Mass partitions via equivariant sections of Stiefel bundles

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 Rn, 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 Zq-equivariant section of the Stiefel bundle Vk(Fn) over S(Fn), where Vk(Fn) is the Stiefel manifold of all orthonormal k-frames in Fn, F = R or C, and S(Fn) is the corresponding unit sphere. For example, the parallelizability of RPn when n = 2,4, or 8 implies that any two masses on Rn can be simultaneously bisected by each of (n-1) pairwise-orthogonal hyperplanes, while when q = 3 or 4, the triviality of the circle bundle V2(C2)=Zq over the standard Lens Spaces L3(q) yields that for any mass on R4, there exist a pair of complex orthogonal regular q-fans, each of which equipartitions the mass.