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

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