2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115341We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space $X$ of bounded geometry with finite asymptotic dimension for which $\as(X\times\R)=\as X$. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe's coarse cohomology is strictly less than its asymptotic dimension.30 pagesMetric GeometryGeometric Topology55M10, 20F69, 51F99, 57M20text