Computational topology of equipartitions by hyperplanes
Computational topology of equipartitions by hyperplanes
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超平面均分的计算拓扑
DOI：
10.12775/tmna.2015.004
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
R. Živaljević
中科院分区：
文献类型：
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作者：
R. Živaljević
在 2d-3j = 1 的情况下，我们通过两个超平面计算 R^d 中 j 质量分布的均分存在性的主要上同调阻碍。核心的新结果是，如果 d=6 2^k +2 且 j=4 2^k+1 则始终存在这样的均分，对于 k=0，这可简化为 P. Mani-Levitska 等人的论文的主要结果，Topology and Combinatorics of Partitions of Mass by超平面，Adv。数学。 207（2006），266-296。这是一个真正的组合几何结果的例子，它本质上涉及 Z_4 扭转，并且不能通过应用 Stiefel-Whitney 类或具有 Z_2 系数的上同调指数理论来获得。该方法开启了开发基于$G$流形复合体的“有效初级阻塞理论”的可能性，并应用于几何组合学、离散和计算几何以及计算代数拓扑。
We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2 and j=4 2^k+1 which for k=0 reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. This is an example of a genuine combinatorial geometric result which involves Z_4-torsion in an essential way and cannot be obtained by the application of either Stiefel-Whitney classes or cohomological index theories with Z_2 coefficients. The method opens a possibility of developing an "effective primary obstruction theory" based on $G$-manifold complexes, with applications in geometric combinatorics, discrete and computational geometry, and computational algebraic topology.