Abstract Nonsense

Crushing one theorem at a time

Compact Riemann Surfaces are Topologically g-holed Tori


Point of Post: In this post we prove that topologically compact Riemann surfaces are just g-holed tori.

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Motivation

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A very natural question when one starts studying Riemann surfaces is “what topological spaces admit a complex structure?” It turns out that while difficult in general, for compact Riemann surfaces these are precisely the g-holed tori. While this sounds pretty deep, to anyone who is familiar with the classification of compact surfaces, we can mask most of the difficulty by just proving that the Riemann surface is orientable. This somewhat surprising fact follows immediately from the Cauchy-Riemann equations. The somewhat surprising part is that the converse is true. Namely, if M is any smooth orientable 2-manifold, then M admits a complex structure. This is somewhat of a deep fact, one that takes some serious (unavailable) machinery to prove and so we’ll just state it.

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Topological Classification of Compact Riemann Surfaces

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Basically we just want to show that every compact Riemann surface X is orientable. The basic idea is simple enough. Suppose that (U,\varphi) and (V,\psi) are any two charts on X (in the maximal atlas!) such that U\cap V\ne\varnothing. We want to show that \psi\circ\varphi^{-1} has the property that it’s derivative at each point (x,y)\in\varphi(U\cap V) has positive determinant. But, the basic idea is simple enough. We know that \psi\circ\varphi^{-1} is holomorphic at each point p in \varphi(U\cap V). So, if we let \psi\circ\varphi^{-1}=u+iv then the derivative of \psi\circ\varphi^{-1} at p is

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\displaystyle \begin{pmatrix}u_x(p) & v_x(p)\\ u_y(p) & v_y(p)\end{pmatrix}

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Thus, the determinant of the derivative of \psi\circ\varphi^{-1} at p is u_x(p)v_y(p)-u_y(p)v_x(p). Now, by the Cauchy-Riemann equations we know that u_x=v_y and u_y=-v_x so that the derivative can be rewritten as u_x(p)^2+u_y(p)^2\geqslant 0. And, since the derivative is non-zero (being a biholomorphism!) this implies that the derivative is strictly positive. Since everything in this discussion was arbitrary we may conclude that:

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Theorem: Every Riemann surface is orientable.

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From this, and the classification of compact surfaces we get the following theorem:

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Theorem: Every compact Riemann surface is homeomorphic to

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\underbrace{\mathbb{T}^2\#\cdots\#\mathbb{T}^2}_{g}

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for some g\geqslant 0 (where, as usual, if g=0 we get \mathbb{S}^2). 

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Since, as is obvious by considering homology (or, hell, we’re already using the classification which tells us the following is true) this integer is a well-defined homeomorphism invariant we can attach to each compact Riemann surface X the g such that X is homeomorphic to the g-holed torus. As usual, we call this the genus of X, and denote it g(X).

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The Converse

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As was discussed in the motivation, there is somewhat of a converse. Namely, one may begin to wonder if orientability is the only condition that a smooth manifold must satisfy to support a complex structure. In fact, even more is true:

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Theorem: Let M be an orientable 2-manifold of class C^k for k\geqslant 2. Then, M admits a complex structure.

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The proof relies on the existence of of ‘isothermal coordinates‘, something that I do not discuss. Thus, I will not attempt to prove this here and instead refer the interested reader to [1]. I, in particular, suggest that one reads the remark following the proof of this theorem (this is on pg. 24) which elucidates not only the proof of this theorem, but how to think of complex manifolds in general.

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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.

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October 2, 2012 - Posted by | Complex Analysis, Riemann Surfaces | , , ,

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