On stellated spheres and a tightness criterion for combinatorial manifolds
On stellated spheres and a tightness criterion for combinatorial manifolds
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关于星球和组合流形的紧性判据
DOI：
10.1016/j.ejc.2013.07.018
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
B. Datta
中科院分区：
文献类型：
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作者：
B. Bagchi;B. Datta
我们引入k-星状球面，并考虑三角化d-流形类Wk（d），其所有顶点链都是k-星状的，以及它的子类Wk（d），由Wk（d）的（k+ 1）-邻域成员组成.我们介绍了任何单纯复形的μ向量，并表明，在2-neighborly单纯复形的情况下，μ向量占主导地位的Betti数分量的向量，这两个向量是相等的紧单纯复形。我们能够估计/计算Wk（d）的任意2-邻域成员的μ-向量的分量的某些交错和，其中d≥ 2k.作为这一理论的结果，我们证明了这类三角流形的一个下界定理，并确定了当d≥ 2k + 2时，Wk <$（d）的成员的积分同调型。作为另一个应用，我们证明了当d <$2k + 1时，Wk <$（d）的所有成员都是紧的.我们还用k阶Betti数刻画了Wk ∈（2k + 1）的紧成员.这些结果或多或少地回答了最近的一个问题的Effenberger，也提供了一个统一的和概念上的紧性证明，所有已知的紧三角流形，除了两个。我们还证明了一个下界定理的同调流形中的成员W1（d）提供平等的情况。这推广了Walkup和Kühnel的一个结果（d= 4的情况）。结果表明，W 1（d）的每一个紧成员都是强极小的，从而为Kühnel和Lutz提出的紧同调流形应该是强极小的猜想提供了有力的证据.
We introduce k-stellated spheres and consider the class W k (d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W k∗(d), consisting of the (k+ 1)-neighbourly members of W k (d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W k (d) for d≥ 2 k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W k∗(d) for d≥ 2 k+ 2. As another application, we prove that, when d≠ 2 k+ 1, all members of W k∗(d) are tight. We also characterize the tight members of W k∗(2 k+ 1) in terms of their k th Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W 1 (d) provide the equality case. This generalizes a result (the d= 4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W 1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight homology manifolds should be strongly minimal.