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Concept

Boundary

The edge of a region — the points lying in neither its inside nor its outside. What a set does at its boundary decides whether it is open or closed, and both fixed-point and extreme-value arguments turn on it.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A Möbius band. A strip joined end to end after a half twist, so it has one side and one edge.

The surface with one side, and what happens when it is cut

A strip of paper, half a twist, and a join. The result has one side and one edge, and cutting it down the middle does not produce two of anything.

topology · Orientability A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

Something always stays put

Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.

topology · Fixed points A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Which side of the line is inside

A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

topology · Jordan curve A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Area by counting dots

Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.

discrete · Pick theorem Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

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.

topology · Fixed points The Klein bottle, drawn where it does not fit. A closed one-sided surface in three dimensions, drawn as a tube with a figure-eight cross-section that turns over once on the way round, with the circle where the drawing passes through itself marked.

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.

topology · Orientability 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.

topology · Surface classification The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

The surface a knot bounds

Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

topology · Knots Every way to cut a genus-2 surface into pairs of pants. 2 thickened graphs, one for each type of pants decomposition of the closed surface of genus 2, each with 2 coloured junctions and 3 cutting circles.

Every surface is sewn from pants

A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.

topology · Surface classification A loop that is a hole, and the same loop filled in. a hollow triangle: 3 points, 3 edges, 0 triangles; holes in each dimension 1, 1, 0; a filled triangle: 3 points, 3 edges, 1 triangles; holes in each dimension 1, 0, 0.

A hole is a cycle that bounds nothing

A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.

algebra · Linear maps

Named alongside it

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

Euler characteristicTopological invariantGluing diagramNon-orientableCounterexampleGenusOrientationBrouwerConnectednessContinuityFixed pointInvariant

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