Codimension two souls and cancellation phenomena

TL;DR

This paper constructs high-dimensional, simply-connected manifolds with nonnegative curvature that contain pairs of non-diffeomorphic, homeomorphic souls of codimension 2, and classifies certain diffeomorphism types involving complex line bundles.

Contribution

It demonstrates the existence of manifolds with nonnegative curvature having non-diffeomorphic souls of codimension 2 and classifies specific pairs involving complex line bundles over 7-sphere and complex projective plane.

Findings

Existence of manifolds with non-diffeomorphic, homeomorphic souls of codimension 2.

Classification of pairs (N, soul) over certain complex line bundles.

Identification of exactly three such diffeomorphism classes.

Abstract

For each nonnegative integer we find an open (4m+9)-dimensional simply-connected manifold admitting complete nonnegatively curved metrics whose souls are non-diffeomorphic, homeomorphic, and have codimension 2. We give a diffeomorphism classification of the pairs (N, soul) when N is a nontrivial complex line bundle over the product of 7-sphere and complex projective plane: up to diffeomorphism there are precisely three such pairs, distinguished by their non-diffeomorphic souls.