|Affiliations:||Mathematical Institute of the Serbian Academy of Sciences and Arts||Title:||Topology and combinatorics of partitions of masses by hyperplanes||Journal:||Advances in Mathematics||Volume:||207||Issue:||1||First page:||266||Last page:||296||Issue Date:||1-Dec-2006||Rank:||M21a||ISSN:||0001-8708||DOI:||10.1016/j.aim.2005.11.013||Abstract:||
An old problem in combinatorial geometry is to determine when one or more measurable sets in R d admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257-1261]. A related topological problem is the question of (non)existence of a map f : (S d ) k → S (U), equivariant with respect to the Weyl group W k = B k : = (Z / 2) ⊕ k ⋊ S k , where U is a representation of W k and S (U) ⊂ U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R 8 [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167]. The obstruction in this case is identified as the element 2 X a b ∈ H 1 (D 8 ; Z) ≅ Z / 4, where X a b is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel-Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems 8 * (1988) 73-85].
|Keywords:||Cohomological index theory | Cyclic words | Equipartitions of masses | Equivariant maps | Obstruction theory||Publisher:||Elsevier||Project:||Serbian Ministry of Science, Grant no. 144026|
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