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Topology

Every surface is a sphere with handles

Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.
15 min read 7 figures The same thing twiceThrowing things away

Worth reading first: The surface with one side, and what happens when it is cut · Every corner pays for itself.

A screen on which a character walking off the right edge reappears on the left, and off the top reappears at the bottom, is not a rectangle. It is a torus, and it has been one since the first such game was written. The rectangle is a set of instructions, and the surface is what the instructions make.

That is the whole method. Take a polygon, pair up its edges, and say for each pair which way round the gluing goes. Everything that follows is a consequence.

The four ways to glue a square's edges in pairsSquares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.abbaabb⁻¹a⁻¹V 3 − E 2 + F 1 = 2two-sideda sphereabababa⁻¹b⁻¹V 1 − E 2 + F 1 = 0two-sideda torusabababab⁻¹V 1 − E 2 + F 1 = 0one-sideda Klein bottleababababV 2 − E 2 + F 1 = 1one-sideda projective planethe corners are coloured by the vertex they become, so the count of colours is the V in V − E + Fthe characteristic and the orientability are computed from the word; the name underneath is looked up from that pair
Fig. 1 Four rules for gluing a square’s four edges in two pairs. The arrows say which direction each edge runs when glued. Under each: the vertices, edges and faces the gluing leaves, and the name those numbers produce.

Four rules, four different surfaces, and the difference between the last two is a single arrow reversed.

Reading the corners

The interesting quantity is not visible in the square; it appears when the gluing is carried out. Consider what happens to the four corners.

The gluing aba⁻¹b⁻¹ makes a torusA polygon whose edges carry the word aba⁻¹b⁻¹, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.ababcorners 1, 2, 3, 4 become one vertexthe 4 corners fall into 1 class, so V − E + F is 1 − 2 + 1 = 0every letter is used once each way round, so a consistent sense of turning survives the gluing
Fig. 2 The torus gluing, drawn large. Each edge is glued to its partner running the same way, and following the identifications round shows that all four corners end up at the same point of the finished surface.

On the torus, all four corners become the same point. Glue the left edge to the right one and the top-left corner meets the top-right; glue the top to the bottom and both of those meet both bottom corners. Four corners, one vertex.

The gluing abab makes a projective planeA polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.ababcorners 1, 3 become one vertexcorners 2, 4 become one vertexthe 4 corners fall into 2 classes, so V − E + F is 2 − 2 + 1 = 1some letter is used twice the same way round, so a sense of turning does not survive and thesurface is one-sided
Fig. 3 The projective plane’s gluing, where both pairs are joined with a flip. The corners fall into two classes rather than one: corners 1 and 3 become one vertex, corners 2 and 4 another.

On the projective plane, they fall into two classes. The figure works this out by pooling: it walks the identifications the word forces and finds which corners are dragged onto which, exactly as colourings were pooled into classes by a group. Nothing is drawn from a picture of the finished surface, which is just as well, because the finished surface does not fit in three dimensions.

The count of classes is V. The count of edges after gluing is the number of distinct letters, because each pair becomes one edge. The count of faces is one, the polygon itself. And V − E + F is the Euler characteristic.

How the pooling is done

The corner classes are not read off a picture, and they could not be — the picture is the square before gluing, in which the four corners are four separate points.

They are found by a rule. Each edge of the polygon runs from one corner to the next, and the word says which direction it is glued in. When two edges carry the same letter, the gluing identifies their starting points with each other and their ending points with each other; and which end counts as the start depends on the direction the arrow points. Applying that to every pair gives a list of forced identifications, and pooling the corners under those identifications gives the classes.

The pooling is the same operation as pooling anything else by an equivalence: repeatedly merge, until nothing is left to merge. It is the operation behind matching two collections one to one and behind every orbit count on this site. What makes it worth naming here is that the answer is not predictable by inspection — the torus gluing and the Klein bottle gluing look almost identical on the page, they differ by one arrow, and they both come out with one vertex, while the projective plane’s word comes out with two.

The number, and the other number

V − E + F = 2, five timesVertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.VEFV − E + Ftetrahedron4642cube81262octahedron61282dodecahedron2030122icosahedron1230202
Fig. 4 The same alternating sum on the five regular solids, where it is 2 every time — because each of them is a sphere cut into pieces, and the characteristic depends on the surface rather than on the cutting.

That alternating sum is the one every corner pays for itself is about, and its virtue is that it does not depend on how the surface was cut up. A cube and a dodecahedron are both spheres carved differently and both give 2. A torus carved into a thousand triangles still gives nothing.

So the characteristic is a property of the surface. It is 2 for the sphere, 0 for the torus, 0 for the Klein bottle, 1 for the projective plane.

