About differential structures, manifolds

Question

I am dealing with the following problem on Differential manifolds and I don't know how to solve it. If anyone could please help me, I would be very thankful. Tanks in advance.

Let $\lbrace (x,y)\in \mathbb{R}^2: y^2-4x^2(1-x^2)=0\rbrace$. Prove that the following applications are bijective: $$P:(0,2\pi)\ni u \rightarrow (\sin u, \sin 2u)\in M$$ $$T: (-\pi, \pi)\ni w \rightarrow (\sin w, \sin 2w)\in M$$ but $\phi=P^{-1} $ and $\psi= T^{-1}$ define different differentiable structures in $M$. ¿Are these structures diffeomorphic? ¿Can $M$ be differentiable manifold with the usual topology of $\mathbb{R}^2$?

Answer 1

First it is useful to understand the picture: M is an eight-shaped variety centered in the origin, and the two charts correspond to the two possible ways to write an 8 without touching the same point twice.

If you consider the induced topology from $\mathbb{R}^2$, $M$ cannot be a manifold as there is a bifurcation at the origin. This answers the last question.

To understand if the two global charts given define the same smooth structure you need to check whether the transition function $T^{-1}\circ P$ is a diffeomorphism of the open interval, but clearly this is not the case as the map is not even continuous (which means that not only the smooth structures but also the topologies induced by the two charts are different).

Finally, observe that the reflection through the y axis is a diffeomorphism between the two smooth structures.