The same loop, unrolled

A loop in a ring either can be pulled tight or it cannot, and the reason is a whole number: how many times it goes round the hole. That number is easy to state and awkward to define, because how many times round is a statement about the whole loop at once, and the natural way to compute it — follow the angle and see how much it has changed — requires the angle to be followed continuously through values it never actually takes.

Making that respectable is what a covering space is for. The helix above is the space in which the angle really does take those values, and the map that sends each of its points straight down onto the circle is the machine that translates between the two.

What a covering is #

A map $p$ from a space $E$ to a space $B$ is a covering when every point of $B$ has a neighbourhood whose preimage is a disjoint union of pieces, each of which $p$ carries onto that neighbourhood by a homeomorphism. Locally, in other words, $E$ looks like several separate copies of $B$ stacked up; globally it need not look like $B$ at all.

The helix is the model case. Take $E$ to be the real line and $B$ the circle, and let $p(t) = (\cos t, \sin t)$. A small arc of the circle has as its preimage infinitely many intervals of the line, each an exact copy of the arc, spaced one turn apart. The fibre over a point — its full preimage — is a copy of the integers.

Two things about the definition are worth flagging. It is a local condition with a global consequence, which is the usual arrangement in topology and the reason the subject can say anything. And nothing in it requires $E$ to be bigger, smaller, connected or simpler than $B$; what it requires is uniformity, that the stacking looks the same everywhere.

Lifting, and why it works #

The property that makes coverings useful is that paths lift. Given a path in $B$ and a chosen point of the fibre above its start, there is exactly one path in $E$ starting there that projects down onto it.

Existence and uniqueness are both easy to believe and both need the covering condition. Follow the path in $B$; whenever it sits inside an evenly covered neighbourhood, the lift is forced, since the sheet it started in determines the copy. Cover the path with finitely many such neighbourhoods and the lift is built piece by piece, each piece determined by the last.

Now take a loop in the base — a path that ends where it started. Its lift need not close, and the failure to close is the invariant. For the helix over the circle, the lift ends at a point of the fibre above the starting point, so it ends a whole number of turns above where it began, and that whole number is the winding number.

Two consequences follow immediately, and both were awkward before.

The number is well defined, because the lift is unique once its starting point is chosen. There is no arbitrary choice of branch and no ambiguity of $2\pi$; the ambiguity has been moved into the choice of which point of the fibre to start at, and changing that shifts both ends of the lift by the same amount.

The number cannot change a little. Deform the loop continuously and its lift deforms continuously, so the endpoint moves continuously within the fibre — and the fibre is a discrete set of points. A continuously moving point in a discrete set does not move. That is the whole proof that the winding number is a homotopy invariant, and it is one sentence.

Finitely many sheets #

The line is not the only cover of the circle. Wrap a circle of $n$ times the circumference around it $n$ times and the result is a covering with $n$ sheets.

In this cover the fibres are finite, and a loop’s lift closes exactly when the loop has gone round a multiple of $n$ times.

The lift’s failure to close is again the whole content, and now it is a failure of a different kind: not an integer, but an integer modulo $n$. Which is to say the $n$-sheeted cover forgets the multiples of $n$ and remembers the rest.

That table is the classification. The loops of the circle up to deformation are the integers; the covers of the circle are the $n$-sheeted ones together with the line; and the cover corresponding to a subgroup $H$ of the integers is the one whose lifts close exactly on $H$. Since every subgroup of the integers is $n\mathbb{Z}$ for some $n$, or the trivial subgroup, the list is complete. The correspondence runs both ways and it is not a coincidence: for any reasonable space, the connected coverings correspond exactly to the subgroups of its fundamental group, with the number of sheets equal to the index.

The universal one, and the moves that shuffle it #

The line is the cover in which no loop closes except the constant one. It is called universal, and there are two reasons for the name.

It is universal in the sense of being simply connected — it has no loops of its own to speak of, so nothing is left to unroll. And it is universal in the sense of covering everything else: the line covers each $n$-sheeted circle, which covers the base, so any cover of the circle is a quotient of the line.

The quotient is by a group of motions. Translating the line by $2\pi$ carries fibres to fibres and commutes with the projection: it is a deck transformation, a symmetry of the cover that leaves the base alone. The deck transformations of the line over the circle are the translations by multiples of $2\pi$, a copy of the integers — which is the fundamental group of the circle again, arriving from a third direction.

That is the pattern worth carrying. For a universal cover, the deck transformations are the fundamental group, and the base is the cover divided by that group. Loops downstairs become paths upstairs; composing loops becomes composing deck transformations; and a question about deformation becomes a question about a group acting on a space.

Where the surfaces come in #

The circle is the smallest interesting example, and the machinery is not about circles.

A Möbius band has a two-sheeted cover: an untwisted cylinder, wrapped round it twice, with the two sheets over each point being the two local choices of which side. Going once round the band swaps the sheets — that is what one-sidedness is — and going round twice comes back. So the band’s loops modulo the ones that lift to loops are the integers modulo two, and the cylinder is the orientation cover.

The same construction works on any surface and settles the question of orientability without hand-waving: a surface is orientable exactly when its orientation cover falls apart into two copies of it, and non-orientable exactly when the cover is connected. The Klein bottle is covered twice by the torus in the same way.

