a 3-edge subcomplex. (This improves a result of Benedetti and Ziegler.) (4) Rudin's ball is non-evasive.
We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space. We show there is a simplicial 3-ball with 8 vertices that is collapsible, but also collapses onto the dunce hat, which is not collapsible. This 3-ball is as small as possible, because all triangulated 3-balls with fewer vertices are extendably collapsible. As we will see, the Alexander dual of the dunce hat is collapsible.
In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all convex simplicial 3-balls are non-evasive. In contrast, we show that many non-evasive 3-balls are not convex.
Given an arbitrary graph G, we study its basic covers algebra, which is the symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the case when G is bipartite, we characterize in a combinatorial fashion the situations when: 1) the basic covers algebra is a domain, and 2) it is a domain and in addition (the edge ideal of) G is unmixed. It turns out that the last result gives a complete characterization of those graphs for which any symbolic power of the edge ideal is generated by monomials of the same degree.
We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain "Relative Morse Inequalities" relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are: -- For each d \ge 3 and for each k \ge 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d-1)-cells. (This solves a problem by Chari.) -- For fixed d and k, there are exponentially many combinatorial types of simplicial d-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d-1)-cells. (This connects discrete Morse theory to enumerative combinatorics/discrete quantum gravity.) -- The barycentric subdivision of any constructible d-ball is collapsible. (This "almost" solves a problem by Hachimori.) -- Every constructible ball collapses onto its boundary minus a facet. (This improves a result by the author and Ziegler.) -- Any 3-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet. (This strengthens a classical result by Bing and a recent result by the author and Ziegler.)
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are: 1.) Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) 2.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.) 3.) All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.) 4.) Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with exactly one facet removed.
We study the basic $k$-covers of a bipartite graph $G$; the algebra $\AG$ they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when $\AG$ is a domain in terms of the combinatorics of $G$; if follows from a result of Hochster that when $\AG$ is a domain, it is also Cohen-Macaulay. We then study the dimension of $\AG$ by introducing a geometric invariant of bipartite graphs, the "graphical dimension". We show that the graphical dimension of $G$ is not larger than $\dim(\AG)$, and equality holds in many cases (e.g. when $G$ is a tree, or a cycle). Finally, we discuss applications of this theory to the arithmetical rank.