Vanishing cycles and singularities of meromorphic functions

Abstract

We study vanishing cycles of meromorphic functions This gives a new and unitary point of view extending the study of the topology of holomorphic germs as initiated by Milnor in the sixties and of the global topology of polynomial functions which has been advanced more recently We dene singularities along the poles with respect to a certain weak stratication and prove local and global bouquet structure in case of isolated singularities In general splitting of vanishing homology at singular points and global PicardLefschetz phenomena occur