Every section of the band must vanish

Orientation is a sign carried a pair of arrows round a loop and read off one number — the sign of the determinant comparing the frame that came back with the frame that left. It ended by naming the place where that idea is actually used: not on surfaces, but on bundles, families of lines or planes standing over the points of a space, of which the Möbius band is the smallest example.

Here is that example on its own. Forget the band’s shape in space. Keep only its structure: over every point of the centre circle there is a line segment across the strip, and as the point moves round the circle, the segment moves with it. After one full turn, the segment comes back turned over — its top end where its bottom end was. That is the whole of a Möbius band, read as a line bundle, and the cylinder is the same thing with the segment coming back as it left.

A section is a choice on every line #

A section of the band is a continuous choice of one point on each of the lines — a curve drawn along the strip that crosses each cross-section exactly once. On the flat rectangle, with the centre circle running from left to right as $0 \le t \le 1$ and the cross-section running up and down from $-1$ to $1$, a section is simply the graph of a continuous function $s(t)$.

The gluing puts one condition on it. On a cylinder the right edge is glued straight to the left, so the section must meet itself: $s(1) = s(0)$. On a Möbius band the right edge is glued after turning it over, so the point at height $s$ on the right is the point at height $-s$ on the left, and the section must satisfy

$$s(1) = -s(0).$$

The zero section is the centre line, $s(t) = 0$ for every $t$. It exists on every bundle, because zero is a point on every line and turning a line over leaves zero where it was. The question is whether there is a section that avoids it.

The intermediate value theorem does the work #

On the band, the answer is no, and the proof is one line. If $s(0)$ is positive, then $s(1) = -s(0)$ is negative, and a continuous function that starts positive and ends negative must pass through zero somewhere between. If $s(0)$ is negative, the same argument runs the other way. If $s(0)$ is zero, the section is already zero there. Every section of the Möbius band vanishes somewhere.

It says more than that, counted properly. Each time the curve crosses the centre line it changes sign, and it must change sign an odd number of times to get from $s(0)$ to $-s(0)$. So a section that crosses the centre line cleanly — not merely touching it — crosses it an odd number of times. The figure above draws three: the most economical, a half cosine that crosses once, and two wandering curves that cross three times and once. The count is taken by the drawing itself, by finding every sign change, and checked to be odd.

On the band in space this is a fact anyone can check with a pen. Draw a line along the strip, staying on one side of the centre, and go round once. The line comes back on the other side of the centre, because the strip has turned over — and to close up it must cross the centre somewhere. That is the same argument as the one-sidedness of the surface with one side, seen from the circle rather than from the surface.

The cylinder has a section that never vanishes #

On the cylinder the condition is $s(1) = s(0)$, and nothing forces a zero.

The constant section, at height $0.62$ all the way round, is never zero. Sections can still cross the centre line — the two wandering curves cross it twice each — but they must cross an even number of times, since they end where they began, and zero is allowed.

That is the difference between the two bundles, stated as a property of sections rather than of shapes: the cylinder has a nowhere-zero section and the Möbius band does not. A bundle with a nowhere-zero section is the plain product of the circle with a line: every line can be given the coordinate in which the section sits at $1$, and the coordinates fit together all the way round. A bundle without one is twisted, and no choice of coordinates untwists it.

Four bands in space, two bundles #

A strip can be joined with any number of half-twists, and in space each number gives a different object.

A band with two half-twists — one full twist — cannot be moved in space onto a plain cylinder: its two edges are linked, once, where the cylinder’s are not. But as a bundle it is the cylinder. Walking once round, the cross-section turns through a full turn and comes back as it left, so the gluing is by $+1$ and the constant section closes up. The linking of the edges is a fact about how the band sits in the room, and the bundle has no room.

