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

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug

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

Evasiveness of Graph Properties and Topological Fixed-Point Theorems

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive -- that is, any algorithm which computes a monotone graph property must check all edges in the worst case. This conjecture is unproven, but a lot of progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in 1984, topological methods have been applied to prove partial results on the Karp conjecture. This text is a tutorial on these topological methods. I give a fully self-contained account of the central proofs from the paper of Kahn, Saks, and Sturtevant, with no prior knowledge of topology assumed. I also briefly survey some of the more recent results on evasiveness.Comment: Book version, 92 page

Collapses, products and LC manifolds

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.Comment: 6 pages; added references; minor changes. Accepted for J. Comb. Theory, Series

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