Open manifolds with non-homeomorphic positively curved souls

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.

Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics

“…There are many examples where the diffeomorphism (or even homeomorphism) type of the soul depends on the metric, see [4,6,7,19,25,32], and if the ambient open manifold V is indecomposable, this gives examples where M K 0 (V ) is not connected, or even has infinitely many connected components.…”

Bounds on Sectional Curvature and Interactions with Topology

“…Rigas' statement (or even stronger versions of it) were shown to hold when one replaces the base space S n by certain classes of homogeneous spaces (see [30,32]) or certain classes of biquotients (see [33]), etc.…”