Dupin hypersurfaces, group actions and the double mapping cylinder (Q1098435)

Dupin hypersurfaces in \(S^{n+1}\) generalize isoparametric ones: The principal curvatures are constant along the corresponding curvature leaves and have constant multiplicities. In this paper the authors provide far-reaching information about the topology of Dupin hypersurfaces. This includes a list of the possible rational homotopy types and fundamental groups and restrictions on the multiplicities. The same methods are applied to study the topology of manifolds which admit a cohomogeneity one action.