Until now, I had thought (intuitively) that two objects were diffeomorphic if and only if they had the same number of peaks, but I think that I have found a counter example for this intuition.
Suppose we are in $\mathbb{C}$ and let $\Delta$ be a triangle and let $\bigcirc$ be its incircle of radious $r$. Without any loss of generality, I might suppose that the center of $\bigcirc$ (the incenter) is $0\in \mathbb{C}$, what will make my reasoning cleaner.
Let $A,B,C$ be the vertices of $\Delta$ and denote $D,E,F$ the tangent points of $\Delta$ and $\bigcirc$. Hence $\phi :\Delta \longrightarrow \bigcirc$ such that $$\phi (a+bi)=r\dfrac{a+bi}{\vert a+bi\vert}$$ sends each point of $\Delta$ to the one in $\bigcirc$ that has the same angle anticlockwise from the real axis. It is obvious that $D,E,F$ are fixed points of $\phi$. Also, $\phi$ is differenciable (as we do not divide by 0).
I shall now work on the inverse of such $\phi$, lets call it $\psi$. Note that $D,E,F$ allow us to reconstruct $\Delta$, as $DO$ must be perpendicular to $BC$, $FO$ must be perpendicular to $AC$ and $EO$ mus be perpendicular to $AB$, therefore points $A,B,c$ and segments $AB,\ AC\, BC$ are well defined.
Now, I can define a function $\psi :\bigcirc \longrightarrow \Delta$ that sends each point on the one in the triangle that has the same angle anticlockwise from the real axis the next way:
Let $z=a+bi$ and let $r_{z}$ denote the straight line that conects $0$ and $z\in \bigcirc$. It is obvious that $r_z\cap \Delta$ has two points, i.e., $$r_z\cap \Delta =\lbrace z_1=a_1+b_1i,\ z_2=a_2+b_2i \rbrace.$$ enter image description here Hence, $\psi (a+bi)=z_i$ such that $sign(a)=sign(a_i),sign(b)=sign(b_i),\ i=1,2$ is well defined. It does satisfy that $\phi ^{-1}=\psi$ and vice versa and, if I am not mistaken, it is continuos, hence, our objects are homeomorphic.
Now, I cannot convince myself of the fact that such $\psi$ is not differentiable, what would make the triangle and the circle diffeomorphic, what would contradict my intuition, that being that two objects are diffeomorphic if and only if they have the same number of non smooth points.
On the light of this argument, could anyone tell me wether my intuition is correct or not, or this example is absolutely falacious and why?
Thank you in advance!