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 of 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 resul t of Poincare in C-2 fails in C-n for n > 2. It is proved that Poincar e's result holds for hypersurfaces in C-n with a 'thin' set of singula rities that are complete intersections.