Three-dimensional normal pseudomanifolds with relatively few edges
Three-dimensional normal pseudomanifolds with relatively few edges
复制标题
具有相对较少边的三维正态赝流形
DOI：
10.1016/j.aim.2020.107035
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发表时间：
2018
影响因子：
1.7
通讯作者：
Ed Swartz
中科院分区：
文献类型：
--
作者：
Biplab Basak;Ed Swartz
设Δ是d维正规伪流形，d≥ 3。Δ中边数的一个相对下界是Δ的g2至少是任意顶点的链环的g2。当这个不等式很尖锐时，Δ有相对最小的g2.例如，只要Δ的单骨架等于顶点的星星的单骨架，则Δ具有相对最小的g 2。在这样的例子中，细分一个面也给出了一个具有相对最小g2的复形。我们证明，在三维这些是唯一的例子。作为应用，当Δ有两个或更少的奇点时，我们确定了具有相对最小g2的三维Δ的组合型和拓扑型.任何这样的复合体的拓扑类型都是伪压缩体，压缩体的伪流形形式。Kalai [12]（g2 = 0）、Nevo和Novinsky [13]（g2 = 1）和Zheng [20]（g2 = 2）给出了Δ满足g2（Δ）≤ 2的完整组合刻划。在这三种情况下，Δ都是单纯多胞形的边界。郑观察到，对于所有d≥ 0有三角形S d <$RP 2与g 2= 3。她问这是否是g 2（Δ）= 3时唯一可能的非球形拓扑。作为相对最小g2的另一个应用，当Δ为3维时，我们给出了肯定的回答.
Let Δ be a d-dimensional normal pseudomanifold, d≥ 3. A relative lower bound for the number of edges in Δ is that g 2 of Δ is at least g 2 of the link of any vertex. When this inequality is sharp Δ has relatively minimal g 2. For example, whenever the one-skeleton of Δ equals the one-skeleton of the star of a vertex, then Δ has relatively minimal g 2. Subdividing a facet in such an example also gives a complex with relatively minimal g 2. We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional Δ with relatively minimal g 2 whenever Δ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body. Complete combinatorial descriptions of Δ with g 2 (Δ)≤ 2 are due to Kalai [12](g 2= 0), Nevo and Novinsky [13](g 2= 1) and Zheng [20](g 2= 2). In all three cases Δ is the boundary of a simplicial polytope. Zheng observed that for all d≥ 0 there are triangulations of S d⁎ R P 2 with g 2= 3. She asked if this is the only nonspherical topology possible for g 2 (Δ)= 3. As another application of relatively minimal g 2 we give an affirmative answer when Δ is 3-dimensional.