# No diffeomorphism that takes unit circle to unit square

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 $\mathbb R^2$. –  Ryan Budney Jul 27 '11 at 21:24
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. –  Ryan Budney Jul 27 '11 at 21:26
@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 help –  Udayana Jul 27 '11 at 21:57
What definition of "tangent space" are you using? –  Ryan Budney Jul 27 '11 at 22:43
Show that $K$ and $\mathbb{S}^1$ are isomorphic in $\textbf{Top}$, but $K \not\in ob(\textbf{Diff})$. –  user1876508 Apr 4 at 21:01

Here's a somewhat rigorous way to see this. Let $\gamma$ be an arc in $C$ such that $F\circ \gamma(0)$ is the corner (1,1). Then (assuming it goes clockwise) there are some functions $x$ and $y$ such that $F \circ \gamma(t) = (1,y(t))$ for $t< 0$ and $F \circ \gamma(t) = (x(t),1)$ for $t > 0$. Thus $F_*\gamma'(t)$ is $(0,y'(t))$ for $t<0$ and $(x'(t),0)$ for $t > 0$. Taking limits, this means that $F_*\gamma'(0) = 0$ contradicting that $\gamma'(0) \ne 0$.