Recall the Soul Theorem of Cheeger and Gromoll: *If \((M,g)\) is a complete and connected Riemannian manifold of non-negative sectional curvature, then there exists a closed, totally convex and totally geodesic embedded submanifold whose normal bundle is diffeomorphic to \(M\).* Such a submanifold is called a soul of \((M,g)\) and the Riemannian metric induced on it has automatically also non-negative sectional curvature.

The above theorem reduces the study of Riemannian manifolds of non-negative sectional curvature to the case of closed manifolds with such a metric. And in this case there was the following conjecture about their topology (**Double Soul Conjecture**): *If \(M\) is a closed manifold admitting a Riemannian metric of non-negative sectional curvature, then \(M\) is a double disc bundle. *This means that there exist closed smooth manifolds \(B, B’\), disc bundles \(D \to B, D’ \to B’\) and a diffeomorphism \(f\colon \partial D \to \partial D’\) between their boundaries such that \(M \cong D \cup_f D’\).

Evidence for this conjecture was that manifolds of cohomogeneity one (which is one of the two main sources of manifolds of non-negative sectional curvature) satisfy it, all known simply-connected positively curved manifolds satisfy it and also some more examples.

Today Jason DeVito posted a preprint on the arXiv:2301.05675 **disproving the conjecture**! But he stresses that all the counter-examples have non-trivial fundamental group, so the conjecture is still open in the simply-connected case.

*edit (25.01.2023):* Unfortunately for Jason, it turned out that he is not the first one to come up with a counter-example to the conjecture. Karsten Grove already provided one in the paper *Geometry of, and via, symmetries*. This is the paper where he stated the double soul conjecture, but the counter-example only appears in the published version of the paper, not in the online available one. (This is the reason why I always update my arxiv preprints to the final version after acceptance.)