  The second de Rham cohomology groups of nilpotent orbits in non-compact real
forms of classical complex simple Lie algebras are explicitly computed.
Furthermore, the first de Rham cohomology groups of nilpotent orbits in
non-compact classical simple Lie algebras are computed; they are proven to be
zero for nilpotent orbits in all the complex simple Lie algebras. A key
component in these computations is a description of the second and first
cohomology groups of homogeneous spaces of general connected Lie groups which
is obtained here. This description, which generalizes a previous theorem of the
first two authors, may be of independent interest.
