This paper introduces a general method for relating
characteristic classes to singularities of a bundle map. The
method is based on the notion of geometric
atomicity. This is a property of bundle maps
$\alpha:E\to F$
which universally guarantees the existence of certain
limits arising in the theory of singular connections. Under
this hypothesis, each characteristic form $\Phi$
of E or F
satisfies an equation of the form
$\Phi=L+dT$ ,
¶ where L is an explicit localization of Φ along the
singularities of α and T is a canonical form with
locally integrable coefficients. The method is constructive
and leads to explicit calculations. For normal maps (those
transversal to the universal singularity sets) it retrieves
classical formulas of R. MacPherson at the level of forms and
currents; see Part I ([HL4]). It also produces such formulas
for direct sum and tensor product mappings. These are new even
at the topological level. The condition of geometric atomicity
is quite broad and holds in essentially every case of
interest, including all real analytic bundle maps. An
important aspect of the theory is that it applies even in
cases of "excess dimension," that is, where the the
singularity sets of α have dimensions greater than
those of the generic map. The method yields explicit number of
examples are worked out detail.