A torus quotient an equivalence relation

A torus is $S^1\times S^1=\{(e^{2\pi ix},e^{2\pi iy})|x,y\in \Bbb R\}$, define an equivalence relation $(w,z) \sim (iw,iz)$. I want to know $S^1\times S^1/\sim$.

I try to rewrite it in $\Bbb R^2$, $(e^{2\pi ix},e^{2\pi iy})=(x,y)$,$(w,z) \sim (iw,iz)$ is $(x,y) \sim (x+1/4,y+1/4)$, but I don't know what it is.

Solution 1:

You made a good start (+1). The following picture should help now:

On the left we have a standard representation of the unit square with opposite edges identified. The equivalence relation generated by $\sim$ will take any point in the square to one (at least) in the bottom quarter. Thus we can model the quotient of the torus as a quotient of the bottom rectangle.

We may break off the left hand triangle from the bottom rectangle, and glue along the vertical identification. We must identify the diagonal edges to record where we made this break.

The only points in the resulting trapezium which are equivalent under $\sim$ are corresponding points on the top and bottom edges.

The end result is just another torus.