2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156090We show that if the Atiyah Jones conjecture holds for a surface $X,$ then it also holds for the blow-up of $X$ at a point. Since the conjecture is known to hold for ${\mathbb P}^2$ and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.final versionAlgebraic GeometryAlgebraic Topology14D20;32G13;58D27text