J. DIFFERENTIAL GEOMETRY 27 (1988) 423-460 THE TOPOLOGY OF ISOPARAMETRIC SUBMANIFOLDS WU-YI HSIANG, RICHARD S. PALAIS & CHUU-LIAN TERNG Abstract It has been known since a famous paper of Bott and Samelson that, using Morse theory, the homology and cohomology of certain homoge- neous 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 the- ory of isoparametric submanifolds of arbitrary codimension in Euclidean space, and these same orbits turn out to be exactly the homogeneous ex- amples. 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 homol- ogy and cohomology of isoparametric submanifolds from their marked Dynkin diagrams. 0. Introduction In 1958 Bott and Samelson introduced the concept of variational complete- ness for isometric group actions [5], and developed powerful Morse theoretic arguments to compute the homology and cohomology of orbits of variationally complete actions. As already noted in their paper, the isotropy representa- tions of symmetric spaces (^-representations) are variationally complete, and from results of L. Conlon [19], [20] an orthogonal representation is variation- ally complete if and only if there is a linear subspace which meets every orbit orthogonally. Such representations (called polar by J. Dadok [21] and repre- sentations admitting sections by Palais and Terng [40]) have been classified by Dadok, who showed that, at least as far as orbit structure is concerned, they are exactly the ^-representations. This class of homogeneous spaces in- cludes all the flag manifolds and Grassmannians, and because of its important roles in geometry, topology, and representation theory it has been intensively studied. We shall see below that isoparametric submanifolds and their focal manifolds are a geometric generalization of these homogeneous spaces. Received January 21, 1986 and, in revised form, April 22, 1987. Research was sup- ported in part by National Science Foundation grants MC 577-23579 (Hsiang), MCS- 8102696 (Palais) and DMS-8301928 (Terng). The second author was also supported by the Mathematical Sciences Research Institute.