Perverse Cohomology and the Vanishing Index Theorem

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.