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

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 α-partition for a probability measure μ if μ(σi) = αi for each i = 1, . . . , k where {σi}ki=1 are conical sectors associated with the k-fan and α = (α1, . . . , αk). The problem whether for a given collection of measures μ1, . . . , μm and given α = (α1, . . . , αk) there exists a simultaneous α-partition by a k-fan was raised and studied in [3] in connection with some partition problems in Discrete and Computational Geometry. The set of all α = (α1, . . . , αm) such that for any collection of probability measures μ1, . . . , μm there exists a common α-partition by a k-fan is denoted by Am,k. It was shown in [3] that the interesting cases of the problem are (m, k) = (3, 2), (2, 3), (2, 4). We prove, as a central result of this paper, that A3,2 = {(s, t) ∈ R | s+t = 1 and s, t > 0}. The result follows from the fact that under mild conditions there does not exist a Q4n-equivariant map f : S 3 → V \A(α) where A(α) is a Q4n-invariant, linear subspace arrangement in a Q4n-representation V , where Q4n is the generalized quaternion group. This fact is established by showing that an appropriate obstruction in the group Ω1(Q4n) of Q4n-bordisms does not vanish.