2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107420In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space.11 pagesGeometric TopologyAlgebraic Geometrytext