Cutting a space to find its group

Every fundamental group on this ladder so far has been read off a picture. The ring’s is $\mathbb{Z}$ because loops can be counted; the figure-eight’s is free on two generators because words in the two loops cannot be shortened. Those readings are correct and they do not scale: nobody looks at a surface of genus three and sees its group.

What scales is a recipe for building the group of a space out of the groups of its pieces. Van Kampen’s theorem is that recipe, and it is close to the only general tool there is.

The statement #

Let $X = U \cup V$ with $U$, $V$ open, both containing the base point, and with $U \cap V$ path-connected. Then $\pi_1(X)$ is the free product $\pi_1(U) * \pi_1(V)$ modulo the relations saying that each loop of the intersection, read into $U$, equals the same loop read into $V$.

In the language of presentations: take all the generators of both pieces, all the relations of both pieces, and one extra relation for each generator of the intersection.

Two informal readings are worth having.

Everything both pieces can do, the whole can do. A loop in $U$ is a loop in $X$; likewise for $V$; and a loop in $X$ can be cut at the moments it crosses between the pieces into a product of loops in one or the other. So the generators of $X$’s group are the generators of the two pieces’ groups. That is the surjectivity half.

And the only new relations come from the overlap. A loop that is contractible in $X$ but not in either piece must be contractible by a homotopy sweeping across the overlap, and the overlap’s loops are what record it. That is the harder half.

Why the overlap must be connected #

The hypothesis is not decoration and the standard counterexample is one line.

Cover the circle with two arcs. Each arc is contractible, so both groups are trivial, and the recipe would give the trivial group. The circle’s group is $\mathbb{Z}$.

What has gone wrong is that two arcs covering a circle meet in two components, not one, and a loop of the circle crosses from one piece to the other and back through different components. The theorem’s proof cuts a loop into pieces and needs a canonical way of joining consecutive pieces at the base point, which is a path in the overlap — and there is none between two components.

A hypothesis whose failure is exhibited by the simplest example in the subject is a hypothesis worth remembering. The version of the theorem that does handle several components exists — it is the groupoid version, due to Philip Higgins and popularised by Ronald Brown — and it computes the circle correctly. Its statement is about a category rather than a group, which is why the group version, with its awkward hypothesis, is still the one everybody quotes.

The three cases in the figure #

The sphere. Cover it with two overlapping caps, each a disc, each with trivial group. The overlap is an annulus, connected. So the group is the free product of two trivial groups modulo some relations: trivial. The sphere is simply connected, and the calculation used nothing about spheres.

The wedge of two circles. Take $U$ to be one circle plus a bit of the other, $V$ the reverse. Each deformation-retracts to a circle, so each group is $\mathbb{Z}$. The overlap is a small contractible neighbourhood of the joining point, so it contributes no relations. The result is the free product $\mathbb{Z} * \mathbb{Z}$ — the free group on two generators.

The torus. Take the square with edges glued. Let $U$ be everything but the centre point; it deformation-retracts onto the boundary, which is the wedge of two circles, so its group is free on $a$ and $b$. Let $V$ be a small disc round the centre; trivial group. The overlap is an annulus, whose group is $\mathbb{Z}$, generated by a loop that goes once round the disc — and that loop, pushed into $U$, is the boundary word $aba^{-1}b^{-1}$.

So the relation is $aba^{-1}b^{-1} = 1$, and the group is the free group on two generators with the two made to commute: $\mathbb{Z}^2$.

The shape of the recipe, in algebra #

The construction the theorem produces has a name and it is worth knowing, because it is what the three cases above are three instances of.

Given groups $A$ and $B$ and a group $C$ with maps into both, the free product with amalgamation $A *_C B$ is the group generated by $A$ and $B$ with the images of each element of $C$ identified. Van Kampen says $\pi_1(X) = \pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V)$.

The three cases sort themselves by what $C$ is.

• $C$ trivial: no identifications, so the answer is the plain free product. That is the wedge.
• $A$ and $B$ trivial: nothing to identify, so the answer is trivial whatever $C$ is. That is the sphere, and it is why the sphere calculation used no property of spheres.
• $B$ trivial and $C$ not: every element of $C$ is identified with the identity, so the answer is $A$ modulo the normal closure of $C$’s image. That is the torus, and it is the case that produces relations rather than merely gluing generators together.

