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Concept

Fundamental group

The loops in a space based at one point, counted up to deformation, composed by running one after another. It separates spaces that no count of pieces can separate, and for a ring it is the whole numbers under addition.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

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.

topology · homotopy The line spiralling over the circle. A circle with a helix drawn above it: the helix is the real line, and the map that sends each of its points straight down onto the circle covers the circle once per turn. Above one marked point sits a column of points, one per turn.

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.

topology · covering spaces A wedge of 2 circles. Several circles all passing through one common point, each labelled with a generator, so that a loop is a word in those letters.

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.

topology · covering spaces the Borromean rings, with every crossing signed. A diagram of the Borromean rings with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

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.

topology · linking number

Named alongside it

The objects these essays reach for when they reach for this one.

Covering spaceLoopWinding numberFree groupHomotopyBorromean ringsBrunnian linkCommutatorContinuityDeck transformationDeformationGenerator

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