Homology reduction of cycles in the complement of an algebraic hypersurface

An example of a 3-dimensional cycle in the complement of an algebraic hypersurface $V\subset\mathbb C^3$ that cannot be deformed into a tube over (is not homologous to the coboundary of) a 2-dimensional cycle in the set of regular points of $V$ is presented. Thus, the corresponding result of Poincare in $\mathbb C^2$ fails in $\mathbb C^n$ for $n>2$. It is proved that Poincare's result holds for hypersurfaces in $\mathbb C^n$ with a 'thin' set of singularities that are complete intersections.