Um teorema tipo Borsuk-Ulam para espaços topológicos gerais em termos do grupo fundamental
AUTOR(ES)
Patrícia Elaine Desideri
DATA DE PUBLICAÇÃO
2008
RESUMO
The celebrate 2-dimensional Borsuk-Ulam theorem says that if f is a continuous map from the 2-dimensional sphere and with values in the euclidean 2-dimensional space then there exists a point x in the 2-dimensional sphere such that f(x) = f(-x). Many generalizations of this result have been studied, in many directions. A line of generalizations consists in replacing the 2-dimensional sphere and the antipodal map by more general spaces X equipped with arbitrary involutions T without fixed points, that is, a continuous map which is its own inverse and does not have fixed points. In this case, we say that the pair (X; T) satisfies BBUT (an abbreviation for 2-dimensional Borsuk-Ulam theorem) if for every continuous map f from X into the 2-dimensional euclidean space there exists a point x in X such that f(x) = f(T(x)). In this setting, we show in this work the following result: Let X be a Hausdor, path connected and locally path connected space, and let T be a free involution on X. Let X=T be the quotient space of X by T and p the quotient map from X into X=T . Consider the homomorphism induced in the fundamental groups of X and X=T by p. Suppose that there exists a torsion element in the fundamental group of X=T that does not belong to the image of such homomorphism. Then (X; T) satisfies BBUT. As a consequence of this result, we prove the following fact: Suppose X and Y are Hausdor, path connected and locally path connected spaces, and suppose that T and S are involutions on X and Y , respectively, with T being free. Suppose that the fundamental group of X is finite. Then the pair formed by the cartesian product of X and Y , equipped with the involution given by the product of the involutions T and S satisfies BBUT. In particular, by taking (Y; S) = (fpointg; Identity), one has the following result, which is a generalization of the 2-dimensional Borsuk-Ulam theorem: If X is a Hausdor, path connected and locally path connected space, and has finite fundamental group then, for every free involution T from into X, the pair (X; T) satisfies BBUT.
ASSUNTO(S)
grupo fundamental características de euler teorema de borsuk-ulam involução topologia algebrica topologia algébrica
