Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions

Complements to discriminants of singularities of smooth functions are far generalizations of the classifying spaces of Artin and Brieskorn braid groups. A group of stable cohomologies (i.e., cohomologies preserved under adjacency of singularities) is described for these spaces. A relationship between these cohomologies and Gauss–Manin connectivity of singularities is indicated. A cellular realization of cohomologies of symmetric groups with coefficients in $Z_2$ is described.