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

Abstract Let T () T (γ) be the total space of the canonical line bundle γ over CP¹ C P 1 and r an integer, which is divisible by an odd prime. We prove that Lᵣ³ T () L r 3 × T (γ) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where Lᵣ³ L r 3 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' S ′ of codimension two such that S and S' S ′ are diffeomorphic whereas the pairs (N, S) and (N, S') (N, S ′) are not diffeomorphic. These results give solutions to two problems posed by Igor Belegradek, Slawomir Kwasik and Reinhard Schultz.