The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes

TL;DR

This paper calculates the Fadell-Husseini index for the dihedral group D_8 acting on a product of spheres, providing cohomological bounds for a mass partition problem involving two hyperplanes.

Contribution

It establishes the complete cohomology bounds for the dihedral group action relevant to the mass partition problem, extending previous bounds based on abelian subgroups.

Findings

Complete cohomology bounds for D_8 action on S^d x S^d

Bounds match those from maximal abelian subgroups

Implications for mass partition by two hyperplanes

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.