Perfect topological charge for asymptotically free theories

The classical equations of motion of the perfect lattice action in asymptotically free $d=2$ spin and $d=4$ gauge models possess scale invariant instanton solutions. This property allows the definition of a topological charge on the lattice which is perfect in the sense that no topological defects exist. The basic construction is illustrated in the $d=2$ O(3) non--linear $\sigma$--model and the topological susceptibility is measured to high precision in the range of correlation lengths $\xi \in (2 - 60)$. Our results strongly suggest that the topological susceptibility is not a physical quantity in this model.