Parametrized Borsuk-Ulam theorems for free involutions on FPm×S3

Let X be a finitistic space with mod 2 cohomology algebra isomorphic to that of FPm×S3, where F=R,C or H. Let (X,E,π,B) be a fibre bundle and (Rk,E′,π′,B) be a k-dimensional real vector bundle with fibre preserving G=Z2 action such that G acts freely on E and E′−{0}, where {0} is the zero section of the vector bundle. We determine lower bounds for the cohomological dimension of the zero set f−1({0}) of a fibre preserving G-equivariant map f:E→E′. As an application of this result, we determine a lower bound for the cohomological dimension of the coincidence sets of continuous maps f:X→Rn. In particular, we estimate the size of the coincidence sets of continuous maps f:Si×S3→Rk relative to any free involution on Si×S3, (i=1,2,4).