Differential Homology

We first construct the differential refinement of singular homology, verifying that it satisfies the axioms dual to their cohomological counterparts. Then, we define the differential cap product, leading to Poincar\'e duality. The essential uniqueness of both differential homology and the cap product is proven. Moreover, we construct the relative, non-compact, and local coefficient versions, so that we can state Poincar\'e and Lefschetz dualities for every smooth manifold with our without boundary. Afterwards, we develop the analogous refinement of any rationally-even homology theory, with the same properties. We also define a "differentiation map" for fibre bundles, dual to the integration map in cohomology. We conclude by sketching some possible applications of this theory in mathematical physics.