On stellated spheres and a tightness criterion for combinatorial manifolds

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• 作者：Bhaskar Bagchi ; Basudeb Datta
• 刊名：European Journal of Combinatorics
• 出版年：February, 2014
• 年：2014
• 卷：36
• 期：Complete
• 页码：294-313
• 全文大小：465 K

文摘

We introduce -stellated spheres and consider the class of triangulated -manifolds, all of whose vertex links are -stellated, and its subclass , consisting of the -neighbourly members of . 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 for . 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 for . As another application, we prove that, when , all members of are tight. We also characterize the tight members of in terms of their 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 provide the equality case. This generalizes a result (the case) due to Walkup and K眉hnel. As a consequence, it is shown that every tight member of is strongly minimal, thus providing substantial evidence in favour of a conjecture of K眉hnel and Lutz asserting that tight homology manifolds should be strongly minimal.