On the topology of positively curved Bazaikin spaces

We study the topology of the 13 dimensional positively curved Bazaikinspaces. We show that there is only one such manifold which is homotopyequivalent to a homogeneous space, the so called Berger space. This is incontrast to the case of the 7 dimensional positively curved Eschenburg spaces.In addition, we compute the Pontryagin classes and the linking form and showthat the first two billion positively curved Bazaikin manifolds arehomeomorphically distinct, raising the question whether this is true ingeneral.