Counterexamples to the non-simply connected Double Soul Conjecture

TL;DR

This paper demonstrates that the generalized Double Soul Conjecture, which removes the simply connected restriction, is false by providing infinitely many counterexamples in three dimensions and beyond.

Contribution

It introduces the first infinite families of counterexamples to the generalized conjecture, expanding understanding of manifold structures beyond simply connected cases.

Findings

Identifies infinitely many 3D counterexamples.

Discovers infinite families of flat counterexamples.

Shows the conjecture does not hold without simple connectivity.

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 non-negative 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 Poincar\'e 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.