Topological regularity theorems for Alexandrov spaces

Since Gromov gave in [G1], [G2] an abstract definition of Hausdorff distance between two compact metric spaces, the Gromov-Hausdorff convergence theory has played an important role in Riemannian geometry. Usually, Gromov-Hausdorff limits of Riemannian manifolds are almost never Riemannian manifolds. This motivates the study of Alexandrov spaces which are more singular than Riemannian manifolds since it is observed in [GP1] that the limit spaces are Alexandrov spaces if the manifolds in the sequence have curvature bounded uniformly from below. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the sense of distance comparison. It is now well known that the topological and geometric properties of Gromov-Hausdorff limits will reveal those of Riemannian manifolds considered in the sequence. For a discussion of this viewpoint, see [W1]. In view of this, the investigation of the topological and geometric properties of Alexandrov spaces has recently attracted a lot of attention; see for example [BGP], [FY], [GP1], [Pe], [Sh] and [Pt]. The structure of Alexandrov spaces is studied in [BGP], [Pe] and [Pt]. In particuar, if $P$ is a point in an Alexandrov space $X$, then the space of directions $\Sigma_{p}$ at $P$ is an Alexandrov space of one less dimension and with curvature Zl. Moreover, a neighborhood of $P$ in $X$ is homeomorphic to the linear cone over $\Sigma_{p}$ . One important implication of this local structure result is that if $\Sigma_{p}$ is a sphere then the point $P$ is a manifold point. However, the converse is not true. This can be seen from the following example from [GP2].