  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.
