In John Lee’s book, I was asked to find unique smooth structure on E to make the quotient map $q: \mathbb{R}^{2}\rightarrow E$ smooth . Where q is the quotient map to a equivalent class in $\mathbb{R}^{2}:$
$\{(x,y)\sim (x+n,(-1)^{n}y)\}$
I want to use the natural embedding in $\mathbb{R}^{3}$ and the projection map, but I don’t know how to proof the uniqueness, and furthermore, I want to find a more intrinsic way, what should I do?
There is a unique differential structure on $E$ such that the quotient map $q:\mathbb{R}^2\to E$ is locally a diffeomorphism. So if you find any smooth structure which makes $q$ smooth, you're done. But this isn't so bad. The idea is that we should look at the universal cover and try to restrict to open sets where the quotient is giving us our charts.
Here's my attempt at this, with maybe some details skipped over: Let $U_1=\lbrace [(x,y)]: 1/3<x<2/3\rbrace$ and $U_2=\lbrace [(x,y)]:1/2<x<3/2\rbrace$ - I'm using $[(x,y)]$ to denote the equivalence class containing the tuple $(x,y)$. Let $\overline{x}$ denote the map $\mathbb{R}\to [0,1)$ so that $x$ is sent to its value mod $1$. Then let $\phi_1:U_1\to\mathbb{R}^2$ be given by $\phi_1([(x,y)])=(\overline{x},(-1)^{x-\overline{x}}y)$ and $\phi_2:U_2\to\mathbb{R}^2$ by $\phi([x,y])=(\overline{x}+1/2,(-1)^{x-(\overline{x}+1/2)}y)$. It is not hard to check these are well-defined. Then note that $U_1\cup U_2=E$ and $U_1\cap U_2=\lbrace [(x,y)]:1/3< x<1/2\text{ or }1/2<x<2/3\rbrace$. It should work out that $\phi_2\circ\phi_1^{-1}$ restricted to $\phi_1(U_1\cap U_2)$ has the formula $(x,y)\mapsto (x+1/2,y)$ and $\phi_1\circ\phi_2^{-1}$ restricted to $\phi_2(U_1\cap U_2)$ has the formula $(x,y)\mapsto \left\lbrace\begin{array}[ll] ((x,y)&x\in(5/6,1) \\ (x-1,-y) &x\in(1,7/6)\end{array}\right.$. So $\lbrace \phi_i\rbrace_{i=1,2}$ is a smooth atlas. Now we want to show that $q:\mathbb{R}^2\to E$ is smooth. Let $(x_0,y_0)\in\mathbb{R}^2$. Then $(x_0,y_0)$ is contained in an integral horizontal translation of $\lbrace (x,y):1/3<x<2/3\rbrace$ or an integral horizontal translation of $\lbrace (x,y):1/2<x<3/2\rbrace$. Suppose we are in the former case so that there is $n\in\mathbb{N}$ with $x-n\in (1/3,2/3)$. Then choose an open ball $V$ around $(x_0,y_0)$ completely contained in the translated set. In this quotient this ball is an open ball in $U_1$ so we can post-compose with $\phi_1$ and we get that $\phi\circ q:V\to\mathbb{R}^2$ is given by $(x,y)\mapsto (x-n,(-1)^ny)$. The other case is similar!
Like I said, there are some missing pieces but this is the general idea.