Confusion with immersions, embeddings, local homeomorphisms, and local diffeomorphisms.

Question

Definitions.

• A local homeo/diffemorphism is a continuous/smooth map $f:X\to Y$ such that there's an open cover $(U_i)$ of $X$ for which $f|_{U_i}:U_i\to fU_i$ is a homeo/diffeomorphism.
• A topological/smooth embedding is a continuous/smooth map $f:X\to Y$ which is a homeo/diffeomorphism onto its image. In other words, the first factor of its image factorization is an isomorphism.

If I understand correctly, a local homeo/diffeomorphism is thus precisely a local topological/smooth embedding: there's an open cover of the domain making the restrictions into topological/smooth embeddings. This leads me to two sources of confusion.

1. This answer involves the words "local diffeomorphism onto its image". This is strange to me - it seems, at least using the my definition, that a smooth map is a local diffeomorphism iff its a local diffeomorphism onto its image. What am I missing here?

2. This answer proves that any immersion (injective differential) is locally a smooth homeomorphism onto its image with injective derivative, i.e a local immersion which is a local topological embedding. Following my (probably frail) reasoning in the paragraph following the definitions, this would imply any immersion is a local homeomorphism. In fact the answer seems to prove any immersion is locally a smooth embedding (following my definition) since the local section constructed seems smooth. But that would mean it's even a local diffeomorphism! (I am not sure which definition of 'embedding' the asker had in mind).

I am confused: (1) makes me think I am corrigibly crazy. (2) makes me think I'm hopelessly crazy, since by the inverse function theorem a smooth map is a local diffeomorphism iff it's an immersion and a submersion, and I also don't think immersions need to be local homeomorphisms.

What are my mistakes?

Answer 1

The definitions of topological/smooth embedding seem to be alright. The suggested definitions of local homeo/diffeomorphism are wrong. The correct definitions require the existence of an open cover $(U_i)$ of the domain on which $f$ restricts to topological/smooth embeddings such that moreover $fU_i\subset Y$ is open. This is in contrast to local topological/smooth embeddings which forgo the latter requirement.

Now we may resolve the confusion.

1. Asking for $f:X\to Y$ to be a local homeo/diffeomorphism onto its image means there's a cover such that $f|_{U_i}$ are topological/smooth embeddings and that $fU_i\subset fX$ is open. If we forgo "onto its image" then we'd want $fU_i\subset Y$ to be open. These are distinct conditions.
2. Indeed an immersion is a local smooth embedding, but there need not exist an open cover whose members are mapped diffeomorphically and also have images open in the codomain.

Answer 2

See these:

What if potential errors in an answer are pointed out in comments but not addressed?

What is/are the definitions of local diffeomorphism onto image?

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Neal says here that immersions are "local diffeomorphisms onto images". If we read "local diffeomorphisms onto images" as "local-(diffeomorphisms onto images)" rather than "(local diffeomorphisms)-onto images", then this is correct because diffeomorphisms onto (submanifold) images are equivalent to embeddings and because immersions are equivalent to local embeddings.

However, "(local diffeomorphisms)-onto images" imply images are regular/embedded submanifolds and not just immersed submanifolds. Therefore, Neal is wrong if Neal claims that immersions are "(local diffeomorphisms)-onto images".

Therefore, reading "local diffeomorphisms onto images" as "local-(diffeomorphisms onto images)", we have

$$\text{local diffeomorphism} \implies \text{local diffeomorphism onto image} \implies \text{immersion and image is submanifold} \implies \text{immersion} \iff \text{local embedding}$$

These are the definitions:

Let $X$ and $Y$ be smooth manifolds with dimensions.

• Local diffeomorphism:

  A map $f:X\to Y$, is a local diffeomorphism, if for each point x in X, there exists an open set $U$ containing $x$, such that $f(U)$ is a submanifold with dimension of $Y$, $f|_{U}:U\to Y$ is an embedding and $f(U)$ is open in $Y$. (So $f(U)$ is a submanifold of codimension 0.)

• Local diffeomorphism onto image:

  A map $f:X\to Y$, is a local diffeomorphism onto image, if for each point x in X, there exists an open set $U$ containing $x$, such that $f(U)$ is a submanifold with dimension of $Y$, $f|_{U}:U\to Y$ is an embedding and $f(U)$ is open in $f(X)$. (This says nothing about $f(X)$ explicitly, but it will turn out $f(X)$, like $f(U)$ is a submanifold of $Y$.)

• Local embedding/Immersion:

  A map $f:X\to Y$, is a local embedding/an immersion, if for each point x in X, there exists an open set $U$ containing $x$, such that $f(U)$ is a submanifold of $Y$ with dimension and $f|_{U}:U\to Y$ is an embedding. (This says nothing about $f(X)$ explicitly, but it will turn out $f(X)$, like $f(U)$ is an immersed submanifold of $Y$. However, $f(X)$, unlike $f(U)$, is not necessarily a regular/an embedded submanifold of $Y$.)

The difference in all these 3 is what $f(U)$ is. In all cases, $f(U)$ is a submanifold of $Y$, so indeed you still get a "diffeomorphism" out of an immersion.

Observe that while local diffeomorphism implies immersion but not conversely, local diffeomorphisms are equivalent to open immersions, to immersions whose domain and range have equal dimensions and to immersions that are also submersions (submersions are open maps).