ISEGORIA / MATH ENCYCLOPEDIA
Riemann surfaces: where many-valued functions live
The square root of a complex number has two values, and if you follow one of them round the origin it turns into the other. Riemann’s answer was to stop treating such a function as many-valued on the plane and to build a new surface on which it is single-valued: several sheets, glued crosswise along cuts. The branch points decide how the sheets join, the topology of the result (its genus) follows from counting them, and the simplest closed Riemann surface of all, the sphere, is where Möbius transformations act as rigid motions and flows.
Before you begin: Complex numbers, the complex logarithm, and the Euler characteristic
Predict, manipulate, then test yourself: every experiment ends with practice questions that are generated fresh at three levels. Graphs illustrate the mathematics; they do not replace a proof.
1. Square roots and logarithms live on surfaces
Pick a starting value of \(\sqrt z\) and follow it continuously as z travels round a loop. If the loop does not go round 0 you come back to the value you started with; if it goes round once, you come back to its negative. So \(\sqrt z\) is really a function on a surface made of two copies of the plane, cut along a ray from 0 and glued crosswise. The cube root needs three sheets, and the logarithm infinitely many, stacked like a spiral staircase because every turn adds \(2\pi i\). The 3D view draws Im w above each point of the plane for all branches at once. Move the loop, change its size, and send it round more than once.
Worked example. Start at \(z=1.3\) with \(w=\sqrt{1.3}=1.140\) and walk once round the circle of radius 0.9 about 0.4, which encloses 0: the argument of z grows by \(2\pi\), so w ends at \(1.140\,e^{i\pi}=-1.140\). Round the same loop the cube root ends at \(e^{2\pi i/3}\sqrt[3]{1.3}\), and after two turns the logarithm has gained \(4\pi i\approx12.566i\).
Watch out. The picture plots \((x,y,\operatorname{Im}w)\), so the square-root surface seems to pass through itself along the negative real axis. That crossing is an artefact of drawing a surface that really lives in four real dimensions in three; the Riemann surface itself has no self-intersection. For the logarithm the height is \(\theta\) shrunk by a factor of 0.16 so that three turns fit.
Why can the loop not shrink to a point without changing the answer?
Because to shrink it you would have to pass it over 0, where the function has no derivative and the sheets meet. Away from the branch point the continued value depends only on how many times the loop winds round it, and that integer cannot change continuously.
2. Gluing sheets: branch points and genus
The curve \(w^2=(z-a_1)\cdots(z-a_n)\) defines a two-sheeted surface over the z-plane: the sheets carry \(+w\) and \(-w\) and meet only at the roots \(a_k\). A loop round one branch point swaps the sheets, round two brings you back, so cuts can join the branch points in pairs (and the last one to \(\infty\) when n is odd). Glue two cut spheres crosswise along their cuts and you get a closed surface whose genus follows from the Riemann–Hurwitz formula. Drag the branch points, change their number, and move the loop.
Worked example. With four branch points there are two cuts, and the two slit spheres glue into a torus: \(B=4\), \(\chi=0\), \(g=1\). A cubic \(w^2=z^3-z\) also has \(g=1\), because \(\infty\) is a fourth branch point; this is an elliptic curve. Five or six roots give \(B=6\) and a surface with two holes.
Watch out. The 3D view draws the sheets as graphs of \(\operatorname{Re}w\), compressed so large values do not dominate, and coloured by \(\operatorname{Im}w\); the slits you see are where the drawing cuts the surface to keep each sheet single-valued, and their position is arbitrary. The genus does not depend on where the cuts go, only on how many branch points there are, and it assumes the roots are distinct.
Why does \(\infty\) become a branch point when n is odd?
Because a very large loop encloses all n roots, and going round it multiplies \(w\) by \((-1)^n\). For odd n that is \(-1\): the sheets swap, so the loop must be going round a branch point, and the only one left outside it is \(\infty\).
3. The Riemann sphere and Möbius maps
Add one point \(\infty\) to the plane and it closes up into a sphere: project each point of the unit sphere from its north pole onto the equatorial plane, and the north pole itself goes to \(\infty\). On this sphere the Möbius transformations \(z\mapsto(az+b)/(cz+d)\) are exactly the conformal one-to-one maps. Each has one or two fixed points, and one number, the trace squared of its normalised matrix, says how it moves everything else: rotating, streaming from one fixed point to the other, spiralling, or sliding along circles that all touch at a single point. Choose a type, its strength and the tilt of its axis, and drag a point on the sphere.
Worked example. The hyperbolic map \(z\mapsto e\,z\) has normalised matrix \(\operatorname{diag}(e^{1/2},e^{-1/2})\) and \(\operatorname{tr}^2=e+2+e^{-1}=5.086>4\): points stream out of 0 and into \(\infty\). The rotation \(z\mapsto iz\) has \(\operatorname{tr}^2=4\cos^2(\pi/4)=2\). Tilting the axis by \(30^\circ\) about the real axis moves the fixed points to \(0.268i\) and \(-3.732i\), a pair of antipodal points on the sphere, and leaves the trace unchanged.
Watch out. Each type is shown in a standard position (fixed points 0 and \(\infty\), or \(\infty\) alone for the parabolic case) and then conjugated by a rotation of the sphere, which is itself a Möbius map; conjugation moves the fixed points but keeps the type and the trace. The flow is the one-parameter family \(f_t\) with \(f_1=f\), played for \(-1\le t\le1\).
Why does a Möbius map send circles to circles?
Because every one is a composition of translations, scalings, rotations and the inversion \(z\mapsto1/z\), and each of these maps circles and lines to circles and lines. On the sphere a line is just a circle through \(\infty\), so the statement is simply that circles go to circles.