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.)