The torus itself has the plane as its universal cover, with the deck transformations the translations by a lattice of vectors — which makes a torus a plane with a grid of identifications, and makes a loop on a torus a pair of whole numbers saying how far the lift travelled in each direction. The classification of loops on a torus is that pair, and the reason there are two numbers rather than one is that the lattice has two generators.

Reading a group off a cover #

The classification table above has a use beyond bookkeeping, and it is the reason covering spaces are a computational tool rather than a picture.

Suppose the fundamental group of a space is wanted and not known. Find a cover with $n$ sheets whose total space is understood; then the subgroup of loops that lift to loops has index $n$ in the group, and the deck transformations give a group of order $n$ when the cover is a regular one. Enough covers pin the group down.

The circle makes the arithmetic visible. Its $n$-sheeted cover corresponds to the subgroup $n\mathbb{Z}$, whose index in $\mathbb{Z}$ is $n$ — the number of sheets — and whose quotient $\mathbb{Z}/n\mathbb{Z}$ is exactly the group of deck transformations, the $n$ rotations that shuffle the sheets. Three numbers that had no reason to be the same number are the same number: sheets, index, order of the symmetry group.

The relation runs the other way as well, and this is where it earns its keep. A subgroup of a group is a subobject and therefore usually harder to picture than the group; a covering space is a space, and spaces can be drawn. So a question about subgroups can be moved to a question about coverings, answered geometrically, and moved back. The standard demonstration is the Nielsen–Schreier theorem — every subgroup of a free group is free — whose slick proof observes that a free group is the fundamental group of a bouquet of circles, that every covering of a graph is a graph, and that the fundamental group of a graph is free. A statement about all subgroups, proved by noticing what covers look like.

Where it came from #

Covering spaces arrived from complex analysis rather than from topology. A function like the logarithm or the square root has no single-valued definition on the plane with the origin removed: going round the origin once returns to the same point with a different value. Riemann’s response, in the 1850s, was to build a new surface on which the function is single valued — a surface that covers the punctured plane, with one sheet for each branch.

The topological abstraction came later, from Poincaré and then from the school of the 1920s and 1930s that turned the fundamental group into a systematic tool. The reversal of emphasis is worth noticing: Riemann built the covering space to make an analytic object behave, and the topologists kept the space and threw the function away, discovering that the space alone carries the information.

The vocabulary still shows its origins. Sheet, branch point and monodromy — the last being the deck transformation induced by going round a loop — are all words from the analytic subject, applied now to spaces with no function on them at all.

Where it fails, and what it needs #

A universal cover requires local good behaviour. The theorem that every space has a simply connected cover needs the space to be connected, locally path-connected and semi-locally simply connected; the last condition rules out spaces whose small loops are unshrinkable at every scale. The standard counterexample is the Hawaiian earring — infinitely many circles shrinking to a point — which has no universal cover, and the failure is not a technicality but a real obstruction.

Lifting maps is harder than lifting paths. A path always lifts. A map from a general space $X$ into the base lifts only when the image of $X$’s fundamental group sits inside the subgroup the cover corresponds to. Paths lift unconditionally because an interval has no loops to worry about.

The correspondence is with subgroups up to conjugacy. The classification stated above is clean for the circle because the integers are abelian and every subgroup is normal. In general, connected coverings correspond to conjugacy classes of subgroups, and only the normal ones have deck transformation groups acting transitively on the fibre. The circle hides that distinction entirely, which is worth knowing before generalising from it.

What the pictures cannot show #

The helix drawn here is a spiral in space, and that is a lie of convenience: the covering space is the real line, not a curve in three dimensions, and the spiral is a picture of the map rather than of the space. Nothing goes wrong in the circle’s case, but the habit is dangerous — the universal cover of a torus is a plane, and no drawing of it as a surface in space can be a covering, since the plane wraps infinitely and the picture cannot.

The finite covers are drawn with their ends glued, and the gluing is exactly the thing a picture cannot do. A three-sheeted cover is a circle, and drawing it as three turns of a spiral with the ends marked as identified is the best available compromise; a reader has to supply the identification, and the figure says so in words.

And no figure here shows the classification being complete. That the list of covers of the circle is the $n$-sheeted ones and the line is a theorem about all covering spaces, and the table drawn is a table of the ones constructed. What the figure verifies is which loops close in each, by lifting them; what it cannot verify is that nothing else exists.

The ladder from here #

Below: the loop that cannot be pulled tight, which is the question this machinery answers, and the one-sided surface, whose two-sheeted cover is the cleanest example after the circle. Sideways: the linking number, which is the same lifting argument with a torus mapping to a sphere, and the classification of surfaces, where orientability is a covering question. Above: the correspondence between coverings and subgroups in general, monodromy, and the Riemann surfaces the whole subject came from.

What is worth carrying away #

The move here is to replace a hard question about a space with an easy question about a bigger one. Can this loop be shrunk? is a question about infinitely many deformations. Where does its lift end? is a question with a point for an answer, and the point lives in a discrete set, so it cannot move under deformation and it cannot be argued about.

Buying that comes at the cost of constructing the bigger space, and the construction is the work. What makes it worth doing is that the same space answers every question of the kind at once: build the universal cover once, and every loop’s class, every finite quotient, and the whole group of symmetries come out of the same object. That is the usual economics of a good abstraction — one expensive construction, then a series of cheap answers.