2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60409We give explicit examples of degree 3 cohomology classes not Poincare dual to submanifolds, and discuss the realisability of homology classes by submanifolds with Spin-C normal bundles.Several changes including an improvement of Theorem 1, our new examples have torsion-free homologyGeometric TopologyHigh Energy Physics - TheoryAlgebraic Topology57R95 ; 81T30text