Does the "Number Doughnut" make sense?

Question

It's well known that is we add a point at infinity to the complex plane then we get the Riemann Sphere, or extended complex plane.

What if we distinguish between real infinity and imaginary infinity?

We identify $a+\mathrm i \infty$ with $a-\mathrm i \infty$ for each $a \in \mathbb R$, and separately we identify $\infty + \mathrm i b$ with $-\infty + \mathrm i b$ for each $b \in \mathbb R$? This would give the product of two projectively extended real lines and, in theory, something homeomorphic to the torus, i.e. a "number doughnut".

Is this object possible, is it well know, are there any references?

Answer 1

You can definitely consider the product of two projectively extended real number lines as a topological space, and it would be a torus. However, it's not a meaningfully "complex" object.

Specifically, we are considering the space $X= (\Bbb{R} \cup \{\infty\})^2$. There is an obvious embedding $\iota: \Bbb{R}^2 \to X$. If we want to consider $X$ as a complex manifold in any reasonable sense, that presumably means identifying $\Bbb{R}^2$ with $\Bbb{C}$ and then putting a complex structure on $X$ which makes $\iota$ holomorphic. Since the image of $\iota$ is contractible, it lifts to the universal cover of $X$, which — by the uniformization theorem — we can take to be $\Bbb{C}$ with its standard complex structure. But then this gives a bounded holomorphic map from $\Bbb{C}$ to itself, which is impossible by Liouville's theorem.

Answer 2

You can define this if you want. But (so far) there are no useful applications of it. Unlike the Riemann sphere, which has some useful applications.

Answer 3

The problem is that there are NO $\infty$ and $-\infty$ in the complex numbers. There are, rather, and infinite number of different "infinities", on as you leave the origin of the complex plane along any straight line.

In the real numbers, we can either add $\infty$ and $-\infty$ to have an "extended" real number system that is topologically equivalent to a closed line segment. Or we can add a single $\infty$ that makes the line topologically equivalent to a circle.

In the complex numbers, we can add an infinite number of "infinities" as above so that the complex plane is topologically equivalent to a disk or we can add a single "infinity" so that it is topologically equivalent to a sphere.