Open manifolds with non-homeomorphic positively curved souls

TL;DR

This paper demonstrates the existence of open manifolds with positively curved souls that are non-homeomorphic, extending known results and providing explicit examples involving Eschenburg spaces.

Contribution

It extends existence results to simply connected positively curved manifolds, showing non-homeomorphic souls and tangentially homotopy equivalent but non-homeomorphic pairs.

Findings

Existence of non-homeomorphic positively curved souls in open manifolds

Explicit examples using Eschenburg spaces

Extension of the stable converse soul question to a broader class of spaces

Abstract

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.