Fibrados, classes de Stiefel-Whitney e resultados de não imersão

Abstract

We present an introductory study of smooth manifolds, bundles and Stiefel- Whitney classes (of real vector bundles). We explained that, given a certain smooth m-dimensional manifold, the Stiefel- Whitney classes of its tangent bundle can be used to ensure that such a manifold does not immerse (smoothly) in certain Euclidean spaces Rj . In this sense, we consider the Grassmann manifold G2;n of the 2-subspaces of Rn+2, and we carry out a detailed study of the following non-immersion theorem, proved by V. Oproiu [Proceedings of the Edinburgh Mathematical Society, 1977]: "Let n > 1 be a natural number and consider s = 2r such that s < ou = 2n < 2s. If n different s - 1, then G2;n does not immerse in R2s-3; if n = s - 1, then G2;n does not immerse in R3s-3."