The third case is the one that does real work, and it is worth naming its shape: attaching a disc to a space kills the loop the disc is attached along. Every calculation of a surface group, every Wirtinger presentation, and the construction that realises an arbitrary presentation are all that one sentence applied repeatedly.

Every surface at once #

The torus computation generalises with no new idea, which is the sign that the tool is the right one.

Every closed orientable surface is a polygon with its edges glued in pairs — a sphere with handles, presented as a $4g$-gon whose boundary word is $a_1b_1a_1^{-1}b_1^{-1} \cdots a_gb_ga_g^{-1}b_g^{-1}$. Running the torus argument on it gives:

$$\pi_1(\Sigma_g) = \langle a_1, b_1, \ldots, a_g, b_g \mid \textstyle\prod_i [a_i, b_i] = 1 \rangle,$$

a group on $2g$ generators with one relation. For $g = 0$ that is the trivial group; for $g = 1$ it is $\mathbb{Z}^2$; for $g \geq 2$ it is a non-commutative group that is not free.

One theorem, one calculation, and every surface’s group. The calculation is the same three lines each time, with $g$ left as a letter — which is what distinguishes a machine from a method. Comparing that against the alternative — finding each group by inspection — is the argument for the machinery. And the presentations are immediately useful: abelianising gives $\mathbb{Z}^{2g}$, which distinguishes the surfaces from each other and is the first homology group, so the classification of surfaces falls out as a corollary.

The non-orientable surfaces come out of the same machine with a different boundary word. The projective plane is a $2$-gon with boundary $a^2$, so its group is $\langle a \mid a^2 = 1\rangle$, of order two — the smallest non-trivial fundamental group on this site, and the reason a loop on the projective plane returns reversed after going round twice. The Klein bottle is a square with boundary $abab^{-1}$, giving a non-commutative group with one relation.

Comparing the surfaces by their presentations is where the classification becomes a theorem rather than a survey. The groups are $\{1\}$, $\mathbb{Z}^2$, and then a sequence of one-relator groups that grow, and no two of them are isomorphic — which is proved by abelianising, where the answers are $\mathbb{Z}^{2g}$ and a rank is a rank. So a hard-looking geometric classification is settled by computing a rank, and the computation is available because the presentations are.

What a presentation does not give #

Having the group as generators and relations is not the same as understanding it, and the gap is larger than it looks.

Deciding whether two words are the same element can be impossible. Not hard: impossible. There is no algorithm that takes a finite presentation and two words and decides equality, a theorem of Novikov and Boone from the 1950s. Surface groups are much better behaved than that — their word problem is solvable, and quickly — but nothing about the method guarantees it.

And every finitely presented group arises. Given any finite presentation, a two-complex can be built with that group: a wedge of circles for the generators, with a disc glued along each relation, and van Kampen computes the group of the result to be exactly the presented one. So the fundamental groups of reasonable spaces are precisely the finitely presented groups, and every undecidability in group theory is transmitted into topology.

That is worth stating as a limit on the whole ladder. The fundamental group is a complete answer to a question and the question it answers is at least as hard as all of combinatorial group theory.

Knots, where the theorem earns its reputation #

The complement of a knot in space has a fundamental group, and it is a much stronger invariant than any colouring or polynomial: for prime knots it determines the knot up to mirror image.

Computing it is van Kampen applied along the strands of a diagram. Cutting space along a plane just above the knot’s projection and reassembling gives the Wirtinger presentation: one generator per arc of the diagram, one relation per crossing saying how the over-strand conjugates the two under-strands. A diagram with $n$ crossings gives $n$ generators and $n$ relations, of which one is redundant.

That is a mechanical procedure with a picture as its input, which is the best kind of computation. The unknot gives $\mathbb{Z}$; the trefoil gives $\langle a, b \mid aba = bab \rangle$, which is the braid group on three strands and is not $\mathbb{Z}$, which proves the trefoil is knotted. Proving those two groups differ is itself a small exercise — map the trefoil group onto the symmetric group on three letters by sending $a$ and $b$ to two different transpositions, check the relation holds, and observe that $\mathbb{Z}$ has no such quotient. That is the same trick as colouring a knot with three colours, arriving as a homomorphism rather than as a colouring.

The catch is the one above: two Wirtinger presentations of the same group can look completely different, and deciding whether they present the same group is where the difficulty relocates rather than disappearing.

Who proved it, and how long it took to be believed #

The theorem carries two names and neither of them proved the statement now quoted.

