  We construct sequences of ``field theoretical'' (analytical) lattice
topological charge density operators which formally approach geometrical
definitions in 2-d $CP^{N-1}$ models and 4-d $SU(N)$ Yang Mills theories. The
analysis of these sequences of operators suggests a new way of looking at the
geometrical method, showing that geometrical charges can be interpreted as
limits of sequences of field theoretical (analytical) operators. In
perturbation theory renormalization effects formally tend to vanish along such
sequences. But, since the perturbative expansion is asymptotic, this does not
necessarily lead to well behaved geometrical limits. It indeed leaves open the
possibility that non-perturbative renormalizations survive.
