Orientation is a sign

Every statement about orientability so far in this ladder has needed a room. One side means one side as seen from somewhere; a normal vector points out of the surface into a space the surface is sitting in; even the drawing crosses itself is a fact about three dimensions rather than about the surface.

None of that is necessary, and the definition that removes it is short. Choose, at every point, which of the two turning senses is positive. The surface is orientable if the choice can be made consistently. Consistently means: neighbouring points agree, all the way round every loop.

What a frame is, and why two arrows #

At a point of a surface, the directions available form a plane — the tangent plane. A frame is an ordered pair of directions in it that do not lie on one line, and it is what a choice of turning sense is made with: first arrow to second arrow, the short way round, is positive.

Two frames at the same point are compared by the matrix that carries one to the other. Its determinant is non-zero, because both are genuine frames, and the determinant’s sign says whether the two agree about turning sense. Positive means they agree; negative means one is the mirror of the other.

That is the whole comparison, and it is worth noticing what it does not use. No lengths, no angles, no third direction, nothing about a surrounding space. Two frames at one point are compared by a $2 \times 2$ determinant, and a sign comes out.

Two classes, and no third #

The claim that a sign is the whole answer rests on there being exactly two classes of frame, and that is a fact about matrices rather than about surfaces.

The frames at a point are in one-to-one correspondence with the invertible $2 \times 2$ matrices — pick a reference frame, and every other frame is that one moved by an invertible matrix. Those matrices fall into two pieces: the ones with positive determinant and the ones with negative. A continuous path of matrices cannot cross from one piece to the other without passing through determinant zero, which is not invertible, so the two pieces are genuinely separate and nothing joins them.

Two pieces, and no more, because the determinant is a single non-zero number and its sign has two values. The two-ness of orientation is the two-ness of the sign of a real number, and that is the reason the double cover of the rung below has two sheets rather than three.

The same statement in a different vocabulary: the parity of a permutation is the determinant’s sign restricted to the matrices that permute coordinates, and it too has exactly two values for exactly this reason. Rearranging a frame by an odd number of swaps reverses its orientation, and by an even number does not.

Carrying it round #

A frame at one point can be carried to a nearby point by moving each arrow a little and keeping it tangent. Doing that along a path carries a frame from one end to the other, and doing it round a closed loop carries a frame back to where it started, where it can be compared with itself.

The comparison gives a sign for every loop. If it is positive for every loop, a consistent choice exists: pick a frame somewhere, carry it everywhere, and the answer does not depend on the route. If it is negative for some loop, no consistent choice exists, because the choice made along one route and the choice made along another disagree at the meeting point.

Orientable means the sign is positive round every loop. That statement is entirely internal to the surface. It has no room in it, and it applies to objects that are not sitting in any space at all.

The band, computed rather than looked at #

On the Möbius band the sign can be computed instead of asserted, and the computation is one line.

The band is a strip whose cross-section turns steadily as the core is walked, by a total of half a turn over one lap. Carry the frame whose first arrow points along the core and whose second points across it. The first arrow returns as it left. The second returns turned by $\pi$, so it returns as its own negative, and the matrix carrying the departing frame to the arriving one is

$$\begin{pmatrix} 1 & 0 \\ 0 & \cos 2\pi t \end{pmatrix} \bigg|_{t = 1} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$

whose determinant is $-1$.

The figure computes exactly that, and asserts it against the parity of the half-twists. Two independent statements — an odd number of half twists and the transported frame has determinant $-1$ — and the assertion is that they agree, which is the check that would fail if the transport had been drawn rather than computed.

Reading the sign off a gluing word #

On a surface given as a polygon with its edges glued, the transport does not have to be computed at all. It can be read.

Walk the polygon’s boundary and note each letter with the direction it is traversed. Crossing an edge takes a frame from one side of the polygon to the other, and whether it arrives mirrored depends on whether the two occurrences of that letter point the same way. A letter used twice the same way round reverses; a letter used once each way does not.

So a word gives the sign of every loop directly, and the surface is orientable exactly when every letter is used once each way. $aba^{-1}b^{-1}$ and $abb^{-1}a^{-1}$ pass; $abab^{-1}$ and $abab$ fail, on the letter $a$ in both cases.

This is the most practical form of the definition, and it is the one that makes the property decidable. A surface presented as a polygon and a word is orientable or not by an inspection of the word, which takes as long as reading it; no transport is integrated, no cover is built, and no picture is needed. That the answer agrees with the other two definitions is the content of this section, and the figure asserts it on all four cases.

What is gained by dropping the room #

Three things, and each of them is a case where the older definition simply does not apply.

Surfaces that are not in a room. The projective plane does not embed in three dimensions. Asking which side of it something is on is not a hard question; it is not a question. Asking whether a frame returns with a positive sign is a question and has an answer.

Objects that are not surfaces. A vector bundle attaches a vector space to every point of a space, and the same question — can a frame be chosen consistently — makes sense with no surface anywhere. The Möbius band is the smallest non-trivial example, viewed as a line attached to every point of a circle, and it is the standard first example for exactly that reason.

Higher dimensions. A frame in $n$ dimensions is $n$ ordered directions, the comparison is an $n \times n$ determinant, and the sign is still one of two. Nothing in the definition mentioned the number two, and nothing changes.

