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

We compute the complete Fadell–Husseini index of the dihedral group D8=(Z2)2⋊Z2D8=(Z2)2⋊Z2 acting on Sd×SdSd×Sd for F2F2 and for ZZ coefficients, that is, the kernels of the maps in equivariant cohomologyHD8⁎(pt,F2)→HD8⁎(Sd×Sd,F2) andHD8⁎(pt,Z)→HD8⁎(Sd×Sd,Z). This establishes the complete cohomological lower bounds, with F2F2 and with ZZ coefficients, for the two-hyperplane case of Grünbaumʼs 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 RdRd? 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 D8D8.