What is the relation between homology groups of a manifold with boundary and space obtained after removing its boundary?

Question

I don't know much about homology theory, but the following question pops up in my mind and it may have some ambiguities.

Suppose $M$ is a finite-dimensional manifold with non-empty boundary $\partial (M)$. Now suppose $N= M-\partial(M)$ with induced topology from $M$. What is the relation between homology groups of $M$ and $N$?

Answer 1

I think I have a sketch that $M \setminus \partial M \hookrightarrow M$ induces an equivalence in Homology, but I didn't check all the details. So please delete it if it's wrong. If someone works out the details, please post your answer, I'll delete mine then.

1. I think for every open subset $U \subset \mathbb{R}_{\geq 0} \times \mathbb{R}^{n-1}$ the inclusion $\mathbb{R}_{> 0} \times \mathbb{R}^{n-1} \hookrightarrow \mathbb{R}_{\geq 0} \times \mathbb{R}^{n-1}$ should induce a homotopy equivalence $U \cap (\mathbb{R}_{> 0} \times \mathbb{R}^{n-1}) \to U \cap (\mathbb{R}_{\geq 0} \times \mathbb{R}^{n-1})$.
2. There is an open cover $M = \bigcup_{i \in I} B_i$ such that each $B_i$ is isomorphic via a map $\phi$ an open subset $U_i$ of $\mathbb{R}_{\geq 0} \times \mathbb{R}^{n-1}$. Then $\phi$ sends $B_i \cap (M\setminus \partial M)$ to $U_i \cap (\mathbb{R}_{> 0} \times \mathbb{R}^{n-1})$ and $B_i \cap \partial M$ to $U_i \cap (\{ 0\} \times \mathbb{R}^{n-1})$.
3. Inductively over $k \in \mathbb{N}$ one may can show that $\iota: (B_{i_1} \cup \cdots \cup B_{i_k}) \cap (M \setminus \partial M) \hookrightarrow B_{i_1} \cup \cdots \cup B_{i_k}$ induces an isomorphism in homology using the Mayer Vietoris seqence $\require{AMScd}$ \begin{CD} H_i((B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \cap (M \setminus \partial M)) \oplus H_i(B_{i_k} \cap (M \setminus \partial M)) @>>> H_i((B_{i_1} \cup \cdots \cup B_{i_{k}}) \cap (M \setminus \partial M)) @>\partial>> H_{i-1}((B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \cap (M \setminus \partial M) \cap B_{i_k}) @>>> H_{i-1}((B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \cap (M \setminus \partial M)) \oplus H_{i-1}(B_{i_k} \cap (M \setminus \partial M))\\ @V V H_i(\iota) \oplus H_i(\iota) V @VV H_i(\iota) V @VV H_i(\iota) V @V V H_{i-1}(\iota) \oplus H_{i-1}(\iota) V \\ H_i(B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \oplus H_i(B_{i_k}) @>>> H_i(B_{i_1} \cup \cdots \cup B_{i_{k}}) @>\partial>> H_{i-1}((B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \cap B_{i_k})) @>>> H_{i-1}(B_{i_1} \cup \cdots \cup B_{i_{k-1}}) \oplus H_{i-1}(B_{i_k} ) \end{CD} and 1. for the subsets of $B_{i_k}$ as well as the five lemma.
4. In singular homolgy every chain lies in a compact subspace covered by finitely many $B_i$'s. That's why 3. is enough to show that the inclusion $M \setminus \partial M \to M$ induces an equivalence in homology.