Two sheets over a one-sided surface

A one-sided surface has no consistent choice of turning sense: carry a small clock face round the right loop and it comes back running the other way. The repair is not to fix the surface but to build a new one whose points carry the choice — a point of the new surface is a point of the old one together with one of its two senses of turning.

The new surface has exactly twice as many points, it is two-sided by construction, and the map that forgets the choice is two to one. It is called the orientation double cover, and every non-orientable surface has one.

What is above a point #

Fix a point of the projective plane. It is a line through the origin, and it meets the unit sphere in two places. Those two places are the two points above it, and they are the two senses of turning: a small circle drawn round one of them on the sphere is carried by the identification to a small circle round the other, traversed the other way.

So the sphere is the double cover of the projective plane, and the covering map is “identify antipodal points”. That is the cleanest instance of the construction and the one worth having in hand before the general statement.

The construction is a quotient, and the property that makes a quotient a surface is that the identification has no fixed points. The antipodal map moves every point of the sphere; the reflection in a plane does not, and the quotient of a sphere by a reflection is a disc with an edge rather than a closed surface. The figure checks the freeness cell by cell — no vertex, edge or face of the icosahedron is carried to itself — because that is the hypothesis the whole construction rests on and it is the one that fails first when the map is chosen carelessly.

Every count halves, and so does the characteristic #

The cover is two sheets, so it has twice the area, twice the cells, and — this is the part worth pausing on — twice the Euler characteristic.

That is not automatic-sounding, because the characteristic is an alternating sum and alternating sums do not always survive doubling. It works here because the covering is cellular: each cell downstairs is covered by exactly two cells upstairs of the same dimension, so $V$, $E$ and $F$ each double and the alternating sum doubles with them.

The figure counts it: $12 - 30 + 20 = 2$ upstairs, $6 - 15 + 10 = 1$ downstairs. The characteristic of a solid is being used here as a count over a cell structure rather than as a fact about polyhedra, which is the reading that makes it an invariant of the surface.

The doubling has an immediate consequence. The Klein bottle has characteristic $0$, so its cover has characteristic $0$ and is two-sided — it is the torus. A sphere with three cross-caps has characteristic $-1$, so its cover has characteristic $-2$ and is the two-handled surface. In general the cover of a sphere with $k$ cross-caps is a sphere with $k - 1$ handles, and the arithmetic is the only thing needed to say which.

Why a two-sided surface has no such cover #

A two-sided surface has a double cover too, in a trivial sense: two disjoint copies of it, one for each choice of turning sense, made consistently over the whole surface.

That is exactly the point. The cover is disconnected precisely when the surface is orientable, because the choice can be made globally and the two global choices never meet. On a one-sided surface the choice cannot be made globally, following a suitable loop swaps the two sheets, and the cover is connected.

Orientable means the double cover falls apart. That is a clean restatement of the property and it is the first time in this ladder that orientability has been defined without mentioning a side, a room, or a normal vector. It is a statement about a map.

The torus over the Klein bottle, written out #

The projective plane’s cover is easy because a symmetry of the sphere happens to do the job. The Klein bottle’s is worth doing because nothing is handed over: the cover has to be assembled from the gluing rule.

Start with the Klein bottle’s square, whose word is $abab^{-1}$. Take two copies of it, and glue them by the rule that an edge used consistently stays on its own sheet while an edge used the same way round twice crosses to the other sheet. The letter $a$ appears twice with the same direction, so it is the one that crosses; $b$ does not.

The result is two squares glued into a rectangle twice as long, with the far ends joined straight across. That is the torus’s gluing, $aba^{-1}b^{-1}$, on a polygon of twice the area.

Every claim the general construction makes can be read off that. The cover has two faces, four edges and two vertex classes rather than one, so its characteristic is $2 \times 0 = 0$; it is orientable, because in the doubled word every letter now appears once each way round; and the deck transformation is “slide by one square”, which has order two because sliding twice returns to the start.

The same recipe applied to the projective plane’s $abab$ gives a square doubled into the sphere’s $abb^{-1}a^{-1}$, which is the sphere-over-projective-plane cover arrived at without any solid at all.

The cover of a surface with an edge #

The construction does not need the surface to be closed, and the Möbius band is the smallest instructive case.

Its double cover is an annulus — a cylinder — and the picture is the band’s own rectangle, doubled. Walk along the band’s core once and the sheet exchanges; walk twice and it does not, so the cover closes up after two laps. The cylinder’s core is exactly the band’s core gone round twice.

Both characteristics are zero, which is the doubling working in the least informative way it can. What separates the two surfaces is not the number but the boundary count: the band has one boundary curve and the cylinder has two, and the covering map sends both of the cylinder’s onto the band’s single one, wrapping each once.

The deck group, and why it has two elements #

The map that exchanges the two sheets — antipodal on the sphere — is a deck transformation: a symmetry of the cover that leaves the covering map alone. Here there are exactly two of them, the exchange and doing nothing, so the deck group has two elements.

Two is not a coincidence and it is not a choice. The number of sheets of the orientation cover is two because there are two senses of turning in a plane, and there are two because the group of linear maps of a plane that preserve area falls into exactly two pieces according to the sign of the determinant. The rung above is about that sign; here it is enough that there are two of it.

