The symmetries a cover has of its own

The rung below treated a covering as an object to count: vertices, edges, a spanning tree, a rank. A covering is also an object with symmetries of its own — ways of rearranging it that leave the space underneath completely undisturbed.

How many such symmetries there are is not a free parameter. It is decided by the subgroup the covering corresponds to, and the count is at most the number of sheets.

What a symmetry is here #

A deck transformation of a covering is a map of the covering space to itself that commutes with the projection down: it may move points around upstairs, and every point must land over the point it started over.

On a covering graph that is a permutation of the vertices sending each sheet to another sheet, carrying edges to edges of the same label. Written in terms of the permutations that define the covering, it is a permutation $\sigma$ of the sheets with

$$\sigma \circ p_x = p_x \circ \sigma$$

for every generator $x$ — so the deck group is the centraliser of the covering’s permutations inside the symmetric group, and for a handful of sheets it can be found by trying all of them, which is what the figures do.

Why there cannot be more than there are sheets #

The bound is the one fact everything else follows from, and its proof is a lifting argument in one line.

A deck transformation is determined by where it sends a single point. Suppose two of them agree at one vertex; then they agree at every vertex reachable from it, because a path upstairs is determined by its starting point and its image downstairs, and both transformations carry that path the same way. On a connected covering every vertex is reachable, so two deck transformations agreeing anywhere agree everywhere.

So the map “where does it send the base vertex” is injective on the deck group, and the group has at most as many elements as there are sheets. The figures check that injectivity directly, by requiring no two of the found permutations to agree on the base.

That is a strong constraint and it is worth appreciating how strong. A covering with a hundred sheets has at most a hundred symmetries, however complicated it is — and a graph with a hundred vertices generally has an enormous automorphism group. The requirement to commute with the projection removes nearly all of it.

The bound is worth comparing against the alternative, since it says what the projection is doing. A graph automorphism may send any vertex to any vertex and then choose freely among the edges; a deck transformation may send the base vertex to any of the sheets and then has no choices at all. One free choice against a whole tree of them, and the difference is that a deck transformation has to respect a labelling that the projection supplies.

Read that way the deck group is the automorphism group of a labelled graph rather than of a graph, and labelled objects have far fewer symmetries than unlabelled ones — which is the same reason a Latin square has fewer symmetries than a grid and a rooted tree fewer than a tree.

Regular, and what it means downstairs #

A covering with exactly as many deck transformations as sheets is called regular or normal, and the second name is the informative one.

The correspondence the first rung set up attaches a subgroup $H$ of the fundamental group to each covering. The theorem is:

The covering is regular exactly when $H$ is a normal subgroup, and in that case the deck group is the quotient $G/H$.

Both halves are worth reading. A normal subgroup is one closed under conjugation — $gHg^{-1} = H$ for every $g$ — and the geometric meaning of that condition is now available: conjugating by $g$ corresponds to moving the base point around the loop $g$, and the subgroup being unchanged means the covering looks the same from every sheet.

A regular covering is one with no distinguished sheet, in the sense the blocks a subgroup cuts out makes precise downstairs. Every sheet is like every other, and the deck group is what moves between them.

That last figure is the case worth having, because the regular case is the one every account draws and it is the special one. Most subgroups are not normal, so most coverings have almost no symmetry at all, and the three-sheeted cover of the figure-eight above has exactly one deck transformation: the identity.

What the deck group is when it is large #

At the other extreme is the universal cover — the covering corresponding to the trivial subgroup — and there the deck group is the whole fundamental group.

For the wedge of two circles the universal cover is the infinite tree in which every vertex meets four edges, and its deck group is the free group on two letters acting on the tree by translations. That action is free — no non-identity element fixes a vertex — and the quotient is the wedge itself.

So the fundamental group is the deck group of the universal cover, which is a completely different description of it from “loops up to deformation”. The rung below quoted Serre’s characterisation of freeness in exactly these terms: a group is free precisely when it acts freely on a tree, and the tree is this one.

