  We prove that if the $m$-th homotopy group for $m \geq 2$ of a closed
manifold has non-trivial invariants or coinvariants under the action of the
fundamental group, then there exist infinitely many geometrically distinct
closed geodesics for a $C^4$-generic Riemannian metric. If moreover there are
infinitely many conjugacy classes in the fundamental group, then the same holds
for every Riemannian metric.
