模元素链与可壳性
Chains of modular elements and shellability
摘要：
设L是一个格，允许一个长度为r的左模链，不一定是极大的。我们证明了如果L是分次的或链是模的，则L的（r-2）-骨架是顶点可分解的（因此是可壳的）。这证明了Hersh的一个猜想。在某些情况下，我们可以找到更高的埃克塞特炮弹。例如，如果左模链由某个最大长度链的每隔一个元素组成，则L本身是可壳的。应用这些结果，我们用子群格的拓扑给出了有限可解群的一个新的刻画。我们的主要工具放宽了EL标记的条件，允许多个升序链，只要它们在字典顺序上在非升序链之前。我们从EL-壳偏序集理论的结果，这样的标签。某些黑猩猩的可剥性就是这样一个结果。另一种观点是，具有这种标记的偏序集同伦等价（根据离散莫尔斯理论）于一个胞腔复形，其中细胞对应于弱下降链。
英文摘要：
Let L be a lattice admitting a left-modular chain of length r, not necessarily maximal. We show that if either L is graded or the chain is modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable). This proves a conjecture of Hersh. Under certain circumstances, we can find shellings of higher skeleta. For instance, if the left-modular chain consists of every other element of some maximum length chain, then L itself is shellable. We apply these results to give a new characterization of finite solvable groups in terms of the topology of subgroup lattices. Our main tool relaxes the conditions for an EL-labeling, allowing multiple ascending chains as long as they are lexicographically before non-ascending chains. We extend results from the theory of EL-shellable posets to such labelings. The shellability of certain skeleta is one such result. Another is that a poset with such a labeling is homotopy equivalent (by discrete Morse theory) to a cell complex with cells in correspondence to weakly descending chains.
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