  We introduce a new spectral sequence for the study of $\mathcal{K}$-manifolds
which arises by restricting the spectral sequence of a Riemannian foliation to
forms invariant under the flows of $\{\xi_1,...,\xi_s\}$. We use this sequence
to generalize a number of theorems from $K$-contact geometry to
$\mathcal{K}$-manifolds. Most importantly we compute the cohomology ring and
harmonic forms of $\mathcal{S}$-manifolds in terms of primitive basic
cohomology and primitive basic harmonic forms (respectively). As an immediate
consequence of this we get that the basic cohomology of $\mathcal{S}$-manifolds
are a topological invariant. We also show that the basic Hodge numbers of
$\mathcal{S}$-manifolds are invariant under deformations. Finally, we provide
similar results for $\mathcal{C}$-manifolds.