That reformulation is why the subject is called geometric group theory. A group presented by generators and relations is hard to reason about; the same group acting on a space it is the deck group of is a geometric object, and the space’s properties become the group’s.

Reading the deck group off the covering #

The two regular examples above have deck groups that can be named, and naming them says what the quotient is.

The two-sheeted cover has deck group of order two, so the subgroup is the words with an even number of $a$’s and the quotient is the two-element group. That subgroup is the kernel of the homomorphism counting $a$’s modulo two, which is why it is normal without any calculation.

The three-sheeted cyclic cover has deck group of order three, the subgroup is the words with $a$-count divisible by three, and the quotient is the three-element group.

In both, the subgroup is a kernel — of a homomorphism from the free group onto a finite group — and that is not a coincidence. A normal subgroup is exactly a kernel, and a regular covering is exactly one whose subgroup is a kernel, so a regular covering is a homomorphism onto a finite group with the sheets as its elements.

That gives a way of building regular coverings to order. Pick a finite group, pick where each generator goes, and the covering is determined: the sheets are the group’s elements and the generator $x$ moves each sheet by multiplication by its image. Every regular covering arises this way and no others do.

The non-regular example has no such description, and could not: its subgroup is not a kernel, so there is no group of sheets for the generators to act on by multiplication. The sheets are a set the group acts on and not a group in their own right, which is exactly the distinction between a coset space and a quotient group.

Where the intermediate cases sit #

Between “no symmetry” and “as much as possible” there is a range, and it is organised by a second subgroup.

The deck group of the covering for $H$ is $N(H)/H$, where $N(H)$ is the normaliser of $H$ — the largest subgroup in which $H$ is normal. When $H$ is normal, $N(H)$ is everything and the deck group is $G/H$, which has as many elements as there are sheets. When $H$ is its own normaliser, the deck group is trivial.

So the count of symmetries measures how close to normal the subgroup is, and it is a divisor of the number of sheets — which the figures check by requiring the deck group’s order to divide the sheet count.

That divisibility is Lagrange’s theorem appearing in a covering, and it is the same statement: the deck group acts on the sheets with all its orbits the same size, so its order divides the number of sheets.

The orbits being equal in size is itself the injectivity argument again. The deck group acts freely — a non-identity deck transformation fixes no sheet, since fixing one would make it agree with the identity somewhere and therefore everywhere — so every orbit is a copy of the group, and the sheets split into orbits of that size. Free actions are what make counting arguments work, and the rung below’s Nielsen–Schreier proof leans on exactly the same property one level up.

The intermediate case is then easy to picture. The deck group’s orbits partition the sheets into blocks of equal size; a regular covering has one block; a covering with trivial deck group has one block per sheet; and the number of blocks is the index of the normaliser.

What the symmetries are worth #

Deck transformations are not merely a classification device, and it is worth saying what they do.

They let a covering be used as a change of coordinates. A regular covering with deck group $G$ turns a problem downstairs into a $G$-equivariant problem upstairs, which is often easier because the upstairs space is simpler — the universal cover of a surface is a disc or a plane, and every question about the surface becomes a question about a group acting on one of those.

They compute quotients. Handed a space with a group acting freely on it, the quotient is a space with the original as a regular covering and the group as its deck group. So the two directions — build a covering from a subgroup, or take a quotient by a group action — are the same construction read forwards and backwards.

And they make a fundamental group into a symmetry group. The fundamental group is defined by loops and deformations, which is an awkward object to compute with. The same group acting on the universal cover is a group of symmetries of a concrete space, and symmetries can be composed, drawn and enumerated.

That last is the whole of what the rung below’s proof of Nielsen–Schreier depends on. A free group is the deck group of a tree; a subgroup acts on the same tree by the same rule; and the argument that the subgroup is free is the observation that acting freely on a tree is what freeness means.

