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My question is related to this post. It should be simple to answer, I must be making a fundamental mistake.

The answers all claim that if you have a submersion $f$ from a compact manifold $M \subset R^n$ to $R^n$ then, because submersions are open maps, $f(M)$ must be open in $R^n$. Therefore (after showing that $f(M)$ is also closed), $f(M)$ is both open and closed thus, because $R^n$ is connected, onto.

The following example is faulty since $M$ is a manifold with boundary:

The problem I have can be illustrated by this example: Let $M$ (a manifold be the closed unit ball in $R^n$ and $f$ be the identity. My understanding is that $f$ is a smooth submersion. Yet $f(M) = M$. I rationalize this by making the distinction between strongly and relatively open subsets. In my example $f(M)$ is relatively open in $R^n$, but not strongly open.

Would someone be kind enough to point out my error (or offer a link that might shed light on its cause)?

Thank you in advance.

Edit: I was going to place this in a comment but it's really too long for that. Thanks to both Moishe Kohan and Ben Steffan for their posts alerting me to the problem with my example.

First, yes, I realize I should have pointed out clearly that my $M$ is a manifold with boundary so that it's not directly comparable to the linked post. However, I know that submersions are defined for manifolds with boundary, even manifolds with corners, and that even in these contexts they are open maps (or so it says in my Differential Topology reference). So my question becomes:

Question Is the issue that indeed, if $M$ is a manifold with boundary (or with corners), then $f(M)$ would not be an open subset of $R^n$ (just relatively open) and the argument in the linked post would not go through? Or am I missing something even more fundamental?

Edit 2 Moishe Kohan has alerted me to, in effect, the fact that submersions between manifolds with boundary/corners may not be open maps. The text I use may have an error or, what is far more likely, I have misunderstood or missed one or more important assumptions the authors make. I will investigate.

Resolution The reference I was using that stated submersions between manifolds-with-corners were open maps was using a non-standard definition of submersion. Again, thanks to Moishe Kohan for pointing this out.

Gary
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user167131
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    What is your definition of a manifold? – Moishe Kohan Jan 21 '25 at 20:48
  • "My understanding is that $f$ is a smooth submersion." You should rethink this. For a start, the closed unit ball is a manifold with boundary. – Ben Steffan Jan 21 '25 at 20:51
  • @Moishe Kohan: Yes, if your point is that $M$ is a manifold with boundary, I understand, sorry for being loose with my language. I will edit the question. – user167131 Jan 21 '25 at 20:59
  • If your differential topology source (which one?) says so, it is not trustworthy. Or you are omitting some assumptions that your source is making. – Moishe Kohan Jan 21 '25 at 21:29
  • I understand your question completely, I wouldn't trust me either. :) My reference is Margalef-Roig & Dominguez, Differential Topology. Proposition 4.1.2 says: "Every submersion of class p is an open map". This is at the beginning of Chapter 4, right after the definition of submersions between manifolds with corners. My guess is I am omitting an important detail (perhaps the definition of submersion, different from what I'm used to). Certainly MUCH more likely than is a mistake in the text. Thank you for alerting me, I will investigate. – user167131 Jan 21 '25 at 21:39
  • Yes, their definition of a submersion (4.1.1) is nonstandard. – Moishe Kohan Jan 21 '25 at 21:47
  • OK, well, thank you for pointing that out. I perhaps should have checked it myself but figured it wouldn't make sense for them to define a common term in an uncommon way. At least unless they stated that explicitly in the text. But now I know, this open map thingie was driving me crazy. – user167131 Jan 21 '25 at 22:02

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