A loop that cannot be pulled tight

A disc with a hole punched out of it is not the same as a disc. Everybody knows this and nobody can say what it means, because the difference is an absence: the two objects differ by a piece of material that is not there in one of them and is not there to be examined in the other either.

The move that turns this into mathematics is to stop looking at the hole and start looking at loops.

A loop is a thing that can be drawn. It has coordinates, it can be measured, and — this is the point — it can be moved around inside the ring and watched to see what survives. What survives is a single whole number, and that number is the hole made visible.

Two loops are the same when one slides onto the other #

The relation that matters here is not equality. Two loops are counted as the same when one can be slid, stretched and squashed onto the other without ever leaving the region. Every intermediate stage has to be a loop lying in the region; the deformation is not allowed to cheat by lifting a strand over the hole.

That relation is called homotopy, and it is an equivalence: a loop is deformable to itself, a deformation can be run backwards, and two deformations run one after the other are a deformation. So the loops in a region fall into classes, and the question becomes how many classes there are.

For a disc, there is one class. Any loop in a disc can be pulled straight in to a point, and the picture of that is the definition of a convex region: the segment between two points of the region is in the region, so every loop can be shrunk along those segments simultaneously.

For a ring the straight-line shrink fails, and the figure shows it failing. That is worth stating carefully, because it is a weaker fact than the theorem needs: what the picture establishes is that this attempt does not work. A reader is entitled to ask whether some cleverer deformation might succeed.

Nothing cleverer works, and the reason is the number under each loop.

The count cannot change a little #

Walk once round a loop and keep a running total of the angle turned through, as seen from the centre of the hole. When the loop closes, the total is a whole number of turns.

That whole number is what the first figure labels. It is the same winding count that decides how many roots a polynomial has inside a circle, computed the same way, and the argument for its being unchangeable is the same too: as a loop is deformed continuously, the accumulated angle varies continuously, and a continuous quantity that is always a whole number is constant.

So a loop of count 1 cannot be deformed into a loop of count 0, since that would require the count to move from one whole number to another without passing through anything in between. The hole is not shown; the impossibility of the deformation is.

The condition hiding inside the phrase “as seen from the centre” is the whole hypothesis. If a stage of the deformation crossed the centre, the angle there would be undefined and the count would be free to jump — which is exactly what the straight-line shrink did, and exactly why it is not allowed.

The classes are the whole numbers, with addition #

Having a class for each whole number is only half a description. The classes also combine: run one loop and then another, starting and finishing at the same point, and the result is a loop whose class is determined by the two.

Running a loop and then its reverse gives a loop that can be pulled tight; running the constant loop before or after anything changes nothing. So the classes have an addition, an identity and negatives, and the count is a perfect dictionary between them and the integers.

That structure is the fundamental group, and for the ring it is the group of whole numbers under addition. Naming it that way is not decoration: it says that everything about loops in a ring is already known to anybody who knows how integers add, and it is why a group turns up in a subject with no arithmetic in it. The same move appears when the symmetries of a square are found by exhaustive search and then recognised as a familiar object; the recognition is what makes the classification useful rather than merely complete.

The count is really a height #

There is a second picture of the same integer, and it is the one that generalises furthest.

Following the angle without reducing it modulo a whole turn produces a path that need not close. It starts at a height and ends at a height, and those two heights differ by a whole number of turns. That is the count, and stated this way it is obviously an integer rather than merely observed to be one.

The picture behind it is that the circle has a spiral lying above it — a helix, with the whole line of heights sitting over each point of the circle. Every path downstairs lifts to a path upstairs once a starting height is chosen, and a loop downstairs lifts to a path whose two ends are a whole number of storeys apart. The helix is the universal cover of the circle, and the number of storeys is the class.

Two properties make the lifting work and both are worth naming, because they are what a covering space is. Each point downstairs has a small neighbourhood whose preimage upstairs is a stack of separate copies of it, so a path can be lifted step by step with no choices to make after the first; and the lift of a path is determined entirely by where it starts, so two lifts of the same path either coincide everywhere or nowhere. Together those give the count its second definition, and the second definition is the one that survives when the region is something a protractor cannot be held against.

The device also explains why the count of a concatenation is a sum without any calculation. Lift the first loop from height zero and it ends at height p; lift the second from height p — which is allowed, since lifts may start anywhere — and it ends q further up, because the covering looks the same at every storey. The joined path therefore ends at p + q, and that is the addition of the previous section, obtained for nothing.

This reframing is what makes the theory computable in cases where no angle is available to accumulate. There is no “angle” on a surface of two holes, but there is still a covering space, and the classes are still the ways a path can fail to close up when lifted.

Two holes, and an arithmetic that stops being commutative #

On a torus the classification needs two numbers rather than one.

A loop on a torus can wind the short way, the long way, or both, and no amount of sliding trades one kind of winding for the other. The classes are pairs of whole numbers, they add coordinate-wise, and the order in which two loops are run makes no difference to the result — the torus’s fundamental group is commutative.

