Resumen de Some generalizations of the Borsuk�Ulam Theorem

Daniel Vendrúscolo, Patricia E. Desideri, Pedro L. Q. Pergher

• Let Sn be the n-dimensional sphere, A : Sn ! Sn the antipodal involution and Rn the n-dimensional euclidean space. The famous Borsuk�Ulam Theorem states that, if f : Sn ! Rn is any continuous map, then there exists a point x 2 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 (Sn;A) by other free involution pairs (X; T), or of the target space Rn by more general topological spaces. For example, we consider the cases where: i) (S2;A) is replaced by a product involution (X; T)(Y; S) = (XY; TS), 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) Rn is replaced by Mr  Ns, where Mr and Ns are closed manifolds with dimensions r and s, respectively, and r+s = n; iii) (S2;A) is replaced by a product involution as described in i), and R2 is replaced by the 2-dimensional torus T2. We remark that i) includes the case in which (X; T)  (Y; S) = (X; T), by taking (Y; S) = (fpointg; identity), and in particular the popular 2-dimensional Borsuk�Ulam Theorem.