What's the quotient space of torus $S^1\times S^1$ under equivalence relation $(z,w)\sim (w,z)$

Question

Consider the quotient space of torus $S^1\times S^1$ under the equivalence relation $(z,w)\sim (w,z)$. I'm trying to visulaize it but find it pretty hard to do. Any hint?

Answer 1

Identify the torus $S^1\times S^1$ with the square $[0,1]^2$ modulo the identification of $(0,w)$ with $(1,w)$ and $(z,0)$ with $(z,1).$ Then you can identify the quotient space by your equivalence relation with $\{(z,w)\in[0,1]^2 : z\ge w\}.$ Now let \begin{align} u & = z+w-1 \\ v & = -z+w+1 \\[12pt] \text{so that } z & = \frac{u-v} 2 + 1 \\[8pt] \text{and } w & = \frac{u+v} 2 \end{align} and reduce $z$ and $w$ modulo $1.$ Then you can view $(u,v)$ as being in the square $[0,1]^2$ modulo the identification of $(0,v)$ with $(1,1-v),$ and that quotient space is a Möbius band.