Vanishing cycle

Summary

A vanishing cycle is a homology class in a smooth fibre of a holomorphic function, see the definition. It shrinks to a point as the fibre approaches a critical fibre. Near a nondegenerate critical point it is represented by an embedded sphere of the middle dimension. Vanishing cycles generate the homology of the Milnor fibre of an isolated singularity, and Picard–Lefschetz theory computes the monodromy in terms of them. The idea was later turned into the vanishing cycle functor on sheaves, used in étale cohomology and in the theory of perverse sheaves.