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I am trying to follow Hatcher's proof of Poincaré Duality on p. 248.

Suppose $M$ is an $R$-oriented manifold. We have first defined $H^k_c(M;R)$ to be the direct limit of groups $H^k(M,M\setminus K;R)$ where we take some compact subset $K\subset M$. We define a duality map $$D_M:H^k_c(M;R)\to H_{n-k}(M;R)$$ which sends $[\phi]$ with $\phi\in H^k(M,M\setminus K;R)$ to $\phi\smallfrown\mu_K$. Where $\mu_K$ is the unique class in $H_n(M,M\setminus K;R)$ which for all $x\in K$ gets mapped to the local orientation at $x$ under the inclusion induced morphism $$H_n(M,M\setminus K;R)\to H_n(M,M\setminus \{x\};R)$$

The noncompact version of Poincaré Duality then says the following:

For an $R$-oriented $n$-manifold $M$ the duality map $$D_M:H^k_c(M;R)\to H_{n-k}(M;R)$$ is an isomorphism for all $k$.

The proof is inductive and begins with the observation that it holds for $M=\mathbb{R}^n$.

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I do not know how one explicitly makes this identification $H^k(\mathbb{R}^n;R)\to H^k_c(\Delta^n,\partial\Delta^n)$ and why the map turns into the cap product with the class represented by the identity.

Another method I looked into was via an isomorphism $$H^k(\mathbb{R}^n,\mathbb{R}^n\setminus B)\to H^k_c(\mathbb{R}^n;R)$$ where $B$ is a closed ball around the origin. The duality map then becomes the cap product with $\mu_K$, but I am not sure how to finish the argument here.

A similar question was asked here, but no answers were given and I was unable to translate the comments into something concrete.

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  • $\begingroup$ For a compact manifold with boundary, $H^k_c(M-\partial M;R)$ is naturally isomorphic to $H^k(M,\partial M;R)$: the former group is isomorphic to $\varinjlim H^k(M-\partial M, M-\partial M-K;R)$ where $K$ varies over compact subsets of $M-\partial M$. Fixing a collar of $\partial M$, one can see that this limit equals $\varinjlim H^k(M-\partial M,\partial M \times (0,\epsilon);R)$ for varying $\epsilon$, and this group is $H^k(M,\partial M;R)$. $\endgroup$
    – user302934
    Jun 1, 2021 at 13:18
  • $\begingroup$ Thanks for the comment. I am not too familiar with collars. Does $\partial M\times (0,1)$ cover $M\setminus\partial M$? $\endgroup$ Jun 1, 2021 at 13:37
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    $\begingroup$ Collars will appear soon in Hatcher's book. Every compact subset of $M-\partial M$ is contained in $M-\partial M\times (0,\epsilon]$ for some $\epsilon \in (0,1)$. $\endgroup$
    – user302934
    Jun 1, 2021 at 14:02
  • $\begingroup$ @user302934 I am still a bit unsure on how to show that the limit with the collar neighborhoods is isomorphic to $H^k(M,\partial M;R)$. And also on why this turns the duality map in the cap product with a unit times the generator $[M]$. $\endgroup$ Jun 2, 2021 at 9:19
  • $\begingroup$ Maybe by regarding them as cohomologies of quotient space? But I cannot explain why that becomes a good pair. $\endgroup$ Jan 28 at 16:22

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