Fundamental group
Named by 16 essays across one field — each of them below, with the objects they name alongside it.
A loop that cannot be pulled tight
A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.
The same loop, unrolled
Spread a circle out into a line spiralling above it, and a loop that closes downstairs becomes a path that does not — so a question about which loops can be shrunk becomes a question about where a path ends, which is easy.
The subgroup that is freer than the group
A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.
Linked, and no two of them are
Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.
Two sheets over a one-sided surface
Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.
The group a space has at a point
The loops of a space form a group once a starting point is fixed, and the fixing looks like an arbitrary choice that ought to be removable. It is removable, but only up to conjugation, and the residue is exactly what makes a non-commutative fundamental group harder to state than to compute.
Every cover is a subgroup
A space can be unrolled, and the ways of unrolling it are not arbitrary. They correspond exactly to the subgroups of its fundamental group — index equals sheets, normality equals symmetry — so a question about a group becomes a question about a picture and back again.
Cutting a space to find its group
A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.
Why the second group commutes
Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.
The symmetries a cover has of its own
A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.
A covering is a permutation
Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.
Covering a surface multiplies its count
A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.
The loops on a torus that never cross themselves
Every loop on a torus is classified by two whole numbers: how often it goes round one way and how often the other. Some classes can be drawn without the loop ever crossing itself and some cannot, and the rule is the oldest in arithmetic — the two numbers must have no common factor. The same two numbers say how often any two loops must meet.
What homology forgets about a loop
Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.
An octagon only the hyperbolic plane can hold
Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.
How a random path winds round a point
A path in the plane with one point removed has a class in the punctured plane's fundamental group: how many times it has wound round the point. For a random path the class grows like the logarithm of time, and its spread has a law. Brownian motion follows Spitzer's Cauchy law, so heavy-tailed that the mean winding does not exist; a walk on a grid follows the hyperbolic secant law instead, with tails that fall exponentially. The difference is made entirely by close passes, which a grid forbids. With two points removed the class becomes a word, and some loops wind round neither point and still cannot be shrunk.
Named alongside it
The objects these essays reach for when they reach for this one.
Covering spaceHomotopyFree groupWinding numberSubgroupGenusIndexDeck transformationEuler characteristicGroupLiftingLoop