Low Dimension Soul Theorems

We study the Soul theorem for low dimensional topologically regular open complete nonnegatively curved Alexandrov spaces and give a topological classification of these spaces. These spaces occurs naturally as the blow-up limits of sequences of Riemannian manifold with a lower curvature bound. This will be used to study the collapsing of 3-dimension manifold as well as of 4-dimension Riemannian manifold with a lower curvature bound. These spaces have also been studied in [SY00] and [Yam02]. Our main tools are critical point theory for distance functions and Perelman's Fibration Theorem.