A covering is a permutation

Every covering in this ladder has been specified the same way: by saying, for each generator, which sheet it sends each sheet to. That is a permutation per generator, and once the permutations are given the covering is completely determined.

Which means the whole apparatus can be replaced by a homomorphism.

A word in the generators acts on the sheets by composing the permutations, so the fundamental group acts on the set of sheets. The covering is that action and nothing else.

The action, and the subgroup it names #

Fix a base sheet. A loop downstairs lifts to a path upstairs starting there and ending somewhere, and where it ends is where the action sends the base sheet. That is the monodromy of the loop, and it depends only on the loop’s class.

Two things follow immediately.

The subgroup is a stabiliser. The first rung attached to a covering the set of loops whose lifts close up. In this language those are the words fixing the base sheet — the stabiliser of a point under the action — and the figures mark them.

The sheets are the cosets. Two words send the base sheet to the same place exactly when one composed with the other’s inverse fixes it, which is exactly when they lie in the same left coset of the stabiliser. So the sheets are in bijection with the cosets, and the number of sheets is the index.

That last is the index theorem the whole ladder has been using, arriving now as the orbit–stabiliser theorem — the size of an orbit is the index of a stabiliser — which is a statement about group actions with nothing topological in it — and which Lagrange’s theorem is the special case of where the action is on the group itself.

What is gained by the translation #

Reading a covering as an action makes several questions immediately answerable that were not obviously questions before.

Is it connected? The covering is connected exactly when the action is transitive — every sheet reachable from every other. The figures check that by requiring every sheet to be landed on by some word.

Is it regular? The rung below found the deck group as a centraliser; in this language a covering is regular exactly when the stabiliser is normal, and equivalently when the action is free on the sheets once the deck group is brought in.

How many coverings are there with $n$ sheets? Exactly as many as there are homomorphisms from the fundamental group to the symmetric group on $n$ letters with transitive image, counted up to relabelling the sheets. For a free group on $k$ generators a homomorphism is just $k$ arbitrary permutations, so the count is a combinatorial question — which is why coverings of a wedge of circles can be enumerated and coverings of a general space cannot.

Counting the coverings #

The enumeration is worth doing because the answer is a number and because it says how many objects the ladder’s dictionary is describing.

A covering of the wedge of two circles with $n$ sheets is a pair of permutations of $n$ things, together generating a transitive group, up to simultaneous relabelling. For $n = 2$ there are four pairs of permutations of two things, three of them transitive, and up to relabelling those give three connected double covers.

For $n = 3$ the count is seven, for $n = 4$ it is twenty-six, and the sequence grows quickly. Each of those is a subgroup of index $n$ in the free group on two letters — up to conjugacy — so the same numbers count the subgroups.

A question about subgroups of a free group has become a question about pairs of permutations, which is finite and can be worked out by a computer for any modest $n$. That is the practical payoff of the translation, and it is the reason subgroup counts in free groups are known exactly while the corresponding counts in most other groups are not.

The word “monodromy” #

The name comes from the subject the whole theory started in, and knowing where it comes from explains the vocabulary.

A multivalued function of a complex variable — a square root, a logarithm — is a covering of the punctured plane, with the sheets being the branches. Following the function round a loop enclosing the puncture returns a different branch: the square root comes back negated, the logarithm comes back with $2\pi i$ added. That change is the loop’s monodromy, and mono-dromy is Greek for “running round singly”.

So the permutation associated to a loop is literally what happens when the loop is run once, and the entire theory of covering spaces is the abstraction of that phenomenon away from complex functions.

The subgroup of loops with trivial monodromy is the set of loops after which the branch is unchanged, and the covering it corresponds to is the Riemann surface of the function. The dictionary this ladder has been building was built to describe that.

Two of the classical examples are worth putting in these terms, because they are the two extremes.

