Homotopy Groups of the Complements to Singular Hypersurfaces, II

Abstract

The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface with a singular locus of dimension $k$ and good behavior at infinity is described using generic pencils. This is analogous to the van Kampen procedure for finding a fundamental group of a plane curve. In addition we use a certain representation generalizing the Burau representation of the braid group. A divisibility theorem is proven that shows the dependence of this homotopy group on the local type of singularities and behavior at infinity. Examples are given showing that this group depends on certain global data in addition to local data on singularities.