The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes
Algebraic Topology
2015-03-13 v4 Combinatorics
Abstract
We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8.
Cite
@article{arxiv.0704.1943,
title = {The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes},
author = {Pavle V. M. Blagojević and Günter M. Ziegler},
journal= {arXiv preprint arXiv:0704.1943},
year = {2015}
}
Comments
new version revised according to referee's comments, 44 pages, many diagrams; a shorter version of this will appear in Topology and its Applications (ATA 2010 proceedings)