That is a fact about the torus and not about surfaces. On a surface with two holes, running loop a then loop b is genuinely different from running b then a, and the group of classes is a non-commutative object that no list of counting numbers describes. The classification of surfaces by handles has a parallel classification by these groups, and the second is finer and much harder.

The place the difference becomes visible is a loop that goes a, b, a backwards, b backwards. On a torus that loop can be pulled tight — the square’s gluing word says so in as many letters. On a two-holed surface the same instruction traces a loop that goes round the join between the handles and cannot be pulled tight at all, which is the first genuinely non-commutative fact in the subject and the reason the theory needs groups rather than counts.

The square-with-glued-edges picture is worth keeping for another reason: it is the same device that turns a gluing word into a surface, and a loop’s two counts can be read straight off the word.

The letters are the loops. In the word for the torus, a is the loop that crosses the side edges once and b the one that crosses the top and bottom, and the word says that going a, then b, then a backwards, then b backwards brings a walker back to the start with nothing left over — which is precisely the statement that the two loops commute.

The Möbius band, where the count is not enough #

The Möbius band has the same classification of loops as an ordinary ring: one integer, counting how many times a loop runs round the band. So the two are indistinguishable by this invariant, and they are certainly not the same surface — one has one side and one edge and the other has two of each.

This is the standard warning about any invariant, and it applies here as much as it does to tricolourability for knots or the Euler characteristic for surfaces. An invariant that agrees on two objects has said nothing about whether they are the same; only disagreement is evidence. What separates the band from the ring is orientability, which is a different question asked with a different tool.

The moral is not that the loop count is weak. It is that a single number extracted from a complicated object is bound to lose information, and knowing which information it loses is part of knowing the invariant.

What it costs #

Computing the class of a drawn loop is cheap: walk it, add angles, divide by a full turn. The figures here do it with several hundred samples and assert, while walking, that no single step turns more than a quarter of a turn — which is the condition under which the accumulated total can be trusted to have followed the curve rather than jumped across it.

Computing the fundamental group of a space presented some other way is not cheap at all. For a surface built by gluing a polygon the group can be written down from the gluing word; for a space presented as a system of equations, or as a knot’s complement, the group can be computed but comparing two such groups is a different matter. Deciding whether two finitely presented groups are the same is not merely difficult but undecidable — there is no procedure at all — so an invariant that is easy to write down can be impossible to compare.

That is a peculiar place for a subject to end up: the classification is complete, the objects are computable, and the comparison is provably beyond any algorithm.

There is a cheaper invariant that gives up exactly enough to be comparable, and it is worth naming here because it is what most of the subject actually uses. Force the group to be commutative — declare that the order of two loops never matters — and what is left is a list of whole numbers, one per independent hole. That is the first homology group, it is computable by linear algebra over the integers, and two of them can be compared by inspection. What it loses is precisely the information the two-holed surface’s non-commuting loops carry, and the trade is made deliberately every time.

What the picture cannot show #

Every loop drawn here is a smooth curve made of a few hundred straight steps, and the theory is about continuous loops, which may be far worse behaved — a loop can be nowhere differentiable, or fill a region entirely. The classification applies to those too, and no figure can suggest what they look like.

Nor can the figures show a deformation that fails for a subtle reason. The straight-line shrink fails visibly, by crossing the hole, and that visibility is exactly what makes it a weak piece of evidence. A deformation that stayed inside the ring for two hundred stages and left it at the two hundred and first would look correct at every drawn frame.

And there is no picture of the group. The claim that the classes form a copy of the integers is a statement about all loops at once, and every figure here draws three of them.

The ladder from here #

Rungs above: the fundamental group defined properly, with base points and the reason they can usually be ignored. Covering spaces, and the dictionary between subgroups and covers. Van Kampen’s theorem, which computes the group of a space glued from pieces whose groups are known. The fundamental group of a knot’s complement, which is the invariant that finally separates knots that colourings cannot. Higher homotopy groups, where the loops become spheres and the arithmetic becomes commutative again for a reason nobody finds obvious. Homology, which throws away enough of the structure to be computable and keeps enough to be useful. And the Poincaré conjecture, which asks whether a three-dimensional space all of whose loops shrink must be a sphere, and which took a century to settle.

The shape of the idea #

The manoeuvre this essay is built on gets used, in one form or another, in every part of the subject: to say something about an object that resists description, attach to it a simpler object that can be computed, and prove that the attachment does not change under the deformations being ignored.

Here the simpler object is a whole number, or a pair of them, or a group. In Euler’s formula it is an alternating sum of counts. For knots it is a colouring rule. In each case the invariant is easy to compute and the theorem is that it survives, and in each case the survival is what makes the computation mean anything at all.

A hole, in the end, is not a thing. It is a fact about loops.