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Let $X$ be a connected, oriented and aspherical (meaning $\pi_i(X) = 0$ for $i\geq 2$) manifold of dimension $n$, and $S$ a local coefficient system (i.e., a $\pi_1(X)$-module). If $X$ is closed, we have the well-known Poincaré duality: $$H^i(X;S) \cong H_{n-i}(X;S)\,.$$

My question is: for $X$ with boundary, do we have $$H^i(X,\partial X;S)\cong H_{n-i}(X;S)\;\textrm{ and }\; H_i(X,\partial X;S)\cong H^{n-i}(X;S)$$ extending the Lefschetz-Poincaré duality? If not, what would be a necessary condition for this to hold?

Thank you.

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It is true if $X$ is compact. One does not need to assume $X$ aspherical.

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  • $\begingroup$ Thank you for the answer; do you have any preferred reference for this? $\endgroup$
    – Qwert Otto
    Commented Aug 7 at 14:29
  • $\begingroup$ I prefer to think about this in terms of Borel-Moore homology/compact support cohomology. If $X$ is compact then $H_i(X,\partial X;S)$ is isomorphic to $H_i^{BM}(X^\circ,S)$ (where $X^\circ = X \setminus \partial X$). A nice version of Poincaré duality for noncompact manifolds is a cap-product isomorphism between cohomology and Borel-Moore homology, which in this case would say $H_i^{BM}(X^\circ,S) \cong H^{n-i}(X^\circ,S)$. But the latter coincides with $H^{n-i}(X,S)$ by homotopy invariance. $\endgroup$
    – Dan Petersen
    Commented Aug 8 at 6:53
  • $\begingroup$ Preferred references for me could be Sheaves in topology (Dimca) or Cohomology of sheaves (Iversen), but you might not be a fan of the sheafy/derived category perspective. $\endgroup$
    – Dan Petersen
    Commented Aug 8 at 6:54
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You may also want to have a look at Spanier's "Singular homology and cohomology with local coefficients and duality for manifolds" in the Pacific Journal of Mathematics.

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