So the half-twists, which count by whole numbers in space, count only modulo two as bundles. Zero and two and four half-twists give the cylinder’s bundle; one and three and five give the Möbius band’s. There is no third line bundle over the circle, and the reason is the one orientation is a sign gave for frames: a line has exactly two directions, so going round a loop can only preserve them or swap them.

An orientation of a bundle is a section that never vanishes #

For a single line, an orientation is a choice of one of its two directions — which way is positive. For a bundle of lines, an orientation is a choice of direction on every line that varies continuously. The two ideas meet here.

A nowhere-zero section chooses a direction on every line: the direction in which the section sits. It varies continuously because the section does. Conversely, a continuous choice of direction on every line gives a nowhere-zero section — take the unit vector in the chosen direction. So a line bundle is orientable exactly when it has a nowhere-zero section, and the Möbius bundle, having none, is the non-orientable one.

The frame from orientation is a sign is the same object. Its arrow along the band points along the circle and never changes; its arrow across the band is a section, drawn at a few stations. It leaves pointing up and arrives pointing down, which is the condition $s(1) = -s(0)$ again. The determinant that essay computed is the product of the along-arrow’s length with the cross-arrow’s height, and its sign is the sign of $s$. The sign of a frame and the sign of a section are one sign.

This also settles a question that essay left open: whether the answer depends on how the frame is carried. It does not, because any other way of carrying the cross arrow is another section, and every section of the band changes sign an odd number of times. However the arrow wanders along the way, it arrives with the opposite sign to the one it left with — because arriving any other way would mean a section with an even number of zeros, and there is none.

Two Möbius bands add up to a product #

Bundles can be added, line to line. The sum of two line bundles over the circle has a plane over each point — the first bundle’s line and the second’s, at right angles — and the gluing acts on each line separately.

Add a Möbius band to itself. The gluing on each line is by $-1$, so on the plane it is by $-1$ in both directions at once: every vector is sent to its opposite. And that map, unlike $-1$ on a line, is a rotation — a half-turn. A pair of arrows can be carried round the circle turning steadily through half a turn, and they arrive exactly reversed, which is what the gluing demands. That gives two sections that are independent at every point: the sum of two Möbius bands is the plain product of the circle with a plane.

The difference between one band and two is the difference between the line and the plane. On a line, $+1$ and $-1$ are the only two directions and no path joins them without passing through zero. In the plane, the directions form a circle, $-1$ is halfway round it, and the path of rotations goes the long way round zero rather than through it. Multiplying is turning is the same fact about complex numbers: multiplying by $-1$ is a half-turn, and a half-turn can be done a little at a time.

So the twist adds like a sign: one band is twisted, two are not, three are again. The line bundles over a circle form a group of two elements under this addition, the same two-element group that recorded whether a loop reverses a surface’s sense of turning.

The lines through a point make a Möbius band #

The Möbius bundle is not an invention; it is the first bundle anyone meets when they look at lines.

Take all the straight lines through the origin in a plane. Each is fixed by its angle, and the angle runs from $0$ to $\pi$, since the line at angle $\pi$ is the line at angle $0$ again. So the lines through a point form a circle. Over each line, put the line itself: the points lying on it. That is a line bundle over the circle of lines, called the tautological bundle, since the line over each point is the point.

Walk once round the circle of lines, turning the line from angle $0$ to angle $\pi$, and carry a vector along it. The vector starts pointing right along the horizontal line and ends pointing left along the same line, because turning through $\pi$ reverses it. So the gluing is by $-1$, and the tautological bundle is the Möbius band. Every section vanishes — which says that there is no way to choose a nonzero vector on every line through the origin, continuously, which anyone who tries to label directions on a set of lines finds out in a few seconds.

Where the sign comes back to surfaces #

The bundle picture explains the surface picture, rather than merely resembling it.

