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 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 Zq-equivariant section of the Stiefel bundle V k (F n ) over S(F n ), where V k (F n ) is the Stiefel manifold of all orthonormal k-frames in F n , F = R or C, and S(F n ) is the corresponding unit sphere. For example, the parallelizability of RP n when n = 2, 4, or 8 implies that any two masses on 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 V2(C 2 )/Zq over the standard Lens Spaces L 3 (q) yields that for any mass on R 4 , there exist a pair of complex orthogonal regular q-fans, each of which equipartitions the mass.