  We study general conditions under which the computations of the index of a
perturbed Dirac operator $D_{s}=D+sZ$ localize to the singular set of the
bundle endomorphism $Z$ in the semi-classical limit $s\to \infty $. We show how
to use Witten's method to compute the index of $D$ by doing a combinatorial
computation involving local data at the nondegenerate singular points of the
operator $Z$. In particular, we provide examples of novel deformations of the
de Rham operator to establish new results relating the Euler characteristic of
a spin$^{c}$ manifold to maps between its even and odd spinor bundles. The
paper contains a list of the current literature on the subject.