Take a loop on a surface and a thin strip around it. Over each point of the loop, the strip gives a short line segment across the loop — the direction on the surface perpendicular to the loop. That is a line bundle over the loop, which is a circle, so it is a cylinder or a Möbius band. The loop reverses the surface’s sense of turning exactly when the strip around it is a Möbius band, because the frame on the surface is the along-arrow and the cross-arrow, the along-arrow never flips, and the cross-arrow is a section of the strip.

That is why every one-sided surface contains a Möbius band: take any loop that reverses the sense, and thicken it. The projective plane is a disc sewn to a Möbius band because its core loop’s strip is one; the Klein bottle contains two. And two copies of the polygon, cross-matched glued its letters across the copies exactly on the loops whose strips are Möbius bands.

A zero that cannot be removed, one dimension up #

The vanishing of every section is the one-dimensional case of a larger pattern.

Over the sphere, the directions tangent to the surface form a bundle of planes. A section is a choice of tangent arrow at every point — a combing of the sphere — and nothing can be combed flat proves that every such section vanishes somewhere. That obstruction is measured by an integer, the Euler characteristic of the sphere, $2$, which counts zeros with signs. On the circle the obstruction is measured by a sign, $\pm 1$, which counts zeros modulo two. One is the Euler class; the other is the first Stiefel–Whitney class, which is the name the sign of this page carries in the general theory.

The one-dimensional version has a second guise too. Sections of the Möbius band are the same as continuous functions on a circle of double length that change sign when moved halfway round — odd functions, $f(\theta + \pi) = -f(\theta)$ — and the statement that every such function vanishes is the circle’s case of the theorem behind two opposite points that agree twice.

What the figures can and cannot show #

The sections are genuine and their zeros are counted. Each curve is built to satisfy the gluing condition exactly — its two ends are checked to be equal, or equal and opposite — and its zeros are found by locating every sign change along two hundred and more steps. A curve that merely touched zero without crossing would be missed, which is why the claim made is about crossings: a touching zero can be removed by a small push, and a crossing cannot.

The bands in space are drawn, the bundles are not. A bundle has no shape; the four bands in the parity figure are four ways of putting the two bundles into a room, and nothing about the room is part of the bundle.

The sum is drawn at seven stations. The rotation by $\pi t$ is checked to have determinant one at every station and to equal $-1$ in both directions at the end; the continuity in between is the continuity of the cosine.

Still open: the smallest room for a projective space #

The signs on this page have a use far beyond the circle: they decide what fits where. The projective plane cannot sit in three-dimensional space without passing through itself, and the proof can be run through bundles. A surface sitting in a room has, beside its own tangent planes, a bundle of normal lines — the directions sticking straight out of it — and the two together make the room’s own bundle, which has no twist at all. So the normal line bundle must carry exactly the twist the surface carries: over a loop that reverses the surface’s sense of turning, the normal line must turn over too. But a closed surface sitting in space without crossing itself separates an inside from an outside, and “pointing outward” is a normal section that never vanishes. The normal bundle is untwisted; the projective plane’s would have to be twisted; so no such surface exists. An immersion, which may cross itself, has no inside and escapes the argument, and Werner Boy found one in 1901.

The same question in every dimension is the immersion problem for projective spaces: for each $n$, what is the smallest dimension of room into which the $n$-dimensional projective space can be immersed? Whitney’s methods give lower bounds from exactly these signs, and constructions give upper bounds. When $n$ is a power of two the answer is known — $2n - 1$ — and for many other $n$ the bounds meet. For general $n$ they do not, and the exact immersion dimension of real projective space remains unknown, after seventy years of work that has used every refinement of these classes the subject has invented.

One sign, three readings #

The habit worth keeping is to ask what a picture is a picture of.

The Möbius band began as a strip of paper with a surprising edge. It became a surface with one side, then a determinant with a negative sign. Here it is a line over each point of a circle, and its one-sidedness is the statement that a continuous choice on those lines must pass through zero. The strip, the sign and the zero are one fact, and it is the bundle reading — the one with no room and no shape — that generalises to every space and every dimension.