Two of those are the same, and the two surfaces are not. Something else is needed, and the word supplies it: is any letter used twice the same way round? On the torus each letter appears once forwards and once backwards; on the Klein bottle one of them appears twice forwards. A letter used twice the same way round is a gluing with a flip in it, and a flip is what destroys any consistent sense of clockwise.

That is orientability, and it is exactly the property the Möbius band is famous for lacking. The figure decides it by inspecting the word, which takes no geometry at all.

The pair that differ by one arrow

The torus and the Klein bottle are worth a paragraph on their own, because they are the case that makes both numbers necessary and because the difference between them is as small as a difference can be.

Both are squares with both pairs of edges glued. In one, both gluings run straight across; in the other, one pair is joined with a flip. Both have one vertex, two edges and one face, so both have characteristic nothing. Every count that can be made by cutting the surface into pieces and counting the pieces gives the same answer for the two.

They are not the same surface. On the torus, a small circle drawn anywhere can be carried all the way round the surface and back to where it started, still turning the same way. On the Klein bottle there is a route round which it comes back turning the other way — which is exactly what happens on a Möbius band, and a Klein bottle contains one.

So the pair (characteristic, orientability) is the minimum that will do, and neither half can be dropped. This is a recurring situation: three legal moves on a knot diagram preserve tricolourability, which tells the trefoil from the unknot and tells almost nothing else apart, and more invariants have to be added until the list separates what needs separating. Surfaces are the pleasant case where two suffice and the list is complete.

Two numbers, and a complete answer

The classification theorem for surfaces says something that very few subjects can say: the characteristic and the orientability, between them, determine the surface completely. Two closed surfaces with the same pair are the same surface, and every possible pair that occurs occurs exactly once.

A sphere, and a sphere with handlesTwo-sided closed surfaces in order: a sphere, then one with a handle, then two, then three, each with the Euler characteristic that counts them.a sphereχ = 2a torusχ = 0a surface with 2 handlesχ = −2a surface with 3 handlesχ = −4a sphere with g handles has characteristic 2 − 2g, so the number falls by two for every handleon the two-sided side of the classification, that number is the whole of the answer: two surfaces with the same characteristic are thesame surface
Fig. 5 The two-sided surfaces, in order: a sphere, a sphere with one handle, with two, with three. Every handle costs two from the characteristic, so the sequence of values is 2, 0, −2, −4, and no two of them coincide.

On the two-sided side the list is a sphere and then spheres with handles, and the characteristic falls by two for each handle: 2, 0, −2, −4. The number of handles is the genus, and it is recoverable from the characteristic by χ = 2 − 2g.

On the one-sided side there is a parallel list — a projective plane, a Klein bottle, and onwards — with the characteristic falling by one at each step: 1, 0, −1, −2. The two lists overlap in their values, which is why orientability cannot be dispensed with, and they do not overlap in their surfaces.

A classification like this is rare enough to be worth dwelling on. It says that a complete inventory exists, that it is indexed by two small pieces of data, and that anyone who builds a closed surface by any method whatsoever has built something already on the list. There is no room for a surprise.

Why the characteristic survives the cutting

Everything above depends on the characteristic being a property of the surface rather than of the polygon it was built from, and that is a claim worth checking rather than assuming.

The check is a move: take any face and cut it in two with a new edge between two of its corners. One face becomes two, one edge is added, no vertices change — so V − E + F is unchanged. Take any edge and put a new vertex in the middle of it: one vertex and one edge are added, and again nothing moves.

Those two moves generate every re-cutting of a surface, so the alternating sum cannot depend on which cutting was used. That is the whole argument, and it is the same argument that makes the number 2 come out for all five regular solids despite their having wildly different counts of everything.

A quantity that survives every re-description of an object is what topology means by an invariant, and this is the oldest one. Its power is entirely in what it cannot see: it does not know how large the surface is, how it is embedded, or whether it has been stretched into an unrecognisable shape. It knows the number of handles, and it took two thousand years for anybody to notice that this was a thing worth knowing.

The octagon, and how the genus arrives

The gluing abcda⁻¹b⁻¹c⁻¹d⁻¹ makes a surface with 2 handlesA polygon whose edges carry the word abcda⁻¹b⁻¹c⁻¹d⁻¹, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.abcdabcdcorners 1, 2, 3, 4, 5, 6, 7, 8 become one vertexthe 8 corners fall into 1 class, so V − E + F is 1 − 4 + 1 = −2every letter is used once each way round, so a consistent sense of turning survives the gluing
Fig. 6 An eight-sided polygon with its edges glued in four pairs, each running the opposite way from its partner. All eight corners become a single vertex, so the characteristic is 1 − 4 + 1 = −2, which is the two-handled surface.

The square gives the torus; an octagon glued the same way gives the two-handled surface. The pattern continues: a 4g-sided polygon with its edges glued in that alternating pattern gives the genus-g surface, and every two-sided closed surface arises this way.

