Computational topology of equipartitions by hyperplanes

TL;DR

This paper establishes new topological conditions guaranteeing the existence of equipartitions of mass distributions by hyperplanes, revealing cases where such partitions always exist and introducing methods involving Z_4-torsion.

Contribution

It introduces a novel cohomological obstruction approach that accounts for Z_4-torsion, advancing the understanding of hyperplane equipartitions beyond traditional Z_2 methods.

Findings

Existence of equipartitions for specific dimensions and mass counts

Identification of Z_4-torsion as essential in these topological obstructions

Development of a new obstruction theory framework

Abstract

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.