Computational topology of equipartitions by hyperplanes

Abstract

We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in ℝd by two hyperplanes in the case 2d‒3j = 1. The central new result is that such an equipartition always exists if d = 6 · 2k + 2 and j = 4 · 2k + 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. The theorem follows from a Borsuk‒Ulam type result claiming the non-existence of a D8-equivariant map f : Sd × Sd ® S(W⊕j) for an associated real D8-module W. 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 or Z 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.