  We show that if $F(M)$ is some space of holonomic solutions with space of
formal solutions $F^f(M)$ that satisfies a certain relative $h$-principle, then
the non-relative map $F(M) \to F^f(M)$ admits a section up to homotopy. We
apply this to the relative $h$-principle for overtwisted contact structures
proved by Borman-Eliashberg-Murphy to find infinite cyclic subgroups in the
homotopy groups of the contactomorphism group of $M$.
