The non-simply connected double soul conjecture

Abstract: Cheeger and Gromoll's Soul theorem asserts that a complete non-compact Riemannian manifold of non-negative sectional curvature has the structure of a vector bundle over a closed totally geodesic submanifold. The double soul conjecture (DSC) predicts an analogous structure on every closed simply connected Riemannian manifold of non-negative sectional curvature: it should decompose as a union of two disk bundles (possible of different ranks).

If one relaxes the hypothesis of the DSC to allow non-simply connected manifolds, then previously only a single counterexample was known. We will discuss two new infinite families of counterexamples, one positively curved and the other flat. In addition, all of our counterexamples are so-called biquotients, quotients of Riemannian homogeneous spaces by free isometric actions. We will also investigate the biquotient structure on the flat examples, finding that, in contrast with the homogeneous case, they do not support a biquotient structure induced from a connected Lie group.

differential geometrygeometric topology

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

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