2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64603We establish a new homological lower bound for the Thurston norm on 1-cohomology of 3-manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1-cohomology of 2-dimensional CW-complexes.32 pages, no figuresGeometric Topology57M27, 57M20, 57M05text