From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Klein bottle question Date: 18 Oct 1998 08:10:25 GMT Kate Orman wrote: >Ah... so the folks living on a Klein bottle wouldn't be three-dimensional >like us, they'd be two-dimensional, like the Flatlanders dwelling on the >Moebius strip - is that right? May I suggest that the ideal of "dimensionality" bandied about here is a bit murky? As has been pointed out in other posts, most of the spaces being discussed are (intrinsically) two dimensional. Roughly speaking this means that from any point in the space one describes the location of nearby points by giving two "coordinates" or "displacements"; note that this only works locally, and that it requires the establishment of a coordinate convention. Let me give some examples of two-dimensional spaces. The plane is two-dimensional. At each point in the plane you can draw two vectors which can be used for directions near that point. When the plane is the blackboard, we might have a vector pointing right and another pointing up. Observe that in fact the coordinates established at one point suffice to determine positions everywhere in the space: every point is unambiguously located by telling how far right and up it is from the origin (or any other pre-assigned Mecca). A torus -- the doughnut (or coffee-cup :-) ) -- is a two-dimensional manifold. At each point we can draw a pair of vectors tangent to the surface -- say, a red one pointing one way parallel to the "rim" of the torus, and a blue one pointing along its "waist". Now, I can pick one location to be the "origin" again, and tell how to reach any other point by travelling a certain distance "around" (along the red) and a certain distance "over" (along the blue). This time, however, a point does not have a unique set of coordinate; the same point can be reached by going 1/2 unit "blue" or by going -1/2 unit "blue" (where the distance around is taken as the unit of measure.) We say the torus is not homeomorphic to the plane (although it is locally homeomorphic to it). The earth is two-dimensional. We give coordinates at most points by specifying displacements "North" and "East", say. Now we have to surrender another feature: it is impossible to establish directions uniformly across the earth (there is neither "north" nor "east" at the poles). This is a fundamental problem: you cannot set up two sets of nonzero-vectors (red and blue, say), one set at each point. Indeed, one cannot even draw a _single_ nonzero vector tangent to the sphere at each point, varying in a continuous way, let alone trying to draw two which never point in parallel directions. This is the "Hairy Ball Theorem" (You can't comb the hair on a sphere). The property which the torus had but the earth lacks is parallelizability. Note however, that we could set up coordinates across the earth something like this: take a snapshot of the earth from far above the north pole, and mark off blue vectors pointing right (towards the Greenwich meridian, say, more or less) and red vectors pointing up (more or less towards the Indian Ocean). Think of how these vectors point at locations a little north of the equator: in Ouagadougou, the blue points south and the red points east; in Calcutta, blue points west and red points south; in Honolulu, blue is roughly northeast and red is roughly northwest. You could extend these definitions of red and blue directions to neighboring points just south of the equator. You could also cover the whole southern hemisphere with a similar set of vectors. This would make for sort of a confusing situation near the equator; ships crossing that line would have to take note of the fact that they are switching from north-hemisphere convention to south-hemisphere convention. But it wouldn't be all that bad: strip maps of the equator would show _two_ sets of arrows, one red and one blue in each set. The key observation is that one could convert one set at a point to the other set simply by _rotating_. (For example the conventions might be in synch at the dateline and prime meridian, but 180-degrees out of phase in Quito.) A Moebius strip is a two dimensional manifold, too. This time, too, we draw a (non-parallel) pair of vectors at each point. We _can_ comb the hair in the strip -- at each point, draw a blue vector along the "center circle" of the strip. We can't draw a second direction uniformly, as in the torus, but certainly at each point we may pick one of the two directions "crosswise" as the red vector. Indeed, with a pair of "crosswise" cuts we can split that pesky Moebius strip into two rectangles, which we can call the "northern" and "southern" ones, each with its own uniform conventions of blue and red directions. Again, maps of the "equator" (that is, the two regions where we made the cuts) would have to have two pairs of colored arrows at each point. Without any real trouble, they could be made to agree completely along one of the two cuts. But, and here's the kicker, along the other part of the equator we would find the sets of arrows to be "flipped": if the two candidate blue vectors are rotated to point in the same direction, the red vectors would still be 180 degrees out of kilter! The property we have lost from the other models is orientability. (Actually we lost another property: edgelessness. If it is possible to move at a uniform