Some generalizations of the Borsuk-Ulam Theorem

Let S-n be the n-dimensional sphere, A : S-n -> S-n the antipodal involution and R-n the n-dimensional euclidean space. The famous Borsuk-Ulam Theorem states that, if f : S-n -> R-n is any continuous map, then there exists a point x E Sn such that f (x) = f (A(x)). In this paper we discuss some generalizations and variants of this theorem concerning the replacement either of the domain (S-n, A) by other free involution pairs (X, T), or of the target space R-n by more general topological spaces. For example, we consider the cases where: i) (S-2, A) is replaced by a product involution (X, T) x (Y,S) = (X x Y,T x S), where X and Y are Hausdorff and pathwise connected topological spaces, the involution T is free and the fundamental group of X is a torsion group; ii) R-n is replaced by M-r x N-s, where M-r and N-s are closed manifolds with dimensions r and s, respectively, and r + s = n; iii) (S-2, A) is replaced by a product involution as described in i), and R-2 is replaced by the 2-dimensional torus T-2. We remark that i) includes the case in which (X,T) x (Y, S) = (X,T), by taking (Y, S) = ({point}, identity), and in particular the popular 2-dimensional Borsuk-Ulam Theorem.