# Abstract Nonsense

## Riemann Surfaces (Pt. III)

Point of Post: This is a continuation of this post.

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‘The’ Riemann Sphere

The last example we shall be a three-in-one. In particular, we shall define three Riemann surfaces $\mathbb{C}_\infty, \mathbb{S}^2$, and $\mathbb{P}^1:=\mathbb{CP}^1$, which shall end up being equivalent (biholomorphic).

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First, we consider $\mathbb{C}_\infty$, which topologically is just the one-point compactification of $\mathbb{C}$. Clearly $(\mathbb{C},\text{id}_\mathbb{C})$ is a chart on $\mathbb{C}_\infty$. We claim that $(\mathbb{C}_\infty-\{0\},\varphi)$ with

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$\displaystyle \varphi(z)=\begin{cases}\displaystyle \frac{1}{z} & \mbox{if}\quad z\ne\infty\\ 0 & \mbox{if}\quad z=\infty\end{cases}$

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is a chart on $\mathbb{C}_\infty$. Indeed, it’s clear that $\varphi$ is bijective, and continuous everywhere but at $\infty$. It’s continuoust at $\infty$, because if $D_r(0)$ is a neighborhood of $0$ then $\varphi^{-1}(D_r(0))$ is just $\mathbb{C}_\infty-\overline{D_r(0)}$, and thus is the complement in $\mathbb{C}_\infty$ of a compact subset of $\mathbb{C}$, and thus open in $\mathbb{C}_\infty$. Now, since $\varphi(\mathbb{C}_\infty-\{0\})=\mathbb{C}$ is open, and $\varphi^{-1}$ is evidently continuous it follows that $(\mathbb{C}_\infty-\{0\},\varphi)$ is indeed a chart on $\mathbb{C}_\infty-\{0\}$. Now, we claim that $\left\{(\mathbb{C},\text{id}_\mathbb{C}),(\mathbb{C}_\infty-\{0\},\varphi)\right\}$ is an atlas for $\mathbb{C}_\infty$. Now, since the domains of these charts cover $\mathbb{C}_\infty$ it really suffices to check that the two charts are compatible. But, this follows since the intersection of their domains is $\mathbb{C}^\times$ and the overlap map (in both directions!) is just the map $\mathbb{C}^\times\to\mathbb{C}^\times$ given by $\displaystyle z\mapsto\frac{1}{z}$, which is a biholomorphism. When we give $\mathbb{C}_\infty$ this complex structure, we call it the extended complex plane or the Riemann sphere.

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Next, we shall imbue the $2$-sphere with the structure of a Riemann surface. Indeed, as usual, let $N=(0,0,1)$ and $S=-N$, and define charts $\varphi_N:U_N\to\mathbb{C}$ and $\varphi_S=U_S\to\mathbb{C}$ (where $U_N=\mathbb{S}^2-N$, and similarly for $U_S$) by the formulae

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$\displaystyle \varphi_N(x,y,z)=\frac{x}{1-z}+i\frac{y}{1-z}\quad \varphi_S(x,y,z)=\frac{x}{1+z}+i\frac{y}{1+z}$

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It’s evident that $\varphi_N,\varphi_S$ are homeomorphisms onto $\mathbb{C}$ and thus we see that $\{\varphi_N,\varphi_S\}$ is a topological atlas for $\mathbb{S}^2$ (since their domains form an open cover) and thus we need merely check that $\varphi_N$ and $\varphi_S$ are compatible. This is easily verified, since one can check that

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$\displaystyle \varphi_N\circ\varphi_S^{-1}(z)=\varphi_S\circ\varphi_N^{-1}(z)=\frac{1}{z}$

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on $\varphi_N(U_N\cap U_S)=\varphi_S(U_N\cap U_S)=\mathbb{C}^\times$, which is a biholomorphism $\mathbb{C}^\times\to\mathbb{C}^\times$. When we give $\mathbb{S}^2$ this complex structure, it is called the Riemann Sphere.

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Lastly, we define the complex projective line or just the projective line, denoted $\mathbb{CP}^1$ or just $\mathbb{P}^1$ for short. Intuitively, we can think of $\mathbb{P}^1$ as being $\mathbb{C}$ along with a “point at infinity” which allows parallel lines to intersect. Rigorously, we define $\mathbb{P}^1$ to be the space of all $1$-dimensional (complex!) subspaces of $\mathbb{C}^2$. Since any line in $\mathbb{C}^2$ through the origin is determined by one non-zero point on it, we may identify $\mathbb{P}^1$ with points of $X=\mathbb{C}^2-\{0\}$, but we must be careful to not differentiate between points which differ by a $\mathbb{C}^\times$ action. To be more precise, we know that $\mathbb{C}^\times$ acts on $X$ via $\lambda\cdot(z_0,z_1)=(\lambda z_0,\lambda z_1)$ and thus we can consider the orbit space $X/\mathbb{C}^\times$ (i.e. the equivalence class of points, where two points are equivalent if they are in the same orbit of the action). We see that the orbits of $X/\mathbb{C}^\times$ are of the form $\{\lambda(z_0,z_1):\lambda\ne0\}$ which are exactly the nonzero points on a line in $\mathbb{C}^2$–thus we can identify $\mathbb{P}^1$ with $X/\mathbb{C}^\times$. The advantage of this is that we now have a projection map $\pi:\mathbb{C}^2-\{0\}\to\mathbb{P}^1$, which using the quotient topology given by this map, allows us to topologize $\mathbb{P}^1$.

