Related papers: On flag spheres with few equators

We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain K\"uhnel-type bounds on the Betti numbers as well as on certain…

Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face…

In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $\Delta$ over $[n]$, shellable in the sense of Bj\"orner and Wachs \cite{BW}. We show that the spanning simplicial complex of…

There is only one type of tilings of the sphere by $12$ congruent pentagons. These tilings are isohedral.

The aim of this paper is to give a survey of the known results concerning centrally symmetric polytopes, spheres, and manifolds. We further enumerate nearly neighborly centrally symmetric spheres and centrally symmetric products of spheres…

The computation of the effect of a simple monodromy defect in the case of a sphere with twisted boundary conditions is revisited and streamlined using earlier calculations for a similar system. Compact and explicit expressions are found for…

Starting from a finite simple graph $G$, for each eigenvalue $\theta$ of its adjacency matrix one can construct a convex polytope $P_G(\theta)$, the so called $\theta$-eigenpolytop of $G$. For some polytopes this technique can be used to…

We describe all possible structures of discrete vector field (discrete Morse functions) with minimal number of critical cells on the regular CW-complex for the 2-disk (1 cell), the 2-sphere (2 cells), the cylinder (2 cells) and Mobius band…

Answering a question of Wright, we show that spheres of any radius are always connected in the curve graph of surfaces $\Sigma_{2,0}, \Sigma_{1,3},$ and $\Sigma_{0,6}$, and the union of two consecutive spheres is always connected for…

The present work considers the properties of classes of generally convex sets in the plane known as $1$-semiconvex and weakly $1$-semiconvex. More specifically, the examples of open and closed weakly $1$-semiconvex but non $1$-semiconvex…

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with…

This paper continues investigation of the class of flag simple polytopes called 2-truncated cubes. It is an extended version of the short note Volodin (2012). A 2-truncated cube is a polytope obtained from a cube by sequence of truncations…

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Sigma: we exhibit a monomial combination of classical 3-term Pl\"ucker relations that yields a sum of products of determinants that are known to be…

For a simplicial complex or more generally Boolean cell complex $\Delta$ we study the behavior of the $f$- and $h$-vector under barycentric subdivision. We show that if $\Delta$ has a non-negative $h$-vector then the $h$-polynomial of its…

The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the…

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex…

A triangulation of a simplicial complex $\Delta$ is called uniform if the $f$-vector of its restriction to a face of $\Delta$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform…

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999…

Most applications of the hard Lefschetz theorem related to combinatorial properties of simplicial complexes involve their $h$-vectors. In the context of positivity properties involving $h$-vectors of flag spheres, $f$-vectors with a…

A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…