A classification of 5-dimensional manifolds, homogeneous souls of codimension two and non-diffeomorphic pairs

Let $T$(y) be the total space of the canonical line bundle $Y$ over $\mathbb{C}$P$^1$ and r an integer, which is divisible by an odd prime. We prove that L$^3_r$ x T(y) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L$^3_r$ is a 3-dimensional lens space with fundamental group of order r. Furthermore, we classify a class of non-simply connected 5-manifolds up to diffeomorphism and use this result to give first examples of manifolds N, which admit two complete metrics of nonnegative sectional curvature with souls S and S' of codimension two such that S and S' are diffeomorphic whereas the pairs (N, S) and (N, S') are not diffeomorphic. These results give solutions to two problems posed by Igor Belegradek, Slawomir Kwasik and Reinhard Schultz.