One-sided surfaces

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Earl Richard, 'Making surfaces',
Topology: A Very Short Introduction (Oxford, 2019)
page 34

At this point, we still can’t identify the peculiar surface from Figure 14 which has an Euler number of –1. So far we’ve only constructed surfaces with even Euler numbers and –1 is odd. In fact, with the tori 𝕋^#g⁠, we’ve only met half the story and half of the closed surfaces. Recall how in Figure 12 we made a cylinder by gluing two edges of a square. We could, instead, have glued those two sides using reversed arrows (Figure 17(a)), introducing a single twist. So the points near v₂ on e₁ are glued to the points near v₄ on e₃ and those near v₁ on e₁ are glued to the points near v₃ on e₃. This would have created a Möbius strip, named after August Möbius who discovered it in 1858. Figure 17.

The Möbius strip (a) A square with identified edges, (b) Runners on a Möbius strip. The Möbius strip is unusual in only having one side—this is apparent in Figure 17(b) as the runners cover the entirety of the strip rather than just one side of it as they would if running around just the outside (or inside) of a cylinder. Or you can imagine painting the outside of a cylinder black and the inside white, but should you begin painting a Möbius strip one colour you would find yourself covering the entire strip in that colour. The Möbius strip is an example of a non-orientable surface. Like the cylinder it is a surface with boundary, but note its boundary is a single circle rather than two separate ones as with the cylinder.

In Figures 18(a)–(d) we see an oriented loop—here a circle—moving around a Möbius strip. By an oriented loop I mean a loop with a given sense of direction, here initially (18(a)) appearing as clockwise to the reader. But as this loop moves around the strip (or equivalently moves left in the square) we see that when the circle returns to its original position (18(d)) that sense has now reversed and appears anti-clockwise. If you are having a little trouble visualizing what’s happening to the loop, note in 18(b) and 18(c) how the points labelled P are glued together and likewise the Qs. In 18(b) most of the loop (on the left) looks to be clockwise running from P to Q, but as the loop appears on the right and continues from Q to P that sense is beginning to appear as anti-clockwise. Figure 18.

Moving an oriented loop around a Möbius strip. Any surface on which it is possible to reverse the sense of an oriented loop is called non-orientable. If it is impossible to reverse a loop’s sense, then the surface is called orientable. Any surface that contains a Möbius strip is non-orientable as we could just send an oriented loop once around that strip to reverse its sense. A surface with an inside and an outside is orientable. To appreciate this, imagine walking around the outside of such a surface. Looking down to your feet on the surface you could draw a circle in a clockwise manner. As you wander around the outside of the surface you can consistently take your notion of clockwise across the whole surface. This means, in particular, that the tori 𝕋^#g⁠, which we met earlier and which each have an inside and outside, are all examples of orientable surfaces.

Returning to Figure 17(a), a partly glued square making a Möbius strip, there remain two unglued edges e₂ and e₄. We could glue these together as in Figure 19(a), but what surface would we make? Certainly a non-orientable one as it contains a Möbius strip (the shaded region). If instead we make this surface by gluing e₂ and e₄ first, we first create a cylinder with e₁ and e₃ as its circular ends. But to complete the surface, rather than bringing those circular ends together as with a torus, one circular end has to be glued backwards on to the other circular end—this is because of the reverse arrows on e₁ and e₃. Figure 19(b) shows how we might try to do this; we could take one circular end back into the cylinder and glue it to the other end from inside, and this way the reverse arrows line up properly. The surface made is called a Klein bottle, after Felix Klein who first described it in 1882. Being non-orientable, the Klein bottle does not have an inside and outside. Figure 19.

The Klein bottle and projective plane (a) A Klein bottle, (b) 3D depiction of a Klein Bottle, (c) A projective plane. There is a subtle problem with the Klein bottle in Figure 19(b). When we take the cylinder back into itself, some single points in space actually represent two distinct points on the Klein bottle. So this image is not a proper representation or embedding of the Klein bottle in 3D. In fact, it is impossible to construct a Klein bottle in 3D without such self-intersections as occur where the cylinder cuts back into itself. The relevant result demonstrating this impossibility can be viewed as a generalization of the Jordan curve theorem. That theorem concerned embedding circles in the plane with a Jordan curve having an inside and an outside. In a like manner when a closed surface is embedded in 3D, the surface again divides the remaining space into an inside and an outside and so the closed surface must be orientable. As the Klein bottle is non-orientable, it cannot be embedded in 3D.

However, the Klein bottle can be embedded in 4D and this isn’t too hard to imagine if we treat the fourth dimension as time. The Klein bottle is two-dimensional (as surfaces are) and so from this 4D viewpoint it is important to consider the Klein bottle as only existing for an instant, a certain ‘now’; for it to have a past or future would give it a third dimension. So when faced with bringing the cylinder back into itself—which would normally cause self-intersections—we can instead move that bit of cylinder gradually into the future (the fourth dimension), where the remainder of the Klein bottle doesn’t exist and then, once the cylinder has passed through the space its present self occupies, we can gradually bring that bit of the cylinder back into the present. The self-intersections no longer occur, as the distinct points of the Klein bottle that became merged in Figure 19(b) instead sit in the same point of space but crucially at different times.

We can also determine the Euler number of the Klein bottle, again being careful to note how edges and vertices are glued together. The square is our only face; e₁ and e₃ are glued together, as are e₂ and e₄, making two rather than four edges; finally v₁ is glued to v₂ which is glued to v₄ which is glued to v₃ and so we have just one vertex, giving $V-E+F=1-2+1=0$ the Euler number of the Klein bottle. Unfortunately, 0 is also the Euler number of the torus, so any hope we might have had that the Euler number alone is information enough to recognize the shape of a surface was simplistic. The torus and Klein bottle are different surfaces—the former is orientable (two-sided), the latter not—and yet they both have the same Euler number.

Another important non-orientable surface, which can be formed from gluing a square’s edges together, is the projective plane $ℙ$. In Figure 19(c) we assign e₂ and e₄ reverse arrows (in contrast to 19(a)). The surface formed is non-orientable, as it again contains a Möbius strip (the shaded region), and we can calculate the Euler number as before: again $F=1$ and $E=2$ but this time v₁ and v₃ are glued together and separately v₂ and v₄ are glued, so that $V=2$. Hence $ℙ$ has Euler number $V-E+F=2-2+1=1⁠$.