Egbert van Kampen published a version in 1933; Herbert Seifert had published a related one in 1931. Both were working on knot and surface groups, and both statements were narrower than the modern one — Seifert’s assumed the pieces were complexes, van Kampen’s had a different set of conditions. The general open-cover statement is a later synthesis, and it is what the name attaches to.

What the two of them supplied that mattered was the observation that the overlap is the whole difficulty. Before it, computing a fundamental group meant an ad hoc argument per space, and every published computation was its own paper. After it, the computation is mechanical given a decomposition, and finding a good decomposition became the skill.

The groupoid version arrived much later — Higgins in the sixties, Brown’s textbook in the seventies — and has never displaced the group version in general use, despite being cleaner and having no awkward hypothesis. That is worth noting as a fact about how mathematics is transmitted rather than about the mathematics: a statement about groups is quotable by people who know what a group is, and a statement about groupoids is not, and quotability wins.

What the pictures cannot show #

The overlap is the whole content and it is the hardest thing to draw. Every picture here shows two pieces; the theorem is about the loops of their intersection, which is a thin region between them, and thin regions do not draw well. The figures label the pieces and leave the reader to imagine the annulus or the point where they meet.

Deformation retraction is asserted, never shown. Each calculation begins “this piece retracts onto that”, and the retraction is a continuous family of maps. A figure can show the piece and the thing it retracts to; the family is prose.

Nothing distinguishes a redundant relation from a necessary one. The Wirtinger presentation of a diagram with $n$ crossings lists $n$ relations, one of which follows from the others, and no drawing says which. Deciding whether a relation in a presentation is redundant is another problem with no algorithm, so the redundancy above is known by a proof rather than by inspection.

And a presentation is a picture of nothing. The output of every calculation on this rung is a list of symbols. That the list determines a group, and that the group is a property of a space, is the content — and there is nothing to draw of a group beyond drawings of some of its elements, which is what every figure here does.

Where the ladder goes next #

The last rung on this pass replaces loops with spheres and asks the same questions, where the answer is commutative for an argument that has nothing to do with any space.

Named here as a debt: the groupoid version of the theorem, which drops the connected-overlap hypothesis and computes the circle correctly, and which is the honest statement of what is going on. This rung uses the group version and its awkward hypothesis, as almost everybody does.

Also unwritten: the Wirtinger presentation given a figure of its own, with the conjugation at each crossing drawn rather than described. That is a rung on the knot ladder rather than this one, and the three-colouring invariant is the shadow of it that this collection currently carries.

Sideways, the presentations produced here are the input to a Cayley graph, the surface case is the classification of surfaces arriving from the algebraic side, and the covers of the last rung are what a subgroup of one of these presentations looks like when drawn as a space.

Choosing the cut #

The theorem is mechanical once a decomposition is chosen, and choosing one is the whole skill. Three habits cover most cases.

Cut where the space is already described. A polygon with edge identifications comes with its cut: the interior is one piece and a neighbourhood of the boundary is the other. Every surface calculation above is that habit applied without thinking.

Cut so that one piece is trivial. The torus calculation puts a disc against a wedge of circles, which is the third case above and produces a relation. Whenever a space is built by attaching a disc to something, the group is the something’s group with one relation added, and no further thought is needed.

And cut so that the overlap is as simple as possible, since the overlap’s group is what has to be understood. An overlap that deformation-retracts to a point contributes nothing; one that retracts to a circle contributes one relation; anything more complicated is usually a sign that a different cut is available.

A bad cut is not wrong, it is unusable. Splitting a genus-two surface into two arbitrary open halves gives a correct instance of the theorem whose three ingredient groups are all harder to compute than the answer. That is the ordinary failure mode of a general tool: the theorem holds for every decomposition and is informative for very few.

What is worth carrying away #

A theorem that computes an invariant of a whole from the invariants of its parts is worth more than any number of individual computations, and its hypotheses are where all the content is.

Van Kampen’s is one relation per generator of the overlap and nothing else, which is as simple as such a statement could be. Its price is that the overlap must be connected, and the price is real — the circle, the simplest space with a non-trivial group, is exactly the case it cannot handle.

The habit worth taking is to read a gluing theorem’s failure case first. The counterexample says what the hypothesis is protecting, and here it says that the theorem is really about a groupoid and has been squeezed into a statement about a group by a condition that makes the squeeze legal.