2002/03/04 by Rade T. Živaljević, Rade T. Zivaljevic, Zivaljevic, Rade T.
Computer Science · Mathematics · #37F20(Primary) #52C35 #55N91 #57R91(Secondary) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.CO #msc:52C35 #msc:55N91
paper · pdf · doi:10.48550/arxiv.math/0203028
16 pages, 2 figures
arxiv created 2002/03/04 · openalex publication_date 2002/03/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\i=1k are conical sectors associated with the k-fan and α= (α1,... ,αk). 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 \cal Am,k. We prove, as a central result of this paper, that \cal A3,2 = \(s,t)∈ ℝ2| s+t=1 \rm and s,t>0\. The result follows from the fact that under mild conditions there does not exist a Q4n-equivariant map f : S3→ V∖ \cal A(α) where \cal 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.