Authors: Mani-Levitska, Peter
Vrećica, Siniša
Živaljević, Rade 
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
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

Show full item record


checked on May 18, 2024

Page view(s)

checked on May 10, 2024

Google ScholarTM




Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.