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

Abstract

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 Zf={x∈E | f(x)=0} be the zero set of f. It is an interesting problem to estimate the dimension of the set Zf. In 1988, Dold [5] obtained a lower bound for the cohomological dimension of the zero set Zf 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.