  In the context of phenomenological models of quantum gravity, it is claimed
that the ultraviolet and infrared natural cutoffs can be realized from local
deformations of the Hamiltonian systems. In this paper, we scrutinize this
hypothesis and formulate a cutoff-regularized Hamiltonian system. The results
show that while local deformations are necessary to have cutoffs, they are not
sufficient. In fact, the cutoffs can be realized from globally-deformed
Hamiltonian systems that are defined on compact symplectic manifolds. By taking
the universality of quantum gravity effects into account, we then conclude that
quantum gravity cutoffs are global (topological) properties of the symplectic
manifolds. We justify our results by considering three well-known examples: The
Moyal, Snyder and polymer deformed Hamiltonian systems.
