My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of the definitions or theorems in Chapters 1 to 10 that might already be found in An Introduction to Manifolds by Loring W. Tu. I mostly assume the concepts are the same until there is evidence otherwise.
I believe I might have come across evidence that disproves one of my assumptions of equivalence of concepts. The assumption I made is:
Let $M$ be a smooth $n$-manifold with dimension. Let $N \subseteq M$. $N$ is what Madsen and Tornehave would call a "domain with smooth boundary" if and only if $N$ is what Tu would call a smooth $n$-manifold with boundary (see context below). $\tag{A}$
Now, this post shows it is not necessarily the case that compact subspaces of $\mathbb R^n$ can be realized as smooth manifolds with boundary. I assume this is not a dimension issue (see context below), so I guess I'll say that not all compact subspaces of $\mathbb R^n$ can be realized as smooth n-manifolds (or k-manifolds) with boundary.
Question: (Definitions given below) Let $K$ be the compact set of Lemma 11.25. How does the proposition (the "(14)") after Lemma 11.25 apply Corollary 11.23 and thus Theorem 11.22 given $K$ is not stated to be a (compact) domain with smooth boundary while Theorem 11.22 assumes a domain with smooth boundary?
Possible answers:
$K$ disproves (A): $K$ is a domain with smooth boundary but not a smooth $n$-manifold with boundary.
$K$ does not disprove (A), but (A) is still false: $K$ is both a domain with smooth boundary and a smooth $n$-manifold with boundary, but, in general, domains with smooth boundary and smooth $n$-manifolds with boundary are not equivalent.
$K = \{||x|| \le 2\}$, and $K$ does not disprove (A), and (A) is true: $K$ is both a domain with smooth boundary and a smooth $n$-manifold with boundary.
$K \ne \{||x|| \le 2\}$, and $K$ does not disprove (A), and (A) is true: $K$ is both a domain with smooth boundary and a smooth $n$-manifold with boundary.
The proposition is incorrect and should have assumed $K$ is a (compact) domain with smooth boundary.
Context:
On (A) and on dimensions:
Definition 10.5, the definition of "domain with smooth boundary" of Madsen and Tornehave.
Tu Definition 22.6 (part 1) and Tu Definition 22.6 (part 2), of $n$-manifold with boundary (and manifold with boundary), where $\mathcal H^n := \{x \in \mathbb R^n | x_n \ge 0\}$.
Note that Madsen and Tornehave have $x_1 \le 0$ while Tu has $x_n \ge 0$.
Also note that Tu's manifolds are not the same as his $n$-manifolds because not all his manifolds' connected components have the same dimension (see this and this), but for Madsen and Tornehave, I believe their manifolds and domains with smooth boundary have (uniform) dimensions.
Definitions of index: Definition 11.16, Definition 11.19, Definition 11.21
