Every cover is a subgroup

The previous rung established that the loops of a space form a group. The trouble with a group is that it is an abstract object: knowing that the ring’s group is $\mathbb{Z}$ says what the answers are and gives no way to see them. This rung supplies the pictures, and it does so in a way that turns out to be exact rather than illustrative.

The construction is unrolling. A ring’s loops wind round a hole; a helix above the ring does not close up, and following a loop of the ring upwards through the helix records how many times it went round as a height. The helix is a covering space, and the whole of this rung is the observation that covering spaces and subgroups are the same information.

What a cover is #

A map $p : \tilde X \to X$ is a covering if every point of $X$ has a neighbourhood whose preimage is a disjoint union of copies of it, each mapped homeomorphically down. Informally: locally, $\tilde X$ looks like several parallel copies of $X$, and globally the copies can be connected up in interesting ways.

The circle has one cover for each positive whole number — wrap a circle round it $n$ times — and one more, the real line wrapped round it infinitely often. Every one of those is locally a disjoint union of arcs and globally something else.

The word covering is doing precise work and is worth separating from the everyday sense. A map onto a space is not a covering merely because it is onto; the local condition — that some neighbourhood of every point is evenly covered — is what fails for the map from the interval to the circle that wraps once round and stops, and it is what fails for the squaring map on the complex plane at the origin. Those maps are surjective and are not coverings, and the theory below applies to neither. The whole strength of the dictionary comes from a purely local requirement, which is the usual bargain in topology: a condition that can be checked one neighbourhood at a time, with global consequences.

The single fact that makes covers useful is unique path lifting. Given a path in $X$ starting at $x_0$ and a point $\tilde x_0$ above $x_0$, there is exactly one path in $\tilde X$ starting at $\tilde x_0$ that projects to it. Existence follows from patching together the local lifts; uniqueness follows because two lifts agreeing anywhere agree on an open and closed set, hence everywhere on a connected interval.

The subgroup #

Fix a base point $x_0$ and a point $\tilde x_0$ above it. Every loop at $x_0$ lifts to a path starting at $\tilde x_0$, and that path may or may not end where it started.

Consider the loops whose lifts are loops. Call that set $H$.

$H$ is a subgroup. If $f$ and $g$ both lift to loops, then the lift of $fg$ is the lift of $f$ followed by the lift of $g$ starting where the first finished — which is $\tilde x_0$ — so it is a loop. The constant loop lifts to a constant. The reverse of a loop lifts to the reverse of its lift. Closure, identity and inverses, all immediately.

And $H$ is well defined on classes, because homotopic loops have homotopic lifts, by lifting the homotopy itself. So $H$ is a subgroup of $\pi_1(X, x_0)$, and it is exactly the image of $\pi_1(\tilde X, \tilde x_0)$ under the map induced by $p$.

That is the correspondence in one direction. A cover gives a subgroup.

The index is the number of sheets #

The other numerical half of the dictionary is as clean.

Take the set of points above $x_0$ — the fibre. Every loop at $x_0$ lifts to a path from $\tilde x_0$ to some point of the fibre, and that point depends only on the class of the loop. So there is a map from $\pi_1(X, x_0)$ to the fibre.

It is onto, if $\tilde X$ is path-connected: any point of the fibre is reachable from $\tilde x_0$ by a path, which projects to a loop. And two classes give the same point exactly when their difference lifts to a loop — that is, exactly when they lie in the same coset of $H$.

So the cosets of $H$ correspond to the points of the fibre, and the index of $H$ is the number of sheets.

For the circle, whose group is $\mathbb{Z}$, the subgroups are $n\mathbb{Z}$ for $n \geq 1$ and the trivial subgroup. The corresponding covers are the $n$-fold wraps and the real line, and there are no others. A complete classification of covers of the circle falls out of a complete classification of subgroups of $\mathbb{Z}$, which is one line of arithmetic.

Worked, on the figure-eight #

The circle’s classification is one line because its group is $\mathbb{Z}$. Doing the same on a space with a non-commutative group is where the dictionary starts to pay, and the figure-eight is the smallest such space.