The arithmetic is worth doing once. The polygon has 4g edges glued into 2g pairs; all its corners become one vertex; so χ = 1 − 2g + 1 = 2 − 2g, which is the formula the classification promised.

What that says in practice is that any two-sided surface, however it was built, can be cut open into a single polygon. The cutting is the reverse of the gluing, and it is the reason the classification is provable at all: reduce any surface to a polygon with a word, then reduce the word by a finite list of moves to one of the standard forms.

What the words can and cannot be

An edge word is a string in which every letter appears exactly twice, and the figures refuse anything else: a letter appearing once has nothing to be glued to, and a letter appearing three times describes an object that is not a surface at all — three sheets meeting along a line, which has no flat neighbourhood anywhere along it.

Within that constraint the words are far from unique. The same surface has many words: the torus is aba⁻¹b⁻¹ and also bab⁻¹a⁻¹ and also a great many longer strings from polygons with more sides. So the map from words to surfaces is many-to-one, and the classification is really a statement about which words describe the same thing.

That is what makes the proof a reduction rather than an enumeration. There is a short list of moves on words — cut the polygon along a diagonal and reglue along an old edge — that change the word without changing the surface, and the proof shows every word can be driven by those moves into one of the standard forms. It is the same shape of argument as reducing a fraction to lowest terms or a matrix to a normal form: not a survey of all cases, but a procedure that terminates.

The count of distinct words is enormous and the count of distinct surfaces is one per pair of numbers. Almost all the information in a word is about the polygon rather than the surface, and the two invariants are what is left after that information is thrown away.

What the conditions are doing

Three words in the theorem are load-bearing and each excludes something.

Closed means no boundary. A Möbius band is a perfectly good surface, it is one-sided, and it is not on the list — because it has an edge. Adding surfaces with boundary to the classification is possible and it needs a third number, the count of boundary circles.

Gluing a strip with a flipA rectangle whose left and right edges are to be identified after reversing one of them.AAjoin the ends, but turn one over→ Möbius band: one side, one edge
Fig. 7 The Möbius band as a gluing: a rectangle with one pair of edges identified after a flip, and the other pair left alone. The unglued pair is the boundary, and it is why the band is not on the list above.

Compact means finite in extent. The plane is a two-sided surface without boundary and it is not a sphere; it is not on the list either, and the classification says nothing about it. The sphere with a point removed is the same object as the plane, which is the cleanest demonstration that removing a single point from a compact surface takes it out of the theorem’s reach entirely.

Connected means one piece. Two disjoint spheres have characteristic 4 and are not on the list, for the boring reason that the list is a list of connected things.

Where it fails to be a picture

Two of the four surfaces on this page cannot be built in three dimensions. The Klein bottle and the projective plane can be immersed — pushed into space with self-intersections — and cannot be embedded. Every drawing of a Klein bottle is a drawing of a surface passing through itself, and the passing-through is an artefact of the drawing rather than a feature of the surface.

That is exactly why the gluing diagram is the right picture. A square with arrows on its edges is a complete and honest description of a surface that has no honest picture, and every number this essay computes is computed from the square rather than from any attempt to draw the result.

The handles in the classification figure are drawn as holes in a flat blob, which is a schematic and not a surface. A genuine picture of a two-handled surface is a solid object seen from one side, and it hides exactly the features the count is about.

And nothing here proves the theorem. The figures compute the two numbers for particular gluings and check them against the names. That the numbers determine the surface is a claim about all surfaces, its proof is the reduction of words to normal forms, and no drawing of four squares establishes it.

Where the ladder goes next

The classification is a rare complete answer and the natural question is why it stops. In three dimensions there is no such list: the classification of three-dimensional manifolds is enormously harder, took a century, and the statement that finally settled it is not a list but a decomposition. The two-dimensional case is easy in a way that turns out to be exceptional.

Closer to home, the same invariants have more to say. The characteristic controls what can live on a surface: a vector field on a sphere must have a zero because the characteristic is 2, and on a torus it need not because the characteristic is nothing. It controls how many colours a map on the surface can need — seven on a torus, four on a sphere, and the torus case is the easier of the two. And it controls which closed curves separate, which is where this essay meets inside and outside: a curve on a genus-g surface may fail to separate it in 2g independent ways, and on a sphere, in none.

Where this goes next

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. VEFV − E + Ftetrahedron4642cube81262octahedron61282dodecahedron2030122icosahedron1230202 Every corner pays for itself Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself. the one point with nowhere to goa circle on the sphere……is a circle on the planethe plane runs on past the edge of the drawing A sphere is a plane plus one point Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.

What links here

Computed from the collection, not written here: the essays that point at this one.

Named objects

A dashed tag is an object no other essay names yet.

ConnectednessEuler characteristicGenusGluing diagramKlein bottleMöbius bandNon orientableOrientationPolyhedronTopological invariant