Zero sets of equivariant maps from products of spheres to Euclidean spaces

Let E -> B be a fiber bundle and E' -> B be a vector bundle. Let G be a compact group acting fiber preservingly and freely on both E and E' - 0, where 0 is the zero section of E' -> B. Let f : E -> E' be a fiber preserving G-equivariant map, and let Z(f) = {x is an element of E vertical bar f (x) = 0} be the zero set of f. It is an interesting problem to estimate the dimension of the set Z(f). In 1988, Dold [5] obtained a lower bound for the cohomological dimension of the zero set Z(f) when E -> B is the sphere bundle associated with a vector bundle which is equipped with the antipodal action of G = Z/2. In this paper, we generalize this result to products of finitely many spheres equipped with the diagonal antipodal action of Z/2. We also prove a Bourgin-Yang type theorem for products of spheres equipped with the diagonal antipodal action of Z/2. (C) 2015 Elsevier B.V. All rights reserved.