Counterexamples to the nonsimply connected double soul conjecture

Jason DeVito

Vol. 325 (2023), No. 2, 239–254

DOI: 10.2140/pjm.2023.325.239

Abstract

A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The double soul conjecture asserts that a closed simply connected manifold admitting a metric of nonnegative sectional curvature is necessarily a double disk bundle. We study a generalization of this conjecture by dropping the requirement that the manifold be simply connected. Previously, a unique counterexample was known to this generalization, the Poincaré dodecahedral space $S^3∕I^{\ast }$. We find infinitely many $3$-dimensional counterexamples, as well as another infinite family of flat counterexamples whose dimensions grow without bound.

Keywords

homogeneous spaces, double soul conjecture, disk bundles

Mathematical Subject Classification

Primary: 53C21, 53C30, 55R25

Milestones

Received: 2 February 2023

Revised: 28 August 2023

Accepted: 27 September 2023

Published: 3 November 2023