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My first time posting in this forum. This is not a homework problem. I am trying to learn my own from John M. Lee Introdcution to Smooth Manifolds.

In Chapter 3, there is the problem 3-4

Let $C \subset \mathbb{R}^2$ be the unit circle, and let $S \subset \mathbb{R}^2$ be the boundary of the square of side 2 centred at orign: $S= \lbrace (x,y) \colon \max(|x|,|y|)=1 \rbrace.$ Show that there is a homeomorphism $F:\mathbb{R}^2 \to \mathbb{R}^2$ such that $F(C)=S$, buth there is no diffeomorhpism with the same property. [Hint: Consider what $F$ does to the tangent vector to a suitable curve in C].

I can construct a homeomorphism (by placing the circle inside the square and then every radial line intersects the square at exactly one point.) But, I dont know how to do the rest of the problem or understand the hint.

I do not know how to write out what tangent space should be for the square. If there were a diffeomorphism than $F_\star$ is isomorphism between any two tangent space. If I show that the tangent space on the corner of square has dimension zero, would it solve problem?

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    Have you proved that $S$ is not a smooth submanifold of $\mathbb R^2$ yet? If so, you could use that as a lemma. Argue that if there was such a diffeomorphism $F$, then $S$ would neccessarily be a smooth submanifold of $\mathbb R^2$.2011-07-27
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    And if you haven't proven that yet, I think that would be a great way to solve your problem. I like to make this argument using hypothetical charts for $S$ and an application of the Implicit Function Theorem.2011-07-27
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    @Ryan Budney -- Thank you for commenting. No, I have not learned submanifolds yet. It is chapter 8 of the book. Does the tangent space at the corner have zero dimensions? I do not know how to prove it. How can I solve this problem with the things I know (defintion of smooth manifolds, maps and tangent space)? Thank you very much for help2011-07-27
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    What definition of "tangent space" are you using?2011-07-27
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    Show that $K$ and $\mathbb{S}^1$ are isomorphic in $\textbf{Top}$, but $K \not\in ob(\textbf{Diff})$.2014-04-04
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    @user1876508, I am afraid that that does not really mean anything :-/2015-04-02

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