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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

MatterFr1122
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1 Answers1

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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.

Lee Mosher
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  • This proof contains concepts that I'm not familiar with, but did I get the result right: It claims that if $\phi_i$ is a homeomorphism from an $M$-open subset onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$ and $x\in\Omega_1\cap\Omega_2$, then $\phi_1(x)\in\partial\mathbb H^k$ implies $\phi_2(x)\in\partial\mathbb H^k$? Is this correct? I'm able to prove this assuming that the $\phi_i$ are $C^1$-diffeomorphisms. – 0xbadf00d Aug 10 '20 at 09:43
  • First thank you for the answer - I coudl figure out the details myself. Although I do not see how 1. trivially linked to that inductive method you mention: here I needed excision first too (as mentioned in 2) to get $H_n(U,U-{x})$ for some open ball $U$ and then I needed the long exact sequence. If there is a more direct argument it would be nice to hear it :) – 0CT0 Mar 22 '21 at 12:08
  • As said, 1 is only a piece of an inductive method for computing homology of spheres. It sounds like you have figured out the other pieces. – Lee Mosher Mar 22 '21 at 14:47