= Poincaré duality for noncompact manifolds
{c}
{title2=$H^k(M;\mathbb Z)\cong H_{n-k}^{\rm BM}(M;\mathcal O_M)$}
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 $k>n$.
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