Topology — page 2

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 ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge.

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.

A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on.

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.

Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge.

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 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.

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.

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.

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.

A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line.

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.

A turn of the sphere, seen from the plane. A square grid in the plane and its image under the map obtained by lifting to the sphere, rotating by 62° about a tilted axis, and coming back down. The lines become arcs of circles and the crossings stay at right angles.

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.

Two charts on one sphere, meeting by the reciprocal. A sphere with its two polar caps marked, each the part missed by one of the two stereographic charts, and the band where both charts are defined shaded between them.

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.

5 circles filling a three-sphere, every pair linked once. Several closed curves in space, nested on tori of different sizes, each pair passing through the other exactly once and none of them touching.

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.

3 symmetries over 3 sheets: a regular covering. A 3-sheeted covering of a wedge of 2 circles, with the permutations of its sheets that commute with every generator. There are 3, against 3 sheets.

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.

3 sheets, 8 of 26 words coming back. A table of reduced words in two generators with the sheet each sends the base sheet to. The words returning to it are the covering's subgroup, and the 3 sheets are its cosets.

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.

6 vertices folded to 4, and a graph that decides. The graph built from 3 generator words, folded until no vertex has two edges of one label leaving it. Reading a word from the base vertex decides membership, and 6 words are tested.

Folding a graph until it decides

A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.

3 sheets over a surface of genus 2: a surface of genus 4. A 3-sheeted covering of the closed surface of genus 2, drawn as 3 copies of its 8-sided face with each side coloured by its generator and numbered with the sheet it glues to. The Euler characteristic −6 is 3 times −2, and the cover has genus 4.

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.

2 sheets branched over 4 points of a sphere: a surface of genus 1. A 2-sheeted branched covering of the sphere with 4 branch points, drawn as 2 rows of sheets over a centre and the branch points, with the sheets joined where each point's permutation cycles them. Counting cells gives Euler characteristic 0, matching the Riemann–Hurwitz formula, and genus 1.

What a branch point subtracts

Let the sheets of a covering meet at a few points and the count stops multiplying — but it fails by an amount that can be read off each point's permutation. Cut the sphere into a star, lift the cells, and the Riemann–Hurwitz formula falls out of a subtraction. The same count then turns out to be necessary and not sufficient.

Two opposite points on a globe with the same temperature and the same pressure. A world map in longitude and latitude with one curve where each point's temperature matches its opposite point's and another where the pressures match, crossing at a pair of opposite points that are marked.

Two opposite points that agree twice

At any moment there are two points on opposite sides of the Earth with the same temperature and the same pressure. On a seeded globe they sit at 11.9°N 44.6°E and 11.9°S 135.4°W. The reason is the circle argument that halved two shapes, run one dimension up: the differences between opposite readings, walked round the equator, wind round zero an odd number of times — and an odd number cannot be zero.

A necklace of 4 orange, 4 blue, 2 green beads, shared fairly with 3 cuts. A row of coloured beads cut at marked places into pieces, each piece labelled with the thief who receives it, so that both thieves get half of every colour.

As many cuts as colours

Two thieves steal a necklace and want half of every colour of bead each. However the beads are strung, they never need more cuts than there are colours — three cuts for three colours, four for four — and sometimes they need every one. The guarantee is the Borsuk–Ulam theorem again, with a point on a sphere read as a way of cutting the necklace, and every necklace of several small kinds has been checked against it.

Reducing abcabc to a standard form. The gluing word abcabc rewritten step by step into one of the classification's standard forms, with the move used and the two invariants recomputed at each step.

Every word driven to a normal form

The classification is usually met as a statement: two numbers name the surface. The proof is a procedure — a short list of cut-and-reglue moves that drive any gluing word to one of the standard forms, with a measure that never rises to say why the procedure stops.

4 surfaces with an edge, and the three numbers they need. Polygons whose gluing words leave some edges unpaired, each with its Euler characteristic, its sidedness and the number of boundary circles the unpaired edges form.

The third number a surface needs

Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.

Six lists of cycle shapes, and how many coverings each has. A table of lists of cycle shapes over a sphere, each with the Euler characteristic the Riemann–Hurwitz count gives, the number of lists of permutations with that product, and the number of those that connect all the sheets.

A count that can say zero

The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.

A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses it.

A map that offers a choice

Brouwer's theorem needs a function, and the object it was most wanted for is not one — a best reply is a whole set whenever a chooser is indifferent. Allow a point to be sent to a set and the fixed point survives, provided the sets are convex, and the convexity is the entire hypothesis.

The Alexander matrix of the trefoil. The trefoil with its 3 arcs numbered and its 3 crossings lettered, beside the 3 by 3 matrix they give. A minor of the matrix is the Alexander polynomial t − 1 + t⁻¹, whose value at −1 is the determinant 3.

A polynomial behind the colourings

The figure-eight knot and the cinquefoil both have determinant five, so they admit exactly the same colourings, and every counting argument treats them as one. Put a variable where the colouring rule has a two and the determinant becomes a polynomial — and the two knots come apart.