Title: The topology of low dimensional cohomogeneity one manifolds

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Abstract: A cohomogeneity one manifold is a manifold with lots of symmetry, in the sense that, under some fixed Riemannian metric, any point can be carried to some point on a fixed line within the manifold by some isometry of the manifold. We will briefly recall the classification of compact simply connected cohomogeneity one manifolds in dimensions 7 and lower. We will then discuss recent progress toward understanding the topology of these manifolds.