Poincaré duality for noncompact manifolds ($H^k(M;\mathbb{Z})\cong H_{n-k}^{\mathrm{BM}}(M;{\mathcal{O}}_M)$)

Cap product with the locally finite fundamental class of an $n$-manifold without boundary gives the displayed duality, including a possibly nontrivial orientation local system ${\mathcal{O}}_M$. It is the noncompact version of Poincare duality; for a compact manifold Borel-Moore homology is ordinary homology. Locally it is the relative fundamental-class calculation in an oriented coordinate ball; compatibility on overlaps and the Mayer–Vietoris sequence give the global statement. Since the chain complex of locally finite homology is zero in negative degrees, this duality implies $H^k(M;\mathbb{Z})=0$ for $k>n$.