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The question is then, why is $\mathbb{P}^1$ ((now a topological space!) naturally a Riemann surface? Well, let’s denote $\pi(z_0,z_1)$ as $[z_0:z_1]$ (the colon is meant to prompt one to think of ratios–the real important characteristic of a line!). Define then $U_0$ to be $\{[1:z]:z\in\mathbb{C}\}$ and $U_1$ to be the set $\{[z:1]:z\in\mathbb{C}\}$–these are going to be the chart neighborhoods of the complex structure on $\mathbb{P}^1$. Wait, what? This seems like a very unmotivated choice of sets used to cover $\mathbb{P}^1$–what gives? Well, let’s explore this a little more.  What does it mean for a point $[z_0:z_1]$ to be equal to $[1:z]$ for some $z\in\mathbb{C}$? Well, evidently if $z_0\ne 0$ then

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$\displaystyle [z_0:z_1]=\left[\frac{z_0}{z_0}:\frac{z_1}{z_0}\right]=\left[1:\frac{z_1}{z_0}\right]$

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Conversely, if $[z_0:z_1]=[1:z]$ for some $z$ then $(z_0,z_1)=\lambda(1,z)$ for some $\lambda\in\mathbb{C}^\times$, and so, in particular, $z_0\ne 0$. Now, noting that $[0:z]=[0:1]$ for any $z\in\mathbb{C}^\times$ (they just differ by $z$ multiplication!) we see that despite the somewhat strange appearance of $U_0$, one has that $U_0=\mathbb{P}^1-\{[0:1]\}$. Similarly, you can convince yourself that $U_1=\mathbb{P}^1-\{[1:0]\}$.

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Now that we have identified what our coordinate neighborhoods are going to be, and figured out what they “really look like” it shouldn’t be hard to see how we’re going to map them homeomorphically into $\mathbb{C}$. In particular, define $\varphi_0:U_0\to\mathbb{C}$ by $[1:z]\mapsto z$ and $\varphi_1:U_1\to\mathbb{C}$ by $[z:1]\mapsto z$. It’s obvious that $\varphi_0,\varphi_1$ are homeomorphisms onto $\mathbb{C}$, and thus $\{\varphi_0,\varphi_1\}$ is a topological atlas for $\mathbb{P}^1$. Now, since $U_0$ and $U_1$ form an intersecting open cover of $\mathbb{P}^1$ and each $U_i$ is connected, it follows that $\mathbb{P}^1$ is connected. It remains to check that these atlases are compatible. It’s a little less obvious how to check this, but the idea is simple enough. We know that $U_0\cap U_1=\mathbb{P}_1-\{[0:1],[1:0]\}$ and $\varphi_0(U_0\cap U_1)=\varphi_1(U_0\cap U_1)=\mathbb{C}^\times$. Now, the question is, what does $\varphi_1\circ\varphi_0^{-1}$ look like? Well, take $z\in\mathbb{C}^\times$, and we see that $\varphi_0^{-1}(z)=[1:z]$. Now, $[1:z]=[\frac{1}{z}:1]$ and thus we see that $\varphi_1(\varphi_0^{-1}(z))=\frac{1}{z}$. Similarly, one can check that $\varphi_0\circ\varphi_1^{-1}$ is just the inversion map $\mathbb{C}^\times\to\mathbb{C}^\times$. Thus, we see that our atlas is compatible and thus defines a complex structure on $\mathbb{P}^1$.

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It should be obvious that these three spaces are homeomorphic. But, the construction of the complex structures on each space is so similar, it really differs only by what symbols we used, it should not be shocking that all three of these spaces are, in fact, biholomorphic. All three of these biholomorphic objects shall be called ‘the’ Riemann sphere, or one of their individual names. The Riemann sphere is one of the most fundamental of all Riemann surfaces. It shall be integral in defining the notion of meromorphic functions. More importantly though, we shall prove that the Riemann sphere is the only simply connected compact Riemann surface. It is obviously one of the simplest examples to work with as well.

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References;

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Markley, Nelson Groh. Topological Groups: An Introduction. Hoboken, NJ: Wiley, 2010. Print.