  Geometry of manifolds with positive sectional curvature has been a central
object dates back to the beginning of Riemannian geometry. Up to homeomorphism,
there are only finitely many examples of simply connected positively curved
manifolds in all dimensions except in dimension 7 and 13, namely, the
Aloff-Wallach spaces and the Eschenburg spaces in dimension 7, and the Bazaikin
spaces in dimension 13. The topological classification modelled on the
7-dimensional examples has been carried out by Kreck-Stolz which leads to a
complete topological classification for the Aloff-Wallach spaces. The main goal
of this paper is to provide the topological classification of 13-dimensional
manifolds modelled on the Bazaikin spaces.
