Combinatorics and Topology of partitions of spherical measures by 2 and 3 fans

Abstract

An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an $\alpha$-partition for a probability measure $\mu$ if $\mu ({\sigma }_i)={\alpha }_i$ for each $i=1,...,k$ where $\{{\sigma }_i{\}}_{i=1}^k$ are conical sectors associated with the k-fan and $\alpha =({\alpha }_1,...,{\alpha }_k)$. The set of all $\alpha =({\alpha }_1,...,{\alpha }_m)$ such that for any collection of probability measures ${\mu }_1,...,{\mu }_m$ there exists a common $\alpha$-partition by a k-fan is denoted by ${\mathcal{A}}_{m,k}$. We prove, as a central result of this paper, that ${\mathcal{A}}_{3,2}=\{(s,t)\in {\mathbb{R}}^2\mid s+t=1\mathrm{and}s,t>0\}$. The result follows from the fact that under mild conditions there does not exist a $Q_{4n}$-equivariant map $f:S^3\to V\setminus \mathcal{A}(\alpha )$ where $\mathcal{A}(\alpha )$ is a $Q_{4n}$-invariant, linear subspace arrangement in a $Q_{4n}$-representation V, where $Q_{4n}$ is the generalized quaternion group. This fact is established by showing that an appropriate obstruction in the group ${\Omega }_1(Q_{4n})$ of $Q_{4n}$-bordisms does not vanish.

16 pages, 2 figures