A Borsuk-Ulam type theorem for a product of spheres

For a map f : M → V, where M is a product of spheres and V is a manifold, we measure the (cohomological) dimension of the set A(φ) = }(x, y) ϵ M z.sfnc φ(−x,−y) = φ(x, y){, using Conner-Miller classes and the Bredon operation. We also examine a parametrized version of a similar problem using characteristic polynomials.