  In the context of orientable circuits and subcomplexes of these as
representing certain singular spaces, we consider characteristic class formulas
generalizing those classical results as seen for the Riemann-Hurwitz formula
for regulating the topology of branched covering maps and that for monoidal
transformations which include the standard blowing-up process. Here the results
are presented as cap product pairings, which will be elements of a suitable
homology theory, rather than characteristic numbers as would be the case when
taking Kronecker products once Poincar\'e duality is defined. We further
consider possible applications and examples including branched covering maps,
singular varieties involving virtual tangent bundles, the
Chern-Schwartz-MacPherson class, the homology L-class, generalized signature,
and the cohomology signature class.
