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?