  This paper concerns the topology of isospectral real manifolds of certain
Jacobi elements associated with real split semisimple Lie algebras. The
manifolds are related to the compactified level sets of the generalized
(nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We
then give a cellular decomposition and the associated chain complex of the
manifold by introducing colored Dynkin diagrams which parametrize the cells in
the decomposition. We also discuss the Morse chain complex of the manifold.
