Related papers: Kalai's squeezed 3-spheres are polytopal

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the…

We consider unimodality and related properties of f-vectors of polytopes in various dimensions. By a result of Kalai (1988), f-vectors of 5-polytopes are unimodal. In higher dimensions much less can be said; we give an overview on current…

We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise…

We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication).

There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of…

Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee…

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well…

In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra. These tail bounds apply when the number of…

In 1992, Kalai and Kleitman proved that the diameter of a $d$-dimensional polyhedron with $n$ facets is at most $n^{2+\log_2 d}$. In 2014, Todd improved the Kalai-Kleitman bound to $(n-d)^{\log_2 d}$. We improve the Todd bound to…

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…

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible…

The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the…

Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later in 1979, Hakimi, Schmeichel and Thomassen conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles.…

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of…

Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for positive d. A d-graph is a type of combinatorial manifold which is inductively defined as a finite simple graph for which every unit sphere is a…

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k.…

In 2009, Calegari constructed smooth homotopy 4-spheres from monodromies of fibered knots. We prove that all these are diffeomorphic to the standard 4-sphere. Our method uses 5-dimensional handlebody techniques and results on mapping class…

Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and…

We describe a construction that takes as input a graph and a basis for its first homology, and returns a triangulation of a 3-dimensional homology sphere. This makes precise an idea of M. Gromov and A. Nabutovski. The immediate application,…

Let $D$ be a strong digraph on $n\geq 4$ vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) $d(x)+d(y)\geq 2n-1$ and $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq…