2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/197601We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold.Approx 26 pages. v2: 4d results substantially extendedQuantum AlgebraGeneral Relativity and Quantum CosmologyMathematical PhysicsGeometric Topologytext