speed across the Moebius strip, then in a finite amount of time we encounter a "boundary of space"; if not, then the Moebius band, like the plane, appears to "go on forever". The torus and sphere, as well as the Klein bottle, are different: they are compact, meaning a traveler on them experiences no boundary, but if travelling long enough will eventually come close to a point previously travelled.) The Klein bottle is in fact nothing more than two Moebius strips joined together. Each of them has a boundary which is a single circle, so two identical Moebius strips can be stitched together as if zipping up a zipper along two identical lengths. Now, I'm a professional -- don't try this at home. There is a wee bit of a problem as you get near the end of this tailoring job. It seems the things the zipper halves are attached to are rather cumbersome, and seem to get into each other's way. This is an embeddability problem, which I'll come to next. Likewise one more space, while I'm at it: simply take a disk, and sew each point in its boundary to the point directly opposite it. This is a two-dimensional manifold, compact, and non-orientable. It's the "projective plane" or "cross-cap". This rounds out my display of two-dimensional surfaces, but there are more: two tori can be stitched together if you remove a little circular patch from each; the result is a "two-holed torus". Similar attachments allow the construction of grander spaces; in fact this is how all the compact surfaces may be constructed. Now let me discuss this issue of "living on" a surface. I have been careful to refer to points _in_ these spaces, rather than _on_ them. That's because the fundamental properties of the spaces have to do with the points of the space and their neighbors in the space; what happens around them ("in the air" if you like) is not part of their topology. That's a slightly different question, called the embedding of the space into another. You may have seen a trefoil: a kind of loop which winds around itself three times in a knotted way, returning to its starting place. Intrinsically, that's just a circle. We say the trefoil and the circle are homeomorphic (the points within the spaces can be paired off appropriately) but differently embedded into space (the points of the Euclidean spaces around the circle and trefoil cannot be paired off in a continuous way). We have described the Klein bottle and cross-cap, but there is no way to embed them into 3-dimensional space. This "instrinsic" view of the space is often described in terms of its inhabitants. We live on a sphere but it looks like the plane near where we stand. The same is true for inhabitants on any other 2-dimensional manifold, if those inhabitants are constrained to looking only along the "red" and "blue" directions and the plane they span; this lets them look only at nearby points in their space. (A space-dweller living in a toroidal space station -- a popular design, I take it -- is assumed to look only left/right or forward/back; looking straight up at the "ceiling" isn't allowed). This is the perspective taken in the classic "Flatland". Notice the Flatlander living on the Moebius strip is without recourse to reconcile the two coordinate systems on the halves of the strip: turned to face the mutually-agreed upon direction of "blue", s/he will have the northerner's "red" to the left and the southerner's "red" to the right. On the other hand,we are in reality 3-dimensional beings. We _could_ look up inside a toroidal space station. We _do_ look down at our spherical earth and see it's not a plane. Forced to walk around a Moebius strip like Escher's ants, we could reconcile the coordinate systems there by (continuing to face "blue" and) standing on our heads when we need to translate from one system to the other. (Actually we would walk on it and find ourselves "on its other side", a concept foreign to the Flatlanders.) And on the Klein bottle ... ...well, what? Here the concept of "walking on", as opposed to "walking in" has no meaning. We cannot really walk on something which is not embedded in R^3. To understand the problem here, imagine the Flatlanders' pet worm. It is accustomed to walking around a circle in Flatland. Yet if we embed that circle into 3-space, the worm no longer has a notion of which way is "out" on the circle, and might spiral around the cross-section as it crawls around the circle, coming back to its starting point on the "inside" of the circle. What was a simple task on the circle in the plane is now harder to carry out the same way on a circle in 3-space. What is needed is a framing, that is, an additional vector at each point of the circle, pointing "out", so the worm knows how to orient its body relative to the circle at each point. So too it is impossible for us to walk "on" a surface embedded in R^4 (say) unless the surface is also equipped with a third direction at each point. Indeed, should you be given a chance to embed the earth in to R^4 and then wander around the earth, you could easily come home staring into your basement instead of your front door, if you don't take the precaution of embedding an "out" vector along with each point. Your poor 2-dimensional retina would undoubtedly be overwhelmed by the opics in R^4 but I imagine you would only see a circle's worth of the spherical earth at each moment, until