Every essay — page 8
Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Ladders Concepts Search
Topology
What survives bending, and what does not.
Seven hundred and twenty degrees of gap
Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.
The solid where the answer is not two
A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.
Where the fixed point escapes
The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.
Two loops and one number
Give each crossing between two closed curves a sign, add them up, halve — and the answer does not depend on how the curves were drawn, how they are pushed about, or which way the picture was projected.
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.
Zero can mean two different things
The linking number counts how often one loop pierces a surface the other one bounds. Two punctures of opposite sign add to nothing, and a loop that never goes through adds to nothing as well — so the answer zero is two pictures wearing one number.
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.
The bottle that needs a fourth dimension
Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.
A disc sewn to a Möbius band
The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.
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.
Orientation is a sign
Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.
A curve that has area
The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.
Every loop is a circle in disguise
Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.
Two pieces, in every dimension
A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.
A ball whose outside is not one
Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.
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.
Angles survive and areas do not
Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.
The sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
One chart is never enough
Stereographic projection matches the sphere minus a point with the whole plane, and the missing point is not a blemish to be tidied away. It is a theorem — no single flat picture covers a sphere — and the repair is two pictures with a rule for passing between them.
The circles that fill a three-sphere
A three-sphere is filled by circles — one through every point, no two meeting, every two linked exactly once. Stereographic projection is the only way anybody sees it, and the projected picture is a nest of circles on tori whose linking can be counted off the drawing.