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 F P m × S 3 , where F = R , C or H . Let ( X , E , π , B ) be a fibre bundle and ( R k , E ′ , π ′ , B ) be a k -dimensional real vector bundle with fibre preserving G = Z 2 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 → R n . In particular, we estimate the size of the coincidence sets of continuous maps f : S i × S 3 → R k relative to any free involution on S i × S 3 , ( i = 1 , 2 , 4 ) .