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 D 8 = ( Z 2 ) 2 ⋊ Z 2 acting on S d × S d for F 2 and for Z coefficients, that is, the kernels of the maps in equivariant cohomology H D 8 ⁎ ( pt , F 2 ) → H D 8 ⁎ ( S d × S d , F 2 ) and H D 8 ⁎ ( pt , Z ) → H D 8 ⁎ ( S d × S d , Z ) . This establishes the complete cohomological lower bounds, with F 2 and with Z 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 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 .