  The characteristic cycle of a complex of sheaves on a complex analytic space
provides weak information about the complex; essentially, it yields the Euler
characteristics of the hypercohomology of normal data to strata. We show how
perverse cohomology actually allows one to extract the individual Betti numbers
of the hypercohomology of normal data to strata, not merely the Euler
characteristics. We apply this to the ``calculation'' of the vanishing cycles
of a complex, and relate this to the work of Parusi\'nski and Brian\c{c}on,
Maisonobe, and Merle on Thom's $a_f$ condition.