The square root on the plane with the origin removed has two branches, and going round the origin swaps them. So the monodromy of the generating loop is a transposition, the covering is the connected double cover of the circle, and the subgroup is the even integers — which is the wrapping cover the first rung of this anchor drew, arriving from complex analysis.

The logarithm on the same space has infinitely many branches and going round adds $2\pi i$ each time, so the monodromy of the generator is a shift of the integers with no finite order at all. The covering is the infinite spiral — the universal cover of the circle — and the subgroup is trivial.

The two together are the whole classification of coverings of the circle, met as two functions rather than as a theorem. A function’s branches and a covering’s sheets are the same objects, and the subject exists because somebody noticed that.

The circle, where the count is one #

The simplest instance is worth working out because it is where the counting has a completely tidy answer.

The fundamental group of the circle is the integers, generated by one element, so a covering is a single permutation of the sheets. Transitive means the permutation is an $n$-cycle, and all $n$-cycles are the same up to relabelling — so there is exactly one connected $n$-sheeted covering of the circle for each $n$, and it is the circle wrapping round $n$ times.

The subgroups match: the subgroups of the integers are the multiples of $n$, one for each $n$, all normal. Every covering of the circle is regular, and the deck group is the cyclic group of order $n$.

That tidiness disappears the moment there are two generators, because two permutations can fail to commute and the transitive pairs are numerous. One generator gives a classification; two give a combinatorial explosion, and the difference is exactly the difference between an abelian fundamental group and a free one.

The same contrast explains why the rank formula of the rung below has the shape it does. With one generator every covering has rank one, so subgroups of the integers are copies of the integers and nothing grows; with two, the rank climbs by one per sheet, and the room for that growth is the room two non-commuting permutations have that one permutation does not.

So the free group’s strangeness — a subgroup needing more generators than the group — is visible in the counting the moment the coverings are counted rather than drawn. A single permutation has one conjugacy class per cycle type; a pair has as many as there are transitive pairs, and the second number is where all the subgroups live.

What a non-transitive action would mean #

The figures all check transitivity, and it is worth saying what fails without it, because the failure is instructive rather than merely disqualifying.

A pair of permutations generating a group with two orbits describes a covering with two components. Each component is a covering in its own right, with its own subgroup, and the whole thing is a covering of the wedge by a disconnected graph.

The index argument breaks immediately. The number of sheets is no longer the index of anything, because the stabiliser of a base sheet has index equal to the size of that sheet’s orbit and knows nothing about the other component. So a four-sheeted disconnected covering may be two two-sheeted ones, and its “subgroup” would have index two rather than four.

That is the reason connectedness is a hypothesis everywhere in this ladder rather than a convenience. A disconnected covering is not one object with a subgroup; it is several objects, and the correspondence applies to each separately.

The action language makes the repair obvious too. Restrict the action to one orbit and it is transitive, so a disconnected covering is a disjoint union of connected ones — which is the orbit decomposition of a group action, stated topologically.

Where the translation has limits #

It works because the fundamental group is free. For a wedge of circles a homomorphism is an arbitrary choice of permutations, one per generator. For a space with relations the permutations must satisfy those relations, and the enumeration becomes a search rather than a count.

The action is on a set with no structure. The sheets are just $n$ things; everything geometric about the covering — how the edges connect, what the space looks like — is recovered from the permutations and is not visible in them. The translation is faithful and it is not illuminating about shape.

And the correspondence is up to relabelling. Two homomorphisms differing by a relabelling of the sheets give the same covering, so the count of coverings is a count of homomorphisms divided by that equivalence, and dividing correctly requires care — the classes are not all the same size, since some homomorphisms have symmetries and some do not.

That last is the deck group again. A homomorphism whose image commutes with a relabelling has a symmetry — the deck transformations of the rung below — and the coverings with many symmetries are exactly the ones counted with a smaller multiplicity, which is Burnside’s counting lemma doing the bookkeeping.

What this makes computable #

The reason the translation is the working description rather than a curiosity is that it turns questions into calculations.

