2022/09/20 by Cattalani, Spencer, Zinger, Aleksey
#55N99 #57R95 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.10036
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.