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mathematics, in pictures
Concept

Homotopy

A continuous family of maps interpolating between two of them, and the relation of being joined by one. Objects related this way are indistinguishable to any invariant defined by continuity alone, which is most of what topology computes.

Named by 6 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 The same loop, seen from two places. An annulus with two marked points and a loop based at the first. Two paths join the points, differing by a full turn round the hole, and each carries the loop to a loop based at the second point.

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.

topology · Homotopy 3 sheets, and the subgroup they name. A circle with its 3-sheeted cover drawn as a spiral above it, beside a table of the winding classes and whether each lifts to a closed loop. The ones that do are exactly the multiples of 3.

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.

topology · Homotopy Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

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.

topology · Homotopy Sliding one square past another. 4 stages of a slide in which two labelled squares inside a larger one exchange positions without ever overlapping. The larger square's boundary is the base point throughout, and the exchange is what makes the composition commutative.

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.

topology · Homotopy

Named alongside it

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

Fundamental groupWinding numberCovering spaceGroupBase pointLiftingLoopCommutativityConjugacyContinuityDeck transformationDeformation

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