Its fundamental group is free on two generators $a$ and $b$. The index-two subgroups of a free group of rank two are the kernels of the three homomorphisms onto the group of order two — send $a$ to the non-identity element and $b$ to the identity, or the reverse, or both — so there are exactly three, and therefore exactly three double covers of the figure-eight.

They can be drawn. Two vertices, and above each generator either two edges joining a vertex to itself, or two edges joining the two vertices. The three choices of which generator swaps the sheets give the three covers, and each is a graph with two vertices and four edges.

The Euler characteristic then does something surprising. A graph with $V$ vertices and $E$ edges has fundamental group free of rank $E - V + 1$. The figure-eight has one vertex and two edges, so its rank is $2 - 1 + 1 = 2$. Its double cover has two vertices and four edges, so its rank is $4 - 2 + 1 = 3$. So a subgroup of index two in a free group of rank two is free of rank three — a subgroup needing more generators than the group that contains it.

That is the Nielsen–Schreier index formula, and its usual proof is exactly this: realise the group as a graph’s fundamental group, realise the subgroup as a cover, and count. A purely algebraic statement whose only short proof is topological is the sort of thing this correspondence is for.

The universal cover #

The trivial subgroup corresponds to the cover whose own fundamental group is trivial — the universal cover — and it exists for any reasonable space.

Its defining property is that it covers every other cover. Since it is simply connected, every loop upstairs is contractible, so no non-trivial class downstairs lifts to a loop, so the subgroup is trivial and the number of sheets is the size of the whole group. Trivial subgroup, index equal to the whole group’s order, and therefore that many sheets — the arithmetic of the previous section, applied to its extreme case.

That last sentence is worth reading twice. For the circle the group is infinite, so the universal cover has infinitely many sheets — it is the real line. For the figure-eight, the group is free on two generators, and the universal cover is the infinite four-valent tree. The tree’s vertices correspond to elements of the group, and its edges to multiplication by a generator, so the universal cover of the figure-eight is the group’s Cayley graph.

That is not an analogy. The deck transformations of the universal cover — the homeomorphisms commuting with the projection — form a group isomorphic to $\pi_1(X)$, acting freely, and the quotient recovers $X$. So a space is its universal cover divided by its own fundamental group, which is the cleanest statement of what the group is for.

Normality is symmetry #

The dictionary has a third entry, and it is the one that makes the correspondence worth calling a dictionary rather than a pair of counts.

A subgroup $H$ is normal exactly when the corresponding cover is regular — meaning its deck transformation group acts transitively on each fibre, so the cover has as much symmetry as it possibly could. And when $H$ is normal, the deck group is the quotient $\pi_1(X)/H$.

The reason is the base-point ambiguity of the previous rung. Choosing a different point $\tilde x_1$ of the fibre gives a different subgroup, and the two subgroups are conjugate — the conjugating element being the class of a loop whose lift joins the two points. So a cover corresponds not to a subgroup but to a conjugacy class of subgroups, and it corresponds to a single subgroup exactly when that subgroup is normal — that is, when conjugating changes nothing, so the class has one member and the choice was never a choice. The word normal on the algebraic side, chosen in the nineteenth century for reasons of its own, turns out to name the covers with the most symmetry, which nobody would have predicted from the definitions of either.

The base point’s residue reappears at exactly the same place and means exactly the same thing, which is a good sign that both statements are about something real.

It is worth being explicit about what has been achieved by this point, because it is easy to read the dictionary as a restatement. Before it, the fundamental group is a set of equivalence classes of maps, defined by a quotient and manipulated symbolically. After it, a subgroup is a space — an object with points, that can be drawn, whose Euler characteristic can be counted and whose symmetries can be seen. Every question about subgroups becomes a question about spaces, and the last section shows one of them being answered that way and not otherwise.

What it costs #

The correspondence needs hypotheses, and they are not vacuous. The space must be path-connected, locally path-connected and semi-locally simply connected — the last meaning that every point has a neighbourhood whose loops die in the whole space. Without it there is no universal cover. The Hawaiian earring, an infinite family of circles shrinking to a point, fails it, and its covering theory is genuinely broken.

