Pseudocycles for Borel—Moore homology

Pseudocycles are geometric representatives for integral homology classes on smooth manifolds that have proved useful in particular for defining gauge-theoretic invariants. The Borel–Moore homology is often a more natural object to work with in the case of non-compact manifolds than the usual homology. We define weaker versions of the standard notions of pseudocycle and pseudocycle equivalence and then describe a natural isomorphism between the set of equivalence classes of these weaker pseudocycles and the Borel–Moore homology. We also include a direct proof of a Poincaré Duality between the singular cohomology of an oriented manifold and its Borel–Moore homology.