Abstract 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.

中文翻译：

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边缘相对较少的三维正态伪流形

摘要 令Δ为d维正态伪流形，d≥3。Δ中边数的相对下限是Δ的g 2 至少是任何顶点的链接的g 2 。当这种不等式很尖锐时，Δ 具有相对最小的 g 2 。例如，每当 Δ 的一个骨架等于一个顶点的星体的一个骨架时，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 g 2 的复合体。我们证明在维度三中，这些是唯一的例子。作为一个应用，当 Δ 具有两个或更少的奇点时，我们用相对最小的 g 2 确定 3 维 Δ 的组合和拓扑类型。任何此类复合体的拓扑类型都是伪压缩体，即压缩体的伪流形版本。Δ 与 g 2 ( Δ ) ≤ 2 的完整组合描述归因于 Kalai [12] ( g 2 = 0 )、Nevo 和 Novinsky [13] ( g 2 = 1 ) 和 Zheng [20] ( g 2 = 2 ) . 在所有三种情况下，Δ 都是单纯多胞体的边界。郑观察到，对于所有 d ≥ 0，存在 S d ⁎ RP 2 与 g 2 = 3 的三角剖分。她问这是否是 g 2 ( Δ ) = 3 唯一可能的非球面拓扑结构。作为相对最小 g 2 的另一个应用，当 Δ 是 3 维时，我们给出肯定的答案。