Description

• Title:
  Computational topology of equipartitions by hyperplanes
• Creator:
  Živaljevic, Rade
• Description:
  We compute a primary cohomological obstruction to the existence of an equipartition for $j$ mass distributions in $\mathbb{R}^d$ by two hyperplanes in the case $2d-3j = 1$. The central new result is that such an equipartition always exists if $d=6\cdot 2^k +2$ and $j=4\cdot 2^k+1$ which for $k=0$ reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. The theorem follows from a Borsuk-Ulam type result claiming the non-existence of a $\mathbb{D}_8$-equivariant map $f \colon S^{d}\times S^d\rightarrow S(W^{\oplus j})$ for an associated real $\mathbb{D}_8$-module $W$. This is an example of a genuine combinatorial geometric result which involves $\mathbb{Z}/4$-torsion in an essential way and cannot be obtained by the application of either Stiefel-Whitney classes or cohomological index theories with $\mathbb{Z}/2$ or $\mathbb{Z}$ coefficients. The method opens a possibility of developing an ``effective primary obstruction theory'' based on $G$-manifold complexes, with applications in geometric combinatorics, discrete and computational geometry, and computational algebraic topology.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2015 Topological Methods in Nonlinear Analysis
• Date:
  2015.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 45, No 1 (March 2015); 63 - 90;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.004/5339
• Subject:
  Equipartitions of masses;
  obstruction theory;
  computational topology;
  Primary 52A37;
  55S35;
  Secondary 52A38;
  55M35
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.004
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other