Membership. Is a given word in the subgroup? Compose its permutations and see whether the base sheet comes back. That is linear in the word’s length and needs nothing stored but the permutations.

Index. How many sheets? Count the orbit of the base sheet, which is a graph search.

Normality. Is the subgroup normal? Test whether every generator’s permutation commutes with the whole deck group, or equivalently whether the stabilisers of all the sheets coincide.

And intersection. The subgroup corresponding to two coverings at once is the stabiliser under the product action on pairs of sheets — so intersecting subgroups is taking a product of permutation actions, which is an operation on tables.

That last one has a consequence worth stating on its own, because it is a theorem that is hard from the algebraic side and immediate from this one. Two subgroups of index $m$ and $n$ intersect in a subgroup of index at most $mn$, because the product action is on at most $mn$ pairs and the intersection is a stabiliser in it. Poincaré’s theorem on the intersection of finite-index subgroups, in one sentence about a product of actions.

And the same construction gives the covering: the pullback of two coverings is the graph whose vertices are pairs of sheets, one from each, connected when both coordinates are. Its components are the coverings corresponding to the various intersections, and the largest of them corresponds to the intersection of the two subgroups. Intersecting subgroups is drawing a product graph, which is a construction rather than a search.

The transitive pairs, counted #

The count of coverings quoted above deserves the working, because it is the one place this ladder produces a sequence of numbers.

For $n = 2$: each generator is one of two permutations, giving four pairs. The pair where both are the identity is not transitive; the other three are. Relabelling the two sheets is a single swap, and it fixes all three of those pairs — so there are three connected double covers of the wedge of two circles, corresponding to the three subgroups of index two, which are the kernels of the three homomorphisms onto the two-element group.

That last description is the cross-check. A subgroup of index two in a free group of rank two is a kernel of a map to the two-element group, such a map is determined by where each generator goes, and there are four such maps of which one is trivial. Three, again, from a completely different count.

Two ways of counting the same three objects, and they agree — which is the habit this collection runs on, applied to a classification rather than to a figure.

Every one of those is a decision procedure, and the rung above turns the first of them into an algorithm that does not need the permutations to be given in advance.

What the pictures cannot show #

The table lists reduced words of length up to three, of which there are twenty-six drawn out of many more. The action is on the whole free group, which is infinite, and every finite table is a sample.

The check that two words land on the same sheet exactly when one undoes the other into the subgroup is made over sixteen hundred pairs and the claim is about all of them. That check is the orbit–stabiliser theorem instanced rather than proved, and the proof is a bijection between cosets and orbit elements that has no picture.

And the covering itself is not drawn in the table. What is drawn is the action, which is the covering’s complete description and looks nothing like it — a table of words and numbers rather than a graph, which is exactly the trade the translation makes.

The ladder from here #

Rungs above: folding a graph until it decides, where the subgroup is given by generators rather than by permutations and the permutations have to be constructed. Branched coverings and the Riemann–Hurwitz formula, where the monodromy is allowed to be trivial at some points and the Euler characteristics stop simply multiplying. The Galois correspondence proper, of which this is the topological half. Coverings of surfaces and the classification of their subgroups by genus. And the monodromy of a differential equation, which is where the word was invented and where the subject is still used.

Replacing a space by a set with an action #

The habit is the one the whole rung is an instance of: when an object is determined by finitely many choices, replace it by those choices.

A covering of a wedge of two circles is a graph, potentially with many vertices and edges, and reasoning about it means reasoning about a graph. It is determined by two permutations, and reasoning about two permutations is arithmetic.

Nothing is lost — the graph is recoverable — and what is gained is that the questions become finite. How many are there, is this one connected, does this word lie in that subgroup: all of them become computations on tables.

The general form is worth naming because it is the standard move in classification. An object with a lot of structure is replaced by the data determining it, the data live in something finite or algebraic, and the classification becomes a count. A covering is a subgroup was the first such replacement in this ladder; this is the second, and it is the one that makes the first computable.