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

2011/03/01 by Ottenburger, Sadeeb
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1103.0099

Abstract

Let T(γ) be the total space of the canonical line bundle γover CP¹ and r an integer which is greater than one and coprime to six. We prove that L_r³× T(γ) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L_r³ is the standard 3-dimensional lens space with fundamental group isomorphic to Z/r. We classify the total spaces of S¹-fibre bundles over S²× S² 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.

Related

• Codimension two souls and cancellation phenomena
• Vector bundles with infinitely many souls
• Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics
• Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds
• Lifting Group Actions and Nonnegative Curvature