The loop that won't lie down
By now you have lived through the uncomfortable discovery of guide 2. You took an innocent function element — say a branch of the square root, single-valued and holomorphic on a little disc away from 0 — and you carried it by continuation along a path once around the origin. When you came home to the same disc, the element had changed: sqrt(z) had become -sqrt(z). The function did not break, and it did not lie to you. It simply refused to remember which value it had started with. The same thing happens to log z, only worse: one loop around 0 adds 2 pi i to the answer, and looping again adds another, forever.
Guide 3 then showed that when continuation is genuinely blocked you can hit a natural boundary — a wall the function can never cross. But the square root and the logarithm have no such wall. They continue everywhere except at a single branch point; the only flaw is that they will not come home single-valued. That is a much gentler ailment, and this guide is about its cure. The cure is not to forbid the loops. It is to admit that the function's natural domain was never the flat plane in the first place.
What the monodromy theorem was really telling us
Recall the verdict of the monodromy theorem from guide 2: if you can continue an element freely throughout a simply connected domain — one with no holes to loop around — then the result is genuinely single-valued, and two paths with the same endpoints always deliver the same value. Multivaluedness only ever creeps in through holes, through loops that cannot be shrunk to a point. The plane minus the origin has exactly one such hole, and that is precisely why sqrt(z) and log z misbehave there.
Read that the other way around and it becomes a recipe. The trouble is not in the function; it is in the topology of the domain we forced the function to live on. A loop with nonzero winding number around the branch point is the messenger of the change. So instead of cutting the plane to forbid such loops (that is the branch-cut trick from earlier rungs, useful but ad hoc), we ask a braver question: is there a new domain, a reshaped one, on which those troublesome loops simply do not return to the same place — so that the function can be single-valued honestly, with nothing forbidden? The answer is yes, and that new domain is the Riemann surface.
Building the surface: stack the sheets and glue the loop
Here is the construction for the square root, done slowly and concretely. The function sqrt(z) has two branches — call them the +sheet and the -sheet — each of which is a perfectly good single-valued copy of the punctured plane. Take two such copies, two flat sheets of the plane minus the origin, and stack one above the other. On the upper sheet, sqrt(z) takes its + value; on the lower sheet, its - value. So far we have just drawn two unrelated pictures. The magic is in how we join them.
Cut each sheet along the same ray — say the negative real axis — so each now has a top lip and a bottom lip along that slit. Then cross-glue: the top lip of the upper sheet joins the bottom lip of the lower sheet, and the top lip of the lower sheet joins the bottom lip of the upper sheet. Now follow a path that loops once around 0. On the flat plane this would bring you back where you started; on the glued surface, when you cross the old cut you slide from the upper sheet onto the lower one. You arrive 'above the same z' but on the other sheet — exactly where sqrt has flipped sign. Loop a second time and the cross-gluing carries you back up to the first sheet, and the value flips back. Two loops, identity restored. The surface has built the sign-flip into its geometry.
z-plane (one hole at 0) Riemann surface of sqrt(z)
. . . . . . upper sheet ( + value )
. loop adds . ____________________
. sign flip . ===glue==> / cross over the cut /
. . /____________________/
. . . . . . lower sheet ( - value )
one z, two values --> two points stacked over each z
loop returns -sqrt(z) --> loop climbs to the other sheetOn the surface, the function is single-valued at last
Step back and look at what we have made. A point of the Riemann surface is not just a number z; it is a number z together with which sheet you are on — equivalently, z together with which value of sqrt you mean. Over each nonzero z there are two such points (the two branches), so the surface is a genuine two-layer cover of the punctured plane. And now the payoff: as a function of a point of the surface, sqrt is perfectly single-valued. Ask 'what is sqrt at this surface-point?' and there is one unambiguous answer, because the point already carries its branch with it. The multivaluedness did not vanish — it was converted into the two-sheetedness of the domain.
What about the branch point at 0 itself? It is the one place the two sheets are pinned together — the single point that belongs to both. Imagine the two sheets joined there like the crossing point of a spiral ramp. The surface has no torn edge and no free lip anywhere; the old cut, which looked like a boundary on the flat plane, is now an invisible seam in the interior where two sheets pass smoothly through each other. This stitched-up object, with its sheets and branch points, is what the function was secretly defined on all along. The flat plane was only ever its shadow.
The logarithm's endless spiral, and the surface as a cover
The logarithm is the cleanest example of all, because its surface never folds back on itself. Recall the multivaluedness of the logarithm: log z = log|z| + i arg z, and arg z is only defined up to adding multiples of 2 pi. Each full loop around 0 nudges arg z up by 2 pi, so it nudges log z up by 2 pi i. There is no n for which n loops bring you home, so unlike the square root the sheets never reconnect. The Riemann surface of the logarithm is therefore an infinite spiral ramp: above each point z of the punctured plane sit infinitely many surface-points, one per choice of branch, stacked like the steps of a helix climbing forever upward and downward. On this ramp, log is a single, beautifully behaved function — climbing the ramp simply is watching arg z increase steadily, with no jumps and no ambiguity.
There is a name for what we built, and it links this rung to topology. A surface that lies over the plane so that locally it looks like an ordinary patch of plane, but globally may have several layers, is a covering space of the punctured plane. The square-root surface is a two-fold cover; the logarithm's is the infinite-fold universal cover, the most spread-out cover of all, the one on which every loop has finally been unwound. This is the deep reason the construction always works: continuing a function element along paths is, in disguise, lifting those paths up onto a covering space, and a covering is built so that lifting a loop need not return you to your starting layer.
Honest caveats, and where this leads
A few honest cautions so the picture does not mislead. First, the words 'cut' and 'glue' are scaffolding, not the object. The Riemann surface itself does not depend on which ray you cut along — a different cut just stitches the same two sheets differently and yields the same surface up to relabelling. The cut is a drawing aid that you throw away once the surface is built. Second, the upward-spiral image of the logarithm's surface is a faithful sketch, but the genuine surface lives in too many dimensions to draw without it appearing to pass through itself; the apparent self-intersections in any flat picture are artifacts of squashing it down, not real crossings.
Third, and most important: a Riemann surface is not merely a topological gadget — it carries its own complex structure, so that 'holomorphic on the surface' makes honest sense, and on it the function and its continuations are a single object, what guide 2's language called a fully continued complete analytic function. The whole tangle of branches collapses into one genuinely single-valued holomorphic function, at the modest price of enlarging the domain. That is the entire idea, and it is one of the most beautiful trades in mathematics: you cannot make sqrt single-valued on the plane, but you can make it single-valued, by giving it the domain it deserved.
In the final guide of this rung we put this machine to work on the two functions that motivated everything: we will trace the square root's two-sheet surface and the logarithm's endless spiral in loving detail, watch arg z climb the ramp, and see exactly how a single branch point can organize an entire surface. With the general principle in hand — branches become sheets, loops become climbs, the branch point becomes the pin — those two surfaces will feel less like exotic constructions and more like the obvious, inevitable homes they always were.