具有指定顶点链接集合的组合流形的构造

Constructions of $d$-spheres from $(d-1)$-spheres and $d$-balls with same set of vertices

Given a combinatorial $(d-1)$-sphere $S$, to construct a combinatorial $d$-sphere $S^{\hspace{.2mm}\prime}$ containing $S$, one usually needs some more vertices. Here we consider the question whether we can do one such construction without the help of any additional vertices. We show that this question has affirmative answer when $S$ is a flag sphere, a stacked sphere or a join of spheres. We also consider the question whether we can construct an $n$-vertex combinatorial $d$-sphere containing a given $n$-vertex combinatorial $d$-ball.

顶点数不超过 $2d+7$ 的离散 $d$-伪流形的完全分类

A Complete Classification of Discrete $d$-Pseudomanifolds with at Most $2d+7$ Vertices

本文完全分类了顶点数不超过2d+7的离散d-伪流形，证明了最小顶点数为2(d+1)，并建立了与旗正规d-伪流形边图的等价性，给出了旗正规d-伪流形为球面的最优刻画，构造了非球面例子。

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the oth…

Combinatorial 3-manifolds with transitive cyclic symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many dis…

Many Triangulated 3-Spheres

We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about d-spheres and (d+1)-polytopes for fixed d >= 4.

Degree-regular triangulations of the double-torus

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic - 2 must contain 12 vertices. In 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in $\RR^3$. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-man…