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Suppose $X$ is a smooth, compact, connected $n$-manifold without boundary which admits an immersion to $S^n$. Show that if $n>1$, then this immersion is a diffeomorphism.

Thanks for the very inspiring mentors, here I got some thoughts

$df_x$ is bijective. Because the tangent planes of the domain has the same dimension as domain; the tangent space of the codomain has the same dimension as codomain. But dim$X = n$, dim$S^n = n$, so $df_x$ maps from dim$n$ to dim$n$. Given immersion, $df_x$ is injective therefore bijective.

On the other hand, $f$ being an immersion told at that $df_x$ is nonsigular, hence a local diffeomorphism. I got stuck extending local diffeomorphism to global diffeomorphism. Is there a general strategy to achieve this(when this is true)?

Thank you.

1LiterTears
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    Pick a name for the immersion to start, $f:X \to S^n$. What properties of $f$ can you deduce first? And next? And next? For example, is $f$ surjective? ... – Lee Mosher Jun 21 '13 at 18:20
  • To start with - $df_x$ is injective $\forall x \in X$. – 1LiterTears Jun 21 '13 at 18:25
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    But you need some global properties of $f$. Hence my question of whether $f$ is surjective. – Lee Mosher Jun 21 '13 at 18:52
  • Yes, so I am trying to deduce whether $f$ is surjective from $df_x$ is injective.... – 1LiterTears Jun 21 '13 at 19:27
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    Pay attention to the fact, Jellyfish, that $\dim X = \dim S^n = n$. So what do you know immediately if $df_x$ is injective? – Ted Shifrin Jun 21 '13 at 20:13
  • @TedShifrin it is bijective! – 1LiterTears Jun 21 '13 at 21:23
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    So $f$ is a local diffeomorphism? – Ted Shifrin Jun 21 '13 at 21:34
  • Yes as long as $df_x$ is nonsingular, which can be obtained by the condition $f$ is an immersion alone. – 1LiterTears Jun 21 '13 at 21:35
  • You need to use compactness of $X$ to prove $f$ is a covering map. And then you'll see why $n>1$ is crucial. – Ted Shifrin Jun 22 '13 at 02:17
  • Thanks @TedShifrin - So I need to show that $f$ is a surjective open map. But I am not certain about how to show this - and perhaps that's why I don't know why $n>1$ is crucial. My thoughts are, first show $f$ is open map, which means it maps open sets to open sets. Given $X$ is boundaryless, it is open - right? Then,I am not sure is $S^n$ open? – 1LiterTears Jun 22 '13 at 02:37
  • Then I want to show $f$ is surjective. By compactness, its open cover has finite subcover. I don't think we can use the fact that $f$ is locally surjective, so I can't prove surjectivity. – 1LiterTears Jun 22 '13 at 02:39
  • For surjectivity, you need the image both open and closed in $S^n$. But for a coverng map you need more: You need to know each point has a neighborhood that is evenly covered. At some point, your background and where this problem is coming from become relevant! It is not just a beginning problem on smooth manifolds. – Ted Shifrin Jun 22 '13 at 03:45

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Once you know that $f$ is a local diffeomorphism, to conclude that it's a global diffeomorphism you just need to show that it's bijective. Surjectivity is pretty easy: Because $X$ is compact, $f(X)$ is also compact, and because $S^n$ is Hausdorff, $f(X)$ is closed in $S^n$. On the other hand, the fact that $f$ is a local diffeomorphism implies that it's an open map, and thus $f(X)$ is open. Since $S^n$ is connected, $f(X)$ is all of $S^n$.

Injectivity is quite a bit harder. The only proof I know uses the theory of covering spaces. Because $f$ is a proper local homeomorphism, it's a covering map (which is another way to prove surjectivity), and because $S^n$ is simply connected, it follows that $f$ is injective.

One place to read about covering spaces is in my book Introduction to Topological Manifolds.

Jack Lee
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  • Thank you so much Jack Lee! – 1LiterTears Jun 22 '13 at 20:57
  • Thanks again Professor Lee! I am really elated getting your help! I just got your book Introduction to Topological Manifolds, and I am very excited to read it! I am also reading your other book, Riemannian manifolds this summer. Could you give me some suggestion reading your three books - for example, in which order should I read them (or concurrently)? Thank you very much. – 1LiterTears Jul 02 '13 at 16:18
  • @Jellyfish -- they're meant to be read in this order: (1) Introduction to Topological Manifolds, (2) Introduction to Smooth Manifolds, and (3) Riemannian Manifolds. Of course, hardly anybody really reads all three from cover to cover. If you prefer, you can start at the beginning, skip through and focus on the parts that most interest you, and then come back and fill in details when you need them. – Jack Lee Jul 02 '13 at 17:41