2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66818We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space $Y$ with isolated singularity is $(dimY-2)$-connected provided that the divisor has normal crossings outside of the singularity of $Y$. We apply this result to obtain a vanishing property for the cohomology of local systems of rank one and we also study vanishing in the case of local systems of higher rank. This second version contains a corrected proof of Corollary 4.1 from the first version.14 pagesAlgebraic GeometryAlgebraic Topology14F17, 14F45, 32S22, 32S55text