Inverse Function Theorem for Manifolds with Boundary as the Domain

Question

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\hookrightarrow\mathbb{R}^n$
My guess is that this is merely a matter of being open rather than really deep problems, or?

Answer 1

Yes, your guess is right -- it's just that you can't find (open) neighborhoods of the given points that are diffeomorphic.

The theorem you're looking at in my book (Thm. 4.5) says

Suppose $M$ and $N$ are smooth manifolds, and $F:M\to N$ is a smooth map. If $p\in M$ is a point such that $dF_p$ is invertible, then there are connected neighborhoods $U_0$ of $p$ and $V_0$ of $F(p)$ such that $F|_{U_0}: U_0\to V_0$ is a diffeomorphism.

What fails in an example like $\iota:\mathbb H^n\to \mathbb R^n$ is that you can't find any neighborhood of $0$ in $\mathbb H^n$ that's diffeomorphic to a neighborhood of $0$ in $\mathbb R^n$.

It boils down to what we mean by the word "locally." To say a map $F:M\to N$ is locally invertible usually means (at least) that for each $p\in M$, there exist a neighborhood $U$ of $p$ and a neighborhood $V$ of $f(p)$ such that $f|_U:U\to V$ is bijective. If you don't require $V$ to be a neighborhood, then every smooth immersion would be locally invertible. I guess you could define the term that way, but it's not the way most people understand it.