A Combinatorial Analogue of Poincaré's Duality Theorem
A Combinatorial Analogue of Poincaré's Duality Theorem
复制标题
庞加莱对偶定理的组合模拟
DOI：
10.4153/cjm-1964-053-0
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发表时间：
1964
期刊：
影响因子：
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通讯作者：
V. Klee
中科院分区：
文献类型：
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作者：
V. Klee
对于非负整数s和有限单纯复形K，设βS（K）表示K的s-维Betti数，fs（K）表示K的s-单形数.我们的定理，像庞加莱的一样，适用于组合流形M，但它涉及的是数fs（M）而不是数βS（M）。下面给出的公式之一被作者在（5）中用来建立n维凸多面体的顶点数的精确上界，这些凸多面体具有给定的i个（n - 1）-面。这相当于估计的大小的计算问题，可能涉及解决一个系统的i线性不等式在n个变量，是我们的研究的原始动机。
For a non-negative integer s and a finite simplicial complex K, let βS (K) denote the s-dimensional Betti number of K and let fs (K) denote the number of s-simplices of K. Our theorem, like Poincaré's, applies to combinatorial manifolds M, but it concerns the numbers fs (M) instead of the numbers βS (M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices of n-dimensional convex poly topes which have a given number i of (n — 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system of i linear inequalities in n variables, and was the original motivation for our study.