  Given a simply connected, closed four manifold, we associate to it a simply
connected, closed, spin five manifold. This leads to several consequences : the
stable and unstable homotopy groups of such a four manifold is determined by
its second Betti number, and the ranks of the homotopy groups can be explicitly
calculated. We show that for a generic metric on such a smooth four manifold
with second Betti number at least three, the number of geometrically distinct
periodic geodesics of length at most l grow exponentially as a function of l.
The number of closed Reeb orbits of length at most l on the spherization of the
cotangent bundle also grow exponentially for any Reeb flow.
