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Given a function $f:\mathbb C\to\mathbb C$ which we will assume is analytic, we have an embedding $f\subseteq\mathbb C\times\mathbb C\cong\mathbb R^4$ of a surface. My question is with regards to how to graphically (not necessarily actually with pictures, but in the same sense as topologists like to fiddle with pictures in their heads) turn such an embedding into a recognizable topology like $(S^1)^2$ or $S^2$ or $RP^2$? Specifically, how does one "deform":

  • poles (single, double)
  • essential singularities
  • polynomial behavior at infinity
  • exponential behavior at infinity
  • root branch points
  • logarithmic branch points
  • zeros (I don't think these need special treatment, but we'll see...)

For example, the behavior of $\sqrt z$ on the unit circle is a lot like the Möbius strip, but I'm not sure how the behavior near zero directs my mental arts-and-crafts, and I'm not sure about $\infty$ either. I know that we want $\infty$ to be identified in all directions, but what about when it $f(z)$ shoots off in different directions for different directions $z\to\alpha\infty$, or worse, when it goes to zero in other directions (as in $e^z$).

I know that this isn't the first question on Riemann surfaces, but I want to emphasize the intuitive and geometrical/graphical aspects of dealing with the topologies of Riemann surfaces. I don't expect that a complete answer will be able to avoid algebraic manipulation of the functions under consideration, but what I don't want to see is "function $\Rightarrow$ algebra $\Rightarrow$ the surface is genus 2" or somesuch.

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The graph of any analytic function $f:\mathbb{C} \to \mathbb{C}$ is homeomorphic to $\mathbb{C}$, where the homeomorphism is given by projection onto the domain. Now if you have poles you get a plane with a bunch of punctures (at the location of the poles), and the behavior at $\infty$ does not really matter unless you compactify the graph somehow.

Now if you allow branch points, you are looking at the graphs of relations, or multivalued functions, and there it gets interesting. E.g., the graph of $w^2 = z^3-z$ (compactified by viewing it as a subset of complex projective space) is an elliptic curve, in particular a topological torus. There is a simple geometric way to see that this is a torus, by gluing together two spheres according to the two-valued function $w=\sqrt{z^3-z}=\sqrt{z(z-1)(z+1)}$. This function has two single-valued branches on $\mathbb{C} \setminus [-1,0] \setminus [1,\infty)$, and continuous continuation across either slit turns one branch into the other. Topologically you take two copies of these double-slit planes and glue the upper part of either slit to the lower part of the same slit on the other copy of the plane. If you fatten these slits to disks, you have two spheres, each with two circles removed, and you glue them together along the circles, creating a torus. Basic examples like this are covered in many introductory Complex Analysis books, e.g., Gamelin, Complex Analysis, with some nice pictures and exercises.

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