The same two-ness turns up whenever a continuous family of choices has to be made and the choices form a set with two elements: which way a permutation’s crossings come out, which of two square roots, which of two normals. In each case the obstruction to choosing globally is a homomorphism to the same two-element group, and the object that resolves it is a double cover. It is one construction wearing several names, which is the usual sign that it is the right construction.

This is the same machinery as unrolling a loop, where a covering of the circle by a helix has a deck group of all the integers because a loop can be gone round any number of times. Here the group is smaller because the thing being remembered is smaller: not how many times, only which of two.

Reading the property off the loops #

Every closed loop on a surface either preserves the sense of turning or reverses it, and carrying a frame round it decides which. That assignment sends a loop to one of two values, it multiplies when loops are joined, and it is unchanged by deforming a loop — so it is a homomorphism from the loop group to a two-element group.

The double cover is that homomorphism made geometric. The loops that preserve the sense are the ones that lift to closed loops upstairs; the ones that reverse it lift to paths that start on one sheet and end on the other. Which loops do which is the whole of the information, and the surface upstairs is the record of it.

That gives the general construction, and it works on any surface at all rather than only on the ones with a convenient symmetry. Every non-orientable surface has a non-trivial such homomorphism, by definition; the covering space it names is the orientation double cover; and the covering it names is connected exactly when the homomorphism is not the trivial one. The subgroup of loops that lift is the kernel, and it is half the group.

Where the construction is used #

Three uses, because a construction with one example is a curiosity.

Integration. A surface integral needs a consistent sense of turning to be defined at all — reverse it and the answer changes sign — so nothing can be integrated over a one-sided surface. Passing to the double cover repairs that: the integral is defined upstairs, and the quantities that were going to be well defined downstairs are the ones invariant under the deck transformation. This is the standard way a calculation on a non-orientable object is actually done.

Physics with a sign. A quantity that comes back reversed after one circuit and right after two is a quantity defined on the double cover rather than on the surface. Half-integer spin is the famous case, where the cover is of the rotation group rather than of a surface, and the quaternions are the cover. The three-dimensional rotations form a projective space for the same reason a plane’s directions do, and the cover is the sphere of unit quaternions.

Classification. Any statement about non-orientable surfaces can be transported upstairs, proved for orientable ones — where the tools are better — and brought back down by keeping only what the deck transformation leaves alone. The classification of one-sided surfaces was first obtained this way.

Two sheets, and no more, and why that is a theorem #

It is worth asking why the cover has two sheets rather than three or four, since covering spaces come in every number.

The answer is that the cover is built from a specific homomorphism to a specific group, and the group has two elements because a two-dimensional frame has two classes. Nothing else is available to remember. A cover with three sheets exists over many surfaces — over the circle, for one, and over the torus — but it is not remembering orientation, because orientation has only two states.

That makes the orientation cover the smallest useful one and gives it a property the others lack: it is canonical. There is no choice anywhere in the construction. A three-sheeted cover of the torus has to be chosen, and different choices give genuinely different covers; the orientation cover is determined by the surface alone, which is what lets it be used in a definition.

The canonical-ness has a consequence worth naming, because it is how the cover is usually invoked. Any construction that needs an orientation can be performed on the cover and then required to be invariant under the deck transformation. What survives is defined on the original surface, with no arbitrary choice smuggled in, and what does not survive is exactly what depends on an orientation and therefore was never going to be defined downstairs. That is a decision procedure rather than a hope, and it is why the cover is a tool rather than an observation.

What the picture cannot show #

The figure draws the cover and the counting, and it does not draw the map. What is on the page is a solid with its opposite faces coloured alike, and the reader is asked to imagine the identification; there is no way to draw a two-to-one map that does not either draw the quotient separately or draw arrows that clutter the solid.

It cannot show a one-sided surface at all, for the reason the previous rung gives: the projective plane does not embed in three dimensions, so what is drawn is the cover and a table of numbers rather than the two surfaces side by side.

And the antipodal identification is drawn on an icosahedron, which has the symmetry needed and is not otherwise special. Nothing in the construction is about that solid; any triangulation of the sphere invariant under the antipodal map would do, and the icosahedron is chosen because it is the smallest one that is.

Where the ladder goes next #

Above this rung: orientation as a sign — the determinant of the frame that comes back, which removes the last dependence on a surrounding space. Below and beside: covering spaces in general, where the number of sheets is any number and the deck group is any group.

One debt. This essay builds the cover for the projective plane by a symmetry that happens to exist, and states the general construction in one paragraph about homomorphisms. The general construction deserves its own drawing — the cover assembled from the surface’s own gluing polygon, two copies of it, with the edges cross-matched wherever a letter is used twice the same way round — and it is not here.

What the cover is remembering #

A point of the cover is a point of the surface together with a choice, and the surface is one-sided exactly when the choice cannot be carried all the way round.

That is the whole construction, and its value is that it converts a property nobody can define without a room — which side — into a property that is entirely internal: whether a certain two-sheeted cover is connected. Everything about the surface’s sidedness is then a fact about a map, and maps are the objects this subject is good at.