Collar Neighborhood in Lee Smooth Manifolds

Question

In John Lee's Introduction to Smooth Manifolds, a collar neighborhood of a smooth manifold $M$ with boundary is defined to be a neighborhood $V$ of $\partial M$ that is the image of a smooth embedding $\alpha: [0,1) \times \partial M \to M$ such that $\alpha(0,x)=x$ for all $x \in \partial M$. However, is it necessarily true that $\alpha$ is a diffeomorphism onto $V$? The following results in Lee seem to assume this, but I can't figure out how to show this.

The issue is that $[0,1) \times \partial M$ and $M$ both have boundaries, so we can't apply results that depend on the rank theorem. I believe this could be remedied by knowing that we can embed any smooth manifold with boundary into a smooth manifold without boundary, but this result in turns depends on what I'm currently asking being true, for Lee makes this diffeomorphism assumption in proving this fact (see Errata to Thm 9.26).

Any help would be much appreciated! Thank you :)

Answer 1

Here is the desired theorem: Let $f: M \to N$ be a smooth map between smooth manifolds with boundary such that $f(\partial M) = \partial N$. If $df_p$ is invertible, then $f$ is a local diffeomorphism at $p \in M$.

Proof. If $p \in \mathrm{int}\, M$, then simply apply the result for manifolds without boundary to the smooth map $f: \mathrm{int}\, M \to \mathrm{int}\, N$. Otherwise, $p \in \partial M$. By taking smooth charts at $p$ and $f(p)$, we can assume that $M = N = \mathbb H^n$ and $p = f(p) = 0$. We can find a smooth function $g: U \to \mathbb R^n$ on a neighborhood $U$ of $0$ in $\mathbb R^n$ such that $g$ extends $f$ on $U \cap \mathbb H^n$. Then $dg_0$ is invertible, so we can assume $U$ is small enough that $g$ is a diffeomorphism onto an open ball $V \ni 0$ in $\mathbb R^n$.

Now let $\pi: \mathbb R^n \to \mathbb R$ denote projection onto the last coordinate, which is an open map. For a subset $S \subseteq \mathbb R^n$, we write $S^{>a} = S \cap \pi^{-1}((a,\infty))$ and similarly define $S^{\geq a}$, $S^{<a}$, $S^{\leq a}$, and $S^a$ (the latter being $S \cap \pi^{-1}(a)$). We know that $g$ maps $U^0$ into $V^0$ and $U^{>0}$ into $V^{>0}$, so it suffices to prove that $g$ maps $U^{<0}$ into $V^{<0}$ (for then $g^{-1}|_{V^{\geq 0}}$ is a local smooth inverse to $f$ at 0). Notice $$ V^{>0} = (V^{>0} \cap g(U^{\leq0})) \sqcup (V^{>0} - g(U^{\leq 0})). $$ Clearly $V^{>0} \cap g(U^{\leq0}) = V^{>0} \cap g(U^{< 0})$ is open, and $V^{>0} - g(U^{\leq 0}) \supseteq g(U^{>0})$ is nonempty open, so $V^{>0} \cap g(U^{\leq0}) = \varnothing$ because $V^{>0}$ is connected. Thus $g(U^{<0}) \subseteq V^{\leq 0}$. But $\pi(g(U^{<0})) \subseteq (-\infty,0]$ is open, so in fact $g(U^{<0}) \subseteq V^{<0}$, as desired. //

Edit: Thanks to Karthik Kannan for pointing out a subtlety in my original answer, which I have corrected by taking $V$ to be a ball. See also Is a map with invertible differential that maps boundary to boundary a local diffeomorphism?, where this subtlety is again overlooked.

Answer 2

As in the previous answer, let $U$ and $V$ be open neighborhoods of $p$ and $f(p)$ respectively in $\mathbb{R}^{n}$, and let $g : U\rightarrow V$ be a diffeomorphism such that $g|_{U\cap \mathbb{H}^{n}} = f|_{U\cap\mathbb{H}^{n}}$. The problem is that even though $f(\partial\mathbb{H}^{n}) = \partial\mathbb{H}^{n}$, it might happen that$ f(U\cap\partial\mathbb{H}^{n}) = g(U\cap\partial\mathbb{H}^{n})$ is a proper subset of $V\cap\partial\mathbb{H}^{n}$. More specifically, it might be the case that for some $q\in V\cap\partial\mathbb{H}^{n}$, we have $f^{-1}(\{q\})\cap (U\cap\partial\mathbb{H}^{n}) = \varnothing$ (and thus $g^{-1}(q)\in U\setminus\partial\mathbb{H}^{n}$).

This situation can be remedied as follows. If $h = f|_{\partial\mathbb{H}^{n}}: \partial\mathbb{H}^{n}\rightarrow\partial\mathbb{H}^{n}$ then $\text{d}h_{p}: T_{p}\partial\mathbb{H}^{n}\rightarrow T_{f(p)}\partial\mathbb{H}^{n}$ is also an isomorphism. By the Inverse Function Theorem, $h$ is a diffeomorphism on a neighborhood of $p$ in $\partial\mathbb{H}^{n}$. Let $C$ be small open cubical neighborhood of $f(p)$ contained in $V$ such that $h: h^{-1}(C\cap\partial\mathbb{H}^{n})\rightarrow C\cap\partial\mathbb{H}^{n}$ is a diffeomorphism. In particular, $g : g^{-1}(C)\rightarrow C$ is a diffeomorphism that maps $g^{-1}(C)\cap\partial\mathbb{H}^{n} = h^{-1}(C\cap\partial\mathbb{H}^{n})$ onto $C\cap\partial\mathbb{H}^{n}$. Using a connectedness argument, we may conclude that $f(g^{-1}(C)\cap\text{Int}(\mathbb{H}^{n})) = C\cap\text{Int}(\mathbb{H}^{n})$.