相关论文: Centrally symmetric manifolds with few vertices

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…

This paper gives a combinatorial description of spin and spin^c-structures on triangulated PL-manifolds of arbitrary dimension. These formulations of spin and spin^c-structures are established primarily for the purpose of aiding in…

In 1995, Josckusch constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently, Novik and Zheng extended Jockusch's construction: for all $d$ and $n>d$, they…

For a given pair of numbers $(d,k)$, we establish the minimal number of vertices in pure $d$-dimensional simplicial complexes with non-trivial homology in dimension $k$. Furthermore, we solve the problem under the additional constraint of…

A stacked $d$-sphere $S$ is the boundary complex of a stacked $(d+1)$-ball, which is obtained by taking cone over a free $d$-face repeatedly from a $(d+1)$-simplex. A stacked sphere $S$ is called linear if every cone is taken over a face…

Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $\Delta$ is that $g_2$ of $\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $\Delta$ has…

A classical result by Pachner states that two $d$-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e.,…

We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n+d) vertices, where k(d,n)=Theta(d/(1+log ((d+n)/d))). We also show that this bound is tight.

The result of Padrol asserts that for every $d\geq 4$, there exist $2^{\Omega(n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that…

We consider causal 3-dimensional triangulations with the topology of $S^2\times [0,1]$ or $D^2\times [0,1]$ where $S^2$ and $D^2$ are the two-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that…

In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5,…

In 1987 Brehm and K\"uhnel showed that any triangulation of a $d$-manifold (without boundary) that is not homeomorphic to the sphere has at least $3d/2+3$ vertices. Moreover, triangulations with exactly $3d/2+3$ vertices may exist only for…

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar…

We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero…

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…

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le \alpha$, we prove that $\left(\sharp_{i=1}^{\alpha} M_{i}\right) \sharp (\alpha{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure,…

What is the maximum number of vertices that a centrally symmetric 2-neighborly polytope of dimension $d$ can have? It is known that the answer does not exceed $2^d$. Here we provide an explicit construction showing that it is at least…

We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the `strong $d$-step Theorem'…

Given a set $\Sigma$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $\rho_1,\rho_2,...,\rho_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(\Sigma)$ of…

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs).…