  The basic cohomology of a Riemannian foliation on a complete manifold with
all leaves closed is the cohomology of the leaf space. In this paper we
introduce various methods to compute the basic cohomology in the presence of
both closed and non-closed leaves in the simply-connected case (or more
generally for Killing foliations): We show that the total basic Betti number of
the union C of the closed leaves is smaller than or equal to the total basic
Betti number of the foliated manifold, and we give sufficient conditions for
equality. If there is a basic Morse-Bott function with critical set equal to C
we can compute the basic cohomology explicitly. Another case in which the basic
cohomology can be determined is if the space of leaf closures is a simple,
convex polytope. Our results are based on Molino's observation that the
existence of non-closed leaves yields a distinguished transverse action on the
foliated manifold with fixed point set C. We introduce equivariant basic
cohomology of transverse actions in analogy to equivariant cohomology of Lie
group actions enabling us to transfer many results from the theory of Lie group
actions to Riemannian foliations. The prominent role of the fixed point set in
the theory of torus actions explains the relevance of the set C in the basic
setting.
