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For any manifold (compact or not), compactly supported cohomology is Poincare dual to (ordinary) homology, via capping with the fundamental class, which is an infinite chain (i.e the sum of all the top simplices in a triangulation). Likewise, (ordinary) cohomology is Poincare dual to homology with locally finite infinite chains. In notation, $ H^{n-k}_{comp}(X)\cong H_{k}(X)$ and $H^{n-k}(X)\cong H_{k, inf}(X) $.