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Abstract: In 1985-86, Lê, Sabbah, and
Ginsburg all proved, independently, that the Euler characteristic of the stalk cohomology
of the vanishing cycles of a complex of sheaves along a complex analytic function,
f, with an isolated critical point, is equal to the intersection multiplicity
of the characteristic cycle and the image of df. This result was a
conjecture of Deligne. We generalize this result to the case of non-isolated
critical points, and show how perverse cohomology allows one to quickly
extract the Betti number information from the Euler characteristic.
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