The topology of isoparametric submanifolds (Q1822388)

It has been known since a famous paper of Bott and Samelson that, using Morse theory, the homology and cohomology of certain homogeneous spaces can be computed algorithmically from Dynkin diagram and multiplicity data. L. Conlon and J. Dadok noted that these spaces are the orbits of the isotropy representations of symmetric spaces. Recently the theory of isoparametric hypersurfaces has been generalized to a theory of isoparametric submanifolds of arbitrary codimension in Euclidean space, and these same orbits turn out to be exactly the homogeneous examples. Even the nonhomogeneous examples have associated to them Weyl groups with Dynkin diagrams marked with multiplicities. We extend and simplify the Bott-Samelson method to compute the homology and cohomology of isoparametric submanifolds from their marked Dynkin diagrams.