Posets, Regular CW Complexes and Bruhat Order
Posets, Regular CW Complexes and Bruhat Order
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Posets、常规 CW 复合体和 Bruhat 阶
DOI：
10.1016/s0195-6698(84)80012-8
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
A. Björner
中科院分区：
文献类型：
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作者：
A. Björner
本文讨论偏序集（偏序集）的拓扑解释中的若干问题。特别是，这类偏序集的组合对应的定期CW复形被挑出来。这类偏序集出现在一些纯粹的组合背景下，我们的主要目的是展示如何目前的考虑可以导致额外的洞察力，这样的例子。通常的做法是将偏序集P与其有限链的单纯复形d（P）联系起来。这样，每个偏序集通过d（P）的实现确定一个拓扑空间。然而，在某些情况下，d（P）具有更内在的细胞结构的细分的性质，其通常不是单纯的。事实上，几个组合的例子表明，能够直接解释的元素P作为拓扑细胞和顺序关系的细胞复杂的关联关系的愿望。当然，这只对有限的偏序集类是可能的，而胞腔复形的拓扑必须由偏序集明确确定的自然要求进一步限制了这类偏序集。在接下来的几节中，我们将讨论前面程序的精确条件。最适合于组合目的的胞腔复形类似乎是“正则CW复形”，我们建议将相应的偏序集称为“CW偏序集”。这样的偏序集由所有下区间都是球面的这一事实定义。讨论了正则CW复形的可壳性的一个概念，并通过相关的偏序集证明了单纯可壳性理论中的大多数重要事实的推广。一些组合标准的拓扑解释的偏序集，然后开发，例如以下：有限偏序集确定一个可壳正规CW分解的拓扑球当且仅当它是薄的和对偶字典序可壳。
In this paper some questions are discussed concerning the topological interpretation of Posets (partially ordered sets). In particular, the class of posets which are the combinatorial counterparts of regular CW complexes is singled out. Posets of this kind arise in some purely combinatorial contexts, and our main purpose is to show how the present considerations can lead to additional insight into such examples. It is common practice to associate with a poset P the simplicial complex, d (P) of its finite chains. In this manner each poset determines a topological space via the realization of, d (P). In some cases, however,, d (P) has the nature of a subdivision of a more intrinsic cell structure, which is in general not simplicial. Indeed, several combinatorial examples suggest the desirability of being able_to directly interpret the elements of P as topological cells and the order relation as the incidence relation of a cell complex. This is of course possible only for a restricted class of posets, and the natural requirement that the topology of the cell complex should be unambiguously determined by the poset restricts the class even further.In the next few sections the precise conditions for the preceding program are discussed. The most suitable class of cell complexes for combinatorial purposes seems to be the'regular CW complexes', and we propose to call the corresponding posets' CW posets'. Such posets are defined by the fact that all lower intervals are spherical. A certain notion of shellability for regular CW complexes is discussed and it is shown by way of the related posets that most of the important facts from simplicial shellability theory generalize. A number of combinatorial criteria for the topological interpretation of posets are then developed, for instance the following: a finite poset determines a shellable regular CW decomposition of a topological sphere if and only if it is thin and dual lexicographically shellable.