Space of nonnegatively curved metrics and pseudoisotopies

TL;DR

This paper investigates the topology of the space of nonnegatively curved metrics on open manifolds, showing that under certain conditions, the space has infinite higher homotopy groups, indicating rich topological complexity.

Contribution

It establishes a connection between the geometry of nonnegatively curved metrics and the topology of pseudoisotopy spaces, revealing new properties of the metric space on open manifolds.

Findings

Souls of nearby metrics are smoothly close

The space of metrics has infinite higher homotopy groups for many manifolds

Topological properties of pseudoisotopies are used to derive these results

Abstract

Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the space of complete nonnegatively curved metrics has infinite higher homotopy groups.