you're suddenly confronted by your laundry room. (Intrigued? There is a film worth watching: "Turning a sphere inside out"!) If the business of embedding is too dis-orient-ing (sorry) we can mimic the Flatlander description of topology by discussing three-dimensional manifolds. These are just spaces in which there are three independent directions, say "red', "blue", and "green" at each point. Your living room is one such example. So is the inside of the space shuttle: note that we are not requiring any preferred direction to be called "up". A thickened Moebius strip is an example of a 3-dimensional manifold: simply take "red" and "blue" as before, and pick "green" at each point to be a vector pointing straight away from the Moebius strip into space. As with "red", this vector cannot be picked in a continuously-varying way across the whole of the space, but that's OK. Now, topologists recognize that the thickened Moebius strip is nothing but a solid torus. Launch one into space and ask people to take a walk, and the worst that would happen is that they would return home upside down relative to when they left. These kinds of things happen all the time in space, and Miss Manners has learned to be patient. Far more interesting is the direct product M x I (M=Moebius strip, I=unit interval). For the cognoscenti: this is almost the same as above, but the identification on I^3 is not (0,s,t)~(1,-s,-t), as in the previous paragraph, but rather (0,s,t)~(1,-s,t). This construction uses a reflection, not a rotation. As a result, this space resembles a long curved indoor racetrack which loops back on itself in a funny way: when you come back to the starting line, the left lane ends where the right one began and so on. Did you start at the left lane and shake you trainer's right hand just before you started down the track? Then upon your return your trainer will appear to be at the right-most lane, extending her _left_ hand for a congratulatory handshake! (Obviously "left" and "right" have no more meaning in this world than "up" and "down" do in deep space, and just as obviously, I suppose, this manifold does not embed into Euclidean 3-space). By the way, if your left hand hurts, just run around the track again: former positions will be regained and you can shake with your right. There are some variants of this construction which are also fun. Projective 3-space has been mentioned. If it's big enough, it feels like Euclidean 3-space, but when you use a good telescope, you see the back of your head, and it's upside-down. Not only that, when you look straight up, you see your shoes, turned to look as if they belonged to someone talking to you, face to face. For more fun, consider the Poincare sphere; this lets you see anything around you from the "outisde" perspective by looking in any of 120 directions. (This space is X=S^3/G where |G|=120 is perfect, so that X is a homology sphere but not even simply connected.) A few respondents mentioned geometry, which I haven't yet since the issues of orientability (etc.) are at the more fundamental topological level. But in fact one can get some interesting geometry here too. One can endow a manifold with a construct known as a Riemannian metric; this can then be used form computing lengths, areas, and volumes. I mention this only to say that the metric need _not_ be inherited from R^3 or any other Euclidean space containing the manifold. For example, there's no difficulty in assuming that the Moebius strip is infinitely long "crosswise". Also curvature is not forced by the topology except at a really basic global level. For example, the torus, which we usually think of as a curved subspace of R^3, may be embedded into R^4 as a "flat" torus (S^1 x S^1). Our understanding of space time is that it is indeed unevenly curved; but we have no evidence that it is not topologically trivial. Topics of this type have arisen before. For differential topology (that's manifolds, knots, etc.) see index/57-XX.html For differential geometry index/53-XX.html dave ============================================================================== From: "Dr. Michael Albert" Newsgroups: sci.math Subject: Re: klein bottle question Date: Tue, 20 Oct 1998 22:41:22 -0400 Dear Friends: A kind person has pointed out to me that if you take a solid ball and identify antipodal points on the surface, the result is indeed orientable. My apologies for the confusion. I had just wanted to point out that non-orientable spaces came in sizes bigger than two dimensions, but the example I gave was of course completely wrong. As this person has not posted, I assume that it was due to a wish for privacy. However, there is little I can do to improve what I was sent, so I am taking the liberty of quoting it at length: > In one of your postings to sci.math (10/17), you described RP^3 > as being an example of a non-orientable compact 3-manifold. > However, it is orientable, since H^3(RP^3, Z) = Z. The difference > between this and RP^2 is that the antipodal map on S^3 is orientation > preserving, whereas the antipodal map on S^2 is orientation reversing. > > (In fact, RP^3 is homeomorphic to the Lie group SO(3), and one can > translate a volume element at the identity around the whole group to > obtain a nowhere-vanishing volume form.) RP^2 x S^1, or Klein bottle x > S^1, would have worked instead.