What Is Topology? The Geometry That Ignores Distance

Imagine a shape made of infinitely flexible rubber. You may stretch it, compress it, bend it and twist it as much as you like. You may not tear it, and you may not glue parts of it together. Two shapes that can be turned into one another by such deformations are considered the same, and the properties that survive are the topological ones. A square and a circle are the same, since one can be pushed into the other without tearing. A coffee cup and a doughnut are the same because each is a lump with exactly one hole, and the cup's hole is its handle. A doughnut and a sphere are not, because making a hole requires tearing, and the number of holes, called the genus, is the simplest topological invariant, a quantity that cannot change under the permitted operations and so distinguishes shapes that cannot be deformed into each other.

The field's origin is usually dated to a puzzle Euler solved in 1736. The Prussian city of Konigsberg sat on two banks and two islands linked by seven bridges, and the question was whether a walk existed crossing each bridge exactly once. Euler's insight was that the distances, the shapes of the islands and the lengths of the bridges were all irrelevant, and that the problem depended only on which land masses connected to which. Reducing it to points and lines, he showed that such a walk requires every point to have an even number of connections except possibly the two endpoints, that Konigsberg had four points with odd counts, and that no route therefore existed. The argument founded graph theory and demonstrated the general move that defines the subject: strip away everything metric and ask what is left.

A few examples carry most of the intuition and are worth knowing by name:

• •The Mobius strip, a band given a half twist before its ends are joined, which has one side and one edge, so a line drawn along it returns to the start on what was the other face
• •The Klein bottle, the same idea one dimension up, a closed surface with no inside or outside, which cannot exist in three dimensions without passing through itself
• •The torus, the doughnut, on which a shape can wind around two independent directions, which is why it classifies differently from a sphere
• •Knots, treated as closed loops in space, which topology can prove genuinely distinct from an unknotted circle, and which are catalogued by invariants such as polynomials computed from a diagram
• •The Euler characteristic, a single number computed from any polyhedron as vertices minus edges plus faces, which comes to 2 for anything sphere-like and 0 for anything torus-like regardless of the shape's details

Several results have physical statements that sound impossible and are provable. The hairy ball theorem says that a continuous vector field on a sphere must vanish somewhere, which means you cannot comb a hairy ball flat without a cowlick, and which implies that at any moment there is a point on Earth with no horizontal wind, so there is always at least one cyclone or calm. The Brouwer fixed point theorem says that any continuous map of a disc to itself leaves at least one point where it started, so if you stir a cup of coffee and let it settle, some point of the liquid is exactly where it began, and a map of a country laid flat on the ground of that country has a point lying directly over the place it represents. The Borsuk-Ulam theorem implies that at any moment there are two points directly opposite each other on the Earth with the same temperature and the same air pressure. None of these depend on measurements; they follow from continuity and shape alone.

It stopped being purely abstract some decades ago. In physics, topological invariants explain why the quantum Hall effect produces conductance in exact integer steps that are insensitive to the impurities in the sample, and topological insulators, materials that conduct only on their surfaces for reasons of shape rather than chemistry, are an active field that took the Nobel Prize in 2016. Quantum computing proposals use braided excitations whose topological character protects information from local noise. In data analysis, persistent homology tracks how holes in a cloud of data points appear and vanish as a scale parameter grows, giving a summary of shape that is robust to noise and is used on everything from brain imaging to materials. Biology uses knot theory on DNA, which becomes tangled during replication and is untangled by enzymes whose action is described topologically. And robotics represents the space of a machine's possible configurations as a surface whose holes correspond to obstacles it cannot pass through.

Topology keeps the properties of a shape that survive stretching, bending and twisting while discarding length, angle and straightness, so a cup and a doughnut are identical and a doughnut and a sphere are not, distinguished by the number of holes. Euler's solution to the Konigsberg bridges in 1736 founded the approach by reducing a problem to which things connect to which. Its theorems have concrete consequences, including a point on Earth with no wind at any moment, and its invariants now underpin work in condensed matter physics, data analysis and DNA biology.