A Borsuk-Ulam type theorem for the product of a projective space and 3-sphere

The classical Borsuk Ulam theorem can be stated as: there exists no equivariant map Sn→Sn−1, relative to the antipodal actions on the spheres. Let G=Z2 act freely on a finitistic space X with mod 2 cohomology ring isomorphic to that of the product of a projective space (real, complex or quaternionic) and the 3-sphere. In this paper, we show that the Volovikov's index of RPm×S3 is any one of the integers 2, 4, m+3 or m+4. In case of CPm×S3, this index is 3, 4 or 2m+4 and that of HPm×S3 is 4, 5, 8 or 9. We apply this to determine the possibilities of nonexistence of equivariant maps X→Sn or Sn→X.