2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73786This article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture towards which this project is directed is that the Rohlin invariant and the gauge theoretic invariant coincide for a homology $S^1\times S^3$. We discuss the implications of this conjecture for some classical problems in low-dimensional topology, and progress we have made towards proving the conjecture.30 pages, 5 figures. To appear in Proceedings of the Fields-McMaster Conference on Geometry and Topology of ManifoldsGeometric Topology57M27; 57R58; 58D27text