Parametrized Borsuk-Ulam theorems for free involutions on F Pm x S3

Let X be a finitistic space with mod 2 cohomology algebra isomorphic to that of FPm x S-3, where F = R, C or H. Let (X, E, pi, B) be a fibre bundle and (R-k, E', pi', 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 x S-3 -> R-k relative to any free involution on S-i x S-3, (i = 1,2,4). (C) 2018 Elsevier B.V. All rights reserved.