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

Abstract

Let T(\gamma) be the total space of the canonical line bundle \gamma over CP^1 and r an integer which is greater than one and coprime to six. We prove that L_r^3\times T(\gamma) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L_r^3 is the standard 3-dimensional lens space with fundamental group isomorphic to Z/r. We classify the total spaces of S^1-fibre bundles over S^2\times S^2 with fundamental group isomorphic to Z/r up to diffeomorphism and use these results to give 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. This solves a problem posed by I. Belegradek, S. Kwasik and R. Schultz.