Why the second group commutes

The fundamental group is built from maps of a circle into a space. Nothing in the construction insists on a circle. Replace it by a sphere and the same definitions go through: maps of the $n$-sphere sending a chosen point to the base point, considered up to homotopy, with composition defined by putting two of them side by side.

The result is $\pi_n(X, x_0)$, the $n$-th homotopy group. And a fact appears at $n = 2$ that has no counterpart at $n = 1$: every one of these groups is commutative.

Two compositions, not one #

The cleanest way to see the commutativity is to notice that the second homotopy group has two natural products, not one.

An element of $\pi_2$ can be represented by a map of the unit square that sends the whole boundary to the base point. Two such maps can be composed by putting them side by side horizontally and shrinking each to half width — or by stacking them vertically. Call the two operations $\cdot$ and $\circ$.

Both are group operations, both have the same identity — the constant map — and they satisfy a compatibility condition:

$$(a \cdot b) \circ (c \cdot d) = (a \circ c) \cdot (b \circ d).$$

The reason is that both sides are the same picture: a two-by-two arrangement of $a$, $b$, $c$ and $d$ in the four quarters of the square, cut into halves either way. Reading it by rows and then combining, or by columns and then combining, gives the two sides.

The argument, which is about nothing #

That is all that is needed, and the conclusion is much stronger than commutativity.

Theorem. A set with two unital, associative operations sharing an identity and satisfying the interchange law above has $\cdot = \circ$, and the common operation is commutative.

The proof is two lines. Write $e$ for the shared identity. Then

$$a \cdot b = (a \circ e) \cdot (e \circ b) = (a \cdot e) \circ (e \cdot b) = a \circ b,$$

so the operations agree. And

$$a \cdot b = (e \circ a) \cdot (b \circ e) = (e \cdot b) \circ (a \cdot e) = b \circ a = b \cdot a,$$

so it is commutative.

Nothing in that argument mentions spheres, spaces, homotopy or topology. It is the Eckmann–Hilton argument, and it applies wherever two compatible operations share a unit: to the multiplication of a monoid object, to the addition of a ring’s underlying group against its multiplication where they interchange, and to the composition of natural transformations in two directions.

So the commutativity of $\pi_2$ is not a topological fact. It is an algebraic fact about a situation topology happens to produce.

Reading the interchange law off the picture #

The equation above is easier to trust after being read off a drawing once, because it is the only step of the argument with any content.

Draw a square and cut it into four quarters. Put $a$ top-left, $b$ top-right, $c$ bottom-left, $d$ bottom-right.

Read it as two rows. The top row is $a$ beside $b$, which is $a \cdot b$; the bottom is $c \cdot d$; and the two rows stacked give $(a \cdot b) \circ (c \cdot d)$.

Read the same picture as two columns. The left column is $a$ above $c$, which is $a \circ c$; the right is $b \circ d$; and the two columns side by side give $(a \circ c) \cdot (b \circ d)$.

One arrangement, two readings, and the two readings are the two sides of the law. Nothing has been proved by rearranging anything; the equality holds because both expressions describe the same map of the same square, at the level of maps and not merely up to homotopy.

That is worth emphasising because the rest of the argument is formal. The single geometric input to the commutativity of every higher homotopy group is a square can be cut into four quarters two ways, and everything after it is symbol-pushing.

Why the first group escapes #

The natural question is why the same argument does not apply to loops, and the answer is geometric and simple.

An element of $\pi_1$ is a map of an interval with both endpoints at the base point. There is only one way to put two intervals side by side, because an interval has one dimension and a dimension supports one direction of concatenation. So there is only one product, no interchange law, and no argument.

A room to slide sideways is exactly what is missing. In the square, the region around a small square can be filled with the constant map, so a small square can be moved anywhere inside a bigger one without meeting the other. In the interval, moving one sub-interval past another means passing through it, and there is nowhere to go round.

That is the whole difference, and it is a good example of a proof whose content is a dimension count. Two of anything can be moved past each other in the plane; in a line they cannot.

There is a second way of saying the same thing that is worth having, because it explains where the pattern stops. The set of loops in a space is itself a space, and the fundamental group of that space is the second homotopy group of the original. So $\pi_2$ is a fundamental group too — of a different space — and it is commutative because it is the fundamental group of something that already carries a multiplication.

