On quaternionic projective product spaces, Bourgin–Yang theorems and parametrized Borsuk–Ulam theorems

Abstract

We compute the cohomology ring for the quaternionic projective product spaces HP_{\overline{n}} = S^(4n_1+3) × · · · × S^(4n_r+3) / S³. Using similar computations by Davis for the cohomology ring for the projective product spaces, and by González and Velasco for the lens product spaces and complex projective product spaces, we prove a Bourgin–Yang Theorem for maps f : S^(2n_1+1) × · · · ×S^(2n_r+1) → R^m with action of Z_p and, for each group G = Z_2, Z_p, S¹ and S³, we prove a parametrized Borsuk–Ulam theorem for G-equivariant bundle maps f : E → E′ where F → E → B is fiber bundle with action of G, E′ → B is a vector bundle with action of G, and F is a product of spheres. Expanding on the techniques used in the proofs, we prove a general parametrized Borsuk–Ulam Theorem, for arbitrary fiber bundles with action of an arbitrary group G. We use this general theorem to obtain several parametrized Borsuk–Ulam and Bourgin–Yang Theorems.