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 ${\cal A}_{m,k}$. We prove, as a central result of this paper, that ${\cal A}_{3,2} = \{(s,t)\in \mathbb{R}^2\mid s+t=1 {\rm and} s,t>0\}$. The result follows from the fact that under mild conditions there does not exist a $Q_{4n}$-equivariant map $f : S^3\to V\setminus {\cal A}(\alpha)$ where ${\cal A}(\alpha)$ 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.