So the deck group is the bridge between an algebraic object and a geometric one, and every result in this ladder crosses it at some point.

Where it needs its hypotheses #

The covering must be connected. A disconnected covering has extra symmetries permuting whole components, and the injectivity argument fails because a vertex in one component reaches nothing in another. Every figure here checks connectedness by walking the graph.

The base must be nice enough. The general theory needs the space below to be connected, locally path-connected and semi-locally simply connected; a graph is all three, comfortably. The whole apparatus fails on spaces that are not, and the standard counterexample is a wedge of infinitely many shrinking circles.

A deck transformation is not any symmetry of the covering graph. The graph above has automorphisms that do not commute with the projection — swapping two edge labels, for instance — and none of them is a deck transformation. The condition is not “is a symmetry” but “is a symmetry over the identity”.

And the correspondence is with conjugacy classes. Two subgroups that are conjugate give isomorphic coverings, so the covering determines $H$ only up to conjugacy — which is why the base point matters and why moving it conjugates the subgroup.

That is the one place where the tidy dictionary needs care, and it is the reason “regular” and “normal” line up. A covering with a chosen base point upstairs determines a subgroup exactly; forget the base point and what is left is its conjugacy class; and the class is a single subgroup exactly when that subgroup is normal. So the ambiguity in the correspondence vanishes precisely on the coverings this rung is about, which is a fair description of why they are the ones with a name.

Where the vocabulary came from #

The word deck is a translation artefact. The German is Decktransformation, from decken, to cover — so it is the covering’s own transformation rather than anything to do with a deck of cards or the deck of a ship, and the English word carries none of that.

The theory is Poincaré’s in outline and Seifert and Threlfall’s in the form it is taught, and the organising insight — that the whole of covering space theory is a dictionary between subgroups and coverings — is worth restating because it is what makes the subject compact:

Every row is a theorem and the table is the subject, which is why the rung below could prove Nielsen–Schreier by counting edges: the counting happens upstairs and the statement is about downstairs.

What the pictures cannot show #

The figures draw coverings of two or three sheets, and the interesting universal cover is infinite — the tree in which every vertex meets four edges — so the case in which the deck group is the whole fundamental group is the one case with no drawing.

The deck transformation is drawn as arrows between vertices, and it is a map of the whole covering space. On a graph, knowing where the vertices go determines the edges; on a general covering it does not, and the arrows would be a shorthand for something with more to it.

And no figure shows the correspondence. That the deck group’s order equals the index exactly for a normal subgroup is a theorem about groups, checked here on the permutations of three sheets by exhaustion, and the exhaustion is a check on those examples rather than on the statement.

The ladder from here #

Rungs above: the covering as a permutation representation, where the sheets become a set the fundamental group acts on and the subgroup becomes a stabiliser. Folding a graph to decide membership, where the correspondence becomes an algorithm. The Galois correspondence for coverings in general, which is the table above stated as an order-reversing bijection. Riemann surfaces and branched coverings, where the deck group becomes a group of automorphisms and the whole subject was born. And the Cayley graph of a group as the universal cover of its presentation complex, which is where geometric group theory starts.

A symmetry is determined by one point #

The habit is the argument this rung turns on, and it is worth extracting because it recurs wherever a lifting property is available.

A deck transformation is a map of an infinite or at least large object, and it is pinned down by a single choice: where it sends one point. Everything else is forced, because a path determines its lift once its start is chosen.

That is a rigidity statement, and rigidity is what makes a group finite. The automorphism group of a graph can be enormous; the subgroup of it respecting a projection is bounded by the number of sheets, and the bound comes entirely from the lifting.

The same shape appears wherever an object is determined by local data plus one choice. An analytic function is determined by its values on any disc; an isometry of a sphere is determined by where it sends three points; a homomorphism from a free group is determined by where it sends the generators. In every case the conclusion is a smallness result — there are few such maps — and the smallness is what makes them countable, classifiable, and worth naming.