There is an isomorphism between the homology of a closed connected manifold and its local homology. I am wondering whether there is a similar isomorphism for manifolds with boundary, but I cannot find any such result.
More specifically, I am thinking about the following result: if M is a closed connected n-manifold that is R-orientable, then the map $H_n(M;R)\to H_n(M,M\setminus\{x\};R)$ is an isomorphism for all $x\in M$ (Theorem 3.26 in Hatcher).
Is there an analog of this result for compact manifolds with boundary? Is it true that $H_n(M,\partial M;R)\to H_n(M,M\setminus\{x\};R)$ is an isomorphism for all $x\in M\setminus\partial M$? And if it is true, where could I find this result?
In Hatcher's book, the proof of Lemma 3.27 and a discussion at the bottom of page 253 suggest that the proof of the result for closed manifolds easily extends to compact manifolds with boundary, as follows.
Let $x_0\in M\setminus\partial M$. There is a natural homomorphism $H_n(M,\partial M)\to H_n(M,M\setminus\{x_0\})$, sending $\alpha$ to $\alpha_{x_0}$. The goal is to prove that it is an isomorphism, which essentially means that $\alpha$ is uniquely determined by $\alpha_{x_0}$.
Let $A=M\setminus(\partial M\times [0,\epsilon))$ with $\epsilon$ small enough so that $x_0\in A$. By deformation retraction and excision, there is an isomorphism between $H_n(M,\partial M)$ and $H_n(M\setminus\partial M,M\setminus A)$. Lemma 3.27 applied to $A$ shows that $\alpha$ is uniquely determined by the map $x\mapsto\alpha_x$ for $x\in A$, which is uniquely determined by $\alpha_{x_0}$ because $M$ is connected.
Is this argument correct? I'm not sure that ``$\alpha$ is uniquely determined by $\alpha_{x_0}$'' indeed implies that we get an isomorphism.