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.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes has been removed and forms a new paper, called "Barycentric subdivisions of convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration of manifolds has also been removed and forms now a third paper, called "A Cheeger-type exponential bound for the number of triangulated manifolds" (arXiv:1710.00130

Knots in collapsible and non-collapsible balls

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.Comment: 25 pages, 5 figures, 11 tables, references update

Collapsing along monotone poset maps

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $\phi:P\to P$, the simplicial complex $\Delta(P)$ {\tt NE}-reduces to $\Delta(Q)$, for any $Q\supseteq{\text{\rm Fix}}\phi$. As a corollary, we prove that for any order-preserving map $\phi:P\to P$ satisfying $\phi(x)\geq x$, for any $x\in P$, the simplicial complex $\Delta(P)$ collapses to $\Delta(\phi(P))$. We also obtain a generalization of Crapo's closure theorem.Comment: To appear in the International Journal of Mathematics and Mathematical Science

One-Point Suspensions and Wreath Products of Polytopes and Spheres

It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.Comment: 17 pages, 8 figure

Random Discrete Morse Theory and a New Library of Triangulations

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 na\"ivet\'e, 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.Comment: 35 pages, 5 figures, 7 table

Computing Optimal Morse Matchings

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results