This is a continuation of last week's talk. A current problem in geometry is the attempt to understand the apparent scarcity of examples of manifolds with positive sectional curvature. One approach is to study manifolds of non-negative curvature. A cohomogeneity one manifold--that is: a closed, connected, smooth manifold with a smooth action by a compact Lie group resulting in a one-dimensional orbit space--often admits non-negative curvature. In this case, the manifold is diffeomorphic to the union of two disk bundles glued together along their boundaries. This double-disk construction can then be used to study the topology of the manifold.