  For each odd integer r greater than one and not divisible by three we give
explicit examples of infinite families of simply and tangentially homotopy
equivalent but pairwise non-homeomorphic closed homogeneous spaces with
fundamental group isomorphic to Z/r. As an application we construct the first
examples of manifolds which possess infinitely many metrics of nonnegative
sectional curvature with pairwise non-homeomorphic homogeneous souls of
codimension three with trivial normal bundle, such that their curvatures and
the diameters of the souls are uniformly bounded. These manifolds are the first
examples of manifolds fulfilling such geometric conditions and they serve as
solutions to a problem posed by I. Belegradek, S. Kwasik and R. Schultz.
