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

Abstract

We compute the complete Fadell-Husseini index of the dihedral group D8=(Z{double-struck}2)2⋊Z{double-struck}2 acting on Sd×Sd for F{double-struck}2 and for Z{double-struck} coefficients, that is, the kernels of the maps in equivariant cohomology. HD8*(pt,F{double-struck}2)→HD8*(Sd×Sd,F{double-struck}2) and. HD8*(pt,Z{double-struck})→HD8*(Sd×Sd,Z{double-struck}). This establishes the complete cohomological lower bounds, with F{double-struck}2 and with Z{double-struck} coefficients, for the two-hyperplane case of Grünbaum'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{double-struck}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 D8.