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.