Definition of boundary in a topological invariant way

Question

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, here:http://paginas.matem.unam.mx/cprieto/index.php/es/archivos-2/libros?download=11:fiber-bundles).

Since I need to look up everything I don't know (and this shows up very much early on) I started to look for a definition of boundary of a topological space.

Page 2 defines it as: $$ Bd(M)=\{x\in M| H_2(M,M-x)=0\} $$ Looking up on books I couldn't manage to find where this comes from. On the internet I've found something similar called excision or relative homology.

Can someone help me find good resources for this definition. Thanks

Answer 1

If $M$ is an $n$-manifold with boundary then there are two relevant formulas:

• If $x$ is an interior point of $M$ then $$H_n(M,M-x) \approx H_n(\mathbb{R}^n,\mathbb{R}^n-0) \approx \mathbb{Z} $$

The first $\approx$ is proved by applying the excision theorem using a manifold coordinate chart near $x$. The second $\approx$ is proved in any algebraic topology textbook around the same place that you find the computation $H_n(S^n) \approx \mathbb{Z}$.

Let $\mathbb{R}^n_{\ge} = \{p \in \mathbb{R}^n \mid p_n \ge 0\}$.

• If $x$ is a boundary point of $M$ then $$H_n(M,M-x) \approx H_n(\mathbb{R}^n_{\ge}, \mathbb{R}^n_{\ge} - 0) \approx 0 $$

Again the first $\approx$ is proved by applying the excision theorem together with a boundary-of-a-manifold-with-boundary coordinate chart near $x$. The second $\approx$ is an exercise (the inclusion of $\mathbb{R}^n_{\ge}-0$ into $\mathbb{R}^n_{\ge}$ is a homotopy equivalence).