That is a general phenomenon: the fundamental group of a space with a compatible multiplication is always abelian, and the proof is the same two lines with the multiplication playing the role of the second operation. Topological groups have abelian fundamental groups for exactly this reason; the circle’s is $\mathbb{Z}$, the torus’s is $\mathbb{Z}^2$, and no topological group has a free non-abelian fundamental group. The wedge of two circles, whose group is free of rank two, therefore cannot be given a continuous multiplication with a unit — which is a genuinely topological conclusion, obtained from an argument with no topology in it.

What the higher groups are worth #

They are a real invariant and they cost a great deal.

They are computable in some cases and famously not in others. $\pi_n$ of a sphere of the same dimension is $\mathbb{Z}$, given by the degree, and the argument is the same lifting-and-counting as the winding number. Below the dimension they vanish. Above the dimension the answers are wild: $\pi_3(S^2) = \mathbb{Z}$, generated by the Hopf map, which was the first indication that a sphere of one dimension can wrap non-trivially round a sphere of a lower dimension — a fact nobody expected before Hopf found it in 1931.

The wildness is worth one concrete example, because the phrase does not convey it. $\pi_n(S^2)$ is $\mathbb{Z}$ at $n=2$ and at $n=3$, then $\mathbb{Z}/2$, then $\mathbb{Z}/2$, then $\mathbb{Z}/12$, then $\mathbb{Z}/2$, then $\mathbb{Z}/2$, then $\mathbb{Z}/3$, then $\mathbb{Z}/15$ — a sequence with no visible pattern, computed one entry at a time by increasingly heavy machinery, and not known beyond a few dozen terms. A two-dimensional surface, the simplest interesting space there is, has a sequence of invariants nobody can predict.

And they are not known in general. The homotopy groups of spheres are the standard open problem of the subject, and they have been computed by enormous effort into ranges that are small compared with what is wanted. There is no analogue of van Kampen: no general theorem computes $\pi_n$ of a union from the pieces, which is the single biggest practical difference from $\pi_1$.

That last figure is the sharpest statement of the asymmetry. A covering map induces an isomorphism on all homotopy groups above the first, so the whole apparatus of the previous rung — the subgroup dictionary, the universal cover, the Nielsen–Schreier counting — is exactly the machinery for the group it cannot help with above dimension one.

Where the difficulty actually lives #

It is worth being precise about why the higher groups are hard, since “no analogue of van Kampen” is a statement about a missing tool rather than about the objects.

The reason there is no gluing theorem is that a map of a sphere into a union cannot in general be cut into pieces landing in one part or the other. A loop can: subdivide the interval finely enough and each small piece lands in one open set, then reassemble. A map of a square subdivides too, but the pieces are squares whose boundaries do not go to the base point, so they are not elements of anything, and there is no way to compose them.

Subdivision works in dimension one because the boundary of an interval is two points and two points are easy to move. In dimension two the boundary of a piece is a loop, and moving a loop to the base point is exactly the problem the first homotopy group is about. So the higher-dimensional gluing problem contains the lower-dimensional homotopy theory inside it, at every stage.

What exists instead of a gluing theorem is a fibration theory: a map with a nice local structure gives a long exact sequence relating the homotopy groups of the total space, the base and the fibre. That is a genuinely powerful tool — the Hopf map’s fibration is what computes $\pi_3(S^2)$ — and it is a tool for propagating known answers rather than for computing from a decomposition.

What is gained anyway #

Three things make the higher groups worth defining despite the cost.

They detect what $\pi_1$ cannot. The sphere is simply connected and is not a point, and $\pi_2(S^2) = \mathbb{Z}$ says so immediately. The same distinction separates the sphere from any space built by gluing a disc to a point, which the fundamental group cannot do. Any invariant that vanishes on the sphere is blind to a large part of topology, and $\pi_1$ is such an invariant.

The action of $\pi_1$ on them is extra information. The higher groups are not merely abelian groups; they carry an action of the fundamental group, coming from carrying a sphere round a loop and seeing where it comes back. A space where that action is trivial is called simple, and the distinction is real — it is what makes the base point matter for higher groups even though the groups are abelian, and it is why the base-point discussion of the second rung does not become vacuous here.

