  We study the renormalization group evolution up to the fixed point of the
lattice topological susceptibility in the 2-d O(3) non-linear sigma-model. We
start with a discretization of the continuum topological charge by a local
charge density, polynomial in the lattice fields. Among the different choices
we propose also a Symanzik--improved lattice topological charge. We check step
by step in the renormalization group iteration the progressive dumping of
quantum fluctuations, which are responsible for the additive and multiplicative
renormalizations of the lattice topological susceptibility with respect to the
continuum definition. We find that already after three iterations these
renormalizations are negligible and an excellent approximation of the fixed
point is achieved. We also check by an explicit calculation that the assumption
of slowly varying fields in iterating the renormalization group does not lead
to a good approximation of the fixed point charge operator.
