Pseudocycles for Borel-Moore Homology

TL;DR

This paper introduces weaker pseudocycles for Borel-Moore homology on manifolds, establishing their equivalence classes and proving a Poincaré Duality with singular cohomology, thus enhancing geometric tools for non-compact manifolds.

Contribution

It defines weaker pseudocycles and equivalences, and proves an isomorphism with Borel-Moore homology along with Poincaré Duality, extending geometric methods to non-compact manifolds.

Findings

Weaker pseudocycles form an equivalence class set isomorphic to Borel-Moore homology.

Established a direct proof of Poincaré Duality for oriented manifolds.

Enhanced geometric representatives for homology in non-compact settings.

Abstract

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 Poincar\'e Duality between the singular cohomology of an oriented manifold and its Borel-Moore homology.