The last is worth stating carefully because the everyday intuition does not survive it. In three dimensions the two classes are called left-handed and right-handed; in four they have no names; and in every dimension there are exactly two, because the invertible matrices fall into two connected pieces separated by the determinant’s zero.

What one-sidedness costs #

A property is worth having only when something depends on it, and three things do.

Integration. The integral of a quantity over a surface changes sign when the turning sense is reversed. On an orientable surface a sense is chosen once and the integral is a number; on a non-orientable one there is no such choice, and the integral is not defined. The divergence theorem and every result of its kind have orientability in their hypotheses, usually silently.

Volume, as a form rather than as a number. An orientable surface carries a nowhere-zero quantity that assigns a signed area to each frame; a non-orientable one carries none, and the reason is the same sign. Area as an unsigned number survives, which is why a Möbius band can be given a size and cannot be integrated over.

Consistent handedness anywhere in a physical theory. If space were non-orientable, a body carried round the right loop would return as its mirror image, and any law distinguishing left from right would have to explain what happened. That is a real constraint rather than a whimsy, and it is one of the things cosmology can in principle measure.

The Möbius band as a bundle, which is where the idea is really used #

The band has been treated as a surface throughout this ladder. There is a second reading of it that is more useful and looks nothing like a surface.

Take a circle, and attach to each point of it a copy of the real line. Doing that consistently all the way round gives a cylinder. Doing it with a flip — so that going once round the circle turns the attached line upside down — gives the Möbius band. In both cases what has been built is a bundle: a space, with a vector space over each of its points, glued along the overlaps.

The question of whether a frame can be chosen consistently is now the question of whether the bundle has a nowhere-zero section — a choice of non-zero vector in each attached line, varying continuously. The cylinder has one; the band does not, because the choice would have to change sign somewhere and a continuous function changing sign passes through zero. That is the intermediate value theorem doing the work, and it is the whole obstruction.

This is why the band appears in every course on bundles as the first example. It is the smallest object for which a choice that is obviously available locally is not available globally, and every later obstruction in the subject — a bundle that will not trivialise, a field that cannot be combed, an angle that will not be defined consistently all the way round — is the same shape at a larger size. The hairy ball theorem is the two-dimensional instance and the same sentence describes it.

The sign, in this reading, is the transition function: the number by which the two descriptions of the line disagree where they overlap. On the cylinder it is $+1$; on the band it is $-1$; and the entire difference between the two objects is that one number.

The three definitions, and that they agree #

This ladder has now given orientability three times, and the three deserve to be laid beside each other because they are proved equivalent rather than obviously the same.

One-sided. A normal vector carried round some loop comes back reversed. Needs the surface to sit in a space, and needs the surface to have a normal — so it applies to a surface in three dimensions and nowhere else.

The double cover is connected. The two-sheeted surface built out of the choices does not fall into two pieces. Needs nothing outside the surface, and is the definition that generalises to any space at all.

Some loop has transport determinant $-1$. Needs nothing outside the surface either, and is the most computable: it is an integral of a rotation rate, or on a gluing polygon it is a letter used twice the same way round.

The equivalence of the second and third is nearly a restatement — the cover is built from the sign — and the equivalence of the first with either needs the separation theorem the Klein bottle essay quotes. That asymmetry is the point: the extrinsic definition is the one that requires an extra theorem, and it is also the one everybody learns first.

What the picture cannot show #

The figure draws a frame at five stations on a flat rectangle, which makes the transport look like a rotation happening in the page. It is not: the frame is tangent to the surface at each point, and on the band the tangent plane is turning in space as the core is walked. Flattening the band to draw it is what makes the turn visible, and it is also what makes it look like something happening to the arrows rather than to the surface.

The determinant is a number and the picture shows a sign, which is the right amount of information and less than the picture appears to offer. Nothing about the magnitude of the transport matters, and the figure asserts that the magnitude is exactly one — the frame comes back the same size, only possibly mirrored.

And the drawing is of a band, which has an edge and sits in three dimensions, so it is a picture of the case where the older definition also works. The surfaces the new definition was written for cannot be drawn, which is a permanent feature of this rung: the definition that applies everywhere is the one with no picture.

Where the ladder goes next #

Above: orientability of manifolds in every dimension, where the argument is unchanged and the pictures stop; the first Stiefel–Whitney class, which is this sign written as a cohomology class and is the form in which it is actually computed; and orientability of vector bundles, where the Möbius band is the smallest example and the pattern of the whole subject.

One debt, and it is the same one the previous rung left. The transport used here is described as “move each arrow a little and keep it tangent”, which is a connection — a choice — and different connections could in principle give different signs. They do not, and the reason is that the sign is a discrete invariant and cannot change continuously, but that argument is not made here.

What the sign was #

Orientability is the statement that a determinant is positive round every loop.

Everything else in this ladder is a consequence: the side that cannot be chosen, the drawing that crosses itself, the cover that does not fall apart, the integral that is not defined. The property was never about paper or about rooms. It is about whether a comparison of two frames, taken all the way round and back, has forgotten which way was positive — and there are only two answers because there are only two signs.