Computational topology of equipartitions by hyperplanes
DOI:
https://doi.org/10.12775/TMNA.2015.004Keywords
Equipartitions of masses, obstruction theory, computational topologyAbstract
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.Downloads
Published
2015-03-01
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1.
ŽIVALJEVIC, Rade. Computational topology of equipartitions by hyperplanes. Topological Methods in Nonlinear Analysis. Online. 1 March 2015. Vol. 45, no. 1, pp. 63 - 90. [Accessed 12 November 2024]. DOI 10.12775/TMNA.2015.004.
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