Some generalizations of the Borsuk–Ulam theorem and applications to realizing homotopy classes by embedded spheres

Abstract

In this paper, some theorems of the Borsuk-Ulam type (1) are given. One of these can be applied to show that certain homotopy classes in manifolds cannot be realized by embedded spheres. The n-dimensional sphere Sn is the subset of the euclidean spaceRn+l consisting of all points (x1, …,xn+1) satisfying . Let be a piecewise linear (PL) involution on Sn without fixed points.