Covers do not see anything above dimension one. Every cover of a sphere is a disjoint union of spheres, because the sphere is simply connected, and the same holds of any space whose loops all shrink. So a technique that converts questions into covering spaces is silent about everything the fundamental group is silent about, which is most of higher-dimensional topology.

The classification is of connected covers, and disconnected ones are a nuisance. A disjoint union of two covers is a cover, and it corresponds to no single subgroup. The statement is therefore about connected covers, and every enumeration has to say so — which is why the double covers of the figure-eight number three rather than four, the fourth candidate being two disjoint copies of the figure-eight itself.

And the universal cover is usually enormous. For a surface of genus two it is the hyperbolic plane; for a knot complement it is a wild non-compact three-manifold. Having it does not mean being able to draw it, and most uses of it are uses of its existence rather than of its shape.

Where the analogy with Galois is exact #

The dictionary above has a twin in algebra, and the two are close enough that each is usually explained by pointing at the other.

A field extension has intermediate fields; a space has covers. The Galois correspondence matches intermediate fields with subgroups of the Galois group, reversing inclusions; the covering correspondence matches covers with subgroups of the fundamental group, also reversing inclusions — a bigger cover has a smaller subgroup. Degree matches index. Normal extensions match regular covers. The splitting field matches the universal cover. Six matched pairs, and the two subjects were developed a century apart with no contact.

The resemblance is not a coincidence and the modern statement of why is that both are instances of one construction: a category of things over a base, with a group acting, and a correspondence between subgroups and quotients. Grothendieck’s formulation makes the fundamental group a Galois group and the two theorems one theorem, which is a substantial piece of machinery and is not needed to notice the pattern.

What is worth noticing without any machinery is the direction reversal, which is the part that surprises people in both subjects. A larger subgroup means a smaller cover and a larger intermediate field, and the reason is the same on both sides: the subgroup is what is fixed, and fixing more leaves less. The lattice that runs the other way is the same observation made about the algebraic half alone.

What the pictures cannot show #

A cover is drawn as a spiral and it is not one. The $n$-sheeted cover of the circle is a circle, mapped to another circle by going round $n$ times. Drawing it as a spiral separates the sheets so they can be counted, at the price of drawing a curve that does not close when the object does. The infinite cover is the one the spiral is honest about.

Lifting is drawn as one path and it is a construction with a choice in it. The lift depends on which point above the base is chosen, and different choices give paths ending at different heights. The figures fix a choice silently, which is the right thing to draw and hides the conjugation that the choice introduces.

And the universal cover of anything interesting is not on this page. The tree covering the figure-eight is infinite and four-valent; the figures show three sheets of it, which is a finite quotient rather than the object. What is drawn is always some finite cover, and the object the theory is really about is the limit of those.

Where the ladder goes next #

The next rung stops describing the group and computes it, by cutting the space into pieces whose groups are known. The rung above that replaces loops by spheres and finds an arithmetic that is commutative for a reason having nothing to do with the space.

Named here as a debt: the monodromy reading of a cover — the homomorphism from the fundamental group to the permutations of the fibre — which is the same information a third time and is the form the correspondence takes in Galois theory and in the study of differential equations. The pictures above draw it and the prose does not develop it.

Sideways, the universal cover as a Cayley graph is the group drawn as a map, the correspondence between subgroups and intermediate objects is the Galois correspondence in another subject, and the lifting argument is what makes the winding number an integer.

What is worth carrying away #

A correspondence between two subjects is worth more than a theorem in either, and the test of one is whether it translates the structure and not merely the objects.

Covers correspond to subgroups; sheets correspond to index; symmetry corresponds to normality; the whole group corresponds to the universal cover; and the ambiguity of a base point corresponds to conjugacy on both sides. Five matched pairs is what makes this a dictionary rather than a coincidence, and it is why a question about either side can be moved to the other without loss.

The habit worth taking is to check whether a correspondence matches the ambiguities as well as the objects. Two constructions that agree on objects and disagree on how those objects fail to be canonical are not the same construction, and the agreement here — conjugacy on the group side, choice of point in the fibre on the space side — is the strongest evidence that the dictionary is real.