Bruno Benedetti

Random Discrete Morse Theory and a New Library of Triangulations

We introduce random discrete Morse theory as a computational scheme to measure the complexity of a triangulation. The idea is to try to quantify the frequency of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naiveté, this approach turns out to be very successful even in the case of huge inputs. 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.

On locally constructible spheres and balls

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:– Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.)There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.)– All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.)– Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)

Subdivisions, shellability, and collapsibility of products

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 Rudins 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 PL approximation theorems 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.

The Hirsch Conjecture Holds for Normal Flag Complexes

Using an intuition from metric geometry, we prove that any flag normal simplicial complex satisfies the nonrevisiting 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.

Tight complexes in 3-space admit perfect discrete Morse functions

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 Chillingworths 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.

Unmixed Graphs that are Domains

We extend a theorem of Villareal on bipartite graphs to the class of all graphs. On the way to this result, we study the basic covers algebra of an arbitrary graph G. We characterize with purely combinatorial methods the cases when 1) is a domain and 2) G is unmixed and is a domain.

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”.