Classificação dos fibrados circulares sobre superfícies compactas e orientáveis

Abstract

The objective of this work is to present a classification, less than homeomorphisms, of the 3-dimensional manifolds that are performed as circle bundles of surfaces. A compact, connected 3-manifold is called a circle bundle if it is orientable, and if there is a projection of the bundle on a compact, connected and orientable 2-manifold (surface), whose fiber is the unit circle in the two-dimensional plane. For classification, each such bundle will be related to a class in the base surface degree two cohomology, called the Euler Class, and to the base surface genus. This will reduce the classification of circle bundles to the classification of base surfaces and the relationships between Euler Classes. There is always a classification for circle bundles on bordered surfaces, but for non-bordered surfaces the Euler class and surface gender will classify the circle bundles only if the Euler classes of bundles do not differ only by one sign.