Topological combinatorics in the view of substructures and minimal counterexamples of simplicial complexes

本文基于子结构和极小反例的结构分析，研究单纯复形的组合性质。特别地，我们研究了在拓扑组合学研究中起重要作用的可壳性、序列Cohen-Macaulay性和可分性，并给出了这些性质在维数至多为2的单纯复形类或旗复形类中与顶点集限制一致的反例.我们还研究了这些反例的纯骨架，并进行了相关的应用研究。

This study aimed analyses of combinatorial properties of simplicial complexes based on the structural analysis of substructures and minimal counterexamples. Especially, we studied shellability, sequential Cohen-Macaulayness, and partitionability, which play important roles in the study of topological combinatorics, and showed the counterexamples for these properties with respect to the restriction of the vertex set coincide for the class of simplicial complexes of dimension at most two, or in the class of flag complexes. We also studied the pure-skeletons of these counterexamples, and also proceeded some related applicational studies.