And the whole tower is an invariant of a strong kind. Two spaces with the same homotopy groups and the same maps between them are, in a precise sense, indistinguishable by any homotopy-theoretic method — this is Whitehead’s theorem, for spaces built out of cells. That is a statement about what the tower measures rather than about how to compute it, and it is why the subject persists despite the computations being out of reach. It is also the reason the classification of surfaces can be settled by the first group alone: for surfaces the higher groups carry nothing the first one does not already imply.

Who found the argument, and where it now lives #

Beno Eckmann and Peter Hilton published the argument in 1962, by which time the commutativity of the higher homotopy groups had been known for thirty years and proved by direct geometric manipulation — sliding the two spheres past each other, drawn as the figures here draw it.

What Eckmann and Hilton supplied was the observation that the geometry is inessential. Their paper is about group-like structures in categories, and the homotopy result is one corollary among several; the others include the fact that the fundamental group of a topological group is abelian, and the fact that a monoid in the category of monoids is a commutative monoid.

The argument has since become one of the standard tools of higher category theory, where it explains a pattern nobody would otherwise expect: a structure with $n$ compatible compositions becomes increasingly commutative as $n$ grows, and the sequence of possibilities stabilises. That stabilisation — braided at one level, symmetric above it — is the Baez–Dolan periodic table, and its first row is this argument.

The pattern is worth recording independently of the mathematics. A theorem proved geometrically for thirty years turned out to have a two-line proof once somebody asked what the proof actually used, and the two-line version generalised to places the geometric one could not reach. Stripping a proof of its subject is not tidying; it is where the generality is.

What the pictures cannot show #

A map of a square into a space is drawn as a square. Every figure here draws the domain — a square with two smaller squares in it — and none draws the space they map into. What is being composed is a pair of maps, and the picture shows only where they live.

The interchange law is drawn as one square and read twice, which is the one thing a picture does perfectly. That is worth saying because it is the exception on this rung: the geometric content of the whole argument is a single subdivision, and a subdivision is exactly the kind of thing a drawing settles. Everything else here is algebra with a picture beside it rather than a picture doing work.

The slide is a homotopy and the frames are stills. Four or five stages of a continuous family is enough to show the arrangement changing and is not the family. The check the figure makes is that the two squares never overlap in the frames drawn, which is the condition that matters and is verified only at the stages shown.

And nothing here shows a non-trivial element of $\pi_2$. Every figure is about the domain and the algebra. Drawing an actual sphere wrapped non-trivially round another — the Hopf map, say — is a picture this collection makes in a different setting and not here, because the argument on this rung genuinely does not need one.

Where the ladder goes next #

Named here as debts, both raised above and not settled. The action of the fundamental group on the higher groups, which is the structure that makes the base point continue to matter. And the Hopf map with a figure of its own — the first non-trivial element of $\pi_3(S^2)$, whose fibres are circles linked once each — which is the picture that makes the higher groups feel like objects rather than notation.

Beyond those, this ladder’s obvious continuation is homology, which throws away enough of the structure to be computable and keeps enough to be useful, and whose relationship to the homotopy groups is the Hurewicz theorem.

Sideways, the degree of a map of spheres is the winding number one dimension up, the impossibility of combing a sphere is an argument about degree, and the machinery that computes the first group and no other is the covering-space dictionary.

One more consequence of the two-line argument deserves stating, because it runs the other way and is the reason the argument is quoted so often outside topology. Any object that carries two unital operations which interchange has, by the same two lines, only one — so a construction that appears to add structure adds none. That is why a “monoid in the category of monoids” is a commutative monoid rather than something new, why an abelian group object in abelian groups is an abelian group, and why the tower of higher structures stabilises rather than growing forever. The argument is more often used to show that something is not new than to show that something is commutative.

What is worth carrying away #

When a theorem holds for a reason that never mentions its subject, the reason is the more valuable half.

$\pi_2$ is commutative, and the argument uses no property of spheres, spaces or continuity — only that two operations share a unit and interchange. That means the result is not really about topology, and it also means the same conclusion is available anywhere the same situation arises, which turns out to be almost everywhere in modern algebra.

The habit worth taking is to strip a proof of its subject and see what is left. If what remains is still a proof, the theorem was an instance of something more general, and the general statement is usually the one worth remembering.