A general calculation of the number of vanishing cycles

We give an algebraic formula for calculating the change in the Euler characteristic of the Milnor fibre of a function f, as one “grows” from a hypersurface section out to the full Milnor fibre of f. This is a general calculation in that we allow the domain of f to be singular, the hypersurface section may have an isolated singularity, and we allow coefficients in an arbitrary bounded, constructible complex of sheaves. Our formula generalizes both a result of Lê for affine hypersurfaces and a result of Looijenga for isolated complete intersections.