Abstract: The question motivating the first part of this talk is the following: What information can one deduce about ordinary (de Rham) cohomology of a manifold using the theory of equivariant cohomology, if the manifold admits a special type of Lie group action?

The class of group actions we will consider is that of cohomogeneity one actions (i.e., those that admit an orbit of codimension one). Among other things, one can derive the following topological obstruction: if a compact manifold with positive Euler characteristic admits an action of cohomogeneity one, then all of its odd Betti numbers vanish. (A result that was previously shown by Grove and Halperin using rational homotopy theory.)

In the second part we will go into a completely different geometric situation and show how one can use similar techniques to derive a link between the basic cohomology of certain Riemannian foliations and the number of closed leaves of the foliation. The main example here will be the Reeb foliation of a K-contact manifold.

(The first part is a joint work with Augustin-Liviu Mare, and the second one with Hiraku Nozawa and Dirk Töben)