Uniqueness of topology and smooth structure on an immersed submanifold

Question

I am reading "Introduction to Smooth Manifolds" by Lee and encountered the following theorem:

If $N,M$ are smooth manifolds and $F: N \rightarrow M $ is an injective smooth immersion, then $S = F(N)$ has a unique topology and smooth structure such that $S$ is an immersed submanifold of $M$, and $F$ is a diffeomorphism onto its image.

The book defines a set $O \subseteq S$ as open if and only if $F^{-1}(O)$ is open in $N$. It defines a smooth structure on $S$ by taking chart $(U, \phi)$ on $M$ and defining a corresponding chart on $S$ as $(F(U), \phi \circ F^{-1})$.

My question is how is the topology and smooth structure unique?

Specifically, I can take the topology on $S$ to be the subspace topology inherited from $M$ and define the corresponding smooth structure on $S$ by taking any chart $(U, \phi)$ on $M$ and defining a corresponding chart $(U \cap S, \phi \circ F^{-1})$ on $S$. This defines a smooth structure since for any two charts $(U \cap S, \phi \circ F^{-1}),$ and $ (V \cap S, \psi \circ F^{-1})$, we have $\psi \circ F^{-1} \circ F \circ \phi^{-1}$ which is smooth, so the charts are compatible.

The inclusion map $\iota: S \rightarrow M$ is also a smooth immersion since each point $s \in S$ has chart $(U \cap S, \phi \circ F^{-1})$ and $(U, \phi)$ about $\iota(s)$, $$ \phi \circ \iota \circ F \circ \phi^{-1} = \phi \circ F \circ \phi^{-1} $$

which is a smooth immersion. So, $S$ is an immersed submanifold.

Finally, $F$ is a diffeomorphism onto its image because $F$ is assumed smooth and injective, and $F^{-1}: S \rightarrow N$ "in coordinates" is $\phi \circ F^{-1} \circ F \circ \phi^{-1}$ which is smooth.

Therefore, this topology and smooth structure satisfies the properties we sought to prove, despite the topology nor smooth structure not being the same as the intial ones proposed in the book?

Answer 1

The concept of an immersed manifold often leads to confusion and I do not see a real benefit to use it.

Let us consider a more general situation: Given a smooth manifold $M$, a set $S$ and a bijection $f : N \to S$. Then it is clear that

1. there exists a unique topology $\tau$ on $S$ making $f : N \to (S,\tau)$ a homeomorphism.

2. there exists a unique smooth structure $\mathfrak D$ on $(S,\tau)$ making $f : N \to (S,\tau,\mathfrak D)$ a diffeomorphism.

Lee uses this with $S = F(N)$. His definition of open sets and of charts is nothing else than the explicit construction of $\tau$ and $\mathfrak D$.

The problem is that $\tau$ in general is not the subspace topology inherited from $M$. As an example take the injective smooth immersion $\beta : N = (-\pi,\pi) \to \mathbb R^2$ decribed here. Its image $S$ is a "figure eight" which is compact and thus cannot be homeomorphic to $N$. Moreover $S$ is not a manifold. Thus, unfortunately, your approach does not work.

You see that the "immersed submanifold structure" of $S$ is a fairly artificial thing which has nothing to do with the subspace $S$ of $\mathbb R^2$.

Also have a look at Restricting the codomain of a smooth map to submanifold is smooth.

Update:

Let us consider the bijective function $\bar F : N \stackrel{F}{\to} F(N)$. We endow $F(N)$ with the unique topology and smooth structure making $\bar F$ a diffeomorphism. If $\iota : F(N) \hookrightarrow M$ denotes inclusion, then clearly $\iota \circ \bar F = F$. Since $\bar F$ is a diffeomorphism, we get $\iota = F \circ \bar F^{-1}$. This shows that $\iota$ is an injective immersion because $F$ is one.