  We give applications of the higher Lefschetz theorems for foliations of
[BH10], primarily involving Haefliger cohomology. These results show that the
transverse structures of foliations carry important topological and geometric
information. This is in the spirit of the passage from the Atiyah-Singer index
theorem for a single compact manifold to their families index theorem,
involving a compact fiber bundle over a compact base. For foliations, Haefliger
cohomology plays the role that the cohomology of the base space plays in the
families index theorem.
  We obtain highly useful numerical invariants by paring with closed holonomy
invariant currents. In particular, we prove that the non-triviality of the
higher A-hat genus of the foliation in Haefliger cohomology can be an
obstruction to the existence of non-trivial leaf-preserving compact connected
group actions. We then construct a large collection of examples for which no
such actions exist. Finally, we relate our results to Connes' spectral triples,
and prove useful integrality results.
