超平面均分的计算拓扑
Computational topology of equipartitions by hyperplanes
摘要：
在2d-3 j = 1的情形下，我们计算了R^d中的j个质量分布被两个超平面均分的存在性的一个主上同调障碍。中心的新结果是，这样的均分总是存在的，如果d=6 2^k +2和j=4 2^k+1，其中k=0减少到主要结果的文件P. Mani-Levitska等人，超平面划分质量的拓扑学和组合学，Adv. Math. 207（2006），266-296。这是一个真正的组合几何结果的例子，它涉及Z_4-挠的本质方式，不能得到任何应用Stiefel-Whitney类或上同调指数理论与Z_2系数。该方法打开了一个“有效的主要障碍理论”的基础上$G$流形复合物，在几何组合学，离散和计算几何，计算代数拓扑的应用的可能性。
英文摘要：
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.
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