Mass Partitions via Equivariant Sections of Stiefel Bundles

TL;DR

This paper links geometric combinatorics and topology by showing how the existence of equivariant sections of Stiefel bundles can guarantee mass partition results using regular q-fans in Euclidean space.

Contribution

It introduces a topological approach using equivariant sections of Stiefel bundles to solve mass partition problems involving regular q-fans.

Findings

Parallelizability of b^n for n=2,4,8 implies bisecting two masses in b^n.

Triviality of circle bundle V_2(b^2)/b_q over Lens Spaces ensures existence of complex orthogonal q-fans.

Topological conditions on bundles relate to the existence of equipartitions in Euclidean spaces.

Abstract

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of $m$ mass distributions on $\mathbb{R}^n$, the existence of $k$ affinely independent regular $q$-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a $\mathbb{Z}_q$-equivariant section of the Stiefel bundle $V_k(\mathbb{F}^n)$ over $S(\mathbb{F}^n)$, where $V_k(\mathbb{F}^n)$ is the Stiefel manifold of all orthonormal $k$-frames in $\mathbb{F}^n,\, \mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, and $S(\mathbb{F}^n)$ is the corresponding unit sphere. For example, the parallelizability of $\mathbb{R}P^n$ when $n = 2,4$, or $8$ implies that any two masses on $\mathbb{R}^n$ can be simultaneously bisected by each of $(n-1)$ pairwise-orthogonal hyperplanes, while when $q=3$ or 4, the triviality of the circle bundle $V_2(\mathbb{C}^2)/\mathbb{Z}_q$ over the standard Lens Spaces $L^3(q)$ yields that for any mass on $\mathbb{R}^4$, there exist a pair of complex orthogonal regular $q$-fans, each of which equipartitions the mass.