Boundary of a topological manifold invariant?

Question

Let $M=(X,\tau)$ be a topological manifold with boundary. One can proof that the interior $Int(M)$ and boundary $\partial M$ of the manifold are distinct sets.

I was wondering if someone knows a good reference to cite the proof (Lee only presents this proof as an exercise)?

Thanks

Answer 1

Assuming $n$ is the dimension of $M$, you can do this with the following homology computations:

1. $x \in \text{Int}(M)$ if and only if $H_n(M,M-x;\mathbb{Z}) \approx \mathbb{Z}$,

2. $x \in \partial M$ if and only if $H_n(M,M-x;\mathbb{Z}) \approx 0$.

The first is a piece of the standard inductive method for proving $H_n(S^n;\mathbb{Z})=\mathbb{Z}$, as seen for example in Section 2.1 of Hatcher.

The second is on the level of an elementary direct application that combines the excision theorem, the long exact sequence of relative homology, and homotopy invariance of homology. Excision is used to make the left hand side isomorphic to $H_n(H^n, H^n - O;\mathbb{Z})$ where $H^n$ denotes the closed upper half space $\{(x_1,…,x_n) \in \mathbb{R}^n \, | \, x_n \ge 0\}$ and $O$ is the origin.